Change to GIF version
|
|
Metamath Proof Explorer |
This is the Unicode version. Change to GIF version |
|
| Ref | Description |
| a1ii 1 | (_Note_: This inference r... |
| idi 2 | This inference, which requ... |
| mp2 9 | A double modus ponens infe... |
| mp2b 10 | A double modus ponens infe... |
| a1i 11 | Inference introducing an a... |
| 2a1i 12 | Inference introducing two ... |
| mp1i 13 | Inference detaching an ant... |
| a2i 14 | Inference distributing an ... |
| mpd 15 | A modus ponens deduction. ... |
| imim2i 16 | Inference adding common an... |
| syl 17 | An inference version of th... |
| 3syl 18 | Inference chaining two syl... |
| 4syl 19 | Inference chaining three s... |
| mpi 20 | A nested modus ponens infe... |
| mpisyl 21 | A syllogism combined with ... |
| id 22 | Principle of identity. Th... |
| idALT 23 | Alternate proof of ~ id . ... |
| idd 24 | Principle of identity ~ id... |
| a1d 25 | Deduction introducing an e... |
| 2a1d 26 | Deduction introducing two ... |
| a1i13 27 | Add two antecedents to a w... |
| 2a1 28 | A double form of ~ ax-1 . ... |
| a2d 29 | Deduction distributing an ... |
| sylcom 30 | Syllogism inference with c... |
| syl5com 31 | Syllogism inference with c... |
| com12 32 | Inference that swaps (comm... |
| syl11 33 | A syllogism inference. Co... |
| syl5 34 | A syllogism rule of infere... |
| syl6 35 | A syllogism rule of infere... |
| syl56 36 | Combine ~ syl5 and ~ syl6 ... |
| syl6com 37 | Syllogism inference with c... |
| mpcom 38 | Modus ponens inference wit... |
| syli 39 | Syllogism inference with c... |
| syl2im 40 | Replace two antecedents. ... |
| syl2imc 41 | A commuted version of ~ sy... |
| pm2.27 42 | This theorem, called "Asse... |
| mpdd 43 | A nested modus ponens dedu... |
| mpid 44 | A nested modus ponens dedu... |
| mpdi 45 | A nested modus ponens dedu... |
| mpii 46 | A doubly nested modus pone... |
| syld 47 | Syllogism deduction. Dedu... |
| syldc 48 | Syllogism deduction. Comm... |
| mp2d 49 | A double modus ponens dedu... |
| a1dd 50 | Double deduction introduci... |
| 2a1dd 51 | Double deduction introduci... |
| pm2.43i 52 | Inference absorbing redund... |
| pm2.43d 53 | Deduction absorbing redund... |
| pm2.43a 54 | Inference absorbing redund... |
| pm2.43b 55 | Inference absorbing redund... |
| pm2.43 56 | Absorption of redundant an... |
| imim2d 57 | Deduction adding nested an... |
| imim2 58 | A closed form of syllogism... |
| embantd 59 | Deduction embedding an ant... |
| 3syld 60 | Triple syllogism deduction... |
| sylsyld 61 | A double syllogism inferen... |
| imim12i 62 | Inference joining two impl... |
| imim1i 63 | Inference adding common co... |
| imim3i 64 | Inference adding three nes... |
| sylc 65 | A syllogism inference comb... |
| syl3c 66 | A syllogism inference comb... |
| syl6mpi 67 | A syllogism inference. (C... |
| mpsyl 68 | Modus ponens combined with... |
| mpsylsyld 69 | Modus ponens combined with... |
| syl6c 70 | Inference combining ~ syl6... |
| syl6ci 71 | A syllogism inference comb... |
| syldd 72 | Nested syllogism deduction... |
| syl5d 73 | A nested syllogism deducti... |
| syl7 74 | A syllogism rule of infere... |
| syl6d 75 | A nested syllogism deducti... |
| syl8 76 | A syllogism rule of infere... |
| syl9 77 | A nested syllogism inferen... |
| syl9r 78 | A nested syllogism inferen... |
| syl10 79 | A nested syllogism inferen... |
| a1ddd 80 | Triple deduction introduci... |
| imim12d 81 | Deduction combining antece... |
| imim1d 82 | Deduction adding nested co... |
| imim1 83 | A closed form of syllogism... |
| pm2.83 84 | Theorem *2.83 of [Whitehea... |
| peirceroll 85 | Over minimal implicational... |
| com23 86 | Commutation of antecedents... |
| com3r 87 | Commutation of antecedents... |
| com13 88 | Commutation of antecedents... |
| com3l 89 | Commutation of antecedents... |
| pm2.04 90 | Swap antecedents. Theorem... |
| com34 91 | Commutation of antecedents... |
| com4l 92 | Commutation of antecedents... |
| com4t 93 | Commutation of antecedents... |
| com4r 94 | Commutation of antecedents... |
| com24 95 | Commutation of antecedents... |
| com14 96 | Commutation of antecedents... |
| com45 97 | Commutation of antecedents... |
| com35 98 | Commutation of antecedents... |
| com25 99 | Commutation of antecedents... |
| com5l 100 | Commutation of antecedents... |
| com15 101 | Commutation of antecedents... |
| com52l 102 | Commutation of antecedents... |
| com52r 103 | Commutation of antecedents... |
| com5r 104 | Commutation of antecedents... |
| imim12 105 | Closed form of ~ imim12i a... |
| jarr 106 | Elimination of a nested an... |
| pm2.86d 107 | Deduction associated with ... |
| pm2.86 108 | Converse of axiom ~ ax-2 .... |
| pm2.86i 109 | Inference associated with ... |
| loolin 110 | The Linearity Axiom of the... |
| loowoz 111 | An alternate for the Linea... |
| con4 112 | Alias for ~ ax-3 to be use... |
| con4i 113 | Inference associated with ... |
| con4d 114 | Deduction associated with ... |
| mt4 115 | The rule of modus tollens.... |
| pm2.21i 116 | A contradiction implies an... |
| pm2.24ii 117 | A contradiction implies an... |
| pm2.21d 118 | A contradiction implies an... |
| pm2.21ddALT 119 | Alternate proof of ~ pm2.2... |
| pm2.21 120 | From a wff and its negatio... |
| pm2.24 121 | Theorem *2.24 of [Whitehea... |
| pm2.18 122 | Proof by contradiction. T... |
| pm2.18i 123 | Inference associated with ... |
| pm2.18d 124 | Deduction based on reducti... |
| notnotr 125 | Double negation eliminatio... |
| notnotri 126 | Inference associated with ... |
| notnotriOLD 127 | Obsolete proof of ~ notnot... |
| notnotrd 128 | Deduction associated with ... |
| con2d 129 | A contraposition deduction... |
| con2 130 | Contraposition. Theorem *... |
| mt2d 131 | Modus tollens deduction. ... |
| mt2i 132 | Modus tollens inference. ... |
| nsyl3 133 | A negated syllogism infere... |
| con2i 134 | A contraposition inference... |
| nsyl 135 | A negated syllogism infere... |
| notnot 136 | Double negation introducti... |
| notnoti 137 | Inference associated with ... |
| notnotd 138 | Deduction associated with ... |
| con1d 139 | A contraposition deduction... |
| mt3d 140 | Modus tollens deduction. ... |
| mt3i 141 | Modus tollens inference. ... |
| nsyl2 142 | A negated syllogism infere... |
| con1 143 | Contraposition. Theorem *... |
| con1i 144 | A contraposition inference... |
| con4iOLD 145 | Obsolete proof of ~ con4i ... |
| pm2.24i 146 | Inference associated with ... |
| pm2.24d 147 | Deduction form of ~ pm2.24... |
| con3d 148 | A contraposition deduction... |
| con3 149 | Contraposition. Theorem *... |
| con3i 150 | A contraposition inference... |
| con3rr3 151 | Rotate through consequent ... |
| mt4d 152 | Modus tollens deduction. ... |
| mt4i 153 | Modus tollens inference. ... |
| nsyld 154 | A negated syllogism deduct... |
| nsyli 155 | A negated syllogism infere... |
| nsyl4 156 | A negated syllogism infere... |
| pm3.2im 157 | Theorem *3.2 of [Whitehead... |
| mth8 158 | Theorem 8 of [Margaris] p.... |
| jc 159 | Deduction joining the cons... |
| impi 160 | An importation inference. ... |
| expi 161 | An exportation inference. ... |
| simprim 162 | Simplification. Similar t... |
| simplim 163 | Simplification. Similar t... |
| pm2.5 164 | Theorem *2.5 of [Whitehead... |
| pm2.51 165 | Theorem *2.51 of [Whitehea... |
| pm2.521 166 | Theorem *2.521 of [Whitehe... |
| pm2.52 167 | Theorem *2.52 of [Whitehea... |
| expt 168 | Exportation theorem ~ ex e... |
| impt 169 | Importation theorem ~ imp ... |
| pm2.61d 170 | Deduction eliminating an a... |
| pm2.61d1 171 | Inference eliminating an a... |
| pm2.61d2 172 | Inference eliminating an a... |
| ja 173 | Inference joining the ante... |
| jad 174 | Deduction form of ~ ja . ... |
| jarl 175 | Elimination of a nested an... |
| pm2.61i 176 | Inference eliminating an a... |
| pm2.61ii 177 | Inference eliminating two ... |
| pm2.61nii 178 | Inference eliminating two ... |
| pm2.61iii 179 | Inference eliminating thre... |
| pm2.01 180 | Reductio ad absurdum. The... |
| pm2.01d 181 | Deduction based on reducti... |
| pm2.6 182 | Theorem *2.6 of [Whitehead... |
| pm2.61 183 | Theorem *2.61 of [Whitehea... |
| pm2.65 184 | Theorem *2.65 of [Whitehea... |
| pm2.65i 185 | Inference rule for proof b... |
| pm2.21dd 186 | A contradiction implies an... |
| pm2.65d 187 | Deduction rule for proof b... |
| mto 188 | The rule of modus tollens.... |
| mtod 189 | Modus tollens deduction. ... |
| mtoi 190 | Modus tollens inference. ... |
| mt2 191 | A rule similar to modus to... |
| mt3 192 | A rule similar to modus to... |
| peirce 193 | Peirce's axiom. This odd-... |
| looinv 194 | The Inversion Axiom of the... |
| bijust 195 | Theorem used to justify de... |
| impbi 198 | Property of the biconditio... |
| impbii 199 | Infer an equivalence from ... |
| impbidd 200 | Deduce an equivalence from... |
| impbid21d 201 | Deduce an equivalence from... |
| impbid 202 | Deduce an equivalence from... |
| dfbi1 203 | Relate the biconditional c... |
| dfbi1ALT 204 | Alternate proof of ~ dfbi1... |
| biimp 205 | Property of the biconditio... |
| biimpi 206 | Infer an implication from ... |
| sylbi 207 | A mixed syllogism inferenc... |
| sylib 208 | A mixed syllogism inferenc... |
| sylbb 209 | A mixed syllogism inferenc... |
| biimpr 210 | Property of the biconditio... |
| bicom1 211 | Commutative law for the bi... |
| bicom 212 | Commutative law for the bi... |
| bicomd 213 | Commute two sides of a bic... |
| bicomi 214 | Inference from commutative... |
| impbid1 215 | Infer an equivalence from ... |
| impbid2 216 | Infer an equivalence from ... |
| impcon4bid 217 | A variation on ~ impbid wi... |
| biimpri 218 | Infer a converse implicati... |
| biimpd 219 | Deduce an implication from... |
| mpbi 220 | An inference from a bicond... |
| mpbir 221 | An inference from a bicond... |
| mpbid 222 | A deduction from a bicondi... |
| mpbii 223 | An inference from a nested... |
| sylibr 224 | A mixed syllogism inferenc... |
| sylbir 225 | A mixed syllogism inferenc... |
| sylbbr 226 | A mixed syllogism inferenc... |
| sylbb1 227 | A mixed syllogism inferenc... |
| sylbb2 228 | A mixed syllogism inferenc... |
| sylibd 229 | A syllogism deduction. (C... |
| sylbid 230 | A syllogism deduction. (C... |
| mpbidi 231 | A deduction from a bicondi... |
| syl5bi 232 | A mixed syllogism inferenc... |
| syl5bir 233 | A mixed syllogism inferenc... |
| syl5ib 234 | A mixed syllogism inferenc... |
| syl5ibcom 235 | A mixed syllogism inferenc... |
| syl5ibr 236 | A mixed syllogism inferenc... |
| syl5ibrcom 237 | A mixed syllogism inferenc... |
| biimprd 238 | Deduce a converse implicat... |
| biimpcd 239 | Deduce a commuted implicat... |
| biimprcd 240 | Deduce a converse commuted... |
| syl6ib 241 | A mixed syllogism inferenc... |
| syl6ibr 242 | A mixed syllogism inferenc... |
| syl6bi 243 | A mixed syllogism inferenc... |
| syl6bir 244 | A mixed syllogism inferenc... |
| syl7bi 245 | A mixed syllogism inferenc... |
| syl8ib 246 | A syllogism rule of infere... |
| mpbird 247 | A deduction from a bicondi... |
| mpbiri 248 | An inference from a nested... |
| sylibrd 249 | A syllogism deduction. (C... |
| sylbird 250 | A syllogism deduction. (C... |
| biid 251 | Principle of identity for ... |
| biidd 252 | Principle of identity with... |
| pm5.1im 253 | Two propositions are equiv... |
| 2th 254 | Two truths are equivalent.... |
| 2thd 255 | Two truths are equivalent ... |
| ibi 256 | Inference that converts a ... |
| ibir 257 | Inference that converts a ... |
| ibd 258 | Deduction that converts a ... |
| pm5.74 259 | Distribution of implicatio... |
| pm5.74i 260 | Distribution of implicatio... |
| pm5.74ri 261 | Distribution of implicatio... |
| pm5.74d 262 | Distribution of implicatio... |
| pm5.74rd 263 | Distribution of implicatio... |
| bitri 264 | An inference from transiti... |
| bitr2i 265 | An inference from transiti... |
| bitr3i 266 | An inference from transiti... |
| bitr4i 267 | An inference from transiti... |
| bitrd 268 | Deduction form of ~ bitri ... |
| bitr2d 269 | Deduction form of ~ bitr2i... |
| bitr3d 270 | Deduction form of ~ bitr3i... |
| bitr4d 271 | Deduction form of ~ bitr4i... |
| syl5bb 272 | A syllogism inference from... |
| syl5rbb 273 | A syllogism inference from... |
| syl5bbr 274 | A syllogism inference from... |
| syl5rbbr 275 | A syllogism inference from... |
| syl6bb 276 | A syllogism inference from... |
| syl6rbb 277 | A syllogism inference from... |
| syl6bbr 278 | A syllogism inference from... |
| syl6rbbr 279 | A syllogism inference from... |
| 3imtr3i 280 | A mixed syllogism inferenc... |
| 3imtr4i 281 | A mixed syllogism inferenc... |
| 3imtr3d 282 | More general version of ~ ... |
| 3imtr4d 283 | More general version of ~ ... |
| 3imtr3g 284 | More general version of ~ ... |
| 3imtr4g 285 | More general version of ~ ... |
| 3bitri 286 | A chained inference from t... |
| 3bitrri 287 | A chained inference from t... |
| 3bitr2i 288 | A chained inference from t... |
| 3bitr2ri 289 | A chained inference from t... |
| 3bitr3i 290 | A chained inference from t... |
| 3bitr3ri 291 | A chained inference from t... |
| 3bitr4i 292 | A chained inference from t... |
| 3bitr4ri 293 | A chained inference from t... |
| 3bitrd 294 | Deduction from transitivit... |
| 3bitrrd 295 | Deduction from transitivit... |
| 3bitr2d 296 | Deduction from transitivit... |
| 3bitr2rd 297 | Deduction from transitivit... |
| 3bitr3d 298 | Deduction from transitivit... |
| 3bitr3rd 299 | Deduction from transitivit... |
| 3bitr4d 300 | Deduction from transitivit... |
| 3bitr4rd 301 | Deduction from transitivit... |
| 3bitr3g 302 | More general version of ~ ... |
| 3bitr4g 303 | More general version of ~ ... |
| notnotb 304 | Double negation. Theorem ... |
| notnotdOLD 305 | Obsolete proof of ~ notnot... |
| con34b 306 | A biconditional form of co... |
| con4bid 307 | A contraposition deduction... |
| notbid 308 | Deduction negating both si... |
| notbi 309 | Contraposition. Theorem *... |
| notbii 310 | Negate both sides of a log... |
| con4bii 311 | A contraposition inference... |
| mtbi 312 | An inference from a bicond... |
| mtbir 313 | An inference from a bicond... |
| mtbid 314 | A deduction from a bicondi... |
| mtbird 315 | A deduction from a bicondi... |
| mtbii 316 | An inference from a bicond... |
| mtbiri 317 | An inference from a bicond... |
| sylnib 318 | A mixed syllogism inferenc... |
| sylnibr 319 | A mixed syllogism inferenc... |
| sylnbi 320 | A mixed syllogism inferenc... |
| sylnbir 321 | A mixed syllogism inferenc... |
| xchnxbi 322 | Replacement of a subexpres... |
| xchnxbir 323 | Replacement of a subexpres... |
| xchbinx 324 | Replacement of a subexpres... |
| xchbinxr 325 | Replacement of a subexpres... |
| imbi2i 326 | Introduce an antecedent to... |
| bibi2i 327 | Inference adding a bicondi... |
| bibi1i 328 | Inference adding a bicondi... |
| bibi12i 329 | The equivalence of two equ... |
| imbi2d 330 | Deduction adding an antece... |
| imbi1d 331 | Deduction adding a consequ... |
| bibi2d 332 | Deduction adding a bicondi... |
| bibi1d 333 | Deduction adding a bicondi... |
| imbi12d 334 | Deduction joining two equi... |
| bibi12d 335 | Deduction joining two equi... |
| imbi12 336 | Closed form of ~ imbi12i .... |
| imbi1 337 | Theorem *4.84 of [Whitehea... |
| imbi2 338 | Theorem *4.85 of [Whitehea... |
| imbi1i 339 | Introduce a consequent to ... |
| imbi12i 340 | Join two logical equivalen... |
| bibi1 341 | Theorem *4.86 of [Whitehea... |
| bitr3 342 | Closed nested implication ... |
| con2bi 343 | Contraposition. Theorem *... |
| con2bid 344 | A contraposition deduction... |
| con1bid 345 | A contraposition deduction... |
| con1bii 346 | A contraposition inference... |
| con2bii 347 | A contraposition inference... |
| con1b 348 | Contraposition. Bidirecti... |
| con2b 349 | Contraposition. Bidirecti... |
| biimt 350 | A wff is equivalent to its... |
| pm5.5 351 | Theorem *5.5 of [Whitehead... |
| a1bi 352 | Inference rule introducing... |
| mt2bi 353 | A false consequent falsifi... |
| mtt 354 | Modus-tollens-like theorem... |
| imnot 355 | If a proposition is false,... |
| pm5.501 356 | Theorem *5.501 of [Whitehe... |
| ibib 357 | Implication in terms of im... |
| ibibr 358 | Implication in terms of im... |
| tbt 359 | A wff is equivalent to its... |
| nbn2 360 | The negation of a wff is e... |
| bibif 361 | Transfer negation via an e... |
| nbn 362 | The negation of a wff is e... |
| nbn3 363 | Transfer falsehood via equ... |
| pm5.21im 364 | Two propositions are equiv... |
| 2false 365 | Two falsehoods are equival... |
| 2falsed 366 | Two falsehoods are equival... |
| pm5.21ni 367 | Two propositions implying ... |
| pm5.21nii 368 | Eliminate an antecedent im... |
| pm5.21ndd 369 | Eliminate an antecedent im... |
| bija 370 | Combine antecedents into a... |
| pm5.18 371 | Theorem *5.18 of [Whitehea... |
| xor3 372 | Two ways to express "exclu... |
| nbbn 373 | Move negation outside of b... |
| biass 374 | Associative law for the bi... |
| pm5.19 375 | Theorem *5.19 of [Whitehea... |
| bi2.04 376 | Logical equivalence of com... |
| pm5.4 377 | Antecedent absorption impl... |
| imdi 378 | Distributive law for impli... |
| pm5.41 379 | Theorem *5.41 of [Whitehea... |
| pm4.8 380 | Theorem *4.8 of [Whitehead... |
| pm4.81 381 | Theorem *4.81 of [Whitehea... |
| imim21b 382 | Simplify an implication be... |
| pm4.64 387 | Theorem *4.64 of [Whitehea... |
| pm2.53 388 | Theorem *2.53 of [Whitehea... |
| pm2.54 389 | Theorem *2.54 of [Whitehea... |
| ori 390 | Infer implication from dis... |
| orri 391 | Infer disjunction from imp... |
| ord 392 | Deduce implication from di... |
| orrd 393 | Deduce disjunction from im... |
| jaoi 394 | Inference disjoining the a... |
| jaod 395 | Deduction disjoining the a... |
| mpjaod 396 | Eliminate a disjunction in... |
| orel1 397 | Elimination of disjunction... |
| orel2 398 | Elimination of disjunction... |
| olc 399 | Introduction of a disjunct... |
| orc 400 | Introduction of a disjunct... |
| pm1.4 401 | Axiom *1.4 of [WhiteheadRu... |
| orcom 402 | Commutative law for disjun... |
| orcomd 403 | Commutation of disjuncts i... |
| orcoms 404 | Commutation of disjuncts i... |
| orci 405 | Deduction introducing a di... |
| olci 406 | Deduction introducing a di... |
| orcd 407 | Deduction introducing a di... |
| olcd 408 | Deduction introducing a di... |
| orcs 409 | Deduction eliminating disj... |
| olcs 410 | Deduction eliminating disj... |
| pm2.07 411 | Theorem *2.07 of [Whitehea... |
| pm2.45 412 | Theorem *2.45 of [Whitehea... |
| pm2.46 413 | Theorem *2.46 of [Whitehea... |
| pm2.47 414 | Theorem *2.47 of [Whitehea... |
| pm2.48 415 | Theorem *2.48 of [Whitehea... |
| pm2.49 416 | Theorem *2.49 of [Whitehea... |
| pm2.67-2 417 | Slight generalization of T... |
| pm2.67 418 | Theorem *2.67 of [Whitehea... |
| pm2.25 419 | Theorem *2.25 of [Whitehea... |
| biorf 420 | A wff is equivalent to its... |
| biortn 421 | A wff is equivalent to its... |
| biorfi 422 | A wff is equivalent to its... |
| biorfiOLD 423 | Obsolete proof of ~ biorfi... |
| pm2.621 424 | Theorem *2.621 of [Whitehe... |
| pm2.62 425 | Theorem *2.62 of [Whitehea... |
| pm2.68 426 | Theorem *2.68 of [Whitehea... |
| dfor2 427 | Logical 'or' expressed in ... |
| imor 428 | Implication in terms of di... |
| imori 429 | Infer disjunction from imp... |
| imorri 430 | Infer implication from dis... |
| exmid 431 | Law of excluded middle, al... |
| exmidd 432 | Law of excluded middle in ... |
| pm2.1 433 | Theorem *2.1 of [Whitehead... |
| pm2.13 434 | Theorem *2.13 of [Whitehea... |
| pm4.62 435 | Theorem *4.62 of [Whitehea... |
| pm4.66 436 | Theorem *4.66 of [Whitehea... |
| pm4.63 437 | Theorem *4.63 of [Whitehea... |
| imnan 438 | Express implication in ter... |
| imnani 439 | Infer implication from neg... |
| iman 440 | Express implication in ter... |
| annim 441 | Express conjunction in ter... |
| pm4.61 442 | Theorem *4.61 of [Whitehea... |
| pm4.65 443 | Theorem *4.65 of [Whitehea... |
| pm4.67 444 | Theorem *4.67 of [Whitehea... |
| imp 445 | Importation inference. (C... |
| impcom 446 | Importation inference with... |
| impd 447 | Importation deduction. (C... |
| imp31 448 | An importation inference. ... |
| imp32 449 | An importation inference. ... |
| ex 450 | Exportation inference. (T... |
| expcom 451 | Exportation inference with... |
| expd 452 | Exportation deduction. (C... |
| expdimp 453 | A deduction version of exp... |
| expcomd 454 | Deduction form of ~ expcom... |
| expdcom 455 | Commuted form of ~ expd . ... |
| impancom 456 | Mixed importation/commutat... |
| con3dimp 457 | Variant of ~ con3d with im... |
| pm2.01da 458 | Deduction based on reducti... |
| pm2.18da 459 | Deduction based on reducti... |
| pm3.3 460 | Theorem *3.3 (Exp) of [Whi... |
| pm3.31 461 | Theorem *3.31 (Imp) of [Wh... |
| impexp 462 | Import-export theorem. Pa... |
| pm3.2 463 | Join antecedents with conj... |
| pm3.21 464 | Join antecedents with conj... |
| pm3.22 465 | Theorem *3.22 of [Whitehea... |
| ancom 466 | Commutative law for conjun... |
| ancomd 467 | Commutation of conjuncts i... |
| ancomst 468 | Closed form of ~ ancoms . ... |
| ancoms 469 | Inference commuting conjun... |
| ancomsd 470 | Deduction commuting conjun... |
| pm3.2i 471 | Infer conjunction of premi... |
| pm3.43i 472 | Nested conjunction of ante... |
| simpl 473 | Elimination of a conjunct.... |
| simpli 474 | Inference eliminating a co... |
| simpld 475 | Deduction eliminating a co... |
| simplbi 476 | Deduction eliminating a co... |
| simpr 477 | Elimination of a conjunct.... |
| simpri 478 | Inference eliminating a co... |
| simprd 479 | Deduction eliminating a co... |
| simprbi 480 | Deduction eliminating a co... |
| adantr 481 | Inference adding a conjunc... |
| adantl 482 | Inference adding a conjunc... |
| adantld 483 | Deduction adding a conjunc... |
| adantrd 484 | Deduction adding a conjunc... |
| impel 485 | An inference for implicati... |
| mpan9 486 | Modus ponens conjoining di... |
| syldan 487 | A syllogism deduction with... |
| sylan 488 | A syllogism inference. (C... |
| sylanb 489 | A syllogism inference. (C... |
| sylanbr 490 | A syllogism inference. (C... |
| sylan2 491 | A syllogism inference. (C... |
| sylan2b 492 | A syllogism inference. (C... |
| sylan2br 493 | A syllogism inference. (C... |
| syl2an 494 | A double syllogism inferen... |
| syl2anr 495 | A double syllogism inferen... |
| syl2anb 496 | A double syllogism inferen... |
| syl2anbr 497 | A double syllogism inferen... |
| syland 498 | A syllogism deduction. (C... |
| sylan2d 499 | A syllogism deduction. (C... |
| syl2and 500 | A syllogism deduction. (C... |
| biimpa 501 | Importation inference from... |
| biimpar 502 | Importation inference from... |
| biimpac 503 | Importation inference from... |
| biimparc 504 | Importation inference from... |
| animorl 505 | Conjunction implies disjun... |
| animorr 506 | Conjunction implies disjun... |
| animorlr 507 | Conjunction implies disjun... |
| animorrl 508 | Conjunction implies disjun... |
| ianor 509 | Negated conjunction in ter... |
| anor 510 | Conjunction in terms of di... |
| ioran 511 | Negated disjunction in ter... |
| pm4.52 512 | Theorem *4.52 of [Whitehea... |
| pm4.53 513 | Theorem *4.53 of [Whitehea... |
| pm4.54 514 | Theorem *4.54 of [Whitehea... |
| pm4.55 515 | Theorem *4.55 of [Whitehea... |
| pm4.56 516 | Theorem *4.56 of [Whitehea... |
| oran 517 | Disjunction in terms of co... |
| pm4.57 518 | Theorem *4.57 of [Whitehea... |
| pm3.1 519 | Theorem *3.1 of [Whitehead... |
| pm3.11 520 | Theorem *3.11 of [Whitehea... |
| pm3.12 521 | Theorem *3.12 of [Whitehea... |
| pm3.13 522 | Theorem *3.13 of [Whitehea... |
| pm3.14 523 | Theorem *3.14 of [Whitehea... |
| iba 524 | Introduction of antecedent... |
| ibar 525 | Introduction of antecedent... |
| biantru 526 | A wff is equivalent to its... |
| biantrur 527 | A wff is equivalent to its... |
| biantrud 528 | A wff is equivalent to its... |
| biantrurd 529 | A wff is equivalent to its... |
| mpbirand 530 | Detach truth from conjunct... |
| jaao 531 | Inference conjoining and d... |
| jaoa 532 | Inference disjoining and c... |
| pm3.44 533 | Theorem *3.44 of [Whitehea... |
| jao 534 | Disjunction of antecedents... |
| pm1.2 535 | Axiom *1.2 of [WhiteheadRu... |
| oridm 536 | Idempotent law for disjunc... |
| pm4.25 537 | Theorem *4.25 of [Whitehea... |
| orim12i 538 | Disjoin antecedents and co... |
| orim1i 539 | Introduce disjunct to both... |
| orim2i 540 | Introduce disjunct to both... |
| orbi2i 541 | Inference adding a left di... |
| orbi1i 542 | Inference adding a right d... |
| orbi12i 543 | Infer the disjunction of t... |
| pm1.5 544 | Axiom *1.5 (Assoc) of [Whi... |
| or12 545 | Swap two disjuncts. (Cont... |
| orass 546 | Associative law for disjun... |
| pm2.31 547 | Theorem *2.31 of [Whitehea... |
| pm2.32 548 | Theorem *2.32 of [Whitehea... |
| or32 549 | A rearrangement of disjunc... |
| or4 550 | Rearrangement of 4 disjunc... |
| or42 551 | Rearrangement of 4 disjunc... |
| orordi 552 | Distribution of disjunctio... |
| orordir 553 | Distribution of disjunctio... |
| jca 554 | Deduce conjunction of the ... |
| jcad 555 | Deduction conjoining the c... |
| jca2 556 | Inference conjoining the c... |
| jca31 557 | Join three consequents. (... |
| jca32 558 | Join three consequents. (... |
| jcai 559 | Deduction replacing implic... |
| jctil 560 | Inference conjoining a the... |
| jctir 561 | Inference conjoining a the... |
| jccir 562 | Inference conjoining a con... |
| jccil 563 | Inference conjoining a con... |
| jctl 564 | Inference conjoining a the... |
| jctr 565 | Inference conjoining a the... |
| jctild 566 | Deduction conjoining a the... |
| jctird 567 | Deduction conjoining a the... |
| syl6an 568 | A syllogism deduction comb... |
| ancl 569 | Conjoin antecedent to left... |
| anclb 570 | Conjoin antecedent to left... |
| pm5.42 571 | Theorem *5.42 of [Whitehea... |
| ancr 572 | Conjoin antecedent to righ... |
| ancrb 573 | Conjoin antecedent to righ... |
| ancli 574 | Deduction conjoining antec... |
| ancri 575 | Deduction conjoining antec... |
| ancld 576 | Deduction conjoining antec... |
| ancrd 577 | Deduction conjoining antec... |
| anc2l 578 | Conjoin antecedent to left... |
| anc2r 579 | Conjoin antecedent to righ... |
| anc2li 580 | Deduction conjoining antec... |
| anc2ri 581 | Deduction conjoining antec... |
| pm3.41 582 | Theorem *3.41 of [Whitehea... |
| pm3.42 583 | Theorem *3.42 of [Whitehea... |
| pm3.4 584 | Conjunction implies implic... |
| pm4.45im 585 | Conjunction with implicati... |
| anim12d 586 | Conjoin antecedents and co... |
| anim12d1 587 | Variant of ~ anim12d where... |
| anim1d 588 | Add a conjunct to right of... |
| anim2d 589 | Add a conjunct to left of ... |
| anim12i 590 | Conjoin antecedents and co... |
| anim12ci 591 | Variant of ~ anim12i with ... |
| anim1i 592 | Introduce conjunct to both... |
| anim2i 593 | Introduce conjunct to both... |
| anim12ii 594 | Conjoin antecedents and co... |
| prth 595 | Conjoin antecedents and co... |
| pm2.3 596 | Theorem *2.3 of [Whitehead... |
| pm2.41 597 | Theorem *2.41 of [Whitehea... |
| pm2.42 598 | Theorem *2.42 of [Whitehea... |
| pm2.4 599 | Theorem *2.4 of [Whitehead... |
| pm2.65da 600 | Deduction rule for proof b... |
| pm4.44 601 | Theorem *4.44 of [Whitehea... |
| pm4.14 602 | Theorem *4.14 of [Whitehea... |
| pm3.37 603 | Theorem *3.37 (Transp) of ... |
| nan 604 | Theorem to move a conjunct... |
| pm4.15 605 | Theorem *4.15 of [Whitehea... |
| pm4.78 606 | Implication distributes ov... |
| pm4.79 607 | Theorem *4.79 of [Whitehea... |
| pm4.87 608 | Theorem *4.87 of [Whitehea... |
| pm3.33 609 | Theorem *3.33 (Syll) of [W... |
| pm3.34 610 | Theorem *3.34 (Syll) of [W... |
| pm3.35 611 | Conjunctive detachment. T... |
| pm5.31 612 | Theorem *5.31 of [Whitehea... |
| imp4b 613 | An importation inference. ... |
| imp4a 614 | An importation inference. ... |
| imp4aOLD 615 | Obsolete proof of ~ imp4a ... |
| imp4bOLD 616 | Obsolete proof of ~ imp4b ... |
| imp4c 617 | An importation inference. ... |
| imp4d 618 | An importation inference. ... |
| imp41 619 | An importation inference. ... |
| imp42 620 | An importation inference. ... |
| imp43 621 | An importation inference. ... |
| imp44 622 | An importation inference. ... |
| imp45 623 | An importation inference. ... |
| imp5a 624 | An importation inference. ... |
| imp5d 625 | An importation inference. ... |
| imp5g 626 | An importation inference. ... |
| imp55 627 | An importation inference. ... |
| imp511 628 | An importation inference. ... |
| expimpd 629 | Exportation followed by a ... |
| exp31 630 | An exportation inference. ... |
| exp32 631 | An exportation inference. ... |
| exp4b 632 | An exportation inference. ... |
| exp4a 633 | An exportation inference. ... |
| exp4aOLD 634 | Obsolete proof of ~ exp4a ... |
| exp4bOLD 635 | Obsolete proof of ~ exp4b ... |
| exp4c 636 | An exportation inference. ... |
| exp4d 637 | An exportation inference. ... |
| exp41 638 | An exportation inference. ... |
| exp42 639 | An exportation inference. ... |
| exp43 640 | An exportation inference. ... |
| exp44 641 | An exportation inference. ... |
| exp45 642 | An exportation inference. ... |
| expr 643 | Export a wff from a right ... |
| exp5c 644 | An exportation inference. ... |
| exp5j 645 | An exportation inference. ... |
| exp5l 646 | An exportation inference. ... |
| exp53 647 | An exportation inference. ... |
| expl 648 | Export a wff from a left c... |
| impr 649 | Import a wff into a right ... |
| impl 650 | Export a wff from a left c... |
| impac 651 | Importation with conjuncti... |
| exbiri 652 | Inference form of ~ exbir ... |
| simprbda 653 | Deduction eliminating a co... |
| simplbda 654 | Deduction eliminating a co... |
| simplbi2 655 | Deduction eliminating a co... |
| simplbi2comt 656 | Closed form of ~ simplbi2c... |
| simplbi2com 657 | A deduction eliminating a ... |
| simpl2im 658 | Implication from an elimin... |
| simplbiim 659 | Implication from an elimin... |
| dfbi2 660 | A theorem similar to the s... |
| dfbi 661 | Definition ~ df-bi rewritt... |
| pm4.71 662 | Implication in terms of bi... |
| pm4.71r 663 | Implication in terms of bi... |
| pm4.71i 664 | Inference converting an im... |
| pm4.71ri 665 | Inference converting an im... |
| pm4.71d 666 | Deduction converting an im... |
| pm4.71rd 667 | Deduction converting an im... |
| pm5.32 668 | Distribution of implicatio... |
| pm5.32i 669 | Distribution of implicatio... |
| pm5.32ri 670 | Distribution of implicatio... |
| pm5.32d 671 | Distribution of implicatio... |
| pm5.32rd 672 | Distribution of implicatio... |
| pm5.32da 673 | Distribution of implicatio... |
| biadan2 674 | Add a conjunction to an eq... |
| pm4.24 675 | Theorem *4.24 of [Whitehea... |
| anidm 676 | Idempotent law for conjunc... |
| anidms 677 | Inference from idempotent ... |
| anidmdbi 678 | Conjunction idempotence wi... |
| anasss 679 | Associative law for conjun... |
| anassrs 680 | Associative law for conjun... |
| anass 681 | Associative law for conjun... |
| sylanl1 682 | A syllogism inference. (C... |
| sylanl2 683 | A syllogism inference. (C... |
| sylanr1 684 | A syllogism inference. (C... |
| sylanr2 685 | A syllogism inference. (C... |
| sylani 686 | A syllogism inference. (C... |
| sylan2i 687 | A syllogism inference. (C... |
| syl2ani 688 | A syllogism inference. (C... |
| sylan9 689 | Nested syllogism inference... |
| sylan9r 690 | Nested syllogism inference... |
| mtand 691 | A modus tollens deduction.... |
| mtord 692 | A modus tollens deduction ... |
| syl2anc 693 | Syllogism inference combin... |
| sylancl 694 | Syllogism inference combin... |
| sylancr 695 | Syllogism inference combin... |
| sylanblc 696 | Syllogism inference combin... |
| sylanblrc 697 | Syllogism inference combin... |
| sylanbrc 698 | Syllogism inference. (Con... |
| sylancb 699 | A syllogism inference comb... |
| sylancbr 700 | A syllogism inference comb... |
| sylancom 701 | Syllogism inference with c... |
| mpdan 702 | An inference based on modu... |
| mpancom 703 | An inference based on modu... |
| mpidan 704 | A deduction which "stacks"... |
| hypstkdOLD 705 | Obsolete proof of ~ mpidan... |
| mpan 706 | An inference based on modu... |
| mpan2 707 | An inference based on modu... |
| mp2an 708 | An inference based on modu... |
| mp4an 709 | An inference based on modu... |
| mpan2d 710 | A deduction based on modus... |
| mpand 711 | A deduction based on modus... |
| mpani 712 | An inference based on modu... |
| mpan2i 713 | An inference based on modu... |
| mp2ani 714 | An inference based on modu... |
| mp2and 715 | A deduction based on modus... |
| mpanl1 716 | An inference based on modu... |
| mpanl2 717 | An inference based on modu... |
| mpanl12 718 | An inference based on modu... |
| mpanr1 719 | An inference based on modu... |
| mpanr2 720 | An inference based on modu... |
| mpanr12 721 | An inference based on modu... |
| mpanlr1 722 | An inference based on modu... |
| pm5.74da 723 | Distribution of implicatio... |
| pm4.45 724 | Theorem *4.45 of [Whitehea... |
| imdistan 725 | Distribution of implicatio... |
| imdistani 726 | Distribution of implicatio... |
| imdistanri 727 | Distribution of implicatio... |
| imdistand 728 | Distribution of implicatio... |
| imdistanda 729 | Distribution of implicatio... |
| anbi2i 730 | Introduce a left conjunct ... |
| anbi1i 731 | Introduce a right conjunct... |
| anbi2ci 732 | Variant of ~ anbi2i with c... |
| anbi12i 733 | Conjoin both sides of two ... |
| anbi12ci 734 | Variant of ~ anbi12i with ... |
| syldanl 735 | A syllogism deduction with... |
| sylan9bb 736 | Nested syllogism inference... |
| sylan9bbr 737 | Nested syllogism inference... |
| orbi2d 738 | Deduction adding a left di... |
| orbi1d 739 | Deduction adding a right d... |
| anbi2d 740 | Deduction adding a left co... |
| anbi1d 741 | Deduction adding a right c... |
| orbi1 742 | Theorem *4.37 of [Whitehea... |
| anbi1 743 | Introduce a right conjunct... |
| anbi2 744 | Introduce a left conjunct ... |
| bitr 745 | Theorem *4.22 of [Whitehea... |
| orbi12d 746 | Deduction joining two equi... |
| anbi12d 747 | Deduction joining two equi... |
| pm5.3 748 | Theorem *5.3 of [Whitehead... |
| pm5.61 749 | Theorem *5.61 of [Whitehea... |
| adantll 750 | Deduction adding a conjunc... |
| adantlr 751 | Deduction adding a conjunc... |
| adantrl 752 | Deduction adding a conjunc... |
| adantrr 753 | Deduction adding a conjunc... |
| adantlll 754 | Deduction adding a conjunc... |
| adantllr 755 | Deduction adding a conjunc... |
| adantlrl 756 | Deduction adding a conjunc... |
| adantlrr 757 | Deduction adding a conjunc... |
| adantrll 758 | Deduction adding a conjunc... |
| adantrlr 759 | Deduction adding a conjunc... |
| adantrrl 760 | Deduction adding a conjunc... |
| adantrrr 761 | Deduction adding a conjunc... |
| ad2antrr 762 | Deduction adding two conju... |
| ad2antlr 763 | Deduction adding two conju... |
| ad2antrl 764 | Deduction adding two conju... |
| ad2antll 765 | Deduction adding conjuncts... |
| ad3antrrr 766 | Deduction adding three con... |
| ad3antlr 767 | Deduction adding three con... |
| ad4antr 768 | Deduction adding 4 conjunc... |
| ad4antlr 769 | Deduction adding 4 conjunc... |
| ad5antr 770 | Deduction adding 5 conjunc... |
| ad5antlr 771 | Deduction adding 5 conjunc... |
| ad6antr 772 | Deduction adding 6 conjunc... |
| ad6antlr 773 | Deduction adding 6 conjunc... |
| ad7antr 774 | Deduction adding 7 conjunc... |
| ad7antlr 775 | Deduction adding 7 conjunc... |
| ad8antr 776 | Deduction adding 8 conjunc... |
| ad8antlr 777 | Deduction adding 8 conjunc... |
| ad9antr 778 | Deduction adding 9 conjunc... |
| ad9antlr 779 | Deduction adding 9 conjunc... |
| ad10antr 780 | Deduction adding 10 conjun... |
| ad10antlr 781 | Deduction adding 10 conjun... |
| ad2ant2l 782 | Deduction adding two conju... |
| ad2ant2r 783 | Deduction adding two conju... |
| ad2ant2lr 784 | Deduction adding two conju... |
| ad2ant2rl 785 | Deduction adding two conju... |
| adantl3r 786 | Deduction adding 1 conjunc... |
| adantl4r 787 | Deduction adding 1 conjunc... |
| adantl5r 788 | Deduction adding 1 conjunc... |
| adantl6r 789 | Deduction adding 1 conjunc... |
| simpll 790 | Simplification of a conjun... |
| simplld 791 | Deduction form of ~ simpll... |
| simplr 792 | Simplification of a conjun... |
| simplrd 793 | Deduction eliminating a do... |
| simprl 794 | Simplification of a conjun... |
| simprld 795 | Deduction eliminating a do... |
| simprr 796 | Simplification of a conjun... |
| simprrd 797 | Deduction form of ~ simprr... |
| simplll 798 | Simplification of a conjun... |
| simpllr 799 | Simplification of a conjun... |
| simplrl 800 | Simplification of a conjun... |
| simplrr 801 | Simplification of a conjun... |
| simprll 802 | Simplification of a conjun... |
| simprlr 803 | Simplification of a conjun... |
| simprrl 804 | Simplification of a conjun... |
| simprrr 805 | Simplification of a conjun... |
| simp-4l 806 | Simplification of a conjun... |
| simp-4r 807 | Simplification of a conjun... |
| simp-5l 808 | Simplification of a conjun... |
| simp-5r 809 | Simplification of a conjun... |
| simp-6l 810 | Simplification of a conjun... |
| simp-6r 811 | Simplification of a conjun... |
| simp-7l 812 | Simplification of a conjun... |
| simp-7r 813 | Simplification of a conjun... |
| simp-8l 814 | Simplification of a conjun... |
| simp-8r 815 | Simplification of a conjun... |
| simp-9l 816 | Simplification of a conjun... |
| simp-9r 817 | Simplification of a conjun... |
| simp-10l 818 | Simplification of a conjun... |
| simp-10r 819 | Simplification of a conjun... |
| simp-11l 820 | Simplification of a conjun... |
| simp-11r 821 | Simplification of a conjun... |
| jaob 822 | Disjunction of antecedents... |
| adant423OLD 823 | Obsolete as of 2-Oct-2021.... |
| jaoian 824 | Inference disjoining the a... |
| jao1i 825 | Add a disjunct in the ante... |
| jaodan 826 | Deduction disjoining the a... |
| mpjaodan 827 | Eliminate a disjunction in... |
| pm4.77 828 | Theorem *4.77 of [Whitehea... |
| pm2.63 829 | Theorem *2.63 of [Whitehea... |
| pm2.64 830 | Theorem *2.64 of [Whitehea... |
| pm2.61ian 831 | Elimination of an antecede... |
| pm2.61dan 832 | Elimination of an antecede... |
| pm2.61ddan 833 | Elimination of two anteced... |
| pm2.61dda 834 | Elimination of two anteced... |
| condan 835 | Proof by contradiction. (... |
| abai 836 | Introduce one conjunct as ... |
| pm5.53 837 | Theorem *5.53 of [Whitehea... |
| an12 838 | Swap two conjuncts. Note ... |
| an32 839 | A rearrangement of conjunc... |
| an13 840 | A rearrangement of conjunc... |
| an31 841 | A rearrangement of conjunc... |
| bianass 842 | An inference to merge two ... |
| an12s 843 | Swap two conjuncts in ante... |
| ancom2s 844 | Inference commuting a nest... |
| an13s 845 | Swap two conjuncts in ante... |
| an32s 846 | Swap two conjuncts in ante... |
| ancom1s 847 | Inference commuting a nest... |
| an31s 848 | Swap two conjuncts in ante... |
| anass1rs 849 | Commutative-associative la... |
| anabs1 850 | Absorption into embedded c... |
| anabs5 851 | Absorption into embedded c... |
| anabs7 852 | Absorption into embedded c... |
| a2and 853 | Deduction distributing a c... |
| anabsan 854 | Absorption of antecedent w... |
| anabss1 855 | Absorption of antecedent i... |
| anabss4 856 | Absorption of antecedent i... |
| anabss5 857 | Absorption of antecedent i... |
| anabsi5 858 | Absorption of antecedent i... |
| anabsi6 859 | Absorption of antecedent i... |
| anabsi7 860 | Absorption of antecedent i... |
| anabsi8 861 | Absorption of antecedent i... |
| anabss7 862 | Absorption of antecedent i... |
| anabsan2 863 | Absorption of antecedent w... |
| anabss3 864 | Absorption of antecedent i... |
| an4 865 | Rearrangement of 4 conjunc... |
| an42 866 | Rearrangement of 4 conjunc... |
| an43 867 | Rearrangement of 4 conjunc... |
| an3 868 | A rearrangement of conjunc... |
| an4s 869 | Inference rearranging 4 co... |
| an42s 870 | Inference rearranging 4 co... |
| anandi 871 | Distribution of conjunctio... |
| anandir 872 | Distribution of conjunctio... |
| anandis 873 | Inference that undistribut... |
| anandirs 874 | Inference that undistribut... |
| syl2an2 875 | ~ syl2an with antecedents ... |
| syl2an2r 876 | ~ syl2anr with antecedents... |
| impbida 877 | Deduce an equivalence from... |
| pm3.48 878 | Theorem *3.48 of [Whitehea... |
| pm3.45 879 | Theorem *3.45 (Fact) of [W... |
| im2anan9 880 | Deduction joining nested i... |
| im2anan9r 881 | Deduction joining nested i... |
| anim12dan 882 | Conjoin antecedents and co... |
| orim12d 883 | Disjoin antecedents and co... |
| orim1d 884 | Disjoin antecedents and co... |
| orim2d 885 | Disjoin antecedents and co... |
| orim2 886 | Axiom *1.6 (Sum) of [White... |
| pm2.38 887 | Theorem *2.38 of [Whitehea... |
| pm2.36 888 | Theorem *2.36 of [Whitehea... |
| pm2.37 889 | Theorem *2.37 of [Whitehea... |
| pm2.73 890 | Theorem *2.73 of [Whitehea... |
| pm2.74 891 | Theorem *2.74 of [Whitehea... |
| orimdi 892 | Disjunction distributes ov... |
| pm2.76 893 | Theorem *2.76 of [Whitehea... |
| pm2.75 894 | Theorem *2.75 of [Whitehea... |
| pm2.8 895 | Theorem *2.8 of [Whitehead... |
| pm2.81 896 | Theorem *2.81 of [Whitehea... |
| pm2.82 897 | Theorem *2.82 of [Whitehea... |
| pm2.85 898 | Theorem *2.85 of [Whitehea... |
| pm3.2ni 899 | Infer negated disjunction ... |
| orabs 900 | Absorption of redundant in... |
| oranabs 901 | Absorb a disjunct into a c... |
| pm5.1 902 | Two propositions are equiv... |
| pm5.21 903 | Two propositions are equiv... |
| norbi 904 | If neither of two proposit... |
| nbior 905 | If two propositions are no... |
| pm3.43 906 | Theorem *3.43 (Comp) of [W... |
| jcab 907 | Distributive law for impli... |
| ordi 908 | Distributive law for disju... |
| ordir 909 | Distributive law for disju... |
| pm4.76 910 | Theorem *4.76 of [Whitehea... |
| andi 911 | Distributive law for conju... |
| andir 912 | Distributive law for conju... |
| orddi 913 | Double distributive law fo... |
| anddi 914 | Double distributive law fo... |
| pm4.39 915 | Theorem *4.39 of [Whitehea... |
| pm4.38 916 | Theorem *4.38 of [Whitehea... |
| bi2anan9 917 | Deduction joining two equi... |
| bi2anan9r 918 | Deduction joining two equi... |
| bi2bian9 919 | Deduction joining two bico... |
| pm4.72 920 | Implication in terms of bi... |
| imimorb 921 | Simplify an implication be... |
| pm5.33 922 | Theorem *5.33 of [Whitehea... |
| pm5.36 923 | Theorem *5.36 of [Whitehea... |
| bianabs 924 | Absorb a hypothesis into t... |
| oibabs 925 | Absorption of disjunction ... |
| pm3.24 926 | Law of noncontradiction. ... |
| pm2.26 927 | Theorem *2.26 of [Whitehea... |
| pm5.11 928 | Theorem *5.11 of [Whitehea... |
| pm5.12 929 | Theorem *5.12 of [Whitehea... |
| pm5.14 930 | Theorem *5.14 of [Whitehea... |
| pm5.13 931 | Theorem *5.13 of [Whitehea... |
| pm5.17 932 | Theorem *5.17 of [Whitehea... |
| pm5.15 933 | Theorem *5.15 of [Whitehea... |
| pm5.16 934 | Theorem *5.16 of [Whitehea... |
| xor 935 | Two ways to express "exclu... |
| nbi2 936 | Two ways to express "exclu... |
| xordi 937 | Conjunction distributes ov... |
| biort 938 | A wff disjoined with truth... |
| pm5.55 939 | Theorem *5.55 of [Whitehea... |
| ornld 940 | Selecting one statement fr... |
| pm5.21nd 941 | Eliminate an antecedent im... |
| pm5.35 942 | Theorem *5.35 of [Whitehea... |
| pm5.54 943 | Theorem *5.54 of [Whitehea... |
| baib 944 | Move conjunction outside o... |
| baibr 945 | Move conjunction outside o... |
| rbaibr 946 | Move conjunction outside o... |
| rbaib 947 | Move conjunction outside o... |
| baibd 948 | Move conjunction outside o... |
| rbaibd 949 | Move conjunction outside o... |
| pm5.44 950 | Theorem *5.44 of [Whitehea... |
| pm5.6 951 | Conjunction in antecedent ... |
| orcanai 952 | Change disjunction in cons... |
| mpbiran 953 | Detach truth from conjunct... |
| mpbiran2 954 | Detach truth from conjunct... |
| mpbir2an 955 | Detach a conjunction of tr... |
| mpbi2and 956 | Detach a conjunction of tr... |
| mpbir2and 957 | Detach a conjunction of tr... |
| pm5.62 958 | Theorem *5.62 of [Whitehea... |
| pm5.63 959 | Theorem *5.63 of [Whitehea... |
| intnan 960 | Introduction of conjunct i... |
| intnanr 961 | Introduction of conjunct i... |
| intnand 962 | Introduction of conjunct i... |
| intnanrd 963 | Introduction of conjunct i... |
| niabn 964 | Miscellaneous inference re... |
| ninba 965 | Miscellaneous inference re... |
| bianfi 966 | A wff conjoined with false... |
| bianfd 967 | A wff conjoined with false... |
| pm4.43 968 | Theorem *4.43 of [Whitehea... |
| pm4.82 969 | Theorem *4.82 of [Whitehea... |
| pm4.83 970 | Theorem *4.83 of [Whitehea... |
| pclem6 971 | Negation inferred from emb... |
| biantr 972 | A transitive law of equiva... |
| orbidi 973 | Disjunction distributes ov... |
| biluk 974 | Lukasiewicz's shortest axi... |
| pm5.7 975 | Disjunction distributes ov... |
| bigolden 976 | Dijkstra-Scholten's Golden... |
| pm5.71 977 | Theorem *5.71 of [Whitehea... |
| pm5.75 978 | Theorem *5.75 of [Whitehea... |
| pm5.75OLD 979 | Obsolete proof of ~ pm5.75... |
| bimsc1 980 | Removal of conjunct from o... |
| ecase2d 981 | Deduction for elimination ... |
| ecase3 982 | Inference for elimination ... |
| ecase 983 | Inference for elimination ... |
| ecase3d 984 | Deduction for elimination ... |
| ecased 985 | Deduction for elimination ... |
| ecase3ad 986 | Deduction for elimination ... |
| ccase 987 | Inference for combining ca... |
| ccased 988 | Deduction for combining ca... |
| ccase2 989 | Inference for combining ca... |
| 4cases 990 | Inference eliminating two ... |
| 4casesdan 991 | Deduction eliminating two ... |
| cases 992 | Case disjunction according... |
| cases2 993 | Case disjunction according... |
| dfbi3 994 | An alternate definition of... |
| dfbi3OLD 995 | Obsolete proof of ~ dfbi3 ... |
| pm5.24 996 | Theorem *5.24 of [Whitehea... |
| 4exmid 997 | The disjunction of the fou... |
| 4exmidOLD 998 | Obsolete proof of ~ 4exmid... |
| consensus 999 | The consensus theorem. Th... |
| dedlem0a 1000 | Lemma for an alternate ver... |
| dedlem0b 1001 | Lemma for an alternate ver... |
| dedlema 1002 | Lemma for weak deduction t... |
| dedlemb 1003 | Lemma for weak deduction t... |
| pm4.42 1004 | Theorem *4.42 of [Whitehea... |
| prlem1 1005 | A specialized lemma for se... |
| prlem2 1006 | A specialized lemma for se... |
| oplem1 1007 | A specialized lemma for se... |
| dn1 1008 | A single axiom for Boolean... |
| bianir 1009 | A closed form of ~ mpbir ,... |
| jaoi2 1010 | Inference removing a negat... |
| jaoi3 1011 | Inference separating a dis... |
| dfifp2 1014 | Alternate definition of th... |
| dfifp3 1015 | Alternate definition of th... |
| dfifp4 1016 | Alternate definition of th... |
| dfifp5 1017 | Alternate definition of th... |
| dfifp6 1018 | Alternate definition of th... |
| dfifp7 1019 | Alternate definition of th... |
| anifp 1020 | The conditional operator i... |
| ifpor 1021 | The conditional operator i... |
| ifpn 1022 | Conditional operator for t... |
| ifptru 1023 | Value of the conditional o... |
| ifpfal 1024 | Value of the conditional o... |
| ifpid 1025 | Value of the conditional o... |
| casesifp 1026 | Version of ~ cases express... |
| ifpbi123d 1027 | Equality deduction for con... |
| ifpimpda 1028 | Separation of the values o... |
| 1fpid3 1029 | The value of the condition... |
| elimh 1030 | Hypothesis builder for the... |
| dedt 1031 | The weak deduction theorem... |
| con3ALT 1032 | Proof of ~ con3 from its a... |
| elimhOLD 1033 | Old version of ~ elimh . ... |
| dedtOLD 1034 | Old version of ~ dedt . O... |
| con3OLD 1035 | Old version of ~ con3ALT .... |
| 3orass 1040 | Associative law for triple... |
| 3orel1 1041 | Partial elimination of a t... |
| 3anass 1042 | Associative law for triple... |
| 3anrot 1043 | Rotation law for triple co... |
| 3orrot 1044 | Rotation law for triple di... |
| 3ancoma 1045 | Commutation law for triple... |
| 3orcoma 1046 | Commutation law for triple... |
| 3ancomb 1047 | Commutation law for triple... |
| 3orcomb 1048 | Commutation law for triple... |
| 3anrev 1049 | Reversal law for triple co... |
| 3anan32 1050 | Convert triple conjunction... |
| 3anan12 1051 | Convert triple conjunction... |
| anandi3 1052 | Distribution of triple con... |
| anandi3r 1053 | Distribution of triple con... |
| 3anor 1054 | Triple conjunction express... |
| 3ianor 1055 | Negated triple conjunction... |
| 3ioran 1056 | Negated triple disjunction... |
| 3oran 1057 | Triple disjunction in term... |
| 3simpa 1058 | Simplification of triple c... |
| 3simpb 1059 | Simplification of triple c... |
| 3simpc 1060 | Simplification of triple c... |
| simp1 1061 | Simplification of triple c... |
| simp2 1062 | Simplification of triple c... |
| simp3 1063 | Simplification of triple c... |
| simpl1 1064 | Simplification rule. (Con... |
| simpl2 1065 | Simplification rule. (Con... |
| simpl3 1066 | Simplification rule. (Con... |
| simpr1 1067 | Simplification rule. (Con... |
| simpr2 1068 | Simplification rule. (Con... |
| simpr3 1069 | Simplification rule. (Con... |
| simp1i 1070 | Infer a conjunct from a tr... |
| simp2i 1071 | Infer a conjunct from a tr... |
| simp3i 1072 | Infer a conjunct from a tr... |
| simp1d 1073 | Deduce a conjunct from a t... |
| simp2d 1074 | Deduce a conjunct from a t... |
| simp3d 1075 | Deduce a conjunct from a t... |
| simp1bi 1076 | Deduce a conjunct from a t... |
| simp2bi 1077 | Deduce a conjunct from a t... |
| simp3bi 1078 | Deduce a conjunct from a t... |
| 3adant1 1079 | Deduction adding a conjunc... |
| 3adant2 1080 | Deduction adding a conjunc... |
| 3adant3 1081 | Deduction adding a conjunc... |
| 3ad2ant1 1082 | Deduction adding conjuncts... |
| 3ad2ant2 1083 | Deduction adding conjuncts... |
| 3ad2ant3 1084 | Deduction adding conjuncts... |
| simp1l 1085 | Simplification of triple c... |
| simp1r 1086 | Simplification of triple c... |
| simp2l 1087 | Simplification of triple c... |
| simp2r 1088 | Simplification of triple c... |
| simp3l 1089 | Simplification of triple c... |
| simp3r 1090 | Simplification of triple c... |
| simp11 1091 | Simplification of doubly t... |
| simp12 1092 | Simplification of doubly t... |
| simp13 1093 | Simplification of doubly t... |
| simp21 1094 | Simplification of doubly t... |
| simp22 1095 | Simplification of doubly t... |
| simp23 1096 | Simplification of doubly t... |
| simp31 1097 | Simplification of doubly t... |
| simp32 1098 | Simplification of doubly t... |
| simp33 1099 | Simplification of doubly t... |
| simpll1 1100 | Simplification of conjunct... |
| simpll2 1101 | Simplification of conjunct... |
| simpll3 1102 | Simplification of conjunct... |
| simplr1 1103 | Simplification of conjunct... |
| simplr2 1104 | Simplification of conjunct... |
| simplr3 1105 | Simplification of conjunct... |
| simprl1 1106 | Simplification of conjunct... |
| simprl2 1107 | Simplification of conjunct... |
| simprl3 1108 | Simplification of conjunct... |
| simprr1 1109 | Simplification of conjunct... |
| simprr2 1110 | Simplification of conjunct... |
| simprr3 1111 | Simplification of conjunct... |
| simpl1l 1112 | Simplification of conjunct... |
| simpl1r 1113 | Simplification of conjunct... |
| simpl2l 1114 | Simplification of conjunct... |
| simpl2r 1115 | Simplification of conjunct... |
| simpl3l 1116 | Simplification of conjunct... |
| simpl3r 1117 | Simplification of conjunct... |
| simpr1l 1118 | Simplification of conjunct... |
| simpr1r 1119 | Simplification of conjunct... |
| simpr2l 1120 | Simplification of conjunct... |
| simpr2r 1121 | Simplification of conjunct... |
| simpr3l 1122 | Simplification of conjunct... |
| simpr3r 1123 | Simplification of conjunct... |
| simp1ll 1124 | Simplification of conjunct... |
| simp1lr 1125 | Simplification of conjunct... |
| simp1rl 1126 | Simplification of conjunct... |
| simp1rr 1127 | Simplification of conjunct... |
| simp2ll 1128 | Simplification of conjunct... |
| simp2lr 1129 | Simplification of conjunct... |
| simp2rl 1130 | Simplification of conjunct... |
| simp2rr 1131 | Simplification of conjunct... |
| simp3ll 1132 | Simplification of conjunct... |
| simp3lr 1133 | Simplification of conjunct... |
| simp3rl 1134 | Simplification of conjunct... |
| simp3rr 1135 | Simplification of conjunct... |
| simpl11 1136 | Simplification of conjunct... |
| simpl12 1137 | Simplification of conjunct... |
| simpl13 1138 | Simplification of conjunct... |
| simpl21 1139 | Simplification of conjunct... |
| simpl22 1140 | Simplification of conjunct... |
| simpl23 1141 | Simplification of conjunct... |
| simpl31 1142 | Simplification of conjunct... |
| simpl32 1143 | Simplification of conjunct... |
| simpl33 1144 | Simplification of conjunct... |
| simpr11 1145 | Simplification of conjunct... |
| simpr12 1146 | Simplification of conjunct... |
| simpr13 1147 | Simplification of conjunct... |
| simpr21 1148 | Simplification of conjunct... |
| simpr22 1149 | Simplification of conjunct... |
| simpr23 1150 | Simplification of conjunct... |
| simpr31 1151 | Simplification of conjunct... |
| simpr32 1152 | Simplification of conjunct... |
| simpr33 1153 | Simplification of conjunct... |
| simp1l1 1154 | Simplification of conjunct... |
| simp1l2 1155 | Simplification of conjunct... |
| simp1l3 1156 | Simplification of conjunct... |
| simp1r1 1157 | Simplification of conjunct... |
| simp1r2 1158 | Simplification of conjunct... |
| simp1r3 1159 | Simplification of conjunct... |
| simp2l1 1160 | Simplification of conjunct... |
| simp2l2 1161 | Simplification of conjunct... |
| simp2l3 1162 | Simplification of conjunct... |
| simp2r1 1163 | Simplification of conjunct... |
| simp2r2 1164 | Simplification of conjunct... |
| simp2r3 1165 | Simplification of conjunct... |
| simp3l1 1166 | Simplification of conjunct... |
| simp3l2 1167 | Simplification of conjunct... |
| simp3l3 1168 | Simplification of conjunct... |
| simp3r1 1169 | Simplification of conjunct... |
| simp3r2 1170 | Simplification of conjunct... |
| simp3r3 1171 | Simplification of conjunct... |
| simp11l 1172 | Simplification of conjunct... |
| simp11r 1173 | Simplification of conjunct... |
| simp12l 1174 | Simplification of conjunct... |
| simp12r 1175 | Simplification of conjunct... |
| simp13l 1176 | Simplification of conjunct... |
| simp13r 1177 | Simplification of conjunct... |
| simp21l 1178 | Simplification of conjunct... |
| simp21r 1179 | Simplification of conjunct... |
| simp22l 1180 | Simplification of conjunct... |
| simp22r 1181 | Simplification of conjunct... |
| simp23l 1182 | Simplification of conjunct... |
| simp23r 1183 | Simplification of conjunct... |
| simp31l 1184 | Simplification of conjunct... |
| simp31r 1185 | Simplification of conjunct... |
| simp32l 1186 | Simplification of conjunct... |
| simp32r 1187 | Simplification of conjunct... |
| simp33l 1188 | Simplification of conjunct... |
| simp33r 1189 | Simplification of conjunct... |
| simp111 1190 | Simplification of conjunct... |
| simp112 1191 | Simplification of conjunct... |
| simp113 1192 | Simplification of conjunct... |
| simp121 1193 | Simplification of conjunct... |
| simp122 1194 | Simplification of conjunct... |
| simp123 1195 | Simplification of conjunct... |
| simp131 1196 | Simplification of conjunct... |
| simp132 1197 | Simplification of conjunct... |
| simp133 1198 | Simplification of conjunct... |
| simp211 1199 | Simplification of conjunct... |
| simp212 1200 | Simplification of conjunct... |
| simp213 1201 | Simplification of conjunct... |
| simp221 1202 | Simplification of conjunct... |
| simp222 1203 | Simplification of conjunct... |
| simp223 1204 | Simplification of conjunct... |
| simp231 1205 | Simplification of conjunct... |
| simp232 1206 | Simplification of conjunct... |
| simp233 1207 | Simplification of conjunct... |
| simp311 1208 | Simplification of conjunct... |
| simp312 1209 | Simplification of conjunct... |
| simp313 1210 | Simplification of conjunct... |
| simp321 1211 | Simplification of conjunct... |
| simp322 1212 | Simplification of conjunct... |
| simp323 1213 | Simplification of conjunct... |
| simp331 1214 | Simplification of conjunct... |
| simp332 1215 | Simplification of conjunct... |
| simp333 1216 | Simplification of conjunct... |
| 3adantl1 1217 | Deduction adding a conjunc... |
| 3adantl2 1218 | Deduction adding a conjunc... |
| 3adantl3 1219 | Deduction adding a conjunc... |
| 3adantr1 1220 | Deduction adding a conjunc... |
| 3adantr2 1221 | Deduction adding a conjunc... |
| 3adantr3 1222 | Deduction adding a conjunc... |
| 3ad2antl1 1223 | Deduction adding conjuncts... |
| 3ad2antl2 1224 | Deduction adding conjuncts... |
| 3ad2antl3 1225 | Deduction adding conjuncts... |
| 3ad2antr1 1226 | Deduction adding conjuncts... |
| 3ad2antr2 1227 | Deduction adding conjuncts... |
| 3ad2antr3 1228 | Deduction adding conjuncts... |
| 3anibar 1229 | Remove a hypothesis from t... |
| 3mix1 1230 | Introduction in triple dis... |
| 3mix2 1231 | Introduction in triple dis... |
| 3mix3 1232 | Introduction in triple dis... |
| 3mix1i 1233 | Introduction in triple dis... |
| 3mix2i 1234 | Introduction in triple dis... |
| 3mix3i 1235 | Introduction in triple dis... |
| 3mix1d 1236 | Deduction introducing trip... |
| 3mix2d 1237 | Deduction introducing trip... |
| 3mix3d 1238 | Deduction introducing trip... |
| 3pm3.2i 1239 | Infer conjunction of premi... |
| pm3.2an3 1240 | Version of ~ pm3.2 for a t... |
| pm3.2an3OLD 1241 | Obsolete proof of ~ pm3.2a... |
| 3jca 1242 | Join consequents with conj... |
| 3jcad 1243 | Deduction conjoining the c... |
| mpbir3an 1244 | Detach a conjunction of tr... |
| mpbir3and 1245 | Detach a conjunction of tr... |
| syl3anbrc 1246 | Syllogism inference. (Con... |
| 3anim123i 1247 | Join antecedents and conse... |
| 3anim1i 1248 | Add two conjuncts to antec... |
| 3anim2i 1249 | Add two conjuncts to antec... |
| 3anim3i 1250 | Add two conjuncts to antec... |
| 3anbi123i 1251 | Join 3 biconditionals with... |
| 3orbi123i 1252 | Join 3 biconditionals with... |
| 3anbi1i 1253 | Inference adding two conju... |
| 3anbi2i 1254 | Inference adding two conju... |
| 3anbi3i 1255 | Inference adding two conju... |
| 3imp 1256 | Importation inference. (C... |
| 3imp31 1257 | The importation inference ... |
| 3imp231 1258 | Importation inference. (C... |
| 3impa 1259 | Importation from double to... |
| 3impb 1260 | Importation from double to... |
| 3impia 1261 | Importation to triple conj... |
| 3impib 1262 | Importation to triple conj... |
| ex3 1263 | Apply ~ ex to a hypothesis... |
| 3exp 1264 | Exportation inference. (C... |
| 3expa 1265 | Exportation from triple to... |
| 3expb 1266 | Exportation from triple to... |
| 3expia 1267 | Exportation from triple co... |
| 3expib 1268 | Exportation from triple co... |
| 3com12 1269 | Commutation in antecedent.... |
| 3com13 1270 | Commutation in antecedent.... |
| 3com23 1271 | Commutation in antecedent.... |
| 3coml 1272 | Commutation in antecedent.... |
| 3comr 1273 | Commutation in antecedent.... |
| 3adant3r1 1274 | Deduction adding a conjunc... |
| 3adant3r2 1275 | Deduction adding a conjunc... |
| 3adant3r3 1276 | Deduction adding a conjunc... |
| 3imp21 1277 | The importation inference ... |
| 3imp3i2an 1278 | An elimination deduction. ... |
| 3an1rs 1279 | Swap conjuncts. (Contribu... |
| 3imp1 1280 | Importation to left triple... |
| 3impd 1281 | Importation deduction for ... |
| 3imp2 1282 | Importation to right tripl... |
| 3exp1 1283 | Exportation from left trip... |
| 3expd 1284 | Exportation deduction for ... |
| 3exp2 1285 | Exportation from right tri... |
| exp5o 1286 | A triple exportation infer... |
| exp516 1287 | A triple exportation infer... |
| exp520 1288 | A triple exportation infer... |
| 3impexp 1289 | Version of ~ impexp for a ... |
| 3anassrs 1290 | Associative law for conjun... |
| 3an4anass 1291 | Associative law for four c... |
| ad4ant13 1292 | Deduction adding conjuncts... |
| ad4ant14 1293 | Deduction adding conjuncts... |
| ad4ant123 1294 | Deduction adding conjuncts... |
| ad4ant124 1295 | Deduction adding conjuncts... |
| ad4ant134 1296 | Deduction adding conjuncts... |
| ad4ant23 1297 | Deduction adding conjuncts... |
| ad4ant24 1298 | Deduction adding conjuncts... |
| ad4ant234 1299 | Deduction adding conjuncts... |
| ad5ant12 1300 | Deduction adding conjuncts... |
| ad5ant13 1301 | Deduction adding conjuncts... |
| ad5ant14 1302 | Deduction adding conjuncts... |
| ad5ant15 1303 | Deduction adding conjuncts... |
| ad5ant23 1304 | Deduction adding conjuncts... |
| ad5ant24 1305 | Deduction adding conjuncts... |
| ad5ant25 1306 | Deduction adding conjuncts... |
| ad5ant245 1307 | Deduction adding conjuncts... |
| ad5ant234 1308 | Deduction adding conjuncts... |
| ad5ant235 1309 | Deduction adding conjuncts... |
| ad5ant123 1310 | Deduction adding conjuncts... |
| ad5ant124 1311 | Deduction adding conjuncts... |
| ad5ant125 1312 | Deduction adding conjuncts... |
| ad5ant134 1313 | Deduction adding conjuncts... |
| ad5ant135 1314 | Deduction adding conjuncts... |
| ad5ant145 1315 | Deduction adding conjuncts... |
| ad5ant1345 1316 | Deduction adding conjuncts... |
| ad5ant2345 1317 | Deduction adding conjuncts... |
| 3adant1l 1318 | Deduction adding a conjunc... |
| 3adant1r 1319 | Deduction adding a conjunc... |
| 3adant2l 1320 | Deduction adding a conjunc... |
| 3adant2r 1321 | Deduction adding a conjunc... |
| 3adant3l 1322 | Deduction adding a conjunc... |
| 3adant3r 1323 | Deduction adding a conjunc... |
| syl12anc 1324 | Syllogism combined with co... |
| syl21anc 1325 | Syllogism combined with co... |
| syl3anc 1326 | Syllogism combined with co... |
| syl22anc 1327 | Syllogism combined with co... |
| syl13anc 1328 | Syllogism combined with co... |
| syl31anc 1329 | Syllogism combined with co... |
| syl112anc 1330 | Syllogism combined with co... |
| syl121anc 1331 | Syllogism combined with co... |
| syl211anc 1332 | Syllogism combined with co... |
| syl23anc 1333 | Syllogism combined with co... |
| syl32anc 1334 | Syllogism combined with co... |
| syl122anc 1335 | Syllogism combined with co... |
| syl212anc 1336 | Syllogism combined with co... |
| syl221anc 1337 | Syllogism combined with co... |
| syl113anc 1338 | Syllogism combined with co... |
| syl131anc 1339 | Syllogism combined with co... |
| syl311anc 1340 | Syllogism combined with co... |
| syl33anc 1341 | Syllogism combined with co... |
| syl222anc 1342 | Syllogism combined with co... |
| syl123anc 1343 | Syllogism combined with co... |
| syl132anc 1344 | Syllogism combined with co... |
| syl213anc 1345 | Syllogism combined with co... |
| syl231anc 1346 | Syllogism combined with co... |
| syl312anc 1347 | Syllogism combined with co... |
| syl321anc 1348 | Syllogism combined with co... |
| syl133anc 1349 | Syllogism combined with co... |
| syl313anc 1350 | Syllogism combined with co... |
| syl331anc 1351 | Syllogism combined with co... |
| syl223anc 1352 | Syllogism combined with co... |
| syl232anc 1353 | Syllogism combined with co... |
| syl322anc 1354 | Syllogism combined with co... |
| syl233anc 1355 | Syllogism combined with co... |
| syl323anc 1356 | Syllogism combined with co... |
| syl332anc 1357 | Syllogism combined with co... |
| syl333anc 1358 | A syllogism inference comb... |
| syl3an1 1359 | A syllogism inference. (C... |
| syl3an2 1360 | A syllogism inference. (C... |
| syl3an3 1361 | A syllogism inference. (C... |
| syl3an1b 1362 | A syllogism inference. (C... |
| syl3an2b 1363 | A syllogism inference. (C... |
| syl3an3b 1364 | A syllogism inference. (C... |
| syl3an1br 1365 | A syllogism inference. (C... |
| syl3an2br 1366 | A syllogism inference. (C... |
| syl3an3br 1367 | A syllogism inference. (C... |
| syl3an 1368 | A triple syllogism inferen... |
| syl3anb 1369 | A triple syllogism inferen... |
| syl3anbr 1370 | A triple syllogism inferen... |
| syld3an3 1371 | A syllogism inference. (C... |
| syld3an1 1372 | A syllogism inference. (C... |
| syld3an2 1373 | A syllogism inference. (C... |
| syl3anl1 1374 | A syllogism inference. (C... |
| syl3anl2 1375 | A syllogism inference. (C... |
| syl3anl3 1376 | A syllogism inference. (C... |
| syl3anl 1377 | A triple syllogism inferen... |
| syl3anr1 1378 | A syllogism inference. (C... |
| syl3anr2 1379 | A syllogism inference. (C... |
| syl3anr3 1380 | A syllogism inference. (C... |
| 3impdi 1381 | Importation inference (und... |
| 3impdir 1382 | Importation inference (und... |
| 3anidm12 1383 | Inference from idempotent ... |
| 3anidm13 1384 | Inference from idempotent ... |
| 3anidm23 1385 | Inference from idempotent ... |
| syl2an3an 1386 | ~ syl3an with antecedents ... |
| syl2an23an 1387 | Deduction related to ~ syl... |
| 3ori 1388 | Infer implication from tri... |
| 3jao 1389 | Disjunction of three antec... |
| 3jaob 1390 | Disjunction of three antec... |
| 3jaoi 1391 | Disjunction of three antec... |
| 3jaod 1392 | Disjunction of three antec... |
| 3jaoian 1393 | Disjunction of three antec... |
| 3jaodan 1394 | Disjunction of three antec... |
| mpjao3dan 1395 | Eliminate a three-way disj... |
| 3jaao 1396 | Inference conjoining and d... |
| syl3an9b 1397 | Nested syllogism inference... |
| 3orbi123d 1398 | Deduction joining 3 equiva... |
| 3anbi123d 1399 | Deduction joining 3 equiva... |
| 3anbi12d 1400 | Deduction conjoining and a... |
| 3anbi13d 1401 | Deduction conjoining and a... |
| 3anbi23d 1402 | Deduction conjoining and a... |
| 3anbi1d 1403 | Deduction adding conjuncts... |
| 3anbi2d 1404 | Deduction adding conjuncts... |
| 3anbi3d 1405 | Deduction adding conjuncts... |
| 3anim123d 1406 | Deduction joining 3 implic... |
| 3orim123d 1407 | Deduction joining 3 implic... |
| an6 1408 | Rearrangement of 6 conjunc... |
| 3an6 1409 | Analogue of ~ an4 for trip... |
| 3or6 1410 | Analogue of ~ or4 for trip... |
| mp3an1 1411 | An inference based on modu... |
| mp3an2 1412 | An inference based on modu... |
| mp3an3 1413 | An inference based on modu... |
| mp3an12 1414 | An inference based on modu... |
| mp3an13 1415 | An inference based on modu... |
| mp3an23 1416 | An inference based on modu... |
| mp3an1i 1417 | An inference based on modu... |
| mp3anl1 1418 | An inference based on modu... |
| mp3anl2 1419 | An inference based on modu... |
| mp3anl3 1420 | An inference based on modu... |
| mp3anr1 1421 | An inference based on modu... |
| mp3anr2 1422 | An inference based on modu... |
| mp3anr3 1423 | An inference based on modu... |
| mp3an 1424 | An inference based on modu... |
| mpd3an3 1425 | An inference based on modu... |
| mpd3an23 1426 | An inference based on modu... |
| mp3and 1427 | A deduction based on modus... |
| mp3an12i 1428 | ~ mp3an with antecedents i... |
| mp3an2i 1429 | ~ mp3an with antecedents i... |
| mp3an3an 1430 | ~ mp3an with antecedents i... |
| mp3an2ani 1431 | An elimination deduction. ... |
| biimp3a 1432 | Infer implication from a l... |
| biimp3ar 1433 | Infer implication from a l... |
| 3anandis 1434 | Inference that undistribut... |
| 3anandirs 1435 | Inference that undistribut... |
| ecase23d 1436 | Deduction for elimination ... |
| 3ecase 1437 | Inference for elimination ... |
| 3bior1fd 1438 | A disjunction is equivalen... |
| 3bior1fand 1439 | A disjunction is equivalen... |
| 3bior2fd 1440 | A wff is equivalent to its... |
| 3biant1d 1441 | A conjunction is equivalen... |
| intn3an1d 1442 | Introduction of a triple c... |
| intn3an2d 1443 | Introduction of a triple c... |
| intn3an3d 1444 | Introduction of a triple c... |
| an3andi 1445 | Distribution of conjunctio... |
| an33rean 1446 | Rearrange a 9-fold conjunc... |
| nanan 1449 | Write 'and' in terms of 'n... |
| nancom 1450 | The 'nand' operator commut... |
| nannan 1451 | Lemma for handling nested ... |
| nanim 1452 | Show equivalence between i... |
| nannot 1453 | Show equivalence between n... |
| nanbi 1454 | Show equivalence between t... |
| nanbi1 1455 | Introduce a right anti-con... |
| nanbi2 1456 | Introduce a left anti-conj... |
| nanbi12 1457 | Join two logical equivalen... |
| nanbi1i 1458 | Introduce a right anti-con... |
| nanbi2i 1459 | Introduce a left anti-conj... |
| nanbi12i 1460 | Join two logical equivalen... |
| nanbi1d 1461 | Introduce a right anti-con... |
| nanbi2d 1462 | Introduce a left anti-conj... |
| nanbi12d 1463 | Join two logical equivalen... |
| xnor 1466 | Two ways to write XNOR. (C... |
| xorcom 1467 | The connector ` \/_ ` is c... |
| xorass 1468 | The connector ` \/_ ` is a... |
| excxor 1469 | This tautology shows that ... |
| xor2 1470 | Two ways to express "exclu... |
| xoror 1471 | XOR implies OR. (Contribut... |
| xornan 1472 | XOR implies NAND. (Contrib... |
| xornan2 1473 | XOR implies NAND (written ... |
| xorneg2 1474 | The connector ` \/_ ` is n... |
| xorneg1 1475 | The connector ` \/_ ` is n... |
| xorneg 1476 | The connector ` \/_ ` is u... |
| xorbi12i 1477 | Equality property for XOR.... |
| xorbi12d 1478 | Equality property for XOR.... |
| anxordi 1479 | Conjunction distributes ov... |
| xorexmid 1480 | Exclusive-or variant of th... |
| trujust 1485 | Soundness justification th... |
| tru 1487 | The truth value ` T. ` is ... |
| fal 1490 | The truth value ` F. ` is ... |
| dftru2 1491 | An alternate definition of... |
| trut 1492 | A proposition is equivalen... |
| trud 1493 | Eliminate ` T. ` as an ant... |
| tbtru 1494 | A proposition is equivalen... |
| nbfal 1495 | The negation of a proposit... |
| bitru 1496 | A theorem is equivalent to... |
| bifal 1497 | A contradiction is equival... |
| falim 1498 | The truth value ` F. ` imp... |
| falimd 1499 | The truth value ` F. ` imp... |
| a1tru 1500 | Anything implies ` T. ` . ... |
| truan 1501 | True can be removed from a... |
| dfnot 1502 | Given falsum ` F. ` , we c... |
| inegd 1503 | Negation introduction rule... |
| efald 1504 | Deduction based on reducti... |
| pm2.21fal 1505 | If a wff and its negation ... |
| truantru 1506 | A ` /\ ` identity. (Contr... |
| truanfal 1507 | A ` /\ ` identity. (Contr... |
| falantru 1508 | A ` /\ ` identity. (Contr... |
| falanfal 1509 | A ` /\ ` identity. (Contr... |
| truortru 1510 | A ` \/ ` identity. (Contr... |
| truorfal 1511 | A ` \/ ` identity. (Contr... |
| falortru 1512 | A ` \/ ` identity. (Contr... |
| falorfal 1513 | A ` \/ ` identity. (Contr... |
| truimtru 1514 | A ` -> ` identity. (Contr... |
| truimfal 1515 | A ` -> ` identity. (Contr... |
| falimtru 1516 | A ` -> ` identity. (Contr... |
| falimfal 1517 | A ` -> ` identity. (Contr... |
| nottru 1518 | A ` -. ` identity. (Contr... |
| notfal 1519 | A ` -. ` identity. (Contr... |
| trubitru 1520 | A ` <-> ` identity. (Cont... |
| falbitru 1521 | A ` <-> ` identity. (Cont... |
| trubifal 1522 | A ` <-> ` identity. (Cont... |
| falbifal 1523 | A ` <-> ` identity. (Cont... |
| trunantru 1524 | A ` -/\ ` identity. (Cont... |
| trunanfal 1525 | A ` -/\ ` identity. (Cont... |
| falnantru 1526 | A ` -/\ ` identity. (Cont... |
| falnanfal 1527 | A ` -/\ ` identity. (Cont... |
| truxortru 1528 | A ` \/_ ` identity. (Cont... |
| truxorfal 1529 | A ` \/_ ` identity. (Cont... |
| falxortru 1530 | A ` \/_ ` identity. (Cont... |
| falxorfal 1531 | A ` \/_ ` identity. (Cont... |
| hadbi123d 1534 | Equality theorem for the a... |
| hadbi123i 1535 | Equality theorem for the a... |
| hadass 1536 | Associative law for the ad... |
| hadbi 1537 | The adder sum is the same ... |
| hadcoma 1538 | Commutative law for the ad... |
| hadcomb 1539 | Commutative law for the ad... |
| hadrot 1540 | Rotation law for the adder... |
| hadnot 1541 | The adder sum distributes ... |
| had1 1542 | If the first input is true... |
| had0 1543 | If the first input is fals... |
| hadifp 1544 | The value of the adder sum... |
| cador 1547 | The adder carry in disjunc... |
| cadan 1548 | The adder carry in conjunc... |
| cadbi123d 1549 | Equality theorem for the a... |
| cadbi123i 1550 | Equality theorem for the a... |
| cadcoma 1551 | Commutative law for the ad... |
| cadcomb 1552 | Commutative law for the ad... |
| cadrot 1553 | Rotation law for the adder... |
| cadnot 1554 | The adder carry distribute... |
| cad1 1555 | If one input is true, then... |
| cad0 1556 | If one input is false, the... |
| cadifp 1557 | The value of the carry is,... |
| cad11 1558 | If (at least) two inputs a... |
| cadtru 1559 | The adder carry is true as... |
| minimp 1560 | A single axiom for minimal... |
| minimp-sylsimp 1561 | Derivation of sylsimp ( ~ ... |
| minimp-ax1 1562 | Derivation of ~ ax-1 from ... |
| minimp-ax2c 1563 | Derivation of a commuted f... |
| minimp-ax2 1564 | Derivation of ~ ax-2 from ... |
| minimp-pm2.43 1565 | Derivation of ~ pm2.43 (al... |
| meredith 1566 | Carew Meredith's sole axio... |
| merlem1 1567 | Step 3 of Meredith's proof... |
| merlem2 1568 | Step 4 of Meredith's proof... |
| merlem3 1569 | Step 7 of Meredith's proof... |
| merlem4 1570 | Step 8 of Meredith's proof... |
| merlem5 1571 | Step 11 of Meredith's proo... |
| merlem6 1572 | Step 12 of Meredith's proo... |
| merlem7 1573 | Between steps 14 and 15 of... |
| merlem8 1574 | Step 15 of Meredith's proo... |
| merlem9 1575 | Step 18 of Meredith's proo... |
| merlem10 1576 | Step 19 of Meredith's proo... |
| merlem11 1577 | Step 20 of Meredith's proo... |
| merlem12 1578 | Step 28 of Meredith's proo... |
| merlem13 1579 | Step 35 of Meredith's proo... |
| luk-1 1580 | 1 of 3 axioms for proposit... |
| luk-2 1581 | 2 of 3 axioms for proposit... |
| luk-3 1582 | 3 of 3 axioms for proposit... |
| luklem1 1583 | Used to rederive standard ... |
| luklem2 1584 | Used to rederive standard ... |
| luklem3 1585 | Used to rederive standard ... |
| luklem4 1586 | Used to rederive standard ... |
| luklem5 1587 | Used to rederive standard ... |
| luklem6 1588 | Used to rederive standard ... |
| luklem7 1589 | Used to rederive standard ... |
| luklem8 1590 | Used to rederive standard ... |
| ax1 1591 | Standard propositional axi... |
| ax2 1592 | Standard propositional axi... |
| ax3 1593 | Standard propositional axi... |
| nic-dfim 1594 | Define implication in term... |
| nic-dfneg 1595 | Define negation in terms o... |
| nic-mp 1596 | Derive Nicod's rule of mod... |
| nic-mpALT 1597 | A direct proof of ~ nic-mp... |
| nic-ax 1598 | Nicod's axiom derived from... |
| nic-axALT 1599 | A direct proof of ~ nic-ax... |
| nic-imp 1600 | Inference for ~ nic-mp usi... |
| nic-idlem1 1601 | Lemma for ~ nic-id . (Con... |
| nic-idlem2 1602 | Lemma for ~ nic-id . Infe... |
| nic-id 1603 | Theorem ~ id expressed wit... |
| nic-swap 1604 | The connector ` -/\ ` is s... |
| nic-isw1 1605 | Inference version of ~ nic... |
| nic-isw2 1606 | Inference for swapping nes... |
| nic-iimp1 1607 | Inference version of ~ nic... |
| nic-iimp2 1608 | Inference version of ~ nic... |
| nic-idel 1609 | Inference to remove the tr... |
| nic-ich 1610 | Chained inference. (Contr... |
| nic-idbl 1611 | Double the terms. Since d... |
| nic-bijust 1612 | Biconditional justificatio... |
| nic-bi1 1613 | Inference to extract one s... |
| nic-bi2 1614 | Inference to extract the o... |
| nic-stdmp 1615 | Derive the standard modus ... |
| nic-luk1 1616 | Proof of ~ luk-1 from ~ ni... |
| nic-luk2 1617 | Proof of ~ luk-2 from ~ ni... |
| nic-luk3 1618 | Proof of ~ luk-3 from ~ ni... |
| lukshef-ax1 1619 | This alternative axiom for... |
| lukshefth1 1620 | Lemma for ~ renicax . (Co... |
| lukshefth2 1621 | Lemma for ~ renicax . (Co... |
| renicax 1622 | A rederivation of ~ nic-ax... |
| tbw-bijust 1623 | Justification for ~ tbw-ne... |
| tbw-negdf 1624 | The definition of negation... |
| tbw-ax1 1625 | The first of four axioms i... |
| tbw-ax2 1626 | The second of four axioms ... |
| tbw-ax3 1627 | The third of four axioms i... |
| tbw-ax4 1628 | The fourth of four axioms ... |
| tbwsyl 1629 | Used to rederive the Lukas... |
| tbwlem1 1630 | Used to rederive the Lukas... |
| tbwlem2 1631 | Used to rederive the Lukas... |
| tbwlem3 1632 | Used to rederive the Lukas... |
| tbwlem4 1633 | Used to rederive the Lukas... |
| tbwlem5 1634 | Used to rederive the Lukas... |
| re1luk1 1635 | ~ luk-1 derived from the T... |
| re1luk2 1636 | ~ luk-2 derived from the T... |
| re1luk3 1637 | ~ luk-3 derived from the T... |
| merco1 1638 | A single axiom for proposi... |
| merco1lem1 1639 | Used to rederive the Tarsk... |
| retbwax4 1640 | ~ tbw-ax4 rederived from ~... |
| retbwax2 1641 | ~ tbw-ax2 rederived from ~... |
| merco1lem2 1642 | Used to rederive the Tarsk... |
| merco1lem3 1643 | Used to rederive the Tarsk... |
| merco1lem4 1644 | Used to rederive the Tarsk... |
| merco1lem5 1645 | Used to rederive the Tarsk... |
| merco1lem6 1646 | Used to rederive the Tarsk... |
| merco1lem7 1647 | Used to rederive the Tarsk... |
| retbwax3 1648 | ~ tbw-ax3 rederived from ~... |
| merco1lem8 1649 | Used to rederive the Tarsk... |
| merco1lem9 1650 | Used to rederive the Tarsk... |
| merco1lem10 1651 | Used to rederive the Tarsk... |
| merco1lem11 1652 | Used to rederive the Tarsk... |
| merco1lem12 1653 | Used to rederive the Tarsk... |
| merco1lem13 1654 | Used to rederive the Tarsk... |
| merco1lem14 1655 | Used to rederive the Tarsk... |
| merco1lem15 1656 | Used to rederive the Tarsk... |
| merco1lem16 1657 | Used to rederive the Tarsk... |
| merco1lem17 1658 | Used to rederive the Tarsk... |
| merco1lem18 1659 | Used to rederive the Tarsk... |
| retbwax1 1660 | ~ tbw-ax1 rederived from ~... |
| merco2 1661 | A single axiom for proposi... |
| mercolem1 1662 | Used to rederive the Tarsk... |
| mercolem2 1663 | Used to rederive the Tarsk... |
| mercolem3 1664 | Used to rederive the Tarsk... |
| mercolem4 1665 | Used to rederive the Tarsk... |
| mercolem5 1666 | Used to rederive the Tarsk... |
| mercolem6 1667 | Used to rederive the Tarsk... |
| mercolem7 1668 | Used to rederive the Tarsk... |
| mercolem8 1669 | Used to rederive the Tarsk... |
| re1tbw1 1670 | ~ tbw-ax1 rederived from ~... |
| re1tbw2 1671 | ~ tbw-ax2 rederived from ~... |
| re1tbw3 1672 | ~ tbw-ax3 rederived from ~... |
| re1tbw4 1673 | ~ tbw-ax4 rederived from ~... |
| rb-bijust 1674 | Justification for ~ rb-imd... |
| rb-imdf 1675 | The definition of implicat... |
| anmp 1676 | Modus ponens for ` \/ ` ` ... |
| rb-ax1 1677 | The first of four axioms i... |
| rb-ax2 1678 | The second of four axioms ... |
| rb-ax3 1679 | The third of four axioms i... |
| rb-ax4 1680 | The fourth of four axioms ... |
| rbsyl 1681 | Used to rederive the Lukas... |
| rblem1 1682 | Used to rederive the Lukas... |
| rblem2 1683 | Used to rederive the Lukas... |
| rblem3 1684 | Used to rederive the Lukas... |
| rblem4 1685 | Used to rederive the Lukas... |
| rblem5 1686 | Used to rederive the Lukas... |
| rblem6 1687 | Used to rederive the Lukas... |
| rblem7 1688 | Used to rederive the Lukas... |
| re1axmp 1689 | ~ ax-mp derived from Russe... |
| re2luk1 1690 | ~ luk-1 derived from Russe... |
| re2luk2 1691 | ~ luk-2 derived from Russe... |
| re2luk3 1692 | ~ luk-3 derived from Russe... |
| mptnan 1693 | Modus ponendo tollens 1, o... |
| mptxor 1694 | Modus ponendo tollens 2, o... |
| mtpor 1695 | Modus tollendo ponens (inc... |
| mtpxor 1696 | Modus tollendo ponens (ori... |
| stoic1a 1697 | Stoic logic Thema 1 (part ... |
| stoic1b 1698 | Stoic logic Thema 1 (part ... |
| stoic2a 1699 | Stoic logic Thema 2 versio... |
| stoic2b 1700 | Stoic logic Thema 2 versio... |
| stoic3 1701 | Stoic logic Thema 3. Stat... |
| stoic4a 1702 | Stoic logic Thema 4 versio... |
| stoic4b 1703 | Stoic logic Thema 4 versio... |
| alnex 1706 | Theorem 19.7 of [Margaris]... |
| eximal 1707 | A utility theorem. An int... |
| nf2 1711 | Alternate definition of no... |
| nf3 1712 | Alternate definition of no... |
| nf4 1713 | Alternate definition of no... |
| nfi 1714 | Deduce that ` x ` is not f... |
| nfri 1715 | Consequence of the definit... |
| nfd 1716 | Deduce that ` x ` is not f... |
| nfrd 1717 | Consequence of the definit... |
| nftht 1718 | Closed form of ~ nfth . (... |
| nfntht 1719 | Closed form of ~ nfnth . ... |
| nfntht2 1720 | Closed form of ~ nfnth . ... |
| gen2 1723 | Generalization applied twi... |
| mpg 1724 | Modus ponens combined with... |
| mpgbi 1725 | Modus ponens on biconditio... |
| mpgbir 1726 | Modus ponens on biconditio... |
| nfth 1727 | No variable is (effectivel... |
| nfnth 1728 | No variable is (effectivel... |
| hbth 1729 | No variable is (effectivel... |
| nftru 1730 | The true constant has no f... |
| nex 1731 | Generalization rule for ne... |
| nffal 1732 | The false constant has no ... |
| sptruw 1733 | Version of ~ sp when ` ph ... |
| nfiOLD 1734 | Obsolete proof of ~ nf5i a... |
| nfthOLD 1735 | Obsolete proof of ~ nfth a... |
| nfnthOLD 1736 | Obsolete proof of ~ nfnth ... |
| alim 1738 | Restatement of Axiom ~ ax-... |
| alimi 1739 | Inference quantifying both... |
| 2alimi 1740 | Inference doubly quantifyi... |
| ala1 1741 | Add an antecedent in a uni... |
| al2im 1742 | Closed form of ~ al2imi . ... |
| al2imi 1743 | Inference quantifying ante... |
| alanimi 1744 | Variant of ~ al2imi with c... |
| alimdh 1745 | Deduction form of Theorem ... |
| albi 1746 | Theorem 19.15 of [Margaris... |
| albii 1747 | Inference adding universal... |
| 2albii 1748 | Inference adding two unive... |
| sylgt 1749 | Closed form of ~ sylg . (... |
| sylg 1750 | A syllogism combined with ... |
| alrimih 1751 | Inference form of Theorem ... |
| hbxfrbi 1752 | A utility lemma to transfe... |
| alex 1753 | Universal quantifier in te... |
| exnal 1754 | Theorem 19.14 of [Margaris... |
| 2nalexn 1755 | Part of theorem *11.5 in [... |
| 2exnaln 1756 | Theorem *11.22 in [Whitehe... |
| 2nexaln 1757 | Theorem *11.25 in [Whitehe... |
| alimex 1758 | A utility theorem. An int... |
| aleximi 1759 | A variant of ~ al2imi : in... |
| alexbii 1760 | Biconditional form of ~ al... |
| exim 1761 | Theorem 19.22 of [Margaris... |
| eximi 1762 | Inference adding existenti... |
| 2eximi 1763 | Inference adding two exist... |
| eximii 1764 | Inference associated with ... |
| exa1 1765 | Add an antecedent in an ex... |
| 19.38 1766 | Theorem 19.38 of [Margaris... |
| 19.38a 1767 | Under a non-freeness hypot... |
| 19.38b 1768 | Under a non-freeness hypot... |
| imnang 1769 | Quantified implication in ... |
| alinexa 1770 | A transformation of quanti... |
| alexn 1771 | A relationship between two... |
| 2exnexn 1772 | Theorem *11.51 in [Whitehe... |
| exbi 1773 | Theorem 19.18 of [Margaris... |
| exbii 1774 | Inference adding existenti... |
| 2exbii 1775 | Inference adding two exist... |
| 3exbii 1776 | Inference adding three exi... |
| nfbiit 1777 | Equivalence theorem for th... |
| nfbii 1778 | Equality theorem for the n... |
| nfxfr 1779 | A utility lemma to transfe... |
| nfxfrd 1780 | A utility lemma to transfe... |
| nfnbi 1781 | A variable is non-free in ... |
| nfnt 1782 | If a variable is non-free ... |
| nfntOLDOLD 1783 | Obsolete proof of ~ nfnt a... |
| nfn 1784 | Inference associated with ... |
| nfnd 1785 | Deduction associated with ... |
| exanali 1786 | A transformation of quanti... |
| exancom 1787 | Commutation of conjunction... |
| exan 1788 | Place a conjunct in the sc... |
| exanOLD 1789 | Obsolete proof of ~ exan a... |
| alrimdh 1790 | Deduction form of Theorem ... |
| eximdh 1791 | Deduction from Theorem 19.... |
| nexdh 1792 | Deduction for generalizati... |
| albidh 1793 | Formula-building rule for ... |
| exbidh 1794 | Formula-building rule for ... |
| exsimpl 1795 | Simplification of an exist... |
| exsimpr 1796 | Simplification of an exist... |
| 19.40 1797 | Theorem 19.40 of [Margaris... |
| 19.26 1798 | Theorem 19.26 of [Margaris... |
| 19.26-2 1799 | Theorem ~ 19.26 with two q... |
| 19.26-3an 1800 | Theorem ~ 19.26 with tripl... |
| 19.29 1801 | Theorem 19.29 of [Margaris... |
| 19.29r 1802 | Variation of ~ 19.29 . (C... |
| 19.29r2 1803 | Variation of ~ 19.29r with... |
| 19.29x 1804 | Variation of ~ 19.29 with ... |
| 19.35 1805 | Theorem 19.35 of [Margaris... |
| 19.35i 1806 | Inference associated with ... |
| 19.35ri 1807 | Inference associated with ... |
| 19.25 1808 | Theorem 19.25 of [Margaris... |
| 19.30 1809 | Theorem 19.30 of [Margaris... |
| 19.43 1810 | Theorem 19.43 of [Margaris... |
| 19.43OLD 1811 | Obsolete proof of ~ 19.43 ... |
| 19.33 1812 | Theorem 19.33 of [Margaris... |
| 19.33b 1813 | The antecedent provides a ... |
| 19.40-2 1814 | Theorem *11.42 in [Whitehe... |
| 19.40b 1815 | The antecedent provides a ... |
| albiim 1816 | Split a biconditional and ... |
| 2albiim 1817 | Split a biconditional and ... |
| exintrbi 1818 | Add/remove a conjunct in t... |
| exintr 1819 | Introduce a conjunct in th... |
| alsyl 1820 | Universally quantified and... |
| nfimt 1821 | Closed form of ~ nfim and ... |
| nfimt2 1822 | Closed form of ~ nfim and ... |
| nfimd 1823 | If in a context ` x ` is n... |
| nfimdOLDOLD 1824 | Obsolete proof of ~ nfimd ... |
| nfim 1825 | If ` x ` is not free in ` ... |
| nfand 1826 | If in a context ` x ` is n... |
| nf3and 1827 | Deduction form of bound-va... |
| nfan 1828 | If ` x ` is not free in ` ... |
| nfanOLD 1829 | Obsolete proof of ~ nfan a... |
| nfnan 1830 | If ` x ` is not free in ` ... |
| nf3an 1831 | If ` x ` is not free in ` ... |
| nfbid 1832 | If in a context ` x ` is n... |
| nfbi 1833 | If ` x ` is not free in ` ... |
| nfor 1834 | If ` x ` is not free in ` ... |
| nf3or 1835 | If ` x ` is not free in ` ... |
| nfbiiOLD 1836 | Obsolete proof of ~ nfbii ... |
| nfxfrOLD 1837 | Obsolete proof of ~ nfxfr ... |
| nfxfrdOLD 1838 | Obsolete proof of ~ nfxfrd... |
| ax5d 1840 | ~ ax-5 with antecedent. U... |
| ax5e 1841 | A rephrasing of ~ ax-5 usi... |
| ax5ea 1842 | If a formula holds for som... |
| nfv 1843 | If ` x ` is not present in... |
| nfvd 1844 | ~ nfv with antecedent. Us... |
| alimdv 1845 | Deduction form of Theorem ... |
| eximdv 1846 | Deduction form of Theorem ... |
| 2alimdv 1847 | Deduction form of Theorem ... |
| 2eximdv 1848 | Deduction form of Theorem ... |
| albidv 1849 | Formula-building rule for ... |
| exbidv 1850 | Formula-building rule for ... |
| 2albidv 1851 | Formula-building rule for ... |
| 2exbidv 1852 | Formula-building rule for ... |
| 3exbidv 1853 | Formula-building rule for ... |
| 4exbidv 1854 | Formula-building rule for ... |
| alrimiv 1855 | Inference form of Theorem ... |
| alrimivv 1856 | Inference form of Theorem ... |
| alrimdv 1857 | Deduction form of Theorem ... |
| exlimiv 1858 | Inference form of Theorem ... |
| exlimiiv 1859 | Inference associated with ... |
| exlimivv 1860 | Inference form of Theorem ... |
| exlimdv 1861 | Deduction form of Theorem ... |
| exlimdvv 1862 | Deduction form of Theorem ... |
| exlimddv 1863 | Existential elimination ru... |
| nexdv 1864 | Deduction for generalizati... |
| nexdvOLD 1865 | Obsolete proof of ~ nexdv ... |
| 2ax5 1866 | Quantification of two vari... |
| stdpc5v 1867 | Version of ~ stdpc5 with a... |
| 19.21v 1868 | Version of ~ 19.21 with a ... |
| 19.32v 1869 | Version of ~ 19.32 with a ... |
| 19.31v 1870 | Version of ~ 19.31 with a ... |
| nfvOLD 1871 | Obsolete proof of ~ nfv as... |
| nfvdOLD 1872 | Obsolete proof of ~ nfvd a... |
| nfdvOLD 1873 | Obsolete proof of ~ nf5dv ... |
| weq 1874 | Extend wff definition to i... |
| equs3 1875 | Lemma used in proofs of su... |
| speimfw 1876 | Specialization, with addit... |
| speimfwALT 1877 | Alternate proof of ~ speim... |
| spimfw 1878 | Specialization, with addit... |
| ax12i 1879 | Inference that has ~ ax-12... |
| sbequ2 1882 | An equality theorem for su... |
| sb1 1883 | One direction of a simplif... |
| spsbe 1884 | A specialization theorem. ... |
| sbequ8 1885 | Elimination of equality fr... |
| sbimi 1886 | Infer substitution into an... |
| sbbii 1887 | Infer substitution into bo... |
| ax6v 1889 | Axiom B7 of [Tarski] p. 75... |
| ax6ev 1890 | At least one individual ex... |
| exiftru 1891 | Rule of existential genera... |
| 19.2 1892 | Theorem 19.2 of [Margaris]... |
| 19.2d 1893 | Deduction associated with ... |
| 19.8w 1894 | Weak version of ~ 19.8a an... |
| 19.8v 1895 | Version of ~ 19.8a with a ... |
| 19.9v 1896 | Version of ~ 19.9 with a d... |
| 19.3v 1897 | Version of ~ 19.3 with a d... |
| spvw 1898 | Version of ~ sp when ` x `... |
| 19.39 1899 | Theorem 19.39 of [Margaris... |
| 19.24 1900 | Theorem 19.24 of [Margaris... |
| 19.34 1901 | Theorem 19.34 of [Margaris... |
| 19.23v 1902 | Version of ~ 19.23 with a ... |
| 19.23vv 1903 | Theorem ~ 19.23v extended ... |
| 19.36v 1904 | Version of ~ 19.36 with a ... |
| 19.36iv 1905 | Inference associated with ... |
| pm11.53v 1906 | Version of ~ pm11.53 with ... |
| 19.12vvv 1907 | Version of ~ 19.12vv with ... |
| 19.27v 1908 | Version of ~ 19.27 with a ... |
| 19.28v 1909 | Version of ~ 19.28 with a ... |
| 19.37v 1910 | Version of ~ 19.37 with a ... |
| 19.37iv 1911 | Inference associated with ... |
| 19.44v 1912 | Version of ~ 19.44 with a ... |
| 19.45v 1913 | Version of ~ 19.45 with a ... |
| 19.41v 1914 | Version of ~ 19.41 with a ... |
| 19.41vv 1915 | Version of ~ 19.41 with tw... |
| 19.41vvv 1916 | Version of ~ 19.41 with th... |
| 19.41vvvv 1917 | Version of ~ 19.41 with fo... |
| 19.42v 1918 | Version of ~ 19.42 with a ... |
| exdistr 1919 | Distribution of existentia... |
| 19.42vv 1920 | Version of ~ 19.42 with tw... |
| 19.42vvv 1921 | Version of ~ 19.42 with th... |
| exdistr2 1922 | Distribution of existentia... |
| 3exdistr 1923 | Distribution of existentia... |
| 4exdistr 1924 | Distribution of existentia... |
| spimeh 1925 | Existential introduction, ... |
| spimw 1926 | Specialization. Lemma 8 o... |
| spimvw 1927 | Specialization. Lemma 8 o... |
| spnfw 1928 | Weak version of ~ sp . Us... |
| spfalw 1929 | Version of ~ sp when ` ph ... |
| equs4v 1930 | Version of ~ equs4 with a ... |
| equsalvw 1931 | Version of ~ equsalv with ... |
| equsexvw 1932 | Version of ~ equsexv with ... |
| cbvaliw 1933 | Change bound variable. Us... |
| cbvalivw 1934 | Change bound variable. Us... |
| ax7v 1936 | Weakened version of ~ ax-7... |
| ax7v1 1937 | First of two weakened vers... |
| ax7v2 1938 | Second of two weakened ver... |
| equid 1939 | Identity law for equality.... |
| nfequid 1940 | Bound-variable hypothesis ... |
| equcomiv 1941 | Weaker form of ~ equcomi w... |
| ax6evr 1942 | A commuted form of ~ ax6ev... |
| ax7 1943 | Proof of ~ ax-7 from ~ ax7... |
| equcomi 1944 | Commutative law for equali... |
| equcom 1945 | Commutative law for equali... |
| equcomd 1946 | Deduction form of ~ equcom... |
| equcoms 1947 | An inference commuting equ... |
| equtr 1948 | A transitive law for equal... |
| equtrr 1949 | A transitive law for equal... |
| equeuclr 1950 | Commuted version of ~ eque... |
| equeucl 1951 | Equality is a left-Euclide... |
| equequ1 1952 | An equivalence law for equ... |
| equequ2 1953 | An equivalence law for equ... |
| equtr2 1954 | Equality is a left-Euclide... |
| equequ2OLD 1955 | Obsolete proof of ~ equequ... |
| equtr2OLD 1956 | Obsolete proof of ~ equtr2... |
| stdpc6 1957 | One of the two equality ax... |
| stdpc7 1958 | One of the two equality ax... |
| equvinv 1959 | A variable introduction la... |
| equviniva 1960 | A modified version of the ... |
| equvinivOLD 1961 | The forward implication of... |
| equvinvOLD 1962 | Obsolete version of ~ equv... |
| equvelv 1963 | A specialized version of ~... |
| ax13b 1964 | An equivalence between two... |
| spfw 1965 | Weak version of ~ sp . Us... |
| spfwOLD 1966 | Obsolete proof of ~ spfw a... |
| spw 1967 | Weak version of the specia... |
| cbvalw 1968 | Change bound variable. Us... |
| cbvalvw 1969 | Change bound variable. Us... |
| cbvexvw 1970 | Change bound variable. Us... |
| alcomiw 1971 | Weak version of ~ alcom . ... |
| hbn1fw 1972 | Weak version of ~ ax-10 fr... |
| hbn1w 1973 | Weak version of ~ hbn1 . ... |
| hba1w 1974 | Weak version of ~ hba1 . ... |
| hba1wOLD 1975 | Obsolete proof of ~ hba1w ... |
| hbe1w 1976 | Weak version of ~ hbe1 . ... |
| hbalw 1977 | Weak version of ~ hbal . ... |
| spaev 1978 | A special instance of ~ sp... |
| cbvaev 1979 | Change bound variable in a... |
| aevlem0 1980 | Lemma for ~ aevlem . Inst... |
| aevlem 1981 | Lemma for ~ aev and ~ axc1... |
| aeveq 1982 | The antecedent ` A. x x = ... |
| aev 1983 | A "distinctor elimination"... |
| hbaevg 1984 | Generalization of ~ hbaev ... |
| hbaev 1985 | Version of ~ hbae with a D... |
| aev2 1986 | A version of ~ aev with tw... |
| aev2ALT 1987 | Alternate proof of ~ aev2 ... |
| axc11nlemOLD2 1988 | Lemma for ~ axc11n . Chan... |
| aevlemOLD 1989 | Old proof of ~ aevlem . O... |
| wel 1991 | Extend wff definition to i... |
| ax8v 1993 | Weakened version of ~ ax-8... |
| ax8v1 1994 | First of two weakened vers... |
| ax8v2 1995 | Second of two weakened ver... |
| ax8 1996 | Proof of ~ ax-8 from ~ ax8... |
| elequ1 1997 | An identity law for the no... |
| cleljust 1998 | When the class variables i... |
| ax9v 2000 | Weakened version of ~ ax-9... |
| ax9v1 2001 | First of two weakened vers... |
| ax9v2 2002 | Second of two weakened ver... |
| ax9 2003 | Proof of ~ ax-9 from ~ ax9... |
| elequ2 2004 | An identity law for the no... |
| ax6dgen 2005 | Tarski's system uses the w... |
| ax10w 2006 | Weak version of ~ ax-10 fr... |
| ax11w 2007 | Weak version of ~ ax-11 fr... |
| ax11dgen 2008 | Degenerate instance of ~ a... |
| ax12wlem 2009 | Lemma for weak version of ... |
| ax12w 2010 | Weak version of ~ ax-12 fr... |
| ax12dgen 2011 | Degenerate instance of ~ a... |
| ax12wdemo 2012 | Example of an application ... |
| ax13w 2013 | Weak version (principal in... |
| ax13dgen1 2014 | Degenerate instance of ~ a... |
| ax13dgen2 2015 | Degenerate instance of ~ a... |
| ax13dgen3 2016 | Degenerate instance of ~ a... |
| ax13dgen4 2017 | Degenerate instance of ~ a... |
| ax13dgen4OLD 2018 | Obsolete proof of ~ ax13dg... |
| hbn1 2020 | Alias for ~ ax-10 to be us... |
| hbe1 2021 | The setvar ` x ` is not fr... |
| hbe1a 2022 | Dual statement of ~ hbe1 .... |
| nf5-1 2023 | One direction of ~ nf5 can... |
| nf5i 2024 | Deduce that ` x ` is not f... |
| nf5dv 2025 | Apply the definition of no... |
| nf5dh 2026 | Deduce that ` x ` is not f... |
| nfe1 2027 | The setvar ` x ` is not fr... |
| nfa1 2028 | The setvar ` x ` is not fr... |
| nfna1 2029 | A convenience theorem part... |
| nfia1 2030 | Lemma 23 of [Monk2] p. 114... |
| nfnf1 2031 | The setvar ` x ` is not fr... |
| modal-5 2032 | The analogue in our predic... |
| nfe1OLD 2033 | Obsolete proof of ~ nfe1 a... |
| alcoms 2035 | Swap quantifiers in an ant... |
| hbal 2036 | If ` x ` is not free in ` ... |
| alcom 2037 | Theorem 19.5 of [Margaris]... |
| alrot3 2038 | Theorem *11.21 in [Whitehe... |
| alrot4 2039 | Rotate four universal quan... |
| nfa2 2040 | Lemma 24 of [Monk2] p. 114... |
| hbald 2041 | Deduction form of bound-va... |
| excom 2042 | Theorem 19.11 of [Margaris... |
| excomim 2043 | One direction of Theorem 1... |
| excom13 2044 | Swap 1st and 3rd existenti... |
| exrot3 2045 | Rotate existential quantif... |
| exrot4 2046 | Rotate existential quantif... |
| ax12v 2048 | This is essentially axiom ... |
| ax12v2 2049 | It is possible to remove a... |
| ax12vOLD 2050 | Obsolete proof of ~ ax12v2... |
| ax12vOLDOLD 2051 | Obsolete proof of ~ ax12v ... |
| 19.8a 2052 | If a wff is true, it is tr... |
| sp 2053 | Specialization. A univers... |
| spi 2054 | Inference rule reversing g... |
| sps 2055 | Generalization of antecede... |
| 2sp 2056 | A double specialization (s... |
| spsd 2057 | Deduction generalizing ant... |
| 19.2g 2058 | Theorem 19.2 of [Margaris]... |
| 19.21bi 2059 | Inference form of ~ 19.21 ... |
| 19.21bbi 2060 | Inference removing double ... |
| 19.23bi 2061 | Inference form of Theorem ... |
| nexr 2062 | Inference associated with ... |
| qexmid 2063 | Quantified excluded middle... |
| nf5r 2064 | Consequence of the definit... |
| nf5ri 2065 | Consequence of the definit... |
| nf5rd 2066 | Consequence of the definit... |
| nfim1 2067 | A closed form of ~ nfim . ... |
| nfan1 2068 | A closed form of ~ nfan . ... |
| 19.3 2069 | A wff may be quantified wi... |
| 19.9d 2070 | A deduction version of one... |
| 19.9t 2071 | A closed version of ~ 19.9... |
| 19.9 2072 | A wff may be existentially... |
| 19.21t 2073 | Closed form of Theorem 19.... |
| 19.21tOLDOLD 2074 | Obsolete proof of ~ 19.21t... |
| 19.21 2075 | Theorem 19.21 of [Margaris... |
| stdpc5 2076 | An axiom scheme of standar... |
| stdpc5OLD 2077 | Obsolete proof of ~ stdpc5... |
| 19.21-2 2078 | Version of ~ 19.21 with tw... |
| 19.23t 2079 | Closed form of Theorem 197... |
| 19.23 2080 | Theorem 19.23 of [Margaris... |
| alimd 2081 | Deduction form of Theorem ... |
| alrimi 2082 | Inference form of Theorem ... |
| alrimdd 2083 | Deduction form of Theorem ... |
| alrimd 2084 | Deduction form of Theorem ... |
| eximd 2085 | Deduction form of Theorem ... |
| exlimi 2086 | Inference associated with ... |
| exlimd 2087 | Deduction form of Theorem ... |
| exlimdd 2088 | Existential elimination ru... |
| nexd 2089 | Deduction for generalizati... |
| albid 2090 | Formula-building rule for ... |
| exbid 2091 | Formula-building rule for ... |
| nfbidf 2092 | An equality theorem for ef... |
| 19.16 2093 | Theorem 19.16 of [Margaris... |
| 19.17 2094 | Theorem 19.17 of [Margaris... |
| 19.27 2095 | Theorem 19.27 of [Margaris... |
| 19.28 2096 | Theorem 19.28 of [Margaris... |
| 19.19 2097 | Theorem 19.19 of [Margaris... |
| 19.36 2098 | Theorem 19.36 of [Margaris... |
| 19.36i 2099 | Inference associated with ... |
| 19.37 2100 | Theorem 19.37 of [Margaris... |
| 19.32 2101 | Theorem 19.32 of [Margaris... |
| 19.31 2102 | Theorem 19.31 of [Margaris... |
| 19.41 2103 | Theorem 19.41 of [Margaris... |
| 19.42-1 2104 | One direction of ~ 19.42 .... |
| 19.42 2105 | Theorem 19.42 of [Margaris... |
| 19.44 2106 | Theorem 19.44 of [Margaris... |
| 19.45 2107 | Theorem 19.45 of [Margaris... |
| equsalv 2108 | Version of ~ equsal with a... |
| equsexv 2109 | Version of ~ equsex with a... |
| sbequ1 2110 | An equality theorem for su... |
| sbequ12 2111 | An equality theorem for su... |
| sbequ12r 2112 | An equality theorem for su... |
| sbequ12a 2113 | An equality theorem for su... |
| sbid 2114 | An identity theorem for su... |
| spimv1 2115 | Version of ~ spim with a d... |
| nf5 2116 | Alternate definition of ~ ... |
| nf6 2117 | An alternate definition of... |
| nf5d 2118 | Deduce that ` x ` is not f... |
| nf5di 2119 | Since the converse holds b... |
| 19.9h 2120 | A wff may be existentially... |
| 19.21h 2121 | Theorem 19.21 of [Margaris... |
| 19.23h 2122 | Theorem 19.23 of [Margaris... |
| equsalhw 2123 | Weaker version of ~ equsal... |
| equsexhv 2124 | Version of ~ equsexh with ... |
| hbim1 2125 | A closed form of ~ hbim . ... |
| hbimd 2126 | Deduction form of bound-va... |
| hbim 2127 | If ` x ` is not free in ` ... |
| hban 2128 | If ` x ` is not free in ` ... |
| hb3an 2129 | If ` x ` is not free in ` ... |
| axc4 2130 | Show that the original axi... |
| axc4i 2131 | Inference version of ~ axc... |
| axc7 2132 | Show that the original axi... |
| axc7e 2133 | Abbreviated version of ~ a... |
| axc16g 2134 | Generalization of ~ axc16 ... |
| axc16 2135 | Proof of older axiom ~ ax-... |
| axc16gb 2136 | Biconditional strengthenin... |
| axc16nf 2137 | If ~ dtru is false, then t... |
| axc11v 2138 | Version of ~ axc11 with a ... |
| axc11rv 2139 | Version of ~ axc11r with a... |
| axc11rvOLD 2140 | Obsolete proof of ~ axc11r... |
| axc11vOLD 2141 | Obsolete proof of ~ axc11v... |
| modal-b 2142 | The analogue in our predic... |
| 19.9ht 2143 | A closed version of ~ 19.9... |
| hbnt 2144 | Closed theorem version of ... |
| hbntOLD 2145 | Obsolete proof of ~ hbnt a... |
| hbn 2146 | If ` x ` is not free in ` ... |
| hbnd 2147 | Deduction form of bound-va... |
| exlimih 2148 | Inference associated with ... |
| exlimdh 2149 | Deduction form of Theorem ... |
| sb56 2150 | Two equivalent ways of exp... |
| hba1 2151 | The setvar ` x ` is not fr... |
| hbexOLD 2152 | Obsolete proof of ~ hbex a... |
| nfal 2153 | If ` x ` is not free in ` ... |
| nfex 2154 | If ` x ` is not free in ` ... |
| nfexOLD 2155 | Obsolete proof of ~ nfex a... |
| hbex 2156 | If ` x ` is not free in ` ... |
| nfa1OLD 2157 | Obsolete proof of ~ nfa1 a... |
| nfnf 2158 | If ` x ` is not free in ` ... |
| nfnf1OLD 2159 | Obsolete proof of ~ nfnf1 ... |
| axc11nlemOLD 2160 | Obsolete proof of ~ axc11n... |
| axc16gOLD 2161 | Obsolete proof of ~ axc16g... |
| aevOLD 2162 | Obsolete proof of ~ aev as... |
| axc16nfOLD 2163 | Obsolete proof of ~ axc16n... |
| 19.12 2164 | Theorem 19.12 of [Margaris... |
| nfald 2165 | Deduction form of ~ nfal .... |
| nfaldOLD 2166 | Obsolete proof of ~ nfald ... |
| nfexd 2167 | If ` x ` is not free in ` ... |
| nfa2OLD 2168 | Obsolete proof of ~ nfa2 a... |
| exanOLDOLD 2169 | Obsolete proof of ~ exan a... |
| aaan 2170 | Rearrange universal quanti... |
| eeor 2171 | Rearrange existential quan... |
| cbv3v 2172 | Version of ~ cbv3 with a d... |
| dvelimhw 2173 | Proof of ~ dvelimh without... |
| cbv3hv 2174 | Version of ~ cbv3h with a ... |
| cbvalv1 2175 | Version of ~ cbval with a ... |
| cbvexv1 2176 | Version of ~ cbvex with a ... |
| equs5aALT 2177 | Alternate proof of ~ equs5... |
| equs5eALT 2178 | Alternate proof of ~ equs5... |
| pm11.53 2179 | Theorem *11.53 in [Whitehe... |
| 19.12vv 2180 | Special case of ~ 19.12 wh... |
| eean 2181 | Rearrange existential quan... |
| eeanv 2182 | Rearrange existential quan... |
| eeeanv 2183 | Rearrange existential quan... |
| ee4anv 2184 | Rearrange existential quan... |
| cleljustALT 2185 | Alternate proof of ~ clelj... |
| cleljustALT2 2186 | Alternate proof of ~ clelj... |
| axc11r 2187 | Same as ~ axc11 but with r... |
| nfrOLD 2188 | Obsolete proof of ~ nf5r a... |
| nfriOLD 2189 | Obsolete proof of ~ nf5ri ... |
| nfrdOLD 2190 | Obsolete proof of ~ nf5rd ... |
| alimdOLD 2191 | Obsolete proof of ~ alimd ... |
| alrimiOLD 2192 | Obsolete proof of ~ alrimi... |
| nfdOLD 2193 | Obsolete proof of ~ nf5d a... |
| nfdhOLD 2194 | Obsolete proof of ~ nf5dh ... |
| alrimddOLD 2195 | Obsolete proof of ~ alrimd... |
| alrimdOLD 2196 | Obsolete proof of ~ alrimd... |
| eximdOLD 2197 | Obsolete proof of ~ eximd ... |
| nexdOLD 2198 | Obsolete proof of ~ nexd a... |
| albidOLD 2199 | Obsolete proof of ~ albid ... |
| exbidOLD 2200 | Obsolete proof of ~ exbid ... |
| nfbidfOLD 2201 | Obsolete proof of ~ nfbidf... |
| 19.3OLD 2202 | Obsolete proof of ~ 19.3 a... |
| 19.9dOLD 2203 | Obsolete proof of ~ 19.9d ... |
| 19.9tOLD 2204 | Obsolete proof of ~ 19.9t ... |
| 19.9OLD 2205 | Obsolete proof of ~ 19.9 a... |
| 19.9hOLD 2206 | Obsolete proof of ~ 19.9h ... |
| nfa1OLDOLD 2207 | Obsolete proof of ~ nfa1 a... |
| nfnf1OLDOLD 2208 | Obsolete proof of ~ nfnf1 ... |
| nfntOLD 2209 | Obsolete proof of ~ nfnt a... |
| nfnOLD 2210 | Obsolete proof of ~ nfn as... |
| nfndOLD 2211 | Obsolete proof of ~ nfnd a... |
| 19.21t-1OLD 2212 | One direction of the bi-co... |
| 19.21tOLD 2213 | Obsolete proof of ~ 19.21t... |
| 19.21OLD 2214 | Obsolete proof of ~ 19.21 ... |
| 19.21-2OLD 2215 | Obsolete proof of ~ 19.21-... |
| 19.21hOLD 2216 | Obsolete proof of ~ 19.21h... |
| stdpc5OLDOLD 2217 | Obsolete proof of ~ stdpc5... |
| 19.23tOLD 2218 | Obsolete proof of ~ 19.23t... |
| 19.23OLD 2219 | Obsolete proof of ~ 19.23 ... |
| 19.23hOLD 2220 | Obsolete proof of ~ 19.23h... |
| exlimiOLD 2221 | Obsolete proof of ~ exlimi... |
| exlimihOLD 2222 | Obsolete proof of ~ exlimi... |
| exlimdOLD 2223 | Obsolete proof of ~ exlimd... |
| exlimdhOLD 2224 | Obsolete proof of ~ exlimd... |
| nfdiOLD 2225 | Obsolete proof of ~ nf5di ... |
| nfimdOLD 2226 | Obsolete proof of ~ nfimd ... |
| hbim1OLD 2227 | Obsolete proof of ~ hbim a... |
| nfim1OLD 2228 | Obsolete proof of ~ nfim1 ... |
| nfimOLD 2229 | Obsolete proof of ~ nfim a... |
| hbimdOLD 2230 | Obsolete proof of ~ hbimd ... |
| hbimOLD 2231 | Obsolete proof of ~ hbim a... |
| nfandOLD 2232 | Obsolete proof of ~ nfand ... |
| nf3andOLD 2233 | Obsolete proof of ~ nf3and... |
| 19.27OLD 2234 | Obsolete proof of ~ 19.27 ... |
| 19.28OLD 2235 | Obsolete proof of ~ 19.28 ... |
| nfan1OLD 2236 | Obsolete proof of ~ nfan1 ... |
| nfanOLDOLD 2237 | Obsolete proof of ~ nfan a... |
| nfnanOLD 2238 | Obsolete proof of ~ nfnan ... |
| nf3anOLD 2239 | Obsolete proof of ~ nf3an ... |
| hbanOLD 2240 | Obsolete proof of ~ hban a... |
| hb3anOLD 2241 | Obsolete proof of ~ hb3an ... |
| nfbidOLD 2242 | Obsolete proof of ~ nfbid ... |
| nfbiOLD 2243 | Obsolete proof of ~ nfbi a... |
| nforOLD 2244 | Obsolete proof of ~ nfor a... |
| nf3orOLD 2245 | Obsolete proof of ~ nf3or ... |
| ax13v 2247 | A weaker version of ~ ax-1... |
| ax13lem1 2248 | A version of ~ ax13v with ... |
| ax13 2249 | Derive ~ ax-13 from ~ ax13... |
| ax6e 2250 | At least one individual ex... |
| ax6 2251 | Theorem showing that ~ ax-... |
| axc10 2252 | Show that the original axi... |
| spimt 2253 | Closed theorem form of ~ s... |
| spim 2254 | Specialization, using impl... |
| spimed 2255 | Deduction version of ~ spi... |
| spime 2256 | Existential introduction, ... |
| spimv 2257 | A version of ~ spim with a... |
| spimvALT 2258 | Alternate proof of ~ spimv... |
| spimev 2259 | Distinct-variable version ... |
| spv 2260 | Specialization, using impl... |
| spei 2261 | Inference from existential... |
| chvar 2262 | Implicit substitution of `... |
| chvarv 2263 | Implicit substitution of `... |
| chvarvOLD 2264 | Obsolete proof of ~ chvarv... |
| cbv3 2265 | Rule used to change bound ... |
| cbv3h 2266 | Rule used to change bound ... |
| cbv1 2267 | Rule used to change bound ... |
| cbv1h 2268 | Rule used to change bound ... |
| cbv2h 2269 | Rule used to change bound ... |
| cbv2 2270 | Rule used to change bound ... |
| cbval 2271 | Rule used to change bound ... |
| cbvex 2272 | Rule used to change bound ... |
| cbvalv 2273 | Rule used to change bound ... |
| cbvalvOLD 2274 | Obsolete proof of ~ cbvalv... |
| cbvexv 2275 | Rule used to change bound ... |
| cbvexvOLD 2276 | Obsolete proof of ~ cbvexv... |
| cbvald 2277 | Deduction used to change b... |
| cbvexd 2278 | Deduction used to change b... |
| cbval2 2279 | Rule used to change bound ... |
| cbvex2 2280 | Rule used to change bound ... |
| cbvaldva 2281 | Rule used to change the bo... |
| cbvaldvaOLD 2282 | Obsolete proof of ~ cbvald... |
| cbvexdva 2283 | Rule used to change the bo... |
| cbvexdvaOLD 2284 | Obsolete proof of ~ cbvexd... |
| cbval2v 2285 | Rule used to change bound ... |
| cbval2vOLD 2286 | Obsolete proof of ~ cbval2... |
| cbvex2v 2287 | Rule used to change bound ... |
| cbvex2vOLD 2288 | Obsolete proof of ~ cbvex2... |
| cbvex4v 2289 | Rule used to change bound ... |
| equs4 2290 | Lemma used in proofs of im... |
| equsal 2291 | An equivalence related to ... |
| equsex 2292 | An equivalence related to ... |
| equsexALT 2293 | Alternate proof of ~ equse... |
| equsalh 2294 | An equivalence related to ... |
| equsexh 2295 | An equivalence related to ... |
| ax13lem2 2296 | Lemma for ~ nfeqf2 . This... |
| nfeqf2 2297 | An equation between setvar... |
| dveeq2 2298 | Quantifier introduction wh... |
| nfeqf1 2299 | An equation between setvar... |
| dveeq1 2300 | Quantifier introduction wh... |
| nfeqf 2301 | A variable is effectively ... |
| axc9 2302 | Derive set.mm's original ~... |
| axc15 2303 | Derivation of set.mm's ori... |
| ax12 2304 | Rederivation of axiom ~ ax... |
| ax13ALT 2305 | Alternate proof of ~ ax13 ... |
| axc11nlemALT 2306 | Alternate version of ~ axc... |
| axc11n 2307 | Derive set.mm's original ~... |
| axc11nOLD 2308 | Obsolete proof of ~ axc11n... |
| axc11nOLDOLD 2309 | Old proof of ~ axc11n . O... |
| axc11nALT 2310 | Alternate proof of ~ axc11... |
| aecom 2311 | Commutation law for identi... |
| aecoms 2312 | A commutation rule for ide... |
| naecoms 2313 | A commutation rule for dis... |
| axc11 2314 | Show that ~ ax-c11 can be ... |
| hbae 2315 | All variables are effectiv... |
| nfae 2316 | All variables are effectiv... |
| hbnae 2317 | All variables are effectiv... |
| nfnae 2318 | All variables are effectiv... |
| hbnaes 2319 | Rule that applies ~ hbnae ... |
| aevlemALTOLD 2320 | Older alternate version of... |
| aevALTOLD 2321 | Older alternate proof of ~... |
| axc16i 2322 | Inference with ~ axc16 as ... |
| axc16nfALT 2323 | Alternate proof of ~ axc16... |
| dral2 2324 | Formula-building lemma for... |
| dral1 2325 | Formula-building lemma for... |
| dral1ALT 2326 | Alternate proof of ~ dral1... |
| drex1 2327 | Formula-building lemma for... |
| drex2 2328 | Formula-building lemma for... |
| drnf1 2329 | Formula-building lemma for... |
| drnf2 2330 | Formula-building lemma for... |
| nfald2 2331 | Variation on ~ nfald which... |
| nfexd2 2332 | Variation on ~ nfexd which... |
| exdistrf 2333 | Distribution of existentia... |
| dvelimf 2334 | Version of ~ dvelimv witho... |
| dvelimdf 2335 | Deduction form of ~ dvelim... |
| dvelimh 2336 | Version of ~ dvelim withou... |
| dvelim 2337 | This theorem can be used t... |
| dvelimv 2338 | Similar to ~ dvelim with f... |
| dvelimnf 2339 | Version of ~ dvelim using ... |
| dveeq2ALT 2340 | Alternate proof of ~ dveeq... |
| ax12OLD 2341 | Obsolete proof of ~ ax12 a... |
| ax12v2OLD 2342 | Obsolete proof of ~ ax12v ... |
| ax12a2OLD 2343 | Obsolete proof of ~ ax12v ... |
| axc15OLD 2344 | Obsolete proof of ~ axc15 ... |
| ax12b 2345 | A bidirectional version of... |
| equvini 2346 | A variable introduction la... |
| equvel 2347 | A variable elimination law... |
| equs5a 2348 | A property related to subs... |
| equs5e 2349 | A property related to subs... |
| equs45f 2350 | Two ways of expressing sub... |
| equs5 2351 | Lemma used in proofs of su... |
| sb2 2352 | One direction of a simplif... |
| stdpc4 2353 | The specialization axiom o... |
| 2stdpc4 2354 | A double specialization us... |
| sb3 2355 | One direction of a simplif... |
| sb4 2356 | One direction of a simplif... |
| sb4a 2357 | A version of ~ sb4 that do... |
| sb4b 2358 | Simplified definition of s... |
| hbsb2 2359 | Bound-variable hypothesis ... |
| nfsb2 2360 | Bound-variable hypothesis ... |
| hbsb2a 2361 | Special case of a bound-va... |
| sb4e 2362 | One direction of a simplif... |
| hbsb2e 2363 | Special case of a bound-va... |
| hbsb3 2364 | If ` y ` is not free in ` ... |
| nfs1 2365 | If ` y ` is not free in ` ... |
| axc16ALT 2366 | Alternate proof of ~ axc16... |
| axc16gALT 2367 | Alternate proof of ~ axc16... |
| equsb1 2368 | Substitution applied to an... |
| equsb2 2369 | Substitution applied to an... |
| dveel1 2370 | Quantifier introduction wh... |
| dveel2 2371 | Quantifier introduction wh... |
| axc14 2372 | Axiom ~ ax-c14 is redundan... |
| dfsb2 2373 | An alternate definition of... |
| dfsb3 2374 | An alternate definition of... |
| sbequi 2375 | An equality theorem for su... |
| sbequ 2376 | An equality theorem for su... |
| drsb1 2377 | Formula-building lemma for... |
| drsb2 2378 | Formula-building lemma for... |
| sbft 2379 | Substitution has no effect... |
| sbf 2380 | Substitution for a variabl... |
| sbh 2381 | Substitution for a variabl... |
| sbf2 2382 | Substitution has no effect... |
| nfs1f 2383 | If ` x ` is not free in ` ... |
| sb6x 2384 | Equivalence involving subs... |
| sb6f 2385 | Equivalence for substituti... |
| sb5f 2386 | Equivalence for substituti... |
| sbequ5 2387 | Substitution does not chan... |
| sbequ6 2388 | Substitution does not chan... |
| nfsb4t 2389 | A variable not free remain... |
| nfsb4 2390 | A variable not free remain... |
| sbn 2391 | Negation inside and outsid... |
| sbi1 2392 | Removal of implication fro... |
| sbi2 2393 | Introduction of implicatio... |
| spsbim 2394 | Specialization of implicat... |
| sbim 2395 | Implication inside and out... |
| sbrim 2396 | Substitution with a variab... |
| sblim 2397 | Substitution with a variab... |
| sbor 2398 | Logical OR inside and outs... |
| sban 2399 | Conjunction inside and out... |
| sb3an 2400 | Conjunction inside and out... |
| sbbi 2401 | Equivalence inside and out... |
| spsbbi 2402 | Specialization of bicondit... |
| sbbid 2403 | Deduction substituting bot... |
| sblbis 2404 | Introduce left bicondition... |
| sbrbis 2405 | Introduce right biconditio... |
| sbrbif 2406 | Introduce right biconditio... |
| sbequ8ALT 2407 | Alternate proof of ~ sbequ... |
| sbie 2408 | Conversion of implicit sub... |
| sbied 2409 | Conversion of implicit sub... |
| sbiedv 2410 | Conversion of implicit sub... |
| sbcom3 2411 | Substituting ` y ` for ` x... |
| sbco 2412 | A composition law for subs... |
| sbid2 2413 | An identity law for substi... |
| sbidm 2414 | An idempotent law for subs... |
| sbco2 2415 | A composition law for subs... |
| sbco2d 2416 | A composition law for subs... |
| sbco3 2417 | A composition law for subs... |
| sbcom 2418 | A commutativity law for su... |
| sbt 2419 | A substitution into a theo... |
| sbtrt 2420 | Partially closed form of ~... |
| sbtr 2421 | A partial converse to ~ sb... |
| sb5rf 2422 | Reversed substitution. (C... |
| sb6rf 2423 | Reversed substitution. (C... |
| sb8 2424 | Substitution of variable i... |
| sb8e 2425 | Substitution of variable i... |
| sb9 2426 | Commutation of quantificat... |
| sb9i 2427 | Commutation of quantificat... |
| ax12vALT 2428 | Alternate proof of ~ ax12v... |
| sb6 2429 | Equivalence for substituti... |
| sb5 2430 | Equivalence for substituti... |
| equsb3lem 2431 | Lemma for ~ equsb3 . (Con... |
| equsb3 2432 | Substitution applied to an... |
| equsb3ALT 2433 | Alternate proof of ~ equsb... |
| elsb3 2434 | Substitution applied to an... |
| elsb4 2435 | Substitution applied to an... |
| hbs1 2436 | The setvar ` x ` is not fr... |
| nfs1v 2437 | The setvar ` x ` is not fr... |
| sbhb 2438 | Two ways of expressing " `... |
| sbnf2 2439 | Two ways of expressing " `... |
| nfsb 2440 | If ` z ` is not free in ` ... |
| hbsb 2441 | If ` z ` is not free in ` ... |
| nfsbd 2442 | Deduction version of ~ nfs... |
| 2sb5 2443 | Equivalence for double sub... |
| 2sb6 2444 | Equivalence for double sub... |
| sbcom2 2445 | Commutativity law for subs... |
| sbcom4 2446 | Commutativity law for subs... |
| pm11.07 2447 | Axiom *11.07 in [Whitehead... |
| sb6a 2448 | Equivalence for substituti... |
| 2ax6elem 2449 | We can always find values ... |
| 2ax6e 2450 | We can always find values ... |
| 2sb5rf 2451 | Reversed double substituti... |
| 2sb6rf 2452 | Reversed double substituti... |
| sb7f 2453 | This version of ~ dfsb7 do... |
| sb7h 2454 | This version of ~ dfsb7 do... |
| dfsb7 2455 | An alternate definition of... |
| sb10f 2456 | Hao Wang's identity axiom ... |
| sbid2v 2457 | An identity law for substi... |
| sbelx 2458 | Elimination of substitutio... |
| sbel2x 2459 | Elimination of double subs... |
| sbal1 2460 | A theorem used in eliminat... |
| sbal2 2461 | Move quantifier in and out... |
| sbal 2462 | Move universal quantifier ... |
| sbex 2463 | Move existential quantifie... |
| sbalv 2464 | Quantify with new variable... |
| sbco4lem 2465 | Lemma for ~ sbco4 . It re... |
| sbco4 2466 | Two ways of exchanging two... |
| 2sb8e 2467 | An equivalent expression f... |
| exsb 2468 | An equivalent expression f... |
| 2exsb 2469 | An equivalent expression f... |
| eujust 2472 | A soundness justification ... |
| eujustALT 2473 | Alternate proof of ~ eujus... |
| euequ1 2476 | Equality has existential u... |
| mo2v 2477 | Alternate definition of "a... |
| euf 2478 | A version of the existenti... |
| mo2 2479 | Alternate definition of "a... |
| nfeu1 2480 | Bound-variable hypothesis ... |
| nfmo1 2481 | Bound-variable hypothesis ... |
| nfeud2 2482 | Bound-variable hypothesis ... |
| nfmod2 2483 | Bound-variable hypothesis ... |
| nfeud 2484 | Deduction version of ~ nfe... |
| nfmod 2485 | Bound-variable hypothesis ... |
| nfeu 2486 | Bound-variable hypothesis ... |
| nfmo 2487 | Bound-variable hypothesis ... |
| eubid 2488 | Formula-building rule for ... |
| mobid 2489 | Formula-building rule for ... |
| eubidv 2490 | Formula-building rule for ... |
| mobidv 2491 | Formula-building rule for ... |
| eubii 2492 | Introduce uniqueness quant... |
| mobii 2493 | Formula-building rule for ... |
| euex 2494 | Existential uniqueness imp... |
| exmo 2495 | Something exists or at mos... |
| eu5 2496 | Uniqueness in terms of "at... |
| exmoeu2 2497 | Existence implies "at most... |
| eu3v 2498 | An alternate way to expres... |
| eumo 2499 | Existential uniqueness imp... |
| eumoi 2500 | "At most one" inferred fro... |
| moabs 2501 | Absorption of existence co... |
| exmoeu 2502 | Existence in terms of "at ... |
| sb8eu 2503 | Variable substitution in u... |
| sb8mo 2504 | Variable substitution for ... |
| cbveu 2505 | Rule used to change bound ... |
| cbvmo 2506 | Rule used to change bound ... |
| mo3 2507 | Alternate definition of "a... |
| mo 2508 | Equivalent definitions of ... |
| eu2 2509 | An alternate way of defini... |
| eu1 2510 | An alternate way to expres... |
| euexALT 2511 | Alternate proof of ~ euex ... |
| euor 2512 | Introduce a disjunct into ... |
| euorv 2513 | Introduce a disjunct into ... |
| euor2 2514 | Introduce or eliminate a d... |
| sbmo 2515 | Substitution into "at most... |
| mo4f 2516 | "At most one" expressed us... |
| mo4 2517 | "At most one" expressed us... |
| eu4 2518 | Uniqueness using implicit ... |
| moim 2519 | "At most one" reverses imp... |
| moimi 2520 | "At most one" reverses imp... |
| moa1 2521 | If an implication holds fo... |
| euimmo 2522 | Uniqueness implies "at mos... |
| euim 2523 | Add existential uniqueness... |
| moan 2524 | "At most one" is still the... |
| moani 2525 | "At most one" is still tru... |
| moor 2526 | "At most one" is still the... |
| mooran1 2527 | "At most one" imports disj... |
| mooran2 2528 | "At most one" exports disj... |
| moanim 2529 | Introduction of a conjunct... |
| euan 2530 | Introduction of a conjunct... |
| moanimv 2531 | Introduction of a conjunct... |
| moanmo 2532 | Nested "at most one" quant... |
| moaneu 2533 | Nested "at most one" and u... |
| euanv 2534 | Introduction of a conjunct... |
| mopick 2535 | "At most one" picks a vari... |
| eupick 2536 | Existential uniqueness "pi... |
| eupicka 2537 | Version of ~ eupick with c... |
| eupickb 2538 | Existential uniqueness "pi... |
| eupickbi 2539 | Theorem *14.26 in [Whitehe... |
| mopick2 2540 | "At most one" can show the... |
| moexex 2541 | "At most one" double quant... |
| moexexv 2542 | "At most one" double quant... |
| 2moex 2543 | Double quantification with... |
| 2euex 2544 | Double quantification with... |
| 2eumo 2545 | Double quantification with... |
| 2eu2ex 2546 | Double existential uniquen... |
| 2moswap 2547 | A condition allowing swap ... |
| 2euswap 2548 | A condition allowing swap ... |
| 2exeu 2549 | Double existential uniquen... |
| 2mo2 2550 | This theorem extends the i... |
| 2mo 2551 | Two equivalent expressions... |
| 2mos 2552 | Double "exists at most one... |
| 2eu1 2553 | Double existential uniquen... |
| 2eu2 2554 | Double existential uniquen... |
| 2eu3 2555 | Double existential uniquen... |
| 2eu4 2556 | This theorem provides us w... |
| 2eu5 2557 | An alternate definition of... |
| 2eu6 2558 | Two equivalent expressions... |
| 2eu7 2559 | Two equivalent expressions... |
| 2eu8 2560 | Two equivalent expressions... |
| exists1 2561 | Two ways to express "only ... |
| exists2 2562 | A condition implying that ... |
| barbara 2563 | "Barbara", one of the fund... |
| celarent 2564 | "Celarent", one of the syl... |
| darii 2565 | "Darii", one of the syllog... |
| ferio 2566 | "Ferio" ("Ferioque"), one ... |
| barbari 2567 | "Barbari", one of the syll... |
| celaront 2568 | "Celaront", one of the syl... |
| cesare 2569 | "Cesare", one of the syllo... |
| camestres 2570 | "Camestres", one of the sy... |
| festino 2571 | "Festino", one of the syll... |
| baroco 2572 | "Baroco", one of the syllo... |
| cesaro 2573 | "Cesaro", one of the syllo... |
| camestros 2574 | "Camestros", one of the sy... |
| datisi 2575 | "Datisi", one of the syllo... |
| disamis 2576 | "Disamis", one of the syll... |
| ferison 2577 | "Ferison", one of the syll... |
| bocardo 2578 | "Bocardo", one of the syll... |
| felapton 2579 | "Felapton", one of the syl... |
| darapti 2580 | "Darapti", one of the syll... |
| calemes 2581 | "Calemes", one of the syll... |
| dimatis 2582 | "Dimatis", one of the syll... |
| fresison 2583 | "Fresison", one of the syl... |
| calemos 2584 | "Calemos", one of the syll... |
| fesapo 2585 | "Fesapo", one of the syllo... |
| bamalip 2586 | "Bamalip", one of the syll... |
| axia1 2587 | Left 'and' elimination (in... |
| axia2 2588 | Right 'and' elimination (i... |
| axia3 2589 | 'And' introduction (intuit... |
| axin1 2590 | 'Not' introduction (intuit... |
| axin2 2591 | 'Not' elimination (intuiti... |
| axio 2592 | Definition of 'or' (intuit... |
| axi4 2593 | Specialization (intuitioni... |
| axi5r 2594 | Converse of ax-c4 (intuiti... |
| axial 2595 | The setvar ` x ` is not fr... |
| axie1 2596 | The setvar ` x ` is not fr... |
| axie2 2597 | A key property of existent... |
| axi9 2598 | Axiom of existence (intuit... |
| axi10 2599 | Axiom of Quantifier Substi... |
| axi12 2600 | Axiom of Quantifier Introd... |
| axbnd 2601 | Axiom of Bundling (intuiti... |
| axext2 2603 | The Axiom of Extensionalit... |
| axext3 2604 | A generalization of the Ax... |
| axext3ALT 2605 | Alternate proof of ~ axext... |
| axext4 2606 | A bidirectional version of... |
| bm1.1 2607 | Any set defined by a prope... |
| abid 2610 | Simplification of class ab... |
| hbab1 2611 | Bound-variable hypothesis ... |
| nfsab1 2612 | Bound-variable hypothesis ... |
| hbab 2613 | Bound-variable hypothesis ... |
| nfsab 2614 | Bound-variable hypothesis ... |
| dfcleq 2616 | The same as ~ df-cleq with... |
| cvjust 2617 | Every set is a class. Pro... |
| eqriv 2619 | Infer equality of classes ... |
| eqrdv 2620 | Deduce equality of classes... |
| eqrdav 2621 | Deduce equality of classes... |
| eqid 2622 | Law of identity (reflexivi... |
| eqidd 2623 | Class identity law with an... |
| eqeq1d 2624 | Deduction from equality to... |
| eqeq1dALT 2625 | Shorter proof of ~ eqeq1d ... |
| eqeq1 2626 | Equality implies equivalen... |
| eqeq1i 2627 | Inference from equality to... |
| eqcomd 2628 | Deduction from commutative... |
| eqcom 2629 | Commutative law for class ... |
| eqcoms 2630 | Inference applying commuta... |
| eqcomi 2631 | Inference from commutative... |
| eqeq2d 2632 | Deduction from equality to... |
| eqeq2 2633 | Equality implies equivalen... |
| eqeq2i 2634 | Inference from equality to... |
| eqeq12 2635 | Equality relationship amon... |
| eqeq12i 2636 | A useful inference for sub... |
| eqeq12d 2637 | A useful inference for sub... |
| eqeqan12d 2638 | A useful inference for sub... |
| eqeqan12dALT 2639 | Alternate proof of ~ eqeqa... |
| eqeqan12rd 2640 | A useful inference for sub... |
| eqtr 2641 | Transitive law for class e... |
| eqtr2 2642 | A transitive law for class... |
| eqtr3 2643 | A transitive law for class... |
| eqtri 2644 | An equality transitivity i... |
| eqtr2i 2645 | An equality transitivity i... |
| eqtr3i 2646 | An equality transitivity i... |
| eqtr4i 2647 | An equality transitivity i... |
| 3eqtri 2648 | An inference from three ch... |
| 3eqtrri 2649 | An inference from three ch... |
| 3eqtr2i 2650 | An inference from three ch... |
| 3eqtr2ri 2651 | An inference from three ch... |
| 3eqtr3i 2652 | An inference from three ch... |
| 3eqtr3ri 2653 | An inference from three ch... |
| 3eqtr4i 2654 | An inference from three ch... |
| 3eqtr4ri 2655 | An inference from three ch... |
| eqtrd 2656 | An equality transitivity d... |
| eqtr2d 2657 | An equality transitivity d... |
| eqtr3d 2658 | An equality transitivity e... |
| eqtr4d 2659 | An equality transitivity e... |
| 3eqtrd 2660 | A deduction from three cha... |
| 3eqtrrd 2661 | A deduction from three cha... |
| 3eqtr2d 2662 | A deduction from three cha... |
| 3eqtr2rd 2663 | A deduction from three cha... |
| 3eqtr3d 2664 | A deduction from three cha... |
| 3eqtr3rd 2665 | A deduction from three cha... |
| 3eqtr4d 2666 | A deduction from three cha... |
| 3eqtr4rd 2667 | A deduction from three cha... |
| syl5eq 2668 | An equality transitivity d... |
| syl5req 2669 | An equality transitivity d... |
| syl5eqr 2670 | An equality transitivity d... |
| syl5reqr 2671 | An equality transitivity d... |
| syl6eq 2672 | An equality transitivity d... |
| syl6req 2673 | An equality transitivity d... |
| syl6eqr 2674 | An equality transitivity d... |
| syl6reqr 2675 | An equality transitivity d... |
| sylan9eq 2676 | An equality transitivity d... |
| sylan9req 2677 | An equality transitivity d... |
| sylan9eqr 2678 | An equality transitivity d... |
| 3eqtr3g 2679 | A chained equality inferen... |
| 3eqtr3a 2680 | A chained equality inferen... |
| 3eqtr4g 2681 | A chained equality inferen... |
| 3eqtr4a 2682 | A chained equality inferen... |
| eq2tri 2683 | A compound transitive infe... |
| eleq1w 2684 | Weaker version of ~ eleq1 ... |
| eleq2w 2685 | Weaker version of ~ eleq2 ... |
| eleq1d 2686 | Deduction from equality to... |
| eleq2d 2687 | Deduction from equality to... |
| eleq2dALT 2688 | Alternate proof of ~ eleq2... |
| eleq1 2689 | Equality implies equivalen... |
| eleq2 2690 | Equality implies equivalen... |
| eleq12 2691 | Equality implies equivalen... |
| eleq1i 2692 | Inference from equality to... |
| eleq2i 2693 | Inference from equality to... |
| eleq12i 2694 | Inference from equality to... |
| eleq12d 2695 | Deduction from equality to... |
| eleq1a 2696 | A transitive-type law rela... |
| eqeltri 2697 | Substitution of equal clas... |
| eqeltrri 2698 | Substitution of equal clas... |
| eleqtri 2699 | Substitution of equal clas... |
| eleqtrri 2700 | Substitution of equal clas... |
| eqeltrd 2701 | Substitution of equal clas... |
| eqeltrrd 2702 | Deduction that substitutes... |
| eleqtrd 2703 | Deduction that substitutes... |
| eleqtrrd 2704 | Deduction that substitutes... |
| syl5eqel 2705 | A membership and equality ... |
| syl5eqelr 2706 | A membership and equality ... |
| syl5eleq 2707 | A membership and equality ... |
| syl5eleqr 2708 | A membership and equality ... |
| syl6eqel 2709 | A membership and equality ... |
| syl6eqelr 2710 | A membership and equality ... |
| syl6eleq 2711 | A membership and equality ... |
| syl6eleqr 2712 | A membership and equality ... |
| 3eltr3i 2713 | Substitution of equal clas... |
| 3eltr4i 2714 | Substitution of equal clas... |
| 3eltr3d 2715 | Substitution of equal clas... |
| 3eltr4d 2716 | Substitution of equal clas... |
| 3eltr3g 2717 | Substitution of equal clas... |
| 3eltr4g 2718 | Substitution of equal clas... |
| eleq2s 2719 | Substitution of equal clas... |
| eqneltrd 2720 | If a class is not an eleme... |
| eqneltrrd 2721 | If a class is not an eleme... |
| neleqtrd 2722 | If a class is not an eleme... |
| neleqtrrd 2723 | If a class is not an eleme... |
| cleqh 2724 | Establish equality between... |
| nelneq 2725 | A way of showing two class... |
| nelneq2 2726 | A way of showing two class... |
| eqsb3lem 2727 | Lemma for ~ eqsb3 . (Cont... |
| eqsb3 2728 | Substitution applied to an... |
| clelsb3 2729 | Substitution applied to an... |
| hbxfreq 2730 | A utility lemma to transfe... |
| hblem 2731 | Change the free variable o... |
| abeq2 2732 | Equality of a class variab... |
| abeq1 2733 | Equality of a class variab... |
| abeq2d 2734 | Equality of a class variab... |
| abeq2i 2735 | Equality of a class variab... |
| abeq1i 2736 | Equality of a class variab... |
| abbi 2737 | Equivalent wff's correspon... |
| abbi2i 2738 | Equality of a class variab... |
| abbii 2739 | Equivalent wff's yield equ... |
| abbid 2740 | Equivalent wff's yield equ... |
| abbidv 2741 | Equivalent wff's yield equ... |
| abbi2dv 2742 | Deduction from a wff to a ... |
| abbi1dv 2743 | Deduction from a wff to a ... |
| abid1 2744 | Every class is equal to a ... |
| abid2 2745 | A simplification of class ... |
| cbvab 2746 | Rule used to change bound ... |
| cbvabv 2747 | Rule used to change bound ... |
| clelab 2748 | Membership of a class vari... |
| clabel 2749 | Membership of a class abst... |
| sbab 2750 | The right-hand side of the... |
| nfcjust 2752 | Justification theorem for ... |
| nfci 2754 | Deduce that a class ` A ` ... |
| nfcii 2755 | Deduce that a class ` A ` ... |
| nfcr 2756 | Consequence of the not-fre... |
| nfcrii 2757 | Consequence of the not-fre... |
| nfcri 2758 | Consequence of the not-fre... |
| nfcd 2759 | Deduce that a class ` A ` ... |
| nfceqdf 2760 | An equality theorem for ef... |
| nfceqi 2761 | Equality theorem for class... |
| nfcxfr 2762 | A utility lemma to transfe... |
| nfcxfrd 2763 | A utility lemma to transfe... |
| nfcv 2764 | If ` x ` is disjoint from ... |
| nfcvd 2765 | If ` x ` is disjoint from ... |
| nfab1 2766 | Bound-variable hypothesis ... |
| nfnfc1 2767 | The setvar ` x ` is bound ... |
| clelsb3f 2768 | Substitution applied to an... |
| nfab 2769 | Bound-variable hypothesis ... |
| nfaba1 2770 | Bound-variable hypothesis ... |
| nfcrd 2771 | Consequence of the not-fre... |
| nfeqd 2772 | Hypothesis builder for equ... |
| nfeld 2773 | Hypothesis builder for ele... |
| nfnfc 2774 | Hypothesis builder for ` F... |
| nfnfcALT 2775 | Alternate proof of ~ nfnfc... |
| nfeq 2776 | Hypothesis builder for equ... |
| nfel 2777 | Hypothesis builder for ele... |
| nfeq1 2778 | Hypothesis builder for equ... |
| nfel1 2779 | Hypothesis builder for ele... |
| nfeq2 2780 | Hypothesis builder for equ... |
| nfel2 2781 | Hypothesis builder for ele... |
| drnfc1 2782 | Formula-building lemma for... |
| drnfc2 2783 | Formula-building lemma for... |
| nfabd2 2784 | Bound-variable hypothesis ... |
| nfabd 2785 | Bound-variable hypothesis ... |
| dvelimdc 2786 | Deduction form of ~ dvelim... |
| dvelimc 2787 | Version of ~ dvelim for cl... |
| nfcvf 2788 | If ` x ` and ` y ` are dis... |
| nfcvf2 2789 | If ` x ` and ` y ` are dis... |
| cleqf 2790 | Establish equality between... |
| abid2f 2791 | A simplification of class ... |
| abeq2f 2792 | Equality of a class variab... |
| sbabel 2793 | Theorem to move a substitu... |
| neii 2796 | Inference associated with ... |
| neir 2797 | Inference associated with ... |
| nne 2798 | Negation of inequality. (... |
| neneqd 2799 | Deduction eliminating ineq... |
| neneq 2800 | From inequality to non equ... |
| neqned 2801 | If it is not the case that... |
| neqne 2802 | From non equality to inequ... |
| neirr 2803 | No class is unequal to its... |
| exmidne 2804 | Excluded middle with equal... |
| eqneqall 2805 | A contradiction concerning... |
| nonconne 2806 | Law of noncontradiction wi... |
| necon3ad 2807 | Contrapositive law deducti... |
| necon3bd 2808 | Contrapositive law deducti... |
| necon2ad 2809 | Contrapositive inference f... |
| necon2bd 2810 | Contrapositive inference f... |
| necon1ad 2811 | Contrapositive deduction f... |
| necon1bd 2812 | Contrapositive deduction f... |
| necon4ad 2813 | Contrapositive inference f... |
| necon4bd 2814 | Contrapositive inference f... |
| necon3d 2815 | Contrapositive law deducti... |
| necon1d 2816 | Contrapositive law deducti... |
| necon2d 2817 | Contrapositive inference f... |
| necon4d 2818 | Contrapositive inference f... |
| necon3ai 2819 | Contrapositive inference f... |
| necon3bi 2820 | Contrapositive inference f... |
| necon1ai 2821 | Contrapositive inference f... |
| necon1bi 2822 | Contrapositive inference f... |
| necon2ai 2823 | Contrapositive inference f... |
| necon2bi 2824 | Contrapositive inference f... |
| necon4ai 2825 | Contrapositive inference f... |
| necon3i 2826 | Contrapositive inference f... |
| necon1i 2827 | Contrapositive inference f... |
| necon2i 2828 | Contrapositive inference f... |
| necon4i 2829 | Contrapositive inference f... |
| necon3abid 2830 | Deduction from equality to... |
| necon3bbid 2831 | Deduction from equality to... |
| necon1abid 2832 | Contrapositive deduction f... |
| necon1bbid 2833 | Contrapositive inference f... |
| necon4abid 2834 | Contrapositive law deducti... |
| necon4bbid 2835 | Contrapositive law deducti... |
| necon2abid 2836 | Contrapositive deduction f... |
| necon2bbid 2837 | Contrapositive deduction f... |
| necon3bid 2838 | Deduction from equality to... |
| necon4bid 2839 | Contrapositive law deducti... |
| necon3abii 2840 | Deduction from equality to... |
| necon3bbii 2841 | Deduction from equality to... |
| necon1abii 2842 | Contrapositive inference f... |
| necon1bbii 2843 | Contrapositive inference f... |
| necon2abii 2844 | Contrapositive inference f... |
| necon2bbii 2845 | Contrapositive inference f... |
| necon3bii 2846 | Inference from equality to... |
| necom 2847 | Commutation of inequality.... |
| necomi 2848 | Inference from commutative... |
| necomd 2849 | Deduction from commutative... |
| nesym 2850 | Characterization of inequa... |
| nesymi 2851 | Inference associated with ... |
| nesymir 2852 | Inference associated with ... |
| neeq1d 2853 | Deduction for inequality. ... |
| neeq2d 2854 | Deduction for inequality. ... |
| neeq12d 2855 | Deduction for inequality. ... |
| neeq1 2856 | Equality theorem for inequ... |
| neeq2 2857 | Equality theorem for inequ... |
| neeq1i 2858 | Inference for inequality. ... |
| neeq2i 2859 | Inference for inequality. ... |
| neeq12i 2860 | Inference for inequality. ... |
| eqnetrd 2861 | Substitution of equal clas... |
| eqnetrrd 2862 | Substitution of equal clas... |
| neeqtrd 2863 | Substitution of equal clas... |
| eqnetri 2864 | Substitution of equal clas... |
| eqnetrri 2865 | Substitution of equal clas... |
| neeqtri 2866 | Substitution of equal clas... |
| neeqtrri 2867 | Substitution of equal clas... |
| neeqtrrd 2868 | Substitution of equal clas... |
| syl5eqner 2869 | A chained equality inferen... |
| 3netr3d 2870 | Substitution of equality i... |
| 3netr4d 2871 | Substitution of equality i... |
| 3netr3g 2872 | Substitution of equality i... |
| 3netr4g 2873 | Substitution of equality i... |
| nebi 2874 | Contraposition law for ine... |
| pm13.18 2875 | Theorem *13.18 in [Whitehe... |
| pm13.181 2876 | Theorem *13.181 in [Whiteh... |
| pm2.61ine 2877 | Inference eliminating an i... |
| pm2.21ddne 2878 | A contradiction implies an... |
| pm2.61ne 2879 | Deduction eliminating an i... |
| pm2.61dne 2880 | Deduction eliminating an i... |
| pm2.61dane 2881 | Deduction eliminating an i... |
| pm2.61da2ne 2882 | Deduction eliminating two ... |
| pm2.61da3ne 2883 | Deduction eliminating thre... |
| pm2.61iine 2884 | Equality version of ~ pm2.... |
| neor 2885 | Logical OR with an equalit... |
| neanior 2886 | A De Morgan's law for ineq... |
| ne3anior 2887 | A De Morgan's law for ineq... |
| neorian 2888 | A De Morgan's law for ineq... |
| nemtbir 2889 | An inference from an inequ... |
| nelne1 2890 | Two classes are different ... |
| nelne2 2891 | Two classes are different ... |
| nelelne 2892 | Two classes are different ... |
| neneor 2893 | If two classes are differe... |
| nfne 2894 | Bound-variable hypothesis ... |
| nfned 2895 | Bound-variable hypothesis ... |
| nabbi 2896 | Not equivalent wff's corre... |
| neli 2899 | Inference associated with ... |
| nelir 2900 | Inference associated with ... |
| neleq12d 2901 | Equality theorem for negat... |
| neleq1 2902 | Equality theorem for negat... |
| neleq2 2903 | Equality theorem for negat... |
| nfnel 2904 | Bound-variable hypothesis ... |
| nfneld 2905 | Bound-variable hypothesis ... |
| nnel 2906 | Negation of negated member... |
| elnelne1 2907 | Two classes are different ... |
| elnelne2 2908 | Two classes are different ... |
| nelcon3d 2909 | Contrapositive law deducti... |
| elnelall 2910 | A contradiction concerning... |
| pm2.61danel 2911 | Deduction eliminating an e... |
| rgen 2922 | Generalization rule for re... |
| ralel 2923 | All elements of a class ar... |
| rgenw 2924 | Generalization rule for re... |
| rgen2w 2925 | Generalization rule for re... |
| mprg 2926 | Modus ponens combined with... |
| mprgbir 2927 | Modus ponens on biconditio... |
| alral 2928 | Universal quantification i... |
| rsp 2929 | Restricted specialization.... |
| rspa 2930 | Restricted specialization.... |
| rspec 2931 | Specialization rule for re... |
| r19.21bi 2932 | Inference from Theorem 19.... |
| r19.21be 2933 | Inference from Theorem 19.... |
| rspec2 2934 | Specialization rule for re... |
| rspec3 2935 | Specialization rule for re... |
| rsp2 2936 | Restricted specialization,... |
| r2allem 2937 | Lemma factoring out common... |
| r2alf 2938 | Double restricted universa... |
| r2al 2939 | Double restricted universa... |
| r3al 2940 | Triple restricted universa... |
| nfra1 2941 | The setvar ` x ` is not fr... |
| hbra1 2942 | The setvar ` x ` is not fr... |
| hbral 2943 | Bound-variable hypothesis ... |
| nfrald 2944 | Deduction version of ~ nfr... |
| nfral 2945 | Bound-variable hypothesis ... |
| nfra2 2946 | Similar to Lemma 24 of [Mo... |
| ral2imi 2947 | Inference quantifying ante... |
| ralim 2948 | Distribution of restricted... |
| ralimi2 2949 | Inference quantifying both... |
| ralimia 2950 | Inference quantifying both... |
| ralimiaa 2951 | Inference quantifying both... |
| ralimi 2952 | Inference quantifying both... |
| 2ralimi 2953 | Inference quantifying both... |
| hbralrimi 2954 | Inference from Theorem 19.... |
| r19.21t 2955 | Restricted quantifier vers... |
| r19.21 2956 | Restricted quantifier vers... |
| ralrimi 2957 | Inference from Theorem 19.... |
| ralimdaa 2958 | Deduction quantifying both... |
| ralrimd 2959 | Inference from Theorem 19.... |
| r19.21v 2960 | Restricted quantifier vers... |
| ralimdv2 2961 | Inference quantifying both... |
| ralimdva 2962 | Deduction quantifying both... |
| ralimdv 2963 | Deduction quantifying both... |
| ralimdvva 2964 | Deduction doubly quantifyi... |
| ralrimiv 2965 | Inference from Theorem 19.... |
| ralrimiva 2966 | Inference from Theorem 19.... |
| ralrimivw 2967 | Inference from Theorem 19.... |
| ralrimdv 2968 | Inference from Theorem 19.... |
| ralrimdva 2969 | Inference from Theorem 19.... |
| ralrimivv 2970 | Inference from Theorem 19.... |
| ralrimivva 2971 | Inference from Theorem 19.... |
| ralrimivvva 2972 | Inference from Theorem 19.... |
| ralrimdvv 2973 | Inference from Theorem 19.... |
| ralrimdvva 2974 | Inference from Theorem 19.... |
| rgen2 2975 | Generalization rule for re... |
| rgen3 2976 | Generalization rule for re... |
| rgen2a 2977 | Generalization rule for re... |
| ralbii2 2978 | Inference adding different... |
| ralbiia 2979 | Inference adding restricte... |
| ralbii 2980 | Inference adding restricte... |
| 2ralbii 2981 | Inference adding two restr... |
| ralbida 2982 | Formula-building rule for ... |
| ralbid 2983 | Formula-building rule for ... |
| ralbidv2 2984 | Formula-building rule for ... |
| ralbidva 2985 | Formula-building rule for ... |
| ralbidv 2986 | Formula-building rule for ... |
| 2ralbida 2987 | Formula-building rule for ... |
| 2ralbidva 2988 | Formula-building rule for ... |
| 2ralbidv 2989 | Formula-building rule for ... |
| raleqbii 2990 | Equality deduction for res... |
| raln 2991 | Restricted universally qua... |
| ralnex 2992 | Relationship between restr... |
| ralnexOLD 2993 | Obsolete proof of ~ ralnex... |
| dfral2 2994 | Relationship between restr... |
| rexnal 2995 | Relationship between restr... |
| dfrex2 2996 | Relationship between restr... |
| ralinexa 2997 | A transformation of restri... |
| rexanali 2998 | A transformation of restri... |
| nrexralim 2999 | Negation of a complex pred... |
| nrex 3000 | Inference adding restricte... |
| nrexdv 3001 | Deduction adding restricte... |
| rexex 3002 | Restricted existence impli... |
| rspe 3003 | Restricted specialization.... |
| rsp2e 3004 | Restricted specialization.... |
| nfre1 3005 | The setvar ` x ` is not fr... |
| nfrexd 3006 | Deduction version of ~ nfr... |
| nfrex 3007 | Bound-variable hypothesis ... |
| rexim 3008 | Theorem 19.22 of [Margaris... |
| reximia 3009 | Inference quantifying both... |
| reximi2 3010 | Inference quantifying both... |
| reximi 3011 | Inference quantifying both... |
| reximdai 3012 | Deduction from Theorem 19.... |
| reximd2a 3013 | Deduction quantifying both... |
| reximdv2 3014 | Deduction quantifying both... |
| reximdvai 3015 | Deduction quantifying both... |
| reximdv 3016 | Deduction from Theorem 19.... |
| reximdva 3017 | Deduction quantifying both... |
| reximddv 3018 | Deduction from Theorem 19.... |
| reximdvva 3019 | Deduction doubly quantifyi... |
| reximddv2 3020 | Double deduction from Theo... |
| r19.23t 3021 | Closed theorem form of ~ r... |
| r19.23 3022 | Restricted quantifier vers... |
| r19.23v 3023 | Restricted quantifier vers... |
| rexlimi 3024 | Restricted quantifier vers... |
| rexlimd2 3025 | Version of ~ rexlimd with ... |
| rexlimd 3026 | Deduction form of ~ rexlim... |
| rexlimiv 3027 | Inference from Theorem 19.... |
| rexlimiva 3028 | Inference from Theorem 19.... |
| rexlimivw 3029 | Weaker version of ~ rexlim... |
| rexlimdv 3030 | Inference from Theorem 19.... |
| rexlimdva 3031 | Inference from Theorem 19.... |
| rexlimdvaa 3032 | Inference from Theorem 19.... |
| rexlimdv3a 3033 | Inference from Theorem 19.... |
| rexlimdvw 3034 | Inference from Theorem 19.... |
| rexlimddv 3035 | Restricted existential eli... |
| rexlimivv 3036 | Inference from Theorem 19.... |
| rexlimdvv 3037 | Inference from Theorem 19.... |
| rexlimdvva 3038 | Inference from Theorem 19.... |
| rexbii2 3039 | Inference adding different... |
| rexbiia 3040 | Inference adding restricte... |
| rexbii 3041 | Inference adding restricte... |
| 2rexbii 3042 | Inference adding two restr... |
| rexnal2 3043 | Relationship between two r... |
| rexnal3 3044 | Relationship between three... |
| ralnex2 3045 | Relationship between two r... |
| ralnex3 3046 | Relationship between three... |
| rexbida 3047 | Formula-building rule for ... |
| rexbidv2 3048 | Formula-building rule for ... |
| rexbidva 3049 | Formula-building rule for ... |
| rexbidvaALT 3050 | Alternate proof of ~ rexbi... |
| rexbid 3051 | Formula-building rule for ... |
| rexbidv 3052 | Formula-building rule for ... |
| rexbidvALT 3053 | Alternate proof of ~ rexbi... |
| rexeqbii 3054 | Equality deduction for res... |
| 2rexbiia 3055 | Inference adding two restr... |
| 2rexbidva 3056 | Formula-building rule for ... |
| 2rexbidv 3057 | Formula-building rule for ... |
| rexralbidv 3058 | Formula-building rule for ... |
| r2exlem 3059 | Lemma factoring out common... |
| r2exf 3060 | Double restricted existent... |
| r2ex 3061 | Double restricted existent... |
| risset 3062 | Two ways to say " ` A ` be... |
| r19.12 3063 | Restricted quantifier vers... |
| r19.26 3064 | Restricted quantifier vers... |
| r19.26-2 3065 | Restricted quantifier vers... |
| r19.26-3 3066 | Version of ~ r19.26 with t... |
| r19.26m 3067 | Version of ~ 19.26 and ~ r... |
| ralbi 3068 | Distribute a restricted un... |
| ralbiim 3069 | Split a biconditional and ... |
| r19.27v 3070 | Restricted quantitifer ver... |
| r19.28v 3071 | Restricted quantifier vers... |
| r19.29 3072 | Restricted quantifier vers... |
| r19.29r 3073 | Restricted quantifier vers... |
| r19.29imd 3074 | Theorem 19.29 of [Margaris... |
| r19.29af2 3075 | A commonly used pattern ba... |
| r19.29af 3076 | A commonly used pattern ba... |
| r19.29an 3077 | A commonly used pattern ba... |
| r19.29a 3078 | A commonly used pattern ba... |
| 2r19.29 3079 | Theorem ~ r19.29 with two ... |
| r19.29d2r 3080 | Theorem 19.29 of [Margaris... |
| r19.29vva 3081 | A commonly used pattern ba... |
| r19.30 3082 | Restricted quantifier vers... |
| r19.32v 3083 | Restricted quantifier vers... |
| r19.35 3084 | Restricted quantifier vers... |
| r19.36v 3085 | Restricted quantifier vers... |
| r19.37 3086 | Restricted quantifier vers... |
| r19.37v 3087 | Restricted quantifier vers... |
| r19.40 3088 | Restricted quantifier vers... |
| r19.41v 3089 | Restricted quantifier vers... |
| r19.41 3090 | Restricted quantifier vers... |
| r19.41vv 3091 | Version of ~ r19.41v with ... |
| r19.42v 3092 | Restricted quantifier vers... |
| r19.43 3093 | Restricted quantifier vers... |
| r19.44v 3094 | One direction of a restric... |
| r19.45v 3095 | Restricted quantifier vers... |
| ralcomf 3096 | Commutation of restricted ... |
| rexcomf 3097 | Commutation of restricted ... |
| ralcom 3098 | Commutation of restricted ... |
| rexcom 3099 | Commutation of restricted ... |
| ralcom13 3100 | Swap first and third restr... |
| rexcom13 3101 | Swap first and third restr... |
| ralrot3 3102 | Rotate three restricted un... |
| rexrot4 3103 | Rotate four restricted exi... |
| ralcom2 3104 | Commutation of restricted ... |
| ralcom3 3105 | A commutation law for rest... |
| reean 3106 | Rearrange restricted exist... |
| reeanv 3107 | Rearrange restricted exist... |
| 3reeanv 3108 | Rearrange three restricted... |
| 2ralor 3109 | Distribute restricted univ... |
| nfreu1 3110 | The setvar ` x ` is not fr... |
| nfrmo1 3111 | The setvar ` x ` is not fr... |
| nfreud 3112 | Deduction version of ~ nfr... |
| nfrmod 3113 | Deduction version of ~ nfr... |
| nfreu 3114 | Bound-variable hypothesis ... |
| nfrmo 3115 | Bound-variable hypothesis ... |
| rabid 3116 | An "identity" law of concr... |
| rabidim1 3117 | Membership in a restricted... |
| rabid2 3118 | An "identity" law for rest... |
| rabid2f 3119 | An "identity" law for rest... |
| rabbi 3120 | Equivalent wff's correspon... |
| rabswap 3121 | Swap with a membership rel... |
| nfrab1 3122 | The abstraction variable i... |
| nfrab 3123 | A variable not free in a w... |
| reubida 3124 | Formula-building rule for ... |
| reubidva 3125 | Formula-building rule for ... |
| reubidv 3126 | Formula-building rule for ... |
| reubiia 3127 | Formula-building rule for ... |
| reubii 3128 | Formula-building rule for ... |
| rmobida 3129 | Formula-building rule for ... |
| rmobidva 3130 | Formula-building rule for ... |
| rmobidv 3131 | Formula-building rule for ... |
| rmobiia 3132 | Formula-building rule for ... |
| rmobii 3133 | Formula-building rule for ... |
| raleqf 3134 | Equality theorem for restr... |
| rexeqf 3135 | Equality theorem for restr... |
| reueq1f 3136 | Equality theorem for restr... |
| rmoeq1f 3137 | Equality theorem for restr... |
| raleq 3138 | Equality theorem for restr... |
| rexeq 3139 | Equality theorem for restr... |
| reueq1 3140 | Equality theorem for restr... |
| rmoeq1 3141 | Equality theorem for restr... |
| raleqi 3142 | Equality inference for res... |
| rexeqi 3143 | Equality inference for res... |
| raleqdv 3144 | Equality deduction for res... |
| rexeqdv 3145 | Equality deduction for res... |
| raleqbi1dv 3146 | Equality deduction for res... |
| rexeqbi1dv 3147 | Equality deduction for res... |
| reueqd 3148 | Equality deduction for res... |
| rmoeqd 3149 | Equality deduction for res... |
| raleqbid 3150 | Equality deduction for res... |
| rexeqbid 3151 | Equality deduction for res... |
| raleqbidv 3152 | Equality deduction for res... |
| rexeqbidv 3153 | Equality deduction for res... |
| raleqbidva 3154 | Equality deduction for res... |
| rexeqbidva 3155 | Equality deduction for res... |
| raleleq 3156 | All elements of a class ar... |
| raleleqALT 3157 | Alternate proof of ~ ralel... |
| mormo 3158 | Unrestricted "at most one"... |
| reu5 3159 | Restricted uniqueness in t... |
| reurex 3160 | Restricted unique existenc... |
| reurmo 3161 | Restricted existential uni... |
| rmo5 3162 | Restricted "at most one" i... |
| nrexrmo 3163 | Nonexistence implies restr... |
| reueubd 3164 | Restricted existential uni... |
| cbvralf 3165 | Rule used to change bound ... |
| cbvrexf 3166 | Rule used to change bound ... |
| cbvral 3167 | Rule used to change bound ... |
| cbvrex 3168 | Rule used to change bound ... |
| cbvreu 3169 | Change the bound variable ... |
| cbvrmo 3170 | Change the bound variable ... |
| cbvralv 3171 | Change the bound variable ... |
| cbvrexv 3172 | Change the bound variable ... |
| cbvreuv 3173 | Change the bound variable ... |
| cbvrmov 3174 | Change the bound variable ... |
| cbvraldva2 3175 | Rule used to change the bo... |
| cbvrexdva2 3176 | Rule used to change the bo... |
| cbvraldva 3177 | Rule used to change the bo... |
| cbvrexdva 3178 | Rule used to change the bo... |
| cbvral2v 3179 | Change bound variables of ... |
| cbvrex2v 3180 | Change bound variables of ... |
| cbvral3v 3181 | Change bound variables of ... |
| cbvralsv 3182 | Change bound variable by u... |
| cbvrexsv 3183 | Change bound variable by u... |
| sbralie 3184 | Implicit to explicit subst... |
| rabbiia 3185 | Equivalent wff's yield equ... |
| rabbidva2 3186 | Equivalent wff's yield equ... |
| rabbia2 3187 | Equivalent wff's yield equ... |
| rabbidva 3188 | Equivalent wff's yield equ... |
| rabbidv 3189 | Equivalent wff's yield equ... |
| rabeqf 3190 | Equality theorem for restr... |
| rabeqif 3191 | Equality theorem for restr... |
| rabeq 3192 | Equality theorem for restr... |
| rabeqi 3193 | Equality theorem for restr... |
| rabeqdv 3194 | Equality of restricted cla... |
| rabeqbidv 3195 | Equality of restricted cla... |
| rabeqbidva 3196 | Equality of restricted cla... |
| rabeq2i 3197 | Inference rule from equali... |
| cbvrab 3198 | Rule to change the bound v... |
| cbvrabv 3199 | Rule to change the bound v... |
| vjust 3201 | Soundness justification th... |
| vex 3203 | All setvar variables are s... |
| eqvf 3204 | The universe contains ever... |
| eqv 3205 | The universe contains ever... |
| abv 3206 | The class of sets verifyin... |
| isset 3207 | Two ways to say " ` A ` is... |
| issetf 3208 | A version of ~ isset that ... |
| isseti 3209 | A way to say " ` A ` is a ... |
| issetri 3210 | A way to say " ` A ` is a ... |
| eqvisset 3211 | A class equal to a variabl... |
| elex 3212 | If a class is a member of ... |
| elexi 3213 | If a class is a member of ... |
| elexd 3214 | If a class is a member of ... |
| elisset 3215 | An element of a class exis... |
| elex2 3216 | If a class contains anothe... |
| elex22 3217 | If two classes each contai... |
| prcnel 3218 | A proper class doesn't bel... |
| ralv 3219 | A universal quantifier res... |
| rexv 3220 | An existential quantifier ... |
| reuv 3221 | A uniqueness quantifier re... |
| rmov 3222 | A uniqueness quantifier re... |
| rabab 3223 | A class abstraction restri... |
| ralcom4 3224 | Commutation of restricted ... |
| rexcom4 3225 | Commutation of restricted ... |
| rexcom4a 3226 | Specialized existential co... |
| rexcom4b 3227 | Specialized existential co... |
| ceqsalt 3228 | Closed theorem version of ... |
| ceqsralt 3229 | Restricted quantifier vers... |
| ceqsalg 3230 | A representation of explic... |
| ceqsalgALT 3231 | Alternate proof of ~ ceqsa... |
| ceqsal 3232 | A representation of explic... |
| ceqsalv 3233 | A representation of explic... |
| ceqsralv 3234 | Restricted quantifier vers... |
| gencl 3235 | Implicit substitution for ... |
| 2gencl 3236 | Implicit substitution for ... |
| 3gencl 3237 | Implicit substitution for ... |
| cgsexg 3238 | Implicit substitution infe... |
| cgsex2g 3239 | Implicit substitution infe... |
| cgsex4g 3240 | An implicit substitution i... |
| ceqsex 3241 | Elimination of an existent... |
| ceqsexv 3242 | Elimination of an existent... |
| ceqsexv2d 3243 | Elimination of an existent... |
| ceqsex2 3244 | Elimination of two existen... |
| ceqsex2v 3245 | Elimination of two existen... |
| ceqsex3v 3246 | Elimination of three exist... |
| ceqsex4v 3247 | Elimination of four existe... |
| ceqsex6v 3248 | Elimination of six existen... |
| ceqsex8v 3249 | Elimination of eight exist... |
| gencbvex 3250 | Change of bound variable u... |
| gencbvex2 3251 | Restatement of ~ gencbvex ... |
| gencbval 3252 | Change of bound variable u... |
| sbhypf 3253 | Introduce an explicit subs... |
| vtoclgft 3254 | Closed theorem form of ~ v... |
| vtoclgftOLD 3255 | Obsolete proof of ~ vtoclg... |
| vtocldf 3256 | Implicit substitution of a... |
| vtocld 3257 | Implicit substitution of a... |
| vtoclf 3258 | Implicit substitution of a... |
| vtocl 3259 | Implicit substitution of a... |
| vtoclALT 3260 | Alternate proof of ~ vtocl... |
| vtocl2 3261 | Implicit substitution of c... |
| vtocl3 3262 | Implicit substitution of c... |
| vtoclb 3263 | Implicit substitution of a... |
| vtoclgf 3264 | Implicit substitution of a... |
| vtoclg1f 3265 | Version of ~ vtoclgf with ... |
| vtoclg 3266 | Implicit substitution of a... |
| vtoclbg 3267 | Implicit substitution of a... |
| vtocl2gf 3268 | Implicit substitution of a... |
| vtocl3gf 3269 | Implicit substitution of a... |
| vtocl2g 3270 | Implicit substitution of 2... |
| vtoclgaf 3271 | Implicit substitution of a... |
| vtoclga 3272 | Implicit substitution of a... |
| vtocl2gaf 3273 | Implicit substitution of 2... |
| vtocl2ga 3274 | Implicit substitution of 2... |
| vtocl3gaf 3275 | Implicit substitution of 3... |
| vtocl3ga 3276 | Implicit substitution of 3... |
| vtocl4g 3277 | Implicit substitution of 4... |
| vtocl4ga 3278 | Implicit substitution of 4... |
| vtocleg 3279 | Implicit substitution of a... |
| vtoclegft 3280 | Implicit substitution of a... |
| vtoclef 3281 | Implicit substitution of a... |
| vtocle 3282 | Implicit substitution of a... |
| vtoclri 3283 | Implicit substitution of a... |
| spcimgft 3284 | A closed version of ~ spci... |
| spcgft 3285 | A closed version of ~ spcg... |
| spcimgf 3286 | Rule of specialization, us... |
| spcimegf 3287 | Existential specialization... |
| spcgf 3288 | Rule of specialization, us... |
| spcegf 3289 | Existential specialization... |
| spcimdv 3290 | Restricted specialization,... |
| spcdv 3291 | Rule of specialization, us... |
| spcimedv 3292 | Restricted existential spe... |
| spcgv 3293 | Rule of specialization, us... |
| spcegv 3294 | Existential specialization... |
| spc2egv 3295 | Existential specialization... |
| spc2gv 3296 | Specialization with two qu... |
| spc3egv 3297 | Existential specialization... |
| spc3gv 3298 | Specialization with three ... |
| spcv 3299 | Rule of specialization, us... |
| spcev 3300 | Existential specialization... |
| spc2ev 3301 | Existential specialization... |
| rspct 3302 | A closed version of ~ rspc... |
| rspc 3303 | Restricted specialization,... |
| rspce 3304 | Restricted existential spe... |
| rspcv 3305 | Restricted specialization,... |
| rspccv 3306 | Restricted specialization,... |
| rspcva 3307 | Restricted specialization,... |
| rspccva 3308 | Restricted specialization,... |
| rspcev 3309 | Restricted existential spe... |
| rspcimdv 3310 | Restricted specialization,... |
| rspcimedv 3311 | Restricted existential spe... |
| rspcdv 3312 | Restricted specialization,... |
| rspcedv 3313 | Restricted existential spe... |
| rspcebdv 3314 | Restricted existential spe... |
| rspcda 3315 | Restricted specialization,... |
| rspcdva 3316 | Restricted specialization,... |
| rspcedvd 3317 | Restricted existential spe... |
| rspcedeq1vd 3318 | Restricted existential spe... |
| rspcedeq2vd 3319 | Restricted existential spe... |
| rspc2 3320 | Restricted specialization ... |
| rspc2gv 3321 | Restricted specialization ... |
| rspc2v 3322 | 2-variable restricted spec... |
| rspc2va 3323 | 2-variable restricted spec... |
| rspc2ev 3324 | 2-variable restricted exis... |
| rspc3v 3325 | 3-variable restricted spec... |
| rspc3ev 3326 | 3-variable restricted exis... |
| ralxpxfr2d 3327 | Transfer a universal quant... |
| rexraleqim 3328 | Statement following from e... |
| eqvincg 3329 | A variable introduction la... |
| eqvinc 3330 | A variable introduction la... |
| eqvincf 3331 | A variable introduction la... |
| alexeqg 3332 | Two ways to express substi... |
| ceqex 3333 | Equality implies equivalen... |
| ceqsexg 3334 | A representation of explic... |
| ceqsexgv 3335 | Elimination of an existent... |
| ceqsrexv 3336 | Elimination of a restricte... |
| ceqsrexbv 3337 | Elimination of a restricte... |
| ceqsrex2v 3338 | Elimination of a restricte... |
| clel2 3339 | An alternate definition of... |
| clel3g 3340 | An alternate definition of... |
| clel3 3341 | An alternate definition of... |
| clel4 3342 | An alternate definition of... |
| clel5 3343 | Alternate definition of cl... |
| pm13.183 3344 | Compare theorem *13.183 in... |
| rr19.3v 3345 | Restricted quantifier vers... |
| rr19.28v 3346 | Restricted quantifier vers... |
| elabgt 3347 | Membership in a class abst... |
| elabgf 3348 | Membership in a class abst... |
| elabf 3349 | Membership in a class abst... |
| elab 3350 | Membership in a class abst... |
| elabg 3351 | Membership in a class abst... |
| elabd 3352 | Explicit demonstration the... |
| elab2g 3353 | Membership in a class abst... |
| elab2 3354 | Membership in a class abst... |
| elab4g 3355 | Membership in a class abst... |
| elab3gf 3356 | Membership in a class abst... |
| elab3g 3357 | Membership in a class abst... |
| elab3 3358 | Membership in a class abst... |
| elrabi 3359 | Implication for the member... |
| elrabf 3360 | Membership in a restricted... |
| rabtru 3361 | Abstract builder using the... |
| elrab3t 3362 | Membership in a restricted... |
| elrab 3363 | Membership in a restricted... |
| elrab3 3364 | Membership in a restricted... |
| elrabd 3365 | Membership in a restricted... |
| elrab2 3366 | Membership in a class abst... |
| ralab 3367 | Universal quantification o... |
| ralrab 3368 | Universal quantification o... |
| rexab 3369 | Existential quantification... |
| rexrab 3370 | Existential quantification... |
| ralab2 3371 | Universal quantification o... |
| ralrab2 3372 | Universal quantification o... |
| rexab2 3373 | Existential quantification... |
| rexrab2 3374 | Existential quantification... |
| abidnf 3375 | Identity used to create cl... |
| dedhb 3376 | A deduction theorem for co... |
| eqeu 3377 | A condition which implies ... |
| eueq 3378 | Equality has existential u... |
| eueq1 3379 | Equality has existential u... |
| eueq2 3380 | Equality has existential u... |
| eueq3 3381 | Equality has existential u... |
| moeq 3382 | There is at most one set e... |
| moeq3 3383 | "At most one" property of ... |
| mosub 3384 | "At most one" remains true... |
| mo2icl 3385 | Theorem for inferring "at ... |
| mob2 3386 | Consequence of "at most on... |
| moi2 3387 | Consequence of "at most on... |
| mob 3388 | Equality implied by "at mo... |
| moi 3389 | Equality implied by "at mo... |
| morex 3390 | Derive membership from uni... |
| euxfr2 3391 | Transfer existential uniqu... |
| euxfr 3392 | Transfer existential uniqu... |
| euind 3393 | Existential uniqueness via... |
| reu2 3394 | A way to express restricte... |
| reu6 3395 | A way to express restricte... |
| reu3 3396 | A way to express restricte... |
| reu6i 3397 | A condition which implies ... |
| eqreu 3398 | A condition which implies ... |
| rmo4 3399 | Restricted "at most one" u... |
| reu4 3400 | Restricted uniqueness usin... |
| reu7 3401 | Restricted uniqueness usin... |
| reu8 3402 | Restricted uniqueness usin... |
| reu2eqd 3403 | Deduce equality from restr... |
| reueq 3404 | Equality has existential u... |
| rmoeq 3405 | Equality's restricted exis... |
| rmoan 3406 | Restricted "at most one" s... |
| rmoim 3407 | Restricted "at most one" i... |
| rmoimia 3408 | Restricted "at most one" i... |
| rmoimi2 3409 | Restricted "at most one" i... |
| 2reuswap 3410 | A condition allowing swap ... |
| reuind 3411 | Existential uniqueness via... |
| 2rmorex 3412 | Double restricted quantifi... |
| 2reu5lem1 3413 | Lemma for ~ 2reu5 . Note ... |
| 2reu5lem2 3414 | Lemma for ~ 2reu5 . (Cont... |
| 2reu5lem3 3415 | Lemma for ~ 2reu5 . This ... |
| 2reu5 3416 | Double restricted existent... |
| nelrdva 3417 | Deduce negative membership... |
| cdeqi 3420 | Deduce conditional equalit... |
| cdeqri 3421 | Property of conditional eq... |
| cdeqth 3422 | Deduce conditional equalit... |
| cdeqnot 3423 | Distribute conditional equ... |
| cdeqal 3424 | Distribute conditional equ... |
| cdeqab 3425 | Distribute conditional equ... |
| cdeqal1 3426 | Distribute conditional equ... |
| cdeqab1 3427 | Distribute conditional equ... |
| cdeqim 3428 | Distribute conditional equ... |
| cdeqcv 3429 | Conditional equality for s... |
| cdeqeq 3430 | Distribute conditional equ... |
| cdeqel 3431 | Distribute conditional equ... |
| nfcdeq 3432 | If we have a conditional e... |
| nfccdeq 3433 | Variation of ~ nfcdeq for ... |
| ru 3434 | Russell's Paradox. Propos... |
| dfsbcq 3437 | Proper substitution of a c... |
| dfsbcq2 3438 | This theorem, which is sim... |
| sbsbc 3439 | Show that ~ df-sb and ~ df... |
| sbceq1d 3440 | Equality theorem for class... |
| sbceq1dd 3441 | Equality theorem for class... |
| sbceqbid 3442 | Equality theorem for class... |
| sbc8g 3443 | This is the closest we can... |
| sbc2or 3444 | The disjunction of two equ... |
| sbcex 3445 | By our definition of prope... |
| sbceq1a 3446 | Equality theorem for class... |
| sbceq2a 3447 | Equality theorem for class... |
| spsbc 3448 | Specialization: if a formu... |
| spsbcd 3449 | Specialization: if a formu... |
| sbcth 3450 | A substitution into a theo... |
| sbcthdv 3451 | Deduction version of ~ sbc... |
| sbcid 3452 | An identity theorem for su... |
| nfsbc1d 3453 | Deduction version of ~ nfs... |
| nfsbc1 3454 | Bound-variable hypothesis ... |
| nfsbc1v 3455 | Bound-variable hypothesis ... |
| nfsbcd 3456 | Deduction version of ~ nfs... |
| nfsbc 3457 | Bound-variable hypothesis ... |
| sbcco 3458 | A composition law for clas... |
| sbcco2 3459 | A composition law for clas... |
| sbc5 3460 | An equivalence for class s... |
| sbc6g 3461 | An equivalence for class s... |
| sbc6 3462 | An equivalence for class s... |
| sbc7 3463 | An equivalence for class s... |
| cbvsbc 3464 | Change bound variables in ... |
| cbvsbcv 3465 | Change the bound variable ... |
| sbciegft 3466 | Conversion of implicit sub... |
| sbciegf 3467 | Conversion of implicit sub... |
| sbcieg 3468 | Conversion of implicit sub... |
| sbcie2g 3469 | Conversion of implicit sub... |
| sbcie 3470 | Conversion of implicit sub... |
| sbciedf 3471 | Conversion of implicit sub... |
| sbcied 3472 | Conversion of implicit sub... |
| sbcied2 3473 | Conversion of implicit sub... |
| elrabsf 3474 | Membership in a restricted... |
| eqsbc3 3475 | Substitution applied to an... |
| sbcng 3476 | Move negation in and out o... |
| sbcimg 3477 | Distribution of class subs... |
| sbcan 3478 | Distribution of class subs... |
| sbcor 3479 | Distribution of class subs... |
| sbcbig 3480 | Distribution of class subs... |
| sbcn1 3481 | Move negation in and out o... |
| sbcim1 3482 | Distribution of class subs... |
| sbcbi1 3483 | Distribution of class subs... |
| sbcbi2 3484 | Substituting into equivale... |
| sbcal 3485 | Move universal quantifier ... |
| sbcex2 3486 | Move existential quantifie... |
| sbceqal 3487 | Set theory version of ~ sb... |
| sbeqalb 3488 | Theorem *14.121 in [Whiteh... |
| sbcbid 3489 | Formula-building deduction... |
| sbcbidv 3490 | Formula-building deduction... |
| sbcbii 3491 | Formula-building inference... |
| eqsbc3r 3492 | ~ eqsbc3 with setvar varia... |
| eqsbc3rOLD 3493 | Obsolete proof of ~ eqsbc3... |
| sbc3an 3494 | Distribution of class subs... |
| sbcel1v 3495 | Class substitution into a ... |
| sbcel2gv 3496 | Class substitution into a ... |
| sbcel21v 3497 | Class substitution into a ... |
| sbcimdv 3498 | Substitution analogue of T... |
| sbcimdvOLD 3499 | Obsolete proof of ~ sbcimd... |
| sbctt 3500 | Substitution for a variabl... |
| sbcgf 3501 | Substitution for a variabl... |
| sbc19.21g 3502 | Substitution for a variabl... |
| sbcg 3503 | Substitution for a variabl... |
| sbc2iegf 3504 | Conversion of implicit sub... |
| sbc2ie 3505 | Conversion of implicit sub... |
| sbc2iedv 3506 | Conversion of implicit sub... |
| sbc3ie 3507 | Conversion of implicit sub... |
| sbccomlem 3508 | Lemma for ~ sbccom . (Con... |
| sbccom 3509 | Commutative law for double... |
| sbcralt 3510 | Interchange class substitu... |
| sbcrext 3511 | Interchange class substitu... |
| sbcrextOLD 3512 | Obsolete proof of ~ sbcrex... |
| sbcralg 3513 | Interchange class substitu... |
| sbcrex 3514 | Interchange class substitu... |
| sbcreu 3515 | Interchange class substitu... |
| reu8nf 3516 | Restricted uniqueness usin... |
| sbcabel 3517 | Interchange class substitu... |
| rspsbc 3518 | Restricted quantifier vers... |
| rspsbca 3519 | Restricted quantifier vers... |
| rspesbca 3520 | Existence form of ~ rspsbc... |
| spesbc 3521 | Existence form of ~ spsbc ... |
| spesbcd 3522 | form of ~ spsbc . (Contri... |
| sbcth2 3523 | A substitution into a theo... |
| ra4v 3524 | Version of ~ ra4 with a dv... |
| ra4 3525 | Restricted quantifier vers... |
| rmo2 3526 | Alternate definition of re... |
| rmo2i 3527 | Condition implying restric... |
| rmo3 3528 | Restricted "at most one" u... |
| rmob 3529 | Consequence of "at most on... |
| rmoi 3530 | Consequence of "at most on... |
| rmob2 3531 | Consequence of "restricted... |
| rmoi2 3532 | Consequence of "restricted... |
| csb2 3535 | Alternate expression for t... |
| csbeq1 3536 | Analogue of ~ dfsbcq for p... |
| csbeq2 3537 | Substituting into equivale... |
| cbvcsb 3538 | Change bound variables in ... |
| cbvcsbv 3539 | Change the bound variable ... |
| csbeq1d 3540 | Equality deduction for pro... |
| csbid 3541 | Analogue of ~ sbid for pro... |
| csbeq1a 3542 | Equality theorem for prope... |
| csbco 3543 | Composition law for chaine... |
| csbtt 3544 | Substitution doesn't affec... |
| csbconstgf 3545 | Substitution doesn't affec... |
| csbconstg 3546 | Substitution doesn't affec... |
| nfcsb1d 3547 | Bound-variable hypothesis ... |
| nfcsb1 3548 | Bound-variable hypothesis ... |
| nfcsb1v 3549 | Bound-variable hypothesis ... |
| nfcsbd 3550 | Deduction version of ~ nfc... |
| nfcsb 3551 | Bound-variable hypothesis ... |
| csbhypf 3552 | Introduce an explicit subs... |
| csbiebt 3553 | Conversion of implicit sub... |
| csbiedf 3554 | Conversion of implicit sub... |
| csbieb 3555 | Bidirectional conversion b... |
| csbiebg 3556 | Bidirectional conversion b... |
| csbiegf 3557 | Conversion of implicit sub... |
| csbief 3558 | Conversion of implicit sub... |
| csbie 3559 | Conversion of implicit sub... |
| csbied 3560 | Conversion of implicit sub... |
| csbied2 3561 | Conversion of implicit sub... |
| csbie2t 3562 | Conversion of implicit sub... |
| csbie2 3563 | Conversion of implicit sub... |
| csbie2g 3564 | Conversion of implicit sub... |
| cbvralcsf 3565 | A more general version of ... |
| cbvrexcsf 3566 | A more general version of ... |
| cbvreucsf 3567 | A more general version of ... |
| cbvrabcsf 3568 | A more general version of ... |
| cbvralv2 3569 | Rule used to change the bo... |
| cbvrexv2 3570 | Rule used to change the bo... |
| difjust 3576 | Soundness justification th... |
| unjust 3578 | Soundness justification th... |
| injust 3580 | Soundness justification th... |
| dfin5 3582 | Alternate definition for t... |
| dfdif2 3583 | Alternate definition of cl... |
| eldif 3584 | Expansion of membership in... |
| eldifd 3585 | If a class is in one class... |
| eldifad 3586 | If a class is in the diffe... |
| eldifbd 3587 | If a class is in the diffe... |
| dfss 3589 | Variant of subclass defini... |
| dfss2 3591 | Alternate definition of th... |
| dfss3 3592 | Alternate definition of su... |
| dfss6 3593 | Alternate definition of su... |
| dfss2f 3594 | Equivalence for subclass r... |
| dfss3f 3595 | Equivalence for subclass r... |
| nfss 3596 | If ` x ` is not free in ` ... |
| ssel 3597 | Membership relationships f... |
| ssel2 3598 | Membership relationships f... |
| sseli 3599 | Membership inference from ... |
| sselii 3600 | Membership inference from ... |
| sseldi 3601 | Membership inference from ... |
| sseld 3602 | Membership deduction from ... |
| sselda 3603 | Membership deduction from ... |
| sseldd 3604 | Membership inference from ... |
| ssneld 3605 | If a class is not in anoth... |
| ssneldd 3606 | If an element is not in a ... |
| ssriv 3607 | Inference rule based on su... |
| ssrd 3608 | Deduction rule based on su... |
| ssrdv 3609 | Deduction rule based on su... |
| sstr2 3610 | Transitivity of subclasses... |
| sstr 3611 | Transitivity of subclasses... |
| sstri 3612 | Subclass transitivity infe... |
| sstrd 3613 | Subclass transitivity dedu... |
| syl5ss 3614 | Subclass transitivity dedu... |
| syl6ss 3615 | Subclass transitivity dedu... |
| sylan9ss 3616 | A subclass transitivity de... |
| sylan9ssr 3617 | A subclass transitivity de... |
| eqss 3618 | The subclass relationship ... |
| eqssi 3619 | Infer equality from two su... |
| eqssd 3620 | Equality deduction from tw... |
| sssseq 3621 | If a class is a subclass o... |
| eqrd 3622 | Deduce equality of classes... |
| eqrdOLD 3623 | Obsolete proof of ~ eqrd a... |
| ssid 3624 | Any class is a subclass of... |
| ssv 3625 | Any class is a subclass of... |
| sseq1 3626 | Equality theorem for subcl... |
| sseq2 3627 | Equality theorem for the s... |
| sseq12 3628 | Equality theorem for the s... |
| sseq1i 3629 | An equality inference for ... |
| sseq2i 3630 | An equality inference for ... |
| sseq12i 3631 | An equality inference for ... |
| sseq1d 3632 | An equality deduction for ... |
| sseq2d 3633 | An equality deduction for ... |
| sseq12d 3634 | An equality deduction for ... |
| eqsstri 3635 | Substitution of equality i... |
| eqsstr3i 3636 | Substitution of equality i... |
| sseqtri 3637 | Substitution of equality i... |
| sseqtr4i 3638 | Substitution of equality i... |
| eqsstrd 3639 | Substitution of equality i... |
| eqsstr3d 3640 | Substitution of equality i... |
| sseqtrd 3641 | Substitution of equality i... |
| sseqtr4d 3642 | Substitution of equality i... |
| 3sstr3i 3643 | Substitution of equality i... |
| 3sstr4i 3644 | Substitution of equality i... |
| 3sstr3g 3645 | Substitution of equality i... |
| 3sstr4g 3646 | Substitution of equality i... |
| 3sstr3d 3647 | Substitution of equality i... |
| 3sstr4d 3648 | Substitution of equality i... |
| syl5eqss 3649 | A chained subclass and equ... |
| syl5eqssr 3650 | A chained subclass and equ... |
| syl6sseq 3651 | A chained subclass and equ... |
| syl6sseqr 3652 | A chained subclass and equ... |
| syl5sseq 3653 | Subclass transitivity dedu... |
| syl5sseqr 3654 | Subclass transitivity dedu... |
| syl6eqss 3655 | A chained subclass and equ... |
| syl6eqssr 3656 | A chained subclass and equ... |
| eqimss 3657 | Equality implies the subcl... |
| eqimss2 3658 | Equality implies the subcl... |
| eqimssi 3659 | Infer subclass relationshi... |
| eqimss2i 3660 | Infer subclass relationshi... |
| nssne1 3661 | Two classes are different ... |
| nssne2 3662 | Two classes are different ... |
| nss 3663 | Negation of subclass relat... |
| nelss 3664 | Demonstrate by witnesses t... |
| ssrexf 3665 | restricted existential qua... |
| ssralv 3666 | Quantification restricted ... |
| ssrexv 3667 | Existential quantification... |
| ralss 3668 | Restricted universal quant... |
| rexss 3669 | Restricted existential qua... |
| ss2ab 3670 | Class abstractions in a su... |
| abss 3671 | Class abstraction in a sub... |
| ssab 3672 | Subclass of a class abstra... |
| ssabral 3673 | The relation for a subclas... |
| ss2abi 3674 | Inference of abstraction s... |
| ss2abdv 3675 | Deduction of abstraction s... |
| abssdv 3676 | Deduction of abstraction s... |
| abssi 3677 | Inference of abstraction s... |
| ss2rab 3678 | Restricted abstraction cla... |
| rabss 3679 | Restricted class abstracti... |
| ssrab 3680 | Subclass of a restricted c... |
| ssrabdv 3681 | Subclass of a restricted c... |
| rabssdv 3682 | Subclass of a restricted c... |
| ss2rabdv 3683 | Deduction of restricted ab... |
| ss2rabi 3684 | Inference of restricted ab... |
| rabss2 3685 | Subclass law for restricte... |
| ssab2 3686 | Subclass relation for the ... |
| ssrab2 3687 | Subclass relation for a re... |
| ssrab3 3688 | Subclass relation for a re... |
| ssrabeq 3689 | If the restricting class o... |
| rabssab 3690 | A restricted class is a su... |
| uniiunlem 3691 | A subset relationship usef... |
| dfpss2 3692 | Alternate definition of pr... |
| dfpss3 3693 | Alternate definition of pr... |
| psseq1 3694 | Equality theorem for prope... |
| psseq2 3695 | Equality theorem for prope... |
| psseq1i 3696 | An equality inference for ... |
| psseq2i 3697 | An equality inference for ... |
| psseq12i 3698 | An equality inference for ... |
| psseq1d 3699 | An equality deduction for ... |
| psseq2d 3700 | An equality deduction for ... |
| psseq12d 3701 | An equality deduction for ... |
| pssss 3702 | A proper subclass is a sub... |
| pssne 3703 | Two classes in a proper su... |
| pssssd 3704 | Deduce subclass from prope... |
| pssned 3705 | Proper subclasses are uneq... |
| sspss 3706 | Subclass in terms of prope... |
| pssirr 3707 | Proper subclass is irrefle... |
| pssn2lp 3708 | Proper subclass has no 2-c... |
| sspsstri 3709 | Two ways of stating tricho... |
| ssnpss 3710 | Partial trichotomy law for... |
| psstr 3711 | Transitive law for proper ... |
| sspsstr 3712 | Transitive law for subclas... |
| psssstr 3713 | Transitive law for subclas... |
| psstrd 3714 | Proper subclass inclusion ... |
| sspsstrd 3715 | Transitivity involving sub... |
| psssstrd 3716 | Transitivity involving sub... |
| npss 3717 | A class is not a proper su... |
| ssnelpss 3718 | A subclass missing a membe... |
| ssnelpssd 3719 | Subclass inclusion with on... |
| ssexnelpss 3720 | If there is an element of ... |
| difeq1 3721 | Equality theorem for class... |
| difeq2 3722 | Equality theorem for class... |
| difeq12 3723 | Equality theorem for class... |
| difeq1i 3724 | Inference adding differenc... |
| difeq2i 3725 | Inference adding differenc... |
| difeq12i 3726 | Equality inference for cla... |
| difeq1d 3727 | Deduction adding differenc... |
| difeq2d 3728 | Deduction adding differenc... |
| difeq12d 3729 | Equality deduction for cla... |
| difeqri 3730 | Inference from membership ... |
| nfdif 3731 | Bound-variable hypothesis ... |
| eldifi 3732 | Implication of membership ... |
| eldifn 3733 | Implication of membership ... |
| elndif 3734 | A set does not belong to a... |
| neldif 3735 | Implication of membership ... |
| difdif 3736 | Double class difference. ... |
| difss 3737 | Subclass relationship for ... |
| difssd 3738 | A difference of two classe... |
| difss2 3739 | If a class is contained in... |
| difss2d 3740 | If a class is contained in... |
| ssdifss 3741 | Preservation of a subclass... |
| ddif 3742 | Double complement under un... |
| ssconb 3743 | Contraposition law for sub... |
| sscon 3744 | Contraposition law for sub... |
| ssdif 3745 | Difference law for subsets... |
| ssdifd 3746 | If ` A ` is contained in `... |
| sscond 3747 | If ` A ` is contained in `... |
| ssdifssd 3748 | If ` A ` is contained in `... |
| ssdif2d 3749 | If ` A ` is contained in `... |
| raldifb 3750 | Restricted universal quant... |
| complss 3751 | Complementation reverses i... |
| compleq 3752 | Two classes are equal if a... |
| elun 3753 | Expansion of membership in... |
| elunnel1 3754 | A member of a union that i... |
| uneqri 3755 | Inference from membership ... |
| unidm 3756 | Idempotent law for union o... |
| uncom 3757 | Commutative law for union ... |
| equncom 3758 | If a class equals the unio... |
| equncomi 3759 | Inference form of ~ equnco... |
| uneq1 3760 | Equality theorem for the u... |
| uneq2 3761 | Equality theorem for the u... |
| uneq12 3762 | Equality theorem for the u... |
| uneq1i 3763 | Inference adding union to ... |
| uneq2i 3764 | Inference adding union to ... |
| uneq12i 3765 | Equality inference for the... |
| uneq1d 3766 | Deduction adding union to ... |
| uneq2d 3767 | Deduction adding union to ... |
| uneq12d 3768 | Equality deduction for the... |
| nfun 3769 | Bound-variable hypothesis ... |
| unass 3770 | Associative law for union ... |
| un12 3771 | A rearrangement of union. ... |
| un23 3772 | A rearrangement of union. ... |
| un4 3773 | A rearrangement of the uni... |
| unundi 3774 | Union distributes over its... |
| unundir 3775 | Union distributes over its... |
| ssun1 3776 | Subclass relationship for ... |
| ssun2 3777 | Subclass relationship for ... |
| ssun3 3778 | Subclass law for union of ... |
| ssun4 3779 | Subclass law for union of ... |
| elun1 3780 | Membership law for union o... |
| elun2 3781 | Membership law for union o... |
| unss1 3782 | Subclass law for union of ... |
| ssequn1 3783 | A relationship between sub... |
| unss2 3784 | Subclass law for union of ... |
| unss12 3785 | Subclass law for union of ... |
| ssequn2 3786 | A relationship between sub... |
| unss 3787 | The union of two subclasse... |
| unssi 3788 | An inference showing the u... |
| unssd 3789 | A deduction showing the un... |
| unssad 3790 | If ` ( A u. B ) ` is conta... |
| unssbd 3791 | If ` ( A u. B ) ` is conta... |
| ssun 3792 | A condition that implies i... |
| rexun 3793 | Restricted existential qua... |
| ralunb 3794 | Restricted quantification ... |
| ralun 3795 | Restricted quantification ... |
| elin 3796 | Expansion of membership in... |
| elini 3797 | Membership in an intersect... |
| elind 3798 | Deduce membership in an in... |
| elinel1 3799 | Membership in an intersect... |
| elinel2 3800 | Membership in an intersect... |
| elin2 3801 | Membership in a class defi... |
| elin1d 3802 | Elementhood in the first s... |
| elin2d 3803 | Elementhood in the first s... |
| elin3 3804 | Membership in a class defi... |
| incom 3805 | Commutative law for inters... |
| ineqri 3806 | Inference from membership ... |
| ineq1 3807 | Equality theorem for inter... |
| ineq2 3808 | Equality theorem for inter... |
| ineq12 3809 | Equality theorem for inter... |
| ineq1i 3810 | Equality inference for int... |
| ineq2i 3811 | Equality inference for int... |
| ineq12i 3812 | Equality inference for int... |
| ineq1d 3813 | Equality deduction for int... |
| ineq2d 3814 | Equality deduction for int... |
| ineq12d 3815 | Equality deduction for int... |
| ineqan12d 3816 | Equality deduction for int... |
| sseqin2 3817 | A relationship between sub... |
| dfss1OLD 3818 | Obsolete as of 22-Jul-2021... |
| dfss5OLD 3819 | Obsolete as of 22-Jul-2021... |
| nfin 3820 | Bound-variable hypothesis ... |
| rabbi2dva 3821 | Deduction from a wff to a ... |
| inidm 3822 | Idempotent law for interse... |
| inass 3823 | Associative law for inters... |
| in12 3824 | A rearrangement of interse... |
| in32 3825 | A rearrangement of interse... |
| in13 3826 | A rearrangement of interse... |
| in31 3827 | A rearrangement of interse... |
| inrot 3828 | Rotate the intersection of... |
| in4 3829 | Rearrangement of intersect... |
| inindi 3830 | Intersection distributes o... |
| inindir 3831 | Intersection distributes o... |
| sseqin2OLD 3832 | Obsolete proof of ~ sseqin... |
| inss1 3833 | The intersection of two cl... |
| inss2 3834 | The intersection of two cl... |
| ssin 3835 | Subclass of intersection. ... |
| ssini 3836 | An inference showing that ... |
| ssind 3837 | A deduction showing that a... |
| ssrin 3838 | Add right intersection to ... |
| sslin 3839 | Add left intersection to s... |
| ss2in 3840 | Intersection of subclasses... |
| ssinss1 3841 | Intersection preserves sub... |
| inss 3842 | Inclusion of an intersecti... |
| symdifcom 3845 | Symmetric difference commu... |
| symdifeq1 3846 | Equality theorem for symme... |
| symdifeq2 3847 | Equality theorem for symme... |
| nfsymdif 3848 | Hypothesis builder for sym... |
| elsymdif 3849 | Membership in a symmetric ... |
| elsymdifxor 3850 | Membership in a symmetric ... |
| dfsymdif2 3851 | Alternate definition of th... |
| symdif2 3852 | Two ways to express symmet... |
| symdifass 3853 | Symmetric difference assoc... |
| unabs 3854 | Absorption law for union. ... |
| inabs 3855 | Absorption law for interse... |
| nssinpss 3856 | Negation of subclass expre... |
| nsspssun 3857 | Negation of subclass expre... |
| dfss4 3858 | Subclass defined in terms ... |
| dfun2 3859 | An alternate definition of... |
| dfin2 3860 | An alternate definition of... |
| difin 3861 | Difference with intersecti... |
| ssdifim 3862 | Implication of a class dif... |
| ssdifsym 3863 | Symmetric class difference... |
| dfss5 3864 | Alternate definition of su... |
| dfun3 3865 | Union defined in terms of ... |
| dfin3 3866 | Intersection defined in te... |
| dfin4 3867 | Alternate definition of th... |
| invdif 3868 | Intersection with universa... |
| indif 3869 | Intersection with class di... |
| indif2 3870 | Bring an intersection in a... |
| indif1 3871 | Bring an intersection in a... |
| indifcom 3872 | Commutation law for inters... |
| indi 3873 | Distributive law for inter... |
| undi 3874 | Distributive law for union... |
| indir 3875 | Distributive law for inter... |
| undir 3876 | Distributive law for union... |
| unineq 3877 | Infer equality from equali... |
| uneqin 3878 | Equality of union and inte... |
| difundi 3879 | Distributive law for class... |
| difundir 3880 | Distributive law for class... |
| difindi 3881 | Distributive law for class... |
| difindir 3882 | Distributive law for class... |
| indifdir 3883 | Distribute intersection ov... |
| difdif2 3884 | Class difference by a clas... |
| undm 3885 | De Morgan's law for union.... |
| indm 3886 | De Morgan's law for inters... |
| difun1 3887 | A relationship involving d... |
| undif3 3888 | An equality involving clas... |
| undif3OLD 3889 | Obsolete proof of ~ undif3... |
| difin2 3890 | Represent a class differen... |
| dif32 3891 | Swap second and third argu... |
| difabs 3892 | Absorption-like law for cl... |
| dfsymdif3 3893 | Alternate definition of th... |
| unab 3894 | Union of two class abstrac... |
| inab 3895 | Intersection of two class ... |
| difab 3896 | Difference of two class ab... |
| notab 3897 | A class builder defined by... |
| unrab 3898 | Union of two restricted cl... |
| inrab 3899 | Intersection of two restri... |
| inrab2 3900 | Intersection with a restri... |
| difrab 3901 | Difference of two restrict... |
| dfrab3 3902 | Alternate definition of re... |
| dfrab2 3903 | Alternate definition of re... |
| notrab 3904 | Complementation of restric... |
| dfrab3ss 3905 | Restricted class abstracti... |
| rabun2 3906 | Abstraction restricted to ... |
| reuss2 3907 | Transfer uniqueness to a s... |
| reuss 3908 | Transfer uniqueness to a s... |
| reuun1 3909 | Transfer uniqueness to a s... |
| reuun2 3910 | Transfer uniqueness to a s... |
| reupick 3911 | Restricted uniqueness "pic... |
| reupick3 3912 | Restricted uniqueness "pic... |
| reupick2 3913 | Restricted uniqueness "pic... |
| euelss 3914 | Transfer uniqueness of an ... |
| dfnul2 3917 | Alternate definition of th... |
| dfnul3 3918 | Alternate definition of th... |
| noel 3919 | The empty set has no eleme... |
| n0i 3920 | If a set has elements, the... |
| ne0i 3921 | If a set has elements, the... |
| n0ii 3922 | If a set has elements, the... |
| ne0ii 3923 | If a set has elements, the... |
| vn0 3924 | The universal class is not... |
| eq0f 3925 | The empty set has no eleme... |
| neq0f 3926 | A nonempty class has at le... |
| n0f 3927 | A nonempty class has at le... |
| n0fOLD 3928 | Obsolete proof of ~ n0f as... |
| eq0 3929 | The empty set has no eleme... |
| neq0 3930 | A nonempty class has at le... |
| n0 3931 | A nonempty class has at le... |
| nel0 3932 | From the general negation ... |
| reximdva0 3933 | Restricted existence deduc... |
| rspn0 3934 | Specialization for restric... |
| n0rex 3935 | There is an element in a n... |
| ssn0rex 3936 | There is an element in a c... |
| n0moeu 3937 | A case of equivalence of "... |
| rex0 3938 | Vacuous existential quanti... |
| 0el 3939 | Membership of the empty se... |
| n0el 3940 | Negated membership of the ... |
| eqeuel 3941 | A condition which implies ... |
| ssdif0 3942 | Subclass expressed in term... |
| difn0 3943 | If the difference of two s... |
| pssdifn0 3944 | A proper subclass has a no... |
| pssdif 3945 | A proper subclass has a no... |
| difin0ss 3946 | Difference, intersection, ... |
| inssdif0 3947 | Intersection, subclass, an... |
| difid 3948 | The difference between a c... |
| difidALT 3949 | Alternate proof of ~ difid... |
| dif0 3950 | The difference between a c... |
| ab0 3951 | The class of sets verifyin... |
| dfnf5 3952 | Characterization of non-fr... |
| ab0orv 3953 | The class builder of a wff... |
| abn0 3954 | Nonempty class abstraction... |
| rab0 3955 | Any restricted class abstr... |
| rab0OLD 3956 | Obsolete proof of ~ rab0 a... |
| rabeq0 3957 | Condition for a restricted... |
| rabn0 3958 | Nonempty restricted class ... |
| rabn0OLD 3959 | Obsolete proof of ~ rabn0 ... |
| rabeq0OLD 3960 | Obsolete proof of ~ rabeq0... |
| rabxm 3961 | Law of excluded middle, in... |
| rabnc 3962 | Law of noncontradiction, i... |
| elneldisj 3963 | The set of elements ` s ` ... |
| elnelun 3964 | The union of the set of el... |
| elneldisjOLD 3965 | Obsolete version of ~ elne... |
| elnelunOLD 3966 | Obsolete version of ~ elne... |
| un0 3967 | The union of a class with ... |
| in0 3968 | The intersection of a clas... |
| 0in 3969 | The intersection of the em... |
| inv1 3970 | The intersection of a clas... |
| unv 3971 | The union of a class with ... |
| 0ss 3972 | The null set is a subset o... |
| ss0b 3973 | Any subset of the empty se... |
| ss0 3974 | Any subset of the empty se... |
| sseq0 3975 | A subclass of an empty cla... |
| ssn0 3976 | A class with a nonempty su... |
| 0dif 3977 | The difference between the... |
| abf 3978 | A class builder with a fal... |
| eq0rdv 3979 | Deduction rule for equalit... |
| csbprc 3980 | The proper substitution of... |
| csbprcOLD 3981 | Obsolete proof of ~ csbprc... |
| csb0 3982 | The proper substitution of... |
| sbcel12 3983 | Distribute proper substitu... |
| sbceqg 3984 | Distribute proper substitu... |
| sbcnel12g 3985 | Distribute proper substitu... |
| sbcne12 3986 | Distribute proper substitu... |
| sbcel1g 3987 | Move proper substitution i... |
| sbceq1g 3988 | Move proper substitution t... |
| sbcel2 3989 | Move proper substitution i... |
| sbceq2g 3990 | Move proper substitution t... |
| csbeq2d 3991 | Formula-building deduction... |
| csbeq2dv 3992 | Formula-building deduction... |
| csbeq2i 3993 | Formula-building inference... |
| csbcom 3994 | Commutative law for double... |
| sbcnestgf 3995 | Nest the composition of tw... |
| csbnestgf 3996 | Nest the composition of tw... |
| sbcnestg 3997 | Nest the composition of tw... |
| csbnestg 3998 | Nest the composition of tw... |
| sbcco3g 3999 | Composition of two substit... |
| csbco3g 4000 | Composition of two class s... |
| csbnest1g 4001 | Nest the composition of tw... |
| csbidm 4002 | Idempotent law for class s... |
| csbvarg 4003 | The proper substitution of... |
| sbccsb 4004 | Substitution into a wff ex... |
| sbccsb2 4005 | Substitution into a wff ex... |
| rspcsbela 4006 | Special case related to ~ ... |
| sbnfc2 4007 | Two ways of expressing " `... |
| csbab 4008 | Move substitution into a c... |
| csbun 4009 | Distribution of class subs... |
| csbin 4010 | Distribute proper substitu... |
| un00 4011 | Two classes are empty iff ... |
| vss 4012 | Only the universal class h... |
| 0pss 4013 | The null set is a proper s... |
| npss0 4014 | No set is a proper subset ... |
| npss0OLD 4015 | Obsolete proof of ~ npss0 ... |
| pssv 4016 | Any non-universal class is... |
| disj 4017 | Two ways of saying that tw... |
| disjr 4018 | Two ways of saying that tw... |
| disj1 4019 | Two ways of saying that tw... |
| reldisj 4020 | Two ways of saying that tw... |
| disj3 4021 | Two ways of saying that tw... |
| disjne 4022 | Members of disjoint sets a... |
| disjel 4023 | A set can't belong to both... |
| disj2 4024 | Two ways of saying that tw... |
| disj4 4025 | Two ways of saying that tw... |
| ssdisj 4026 | Intersection with a subcla... |
| ssdisjOLD 4027 | Obsolete proof of ~ ssdisj... |
| disjpss 4028 | A class is a proper subset... |
| undisj1 4029 | The union of disjoint clas... |
| undisj2 4030 | The union of disjoint clas... |
| ssindif0 4031 | Subclass expressed in term... |
| inelcm 4032 | The intersection of classe... |
| minel 4033 | A minimum element of a cla... |
| minelOLD 4034 | Obsolete proof of ~ minel ... |
| undif4 4035 | Distribute union over diff... |
| disjssun 4036 | Subset relation for disjoi... |
| vdif0 4037 | Universal class equality i... |
| difrab0eq 4038 | If the difference between ... |
| pssnel 4039 | A proper subclass has a me... |
| disjdif 4040 | A class and its relative c... |
| difin0 4041 | The difference of a class ... |
| unvdif 4042 | The union of a class and i... |
| undif1 4043 | Absorption of difference b... |
| undif2 4044 | Absorption of difference b... |
| undifabs 4045 | Absorption of difference b... |
| inundif 4046 | The intersection and class... |
| disjdif2 4047 | The difference of a class ... |
| difun2 4048 | Absorption of union by dif... |
| undif 4049 | Union of complementary par... |
| ssdifin0 4050 | A subset of a difference d... |
| ssdifeq0 4051 | A class is a subclass of i... |
| ssundif 4052 | A condition equivalent to ... |
| difcom 4053 | Swap the arguments of a cl... |
| pssdifcom1 4054 | Two ways to express overla... |
| pssdifcom2 4055 | Two ways to express non-co... |
| difdifdir 4056 | Distributive law for class... |
| uneqdifeq 4057 | Two ways to say that ` A `... |
| uneqdifeqOLD 4058 | Obsolete proof of ~ uneqdi... |
| raldifeq 4059 | Equality theorem for restr... |
| r19.2z 4060 | Theorem 19.2 of [Margaris]... |
| r19.2zb 4061 | A response to the notion t... |
| r19.3rz 4062 | Restricted quantification ... |
| r19.28z 4063 | Restricted quantifier vers... |
| r19.3rzv 4064 | Restricted quantification ... |
| r19.9rzv 4065 | Restricted quantification ... |
| r19.28zv 4066 | Restricted quantifier vers... |
| r19.37zv 4067 | Restricted quantifier vers... |
| r19.45zv 4068 | Restricted version of Theo... |
| r19.44zv 4069 | Restricted version of Theo... |
| r19.27z 4070 | Restricted quantifier vers... |
| r19.27zv 4071 | Restricted quantifier vers... |
| r19.36zv 4072 | Restricted quantifier vers... |
| rzal 4073 | Vacuous quantification is ... |
| rexn0 4074 | Restricted existential qua... |
| ralidm 4075 | Idempotent law for restric... |
| ral0 4076 | Vacuous universal quantifi... |
| rgenzOLD 4077 | Obsolete as of 22-Jul-2021... |
| ralf0 4078 | The quantification of a fa... |
| ralf0OLD 4079 | Obsolete proof of ~ ralf0 ... |
| ralnralall 4080 | A contradiction concerning... |
| falseral0 4081 | A false statement can only... |
| raaan 4082 | Rearrange restricted quant... |
| raaanv 4083 | Rearrange restricted quant... |
| sbss 4084 | Set substitution into the ... |
| sbcssg 4085 | Distribute proper substitu... |
| dfif2 4088 | An alternate definition of... |
| dfif6 4089 | An alternate definition of... |
| ifeq1 4090 | Equality theorem for condi... |
| ifeq2 4091 | Equality theorem for condi... |
| iftrue 4092 | Value of the conditional o... |
| iftruei 4093 | Inference associated with ... |
| iftrued 4094 | Value of the conditional o... |
| iffalse 4095 | Value of the conditional o... |
| iffalsei 4096 | Inference associated with ... |
| iffalsed 4097 | Value of the conditional o... |
| ifnefalse 4098 | When values are unequal, b... |
| ifsb 4099 | Distribute a function over... |
| dfif3 4100 | Alternate definition of th... |
| dfif4 4101 | Alternate definition of th... |
| dfif5 4102 | Alternate definition of th... |
| ifeq12 4103 | Equality theorem for condi... |
| ifeq1d 4104 | Equality deduction for con... |
| ifeq2d 4105 | Equality deduction for con... |
| ifeq12d 4106 | Equality deduction for con... |
| ifbi 4107 | Equivalence theorem for co... |
| ifbid 4108 | Equivalence deduction for ... |
| ifbieq1d 4109 | Equivalence/equality deduc... |
| ifbieq2i 4110 | Equivalence/equality infer... |
| ifbieq2d 4111 | Equivalence/equality deduc... |
| ifbieq12i 4112 | Equivalence deduction for ... |
| ifbieq12d 4113 | Equivalence deduction for ... |
| nfifd 4114 | Deduction version of ~ nfi... |
| nfif 4115 | Bound-variable hypothesis ... |
| ifeq1da 4116 | Conditional equality. (Co... |
| ifeq2da 4117 | Conditional equality. (Co... |
| ifeq12da 4118 | Equivalence deduction for ... |
| ifbieq12d2 4119 | Equivalence deduction for ... |
| ifclda 4120 | Conditional closure. (Con... |
| ifeqda 4121 | Separation of the values o... |
| elimif 4122 | Elimination of a condition... |
| ifbothda 4123 | A wff ` th ` containing a ... |
| ifboth 4124 | A wff ` th ` containing a ... |
| ifid 4125 | Identical true and false a... |
| eqif 4126 | Expansion of an equality w... |
| ifval 4127 | Another expression of the ... |
| elif 4128 | Membership in a conditiona... |
| ifel 4129 | Membership of a conditiona... |
| ifcl 4130 | Membership (closure) of a ... |
| ifcld 4131 | Membership (closure) of a ... |
| ifeqor 4132 | The possible values of a c... |
| ifnot 4133 | Negating the first argumen... |
| ifan 4134 | Rewrite a conjunction in a... |
| ifor 4135 | Rewrite a disjunction in a... |
| 2if2 4136 | Resolve two nested conditi... |
| ifcomnan 4137 | Commute the conditions in ... |
| csbif 4138 | Distribute proper substitu... |
| dedth 4139 | Weak deduction theorem tha... |
| dedth2h 4140 | Weak deduction theorem eli... |
| dedth3h 4141 | Weak deduction theorem eli... |
| dedth4h 4142 | Weak deduction theorem eli... |
| dedth2v 4143 | Weak deduction theorem for... |
| dedth3v 4144 | Weak deduction theorem for... |
| dedth4v 4145 | Weak deduction theorem for... |
| elimhyp 4146 | Eliminate a hypothesis con... |
| elimhyp2v 4147 | Eliminate a hypothesis con... |
| elimhyp3v 4148 | Eliminate a hypothesis con... |
| elimhyp4v 4149 | Eliminate a hypothesis con... |
| elimel 4150 | Eliminate a membership hyp... |
| elimdhyp 4151 | Version of ~ elimhyp where... |
| keephyp 4152 | Transform a hypothesis ` p... |
| keephyp2v 4153 | Keep a hypothesis containi... |
| keephyp3v 4154 | Keep a hypothesis containi... |
| keepel 4155 | Keep a membership hypothes... |
| ifex 4156 | Conditional operator exist... |
| ifexg 4157 | Conditional operator exist... |
| pwjust 4159 | Soundness justification th... |
| pweq 4161 | Equality theorem for power... |
| pweqi 4162 | Equality inference for pow... |
| pweqd 4163 | Equality deduction for pow... |
| elpw 4164 | Membership in a power clas... |
| selpw 4165 | Setvar variable membership... |
| elpwg 4166 | Membership in a power clas... |
| elpwd 4167 | Membership in a power clas... |
| elpwi 4168 | Subset relation implied by... |
| elpwb 4169 | Characterization of the el... |
| elpwid 4170 | An element of a power clas... |
| elelpwi 4171 | If ` A ` belongs to a part... |
| nfpw 4172 | Bound-variable hypothesis ... |
| pwidg 4173 | Membership of the original... |
| pwid 4174 | A set is a member of its p... |
| pwss 4175 | Subclass relationship for ... |
| snjust 4176 | Soundness justification th... |
| sneq 4187 | Equality theorem for singl... |
| sneqi 4188 | Equality inference for sin... |
| sneqd 4189 | Equality deduction for sin... |
| dfsn2 4190 | Alternate definition of si... |
| elsng 4191 | There is exactly one eleme... |
| elsn 4192 | There is exactly one eleme... |
| velsn 4193 | There is only one element ... |
| elsni 4194 | There is only one element ... |
| dfpr2 4195 | Alternate definition of un... |
| elprg 4196 | A member of an unordered p... |
| elpri 4197 | If a class is an element o... |
| elpr 4198 | A member of an unordered p... |
| elpr2 4199 | A member of an unordered p... |
| elpr2OLD 4200 | Obsolete proof of ~ elpr2 ... |
| nelpri 4201 | If an element doesn't matc... |
| prneli 4202 | If an element doesn't matc... |
| nelprd 4203 | If an element doesn't matc... |
| eldifpr 4204 | Membership in a set with t... |
| rexdifpr 4205 | Restricted existential qua... |
| snidg 4206 | A set is a member of its s... |
| snidb 4207 | A class is a set iff it is... |
| snid 4208 | A set is a member of its s... |
| vsnid 4209 | A setvar variable is a mem... |
| elsn2g 4210 | There is exactly one eleme... |
| elsn2 4211 | There is exactly one eleme... |
| nelsn 4212 | If a class is not equal to... |
| nelsnOLD 4213 | Obsolete proof of ~ nelsn ... |
| rabeqsn 4214 | Conditions for a restricte... |
| rabsssn 4215 | Conditions for a restricte... |
| ralsnsg 4216 | Substitution expressed in ... |
| rexsns 4217 | Restricted existential qua... |
| ralsng 4218 | Substitution expressed in ... |
| rexsng 4219 | Restricted existential qua... |
| 2ralsng 4220 | Substitution expressed in ... |
| exsnrex 4221 | There is a set being the e... |
| ralsn 4222 | Convert a quantification o... |
| rexsn 4223 | Restricted existential qua... |
| elpwunsn 4224 | Membership in an extension... |
| eqoreldif 4225 | An element of a set is eit... |
| eqoreldifOLD 4226 | Obsolete proof of ~ eqorel... |
| eltpg 4227 | Members of an unordered tr... |
| eldiftp 4228 | Membership in a set with t... |
| eltpi 4229 | A member of an unordered t... |
| eltp 4230 | A member of an unordered t... |
| dftp2 4231 | Alternate definition of un... |
| nfpr 4232 | Bound-variable hypothesis ... |
| ifpr 4233 | Membership of a conditiona... |
| ralprg 4234 | Convert a quantification o... |
| rexprg 4235 | Convert a quantification o... |
| raltpg 4236 | Convert a quantification o... |
| rextpg 4237 | Convert a quantification o... |
| ralpr 4238 | Convert a quantification o... |
| rexpr 4239 | Convert an existential qua... |
| raltp 4240 | Convert a quantification o... |
| rextp 4241 | Convert a quantification o... |
| nfsn 4242 | Bound-variable hypothesis ... |
| csbsng 4243 | Distribute proper substitu... |
| csbprg 4244 | Distribute proper substitu... |
| elinsn 4245 | If the intersection of two... |
| disjsn 4246 | Intersection with the sing... |
| disjsn2 4247 | Two distinct singletons ar... |
| disjpr2 4248 | Two completely distinct un... |
| disjpr2OLD 4249 | Obsolete proof of ~ disjpr... |
| disjprsn 4250 | The disjoint intersection ... |
| disjtpsn 4251 | The disjoint intersection ... |
| disjtp2 4252 | Two completely distinct un... |
| snprc 4253 | The singleton of a proper ... |
| snnzb 4254 | A singleton is nonempty if... |
| r19.12sn 4255 | Special case of ~ r19.12 w... |
| rabsn 4256 | Condition where a restrict... |
| rabsnifsb 4257 | A restricted class abstrac... |
| rabsnif 4258 | A restricted class abstrac... |
| rabrsn 4259 | A restricted class abstrac... |
| euabsn2 4260 | Another way to express exi... |
| euabsn 4261 | Another way to express exi... |
| reusn 4262 | A way to express restricte... |
| absneu 4263 | Restricted existential uni... |
| rabsneu 4264 | Restricted existential uni... |
| eusn 4265 | Two ways to express " ` A ... |
| rabsnt 4266 | Truth implied by equality ... |
| prcom 4267 | Commutative law for unorde... |
| preq1 4268 | Equality theorem for unord... |
| preq2 4269 | Equality theorem for unord... |
| preq12 4270 | Equality theorem for unord... |
| preq1i 4271 | Equality inference for uno... |
| preq2i 4272 | Equality inference for uno... |
| preq12i 4273 | Equality inference for uno... |
| preq1d 4274 | Equality deduction for uno... |
| preq2d 4275 | Equality deduction for uno... |
| preq12d 4276 | Equality deduction for uno... |
| tpeq1 4277 | Equality theorem for unord... |
| tpeq2 4278 | Equality theorem for unord... |
| tpeq3 4279 | Equality theorem for unord... |
| tpeq1d 4280 | Equality theorem for unord... |
| tpeq2d 4281 | Equality theorem for unord... |
| tpeq3d 4282 | Equality theorem for unord... |
| tpeq123d 4283 | Equality theorem for unord... |
| tprot 4284 | Rotation of the elements o... |
| tpcoma 4285 | Swap 1st and 2nd members o... |
| tpcomb 4286 | Swap 2nd and 3rd members o... |
| tpass 4287 | Split off the first elemen... |
| qdass 4288 | Two ways to write an unord... |
| qdassr 4289 | Two ways to write an unord... |
| tpidm12 4290 | Unordered triple ` { A , A... |
| tpidm13 4291 | Unordered triple ` { A , B... |
| tpidm23 4292 | Unordered triple ` { A , B... |
| tpidm 4293 | Unordered triple ` { A , A... |
| tppreq3 4294 | An unordered triple is an ... |
| prid1g 4295 | An unordered pair contains... |
| prid2g 4296 | An unordered pair contains... |
| prid1 4297 | An unordered pair contains... |
| prid2 4298 | An unordered pair contains... |
| ifpprsnss 4299 | An unordered pair is a sin... |
| prprc1 4300 | A proper class vanishes in... |
| prprc2 4301 | A proper class vanishes in... |
| prprc 4302 | An unordered pair containi... |
| tpid1 4303 | One of the three elements ... |
| tpid2 4304 | One of the three elements ... |
| tpid3g 4305 | Closed theorem form of ~ t... |
| tpid3gOLD 4306 | Obsolete proof of ~ tpid3g... |
| tpid3 4307 | One of the three elements ... |
| snnzg 4308 | The singleton of a set is ... |
| snnz 4309 | The singleton of a set is ... |
| prnz 4310 | A pair containing a set is... |
| prnzg 4311 | A pair containing a set is... |
| prnzgOLD 4312 | Obsolete proof of ~ prnzg ... |
| tpnz 4313 | A triplet containing a set... |
| tpnzd 4314 | A triplet containing a set... |
| raltpd 4315 | Convert a quantification o... |
| snss 4316 | The singleton of an elemen... |
| eldifsn 4317 | Membership in a set with a... |
| ssdifsn 4318 | Subset of a set with an el... |
| elpwdifsn 4319 | A subset of a set is an el... |
| eldifsni 4320 | Membership in a set with a... |
| neldifsn 4321 | The class ` A ` is not in ... |
| neldifsnd 4322 | The class ` A ` is not in ... |
| rexdifsn 4323 | Restricted existential qua... |
| raldifsni 4324 | Rearrangement of a propert... |
| raldifsnb 4325 | Restricted universal quant... |
| eldifvsn 4326 | A set is an element of the... |
| snssg 4327 | The singleton of an elemen... |
| difsn 4328 | An element not in a set ca... |
| difprsnss 4329 | Removal of a singleton fro... |
| difprsn1 4330 | Removal of a singleton fro... |
| difprsn2 4331 | Removal of a singleton fro... |
| diftpsn3 4332 | Removal of a singleton fro... |
| diftpsn3OLD 4333 | Obsolete proof of ~ diftps... |
| difpr 4334 | Removing two elements as p... |
| tpprceq3 4335 | An unordered triple is an ... |
| tppreqb 4336 | An unordered triple is an ... |
| difsnb 4337 | ` ( B \ { A } ) ` equals `... |
| difsnpss 4338 | ` ( B \ { A } ) ` is a pro... |
| snssi 4339 | The singleton of an elemen... |
| snssd 4340 | The singleton of an elemen... |
| difsnid 4341 | If we remove a single elem... |
| eldifeldifsn 4342 | An element of a difference... |
| pw0 4343 | Compute the power set of t... |
| pwpw0 4344 | Compute the power set of t... |
| snsspr1 4345 | A singleton is a subset of... |
| snsspr2 4346 | A singleton is a subset of... |
| snsstp1 4347 | A singleton is a subset of... |
| snsstp2 4348 | A singleton is a subset of... |
| snsstp3 4349 | A singleton is a subset of... |
| prssg 4350 | A pair of elements of a cl... |
| prss 4351 | A pair of elements of a cl... |
| prssOLD 4352 | Obsolete proof of ~ prss a... |
| prssi 4353 | A pair of elements of a cl... |
| prssd 4354 | Deduction version of ~ prs... |
| prsspwg 4355 | An unordered pair belongs ... |
| ssprss 4356 | A pair as subset of a pair... |
| ssprsseq 4357 | A proper pair is a subset ... |
| sssn 4358 | The subsets of a singleton... |
| ssunsn2 4359 | The property of being sand... |
| ssunsn 4360 | Possible values for a set ... |
| eqsn 4361 | Two ways to express that a... |
| eqsnOLD 4362 | Obsolete proof of ~ eqsn a... |
| issn 4363 | A sufficient condition for... |
| n0snor2el 4364 | A nonempty set is either a... |
| ssunpr 4365 | Possible values for a set ... |
| sspr 4366 | The subsets of a pair. (C... |
| sstp 4367 | The subsets of a triple. ... |
| tpss 4368 | A triplet of elements of a... |
| tpssi 4369 | A triple of elements of a ... |
| sneqrg 4370 | Closed form of ~ sneqr . ... |
| sneqr 4371 | If the singletons of two s... |
| snsssn 4372 | If a singleton is a subset... |
| sneqrgOLD 4373 | Obsolete proof of ~ sneqrg... |
| sneqbg 4374 | Two singletons of sets are... |
| snsspw 4375 | The singleton of a class i... |
| prsspw 4376 | An unordered pair belongs ... |
| preq1b 4377 | Biconditional equality lem... |
| preq2b 4378 | Biconditional equality lem... |
| preqr1 4379 | Reverse equality lemma for... |
| preqr1OLD 4380 | Reverse equality lemma for... |
| preqr2 4381 | Reverse equality lemma for... |
| preq12b 4382 | Equality relationship for ... |
| prel12 4383 | Equality of two unordered ... |
| opthpr 4384 | An unordered pair has the ... |
| preqr1g 4385 | Reverse equality lemma for... |
| preq12bg 4386 | Closed form of ~ preq12b .... |
| prel12g 4387 | Closed form of ~ prel12 . ... |
| prneimg 4388 | Two pairs are not equal if... |
| prnebg 4389 | A (proper) pair is not equ... |
| pr1eqbg 4390 | A (proper) pair is equal t... |
| pr1nebg 4391 | A (proper) pair is not equ... |
| preqsnd 4392 | Equivalence for a pair equ... |
| preqsn 4393 | Equivalence for a pair equ... |
| preqsnOLD 4394 | Obsolete proof of ~ preqsn... |
| elpreqprlem 4395 | Lemma for ~ elpreqpr . (C... |
| elpreqpr 4396 | Equality and membership ru... |
| elpreqprb 4397 | A set is an element of an ... |
| elpr2elpr 4398 | For an element ` A ` of an... |
| dfopif 4399 | Rewrite ~ df-op using ` if... |
| dfopg 4400 | Value of the ordered pair ... |
| dfop 4401 | Value of an ordered pair w... |
| opeq1 4402 | Equality theorem for order... |
| opeq2 4403 | Equality theorem for order... |
| opeq12 4404 | Equality theorem for order... |
| opeq1i 4405 | Equality inference for ord... |
| opeq2i 4406 | Equality inference for ord... |
| opeq12i 4407 | Equality inference for ord... |
| opeq1d 4408 | Equality deduction for ord... |
| opeq2d 4409 | Equality deduction for ord... |
| opeq12d 4410 | Equality deduction for ord... |
| oteq1 4411 | Equality theorem for order... |
| oteq2 4412 | Equality theorem for order... |
| oteq3 4413 | Equality theorem for order... |
| oteq1d 4414 | Equality deduction for ord... |
| oteq2d 4415 | Equality deduction for ord... |
| oteq3d 4416 | Equality deduction for ord... |
| oteq123d 4417 | Equality deduction for ord... |
| nfop 4418 | Bound-variable hypothesis ... |
| nfopd 4419 | Deduction version of bound... |
| csbopg 4420 | Distribution of class subs... |
| opid 4421 | The ordered pair ` <. A , ... |
| ralunsn 4422 | Restricted quantification ... |
| 2ralunsn 4423 | Double restricted quantifi... |
| opprc 4424 | Expansion of an ordered pa... |
| opprc1 4425 | Expansion of an ordered pa... |
| opprc2 4426 | Expansion of an ordered pa... |
| oprcl 4427 | If an ordered pair has an ... |
| pwsn 4428 | The power set of a singlet... |
| pwsnALT 4429 | Alternate proof of ~ pwsn ... |
| pwpr 4430 | The power set of an unorde... |
| pwtp 4431 | The power set of an unorde... |
| pwpwpw0 4432 | Compute the power set of t... |
| pwv 4433 | The power class of the uni... |
| prproe 4434 | For an element of a proper... |
| 3elpr2eq 4435 | If there are three element... |
| dfuni2 4438 | Alternate definition of cl... |
| eluni 4439 | Membership in class union.... |
| eluni2 4440 | Membership in class union.... |
| elunii 4441 | Membership in class union.... |
| nfuni 4442 | Bound-variable hypothesis ... |
| nfunid 4443 | Deduction version of ~ nfu... |
| unieq 4444 | Equality theorem for class... |
| unieqi 4445 | Inference of equality of t... |
| unieqd 4446 | Deduction of equality of t... |
| eluniab 4447 | Membership in union of a c... |
| elunirab 4448 | Membership in union of a c... |
| unipr 4449 | The union of a pair is the... |
| uniprg 4450 | The union of a pair is the... |
| unisn 4451 | A set equals the union of ... |
| unisng 4452 | A set equals the union of ... |
| unisn3 4453 | Union of a singleton in th... |
| dfnfc2 4454 | An alternative statement o... |
| dfnfc2OLD 4455 | Obsolete proof of ~ dfnfc2... |
| uniun 4456 | The class union of the uni... |
| uniin 4457 | The class union of the int... |
| uniss 4458 | Subclass relationship for ... |
| ssuni 4459 | Subclass relationship for ... |
| ssuniOLD 4460 | Obsolete proof of ~ ssuni ... |
| unissi 4461 | Subclass relationship for ... |
| unissd 4462 | Subclass relationship for ... |
| uni0b 4463 | The union of a set is empt... |
| uni0c 4464 | The union of a set is empt... |
| uni0 4465 | The union of the empty set... |
| csbuni 4466 | Distribute proper substitu... |
| elssuni 4467 | An element of a class is a... |
| unissel 4468 | Condition turning a subcla... |
| unissb 4469 | Relationship involving mem... |
| uniss2 4470 | A subclass condition on th... |
| unidif 4471 | If the difference ` A \ B ... |
| ssunieq 4472 | Relationship implying unio... |
| unimax 4473 | Any member of a class is t... |
| pwuni 4474 | A class is a subclass of t... |
| dfint2 4477 | Alternate definition of cl... |
| inteq 4478 | Equality law for intersect... |
| inteqi 4479 | Equality inference for cla... |
| inteqd 4480 | Equality deduction for cla... |
| elint 4481 | Membership in class inters... |
| elint2 4482 | Membership in class inters... |
| elintg 4483 | Membership in class inters... |
| elintgOLD 4484 | Obsolete proof of ~ elintg... |
| elinti 4485 | Membership in class inters... |
| nfint 4486 | Bound-variable hypothesis ... |
| elintab 4487 | Membership in the intersec... |
| elintrab 4488 | Membership in the intersec... |
| elintrabg 4489 | Membership in the intersec... |
| int0 4490 | The intersection of the em... |
| int0OLD 4491 | Obsolete proof of ~ int0 a... |
| intss1 4492 | An element of a class incl... |
| ssint 4493 | Subclass of a class inters... |
| ssintab 4494 | Subclass of the intersecti... |
| ssintub 4495 | Subclass of the least uppe... |
| ssmin 4496 | Subclass of the minimum va... |
| intmin 4497 | Any member of a class is t... |
| intss 4498 | Intersection of subclasses... |
| intssuni 4499 | The intersection of a none... |
| ssintrab 4500 | Subclass of the intersecti... |
| unissint 4501 | If the union of a class is... |
| intssuni2 4502 | Subclass relationship for ... |
| intminss 4503 | Under subset ordering, the... |
| intmin2 4504 | Any set is the smallest of... |
| intmin3 4505 | Under subset ordering, the... |
| intmin4 4506 | Elimination of a conjunct ... |
| intab 4507 | The intersection of a spec... |
| int0el 4508 | The intersection of a clas... |
| intun 4509 | The class intersection of ... |
| intpr 4510 | The intersection of a pair... |
| intprg 4511 | The intersection of a pair... |
| intsng 4512 | Intersection of a singleto... |
| intsn 4513 | The intersection of a sing... |
| uniintsn 4514 | Two ways to express " ` A ... |
| uniintab 4515 | The union and the intersec... |
| intunsn 4516 | Theorem joining a singleto... |
| rint0 4517 | Relative intersection of a... |
| elrint 4518 | Membership in a restricted... |
| elrint2 4519 | Membership in a restricted... |
| eliun 4524 | Membership in indexed unio... |
| eliin 4525 | Membership in indexed inte... |
| eliuni 4526 | Membership in an indexed u... |
| iuncom 4527 | Commutation of indexed uni... |
| iuncom4 4528 | Commutation of union with ... |
| iunconst 4529 | Indexed union of a constan... |
| iinconst 4530 | Indexed intersection of a ... |
| iuniin 4531 | Law combining indexed unio... |
| iunss1 4532 | Subclass theorem for index... |
| iinss1 4533 | Subclass theorem for index... |
| iuneq1 4534 | Equality theorem for index... |
| iineq1 4535 | Equality theorem for index... |
| ss2iun 4536 | Subclass theorem for index... |
| iuneq2 4537 | Equality theorem for index... |
| iineq2 4538 | Equality theorem for index... |
| iuneq2i 4539 | Equality inference for ind... |
| iineq2i 4540 | Equality inference for ind... |
| iineq2d 4541 | Equality deduction for ind... |
| iuneq2dv 4542 | Equality deduction for ind... |
| iineq2dv 4543 | Equality deduction for ind... |
| iuneq12df 4544 | Equality deduction for ind... |
| iuneq1d 4545 | Equality theorem for index... |
| iuneq12d 4546 | Equality deduction for ind... |
| iuneq2d 4547 | Equality deduction for ind... |
| nfiun 4548 | Bound-variable hypothesis ... |
| nfiin 4549 | Bound-variable hypothesis ... |
| nfiu1 4550 | Bound-variable hypothesis ... |
| nfii1 4551 | Bound-variable hypothesis ... |
| dfiun2g 4552 | Alternate definition of in... |
| dfiin2g 4553 | Alternate definition of in... |
| dfiun2 4554 | Alternate definition of in... |
| dfiin2 4555 | Alternate definition of in... |
| dfiunv2 4556 | Define double indexed unio... |
| cbviun 4557 | Rule used to change the bo... |
| cbviin 4558 | Change bound variables in ... |
| cbviunv 4559 | Rule used to change the bo... |
| cbviinv 4560 | Change bound variables in ... |
| iunss 4561 | Subset theorem for an inde... |
| ssiun 4562 | Subset implication for an ... |
| ssiun2 4563 | Identity law for subset of... |
| ssiun2s 4564 | Subset relationship for an... |
| iunss2 4565 | A subclass condition on th... |
| iunab 4566 | The indexed union of a cla... |
| iunrab 4567 | The indexed union of a res... |
| iunxdif2 4568 | Indexed union with a class... |
| ssiinf 4569 | Subset theorem for an inde... |
| ssiin 4570 | Subset theorem for an inde... |
| iinss 4571 | Subset implication for an ... |
| iinss2 4572 | An indexed intersection is... |
| uniiun 4573 | Class union in terms of in... |
| intiin 4574 | Class intersection in term... |
| iunid 4575 | An indexed union of single... |
| iun0 4576 | An indexed union of the em... |
| 0iun 4577 | An empty indexed union is ... |
| 0iin 4578 | An empty indexed intersect... |
| viin 4579 | Indexed intersection with ... |
| iunn0 4580 | There is a nonempty class ... |
| iinab 4581 | Indexed intersection of a ... |
| iinrab 4582 | Indexed intersection of a ... |
| iinrab2 4583 | Indexed intersection of a ... |
| iunin2 4584 | Indexed union of intersect... |
| iunin1 4585 | Indexed union of intersect... |
| iinun2 4586 | Indexed intersection of un... |
| iundif2 4587 | Indexed union of class dif... |
| 2iunin 4588 | Rearrange indexed unions o... |
| iindif2 4589 | Indexed intersection of cl... |
| iinin2 4590 | Indexed intersection of in... |
| iinin1 4591 | Indexed intersection of in... |
| iinvdif 4592 | The indexed intersection o... |
| elriin 4593 | Elementhood in a relative ... |
| riin0 4594 | Relative intersection of a... |
| riinn0 4595 | Relative intersection of a... |
| riinrab 4596 | Relative intersection of a... |
| symdif0 4597 | Symmetric difference with ... |
| symdifv 4598 | Symmetric difference with ... |
| symdifid 4599 | Symmetric difference with ... |
| iinxsng 4600 | A singleton index picks ou... |
| iinxprg 4601 | Indexed intersection with ... |
| iunxsng 4602 | A singleton index picks ou... |
| iunxsn 4603 | A singleton index picks ou... |
| iunun 4604 | Separate a union in an ind... |
| iunxun 4605 | Separate a union in the in... |
| iunxdif3 4606 | An indexed union where som... |
| iunxprg 4607 | A pair index picks out two... |
| iunxiun 4608 | Separate an indexed union ... |
| iinuni 4609 | A relationship involving u... |
| iununi 4610 | A relationship involving u... |
| sspwuni 4611 | Subclass relationship for ... |
| pwssb 4612 | Two ways to express a coll... |
| elpwpw 4613 | Characterization of the el... |
| pwpwab 4614 | The double power class wri... |
| pwpwssunieq 4615 | The class of sets whose un... |
| elpwuni 4616 | Relationship for power cla... |
| iinpw 4617 | The power class of an inte... |
| iunpwss 4618 | Inclusion of an indexed un... |
| rintn0 4619 | Relative intersection of a... |
| dfdisj2 4622 | Alternate definition for d... |
| disjss2 4623 | If each element of a colle... |
| disjeq2 4624 | Equality theorem for disjo... |
| disjeq2dv 4625 | Equality deduction for dis... |
| disjss1 4626 | A subset of a disjoint col... |
| disjeq1 4627 | Equality theorem for disjo... |
| disjeq1d 4628 | Equality theorem for disjo... |
| disjeq12d 4629 | Equality theorem for disjo... |
| cbvdisj 4630 | Change bound variables in ... |
| cbvdisjv 4631 | Change bound variables in ... |
| nfdisj 4632 | Bound-variable hypothesis ... |
| nfdisj1 4633 | Bound-variable hypothesis ... |
| disjor 4634 | Two ways to say that a col... |
| disjors 4635 | Two ways to say that a col... |
| disji2 4636 | Property of a disjoint col... |
| disji 4637 | Property of a disjoint col... |
| invdisj 4638 | If there is a function ` C... |
| invdisjrab 4639 | The restricted class abstr... |
| disjiun 4640 | A disjoint collection yiel... |
| disjord 4641 | Conditions for a collectio... |
| disjiunb 4642 | Two ways to say that a col... |
| disjiund 4643 | Conditions for a collectio... |
| sndisj 4644 | Any collection of singleto... |
| 0disj 4645 | Any collection of empty se... |
| disjxsn 4646 | A singleton collection is ... |
| disjx0 4647 | An empty collection is dis... |
| disjprg 4648 | A pair collection is disjo... |
| disjxiun 4649 | An indexed union of a disj... |
| disjxiunOLD 4650 | Obsolete proof of ~ disjxi... |
| disjxun 4651 | The union of two disjoint ... |
| disjss3 4652 | Expand a disjoint collecti... |
| breq 4655 | Equality theorem for binar... |
| breq1 4656 | Equality theorem for a bin... |
| breq2 4657 | Equality theorem for a bin... |
| breq12 4658 | Equality theorem for a bin... |
| breqi 4659 | Equality inference for bin... |
| breq1i 4660 | Equality inference for a b... |
| breq2i 4661 | Equality inference for a b... |
| breq12i 4662 | Equality inference for a b... |
| breq1d 4663 | Equality deduction for a b... |
| breqd 4664 | Equality deduction for a b... |
| breq2d 4665 | Equality deduction for a b... |
| breq12d 4666 | Equality deduction for a b... |
| breq123d 4667 | Equality deduction for a b... |
| breqdi 4668 | Equality deduction for a b... |
| breqan12d 4669 | Equality deduction for a b... |
| breqan12rd 4670 | Equality deduction for a b... |
| eqnbrtrd 4671 | Substitution of equal clas... |
| nbrne1 4672 | Two classes are different ... |
| nbrne2 4673 | Two classes are different ... |
| eqbrtri 4674 | Substitution of equal clas... |
| eqbrtrd 4675 | Substitution of equal clas... |
| eqbrtrri 4676 | Substitution of equal clas... |
| eqbrtrrd 4677 | Substitution of equal clas... |
| breqtri 4678 | Substitution of equal clas... |
| breqtrd 4679 | Substitution of equal clas... |
| breqtrri 4680 | Substitution of equal clas... |
| breqtrrd 4681 | Substitution of equal clas... |
| 3brtr3i 4682 | Substitution of equality i... |
| 3brtr4i 4683 | Substitution of equality i... |
| 3brtr3d 4684 | Substitution of equality i... |
| 3brtr4d 4685 | Substitution of equality i... |
| 3brtr3g 4686 | Substitution of equality i... |
| 3brtr4g 4687 | Substitution of equality i... |
| syl5eqbr 4688 | A chained equality inferen... |
| syl5eqbrr 4689 | A chained equality inferen... |
| syl5breq 4690 | A chained equality inferen... |
| syl5breqr 4691 | A chained equality inferen... |
| syl6eqbr 4692 | A chained equality inferen... |
| syl6eqbrr 4693 | A chained equality inferen... |
| syl6breq 4694 | A chained equality inferen... |
| syl6breqr 4695 | A chained equality inferen... |
| ssbrd 4696 | Deduction from a subclass ... |
| ssbri 4697 | Inference from a subclass ... |
| nfbrd 4698 | Deduction version of bound... |
| nfbr 4699 | Bound-variable hypothesis ... |
| brab1 4700 | Relationship between a bin... |
| br0 4701 | The empty binary relation ... |
| brne0 4702 | If two sets are in a binar... |
| brun 4703 | The union of two binary re... |
| brin 4704 | The intersection of two re... |
| brdif 4705 | The difference of two bina... |
| sbcbr123 4706 | Move substitution in and o... |
| sbcbr 4707 | Move substitution in and o... |
| sbcbr12g 4708 | Move substitution in and o... |
| sbcbr1g 4709 | Move substitution in and o... |
| sbcbr2g 4710 | Move substitution in and o... |
| brsymdif 4711 | Characterization of the sy... |
| opabss 4714 | The collection of ordered ... |
| opabbid 4715 | Equivalent wff's yield equ... |
| opabbidv 4716 | Equivalent wff's yield equ... |
| opabbii 4717 | Equivalent wff's yield equ... |
| nfopab 4718 | Bound-variable hypothesis ... |
| nfopab1 4719 | The first abstraction vari... |
| nfopab2 4720 | The second abstraction var... |
| cbvopab 4721 | Rule used to change bound ... |
| cbvopabv 4722 | Rule used to change bound ... |
| cbvopab1 4723 | Change first bound variabl... |
| cbvopab2 4724 | Change second bound variab... |
| cbvopab1s 4725 | Change first bound variabl... |
| cbvopab1v 4726 | Rule used to change the fi... |
| cbvopab2v 4727 | Rule used to change the se... |
| unopab 4728 | Union of two ordered pair ... |
| mpteq12f 4731 | An equality theorem for th... |
| mpteq12dva 4732 | An equality inference for ... |
| mpteq12dv 4733 | An equality inference for ... |
| mpteq12d 4734 | An equality inference for ... |
| mpteq12df 4735 | An equality theorem for th... |
| mpteq12 4736 | An equality theorem for th... |
| mpteq1 4737 | An equality theorem for th... |
| mpteq1d 4738 | An equality theorem for th... |
| mpteq1i 4739 | An equality theorem for th... |
| mpteq2ia 4740 | An equality inference for ... |
| mpteq2i 4741 | An equality inference for ... |
| mpteq12i 4742 | An equality inference for ... |
| mpteq2da 4743 | Slightly more general equa... |
| mpteq2dva 4744 | Slightly more general equa... |
| mpteq2dv 4745 | An equality inference for ... |
| nfmpt 4746 | Bound-variable hypothesis ... |
| nfmpt1 4747 | Bound-variable hypothesis ... |
| cbvmptf 4748 | Rule to change the bound v... |
| cbvmpt 4749 | Rule to change the bound v... |
| cbvmptv 4750 | Rule to change the bound v... |
| mptv 4751 | Function with universal do... |
| dftr2 4754 | An alternate way of defini... |
| dftr5 4755 | An alternate way of defini... |
| dftr3 4756 | An alternate way of defini... |
| dftr4 4757 | An alternate way of defini... |
| treq 4758 | Equality theorem for the t... |
| trel 4759 | In a transitive class, the... |
| trel3 4760 | In a transitive class, the... |
| trss 4761 | An element of a transitive... |
| trssOLD 4762 | Obsolete proof of ~ trss a... |
| trin 4763 | The intersection of transi... |
| tr0 4764 | The empty set is transitiv... |
| trv 4765 | The universe is transitive... |
| triun 4766 | The indexed union of a cla... |
| truni 4767 | The union of a class of tr... |
| trint 4768 | The intersection of a clas... |
| trintss 4769 | Any nonempty transitive cl... |
| trintssOLD 4770 | Obsolete version of ~ trin... |
| axrep1 4772 | The version of the Axiom o... |
| axrep2 4773 | Axiom of Replacement expre... |
| axrep3 4774 | Axiom of Replacement sligh... |
| axrep4 4775 | A more traditional version... |
| axrep5 4776 | Axiom of Replacement (simi... |
| zfrepclf 4777 | An inference rule based on... |
| zfrep3cl 4778 | An inference rule based on... |
| zfrep4 4779 | A version of Replacement u... |
| axsep 4780 | Separation Scheme, which i... |
| axsep2 4782 | A less restrictive version... |
| zfauscl 4783 | Separation Scheme (Aussond... |
| bm1.3ii 4784 | Convert implication to equ... |
| ax6vsep 4785 | Derive ~ ax6v (a weakened ... |
| zfnuleu 4786 | Show the uniqueness of the... |
| axnulALT 4787 | Alternate proof of ~ axnul... |
| axnul 4788 | The Null Set Axiom of ZF s... |
| 0ex 4790 | The Null Set Axiom of ZF s... |
| sseliALT 4791 | Alternate proof of ~ sseli... |
| csbexg 4792 | The existence of proper su... |
| csbex 4793 | The existence of proper su... |
| unisn2 4794 | A version of ~ unisn witho... |
| nalset 4795 | No set contains all sets. ... |
| vprc 4796 | The universal class is not... |
| nvel 4797 | The universal class doesn'... |
| vnex 4798 | The universal class does n... |
| inex1 4799 | Separation Scheme (Aussond... |
| inex2 4800 | Separation Scheme (Aussond... |
| inex1g 4801 | Closed-form, generalized S... |
| ssex 4802 | The subset of a set is als... |
| ssexi 4803 | The subset of a set is als... |
| ssexg 4804 | The subset of a set is als... |
| ssexd 4805 | A subclass of a set is a s... |
| prcssprc 4806 | The superclass of a proper... |
| sselpwd 4807 | Elementhood to a power set... |
| difexg 4808 | Existence of a difference.... |
| difexi 4809 | Existence of a difference,... |
| difexOLD 4810 | Obsolete version of ~ dife... |
| zfausab 4811 | Separation Scheme (Aussond... |
| rabexg 4812 | Separation Scheme in terms... |
| rabex 4813 | Separation Scheme in terms... |
| rabexd 4814 | Separation Scheme in terms... |
| rabex2 4815 | Separation Scheme in terms... |
| rab2ex 4816 | A class abstraction based ... |
| rabex2OLD 4817 | Obsolete version of ~ rabe... |
| rab2exOLD 4818 | Obsolete version of ~ rab2... |
| elssabg 4819 | Membership in a class abst... |
| intex 4820 | The intersection of a none... |
| intnex 4821 | If a class intersection is... |
| intexab 4822 | The intersection of a none... |
| intexrab 4823 | The intersection of a none... |
| iinexg 4824 | The existence of a class i... |
| intabs 4825 | Absorption of a redundant ... |
| inuni 4826 | The intersection of a unio... |
| elpw2g 4827 | Membership in a power clas... |
| elpw2 4828 | Membership in a power clas... |
| elpwi2 4829 | Membership in a power clas... |
| pwnss 4830 | The power set of a set is ... |
| pwne 4831 | No set equals its power se... |
| class2set 4832 | Construct, from any class ... |
| class2seteq 4833 | Equality theorem based on ... |
| 0elpw 4834 | Every power class contains... |
| pwne0 4835 | A power class is never emp... |
| 0nep0 4836 | The empty set and its powe... |
| 0inp0 4837 | Something cannot be equal ... |
| unidif0 4838 | The removal of the empty s... |
| iin0 4839 | An indexed intersection of... |
| notzfaus 4840 | In the Separation Scheme ~... |
| intv 4841 | The intersection of the un... |
| axpweq 4842 | Two equivalent ways to exp... |
| zfpow 4844 | Axiom of Power Sets expres... |
| axpow2 4845 | A variant of the Axiom of ... |
| axpow3 4846 | A variant of the Axiom of ... |
| el 4847 | Every set is an element of... |
| pwex 4848 | Power set axiom expressed ... |
| vpwex 4849 | The powerset of a setvar i... |
| pwexg 4850 | Power set axiom expressed ... |
| abssexg 4851 | Existence of a class of su... |
| snexALT 4852 | Alternate proof of ~ snex ... |
| p0ex 4853 | The power set of the empty... |
| p0exALT 4854 | Alternate proof of ~ p0ex ... |
| pp0ex 4855 | The power set of the power... |
| ord3ex 4856 | The ordinal number 3 is a ... |
| dtru 4857 | At least two sets exist (o... |
| axc16b 4858 | This theorem shows that ax... |
| eunex 4859 | Existential uniqueness imp... |
| eusv1 4860 | Two ways to express single... |
| eusvnf 4861 | Even if ` x ` is free in `... |
| eusvnfb 4862 | Two ways to say that ` A (... |
| eusv2i 4863 | Two ways to express single... |
| eusv2nf 4864 | Two ways to express single... |
| eusv2 4865 | Two ways to express single... |
| reusv1 4866 | Two ways to express single... |
| reusv1OLD 4867 | Obsolete proof of ~ reusv1... |
| reusv2lem1 4868 | Lemma for ~ reusv2 . (Con... |
| reusv2lem2 4869 | Lemma for ~ reusv2 . (Con... |
| reusv2lem2OLD 4870 | Obsolete proof of ~ reusv2... |
| reusv2lem3 4871 | Lemma for ~ reusv2 . (Con... |
| reusv2lem4 4872 | Lemma for ~ reusv2 . (Con... |
| reusv2lem5 4873 | Lemma for ~ reusv2 . (Con... |
| reusv2 4874 | Two ways to express single... |
| reusv3i 4875 | Two ways of expressing exi... |
| reusv3 4876 | Two ways to express single... |
| eusv4 4877 | Two ways to express single... |
| alxfr 4878 | Transfer universal quantif... |
| ralxfrd 4879 | Transfer universal quantif... |
| ralxfrdOLD 4880 | Obsolete proof of ~ ralxfr... |
| rexxfrd 4881 | Transfer universal quantif... |
| ralxfr2d 4882 | Transfer universal quantif... |
| rexxfr2d 4883 | Transfer universal quantif... |
| ralxfrd2 4884 | Transfer universal quantif... |
| rexxfrd2 4885 | Transfer existence from a ... |
| ralxfr 4886 | Transfer universal quantif... |
| ralxfrALT 4887 | Alternate proof of ~ ralxf... |
| rexxfr 4888 | Transfer existence from a ... |
| rabxfrd 4889 | Class builder membership a... |
| rabxfr 4890 | Class builder membership a... |
| reuxfr2d 4891 | Transfer existential uniqu... |
| reuxfr2 4892 | Transfer existential uniqu... |
| reuxfrd 4893 | Transfer existential uniqu... |
| reuxfr 4894 | Transfer existential uniqu... |
| reuhypd 4895 | A theorem useful for elimi... |
| reuhyp 4896 | A theorem useful for elimi... |
| nfnid 4897 | A setvar variable is not f... |
| nfcvb 4898 | The "distinctor" expressio... |
| dtruALT 4899 | Alternate proof of ~ dtru ... |
| dtrucor 4900 | Corollary of ~ dtru . Thi... |
| dtrucor2 4901 | The theorem form of the de... |
| dvdemo1 4902 | Demonstration of a theorem... |
| dvdemo2 4903 | Demonstration of a theorem... |
| zfpair 4904 | The Axiom of Pairing of Ze... |
| axpr 4905 | Unabbreviated version of t... |
| zfpair2 4907 | Derive the abbreviated ver... |
| snex 4908 | A singleton is a set. The... |
| prex 4909 | The Axiom of Pairing using... |
| elALT 4910 | Alternate proof of ~ el , ... |
| dtruALT2 4911 | Alternate proof of ~ dtru ... |
| snelpwi 4912 | A singleton of a set belon... |
| snelpw 4913 | A singleton of a set belon... |
| prelpw 4914 | A pair of two sets belongs... |
| prelpwi 4915 | A pair of two sets belongs... |
| rext 4916 | A theorem similar to exten... |
| sspwb 4917 | Classes are subclasses if ... |
| unipw 4918 | A class equals the union o... |
| univ 4919 | The union of the universe ... |
| pwel 4920 | Membership of a power clas... |
| pwtr 4921 | A class is transitive iff ... |
| ssextss 4922 | An extensionality-like pri... |
| ssext 4923 | An extensionality-like pri... |
| nssss 4924 | Negation of subclass relat... |
| pweqb 4925 | Classes are equal if and o... |
| intid 4926 | The intersection of all se... |
| moabex 4927 | "At most one" existence im... |
| rmorabex 4928 | Restricted "at most one" e... |
| euabex 4929 | The abstraction of a wff w... |
| nnullss 4930 | A nonempty class (even if ... |
| exss 4931 | Restricted existence in a ... |
| opex 4932 | An ordered pair of classes... |
| otex 4933 | An ordered triple of class... |
| elopg 4934 | Characterization of the el... |
| elop 4935 | Characterization of the el... |
| elopOLD 4936 | Obsolete version of ~ elop... |
| opi1 4937 | One of the two elements in... |
| opi2 4938 | One of the two elements of... |
| opeluu 4939 | Each member of an ordered ... |
| op1stb 4940 | Extract the first member o... |
| brv 4941 | Two classes are always in ... |
| opnz 4942 | An ordered pair is nonempt... |
| opnzi 4943 | An ordered pair is nonempt... |
| opth1 4944 | Equality of the first memb... |
| opth 4945 | The ordered pair theorem. ... |
| opthg 4946 | Ordered pair theorem. ` C ... |
| opth1g 4947 | Equality of the first memb... |
| opthg2 4948 | Ordered pair theorem. (Co... |
| opth2 4949 | Ordered pair theorem. (Co... |
| opthneg 4950 | Two ordered pairs are not ... |
| opthne 4951 | Two ordered pairs are not ... |
| otth2 4952 | Ordered triple theorem, wi... |
| otth 4953 | Ordered triple theorem. (... |
| otthg 4954 | Ordered triple theorem, cl... |
| eqvinop 4955 | A variable introduction la... |
| copsexg 4956 | Substitution of class ` A ... |
| copsex2t 4957 | Closed theorem form of ~ c... |
| copsex2g 4958 | Implicit substitution infe... |
| copsex4g 4959 | An implicit substitution i... |
| 0nelop 4960 | A property of ordered pair... |
| opwo0id 4961 | An ordered pair is equal t... |
| opeqex 4962 | Equivalence of existence i... |
| oteqex2 4963 | Equivalence of existence i... |
| oteqex 4964 | Equivalence of existence i... |
| opcom 4965 | An ordered pair commutes i... |
| moop2 4966 | "At most one" property of ... |
| opeqsn 4967 | Equivalence for an ordered... |
| opeqpr 4968 | Equivalence for an ordered... |
| snopeqop 4969 | Equivalence for an ordered... |
| propeqop 4970 | Equivalence for an ordered... |
| propssopi 4971 | If a pair of ordered pairs... |
| mosubopt 4972 | "At most one" remains true... |
| mosubop 4973 | "At most one" remains true... |
| euop2 4974 | Transfer existential uniqu... |
| euotd 4975 | Prove existential uniquene... |
| opthwiener 4976 | Justification theorem for ... |
| uniop 4977 | The union of an ordered pa... |
| uniopel 4978 | Ordered pair membership is... |
| otsndisj 4979 | The singletons consisting ... |
| otiunsndisj 4980 | The union of singletons co... |
| iunopeqop 4981 | Implication of an ordered ... |
| opabid 4982 | The law of concretion. Sp... |
| elopab 4983 | Membership in a class abst... |
| opelopabsbALT 4984 | The law of concretion in t... |
| opelopabsb 4985 | The law of concretion in t... |
| brabsb 4986 | The law of concretion in t... |
| opelopabt 4987 | Closed theorem form of ~ o... |
| opelopabga 4988 | The law of concretion. Th... |
| brabga 4989 | The law of concretion for ... |
| opelopab2a 4990 | Ordered pair membership in... |
| opelopaba 4991 | The law of concretion. Th... |
| braba 4992 | The law of concretion for ... |
| opelopabg 4993 | The law of concretion. Th... |
| brabg 4994 | The law of concretion for ... |
| opelopabgf 4995 | The law of concretion. Th... |
| opelopab2 4996 | Ordered pair membership in... |
| opelopab 4997 | The law of concretion. Th... |
| brab 4998 | The law of concretion for ... |
| opelopabaf 4999 | The law of concretion. Th... |
| opelopabf 5000 | The law of concretion. Th... |
| ssopab2 5001 | Equivalence of ordered pai... |
| ssopab2b 5002 | Equivalence of ordered pai... |
| ssopab2i 5003 | Inference of ordered pair ... |
| ssopab2dv 5004 | Inference of ordered pair ... |
| eqopab2b 5005 | Equivalence of ordered pai... |
| opabn0 5006 | Nonempty ordered pair clas... |
| opab0 5007 | Empty ordered pair class a... |
| csbopab 5008 | Move substitution into a c... |
| csbopabgALT 5009 | Move substitution into a c... |
| csbmpt12 5010 | Move substitution into a m... |
| csbmpt2 5011 | Move substitution into the... |
| iunopab 5012 | Move indexed union inside ... |
| elopabr 5013 | Membership in a class abst... |
| elopabran 5014 | Membership in a class abst... |
| rbropapd 5015 | Properties of a pair in an... |
| rbropap 5016 | Properties of a pair in a ... |
| 2rbropap 5017 | Properties of a pair in a ... |
| pwin 5018 | The power class of the int... |
| pwunss 5019 | The power class of the uni... |
| pwssun 5020 | The power class of the uni... |
| pwundif 5021 | Break up the power class o... |
| pwun 5022 | The power class of the uni... |
| dfid3 5025 | A stronger version of ~ df... |
| dfid4 5026 | The identity function usin... |
| dfid2 5027 | Alternate definition of th... |
| epelg 5030 | The epsilon relation and m... |
| epelc 5031 | The epsilon relationship a... |
| epel 5032 | The epsilon relation and t... |
| poss 5037 | Subset theorem for the par... |
| poeq1 5038 | Equality theorem for parti... |
| poeq2 5039 | Equality theorem for parti... |
| nfpo 5040 | Bound-variable hypothesis ... |
| nfso 5041 | Bound-variable hypothesis ... |
| pocl 5042 | Properties of partial orde... |
| ispod 5043 | Sufficient conditions for ... |
| swopolem 5044 | Perform the substitutions ... |
| swopo 5045 | A strict weak order is a p... |
| poirr 5046 | A partial order relation i... |
| potr 5047 | A partial order relation i... |
| po2nr 5048 | A partial order relation h... |
| po3nr 5049 | A partial order relation h... |
| po0 5050 | Any relation is a partial ... |
| pofun 5051 | A function preserves a par... |
| sopo 5052 | A strict linear order is a... |
| soss 5053 | Subset theorem for the str... |
| soeq1 5054 | Equality theorem for the s... |
| soeq2 5055 | Equality theorem for the s... |
| sonr 5056 | A strict order relation is... |
| sotr 5057 | A strict order relation is... |
| solin 5058 | A strict order relation is... |
| so2nr 5059 | A strict order relation ha... |
| so3nr 5060 | A strict order relation ha... |
| sotric 5061 | A strict order relation sa... |
| sotrieq 5062 | Trichotomy law for strict ... |
| sotrieq2 5063 | Trichotomy law for strict ... |
| sotr2 5064 | A transitivity relation. ... |
| issod 5065 | An irreflexive, transitive... |
| issoi 5066 | An irreflexive, transitive... |
| isso2i 5067 | Deduce strict ordering fro... |
| so0 5068 | Any relation is a strict o... |
| somo 5069 | A totally ordered set has ... |
| fri 5076 | Property of well-founded r... |
| seex 5077 | The ` R ` -preimage of an ... |
| exse 5078 | Any relation on a set is s... |
| dffr2 5079 | Alternate definition of we... |
| frc 5080 | Property of well-founded r... |
| frss 5081 | Subset theorem for the wel... |
| sess1 5082 | Subset theorem for the set... |
| sess2 5083 | Subset theorem for the set... |
| freq1 5084 | Equality theorem for the w... |
| freq2 5085 | Equality theorem for the w... |
| seeq1 5086 | Equality theorem for the s... |
| seeq2 5087 | Equality theorem for the s... |
| nffr 5088 | Bound-variable hypothesis ... |
| nfse 5089 | Bound-variable hypothesis ... |
| nfwe 5090 | Bound-variable hypothesis ... |
| frirr 5091 | A well-founded relation is... |
| fr2nr 5092 | A well-founded relation ha... |
| fr0 5093 | Any relation is well-found... |
| frminex 5094 | If an element of a well-fo... |
| efrirr 5095 | Irreflexivity of the epsil... |
| efrn2lp 5096 | A set founded by epsilon c... |
| epse 5097 | The epsilon relation is se... |
| tz7.2 5098 | Similar to Theorem 7.2 of ... |
| dfepfr 5099 | An alternate way of saying... |
| epfrc 5100 | A subset of an epsilon-fou... |
| wess 5101 | Subset theorem for the wel... |
| weeq1 5102 | Equality theorem for the w... |
| weeq2 5103 | Equality theorem for the w... |
| wefr 5104 | A well-ordering is well-fo... |
| weso 5105 | A well-ordering is a stric... |
| wecmpep 5106 | The elements of an epsilon... |
| wetrep 5107 | An epsilon well-ordering i... |
| wefrc 5108 | A nonempty (possibly prope... |
| we0 5109 | Any relation is a well-ord... |
| wereu 5110 | A subset of a well-ordered... |
| wereu2 5111 | All nonempty (possibly pro... |
| xpeq1 5128 | Equality theorem for Carte... |
| xpeq2 5129 | Equality theorem for Carte... |
| elxpi 5130 | Membership in a Cartesian ... |
| elxp 5131 | Membership in a Cartesian ... |
| elxp2 5132 | Membership in a Cartesian ... |
| elxp2OLD 5133 | Obsolete proof of ~ elxp2 ... |
| xpeq12 5134 | Equality theorem for Carte... |
| xpeq1i 5135 | Equality inference for Car... |
| xpeq2i 5136 | Equality inference for Car... |
| xpeq12i 5137 | Equality inference for Car... |
| xpeq1d 5138 | Equality deduction for Car... |
| xpeq2d 5139 | Equality deduction for Car... |
| xpeq12d 5140 | Equality deduction for Car... |
| sqxpeqd 5141 | Equality deduction for a C... |
| nfxp 5142 | Bound-variable hypothesis ... |
| 0nelxp 5143 | The empty set is not a mem... |
| 0nelxpOLD 5144 | Obsolete proof of ~ 0nelxp... |
| 0nelelxp 5145 | A member of a Cartesian pr... |
| opelxp 5146 | Ordered pair membership in... |
| brxp 5147 | Binary relation on a Carte... |
| opelxpi 5148 | Ordered pair membership in... |
| opelxpd 5149 | Ordered pair membership in... |
| opelxp1 5150 | The first member of an ord... |
| opelxp2 5151 | The second member of an or... |
| otelxp1 5152 | The first member of an ord... |
| otel3xp 5153 | An ordered triple is an el... |
| rabxp 5154 | Membership in a class buil... |
| brrelex12 5155 | A true binary relation on ... |
| brrelex 5156 | A true binary relation on ... |
| brrelex2 5157 | A true binary relation on ... |
| brrelexi 5158 | The first argument of a bi... |
| brrelex2i 5159 | The second argument of a b... |
| nprrel12 5160 | Proper classes are not rel... |
| nprrel 5161 | No proper class is related... |
| 0nelrel 5162 | A binary relation does not... |
| fconstmpt 5163 | Representation of a consta... |
| vtoclr 5164 | Variable to class conversi... |
| opelvvg 5165 | Ordered pair membership in... |
| opelvv 5166 | Ordered pair membership in... |
| opthprc 5167 | Justification theorem for ... |
| brel 5168 | Two things in a binary rel... |
| elxp3 5169 | Membership in a Cartesian ... |
| opeliunxp 5170 | Membership in a union of C... |
| xpundi 5171 | Distributive law for Carte... |
| xpundir 5172 | Distributive law for Carte... |
| xpiundi 5173 | Distributive law for Carte... |
| xpiundir 5174 | Distributive law for Carte... |
| iunxpconst 5175 | Membership in a union of C... |
| xpun 5176 | The Cartesian product of t... |
| elvv 5177 | Membership in universal cl... |
| elvvv 5178 | Membership in universal cl... |
| elvvuni 5179 | An ordered pair contains i... |
| brinxp2 5180 | Intersection of binary rel... |
| brinxp 5181 | Intersection of binary rel... |
| poinxp 5182 | Intersection of partial or... |
| soinxp 5183 | Intersection of total orde... |
| frinxp 5184 | Intersection of well-found... |
| seinxp 5185 | Intersection of set-like r... |
| weinxp 5186 | Intersection of well-order... |
| posn 5187 | Partial ordering of a sing... |
| sosn 5188 | Strict ordering on a singl... |
| frsn 5189 | Founded relation on a sing... |
| wesn 5190 | Well-ordering of a singlet... |
| elopaelxp 5191 | Membership in an ordered p... |
| bropaex12 5192 | Two classes related by an ... |
| opabssxp 5193 | An abstraction relation is... |
| brab2a 5194 | The law of concretion for ... |
| optocl 5195 | Implicit substitution of c... |
| 2optocl 5196 | Implicit substitution of c... |
| 3optocl 5197 | Implicit substitution of c... |
| opbrop 5198 | Ordered pair membership in... |
| 0xp 5199 | The Cartesian product with... |
| csbxp 5200 | Distribute proper substitu... |
| releq 5201 | Equality theorem for the r... |
| releqi 5202 | Equality inference for the... |
| releqd 5203 | Equality deduction for the... |
| nfrel 5204 | Bound-variable hypothesis ... |
| sbcrel 5205 | Distribute proper substitu... |
| relss 5206 | Subclass theorem for relat... |
| ssrel 5207 | A subclass relationship de... |
| ssrelOLD 5208 | Obsolete proof of ~ ssrel ... |
| eqrel 5209 | Extensionality principle f... |
| ssrel2 5210 | A subclass relationship de... |
| relssi 5211 | Inference from subclass pr... |
| relssdv 5212 | Deduction from subclass pr... |
| eqrelriv 5213 | Inference from extensional... |
| eqrelriiv 5214 | Inference from extensional... |
| eqbrriv 5215 | Inference from extensional... |
| eqrelrdv 5216 | Deduce equality of relatio... |
| eqbrrdv 5217 | Deduction from extensional... |
| eqbrrdiv 5218 | Deduction from extensional... |
| eqrelrdv2 5219 | A version of ~ eqrelrdv . ... |
| ssrelrel 5220 | A subclass relationship de... |
| eqrelrel 5221 | Extensionality principle f... |
| elrel 5222 | A member of a relation is ... |
| relsn 5223 | A singleton is a relation ... |
| relsnop 5224 | A singleton of an ordered ... |
| xpss12 5225 | Subset theorem for Cartesi... |
| xpss 5226 | A Cartesian product is inc... |
| relxp 5227 | A Cartesian product is a r... |
| xpss1 5228 | Subset relation for Cartes... |
| xpss2 5229 | Subset relation for Cartes... |
| copsex2gb 5230 | Implicit substitution infe... |
| copsex2ga 5231 | Implicit substitution infe... |
| elopaba 5232 | Membership in an ordered p... |
| xpsspw 5233 | A Cartesian product is inc... |
| unixpss 5234 | The double class union of ... |
| relun 5235 | The union of two relations... |
| relin1 5236 | The intersection with a re... |
| relin2 5237 | The intersection with a re... |
| reldif 5238 | A difference cutting down ... |
| reliun 5239 | An indexed union is a rela... |
| reliin 5240 | An indexed intersection is... |
| reluni 5241 | The union of a class is a ... |
| relint 5242 | The intersection of a clas... |
| rel0 5243 | The empty set is a relatio... |
| nrelv 5244 | The universal class is not... |
| relopabi 5245 | A class of ordered pairs i... |
| relopabiALT 5246 | Alternate proof of ~ relop... |
| relopab 5247 | A class of ordered pairs i... |
| mptrel 5248 | The maps-to notation alway... |
| reli 5249 | The identity relation is a... |
| rele 5250 | The membership relation is... |
| opabid2 5251 | A relation expressed as an... |
| inopab 5252 | Intersection of two ordere... |
| difopab 5253 | The difference of two orde... |
| inxp 5254 | The intersection of two Ca... |
| xpindi 5255 | Distributive law for Carte... |
| xpindir 5256 | Distributive law for Carte... |
| xpiindi 5257 | Distributive law for Carte... |
| xpriindi 5258 | Distributive law for Carte... |
| eliunxp 5259 | Membership in a union of C... |
| opeliunxp2 5260 | Membership in a union of C... |
| raliunxp 5261 | Write a double restricted ... |
| rexiunxp 5262 | Write a double restricted ... |
| ralxp 5263 | Universal quantification r... |
| rexxp 5264 | Existential quantification... |
| exopxfr 5265 | Transfer ordered-pair exis... |
| exopxfr2 5266 | Transfer ordered-pair exis... |
| djussxp 5267 | Disjoint union is a subset... |
| ralxpf 5268 | Version of ~ ralxp with bo... |
| rexxpf 5269 | Version of ~ rexxp with bo... |
| iunxpf 5270 | Indexed union on a Cartesi... |
| opabbi2dv 5271 | Deduce equality of a relat... |
| relop 5272 | A necessary and sufficient... |
| ideqg 5273 | For sets, the identity rel... |
| ideq 5274 | For sets, the identity rel... |
| ididg 5275 | A set is identical to itse... |
| issetid 5276 | Two ways of expressing set... |
| coss1 5277 | Subclass theorem for compo... |
| coss2 5278 | Subclass theorem for compo... |
| coeq1 5279 | Equality theorem for compo... |
| coeq2 5280 | Equality theorem for compo... |
| coeq1i 5281 | Equality inference for com... |
| coeq2i 5282 | Equality inference for com... |
| coeq1d 5283 | Equality deduction for com... |
| coeq2d 5284 | Equality deduction for com... |
| coeq12i 5285 | Equality inference for com... |
| coeq12d 5286 | Equality deduction for com... |
| nfco 5287 | Bound-variable hypothesis ... |
| brcog 5288 | Ordered pair membership in... |
| opelco2g 5289 | Ordered pair membership in... |
| brcogw 5290 | Ordered pair membership in... |
| eqbrrdva 5291 | Deduction from extensional... |
| brco 5292 | Binary relation on a compo... |
| opelco 5293 | Ordered pair membership in... |
| cnvss 5294 | Subset theorem for convers... |
| cnvssOLD 5295 | Obsolete proof of ~ cnvss ... |
| cnveq 5296 | Equality theorem for conve... |
| cnveqi 5297 | Equality inference for con... |
| cnveqd 5298 | Equality deduction for con... |
| elcnv 5299 | Membership in a converse. ... |
| elcnv2 5300 | Membership in a converse. ... |
| nfcnv 5301 | Bound-variable hypothesis ... |
| opelcnvg 5302 | Ordered-pair membership in... |
| brcnvg 5303 | The converse of a binary r... |
| opelcnv 5304 | Ordered-pair membership in... |
| brcnv 5305 | The converse of a binary r... |
| csbcnv 5306 | Move class substitution in... |
| csbcnvgALT 5307 | Move class substitution in... |
| cnvco 5308 | Distributive law of conver... |
| cnvuni 5309 | The converse of a class un... |
| dfdm3 5310 | Alternate definition of do... |
| dfrn2 5311 | Alternate definition of ra... |
| dfrn3 5312 | Alternate definition of ra... |
| elrn2g 5313 | Membership in a range. (C... |
| elrng 5314 | Membership in a range. (C... |
| ssrelrn 5315 | If a relation is a subset ... |
| dfdm4 5316 | Alternate definition of do... |
| dfdmf 5317 | Definition of domain, usin... |
| csbdm 5318 | Distribute proper substitu... |
| eldmg 5319 | Domain membership. Theore... |
| eldm2g 5320 | Domain membership. Theore... |
| eldm 5321 | Membership in a domain. T... |
| eldm2 5322 | Membership in a domain. T... |
| dmss 5323 | Subset theorem for domain.... |
| dmeq 5324 | Equality theorem for domai... |
| dmeqi 5325 | Equality inference for dom... |
| dmeqd 5326 | Equality deduction for dom... |
| opeldmd 5327 | Membership of first of an ... |
| opeldm 5328 | Membership of first of an ... |
| breldm 5329 | Membership of first of a b... |
| breldmg 5330 | Membership of first of a b... |
| dmun 5331 | The domain of a union is t... |
| dmin 5332 | The domain of an intersect... |
| dmiun 5333 | The domain of an indexed u... |
| dmuni 5334 | The domain of a union. Pa... |
| dmopab 5335 | The domain of a class of o... |
| dmopabss 5336 | Upper bound for the domain... |
| dmopab3 5337 | The domain of a restricted... |
| opabssxpd 5338 | An ordered-pair class abst... |
| dm0 5339 | The domain of the empty se... |
| dmi 5340 | The domain of the identity... |
| dmv 5341 | The domain of the universe... |
| dm0rn0 5342 | An empty domain is equival... |
| reldm0 5343 | A relation is empty iff it... |
| dmxp 5344 | The domain of a Cartesian ... |
| dmxpid 5345 | The domain of a square Car... |
| dmxpin 5346 | The domain of the intersec... |
| xpid11 5347 | The Cartesian product of a... |
| dmcnvcnv 5348 | The domain of the double c... |
| rncnvcnv 5349 | The range of the double co... |
| elreldm 5350 | The first member of an ord... |
| rneq 5351 | Equality theorem for range... |
| rneqi 5352 | Equality inference for ran... |
| rneqd 5353 | Equality deduction for ran... |
| rnss 5354 | Subset theorem for range. ... |
| brelrng 5355 | The second argument of a b... |
| brelrn 5356 | The second argument of a b... |
| opelrn 5357 | Membership of second membe... |
| releldm 5358 | The first argument of a bi... |
| relelrn 5359 | The second argument of a b... |
| releldmb 5360 | Membership in a domain. (... |
| relelrnb 5361 | Membership in a range. (C... |
| releldmi 5362 | The first argument of a bi... |
| relelrni 5363 | The second argument of a b... |
| dfrnf 5364 | Definition of range, using... |
| elrn2 5365 | Membership in a range. (C... |
| elrn 5366 | Membership in a range. (C... |
| nfdm 5367 | Bound-variable hypothesis ... |
| nfrn 5368 | Bound-variable hypothesis ... |
| dmiin 5369 | Domain of an intersection.... |
| rnopab 5370 | The range of a class of or... |
| rnmpt 5371 | The range of a function in... |
| elrnmpt 5372 | The range of a function in... |
| elrnmpt1s 5373 | Elementhood in an image se... |
| elrnmpt1 5374 | Elementhood in an image se... |
| elrnmptg 5375 | Membership in the range of... |
| elrnmpti 5376 | Membership in the range of... |
| rn0 5377 | The range of the empty set... |
| dfiun3g 5378 | Alternate definition of in... |
| dfiin3g 5379 | Alternate definition of in... |
| dfiun3 5380 | Alternate definition of in... |
| dfiin3 5381 | Alternate definition of in... |
| riinint 5382 | Express a relative indexed... |
| relrn0 5383 | A relation is empty iff it... |
| dmrnssfld 5384 | The domain and range of a ... |
| dmcoss 5385 | Domain of a composition. ... |
| rncoss 5386 | Range of a composition. (... |
| dmcosseq 5387 | Domain of a composition. ... |
| dmcoeq 5388 | Domain of a composition. ... |
| rncoeq 5389 | Range of a composition. (... |
| reseq1 5390 | Equality theorem for restr... |
| reseq2 5391 | Equality theorem for restr... |
| reseq1i 5392 | Equality inference for res... |
| reseq2i 5393 | Equality inference for res... |
| reseq12i 5394 | Equality inference for res... |
| reseq1d 5395 | Equality deduction for res... |
| reseq2d 5396 | Equality deduction for res... |
| reseq12d 5397 | Equality deduction for res... |
| nfres 5398 | Bound-variable hypothesis ... |
| csbres 5399 | Distribute proper substitu... |
| res0 5400 | A restriction to the empty... |
| opelres 5401 | Ordered pair membership in... |
| brres 5402 | Binary relation on a restr... |
| dfres3 5403 | Alternate definition of re... |
| opelresg 5404 | Ordered pair membership in... |
| brresg 5405 | Binary relation on a restr... |
| opres 5406 | Ordered pair membership in... |
| resieq 5407 | A restricted identity rela... |
| opelresi 5408 | ` <. A , A >. ` belongs to... |
| resres 5409 | The restriction of a restr... |
| resundi 5410 | Distributive law for restr... |
| resundir 5411 | Distributive law for restr... |
| resindi 5412 | Class restriction distribu... |
| resindir 5413 | Class restriction distribu... |
| inres 5414 | Move intersection into cla... |
| resdifcom 5415 | Commutative law for restri... |
| resiun1 5416 | Distribution of restrictio... |
| resiun1OLD 5417 | Obsolete proof of ~ resiun... |
| resiun2 5418 | Distribution of restrictio... |
| dmres 5419 | The domain of a restrictio... |
| ssdmres 5420 | A domain restricted to a s... |
| dmresexg 5421 | The domain of a restrictio... |
| resss 5422 | A class includes its restr... |
| rescom 5423 | Commutative law for restri... |
| ssres 5424 | Subclass theorem for restr... |
| ssres2 5425 | Subclass theorem for restr... |
| relres 5426 | A restriction is a relatio... |
| resabs1 5427 | Absorption law for restric... |
| resabs1d 5428 | Absorption law for restric... |
| resabs2 5429 | Absorption law for restric... |
| residm 5430 | Idempotent law for restric... |
| resima 5431 | A restriction to an image.... |
| resima2 5432 | Image under a restricted c... |
| resima2OLD 5433 | Obsolete proof of ~ resima... |
| xpssres 5434 | Restriction of a constant ... |
| elres 5435 | Membership in a restrictio... |
| elsnres 5436 | Membership in restriction ... |
| relssres 5437 | Simplification law for res... |
| dmressnsn 5438 | The domain of a restrictio... |
| eldmressnsn 5439 | The element of the domain ... |
| eldmeldmressn 5440 | An element of the domain (... |
| resdm 5441 | A relation restricted to i... |
| resexg 5442 | The restriction of a set i... |
| resex 5443 | The restriction of a set i... |
| resindm 5444 | When restricting a relatio... |
| resdmdfsn 5445 | Restricting a relation to ... |
| resopab 5446 | Restriction of a class abs... |
| iss 5447 | A subclass of the identity... |
| resopab2 5448 | Restriction of a class abs... |
| resmpt 5449 | Restriction of the mapping... |
| resmpt3 5450 | Unconditional restriction ... |
| resmptf 5451 | Restriction of the mapping... |
| resmptd 5452 | Restriction of the mapping... |
| dfres2 5453 | Alternate definition of th... |
| mptss 5454 | Sufficient condition for i... |
| opabresid 5455 | The restricted identity ex... |
| mptresid 5456 | The restricted identity ex... |
| dmresi 5457 | The domain of a restricted... |
| restidsing 5458 | Restriction of the identit... |
| restidsingOLD 5459 | Obsolete proof of ~ restid... |
| resid 5460 | Any relation restricted to... |
| imaeq1 5461 | Equality theorem for image... |
| imaeq2 5462 | Equality theorem for image... |
| imaeq1i 5463 | Equality theorem for image... |
| imaeq2i 5464 | Equality theorem for image... |
| imaeq1d 5465 | Equality theorem for image... |
| imaeq2d 5466 | Equality theorem for image... |
| imaeq12d 5467 | Equality theorem for image... |
| dfima2 5468 | Alternate definition of im... |
| dfima3 5469 | Alternate definition of im... |
| elimag 5470 | Membership in an image. T... |
| elima 5471 | Membership in an image. T... |
| elima2 5472 | Membership in an image. T... |
| elima3 5473 | Membership in an image. T... |
| nfima 5474 | Bound-variable hypothesis ... |
| nfimad 5475 | Deduction version of bound... |
| imadmrn 5476 | The image of the domain of... |
| imassrn 5477 | The image of a class is a ... |
| imai 5478 | Image under the identity r... |
| rnresi 5479 | The range of the restricte... |
| resiima 5480 | The image of a restriction... |
| ima0 5481 | Image of the empty set. T... |
| 0ima 5482 | Image under the empty rela... |
| csbima12 5483 | Move class substitution in... |
| imadisj 5484 | A class whose image under ... |
| cnvimass 5485 | A preimage under any class... |
| cnvimarndm 5486 | The preimage of the range ... |
| imasng 5487 | The image of a singleton. ... |
| relimasn 5488 | The image of a singleton. ... |
| elrelimasn 5489 | Elementhood in the image o... |
| elimasn 5490 | Membership in an image of ... |
| elimasng 5491 | Membership in an image of ... |
| elimasni 5492 | Membership in an image of ... |
| args 5493 | Two ways to express the cl... |
| eliniseg 5494 | Membership in an initial s... |
| epini 5495 | Any set is equal to its pr... |
| iniseg 5496 | An idiom that signifies an... |
| inisegn0 5497 | Nonemptiness of an initial... |
| dffr3 5498 | Alternate definition of we... |
| dfse2 5499 | Alternate definition of se... |
| imass1 5500 | Subset theorem for image. ... |
| imass2 5501 | Subset theorem for image. ... |
| ndmima 5502 | The image of a singleton o... |
| relcnv 5503 | A converse is a relation. ... |
| relbrcnvg 5504 | When ` R ` is a relation, ... |
| eliniseg2 5505 | Eliminate the class existe... |
| relbrcnv 5506 | When ` R ` is a relation, ... |
| cotrg 5507 | Two ways of saying that th... |
| cotr 5508 | Two ways of saying a relat... |
| issref 5509 | Two ways to state a relati... |
| cnvsym 5510 | Two ways of saying a relat... |
| intasym 5511 | Two ways of saying a relat... |
| asymref 5512 | Two ways of saying a relat... |
| asymref2 5513 | Two ways of saying a relat... |
| intirr 5514 | Two ways of saying a relat... |
| brcodir 5515 | Two ways of saying that tw... |
| codir 5516 | Two ways of saying a relat... |
| qfto 5517 | A quantifier-free way of e... |
| xpidtr 5518 | A square Cartesian product... |
| trin2 5519 | The intersection of two tr... |
| poirr2 5520 | A partial order relation i... |
| trinxp 5521 | The relation induced by a ... |
| soirri 5522 | A strict order relation is... |
| sotri 5523 | A strict order relation is... |
| son2lpi 5524 | A strict order relation ha... |
| sotri2 5525 | A transitivity relation. ... |
| sotri3 5526 | A transitivity relation. ... |
| poleloe 5527 | Express "less than or equa... |
| poltletr 5528 | Transitive law for general... |
| somin1 5529 | Property of a minimum in a... |
| somincom 5530 | Commutativity of minimum i... |
| somin2 5531 | Property of a minimum in a... |
| soltmin 5532 | Being less than a minimum,... |
| cnvopab 5533 | The converse of a class ab... |
| mptcnv 5534 | The converse of a mapping ... |
| cnv0 5535 | The converse of the empty ... |
| cnv0OLD 5536 | Obsolete version of ~ cnv0... |
| cnvi 5537 | The converse of the identi... |
| cnvun 5538 | The converse of a union is... |
| cnvdif 5539 | Distributive law for conve... |
| cnvin 5540 | Distributive law for conve... |
| rnun 5541 | Distributive law for range... |
| rnin 5542 | The range of an intersecti... |
| rniun 5543 | The range of an indexed un... |
| rnuni 5544 | The range of a union. Par... |
| imaundi 5545 | Distributive law for image... |
| imaundir 5546 | The image of a union. (Co... |
| dminss 5547 | An upper bound for interse... |
| imainss 5548 | An upper bound for interse... |
| inimass 5549 | The image of an intersecti... |
| inimasn 5550 | The intersection of the im... |
| cnvxp 5551 | The converse of a Cartesia... |
| xp0 5552 | The Cartesian product with... |
| xpnz 5553 | The Cartesian product of n... |
| xpeq0 5554 | At least one member of an ... |
| xpdisj1 5555 | Cartesian products with di... |
| xpdisj2 5556 | Cartesian products with di... |
| xpsndisj 5557 | Cartesian products with tw... |
| difxp 5558 | Difference of Cartesian pr... |
| difxp1 5559 | Difference law for Cartesi... |
| difxp2 5560 | Difference law for Cartesi... |
| djudisj 5561 | Disjoint unions with disjo... |
| xpdifid 5562 | The set of distinct couple... |
| resdisj 5563 | A double restriction to di... |
| rnxp 5564 | The range of a Cartesian p... |
| dmxpss 5565 | The domain of a Cartesian ... |
| rnxpss 5566 | The range of a Cartesian p... |
| rnxpid 5567 | The range of a square Cart... |
| ssxpb 5568 | A Cartesian product subcla... |
| xp11 5569 | The Cartesian product of n... |
| xpcan 5570 | Cancellation law for Carte... |
| xpcan2 5571 | Cancellation law for Carte... |
| ssrnres 5572 | Subset of the range of a r... |
| rninxp 5573 | Range of the intersection ... |
| dminxp 5574 | Domain of the intersection... |
| imainrect 5575 | Image of a relation restri... |
| xpima 5576 | The image by a constant fu... |
| xpima1 5577 | The image by a Cartesian p... |
| xpima2 5578 | The image by a Cartesian p... |
| xpimasn 5579 | The image of a singleton b... |
| sossfld 5580 | The base set of a strict o... |
| sofld 5581 | The base set of a nonempty... |
| cnvcnv3 5582 | The set of all ordered pai... |
| dfrel2 5583 | Alternate definition of re... |
| dfrel4v 5584 | A relation can be expresse... |
| dfrel4 5585 | A relation can be expresse... |
| cnvcnv 5586 | The double converse of a c... |
| cnvcnvOLD 5587 | Obsolete proof of ~ cnvcnv... |
| cnvcnv2 5588 | The double converse of a c... |
| cnvcnvss 5589 | The double converse of a c... |
| cnveqb 5590 | Equality theorem for conve... |
| cnveq0 5591 | A relation empty iff its c... |
| dfrel3 5592 | Alternate definition of re... |
| dmresv 5593 | The domain of a universal ... |
| rnresv 5594 | The range of a universal r... |
| dfrn4 5595 | Range defined in terms of ... |
| csbrn 5596 | Distribute proper substitu... |
| rescnvcnv 5597 | The restriction of the dou... |
| cnvcnvres 5598 | The double converse of the... |
| imacnvcnv 5599 | The image of the double co... |
| dmsnn0 5600 | The domain of a singleton ... |
| rnsnn0 5601 | The range of a singleton i... |
| dmsn0 5602 | The domain of the singleto... |
| cnvsn0 5603 | The converse of the single... |
| dmsn0el 5604 | The domain of a singleton ... |
| relsn2 5605 | A singleton is a relation ... |
| dmsnopg 5606 | The domain of a singleton ... |
| dmsnopss 5607 | The domain of a singleton ... |
| dmpropg 5608 | The domain of an unordered... |
| dmsnop 5609 | The domain of a singleton ... |
| dmprop 5610 | The domain of an unordered... |
| dmtpop 5611 | The domain of an unordered... |
| cnvcnvsn 5612 | Double converse of a singl... |
| dmsnsnsn 5613 | The domain of the singleto... |
| rnsnopg 5614 | The range of a singleton o... |
| rnpropg 5615 | The range of a pair of ord... |
| rnsnop 5616 | The range of a singleton o... |
| op1sta 5617 | Extract the first member o... |
| cnvsn 5618 | Converse of a singleton of... |
| op2ndb 5619 | Extract the second member ... |
| op2nda 5620 | Extract the second member ... |
| cnvsng 5621 | Converse of a singleton of... |
| opswap 5622 | Swap the members of an ord... |
| cnvresima 5623 | An image under the convers... |
| resdm2 5624 | A class restricted to its ... |
| resdmres 5625 | Restriction to the domain ... |
| resresdm 5626 | A restriction by an arbitr... |
| imadmres 5627 | The image of the domain of... |
| mptpreima 5628 | The preimage of a function... |
| mptiniseg 5629 | Converse singleton image o... |
| dmmpt 5630 | The domain of the mapping ... |
| dmmptss 5631 | The domain of a mapping is... |
| dmmptg 5632 | The domain of the mapping ... |
| relco 5633 | A composition is a relatio... |
| dfco2 5634 | Alternate definition of a ... |
| dfco2a 5635 | Generalization of ~ dfco2 ... |
| coundi 5636 | Class composition distribu... |
| coundir 5637 | Class composition distribu... |
| cores 5638 | Restricted first member of... |
| resco 5639 | Associative law for the re... |
| imaco 5640 | Image of the composition o... |
| rnco 5641 | The range of the compositi... |
| rnco2 5642 | The range of the compositi... |
| dmco 5643 | The domain of a compositio... |
| coeq0 5644 | A composition of two relat... |
| coiun 5645 | Composition with an indexe... |
| cocnvcnv1 5646 | A composition is not affec... |
| cocnvcnv2 5647 | A composition is not affec... |
| cores2 5648 | Absorption of a reverse (p... |
| co02 5649 | Composition with the empty... |
| co01 5650 | Composition with the empty... |
| coi1 5651 | Composition with the ident... |
| coi2 5652 | Composition with the ident... |
| coires1 5653 | Composition with a restric... |
| coass 5654 | Associative law for class ... |
| relcnvtr 5655 | A relation is transitive i... |
| relssdmrn 5656 | A relation is included in ... |
| cnvssrndm 5657 | The converse is a subset o... |
| cossxp 5658 | Composition as a subset of... |
| relrelss 5659 | Two ways to describe the s... |
| unielrel 5660 | The membership relation fo... |
| relfld 5661 | The double union of a rela... |
| relresfld 5662 | Restriction of a relation ... |
| relcoi2 5663 | Composition with the ident... |
| relcoi1 5664 | Composition with the ident... |
| unidmrn 5665 | The double union of the co... |
| relcnvfld 5666 | if ` R ` is a relation, it... |
| dfdm2 5667 | Alternate definition of do... |
| unixp 5668 | The double class union of ... |
| unixp0 5669 | A Cartesian product is emp... |
| unixpid 5670 | Field of a square Cartesia... |
| ressn 5671 | Restriction of a class to ... |
| cnviin 5672 | The converse of an interse... |
| cnvpo 5673 | The converse of a partial ... |
| cnvso 5674 | The converse of a strict o... |
| xpco 5675 | Composition of two Cartesi... |
| xpcoid 5676 | Composition of two square ... |
| elsnxp 5677 | Elementhood to a cartesian... |
| elsnxpOLD 5678 | Obsolete proof of ~ elsnxp... |
| predeq123 5681 | Equality theorem for the p... |
| predeq1 5682 | Equality theorem for the p... |
| predeq2 5683 | Equality theorem for the p... |
| predeq3 5684 | Equality theorem for the p... |
| nfpred 5685 | Bound-variable hypothesis ... |
| predpredss 5686 | If ` A ` is a subset of ` ... |
| predss 5687 | The predecessor class of `... |
| sspred 5688 | Another subset/predecessor... |
| dfpred2 5689 | An alternate definition of... |
| dfpred3 5690 | An alternate definition of... |
| dfpred3g 5691 | An alternate definition of... |
| elpredim 5692 | Membership in a predecesso... |
| elpred 5693 | Membership in a predecesso... |
| elpredg 5694 | Membership in a predecesso... |
| predasetex 5695 | The predecessor class exis... |
| dffr4 5696 | Alternate definition of we... |
| predel 5697 | Membership in the predeces... |
| predpo 5698 | Property of the precessor ... |
| predso 5699 | Property of the predecesso... |
| predbrg 5700 | Closed form of ~ elpredim ... |
| setlikespec 5701 | If ` R ` is set-like in ` ... |
| predidm 5702 | Idempotent law for the pre... |
| predin 5703 | Intersection law for prede... |
| predun 5704 | Union law for predecessor ... |
| preddif 5705 | Difference law for predece... |
| predep 5706 | The predecessor under the ... |
| preddowncl 5707 | A property of classes that... |
| predpoirr 5708 | Given a partial ordering, ... |
| predfrirr 5709 | Given a well-founded relat... |
| pred0 5710 | The predecessor class over... |
| tz6.26 5711 | All nonempty (possibly pro... |
| tz6.26i 5712 | All nonempty (possibly pro... |
| wfi 5713 | The Principle of Well-Foun... |
| wfii 5714 | The Principle of Well-Foun... |
| wfisg 5715 | Well-Founded Induction Sch... |
| wfis 5716 | Well-Founded Induction Sch... |
| wfis2fg 5717 | Well-Founded Induction Sch... |
| wfis2f 5718 | Well Founded Induction sch... |
| wfis2g 5719 | Well-Founded Induction Sch... |
| wfis2 5720 | Well Founded Induction sch... |
| wfis3 5721 | Well Founded Induction sch... |
| ordeq 5730 | Equality theorem for the o... |
| elong 5731 | An ordinal number is an or... |
| elon 5732 | An ordinal number is an or... |
| eloni 5733 | An ordinal number has the ... |
| elon2 5734 | An ordinal number is an or... |
| limeq 5735 | Equality theorem for the l... |
| ordwe 5736 | Epsilon well-orders every ... |
| ordtr 5737 | An ordinal class is transi... |
| ordfr 5738 | Epsilon is well-founded on... |
| ordelss 5739 | An element of an ordinal c... |
| trssord 5740 | A transitive subclass of a... |
| ordirr 5741 | Epsilon irreflexivity of o... |
| nordeq 5742 | A member of an ordinal cla... |
| ordn2lp 5743 | An ordinal class cannot be... |
| tz7.5 5744 | A nonempty subclass of an ... |
| ordelord 5745 | An element of an ordinal c... |
| tron 5746 | The class of all ordinal n... |
| ordelon 5747 | An element of an ordinal c... |
| onelon 5748 | An element of an ordinal n... |
| tz7.7 5749 | A transitive class belongs... |
| ordelssne 5750 | For ordinal classes, membe... |
| ordelpss 5751 | For ordinal classes, membe... |
| ordsseleq 5752 | For ordinal classes, inclu... |
| ordin 5753 | The intersection of two or... |
| onin 5754 | The intersection of two or... |
| ordtri3or 5755 | A trichotomy law for ordin... |
| ordtri1 5756 | A trichotomy law for ordin... |
| ontri1 5757 | A trichotomy law for ordin... |
| ordtri2 5758 | A trichotomy law for ordin... |
| ordtri3 5759 | A trichotomy law for ordin... |
| ordtri3OLD 5760 | Obsolete proof of ~ ordtri... |
| ordtri4 5761 | A trichotomy law for ordin... |
| orddisj 5762 | An ordinal class and its s... |
| onfr 5763 | The ordinal class is well-... |
| onelpss 5764 | Relationship between membe... |
| onsseleq 5765 | Relationship between subse... |
| onelss 5766 | An element of an ordinal n... |
| ordtr1 5767 | Transitive law for ordinal... |
| ordtr2 5768 | Transitive law for ordinal... |
| ordtr3 5769 | Transitive law for ordinal... |
| ordtr3OLD 5770 | Obsolete proof of ~ ordtr3... |
| ontr1 5771 | Transitive law for ordinal... |
| ontr2 5772 | Transitive law for ordinal... |
| ordunidif 5773 | The union of an ordinal st... |
| ordintdif 5774 | If ` B ` is smaller than `... |
| onintss 5775 | If a property is true for ... |
| oneqmini 5776 | A way to show that an ordi... |
| ord0 5777 | The empty set is an ordina... |
| 0elon 5778 | The empty set is an ordina... |
| ord0eln0 5779 | A nonempty ordinal contain... |
| on0eln0 5780 | An ordinal number contains... |
| dflim2 5781 | An alternate definition of... |
| inton 5782 | The intersection of the cl... |
| nlim0 5783 | The empty set is not a lim... |
| limord 5784 | A limit ordinal is ordinal... |
| limuni 5785 | A limit ordinal is its own... |
| limuni2 5786 | The union of a limit ordin... |
| 0ellim 5787 | A limit ordinal contains t... |
| limelon 5788 | A limit ordinal class that... |
| onn0 5789 | The class of all ordinal n... |
| suceq 5790 | Equality of successors. (... |
| elsuci 5791 | Membership in a successor.... |
| elsucg 5792 | Membership in a successor.... |
| elsuc2g 5793 | Variant of membership in a... |
| elsuc 5794 | Membership in a successor.... |
| elsuc2 5795 | Membership in a successor.... |
| nfsuc 5796 | Bound-variable hypothesis ... |
| elelsuc 5797 | Membership in a successor.... |
| sucel 5798 | Membership of a successor ... |
| suc0 5799 | The successor of the empty... |
| sucprc 5800 | A proper class is its own ... |
| unisuc 5801 | A transitive class is equa... |
| sssucid 5802 | A class is included in its... |
| sucidg 5803 | Part of Proposition 7.23 o... |
| sucid 5804 | A set belongs to its succe... |
| nsuceq0 5805 | No successor is empty. (C... |
| eqelsuc 5806 | A set belongs to the succe... |
| iunsuc 5807 | Inductive definition for t... |
| suctr 5808 | The successor of a transit... |
| suctrOLD 5809 | Obsolete proof of ~ suctr ... |
| trsuc 5810 | A set whose successor belo... |
| trsucss 5811 | A member of the successor ... |
| ordsssuc 5812 | A subset of an ordinal bel... |
| onsssuc 5813 | A subset of an ordinal num... |
| ordsssuc2 5814 | An ordinal subset of an or... |
| onmindif 5815 | When its successor is subt... |
| ordnbtwn 5816 | There is no set between an... |
| ordnbtwnOLD 5817 | Obsolete proof of ~ ordnbt... |
| onnbtwn 5818 | There is no set between an... |
| sucssel 5819 | A set whose successor is a... |
| orddif 5820 | Ordinal derived from its s... |
| orduniss 5821 | An ordinal class includes ... |
| ordtri2or 5822 | A trichotomy law for ordin... |
| ordtri2or2 5823 | A trichotomy law for ordin... |
| ordtri2or3 5824 | A consequence of total ord... |
| ordelinel 5825 | The intersection of two or... |
| ordelinelOLD 5826 | Obsolete proof of ~ ordeli... |
| ordssun 5827 | Property of a subclass of ... |
| ordequn 5828 | The maximum (i.e. union) o... |
| ordun 5829 | The maximum (i.e. union) o... |
| ordunisssuc 5830 | A subclass relationship fo... |
| suc11 5831 | The successor operation be... |
| onordi 5832 | An ordinal number is an or... |
| ontrci 5833 | An ordinal number is a tra... |
| onirri 5834 | An ordinal number is not a... |
| oneli 5835 | A member of an ordinal num... |
| onelssi 5836 | A member of an ordinal num... |
| onssneli 5837 | An ordering law for ordina... |
| onssnel2i 5838 | An ordering law for ordina... |
| onelini 5839 | An element of an ordinal n... |
| oneluni 5840 | An ordinal number equals i... |
| onunisuci 5841 | An ordinal number is equal... |
| onsseli 5842 | Subset is equivalent to me... |
| onun2i 5843 | The union of two ordinal n... |
| unizlim 5844 | An ordinal equal to its ow... |
| on0eqel 5845 | An ordinal number either e... |
| snsn0non 5846 | The singleton of the singl... |
| onxpdisj 5847 | Ordinal numbers and ordere... |
| onnev 5848 | The class of ordinal numbe... |
| iotajust 5850 | Soundness justification th... |
| dfiota2 5852 | Alternate definition for d... |
| nfiota1 5853 | Bound-variable hypothesis ... |
| nfiotad 5854 | Deduction version of ~ nfi... |
| nfiota 5855 | Bound-variable hypothesis ... |
| cbviota 5856 | Change bound variables in ... |
| cbviotav 5857 | Change bound variables in ... |
| sb8iota 5858 | Variable substitution in d... |
| iotaeq 5859 | Equality theorem for descr... |
| iotabi 5860 | Equivalence theorem for de... |
| uniabio 5861 | Part of Theorem 8.17 in [Q... |
| iotaval 5862 | Theorem 8.19 in [Quine] p.... |
| iotauni 5863 | Equivalence between two di... |
| iotaint 5864 | Equivalence between two di... |
| iota1 5865 | Property of iota. (Contri... |
| iotanul 5866 | Theorem 8.22 in [Quine] p.... |
| iotassuni 5867 | The ` iota ` class is a su... |
| iotaex 5868 | Theorem 8.23 in [Quine] p.... |
| iota4 5869 | Theorem *14.22 in [Whitehe... |
| iota4an 5870 | Theorem *14.23 in [Whitehe... |
| iota5 5871 | A method for computing iot... |
| iotabidv 5872 | Formula-building deduction... |
| iotabii 5873 | Formula-building deduction... |
| iotacl 5874 | Membership law for descrip... |
| iota2df 5875 | A condition that allows us... |
| iota2d 5876 | A condition that allows us... |
| iota2 5877 | The unique element such th... |
| sniota 5878 | A class abstraction with a... |
| dfiota4 5879 | The ` iota ` operation usi... |
| dfiota4OLD 5880 | Obsolete proof of ~ dfiota... |
| csbiota 5881 | Class substitution within ... |
| dffun2 5898 | Alternate definition of a ... |
| dffun3 5899 | Alternate definition of fu... |
| dffun4 5900 | Alternate definition of a ... |
| dffun5 5901 | Alternate definition of fu... |
| dffun6f 5902 | Definition of function, us... |
| dffun6 5903 | Alternate definition of a ... |
| funmo 5904 | A function has at most one... |
| funrel 5905 | A function is a relation. ... |
| 0nelfun 5906 | A function does not contai... |
| funss 5907 | Subclass theorem for funct... |
| funeq 5908 | Equality theorem for funct... |
| funeqi 5909 | Equality inference for the... |
| funeqd 5910 | Equality deduction for the... |
| nffun 5911 | Bound-variable hypothesis ... |
| sbcfung 5912 | Distribute proper substitu... |
| funeu 5913 | There is exactly one value... |
| funeu2 5914 | There is exactly one value... |
| dffun7 5915 | Alternate definition of a ... |
| dffun8 5916 | Alternate definition of a ... |
| dffun9 5917 | Alternate definition of a ... |
| funfn 5918 | An equivalence for the fun... |
| funfnd 5919 | A function is a function o... |
| funi 5920 | The identity relation is a... |
| nfunv 5921 | The universe is not a func... |
| funopg 5922 | A Kuratowski ordered pair ... |
| funopab 5923 | A class of ordered pairs i... |
| funopabeq 5924 | A class of ordered pairs o... |
| funopab4 5925 | A class of ordered pairs o... |
| funmpt 5926 | A function in maps-to nota... |
| funmpt2 5927 | Functionality of a class g... |
| funco 5928 | The composition of two fun... |
| funres 5929 | A restriction of a functio... |
| funssres 5930 | The restriction of a funct... |
| fun2ssres 5931 | Equality of restrictions o... |
| funun 5932 | The union of functions wit... |
| fununmo 5933 | If the union of classes is... |
| fununfun 5934 | If the union of classes is... |
| fundif 5935 | A function with removed el... |
| funcnvsn 5936 | The converse singleton of ... |
| funsng 5937 | A singleton of an ordered ... |
| fnsng 5938 | Functionality and domain o... |
| funsn 5939 | A singleton of an ordered ... |
| funprg 5940 | A set of two pairs is a fu... |
| funprgOLD 5941 | Obsolete proof of ~ funprg... |
| funtpg 5942 | A set of three pairs is a ... |
| funtpgOLD 5943 | Obsolete proof of ~ funtpg... |
| funpr 5944 | A function with a domain o... |
| funtp 5945 | A function with a domain o... |
| fnsn 5946 | Functionality and domain o... |
| fnprg 5947 | Function with a domain of ... |
| fntpg 5948 | Function with a domain of ... |
| fntp 5949 | A function with a domain o... |
| funcnvpr 5950 | The converse pair of order... |
| funcnvtp 5951 | The converse triple of ord... |
| funcnvqp 5952 | The converse quadruple of ... |
| funcnvqpOLD 5953 | Obsolete proof of ~ funcnv... |
| fun0 5954 | The empty set is a functio... |
| funcnv0 5955 | The converse of the empty ... |
| funcnvcnv 5956 | The double converse of a f... |
| funcnv2 5957 | A simpler equivalence for ... |
| funcnv 5958 | The converse of a class is... |
| funcnv3 5959 | A condition showing a clas... |
| fun2cnv 5960 | The double converse of a c... |
| svrelfun 5961 | A single-valued relation i... |
| fncnv 5962 | Single-rootedness (see ~ f... |
| fun11 5963 | Two ways of stating that `... |
| fununi 5964 | The union of a chain (with... |
| funin 5965 | The intersection with a fu... |
| funres11 5966 | The restriction of a one-t... |
| funcnvres 5967 | The converse of a restrict... |
| cnvresid 5968 | Converse of a restricted i... |
| funcnvres2 5969 | The converse of a restrict... |
| funimacnv 5970 | The image of the preimage ... |
| funimass1 5971 | A kind of contraposition l... |
| funimass2 5972 | A kind of contraposition l... |
| imadif 5973 | The image of a difference ... |
| imain 5974 | The image of an intersecti... |
| funimaexg 5975 | Axiom of Replacement using... |
| funimaex 5976 | The image of a set under a... |
| isarep1 5977 | Part of a study of the Axi... |
| isarep2 5978 | Part of a study of the Axi... |
| fneq1 5979 | Equality theorem for funct... |
| fneq2 5980 | Equality theorem for funct... |
| fneq1d 5981 | Equality deduction for fun... |
| fneq2d 5982 | Equality deduction for fun... |
| fneq12d 5983 | Equality deduction for fun... |
| fneq12 5984 | Equality theorem for funct... |
| fneq1i 5985 | Equality inference for fun... |
| fneq2i 5986 | Equality inference for fun... |
| nffn 5987 | Bound-variable hypothesis ... |
| fnfun 5988 | A function with domain is ... |
| fnrel 5989 | A function with domain is ... |
| fndm 5990 | The domain of a function. ... |
| funfni 5991 | Inference to convert a fun... |
| fndmu 5992 | A function has a unique do... |
| fnbr 5993 | The first argument of bina... |
| fnop 5994 | The first argument of an o... |
| fneu 5995 | There is exactly one value... |
| fneu2 5996 | There is exactly one value... |
| fnun 5997 | The union of two functions... |
| fnunsn 5998 | Extension of a function wi... |
| fnco 5999 | Composition of two functio... |
| fnresdm 6000 | A function does not change... |
| fnresdisj 6001 | A function restricted to a... |
| 2elresin 6002 | Membership in two function... |
| fnssresb 6003 | Restriction of a function ... |
| fnssres 6004 | Restriction of a function ... |
| fnresin1 6005 | Restriction of a function'... |
| fnresin2 6006 | Restriction of a function'... |
| fnres 6007 | An equivalence for functio... |
| fnresi 6008 | Functionality and domain o... |
| idssxp 6009 | A diagonal set as a subset... |
| fnima 6010 | The image of a function's ... |
| fn0 6011 | A function with empty doma... |
| fnimadisj 6012 | A class that is disjoint w... |
| fnimaeq0 6013 | Images under a function ne... |
| dfmpt3 6014 | Alternate definition for t... |
| mptfnf 6015 | The maps-to notation defin... |
| fnmptf 6016 | The maps-to notation defin... |
| fnopabg 6017 | Functionality and domain o... |
| fnopab 6018 | Functionality and domain o... |
| mptfng 6019 | The maps-to notation defin... |
| fnmpt 6020 | The maps-to notation defin... |
| mpt0 6021 | A mapping operation with e... |
| fnmpti 6022 | Functionality and domain o... |
| dmmpti 6023 | Domain of the mapping oper... |
| dmmptd 6024 | The domain of the mapping ... |
| mptun 6025 | Union of mappings which ar... |
| feq1 6026 | Equality theorem for funct... |
| feq2 6027 | Equality theorem for funct... |
| feq3 6028 | Equality theorem for funct... |
| feq23 6029 | Equality theorem for funct... |
| feq1d 6030 | Equality deduction for fun... |
| feq2d 6031 | Equality deduction for fun... |
| feq3d 6032 | Equality deduction for fun... |
| feq12d 6033 | Equality deduction for fun... |
| feq123d 6034 | Equality deduction for fun... |
| feq123 6035 | Equality theorem for funct... |
| feq1i 6036 | Equality inference for fun... |
| feq2i 6037 | Equality inference for fun... |
| feq12i 6038 | Equality inference for fun... |
| feq23i 6039 | Equality inference for fun... |
| feq23d 6040 | Equality deduction for fun... |
| nff 6041 | Bound-variable hypothesis ... |
| sbcfng 6042 | Distribute proper substitu... |
| sbcfg 6043 | Distribute proper substitu... |
| elimf 6044 | Eliminate a mapping hypoth... |
| ffn 6045 | A mapping is a function wi... |
| ffnd 6046 | A mapping is a function wi... |
| dffn2 6047 | Any function is a mapping ... |
| ffun 6048 | A mapping is a function. ... |
| ffund 6049 | A mapping is a function, d... |
| frel 6050 | A mapping is a relation. ... |
| fdm 6051 | The domain of a mapping. ... |
| fdmi 6052 | The domain of a mapping. ... |
| frn 6053 | The range of a mapping. (... |
| dffn3 6054 | A function maps to its ran... |
| ffrn 6055 | A function maps to its ran... |
| fss 6056 | Expanding the codomain of ... |
| fssd 6057 | Expanding the codomain of ... |
| fco 6058 | Composition of two mapping... |
| fco2 6059 | Functionality of a composi... |
| fssxp 6060 | A mapping is a class of or... |
| funssxp 6061 | Two ways of specifying a p... |
| ffdm 6062 | A mapping is a partial fun... |
| ffdmd 6063 | The domain of a function. ... |
| fdmrn 6064 | A different way to write `... |
| opelf 6065 | The members of an ordered ... |
| fun 6066 | The union of two functions... |
| fun2 6067 | The union of two functions... |
| fun2d 6068 | The union of functions wit... |
| fnfco 6069 | Composition of two functio... |
| fssres 6070 | Restriction of a function ... |
| fssresd 6071 | Restriction of a function ... |
| fssres2 6072 | Restriction of a restricte... |
| fresin 6073 | An identity for the mappin... |
| resasplit 6074 | If two functions agree on ... |
| fresaun 6075 | The union of two functions... |
| fresaunres2 6076 | From the union of two func... |
| fresaunres1 6077 | From the union of two func... |
| fcoi1 6078 | Composition of a mapping a... |
| fcoi2 6079 | Composition of restricted ... |
| feu 6080 | There is exactly one value... |
| fimass 6081 | The image of a class is a ... |
| fcnvres 6082 | The converse of a restrict... |
| fimacnvdisj 6083 | The preimage of a class di... |
| fint 6084 | Function into an intersect... |
| fin 6085 | Mapping into an intersecti... |
| f0 6086 | The empty function. (Cont... |
| f00 6087 | A class is a function with... |
| f0bi 6088 | A function with empty doma... |
| f0dom0 6089 | A function is empty iff it... |
| f0rn0 6090 | If there is no element in ... |
| fconst 6091 | A Cartesian product with a... |
| fconstg 6092 | A Cartesian product with a... |
| fnconstg 6093 | A Cartesian product with a... |
| fconst6g 6094 | Constant function with loo... |
| fconst6 6095 | A constant function as a m... |
| f1eq1 6096 | Equality theorem for one-t... |
| f1eq2 6097 | Equality theorem for one-t... |
| f1eq3 6098 | Equality theorem for one-t... |
| nff1 6099 | Bound-variable hypothesis ... |
| dff12 6100 | Alternate definition of a ... |
| f1f 6101 | A one-to-one mapping is a ... |
| f1fn 6102 | A one-to-one mapping is a ... |
| f1fun 6103 | A one-to-one mapping is a ... |
| f1rel 6104 | A one-to-one onto mapping ... |
| f1dm 6105 | The domain of a one-to-one... |
| f1ss 6106 | A function that is one-to-... |
| f1ssr 6107 | A function that is one-to-... |
| f1ssres 6108 | A function that is one-to-... |
| f1cnvcnv 6109 | Two ways to express that a... |
| f1co 6110 | Composition of one-to-one ... |
| foeq1 6111 | Equality theorem for onto ... |
| foeq2 6112 | Equality theorem for onto ... |
| foeq3 6113 | Equality theorem for onto ... |
| nffo 6114 | Bound-variable hypothesis ... |
| fof 6115 | An onto mapping is a mappi... |
| fofun 6116 | An onto mapping is a funct... |
| fofn 6117 | An onto mapping is a funct... |
| forn 6118 | The codomain of an onto fu... |
| dffo2 6119 | Alternate definition of an... |
| foima 6120 | The image of the domain of... |
| dffn4 6121 | A function maps onto its r... |
| funforn 6122 | A function maps its domain... |
| fodmrnu 6123 | An onto function has uniqu... |
| fores 6124 | Restriction of an onto fun... |
| foco 6125 | Composition of onto functi... |
| foconst 6126 | A nonzero constant functio... |
| f1oeq1 6127 | Equality theorem for one-t... |
| f1oeq2 6128 | Equality theorem for one-t... |
| f1oeq3 6129 | Equality theorem for one-t... |
| f1oeq23 6130 | Equality theorem for one-t... |
| f1eq123d 6131 | Equality deduction for one... |
| foeq123d 6132 | Equality deduction for ont... |
| f1oeq123d 6133 | Equality deduction for one... |
| f1oeq3d 6134 | Equality deduction for one... |
| nff1o 6135 | Bound-variable hypothesis ... |
| f1of1 6136 | A one-to-one onto mapping ... |
| f1of 6137 | A one-to-one onto mapping ... |
| f1ofn 6138 | A one-to-one onto mapping ... |
| f1ofun 6139 | A one-to-one onto mapping ... |
| f1orel 6140 | A one-to-one onto mapping ... |
| f1odm 6141 | The domain of a one-to-one... |
| dff1o2 6142 | Alternate definition of on... |
| dff1o3 6143 | Alternate definition of on... |
| f1ofo 6144 | A one-to-one onto function... |
| dff1o4 6145 | Alternate definition of on... |
| dff1o5 6146 | Alternate definition of on... |
| f1orn 6147 | A one-to-one function maps... |
| f1f1orn 6148 | A one-to-one function maps... |
| f1ocnv 6149 | The converse of a one-to-o... |
| f1ocnvb 6150 | A relation is a one-to-one... |
| f1ores 6151 | The restriction of a one-t... |
| f1orescnv 6152 | The converse of a one-to-o... |
| f1imacnv 6153 | Preimage of an image. (Co... |
| foimacnv 6154 | A reverse version of ~ f1i... |
| foun 6155 | The union of two onto func... |
| f1oun 6156 | The union of two one-to-on... |
| resdif 6157 | The restriction of a one-t... |
| resin 6158 | The restriction of a one-t... |
| f1oco 6159 | Composition of one-to-one ... |
| f1cnv 6160 | The converse of an injecti... |
| funcocnv2 6161 | Composition with the conve... |
| fococnv2 6162 | The composition of an onto... |
| f1ococnv2 6163 | The composition of a one-t... |
| f1cocnv2 6164 | Composition of an injectiv... |
| f1ococnv1 6165 | The composition of a one-t... |
| f1cocnv1 6166 | Composition of an injectiv... |
| funcoeqres 6167 | Re-express a constraint on... |
| f1ssf1 6168 | A subset of an injective f... |
| f10 6169 | The empty set maps one-to-... |
| f10d 6170 | The empty set maps one-to-... |
| f1o00 6171 | One-to-one onto mapping of... |
| fo00 6172 | Onto mapping of the empty ... |
| f1o0 6173 | One-to-one onto mapping of... |
| f1oi 6174 | A restriction of the ident... |
| f1ovi 6175 | The identity relation is a... |
| f1osn 6176 | A singleton of an ordered ... |
| f1osng 6177 | A singleton of an ordered ... |
| f1sng 6178 | A singleton of an ordered ... |
| fsnd 6179 | A singleton of an ordered ... |
| f1oprswap 6180 | A two-element swap is a bi... |
| f1oprg 6181 | An unordered pair of order... |
| tz6.12-2 6182 | Function value when ` F ` ... |
| fveu 6183 | The value of a function at... |
| brprcneu 6184 | If ` A ` is a proper class... |
| fvprc 6185 | A function's value at a pr... |
| fv2 6186 | Alternate definition of fu... |
| dffv3 6187 | A definition of function v... |
| dffv4 6188 | The previous definition of... |
| elfv 6189 | Membership in a function v... |
| fveq1 6190 | Equality theorem for funct... |
| fveq2 6191 | Equality theorem for funct... |
| fveq1i 6192 | Equality inference for fun... |
| fveq1d 6193 | Equality deduction for fun... |
| fveq2i 6194 | Equality inference for fun... |
| fveq2d 6195 | Equality deduction for fun... |
| fveq12i 6196 | Equality deduction for fun... |
| fveq12d 6197 | Equality deduction for fun... |
| nffv 6198 | Bound-variable hypothesis ... |
| nffvmpt1 6199 | Bound-variable hypothesis ... |
| nffvd 6200 | Deduction version of bound... |
| fvex 6201 | The value of a class exist... |
| fvexi 6202 | The value of a class exist... |
| fvexd 6203 | The value of a class exist... |
| fvif 6204 | Move a conditional outside... |
| iffv 6205 | Move a conditional outside... |
| fv3 6206 | Alternate definition of th... |
| fvres 6207 | The value of a restricted ... |
| fvresd 6208 | The value of a restricted ... |
| funssfv 6209 | The value of a member of t... |
| tz6.12-1 6210 | Function value. Theorem 6... |
| tz6.12 6211 | Function value. Theorem 6... |
| tz6.12f 6212 | Function value, using boun... |
| tz6.12c 6213 | Corollary of Theorem 6.12(... |
| tz6.12i 6214 | Corollary of Theorem 6.12(... |
| fvbr0 6215 | Two possibilities for the ... |
| fvrn0 6216 | A function value is a memb... |
| fvssunirn 6217 | The result of a function v... |
| ndmfv 6218 | The value of a class outsi... |
| ndmfvrcl 6219 | Reverse closure law for fu... |
| elfvdm 6220 | If a function value has a ... |
| elfvex 6221 | If a function value has a ... |
| elfvexd 6222 | If a function value is non... |
| eliman0 6223 | A non-nul function value i... |
| nfvres 6224 | The value of a non-member ... |
| nfunsn 6225 | If the restriction of a cl... |
| fvfundmfvn0 6226 | If a class' value at an ar... |
| 0fv 6227 | Function value of the empt... |
| fv2prc 6228 | A function's value at a fu... |
| elfv2ex 6229 | If a function value of a f... |
| fveqres 6230 | Equal values imply equal v... |
| csbfv12 6231 | Move class substitution in... |
| csbfv2g 6232 | Move class substitution in... |
| csbfv 6233 | Substitution for a functio... |
| funbrfv 6234 | The second argument of a b... |
| funopfv 6235 | The second element in an o... |
| fnbrfvb 6236 | Equivalence of function va... |
| fnopfvb 6237 | Equivalence of function va... |
| funbrfvb 6238 | Equivalence of function va... |
| funopfvb 6239 | Equivalence of function va... |
| funbrfv2b 6240 | Function value in terms of... |
| dffn5 6241 | Representation of a functi... |
| fnrnfv 6242 | The range of a function ex... |
| fvelrnb 6243 | A member of a function's r... |
| foelrni 6244 | A member of a surjective f... |
| dfimafn 6245 | Alternate definition of th... |
| dfimafn2 6246 | Alternate definition of th... |
| funimass4 6247 | Membership relation for th... |
| fvelima 6248 | Function value in an image... |
| feqmptd 6249 | Deduction form of ~ dffn5 ... |
| feqresmpt 6250 | Express a restricted funct... |
| feqmptdf 6251 | Deduction form of ~ dffn5f... |
| dffn5f 6252 | Representation of a functi... |
| fvelimab 6253 | Function value in an image... |
| fvelimabd 6254 | Deduction form of ~ fvelim... |
| fvi 6255 | The value of the identity ... |
| fviss 6256 | The value of the identity ... |
| fniinfv 6257 | The indexed intersection o... |
| fnsnfv 6258 | Singleton of function valu... |
| opabiotafun 6259 | Define a function whose va... |
| opabiotadm 6260 | Define a function whose va... |
| opabiota 6261 | Define a function whose va... |
| fnimapr 6262 | The image of a pair under ... |
| ssimaex 6263 | The existence of a subimag... |
| ssimaexg 6264 | The existence of a subimag... |
| funfv 6265 | A simplified expression fo... |
| funfv2 6266 | The value of a function. ... |
| funfv2f 6267 | The value of a function. ... |
| fvun 6268 | Value of the union of two ... |
| fvun1 6269 | The value of a union when ... |
| fvun2 6270 | The value of a union when ... |
| dffv2 6271 | Alternate definition of fu... |
| dmfco 6272 | Domains of a function comp... |
| fvco2 6273 | Value of a function compos... |
| fvco 6274 | Value of a function compos... |
| fvco3 6275 | Value of a function compos... |
| fvco4i 6276 | Conditions for a compositi... |
| fvopab3g 6277 | Value of a function given ... |
| fvopab3ig 6278 | Value of a function given ... |
| brfvopabrbr 6279 | The binary relation of a f... |
| fvmptg 6280 | Value of a function given ... |
| fvmpti 6281 | Value of a function given ... |
| fvmpt 6282 | Value of a function given ... |
| fvmpt2f 6283 | Value of a function given ... |
| fvtresfn 6284 | Functionality of a tuple-r... |
| fvmpts 6285 | Value of a function given ... |
| fvmpt3 6286 | Value of a function given ... |
| fvmpt3i 6287 | Value of a function given ... |
| fvmptd 6288 | Deduction version of ~ fvm... |
| mptrcl 6289 | Reverse closure for a mapp... |
| fvmpt2i 6290 | Value of a function given ... |
| fvmpt2 6291 | Value of a function given ... |
| fvmptss 6292 | If all the values of the m... |
| fvmpt2d 6293 | Deduction version of ~ fvm... |
| fvmptex 6294 | Express a function ` F ` w... |
| fvmptd3f 6295 | Alternate deduction versio... |
| fvmptdf 6296 | Alternate deduction versio... |
| fvmptdv 6297 | Alternate deduction versio... |
| fvmptdv2 6298 | Alternate deduction versio... |
| mpteqb 6299 | Bidirectional equality the... |
| fvmptt 6300 | Closed theorem form of ~ f... |
| fvmptf 6301 | Value of a function given ... |
| fvmptnf 6302 | The value of a function gi... |
| fvmptn 6303 | This somewhat non-intuitiv... |
| fvmptss2 6304 | A mapping always evaluates... |
| elfvmptrab1 6305 | Implications for the value... |
| elfvmptrab 6306 | Implications for the value... |
| fvopab4ndm 6307 | Value of a function given ... |
| fvmptndm 6308 | Value of a function given ... |
| fvopab5 6309 | The value of a function th... |
| fvopab6 6310 | Value of a function given ... |
| eqfnfv 6311 | Equality of functions is d... |
| eqfnfv2 6312 | Equality of functions is d... |
| eqfnfv3 6313 | Derive equality of functio... |
| eqfnfvd 6314 | Deduction for equality of ... |
| eqfnfv2f 6315 | Equality of functions is d... |
| eqfunfv 6316 | Equality of functions is d... |
| fvreseq0 6317 | Equality of restricted fun... |
| fvreseq1 6318 | Equality of a function res... |
| fvreseq 6319 | Equality of restricted fun... |
| fnmptfvd 6320 | A function with a given do... |
| fndmdif 6321 | Two ways to express the lo... |
| fndmdifcom 6322 | The difference set between... |
| fndmdifeq0 6323 | The difference set of two ... |
| fndmin 6324 | Two ways to express the lo... |
| fneqeql 6325 | Two functions are equal if... |
| fneqeql2 6326 | Two functions are equal if... |
| fnreseql 6327 | Two functions are equal on... |
| chfnrn 6328 | The range of a choice func... |
| funfvop 6329 | Ordered pair with function... |
| funfvbrb 6330 | Two ways to say that ` A `... |
| fvimacnvi 6331 | A member of a preimage is ... |
| fvimacnv 6332 | The argument of a function... |
| funimass3 6333 | A kind of contraposition l... |
| funimass5 6334 | A subclass of a preimage i... |
| funconstss 6335 | Two ways of specifying tha... |
| fvimacnvALT 6336 | Alternate proof of ~ fvima... |
| elpreima 6337 | Membership in the preimage... |
| fniniseg 6338 | Membership in the preimage... |
| fncnvima2 6339 | Inverse images under funct... |
| fniniseg2 6340 | Inverse point images under... |
| unpreima 6341 | Preimage of a union. (Con... |
| inpreima 6342 | Preimage of an intersectio... |
| difpreima 6343 | Preimage of a difference. ... |
| respreima 6344 | The preimage of a restrict... |
| iinpreima 6345 | Preimage of an intersectio... |
| intpreima 6346 | Preimage of an intersectio... |
| fimacnv 6347 | The preimage of the codoma... |
| fimacnvinrn 6348 | Taking the converse image ... |
| fimacnvinrn2 6349 | Taking the converse image ... |
| fvn0ssdmfun 6350 | If a class' function value... |
| fnopfv 6351 | Ordered pair with function... |
| fvelrn 6352 | A function's value belongs... |
| nelrnfvne 6353 | A function value cannot be... |
| fveqdmss 6354 | If the empty set is not co... |
| fveqressseq 6355 | If the empty set is not co... |
| fnfvelrn 6356 | A function's value belongs... |
| ffvelrn 6357 | A function's value belongs... |
| ffvelrni 6358 | A function's value belongs... |
| ffvelrnda 6359 | A function's value belongs... |
| ffvelrnd 6360 | A function's value belongs... |
| rexrn 6361 | Restricted existential qua... |
| ralrn 6362 | Restricted universal quant... |
| elrnrexdm 6363 | For any element in the ran... |
| elrnrexdmb 6364 | For any element in the ran... |
| eldmrexrn 6365 | For any element in the dom... |
| eldmrexrnb 6366 | For any element in the dom... |
| fvcofneq 6367 | The values of two function... |
| ralrnmpt 6368 | A restricted quantifier ov... |
| rexrnmpt 6369 | A restricted quantifier ov... |
| f0cli 6370 | Unconditional closure of a... |
| dff2 6371 | Alternate definition of a ... |
| dff3 6372 | Alternate definition of a ... |
| dff4 6373 | Alternate definition of a ... |
| dffo3 6374 | An onto mapping expressed ... |
| dffo4 6375 | Alternate definition of an... |
| dffo5 6376 | Alternate definition of an... |
| exfo 6377 | A relation equivalent to t... |
| foelrn 6378 | Property of a surjective f... |
| foco2 6379 | If a composition of two fu... |
| foco2OLD 6380 | Obsolete proof of ~ foco2 ... |
| fmpt 6381 | Functionality of the mappi... |
| f1ompt 6382 | Express bijection for a ma... |
| fmpti 6383 | Functionality of the mappi... |
| mptex2 6384 | If a class given as a map-... |
| fmptd 6385 | Domain and codomain of the... |
| fmpt3d 6386 | Domain and co-domain of th... |
| fmptdf 6387 | A version of ~ fmptd using... |
| ffnfv 6388 | A function maps to a class... |
| ffnfvf 6389 | A function maps to a class... |
| fnfvrnss 6390 | An upper bound for range d... |
| frnssb 6391 | A function is a function i... |
| rnmptss 6392 | The range of an operation ... |
| fmpt2d 6393 | Domain and codomain of the... |
| ffvresb 6394 | A necessary and sufficient... |
| f1oresrab 6395 | Build a bijection between ... |
| fmptco 6396 | Composition of two functio... |
| fmptcof 6397 | Version of ~ fmptco where ... |
| fmptcos 6398 | Composition of two functio... |
| cofmpt 6399 | Express composition of a m... |
| fcompt 6400 | Express composition of two... |
| fcoconst 6401 | Composition with a constan... |
| fsn 6402 | A function maps a singleto... |
| fsn2 6403 | A function that maps a sin... |
| fsng 6404 | A function maps a singleto... |
| fsn2g 6405 | A function that maps a sin... |
| xpsng 6406 | The Cartesian product of t... |
| xpsn 6407 | The Cartesian product of t... |
| f1o2sn 6408 | A singleton with a nested ... |
| residpr 6409 | Restriction of the identit... |
| dfmpt 6410 | Alternate definition for t... |
| fnasrn 6411 | A function expressed as th... |
| funiun 6412 | A function is a union of s... |
| funopsn 6413 | If a function is an ordere... |
| funop 6414 | An ordered pair is a funct... |
| funopdmsn 6415 | The domain of a function w... |
| funsndifnop 6416 | A singleton of an ordered ... |
| funsneqopsn 6417 | A singleton of an ordered ... |
| funsneqop 6418 | A singleton of an ordered ... |
| funsneqopb 6419 | A singleton of an ordered ... |
| ressnop0 6420 | If ` A ` is not in ` C ` ,... |
| fpr 6421 | A function with a domain o... |
| fprg 6422 | A function with a domain o... |
| ftpg 6423 | A function with a domain o... |
| ftp 6424 | A function with a domain o... |
| fnressn 6425 | A function restricted to a... |
| funressn 6426 | A function restricted to a... |
| fressnfv 6427 | The value of a function re... |
| fvrnressn 6428 | If the value of a function... |
| fvressn 6429 | The value of a function re... |
| fvn0fvelrn 6430 | If the value of a function... |
| fvconst 6431 | The value of a constant fu... |
| fnsnb 6432 | A function whose domain is... |
| fmptsn 6433 | Express a singleton functi... |
| fmptsng 6434 | Express a singleton functi... |
| fmptsnd 6435 | Express a singleton functi... |
| fmptap 6436 | Append an additional value... |
| fmptapd 6437 | Append an additional value... |
| fmptpr 6438 | Express a pair function in... |
| fvresi 6439 | The value of a restricted ... |
| fninfp 6440 | Express the class of fixed... |
| fnelfp 6441 | Property of a fixed point ... |
| fndifnfp 6442 | Express the class of non-f... |
| fnelnfp 6443 | Property of a non-fixed po... |
| fnnfpeq0 6444 | A function is the identity... |
| fvunsn 6445 | Remove an ordered pair not... |
| fvsn 6446 | The value of a singleton o... |
| fvsng 6447 | The value of a singleton o... |
| fvsnun1 6448 | The value of a function wi... |
| fvsnun2 6449 | The value of a function wi... |
| fnsnsplit 6450 | Split a function into a si... |
| fsnunf 6451 | Adjoining a point to a fun... |
| fsnunf2 6452 | Adjoining a point to a pun... |
| fsnunfv 6453 | Recover the added point fr... |
| fsnunres 6454 | Recover the original funct... |
| funresdfunsn 6455 | Restricting a function to ... |
| fvpr1 6456 | The value of a function wi... |
| fvpr2 6457 | The value of a function wi... |
| fvpr1g 6458 | The value of a function wi... |
| fvpr2g 6459 | The value of a function wi... |
| fvtp1 6460 | The first value of a funct... |
| fvtp2 6461 | The second value of a func... |
| fvtp3 6462 | The third value of a funct... |
| fvtp1g 6463 | The value of a function wi... |
| fvtp2g 6464 | The value of a function wi... |
| fvtp3g 6465 | The value of a function wi... |
| tpres 6466 | An unordered triple of ord... |
| fvconst2g 6467 | The value of a constant fu... |
| fconst2g 6468 | A constant function expres... |
| fvconst2 6469 | The value of a constant fu... |
| fconst2 6470 | A constant function expres... |
| fconst5 6471 | Two ways to express that a... |
| fnprb 6472 | A function whose domain ha... |
| fntpb 6473 | A function whose domain ha... |
| fnpr2g 6474 | A function whose domain ha... |
| fpr2g 6475 | A function that maps a pai... |
| fconstfv 6476 | A constant function expres... |
| fconst3 6477 | Two ways to express a cons... |
| fconst4 6478 | Two ways to express a cons... |
| resfunexg 6479 | The restriction of a funct... |
| resiexd 6480 | The restriction of the ide... |
| fnex 6481 | If the domain of a functio... |
| funex 6482 | If the domain of a functio... |
| opabex 6483 | Existence of a function ex... |
| mptexg 6484 | If the domain of a functio... |
| mptexgf 6485 | If the domain of a functio... |
| mptex 6486 | If the domain of a functio... |
| mptexd 6487 | If the domain of a functio... |
| mptrabex 6488 | If the domain of a functio... |
| mptrabexOLD 6489 | Obsolete version of ~ mptr... |
| fex 6490 | If the domain of a mapping... |
| eufnfv 6491 | A function is uniquely det... |
| funfvima 6492 | A function's value in a pr... |
| funfvima2 6493 | A function's value in an i... |
| resfvresima 6494 | The value of the function ... |
| funfvima3 6495 | A class including a functi... |
| fnfvima 6496 | The function value of an o... |
| rexima 6497 | Existential quantification... |
| ralima 6498 | Universal quantification u... |
| idref 6499 | TODO: This is the same as... |
| fvclss 6500 | Upper bound for the class ... |
| elabrex 6501 | Elementhood in an image se... |
| abrexco 6502 | Composition of two image m... |
| imaiun 6503 | The image of an indexed un... |
| imauni 6504 | The image of a union is th... |
| fniunfv 6505 | The indexed union of a fun... |
| funiunfv 6506 | The indexed union of a fun... |
| funiunfvf 6507 | The indexed union of a fun... |
| eluniima 6508 | Membership in the union of... |
| elunirn 6509 | Membership in the union of... |
| elunirnALT 6510 | Alternate proof of ~ eluni... |
| fnunirn 6511 | Membership in a union of s... |
| dff13 6512 | A one-to-one function in t... |
| dff13f 6513 | A one-to-one function in t... |
| f1veqaeq 6514 | If the values of a one-to-... |
| f1cofveqaeq 6515 | If the values of a composi... |
| f1cofveqaeqALT 6516 | Alternate proof of ~ f1cof... |
| 2f1fvneq 6517 | If two one-to-one function... |
| f1mpt 6518 | Express injection for a ma... |
| f1fveq 6519 | Equality of function value... |
| f1elima 6520 | Membership in the image of... |
| f1imass 6521 | Taking images under a one-... |
| f1imaeq 6522 | Taking images under a one-... |
| f1imapss 6523 | Taking images under a one-... |
| fpropnf1 6524 | A function, given by an un... |
| f1dom3fv3dif 6525 | The function values for a ... |
| f1dom3el3dif 6526 | The range of a 1-1 functio... |
| dff14a 6527 | A one-to-one function in t... |
| dff14b 6528 | A one-to-one function in t... |
| f12dfv 6529 | A one-to-one function with... |
| f13dfv 6530 | A one-to-one function with... |
| dff1o6 6531 | A one-to-one onto function... |
| f1ocnvfv1 6532 | The converse value of the ... |
| f1ocnvfv2 6533 | The value of the converse ... |
| f1ocnvfv 6534 | Relationship between the v... |
| f1ocnvfvb 6535 | Relationship between the v... |
| nvof1o 6536 | An involution is a bijecti... |
| nvocnv 6537 | The converse of an involut... |
| fsnex 6538 | Relate a function with a s... |
| f1prex 6539 | Relate a one-to-one functi... |
| f1ocnvdm 6540 | The value of the converse ... |
| f1ocnvfvrneq 6541 | If the values of a one-to-... |
| fcof1 6542 | An application is injectiv... |
| fcofo 6543 | An application is surjecti... |
| cbvfo 6544 | Change bound variable betw... |
| cbvexfo 6545 | Change bound variable betw... |
| cocan1 6546 | An injection is left-cance... |
| cocan2 6547 | A surjection is right-canc... |
| fcof1oinvd 6548 | Show that a function is th... |
| fcof1od 6549 | A function is bijective if... |
| 2fcoidinvd 6550 | Show that a function is th... |
| fcof1o 6551 | Show that two functions ar... |
| 2fvcoidd 6552 | Show that the composition ... |
| 2fvidf1od 6553 | A function is bijective if... |
| 2fvidinvd 6554 | Show that two functions ar... |
| foeqcnvco 6555 | Condition for function equ... |
| f1eqcocnv 6556 | Condition for function equ... |
| fveqf1o 6557 | Given a bijection ` F ` , ... |
| fliftrel 6558 | ` F ` , a function lift, i... |
| fliftel 6559 | Elementhood in the relatio... |
| fliftel1 6560 | Elementhood in the relatio... |
| fliftcnv 6561 | Converse of the relation `... |
| fliftfun 6562 | The function ` F ` is the ... |
| fliftfund 6563 | The function ` F ` is the ... |
| fliftfuns 6564 | The function ` F ` is the ... |
| fliftf 6565 | The domain and range of th... |
| fliftval 6566 | The value of the function ... |
| isoeq1 6567 | Equality theorem for isomo... |
| isoeq2 6568 | Equality theorem for isomo... |
| isoeq3 6569 | Equality theorem for isomo... |
| isoeq4 6570 | Equality theorem for isomo... |
| isoeq5 6571 | Equality theorem for isomo... |
| nfiso 6572 | Bound-variable hypothesis ... |
| isof1o 6573 | An isomorphism is a one-to... |
| isof1oidb 6574 | A function is a bijection ... |
| isof1oopb 6575 | A function is a bijection ... |
| isorel 6576 | An isomorphism connects bi... |
| soisores 6577 | Express the condition of i... |
| soisoi 6578 | Infer isomorphism from one... |
| isoid 6579 | Identity law for isomorphi... |
| isocnv 6580 | Converse law for isomorphi... |
| isocnv2 6581 | Converse law for isomorphi... |
| isocnv3 6582 | Complementation law for is... |
| isores2 6583 | An isomorphism from one we... |
| isores1 6584 | An isomorphism from one we... |
| isores3 6585 | Induced isomorphism on a s... |
| isotr 6586 | Composition (transitive) l... |
| isomin 6587 | Isomorphisms preserve mini... |
| isoini 6588 | Isomorphisms preserve init... |
| isoini2 6589 | Isomorphisms are isomorphi... |
| isofrlem 6590 | Lemma for ~ isofr . (Cont... |
| isoselem 6591 | Lemma for ~ isose . (Cont... |
| isofr 6592 | An isomorphism preserves w... |
| isose 6593 | An isomorphism preserves s... |
| isofr2 6594 | A weak form of ~ isofr tha... |
| isopolem 6595 | Lemma for ~ isopo . (Cont... |
| isopo 6596 | An isomorphism preserves p... |
| isosolem 6597 | Lemma for ~ isoso . (Cont... |
| isoso 6598 | An isomorphism preserves s... |
| isowe 6599 | An isomorphism preserves w... |
| isowe2 6600 | A weak form of ~ isowe tha... |
| f1oiso 6601 | Any one-to-one onto functi... |
| f1oiso2 6602 | Any one-to-one onto functi... |
| f1owe 6603 | Well-ordering of isomorphi... |
| weniso 6604 | A set-like well-ordering h... |
| weisoeq 6605 | Thus, there is at most one... |
| weisoeq2 6606 | Thus, there is at most one... |
| knatar 6607 | The Knaster-Tarski theorem... |
| canth 6608 | No set ` A ` is equinumero... |
| ncanth 6609 | Cantor's theorem fails for... |
| riotaeqdv 6612 | Formula-building deduction... |
| riotabidv 6613 | Formula-building deduction... |
| riotaeqbidv 6614 | Equality deduction for res... |
| riotaex 6615 | Restricted iota is a set. ... |
| riotav 6616 | An iota restricted to the ... |
| riotauni 6617 | Restricted iota in terms o... |
| nfriota1 6618 | The abstraction variable i... |
| nfriotad 6619 | Deduction version of ~ nfr... |
| nfriota 6620 | A variable not free in a w... |
| cbvriota 6621 | Change bound variable in a... |
| cbvriotav 6622 | Change bound variable in a... |
| csbriota 6623 | Interchange class substitu... |
| riotacl2 6624 | Membership law for "the un... |
| riotacl 6625 | Closure of restricted iota... |
| riotasbc 6626 | Substitution law for descr... |
| riotabidva 6627 | Equivalent wff's yield equ... |
| riotabiia 6628 | Equivalent wff's yield equ... |
| riota1 6629 | Property of restricted iot... |
| riota1a 6630 | Property of iota. (Contri... |
| riota2df 6631 | A deduction version of ~ r... |
| riota2f 6632 | This theorem shows a condi... |
| riota2 6633 | This theorem shows a condi... |
| riotaeqimp 6634 | If two restricted iota des... |
| riotaprop 6635 | Properties of a restricted... |
| riota5f 6636 | A method for computing res... |
| riota5 6637 | A method for computing res... |
| riotass2 6638 | Restriction of a unique el... |
| riotass 6639 | Restriction of a unique el... |
| moriotass 6640 | Restriction of a unique el... |
| snriota 6641 | A restricted class abstrac... |
| riotaxfrd 6642 | Change the variable ` x ` ... |
| eusvobj2 6643 | Specify the same property ... |
| eusvobj1 6644 | Specify the same object in... |
| f1ofveu 6645 | There is one domain elemen... |
| f1ocnvfv3 6646 | Value of the converse of a... |
| riotaund 6647 | Restricted iota equals the... |
| riotassuni 6648 | The restricted iota class ... |
| riotaclb 6649 | Bidirectional closure of r... |
| oveq 6656 | Equality theorem for opera... |
| oveq1 6657 | Equality theorem for opera... |
| oveq2 6658 | Equality theorem for opera... |
| oveq12 6659 | Equality theorem for opera... |
| oveq1i 6660 | Equality inference for ope... |
| oveq2i 6661 | Equality inference for ope... |
| oveq12i 6662 | Equality inference for ope... |
| oveqi 6663 | Equality inference for ope... |
| oveq123i 6664 | Equality inference for ope... |
| oveq1d 6665 | Equality deduction for ope... |
| oveq2d 6666 | Equality deduction for ope... |
| oveqd 6667 | Equality deduction for ope... |
| oveq12d 6668 | Equality deduction for ope... |
| oveqan12d 6669 | Equality deduction for ope... |
| oveqan12rd 6670 | Equality deduction for ope... |
| oveq123d 6671 | Equality deduction for ope... |
| ovrspc2v 6672 | If an operation value is e... |
| oveqrspc2v 6673 | Restricted specialization ... |
| oveqdr 6674 | Equality of two operations... |
| nfovd 6675 | Deduction version of bound... |
| nfov 6676 | Bound-variable hypothesis ... |
| oprabid 6677 | The law of concretion. Sp... |
| ovex 6678 | The result of an operation... |
| ovexi 6679 | The result of an operation... |
| ovexd 6680 | The result of an operation... |
| ovssunirn 6681 | The result of an operation... |
| 0ov 6682 | Operation value of the emp... |
| ovprc 6683 | The value of an operation ... |
| ovprc1 6684 | The value of an operation ... |
| ovprc2 6685 | The value of an operation ... |
| ovrcl 6686 | Reverse closure for an ope... |
| csbov123 6687 | Move class substitution in... |
| csbov 6688 | Move class substitution in... |
| csbov12g 6689 | Move class substitution in... |
| csbov1g 6690 | Move class substitution in... |
| csbov2g 6691 | Move class substitution in... |
| rspceov 6692 | A frequently used special ... |
| elovimad 6693 | Elementhood of the image s... |
| fnotovb 6694 | Equivalence of operation v... |
| opabbrex 6695 | A collection of ordered pa... |
| opabresex2d 6696 | Restrictions of a collecti... |
| fvmptopab 6697 | The function value of a ma... |
| 0neqopab 6698 | The empty set is never an ... |
| brabv 6699 | If two classes are in a re... |
| brfvopab 6700 | The classes involved in a ... |
| dfoprab2 6701 | Class abstraction for oper... |
| reloprab 6702 | An operation class abstrac... |
| oprabv 6703 | If a pair and a class are ... |
| nfoprab1 6704 | The abstraction variables ... |
| nfoprab2 6705 | The abstraction variables ... |
| nfoprab3 6706 | The abstraction variables ... |
| nfoprab 6707 | Bound-variable hypothesis ... |
| oprabbid 6708 | Equivalent wff's yield equ... |
| oprabbidv 6709 | Equivalent wff's yield equ... |
| oprabbii 6710 | Equivalent wff's yield equ... |
| ssoprab2 6711 | Equivalence of ordered pai... |
| ssoprab2b 6712 | Equivalence of ordered pai... |
| eqoprab2b 6713 | Equivalence of ordered pai... |
| mpt2eq123 6714 | An equality theorem for th... |
| mpt2eq12 6715 | An equality theorem for th... |
| mpt2eq123dva 6716 | An equality deduction for ... |
| mpt2eq123dv 6717 | An equality deduction for ... |
| mpt2eq123i 6718 | An equality inference for ... |
| mpt2eq3dva 6719 | Slightly more general equa... |
| mpt2eq3ia 6720 | An equality inference for ... |
| mpt2eq3dv 6721 | An equality deduction for ... |
| nfmpt21 6722 | Bound-variable hypothesis ... |
| nfmpt22 6723 | Bound-variable hypothesis ... |
| nfmpt2 6724 | Bound-variable hypothesis ... |
| mpt20 6725 | A mapping operation with e... |
| oprab4 6726 | Two ways to state the doma... |
| cbvoprab1 6727 | Rule used to change first ... |
| cbvoprab2 6728 | Change the second bound va... |
| cbvoprab12 6729 | Rule used to change first ... |
| cbvoprab12v 6730 | Rule used to change first ... |
| cbvoprab3 6731 | Rule used to change the th... |
| cbvoprab3v 6732 | Rule used to change the th... |
| cbvmpt2x 6733 | Rule to change the bound v... |
| cbvmpt2 6734 | Rule to change the bound v... |
| cbvmpt2v 6735 | Rule to change the bound v... |
| elimdelov 6736 | Eliminate a hypothesis whi... |
| ovif 6737 | Move a conditional outside... |
| ovif2 6738 | Move a conditional outside... |
| ovif12 6739 | Move a conditional outside... |
| ifov 6740 | Move a conditional outside... |
| dmoprab 6741 | The domain of an operation... |
| dmoprabss 6742 | The domain of an operation... |
| rnoprab 6743 | The range of an operation ... |
| rnoprab2 6744 | The range of a restricted ... |
| reldmoprab 6745 | The domain of an operation... |
| oprabss 6746 | Structure of an operation ... |
| eloprabga 6747 | The law of concretion for ... |
| eloprabg 6748 | The law of concretion for ... |
| ssoprab2i 6749 | Inference of operation cla... |
| mpt2v 6750 | Operation with universal d... |
| mpt2mptx 6751 | Express a two-argument fun... |
| mpt2mpt 6752 | Express a two-argument fun... |
| mpt2difsnif 6753 | A mapping with two argumen... |
| mpt2snif 6754 | A mapping with two argumen... |
| fconstmpt2 6755 | Representation of a consta... |
| resoprab 6756 | Restriction of an operatio... |
| resoprab2 6757 | Restriction of an operator... |
| resmpt2 6758 | Restriction of the mapping... |
| funoprabg 6759 | "At most one" is a suffici... |
| funoprab 6760 | "At most one" is a suffici... |
| fnoprabg 6761 | Functionality and domain o... |
| mpt2fun 6762 | The maps-to notation for a... |
| fnoprab 6763 | Functionality and domain o... |
| ffnov 6764 | An operation maps to a cla... |
| fovcl 6765 | Closure law for an operati... |
| eqfnov 6766 | Equality of two operations... |
| eqfnov2 6767 | Two operators with the sam... |
| fnov 6768 | Representation of a functi... |
| mpt22eqb 6769 | Bidirectional equality the... |
| rnmpt2 6770 | The range of an operation ... |
| reldmmpt2 6771 | The domain of an operation... |
| elrnmpt2g 6772 | Membership in the range of... |
| elrnmpt2 6773 | Membership in the range of... |
| elrnmpt2res 6774 | Membership in the range of... |
| ralrnmpt2 6775 | A restricted quantifier ov... |
| rexrnmpt2 6776 | A restricted quantifier ov... |
| ovid 6777 | The value of an operation ... |
| ovidig 6778 | The value of an operation ... |
| ovidi 6779 | The value of an operation ... |
| ov 6780 | The value of an operation ... |
| ovigg 6781 | The value of an operation ... |
| ovig 6782 | The value of an operation ... |
| ovmpt4g 6783 | Value of a function given ... |
| ovmpt2s 6784 | Value of a function given ... |
| ov2gf 6785 | The value of an operation ... |
| ovmpt2dxf 6786 | Value of an operation give... |
| ovmpt2dx 6787 | Value of an operation give... |
| ovmpt2d 6788 | Value of an operation give... |
| ovmpt2x 6789 | The value of an operation ... |
| ovmpt2ga 6790 | Value of an operation give... |
| ovmpt2a 6791 | Value of an operation give... |
| ovmpt2df 6792 | Alternate deduction versio... |
| ovmpt2dv 6793 | Alternate deduction versio... |
| ovmpt2dv2 6794 | Alternate deduction versio... |
| ovmpt2g 6795 | Value of an operation give... |
| ovmpt2 6796 | Value of an operation give... |
| ov3 6797 | The value of an operation ... |
| ov6g 6798 | The value of an operation ... |
| ovg 6799 | The value of an operation ... |
| ovres 6800 | The value of a restricted ... |
| ovresd 6801 | Lemma for converting metri... |
| oprres 6802 | The restriction of an oper... |
| oprssov 6803 | The value of a member of t... |
| fovrn 6804 | An operation's value belon... |
| fovrnda 6805 | An operation's value belon... |
| fovrnd 6806 | An operation's value belon... |
| fnrnov 6807 | The range of an operation ... |
| foov 6808 | An onto mapping of an oper... |
| fnovrn 6809 | An operation's value belon... |
| ovelrn 6810 | A member of an operation's... |
| funimassov 6811 | Membership relation for th... |
| ovelimab 6812 | Operation value in an imag... |
| ovima0 6813 | An operation value is a me... |
| ovconst2 6814 | The value of a constant op... |
| oprssdm 6815 | Domain of closure of an op... |
| nssdmovg 6816 | The value of an operation ... |
| ndmovg 6817 | The value of an operation ... |
| ndmov 6818 | The value of an operation ... |
| ndmovcl 6819 | The closure of an operatio... |
| ndmovrcl 6820 | Reverse closure law, when ... |
| ndmovcom 6821 | Any operation is commutati... |
| ndmovass 6822 | Any operation is associati... |
| ndmovdistr 6823 | Any operation is distribut... |
| ndmovord 6824 | Elimination of redundant a... |
| ndmovordi 6825 | Elimination of redundant a... |
| caovclg 6826 | Convert an operation closu... |
| caovcld 6827 | Convert an operation closu... |
| caovcl 6828 | Convert an operation closu... |
| caovcomg 6829 | Convert an operation commu... |
| caovcomd 6830 | Convert an operation commu... |
| caovcom 6831 | Convert an operation commu... |
| caovassg 6832 | Convert an operation assoc... |
| caovassd 6833 | Convert an operation assoc... |
| caovass 6834 | Convert an operation assoc... |
| caovcang 6835 | Convert an operation cance... |
| caovcand 6836 | Convert an operation cance... |
| caovcanrd 6837 | Commute the arguments of a... |
| caovcan 6838 | Convert an operation cance... |
| caovordig 6839 | Convert an operation order... |
| caovordid 6840 | Convert an operation order... |
| caovordg 6841 | Convert an operation order... |
| caovordd 6842 | Convert an operation order... |
| caovord2d 6843 | Operation ordering law wit... |
| caovord3d 6844 | Ordering law. (Contribute... |
| caovord 6845 | Convert an operation order... |
| caovord2 6846 | Operation ordering law wit... |
| caovord3 6847 | Ordering law. (Contribute... |
| caovdig 6848 | Convert an operation distr... |
| caovdid 6849 | Convert an operation distr... |
| caovdir2d 6850 | Convert an operation distr... |
| caovdirg 6851 | Convert an operation rever... |
| caovdird 6852 | Convert an operation distr... |
| caovdi 6853 | Convert an operation distr... |
| caov32d 6854 | Rearrange arguments in a c... |
| caov12d 6855 | Rearrange arguments in a c... |
| caov31d 6856 | Rearrange arguments in a c... |
| caov13d 6857 | Rearrange arguments in a c... |
| caov4d 6858 | Rearrange arguments in a c... |
| caov411d 6859 | Rearrange arguments in a c... |
| caov42d 6860 | Rearrange arguments in a c... |
| caov32 6861 | Rearrange arguments in a c... |
| caov12 6862 | Rearrange arguments in a c... |
| caov31 6863 | Rearrange arguments in a c... |
| caov13 6864 | Rearrange arguments in a c... |
| caov4 6865 | Rearrange arguments in a c... |
| caov411 6866 | Rearrange arguments in a c... |
| caov42 6867 | Rearrange arguments in a c... |
| caovdir 6868 | Reverse distributive law. ... |
| caovdilem 6869 | Lemma used by real number ... |
| caovlem2 6870 | Lemma used in real number ... |
| caovmo 6871 | Uniqueness of inverse elem... |
| grprinvlem 6872 | Lemma for ~ grprinvd . (C... |
| grprinvd 6873 | Deduce right inverse from ... |
| grpridd 6874 | Deduce right identity from... |
| mpt2ndm0 6875 | The value of an operation ... |
| elmpt2cl 6876 | If a two-parameter class i... |
| elmpt2cl1 6877 | If a two-parameter class i... |
| elmpt2cl2 6878 | If a two-parameter class i... |
| elovmpt2 6879 | Utility lemma for two-para... |
| elovmpt2rab 6880 | Implications for the value... |
| elovmpt2rab1 6881 | Implications for the value... |
| 2mpt20 6882 | If the operation value of ... |
| relmptopab 6883 | Any function to sets of or... |
| f1ocnvd 6884 | Describe an implicit one-t... |
| f1od 6885 | Describe an implicit one-t... |
| f1ocnv2d 6886 | Describe an implicit one-t... |
| f1o2d 6887 | Describe an implicit one-t... |
| f1opw2 6888 | A one-to-one mapping induc... |
| f1opw 6889 | A one-to-one mapping induc... |
| elovmpt3imp 6890 | If the value of a function... |
| ovmpt3rab1 6891 | The value of an operation ... |
| ovmpt3rabdm 6892 | If the value of a function... |
| elovmpt3rab1 6893 | Implications for the value... |
| elovmpt3rab 6894 | Implications for the value... |
| ofeq 6899 | Equality theorem for funct... |
| ofreq 6900 | Equality theorem for funct... |
| ofexg 6901 | A function operation restr... |
| nfof 6902 | Hypothesis builder for fun... |
| nfofr 6903 | Hypothesis builder for fun... |
| offval 6904 | Value of an operation appl... |
| ofrfval 6905 | Value of a relation applie... |
| ofval 6906 | Evaluate a function operat... |
| ofrval 6907 | Exhibit a function relatio... |
| offn 6908 | The function operation pro... |
| offval2f 6909 | The function operation exp... |
| ofmresval 6910 | Value of a restriction of ... |
| fnfvof 6911 | Function value of a pointw... |
| off 6912 | The function operation pro... |
| ofres 6913 | Restrict the operands of a... |
| offval2 6914 | The function operation exp... |
| ofrfval2 6915 | The function relation acti... |
| ofmpteq 6916 | Value of a pointwise opera... |
| ofco 6917 | The composition of a funct... |
| offveq 6918 | Convert an identity of the... |
| offveqb 6919 | Equivalent expressions for... |
| ofc1 6920 | Left operation by a consta... |
| ofc2 6921 | Right operation by a const... |
| ofc12 6922 | Function operation on two ... |
| caofref 6923 | Transfer a reflexive law t... |
| caofinvl 6924 | Transfer a left inverse la... |
| caofid0l 6925 | Transfer a left identity l... |
| caofid0r 6926 | Transfer a right identity ... |
| caofid1 6927 | Transfer a right absorptio... |
| caofid2 6928 | Transfer a right absorptio... |
| caofcom 6929 | Transfer a commutative law... |
| caofrss 6930 | Transfer a relation subset... |
| caofass 6931 | Transfer an associative la... |
| caoftrn 6932 | Transfer a transitivity la... |
| caofdi 6933 | Transfer a distributive la... |
| caofdir 6934 | Transfer a reverse distrib... |
| caonncan 6935 | Transfer ~ nncan -shaped l... |
| relrpss 6938 | The proper subset relation... |
| brrpssg 6939 | The proper subset relation... |
| brrpss 6940 | The proper subset relation... |
| porpss 6941 | Every class is partially o... |
| sorpss 6942 | Express strict ordering un... |
| sorpssi 6943 | Property of a chain of set... |
| sorpssun 6944 | A chain of sets is closed ... |
| sorpssin 6945 | A chain of sets is closed ... |
| sorpssuni 6946 | In a chain of sets, a maxi... |
| sorpssint 6947 | In a chain of sets, a mini... |
| sorpsscmpl 6948 | The componentwise compleme... |
| zfun 6950 | Axiom of Union expressed w... |
| axun2 6951 | A variant of the Axiom of ... |
| uniex2 6952 | The Axiom of Union using t... |
| uniex 6953 | The Axiom of Union in clas... |
| vuniex 6954 | The union of a setvar is a... |
| uniexg 6955 | The ZF Axiom of Union in c... |
| unex 6956 | The union of two sets is a... |
| tpex 6957 | An unordered triple of cla... |
| unexb 6958 | Existence of union is equi... |
| unexg 6959 | A union of two sets is a s... |
| xpexg 6960 | The Cartesian product of t... |
| 3xpexg 6961 | The Cartesian product of t... |
| xpex 6962 | The Cartesian product of t... |
| sqxpexg 6963 | The Cartesian square of a ... |
| abnexg 6964 | Sufficient condition for a... |
| abnex 6965 | Sufficient condition for a... |
| snnex 6966 | The class of all singleton... |
| snnexOLD 6967 | Obsolete proof of ~ snnex ... |
| pwnex 6968 | The class of all power set... |
| difex2 6969 | If the subtrahend of a cla... |
| difsnexi 6970 | If the difference of a cla... |
| uniuni 6971 | Expression for double unio... |
| uniexr 6972 | Converse of the Axiom of U... |
| uniexb 6973 | The Axiom of Union and its... |
| pwexr 6974 | Converse of the Axiom of P... |
| pwexb 6975 | The Axiom of Power Sets an... |
| eldifpw 6976 | Membership in a power clas... |
| elpwun 6977 | Membership in the power cl... |
| iunpw 6978 | An indexed union of a powe... |
| fr3nr 6979 | A well-founded relation ha... |
| epne3 6980 | A set well-founded by epsi... |
| dfwe2 6981 | Alternate definition of we... |
| ordon 6982 | The class of all ordinal n... |
| epweon 6983 | The epsilon relation well-... |
| onprc 6984 | No set contains all ordina... |
| ssorduni 6985 | The union of a class of or... |
| ssonuni 6986 | The union of a set of ordi... |
| ssonunii 6987 | The union of a set of ordi... |
| ordeleqon 6988 | A way to express the ordin... |
| ordsson 6989 | Any ordinal class is a sub... |
| onss 6990 | An ordinal number is a sub... |
| predon 6991 | For an ordinal, the predec... |
| ssonprc 6992 | Two ways of saying a class... |
| onuni 6993 | The union of an ordinal nu... |
| orduni 6994 | The union of an ordinal cl... |
| onint 6995 | The intersection (infimum)... |
| onint0 6996 | The intersection of a clas... |
| onssmin 6997 | A nonempty class of ordina... |
| onminesb 6998 | If a property is true for ... |
| onminsb 6999 | If a property is true for ... |
| oninton 7000 | The intersection of a none... |
| onintrab 7001 | The intersection of a clas... |
| onintrab2 7002 | An existence condition equ... |
| onnmin 7003 | No member of a set of ordi... |
| onnminsb 7004 | An ordinal number smaller ... |
| oneqmin 7005 | A way to show that an ordi... |
| bm2.5ii 7006 | Problem 2.5(ii) of [BellMa... |
| onminex 7007 | If a wff is true for an or... |
| sucon 7008 | The class of all ordinal n... |
| sucexb 7009 | A successor exists iff its... |
| sucexg 7010 | The successor of a set is ... |
| sucex 7011 | The successor of a set is ... |
| onmindif2 7012 | The minimum of a class of ... |
| suceloni 7013 | The successor of an ordina... |
| ordsuc 7014 | The successor of an ordina... |
| ordpwsuc 7015 | The collection of ordinals... |
| onpwsuc 7016 | The collection of ordinal ... |
| sucelon 7017 | The successor of an ordina... |
| ordsucss 7018 | The successor of an elemen... |
| onpsssuc 7019 | An ordinal number is a pro... |
| ordelsuc 7020 | A set belongs to an ordina... |
| onsucmin 7021 | The successor of an ordina... |
| ordsucelsuc 7022 | Membership is inherited by... |
| ordsucsssuc 7023 | The subclass relationship ... |
| ordsucuniel 7024 | Given an element ` A ` of ... |
| ordsucun 7025 | The successor of the maxim... |
| ordunpr 7026 | The maximum of two ordinal... |
| ordunel 7027 | The maximum of two ordinal... |
| onsucuni 7028 | A class of ordinal numbers... |
| ordsucuni 7029 | An ordinal class is a subc... |
| orduniorsuc 7030 | An ordinal class is either... |
| unon 7031 | The class of all ordinal n... |
| ordunisuc 7032 | An ordinal class is equal ... |
| orduniss2 7033 | The union of the ordinal s... |
| onsucuni2 7034 | A successor ordinal is the... |
| 0elsuc 7035 | The successor of an ordina... |
| limon 7036 | The class of ordinal numbe... |
| onssi 7037 | An ordinal number is a sub... |
| onsuci 7038 | The successor of an ordina... |
| onuniorsuci 7039 | An ordinal number is eithe... |
| onuninsuci 7040 | A limit ordinal is not a s... |
| onsucssi 7041 | A set belongs to an ordina... |
| nlimsucg 7042 | A successor is not a limit... |
| orduninsuc 7043 | An ordinal equal to its un... |
| ordunisuc2 7044 | An ordinal equal to its un... |
| ordzsl 7045 | An ordinal is zero, a succ... |
| onzsl 7046 | An ordinal number is zero,... |
| dflim3 7047 | An alternate definition of... |
| dflim4 7048 | An alternate definition of... |
| limsuc 7049 | The successor of a member ... |
| limsssuc 7050 | A class includes a limit o... |
| nlimon 7051 | Two ways to express the cl... |
| limuni3 7052 | The union of a nonempty cl... |
| tfi 7053 | The Principle of Transfini... |
| tfis 7054 | Transfinite Induction Sche... |
| tfis2f 7055 | Transfinite Induction Sche... |
| tfis2 7056 | Transfinite Induction Sche... |
| tfis3 7057 | Transfinite Induction Sche... |
| tfisi 7058 | A transfinite induction sc... |
| tfinds 7059 | Principle of Transfinite I... |
| tfindsg 7060 | Transfinite Induction (inf... |
| tfindsg2 7061 | Transfinite Induction (inf... |
| tfindes 7062 | Transfinite Induction with... |
| tfinds2 7063 | Transfinite Induction (inf... |
| tfinds3 7064 | Principle of Transfinite I... |
| dfom2 7067 | An alternate definition of... |
| elom 7068 | Membership in omega. The ... |
| omsson 7069 | Omega is a subset of ` On ... |
| limomss 7070 | The class of natural numbe... |
| nnon 7071 | A natural number is an ord... |
| nnoni 7072 | A natural number is an ord... |
| nnord 7073 | A natural number is ordina... |
| ordom 7074 | Omega is ordinal. Theorem... |
| elnn 7075 | A member of a natural numb... |
| omon 7076 | The class of natural numbe... |
| omelon2 7077 | Omega is an ordinal number... |
| nnlim 7078 | A natural number is not a ... |
| omssnlim 7079 | The class of natural numbe... |
| limom 7080 | Omega is a limit ordinal. ... |
| peano2b 7081 | A class belongs to omega i... |
| nnsuc 7082 | A nonzero natural number i... |
| ssnlim 7083 | An ordinal subclass of non... |
| omsinds 7084 | Strong (or "total") induct... |
| peano1 7085 | Zero is a natural number. ... |
| peano2 7086 | The successor of any natur... |
| peano3 7087 | The successor of any natur... |
| peano4 7088 | Two natural numbers are eq... |
| peano5 7089 | The induction postulate: a... |
| nn0suc 7090 | A natural number is either... |
| find 7091 | The Principle of Finite In... |
| finds 7092 | Principle of Finite Induct... |
| findsg 7093 | Principle of Finite Induct... |
| finds2 7094 | Principle of Finite Induct... |
| finds1 7095 | Principle of Finite Induct... |
| findes 7096 | Finite induction with expl... |
| dmexg 7097 | The domain of a set is a s... |
| rnexg 7098 | The range of a set is a se... |
| dmex 7099 | The domain of a set is a s... |
| rnex 7100 | The range of a set is a se... |
| iprc 7101 | The identity function is a... |
| resiexg 7102 | The existence of a restric... |
| imaexg 7103 | The image of a set is a se... |
| imaex 7104 | The image of a set is a se... |
| exse2 7105 | Any set relation is set-li... |
| xpexr 7106 | If a Cartesian product is ... |
| xpexr2 7107 | If a nonempty Cartesian pr... |
| xpexcnv 7108 | A condition where the conv... |
| soex 7109 | If the relation in a stric... |
| elxp4 7110 | Membership in a Cartesian ... |
| elxp5 7111 | Membership in a Cartesian ... |
| cnvexg 7112 | The converse of a set is a... |
| cnvex 7113 | The converse of a set is a... |
| relcnvexb 7114 | A relation is a set iff it... |
| f1oexrnex 7115 | If the range of a 1-1 onto... |
| f1oexbi 7116 | There is a one-to-one onto... |
| coexg 7117 | The composition of two set... |
| coex 7118 | The composition of two set... |
| funcnvuni 7119 | The union of a chain (with... |
| fun11uni 7120 | The union of a chain (with... |
| fex2 7121 | A function with bounded do... |
| fabexg 7122 | Existence of a set of func... |
| fabex 7123 | Existence of a set of func... |
| dmfex 7124 | If a mapping is a set, its... |
| f1oabexg 7125 | The class of all 1-1-onto ... |
| fun11iun 7126 | The union of a chain (with... |
| ffoss 7127 | Relationship between a map... |
| f11o 7128 | Relationship between one-t... |
| resfunexgALT 7129 | Alternate proof of ~ resfu... |
| cofunexg 7130 | Existence of a composition... |
| cofunex2g 7131 | Existence of a composition... |
| fnexALT 7132 | Alternate proof of ~ fnex ... |
| funrnex 7133 | If the domain of a functio... |
| zfrep6 7134 | A version of the Axiom of ... |
| fornex 7135 | If the domain of an onto f... |
| f1dmex 7136 | If the codomain of a one-t... |
| f1ovv 7137 | The range of a 1-1 onto fu... |
| fvclex 7138 | Existence of the class of ... |
| fvresex 7139 | Existence of the class of ... |
| abrexexg 7140 | Existence of a class abstr... |
| abrexex 7141 | Existence of a class abstr... |
| abrexexOLD 7142 | Obsolete proof of ~ abrexe... |
| iunexg 7143 | The existence of an indexe... |
| abrexex2g 7144 | Existence of an existentia... |
| opabex3d 7145 | Existence of an ordered pa... |
| opabex3 7146 | Existence of an ordered pa... |
| iunex 7147 | The existence of an indexe... |
| abrexex2 7148 | Existence of an existentia... |
| abexssex 7149 | Existence of a class abstr... |
| abrexex2OLD 7150 | Obsolete proof of ~ abrexe... |
| abexex 7151 | A condition where a class ... |
| f1oweALT 7152 | Alternate proof of ~ f1owe... |
| wemoiso 7153 | Thus, there is at most one... |
| wemoiso2 7154 | Thus, there is at most one... |
| oprabexd 7155 | Existence of an operator a... |
| oprabex 7156 | Existence of an operation ... |
| oprabex3 7157 | Existence of an operation ... |
| oprabrexex2 7158 | Existence of an existentia... |
| ab2rexex 7159 | Existence of a class abstr... |
| ab2rexex2 7160 | Existence of an existentia... |
| xpexgALT 7161 | Alternate proof of ~ xpexg... |
| offval3 7162 | General value of ` ( F oF ... |
| offres 7163 | Pointwise combination comm... |
| ofmres 7164 | Equivalent expressions for... |
| ofmresex 7165 | Existence of a restriction... |
| 1stval 7170 | The value of the function ... |
| 2ndval 7171 | The value of the function ... |
| 1stnpr 7172 | Value of the first-member ... |
| 2ndnpr 7173 | Value of the second-member... |
| 1st0 7174 | The value of the first-mem... |
| 2nd0 7175 | The value of the second-me... |
| op1st 7176 | Extract the first member o... |
| op2nd 7177 | Extract the second member ... |
| op1std 7178 | Extract the first member o... |
| op2ndd 7179 | Extract the second member ... |
| op1stg 7180 | Extract the first member o... |
| op2ndg 7181 | Extract the second member ... |
| ot1stg 7182 | Extract the first member o... |
| ot2ndg 7183 | Extract the second member ... |
| ot3rdg 7184 | Extract the third member o... |
| 1stval2 7185 | Alternate value of the fun... |
| 2ndval2 7186 | Alternate value of the fun... |
| oteqimp 7187 | The components of an order... |
| fo1st 7188 | The ` 1st ` function maps ... |
| fo2nd 7189 | The ` 2nd ` function maps ... |
| f1stres 7190 | Mapping of a restriction o... |
| f2ndres 7191 | Mapping of a restriction o... |
| fo1stres 7192 | Onto mapping of a restrict... |
| fo2ndres 7193 | Onto mapping of a restrict... |
| 1st2val 7194 | Value of an alternate defi... |
| 2nd2val 7195 | Value of an alternate defi... |
| 1stcof 7196 | Composition of the first m... |
| 2ndcof 7197 | Composition of the second ... |
| xp1st 7198 | Location of the first elem... |
| xp2nd 7199 | Location of the second ele... |
| elxp6 7200 | Membership in a Cartesian ... |
| elxp7 7201 | Membership in a Cartesian ... |
| eqopi 7202 | Equality with an ordered p... |
| xp2 7203 | Representation of Cartesia... |
| unielxp 7204 | The membership relation fo... |
| 1st2nd2 7205 | Reconstruction of a member... |
| 1st2ndb 7206 | Reconstruction of an order... |
| xpopth 7207 | An ordered pair theorem fo... |
| eqop 7208 | Two ways to express equali... |
| eqop2 7209 | Two ways to express equali... |
| op1steq 7210 | Two ways of expressing tha... |
| el2xptp 7211 | A member of a nested Carte... |
| el2xptp0 7212 | A member of a nested Carte... |
| 2nd1st 7213 | Swap the members of an ord... |
| 1st2nd 7214 | Reconstruction of a member... |
| 1stdm 7215 | The first ordered pair com... |
| 2ndrn 7216 | The second ordered pair co... |
| 1st2ndbr 7217 | Express an element of a re... |
| releldm2 7218 | Two ways of expressing mem... |
| reldm 7219 | An expression for the doma... |
| sbcopeq1a 7220 | Equality theorem for subst... |
| csbopeq1a 7221 | Equality theorem for subst... |
| dfopab2 7222 | A way to define an ordered... |
| dfoprab3s 7223 | A way to define an operati... |
| dfoprab3 7224 | Operation class abstractio... |
| dfoprab4 7225 | Operation class abstractio... |
| dfoprab4f 7226 | Operation class abstractio... |
| opabex2 7227 | Condition for an operation... |
| opabn1stprc 7228 | An ordered-pair class abst... |
| opiota 7229 | The property of a uniquely... |
| dfxp3 7230 | Define the Cartesian produ... |
| elopabi 7231 | A consequence of membershi... |
| eloprabi 7232 | A consequence of membershi... |
| mpt2mptsx 7233 | Express a two-argument fun... |
| mpt2mpts 7234 | Express a two-argument fun... |
| dmmpt2ssx 7235 | The domain of a mapping is... |
| fmpt2x 7236 | Functionality, domain and ... |
| fmpt2 7237 | Functionality, domain and ... |
| fnmpt2 7238 | Functionality and domain o... |
| fnmpt2i 7239 | Functionality and domain o... |
| dmmpt2 7240 | Domain of a class given by... |
| ovmpt2elrn 7241 | An operation's value belon... |
| dmmpt2ga 7242 | Domain of an operation giv... |
| dmmpt2g 7243 | Domain of an operation giv... |
| mpt2exxg 7244 | Existence of an operation ... |
| mpt2exg 7245 | Existence of an operation ... |
| mpt2exga 7246 | If the domain of a functio... |
| mpt2ex 7247 | If the domain of a functio... |
| mptmpt2opabbrd 7248 | The operation value of a f... |
| mptmpt2opabovd 7249 | The operation value of a f... |
| el2mpt2csbcl 7250 | If the operation value of ... |
| el2mpt2cl 7251 | If the operation value of ... |
| fnmpt2ovd 7252 | A function with a Cartesia... |
| offval22 7253 | The function operation exp... |
| brovpreldm 7254 | If a binary relation holds... |
| bropopvvv 7255 | If a binary relation holds... |
| bropfvvvvlem 7256 | Lemma for ~ bropfvvvv . (... |
| bropfvvvv 7257 | If a binary relation holds... |
| ovmptss 7258 | If all the values of the m... |
| relmpt2opab 7259 | Any function to sets of or... |
| fmpt2co 7260 | Composition of two functio... |
| oprabco 7261 | Composition of a function ... |
| oprab2co 7262 | Composition of operator ab... |
| df1st2 7263 | An alternate possible defi... |
| df2nd2 7264 | An alternate possible defi... |
| 1stconst 7265 | The mapping of a restricti... |
| 2ndconst 7266 | The mapping of a restricti... |
| dfmpt2 7267 | Alternate definition for t... |
| mpt2sn 7268 | An operation (in maps-to n... |
| curry1 7269 | Composition with ` ``' ( 2... |
| curry1val 7270 | The value of a curried fun... |
| curry1f 7271 | Functionality of a curried... |
| curry2 7272 | Composition with ` ``' ( 1... |
| curry2f 7273 | Functionality of a curried... |
| curry2val 7274 | The value of a curried fun... |
| cnvf1olem 7275 | Lemma for ~ cnvf1o . (Con... |
| cnvf1o 7276 | Describe a function that m... |
| fparlem1 7277 | Lemma for ~ fpar . (Contr... |
| fparlem2 7278 | Lemma for ~ fpar . (Contr... |
| fparlem3 7279 | Lemma for ~ fpar . (Contr... |
| fparlem4 7280 | Lemma for ~ fpar . (Contr... |
| fpar 7281 | Merge two functions in par... |
| fsplit 7282 | A function that can be use... |
| f2ndf 7283 | The ` 2nd ` (second member... |
| fo2ndf 7284 | The ` 2nd ` (second member... |
| f1o2ndf1 7285 | The ` 2nd ` (second member... |
| algrflem 7286 | Lemma for ~ algrf and rela... |
| frxp 7287 | A lexicographical ordering... |
| xporderlem 7288 | Lemma for lexicographical ... |
| poxp 7289 | A lexicographical ordering... |
| soxp 7290 | A lexicographical ordering... |
| wexp 7291 | A lexicographical ordering... |
| fnwelem 7292 | Lemma for ~ fnwe . (Contr... |
| fnwe 7293 | A variant on lexicographic... |
| fnse 7294 | Condition for the well-ord... |
| suppval 7297 | The value of the operation... |
| supp0prc 7298 | The support of a class is ... |
| suppvalbr 7299 | The value of the operation... |
| supp0 7300 | The support of the empty s... |
| suppval1 7301 | The value of the operation... |
| suppvalfn 7302 | The value of the operation... |
| elsuppfn 7303 | An element of the support ... |
| cnvimadfsn 7304 | The support of functions "... |
| suppimacnvss 7305 | The support of functions "... |
| suppimacnv 7306 | Support sets of functions ... |
| frnsuppeq 7307 | Two ways of writing the su... |
| suppssdm 7308 | The support of a function ... |
| suppsnop 7309 | The support of a singleton... |
| snopsuppss 7310 | The support of a singleton... |
| fvn0elsupp 7311 | If the function value for ... |
| fvn0elsuppb 7312 | The function value for a g... |
| rexsupp 7313 | Existential quantification... |
| ressuppss 7314 | The support of the restric... |
| suppun 7315 | The support of a class/fun... |
| ressuppssdif 7316 | The support of the restric... |
| mptsuppdifd 7317 | The support of a function ... |
| mptsuppd 7318 | The support of a function ... |
| extmptsuppeq 7319 | The support of an extended... |
| suppfnss 7320 | The support of a function ... |
| funsssuppss 7321 | The support of a function ... |
| fnsuppres 7322 | Two ways to express restri... |
| fnsuppeq0 7323 | The support of a function ... |
| fczsupp0 7324 | The support of a constant ... |
| suppss 7325 | Show that the support of a... |
| suppssr 7326 | A function is zero outside... |
| suppssov1 7327 | Formula building theorem f... |
| suppssof1 7328 | Formula building theorem f... |
| suppss2 7329 | Show that the support of a... |
| suppsssn 7330 | Show that the support of a... |
| suppssfv 7331 | Formula building theorem f... |
| suppofss1d 7332 | Condition for the support ... |
| suppofss2d 7333 | Condition for the support ... |
| supp0cosupp0 7334 | The support of the composi... |
| imacosupp 7335 | The image of the support o... |
| opeliunxp2f 7336 | Membership in a union of C... |
| mpt2xeldm 7337 | If there is an element of ... |
| mpt2xneldm 7338 | If the first argument of a... |
| mpt2xopn0yelv 7339 | If there is an element of ... |
| mpt2xopynvov0g 7340 | If the second argument of ... |
| mpt2xopxnop0 7341 | If the first argument of a... |
| mpt2xopx0ov0 7342 | If the first argument of a... |
| mpt2xopxprcov0 7343 | If the components of the f... |
| mpt2xopynvov0 7344 | If the second argument of ... |
| mpt2xopoveq 7345 | Value of an operation give... |
| mpt2xopovel 7346 | Element of the value of an... |
| mpt2xopoveqd 7347 | Value of an operation give... |
| brovex 7348 | A binary relation of the v... |
| brovmpt2ex 7349 | A binary relation of the v... |
| sprmpt2d 7350 | The extension of a binary ... |
| tposss 7353 | Subset theorem for transpo... |
| tposeq 7354 | Equality theorem for trans... |
| tposeqd 7355 | Equality theorem for trans... |
| tposssxp 7356 | The transposition is a sub... |
| reltpos 7357 | The transposition is a rel... |
| brtpos2 7358 | Value of the transposition... |
| brtpos0 7359 | The behavior of ` tpos ` w... |
| reldmtpos 7360 | Necessary and sufficient c... |
| brtpos 7361 | The transposition swaps ar... |
| ottpos 7362 | The transposition swaps th... |
| relbrtpos 7363 | The transposition swaps ar... |
| dmtpos 7364 | The domain of ` tpos F ` w... |
| rntpos 7365 | The range of ` tpos F ` wh... |
| tposexg 7366 | The transposition of a set... |
| ovtpos 7367 | The transposition swaps th... |
| tposfun 7368 | The transposition of a fun... |
| dftpos2 7369 | Alternate definition of ` ... |
| dftpos3 7370 | Alternate definition of ` ... |
| dftpos4 7371 | Alternate definition of ` ... |
| tpostpos 7372 | Value of the double transp... |
| tpostpos2 7373 | Value of the double transp... |
| tposfn2 7374 | The domain of a transposit... |
| tposfo2 7375 | Condition for a surjective... |
| tposf2 7376 | The domain and range of a ... |
| tposf12 7377 | Condition for an injective... |
| tposf1o2 7378 | Condition of a bijective t... |
| tposfo 7379 | The domain and range of a ... |
| tposf 7380 | The domain and range of a ... |
| tposfn 7381 | Functionality of a transpo... |
| tpos0 7382 | Transposition of the empty... |
| tposco 7383 | Transposition of a composi... |
| tpossym 7384 | Two ways to say a function... |
| tposeqi 7385 | Equality theorem for trans... |
| tposex 7386 | A transposition is a set. ... |
| nftpos 7387 | Hypothesis builder for tra... |
| tposoprab 7388 | Transposition of a class o... |
| tposmpt2 7389 | Transposition of a two-arg... |
| tposconst 7390 | The transposition of a con... |
| mpt2curryd 7395 | The currying of an operati... |
| mpt2curryvald 7396 | The value of a curried ope... |
| fvmpt2curryd 7397 | The value of the value of ... |
| pwuninel2 7400 | Direct proof of ~ pwuninel... |
| pwuninel 7401 | The power set of the union... |
| undefval 7402 | Value of the undefined val... |
| undefnel2 7403 | The undefined value genera... |
| undefnel 7404 | The undefined value genera... |
| undefne0 7405 | The undefined value genera... |
| wrecseq123 7408 | General equality theorem f... |
| nfwrecs 7409 | Bound-variable hypothesis ... |
| wrecseq1 7410 | Equality theorem for the w... |
| wrecseq2 7411 | Equality theorem for the w... |
| wrecseq3 7412 | Equality theorem for the w... |
| wfr3g 7413 | Functions defined by well-... |
| wfrlem1 7414 | Lemma for well-founded rec... |
| wfrlem2 7415 | Lemma for well-founded rec... |
| wfrlem3 7416 | Lemma for well-founded rec... |
| wfrlem3a 7417 | Lemma for well-founded rec... |
| wfrlem4 7418 | Lemma for well-founded rec... |
| wfrlem5 7419 | Lemma for well-founded rec... |
| wfrrel 7420 | The well-founded recursion... |
| wfrdmss 7421 | The domain of the well-fou... |
| wfrlem8 7422 | Lemma for well-founded rec... |
| wfrdmcl 7423 | Given ` F = wrecs ( R , A ... |
| wfrlem10 7424 | Lemma for well-founded rec... |
| wfrfun 7425 | The well-founded function ... |
| wfrlem12 7426 | Lemma for well-founded rec... |
| wfrlem13 7427 | Lemma for well-founded rec... |
| wfrlem14 7428 | Lemma for well-founded rec... |
| wfrlem15 7429 | Lemma for well-founded rec... |
| wfrlem16 7430 | Lemma for well-founded rec... |
| wfrlem17 7431 | Without using ~ ax-rep , s... |
| wfr2a 7432 | A weak version of ~ wfr2 w... |
| wfr1 7433 | The Principle of Well-Foun... |
| wfr2 7434 | The Principle of Well-Foun... |
| wfr3 7435 | The principle of Well-Foun... |
| iunon 7436 | The indexed union of a set... |
| iinon 7437 | The nonempty indexed inter... |
| onfununi 7438 | A property of functions on... |
| onovuni 7439 | A variant of ~ onfununi fo... |
| onoviun 7440 | A variant of ~ onovuni wit... |
| onnseq 7441 | There are no length ` _om ... |
| dfsmo2 7444 | Alternate definition of a ... |
| issmo 7445 | Conditions for which ` A `... |
| issmo2 7446 | Alternate definition of a ... |
| smoeq 7447 | Equality theorem for stric... |
| smodm 7448 | The domain of a strictly m... |
| smores 7449 | A strictly monotone functi... |
| smores3 7450 | A strictly monotone functi... |
| smores2 7451 | A strictly monotone ordina... |
| smodm2 7452 | The domain of a strictly m... |
| smofvon2 7453 | The function values of a s... |
| iordsmo 7454 | The identity relation rest... |
| smo0 7455 | The null set is a strictly... |
| smofvon 7456 | If ` B ` is a strictly mon... |
| smoel 7457 | If ` x ` is less than ` y ... |
| smoiun 7458 | The value of a strictly mo... |
| smoiso 7459 | If ` F ` is an isomorphism... |
| smoel2 7460 | A strictly monotone ordina... |
| smo11 7461 | A strictly monotone ordina... |
| smoord 7462 | A strictly monotone ordina... |
| smoword 7463 | A strictly monotone ordina... |
| smogt 7464 | A strictly monotone ordina... |
| smorndom 7465 | The range of a strictly mo... |
| smoiso2 7466 | The strictly monotone ordi... |
| dfrecs3 7469 | The old definition of tran... |
| recseq 7470 | Equality theorem for ` rec... |
| nfrecs 7471 | Bound-variable hypothesis ... |
| tfrlem1 7472 | A technical lemma for tran... |
| tfrlem3a 7473 | Lemma for transfinite recu... |
| tfrlem3 7474 | Lemma for transfinite recu... |
| tfrlem4 7475 | Lemma for transfinite recu... |
| tfrlem5 7476 | Lemma for transfinite recu... |
| recsfval 7477 | Lemma for transfinite recu... |
| tfrlem6 7478 | Lemma for transfinite recu... |
| tfrlem7 7479 | Lemma for transfinite recu... |
| tfrlem8 7480 | Lemma for transfinite recu... |
| tfrlem9 7481 | Lemma for transfinite recu... |
| tfrlem9a 7482 | Lemma for transfinite recu... |
| tfrlem10 7483 | Lemma for transfinite recu... |
| tfrlem11 7484 | Lemma for transfinite recu... |
| tfrlem12 7485 | Lemma for transfinite recu... |
| tfrlem13 7486 | Lemma for transfinite recu... |
| tfrlem14 7487 | Lemma for transfinite recu... |
| tfrlem15 7488 | Lemma for transfinite recu... |
| tfrlem16 7489 | Lemma for finite recursion... |
| tfr1a 7490 | A weak version of ~ tfr1 w... |
| tfr2a 7491 | A weak version of ~ tfr2 w... |
| tfr2b 7492 | Without assuming ~ ax-rep ... |
| tfr1 7493 | Principle of Transfinite R... |
| tfr2 7494 | Principle of Transfinite R... |
| tfr3 7495 | Principle of Transfinite R... |
| tfr1ALT 7496 | Alternate proof of ~ tfr1 ... |
| tfr2ALT 7497 | Alternate proof of ~ tfr2 ... |
| tfr3ALT 7498 | Alternate proof of ~ tfr3 ... |
| recsfnon 7499 | Strong transfinite recursi... |
| recsval 7500 | Strong transfinite recursi... |
| tz7.44lem1 7501 | ` G ` is a function. Lemm... |
| tz7.44-1 7502 | The value of ` F ` at ` (/... |
| tz7.44-2 7503 | The value of ` F ` at a su... |
| tz7.44-3 7504 | The value of ` F ` at a li... |
| rdgeq1 7507 | Equality theorem for the r... |
| rdgeq2 7508 | Equality theorem for the r... |
| rdgeq12 7509 | Equality theorem for the r... |
| nfrdg 7510 | Bound-variable hypothesis ... |
| rdglem1 7511 | Lemma used with the recurs... |
| rdgfun 7512 | The recursive definition g... |
| rdgdmlim 7513 | The domain of the recursiv... |
| rdgfnon 7514 | The recursive definition g... |
| rdgvalg 7515 | Value of the recursive def... |
| rdgval 7516 | Value of the recursive def... |
| rdg0 7517 | The initial value of the r... |
| rdgseg 7518 | The initial segments of th... |
| rdgsucg 7519 | The value of the recursive... |
| rdgsuc 7520 | The value of the recursive... |
| rdglimg 7521 | The value of the recursive... |
| rdglim 7522 | The value of the recursive... |
| rdg0g 7523 | The initial value of the r... |
| rdgsucmptf 7524 | The value of the recursive... |
| rdgsucmptnf 7525 | The value of the recursive... |
| rdgsucmpt2 7526 | This version of ~ rdgsucmp... |
| rdgsucmpt 7527 | The value of the recursive... |
| rdglim2 7528 | The value of the recursive... |
| rdglim2a 7529 | The value of the recursive... |
| frfnom 7530 | The function generated by ... |
| fr0g 7531 | The initial value resultin... |
| frsuc 7532 | The successor value result... |
| frsucmpt 7533 | The successor value result... |
| frsucmptn 7534 | The value of the finite re... |
| frsucmpt2 7535 | The successor value result... |
| tz7.48lem 7536 | A way of showing an ordina... |
| tz7.48-2 7537 | Proposition 7.48(2) of [Ta... |
| tz7.48-1 7538 | Proposition 7.48(1) of [Ta... |
| tz7.48-3 7539 | Proposition 7.48(3) of [Ta... |
| tz7.49 7540 | Proposition 7.49 of [Takeu... |
| tz7.49c 7541 | Corollary of Proposition 7... |
| seqomlem0 7544 | Lemma for ` seqom ` . Cha... |
| seqomlem1 7545 | Lemma for ` seqom ` . The... |
| seqomlem2 7546 | Lemma for ` seqom ` . (Co... |
| seqomlem3 7547 | Lemma for ` seqom ` . (Co... |
| seqomlem4 7548 | Lemma for ` seqom ` . (Co... |
| seqomeq12 7549 | Equality theorem for ` seq... |
| fnseqom 7550 | An index-aware recursive d... |
| seqom0g 7551 | Value of an index-aware re... |
| seqomsuc 7552 | Value of an index-aware re... |
| 1on 7567 | Ordinal 1 is an ordinal nu... |
| 2on 7568 | Ordinal 2 is an ordinal nu... |
| 2on0 7569 | Ordinal two is not zero. ... |
| 3on 7570 | Ordinal 3 is an ordinal nu... |
| 4on 7571 | Ordinal 3 is an ordinal nu... |
| df1o2 7572 | Expanded value of the ordi... |
| df2o3 7573 | Expanded value of the ordi... |
| df2o2 7574 | Expanded value of the ordi... |
| 1n0 7575 | Ordinal one is not equal t... |
| xp01disj 7576 | Cartesian products with th... |
| ordgt0ge1 7577 | Two ways to express that a... |
| ordge1n0 7578 | An ordinal greater than or... |
| el1o 7579 | Membership in ordinal one.... |
| dif1o 7580 | Two ways to say that ` A `... |
| ondif1 7581 | Two ways to say that ` A `... |
| ondif2 7582 | Two ways to say that ` A `... |
| 2oconcl 7583 | Closure of the pair swappi... |
| 0lt1o 7584 | Ordinal zero is less than ... |
| dif20el 7585 | An ordinal greater than on... |
| 0we1 7586 | The empty set is a well-or... |
| brwitnlem 7587 | Lemma for relations which ... |
| fnoa 7588 | Functionality and domain o... |
| fnom 7589 | Functionality and domain o... |
| fnoe 7590 | Functionality and domain o... |
| oav 7591 | Value of ordinal addition.... |
| omv 7592 | Value of ordinal multiplic... |
| oe0lem 7593 | A helper lemma for ~ oe0 a... |
| oev 7594 | Value of ordinal exponenti... |
| oevn0 7595 | Value of ordinal exponenti... |
| oa0 7596 | Addition with zero. Propo... |
| om0 7597 | Ordinal multiplication wit... |
| oe0m 7598 | Ordinal exponentiation wit... |
| om0x 7599 | Ordinal multiplication wit... |
| oe0m0 7600 | Ordinal exponentiation wit... |
| oe0m1 7601 | Ordinal exponentiation wit... |
| oe0 7602 | Ordinal exponentiation wit... |
| oev2 7603 | Alternate value of ordinal... |
| oasuc 7604 | Addition with successor. ... |
| oesuclem 7605 | Lemma for ~ oesuc . (Cont... |
| omsuc 7606 | Multiplication with succes... |
| oesuc 7607 | Ordinal exponentiation wit... |
| onasuc 7608 | Addition with successor. ... |
| onmsuc 7609 | Multiplication with succes... |
| onesuc 7610 | Exponentiation with a succ... |
| oa1suc 7611 | Addition with 1 is same as... |
| oalim 7612 | Ordinal addition with a li... |
| omlim 7613 | Ordinal multiplication wit... |
| oelim 7614 | Ordinal exponentiation wit... |
| oacl 7615 | Closure law for ordinal ad... |
| omcl 7616 | Closure law for ordinal mu... |
| oecl 7617 | Closure law for ordinal ex... |
| oa0r 7618 | Ordinal addition with zero... |
| om0r 7619 | Ordinal multiplication wit... |
| o1p1e2 7620 | 1 + 1 = 2 for ordinal numb... |
| o2p2e4 7621 | 2 + 2 = 4 for ordinal numb... |
| om1 7622 | Ordinal multiplication wit... |
| om1r 7623 | Ordinal multiplication wit... |
| oe1 7624 | Ordinal exponentiation wit... |
| oe1m 7625 | Ordinal exponentiation wit... |
| oaordi 7626 | Ordering property of ordin... |
| oaord 7627 | Ordering property of ordin... |
| oacan 7628 | Left cancellation law for ... |
| oaword 7629 | Weak ordering property of ... |
| oawordri 7630 | Weak ordering property of ... |
| oaord1 7631 | An ordinal is less than it... |
| oaword1 7632 | An ordinal is less than or... |
| oaword2 7633 | An ordinal is less than or... |
| oawordeulem 7634 | Lemma for ~ oawordex . (C... |
| oawordeu 7635 | Existence theorem for weak... |
| oawordexr 7636 | Existence theorem for weak... |
| oawordex 7637 | Existence theorem for weak... |
| oaordex 7638 | Existence theorem for orde... |
| oa00 7639 | An ordinal sum is zero iff... |
| oalimcl 7640 | The ordinal sum with a lim... |
| oaass 7641 | Ordinal addition is associ... |
| oarec 7642 | Recursive definition of or... |
| oaf1o 7643 | Left addition by a constan... |
| oacomf1olem 7644 | Lemma for ~ oacomf1o . (C... |
| oacomf1o 7645 | Define a bijection from ` ... |
| omordi 7646 | Ordering property of ordin... |
| omord2 7647 | Ordering property of ordin... |
| omord 7648 | Ordering property of ordin... |
| omcan 7649 | Left cancellation law for ... |
| omword 7650 | Weak ordering property of ... |
| omwordi 7651 | Weak ordering property of ... |
| omwordri 7652 | Weak ordering property of ... |
| omword1 7653 | An ordinal is less than or... |
| omword2 7654 | An ordinal is less than or... |
| om00 7655 | The product of two ordinal... |
| om00el 7656 | The product of two nonzero... |
| omordlim 7657 | Ordering involving the pro... |
| omlimcl 7658 | The product of any nonzero... |
| odi 7659 | Distributive law for ordin... |
| omass 7660 | Multiplication of ordinal ... |
| oneo 7661 | If an ordinal number is ev... |
| omeulem1 7662 | Lemma for ~ omeu : existen... |
| omeulem2 7663 | Lemma for ~ omeu : uniquen... |
| omopth2 7664 | An ordered pair-like theor... |
| omeu 7665 | The division algorithm for... |
| oen0 7666 | Ordinal exponentiation wit... |
| oeordi 7667 | Ordering law for ordinal e... |
| oeord 7668 | Ordering property of ordin... |
| oecan 7669 | Left cancellation law for ... |
| oeword 7670 | Weak ordering property of ... |
| oewordi 7671 | Weak ordering property of ... |
| oewordri 7672 | Weak ordering property of ... |
| oeworde 7673 | Ordinal exponentiation com... |
| oeordsuc 7674 | Ordering property of ordin... |
| oelim2 7675 | Ordinal exponentiation wit... |
| oeoalem 7676 | Lemma for ~ oeoa . (Contr... |
| oeoa 7677 | Sum of exponents law for o... |
| oeoelem 7678 | Lemma for ~ oeoe . (Contr... |
| oeoe 7679 | Product of exponents law f... |
| oelimcl 7680 | The ordinal exponential wi... |
| oeeulem 7681 | Lemma for ~ oeeu . (Contr... |
| oeeui 7682 | The division algorithm for... |
| oeeu 7683 | The division algorithm for... |
| nna0 7684 | Addition with zero. Theor... |
| nnm0 7685 | Multiplication with zero. ... |
| nnasuc 7686 | Addition with successor. ... |
| nnmsuc 7687 | Multiplication with succes... |
| nnesuc 7688 | Exponentiation with a succ... |
| nna0r 7689 | Addition to zero. Remark ... |
| nnm0r 7690 | Multiplication with zero. ... |
| nnacl 7691 | Closure of addition of nat... |
| nnmcl 7692 | Closure of multiplication ... |
| nnecl 7693 | Closure of exponentiation ... |
| nnacli 7694 | ` _om ` is closed under ad... |
| nnmcli 7695 | ` _om ` is closed under mu... |
| nnarcl 7696 | Reverse closure law for ad... |
| nnacom 7697 | Addition of natural number... |
| nnaordi 7698 | Ordering property of addit... |
| nnaord 7699 | Ordering property of addit... |
| nnaordr 7700 | Ordering property of addit... |
| nnawordi 7701 | Adding to both sides of an... |
| nnaass 7702 | Addition of natural number... |
| nndi 7703 | Distributive law for natur... |
| nnmass 7704 | Multiplication of natural ... |
| nnmsucr 7705 | Multiplication with succes... |
| nnmcom 7706 | Multiplication of natural ... |
| nnaword 7707 | Weak ordering property of ... |
| nnacan 7708 | Cancellation law for addit... |
| nnaword1 7709 | Weak ordering property of ... |
| nnaword2 7710 | Weak ordering property of ... |
| nnmordi 7711 | Ordering property of multi... |
| nnmord 7712 | Ordering property of multi... |
| nnmword 7713 | Weak ordering property of ... |
| nnmcan 7714 | Cancellation law for multi... |
| nnmwordi 7715 | Weak ordering property of ... |
| nnmwordri 7716 | Weak ordering property of ... |
| nnawordex 7717 | Equivalence for weak order... |
| nnaordex 7718 | Equivalence for ordering. ... |
| 1onn 7719 | One is a natural number. ... |
| 2onn 7720 | The ordinal 2 is a natural... |
| 3onn 7721 | The ordinal 3 is a natural... |
| 4onn 7722 | The ordinal 4 is a natural... |
| oaabslem 7723 | Lemma for ~ oaabs . (Cont... |
| oaabs 7724 | Ordinal addition absorbs a... |
| oaabs2 7725 | The absorption law ~ oaabs... |
| omabslem 7726 | Lemma for ~ omabs . (Cont... |
| omabs 7727 | Ordinal multiplication is ... |
| nnm1 7728 | Multiply an element of ` _... |
| nnm2 7729 | Multiply an element of ` _... |
| nn2m 7730 | Multiply an element of ` _... |
| nnneo 7731 | If a natural number is eve... |
| nneob 7732 | A natural number is even i... |
| omsmolem 7733 | Lemma for ~ omsmo . (Cont... |
| omsmo 7734 | A strictly monotonic ordin... |
| omopthlem1 7735 | Lemma for ~ omopthi . (Co... |
| omopthlem2 7736 | Lemma for ~ omopthi . (Co... |
| omopthi 7737 | An ordered pair theorem fo... |
| omopth 7738 | An ordered pair theorem fo... |
| dfer2 7743 | Alternate definition of eq... |
| dfec2 7745 | Alternate definition of ` ... |
| ecexg 7746 | An equivalence class modul... |
| ecexr 7747 | A nonempty equivalence cla... |
| ereq1 7749 | Equality theorem for equiv... |
| ereq2 7750 | Equality theorem for equiv... |
| errel 7751 | An equivalence relation is... |
| erdm 7752 | The domain of an equivalen... |
| ercl 7753 | Elementhood in the field o... |
| ersym 7754 | An equivalence relation is... |
| ercl2 7755 | Elementhood in the field o... |
| ersymb 7756 | An equivalence relation is... |
| ertr 7757 | An equivalence relation is... |
| ertrd 7758 | A transitivity relation fo... |
| ertr2d 7759 | A transitivity relation fo... |
| ertr3d 7760 | A transitivity relation fo... |
| ertr4d 7761 | A transitivity relation fo... |
| erref 7762 | An equivalence relation is... |
| ercnv 7763 | The converse of an equival... |
| errn 7764 | The range and domain of an... |
| erssxp 7765 | An equivalence relation is... |
| erex 7766 | An equivalence relation is... |
| erexb 7767 | An equivalence relation is... |
| iserd 7768 | A reflexive, symmetric, tr... |
| iseri 7769 | A reflexive, symmetric, tr... |
| iseriALT 7770 | Alternate proof of ~ iseri... |
| brdifun 7771 | Evaluate the incomparabili... |
| swoer 7772 | Incomparability under a st... |
| swoord1 7773 | The incomparability equiva... |
| swoord2 7774 | The incomparability equiva... |
| swoso 7775 | If the incomparability rel... |
| eqerlem 7776 | Lemma for ~ eqer . (Contr... |
| eqer 7777 | Equivalence relation invol... |
| eqerOLD 7778 | Obsolete proof of ~ eqer a... |
| ider 7779 | The identity relation is a... |
| 0er 7780 | The empty set is an equiva... |
| 0erOLD 7781 | Obsolete proof of ~ 0er as... |
| eceq1 7782 | Equality theorem for equiv... |
| eceq1d 7783 | Equality theorem for equiv... |
| eceq2 7784 | Equality theorem for equiv... |
| elecg 7785 | Membership in an equivalen... |
| elec 7786 | Membership in an equivalen... |
| relelec 7787 | Membership in an equivalen... |
| ecss 7788 | An equivalence class is a ... |
| ecdmn0 7789 | A representative of a none... |
| ereldm 7790 | Equality of equivalence cl... |
| erth 7791 | Basic property of equivale... |
| erth2 7792 | Basic property of equivale... |
| erthi 7793 | Basic property of equivale... |
| erdisj 7794 | Equivalence classes do not... |
| ecidsn 7795 | An equivalence class modul... |
| qseq1 7796 | Equality theorem for quoti... |
| qseq2 7797 | Equality theorem for quoti... |
| elqsg 7798 | Closed form of ~ elqs . (... |
| elqs 7799 | Membership in a quotient s... |
| elqsi 7800 | Membership in a quotient s... |
| elqsecl 7801 | Membership in a quotient s... |
| ecelqsg 7802 | Membership of an equivalen... |
| ecelqsi 7803 | Membership of an equivalen... |
| ecopqsi 7804 | "Closure" law for equivale... |
| qsexg 7805 | A quotient set exists. (C... |
| qsex 7806 | A quotient set exists. (C... |
| uniqs 7807 | The union of a quotient se... |
| qsss 7808 | A quotient set is a set of... |
| uniqs2 7809 | The union of a quotient se... |
| snec 7810 | The singleton of an equiva... |
| ecqs 7811 | Equivalence class in terms... |
| ecid 7812 | A set is equal to its conv... |
| qsid 7813 | A set is equal to its quot... |
| ectocld 7814 | Implicit substitution of c... |
| ectocl 7815 | Implicit substitution of c... |
| elqsn0 7816 | A quotient set doesn't con... |
| ecelqsdm 7817 | Membership of an equivalen... |
| xpider 7818 | A square Cartesian product... |
| iiner 7819 | The intersection of a none... |
| riiner 7820 | The relative intersection ... |
| erinxp 7821 | A restricted equivalence r... |
| ecinxp 7822 | Restrict the relation in a... |
| qsinxp 7823 | Restrict the equivalence r... |
| qsdisj 7824 | Members of a quotient set ... |
| qsdisj2 7825 | A quotient set is a disjoi... |
| qsel 7826 | If an element of a quotien... |
| uniinqs 7827 | Class union distributes ov... |
| qliftlem 7828 | ` F ` , a function lift, i... |
| qliftrel 7829 | ` F ` , a function lift, i... |
| qliftel 7830 | Elementhood in the relatio... |
| qliftel1 7831 | Elementhood in the relatio... |
| qliftfun 7832 | The function ` F ` is the ... |
| qliftfund 7833 | The function ` F ` is the ... |
| qliftfuns 7834 | The function ` F ` is the ... |
| qliftf 7835 | The domain and range of th... |
| qliftval 7836 | The value of the function ... |
| ecoptocl 7837 | Implicit substitution of c... |
| 2ecoptocl 7838 | Implicit substitution of c... |
| 3ecoptocl 7839 | Implicit substitution of c... |
| brecop 7840 | Binary relation on a quoti... |
| brecop2 7841 | Binary relation on a quoti... |
| eroveu 7842 | Lemma for ~ erov and ~ ero... |
| erovlem 7843 | Lemma for ~ erov and ~ ero... |
| erov 7844 | The value of an operation ... |
| eroprf 7845 | Functionality of an operat... |
| erov2 7846 | The value of an operation ... |
| eroprf2 7847 | Functionality of an operat... |
| ecopoveq 7848 | This is the first of sever... |
| ecopovsym 7849 | Assuming the operation ` F... |
| ecopovtrn 7850 | Assuming that operation ` ... |
| ecopover 7851 | Assuming that operation ` ... |
| ecopoverOLD 7852 | Obsolete proof of ~ ecopov... |
| eceqoveq 7853 | Equality of equivalence re... |
| ecovcom 7854 | Lemma used to transfer a c... |
| ecovass 7855 | Lemma used to transfer an ... |
| ecovdi 7856 | Lemma used to transfer a d... |
| mapprc 7861 | When ` A ` is a proper cla... |
| pmex 7862 | The class of all partial f... |
| mapex 7863 | The class of all functions... |
| fnmap 7864 | Set exponentiation has a u... |
| fnpm 7865 | Partial function exponenti... |
| reldmmap 7866 | Set exponentiation is a we... |
| mapvalg 7867 | The value of set exponenti... |
| pmvalg 7868 | The value of the partial m... |
| mapval 7869 | The value of set exponenti... |
| elmapg 7870 | Membership relation for se... |
| elmapd 7871 | Deduction form of ~ elmapg... |
| mapdm0 7872 | The empty set is the only ... |
| elpmg 7873 | The predicate "is a partia... |
| elpm2g 7874 | The predicate "is a partia... |
| elpm2r 7875 | Sufficient condition for b... |
| elpmi 7876 | A partial function is a fu... |
| pmfun 7877 | A partial function is a fu... |
| elmapex 7878 | Eliminate antecedent for m... |
| elmapi 7879 | A mapping is a function, f... |
| elmapfn 7880 | A mapping is a function wi... |
| elmapfun 7881 | A mapping is always a func... |
| elmapssres 7882 | A restricted mapping is a ... |
| fpmg 7883 | A total function is a part... |
| pmss12g 7884 | Subset relation for the se... |
| pmresg 7885 | Elementhood of a restricte... |
| elmap 7886 | Membership relation for se... |
| mapval2 7887 | Alternate expression for t... |
| elpm 7888 | The predicate "is a partia... |
| elpm2 7889 | The predicate "is a partia... |
| fpm 7890 | A total function is a part... |
| mapsspm 7891 | Set exponentiation is a su... |
| pmsspw 7892 | Partial maps are a subset ... |
| mapsspw 7893 | Set exponentiation is a su... |
| fvmptmap 7894 | Special case of ~ fvmpt fo... |
| map0e 7895 | Set exponentiation with an... |
| map0b 7896 | Set exponentiation with an... |
| map0g 7897 | Set exponentiation is empt... |
| map0 7898 | Set exponentiation is empt... |
| mapsn 7899 | The value of set exponenti... |
| mapss 7900 | Subset inheritance for set... |
| fdiagfn 7901 | Functionality of the diago... |
| fvdiagfn 7902 | Functionality of the diago... |
| mapsnconst 7903 | Every singleton map is a c... |
| mapsncnv 7904 | Expression for the inverse... |
| mapsnf1o2 7905 | Explicit bijection between... |
| mapsnf1o3 7906 | Explicit bijection in the ... |
| ralxpmap 7907 | Quantification over functi... |
| dfixp 7910 | Eliminate the expression `... |
| ixpsnval 7911 | The value of an infinite C... |
| elixp2 7912 | Membership in an infinite ... |
| fvixp 7913 | Projection of a factor of ... |
| ixpfn 7914 | A nuple is a function. (C... |
| elixp 7915 | Membership in an infinite ... |
| elixpconst 7916 | Membership in an infinite ... |
| ixpconstg 7917 | Infinite Cartesian product... |
| ixpconst 7918 | Infinite Cartesian product... |
| ixpeq1 7919 | Equality theorem for infin... |
| ixpeq1d 7920 | Equality theorem for infin... |
| ss2ixp 7921 | Subclass theorem for infin... |
| ixpeq2 7922 | Equality theorem for infin... |
| ixpeq2dva 7923 | Equality theorem for infin... |
| ixpeq2dv 7924 | Equality theorem for infin... |
| cbvixp 7925 | Change bound variable in a... |
| cbvixpv 7926 | Change bound variable in a... |
| nfixp 7927 | Bound-variable hypothesis ... |
| nfixp1 7928 | The index variable in an i... |
| ixpprc 7929 | A cartesian product of pro... |
| ixpf 7930 | A member of an infinite Ca... |
| uniixp 7931 | The union of an infinite C... |
| ixpexg 7932 | The existence of an infini... |
| ixpin 7933 | The intersection of two in... |
| ixpiin 7934 | The indexed intersection o... |
| ixpint 7935 | The intersection of a coll... |
| ixp0x 7936 | An infinite Cartesian prod... |
| ixpssmap2g 7937 | An infinite Cartesian prod... |
| ixpssmapg 7938 | An infinite Cartesian prod... |
| 0elixp 7939 | Membership of the empty se... |
| ixpn0 7940 | The infinite Cartesian pro... |
| ixp0 7941 | The infinite Cartesian pro... |
| ixpssmap 7942 | An infinite Cartesian prod... |
| resixp 7943 | Restriction of an element ... |
| undifixp 7944 | Union of two projections o... |
| mptelixpg 7945 | Condition for an explicit ... |
| resixpfo 7946 | Restriction of elements of... |
| elixpsn 7947 | Membership in a class of s... |
| ixpsnf1o 7948 | A bijection between a clas... |
| mapsnf1o 7949 | A bijection between a set ... |
| boxriin 7950 | A rectangular subset of a ... |
| boxcutc 7951 | The relative complement of... |
| relen 7960 | Equinumerosity is a relati... |
| reldom 7961 | Dominance is a relation. ... |
| relsdom 7962 | Strict dominance is a rela... |
| encv 7963 | If two classes are equinum... |
| bren 7964 | Equinumerosity relation. ... |
| brdomg 7965 | Dominance relation. (Cont... |
| brdomi 7966 | Dominance relation. (Cont... |
| brdom 7967 | Dominance relation. (Cont... |
| domen 7968 | Dominance in terms of equi... |
| domeng 7969 | Dominance in terms of equi... |
| ctex 7970 | A countable set is a set. ... |
| f1oen3g 7971 | The domain and range of a ... |
| f1oen2g 7972 | The domain and range of a ... |
| f1dom2g 7973 | The domain of a one-to-one... |
| f1oeng 7974 | The domain and range of a ... |
| f1domg 7975 | The domain of a one-to-one... |
| f1oen 7976 | The domain and range of a ... |
| f1dom 7977 | The domain of a one-to-one... |
| brsdom 7978 | Strict dominance relation,... |
| isfi 7979 | Express " ` A ` is finite.... |
| enssdom 7980 | Equinumerosity implies dom... |
| dfdom2 7981 | Alternate definition of do... |
| endom 7982 | Equinumerosity implies dom... |
| sdomdom 7983 | Strict dominance implies d... |
| sdomnen 7984 | Strict dominance implies n... |
| brdom2 7985 | Dominance in terms of stri... |
| bren2 7986 | Equinumerosity expressed i... |
| enrefg 7987 | Equinumerosity is reflexiv... |
| enref 7988 | Equinumerosity is reflexiv... |
| eqeng 7989 | Equality implies equinumer... |
| domrefg 7990 | Dominance is reflexive. (... |
| en2d 7991 | Equinumerosity inference f... |
| en3d 7992 | Equinumerosity inference f... |
| en2i 7993 | Equinumerosity inference f... |
| en3i 7994 | Equinumerosity inference f... |
| dom2lem 7995 | A mapping (first hypothesi... |
| dom2d 7996 | A mapping (first hypothesi... |
| dom3d 7997 | A mapping (first hypothesi... |
| dom2 7998 | A mapping (first hypothesi... |
| dom3 7999 | A mapping (first hypothesi... |
| idssen 8000 | Equality implies equinumer... |
| ssdomg 8001 | A set dominates its subset... |
| ener 8002 | Equinumerosity is an equiv... |
| enerOLD 8003 | Obsolete proof of ~ ener a... |
| ensymb 8004 | Symmetry of equinumerosity... |
| ensym 8005 | Symmetry of equinumerosity... |
| ensymi 8006 | Symmetry of equinumerosity... |
| ensymd 8007 | Symmetry of equinumerosity... |
| entr 8008 | Transitivity of equinumero... |
| domtr 8009 | Transitivity of dominance ... |
| entri 8010 | A chained equinumerosity i... |
| entr2i 8011 | A chained equinumerosity i... |
| entr3i 8012 | A chained equinumerosity i... |
| entr4i 8013 | A chained equinumerosity i... |
| endomtr 8014 | Transitivity of equinumero... |
| domentr 8015 | Transitivity of dominance ... |
| f1imaeng 8016 | A one-to-one function's im... |
| f1imaen2g 8017 | A one-to-one function's im... |
| f1imaen 8018 | A one-to-one function's im... |
| en0 8019 | The empty set is equinumer... |
| ensn1 8020 | A singleton is equinumerou... |
| ensn1g 8021 | A singleton is equinumerou... |
| enpr1g 8022 | ` { A , A } ` has only one... |
| en1 8023 | A set is equinumerous to o... |
| en1b 8024 | A set is equinumerous to o... |
| reuen1 8025 | Two ways to express "exact... |
| euen1 8026 | Two ways to express "exact... |
| euen1b 8027 | Two ways to express " ` A ... |
| en1uniel 8028 | A singleton contains its s... |
| 2dom 8029 | A set that dominates ordin... |
| fundmen 8030 | A function is equinumerous... |
| fundmeng 8031 | A function is equinumerous... |
| cnven 8032 | A relational set is equinu... |
| cnvct 8033 | If a set is countable, so ... |
| fndmeng 8034 | A function is equinumerate... |
| mapsnen 8035 | Set exponentiation to a si... |
| map1 8036 | Set exponentiation: ordina... |
| en2sn 8037 | Two singletons are equinum... |
| snfi 8038 | A singleton is finite. (C... |
| fiprc 8039 | The class of finite sets i... |
| unen 8040 | Equinumerosity of union of... |
| ssct 8041 | Any subset of a countable ... |
| difsnen 8042 | All decrements of a set ar... |
| domdifsn 8043 | Dominance over a set with ... |
| xpsnen 8044 | A set is equinumerous to i... |
| xpsneng 8045 | A set is equinumerous to i... |
| xp1en 8046 | One times a cardinal numbe... |
| endisj 8047 | Any two sets are equinumer... |
| undom 8048 | Dominance law for union. ... |
| xpcomf1o 8049 | The canonical bijection fr... |
| xpcomco 8050 | Composition with the bijec... |
| xpcomen 8051 | Commutative law for equinu... |
| xpcomeng 8052 | Commutative law for equinu... |
| xpsnen2g 8053 | A set is equinumerous to i... |
| xpassen 8054 | Associative law for equinu... |
| xpdom2 8055 | Dominance law for Cartesia... |
| xpdom2g 8056 | Dominance law for Cartesia... |
| xpdom1g 8057 | Dominance law for Cartesia... |
| xpdom3 8058 | A set is dominated by its ... |
| xpdom1 8059 | Dominance law for Cartesia... |
| domunsncan 8060 | A singleton cancellation l... |
| omxpenlem 8061 | Lemma for ~ omxpen . (Con... |
| omxpen 8062 | The cardinal and ordinal p... |
| omf1o 8063 | Construct an explicit bije... |
| pw2f1olem 8064 | Lemma for ~ pw2f1o . (Con... |
| pw2f1o 8065 | The power set of a set is ... |
| pw2eng 8066 | The power set of a set is ... |
| pw2en 8067 | The power set of a set is ... |
| fopwdom 8068 | Covering implies injection... |
| enfixsn 8069 | Given two equipollent sets... |
| sbthlem1 8070 | Lemma for ~ sbth . (Contr... |
| sbthlem2 8071 | Lemma for ~ sbth . (Contr... |
| sbthlem3 8072 | Lemma for ~ sbth . (Contr... |
| sbthlem4 8073 | Lemma for ~ sbth . (Contr... |
| sbthlem5 8074 | Lemma for ~ sbth . (Contr... |
| sbthlem6 8075 | Lemma for ~ sbth . (Contr... |
| sbthlem7 8076 | Lemma for ~ sbth . (Contr... |
| sbthlem8 8077 | Lemma for ~ sbth . (Contr... |
| sbthlem9 8078 | Lemma for ~ sbth . (Contr... |
| sbthlem10 8079 | Lemma for ~ sbth . (Contr... |
| sbth 8080 | Schroeder-Bernstein Theore... |
| sbthb 8081 | Schroeder-Bernstein Theore... |
| sbthcl 8082 | Schroeder-Bernstein Theore... |
| dfsdom2 8083 | Alternate definition of st... |
| brsdom2 8084 | Alternate definition of st... |
| sdomnsym 8085 | Strict dominance is asymme... |
| domnsym 8086 | Theorem 22(i) of [Suppes] ... |
| 0domg 8087 | Any set dominates the empt... |
| dom0 8088 | A set dominated by the emp... |
| 0sdomg 8089 | A set strictly dominates t... |
| 0dom 8090 | Any set dominates the empt... |
| 0sdom 8091 | A set strictly dominates t... |
| sdom0 8092 | The empty set does not str... |
| sdomdomtr 8093 | Transitivity of strict dom... |
| sdomentr 8094 | Transitivity of strict dom... |
| domsdomtr 8095 | Transitivity of dominance ... |
| ensdomtr 8096 | Transitivity of equinumero... |
| sdomirr 8097 | Strict dominance is irrefl... |
| sdomtr 8098 | Strict dominance is transi... |
| sdomn2lp 8099 | Strict dominance has no 2-... |
| enen1 8100 | Equality-like theorem for ... |
| enen2 8101 | Equality-like theorem for ... |
| domen1 8102 | Equality-like theorem for ... |
| domen2 8103 | Equality-like theorem for ... |
| sdomen1 8104 | Equality-like theorem for ... |
| sdomen2 8105 | Equality-like theorem for ... |
| domtriord 8106 | Dominance is trichotomous ... |
| sdomel 8107 | Strict dominance implies o... |
| sdomdif 8108 | The difference of a set fr... |
| onsdominel 8109 | An ordinal with more eleme... |
| domunsn 8110 | Dominance over a set with ... |
| fodomr 8111 | There exists a mapping fro... |
| pwdom 8112 | Injection of sets implies ... |
| canth2 8113 | Cantor's Theorem. No set ... |
| canth2g 8114 | Cantor's theorem with the ... |
| 2pwuninel 8115 | The power set of the power... |
| 2pwne 8116 | No set equals the power se... |
| disjen 8117 | A stronger form of ~ pwuni... |
| disjenex 8118 | Existence version of ~ dis... |
| domss2 8119 | A corollary of ~ disjenex ... |
| domssex2 8120 | A corollary of ~ disjenex ... |
| domssex 8121 | Weakening of ~ domssex to ... |
| xpf1o 8122 | Construct a bijection on a... |
| xpen 8123 | Equinumerosity law for Car... |
| mapen 8124 | Two set exponentiations ar... |
| mapdom1 8125 | Order-preserving property ... |
| mapxpen 8126 | Equinumerosity law for dou... |
| xpmapenlem 8127 | Lemma for ~ xpmapen . (Co... |
| xpmapen 8128 | Equinumerosity law for set... |
| mapunen 8129 | Equinumerosity law for set... |
| map2xp 8130 | A cardinal power with expo... |
| mapdom2 8131 | Order-preserving property ... |
| mapdom3 8132 | Set exponentiation dominat... |
| pwen 8133 | If two sets are equinumero... |
| ssenen 8134 | Equinumerosity of equinume... |
| limenpsi 8135 | A limit ordinal is equinum... |
| limensuci 8136 | A limit ordinal is equinum... |
| limensuc 8137 | A limit ordinal is equinum... |
| infensuc 8138 | Any infinite ordinal is eq... |
| phplem1 8139 | Lemma for Pigeonhole Princ... |
| phplem2 8140 | Lemma for Pigeonhole Princ... |
| phplem3 8141 | Lemma for Pigeonhole Princ... |
| phplem4 8142 | Lemma for Pigeonhole Princ... |
| nneneq 8143 | Two equinumerous natural n... |
| php 8144 | Pigeonhole Principle. A n... |
| php2 8145 | Corollary of Pigeonhole Pr... |
| php3 8146 | Corollary of Pigeonhole Pr... |
| php4 8147 | Corollary of the Pigeonhol... |
| php5 8148 | Corollary of the Pigeonhol... |
| snnen2o 8149 | A singleton ` { A } ` is n... |
| onomeneq 8150 | An ordinal number equinume... |
| onfin 8151 | An ordinal number is finit... |
| onfin2 8152 | A set is a natural number ... |
| nnfi 8153 | Natural numbers are finite... |
| nndomo 8154 | Cardinal ordering agrees w... |
| nnsdomo 8155 | Cardinal ordering agrees w... |
| sucdom2 8156 | Strict dominance of a set ... |
| sucdom 8157 | Strict dominance of a set ... |
| 0sdom1dom 8158 | Strict dominance over zero... |
| 1sdom2 8159 | Ordinal 1 is strictly domi... |
| sdom1 8160 | A set has less than one me... |
| modom 8161 | Two ways to express "at mo... |
| modom2 8162 | Two ways to express "at mo... |
| 1sdom 8163 | A set that strictly domina... |
| unxpdomlem1 8164 | Lemma for ~ unxpdom . (Tr... |
| unxpdomlem2 8165 | Lemma for ~ unxpdom . (Co... |
| unxpdomlem3 8166 | Lemma for ~ unxpdom . (Co... |
| unxpdom 8167 | Cartesian product dominate... |
| unxpdom2 8168 | Corollary of ~ unxpdom . ... |
| sucxpdom 8169 | Cartesian product dominate... |
| pssinf 8170 | A set equinumerous to a pr... |
| fisseneq 8171 | A finite set is equal to i... |
| ominf 8172 | The set of natural numbers... |
| isinf 8173 | Any set that is not finite... |
| fineqvlem 8174 | Lemma for ~ fineqv . (Con... |
| fineqv 8175 | If the Axiom of Infinity i... |
| enfi 8176 | Equinumerous sets have the... |
| enfii 8177 | A set equinumerous to a fi... |
| pssnn 8178 | A proper subset of a natur... |
| ssnnfi 8179 | A subset of a natural numb... |
| ssfi 8180 | A subset of a finite set i... |
| domfi 8181 | A set dominated by a finit... |
| xpfir 8182 | The components of a nonemp... |
| ssfid 8183 | A subset of a finite set i... |
| infi 8184 | The intersection of two se... |
| rabfi 8185 | A restricted class built f... |
| finresfin 8186 | The restriction of a finit... |
| f1finf1o 8187 | Any injection from one fin... |
| 0fin 8188 | The empty set is finite. ... |
| nfielex 8189 | If a class is not finite, ... |
| en1eqsn 8190 | A set with one element is ... |
| en1eqsnbi 8191 | A set containing an elemen... |
| diffi 8192 | If ` A ` is finite, ` ( A ... |
| dif1en 8193 | If a set ` A ` is equinume... |
| enp1ilem 8194 | Lemma for uses of ~ enp1i ... |
| enp1i 8195 | Proof induction for ~ en2i... |
| en2 8196 | A set equinumerous to ordi... |
| en3 8197 | A set equinumerous to ordi... |
| en4 8198 | A set equinumerous to ordi... |
| findcard 8199 | Schema for induction on th... |
| findcard2 8200 | Schema for induction on th... |
| findcard2s 8201 | Variation of ~ findcard2 r... |
| findcard2d 8202 | Deduction version of ~ fin... |
| findcard3 8203 | Schema for strong inductio... |
| ac6sfi 8204 | A version of ~ ac6s for fi... |
| frfi 8205 | A partial order is well-fo... |
| fimax2g 8206 | A finite set has a maximum... |
| fimaxg 8207 | A finite set has a maximum... |
| fisupg 8208 | Lemma showing existence an... |
| wofi 8209 | A total order on a finite ... |
| ordunifi 8210 | The maximum of a finite co... |
| nnunifi 8211 | The union (supremum) of a ... |
| unblem1 8212 | Lemma for ~ unbnn . After... |
| unblem2 8213 | Lemma for ~ unbnn . The v... |
| unblem3 8214 | Lemma for ~ unbnn . The v... |
| unblem4 8215 | Lemma for ~ unbnn . The f... |
| unbnn 8216 | Any unbounded subset of na... |
| unbnn2 8217 | Version of ~ unbnn that do... |
| isfinite2 8218 | Any set strictly dominated... |
| nnsdomg 8219 | Omega strictly dominates a... |
| isfiniteg 8220 | A set is finite iff it is ... |
| infsdomnn 8221 | An infinite set strictly d... |
| infn0 8222 | An infinite set is not emp... |
| fin2inf 8223 | This (useless) theorem, wh... |
| unfilem1 8224 | Lemma for proving that the... |
| unfilem2 8225 | Lemma for proving that the... |
| unfilem3 8226 | Lemma for proving that the... |
| unfi 8227 | The union of two finite se... |
| unfir 8228 | If a union is finite, the ... |
| unfi2 8229 | The union of two finite se... |
| difinf 8230 | An infinite set ` A ` minu... |
| xpfi 8231 | The Cartesian product of t... |
| 3xpfi 8232 | The Cartesian product of t... |
| domunfican 8233 | A finite set union cancell... |
| infcntss 8234 | Every infinite set has a d... |
| prfi 8235 | An unordered pair is finit... |
| tpfi 8236 | An unordered triple is fin... |
| fiint 8237 | Equivalent ways of stating... |
| fnfi 8238 | A version of ~ fnex for fi... |
| fodomfi 8239 | An onto function implies d... |
| fodomfib 8240 | Equivalence of an onto map... |
| fofinf1o 8241 | Any surjection from one fi... |
| rneqdmfinf1o 8242 | Any function from a finite... |
| fidomdm 8243 | Any finite set dominates i... |
| dmfi 8244 | The domain of a finite set... |
| fundmfibi 8245 | A function is finite if an... |
| resfnfinfin 8246 | The restriction of a funct... |
| residfi 8247 | A restricted identity func... |
| cnvfi 8248 | If a set is finite, its co... |
| rnfi 8249 | The range of a finite set ... |
| f1dmvrnfibi 8250 | A one-to-one function whos... |
| f1vrnfibi 8251 | A one-to-one function whic... |
| fofi 8252 | If a function has a finite... |
| f1fi 8253 | If a 1-to-1 function has a... |
| iunfi 8254 | The finite union of finite... |
| unifi 8255 | The finite union of finite... |
| unifi2 8256 | The finite union of finite... |
| infssuni 8257 | If an infinite set ` A ` i... |
| unirnffid 8258 | The union of the range of ... |
| imafi 8259 | Images of finite sets are ... |
| pwfilem 8260 | Lemma for ~ pwfi . (Contr... |
| pwfi 8261 | The power set of a finite ... |
| mapfi 8262 | Set exponentiation of fini... |
| ixpfi 8263 | A Cartesian product of fin... |
| ixpfi2 8264 | A Cartesian product of fin... |
| mptfi 8265 | A finite mapping set is fi... |
| abrexfi 8266 | An image set from a finite... |
| cnvimamptfin 8267 | A preimage of a mapping wi... |
| elfpw 8268 | Membership in a class of f... |
| unifpw 8269 | A set is the union of its ... |
| f1opwfi 8270 | A one-to-one mapping induc... |
| fissuni 8271 | A finite subset of a union... |
| fipreima 8272 | Given a finite subset ` A ... |
| finsschain 8273 | A finite subset of the uni... |
| indexfi 8274 | If for every element of a ... |
| relfsupp 8277 | The property of a function... |
| relprcnfsupp 8278 | A proper class is never fi... |
| isfsupp 8279 | The property of a class to... |
| funisfsupp 8280 | The property of a function... |
| fsuppimp 8281 | Implications of a class be... |
| fsuppimpd 8282 | A finitely supported funct... |
| fisuppfi 8283 | A function on a finite set... |
| fdmfisuppfi 8284 | The support of a function ... |
| fdmfifsupp 8285 | A function with a finite d... |
| fsuppmptdm 8286 | A mapping with a finite do... |
| fndmfisuppfi 8287 | The support of a function ... |
| fndmfifsupp 8288 | A function with a finite d... |
| suppeqfsuppbi 8289 | If two functions have the ... |
| suppssfifsupp 8290 | If the support of a functi... |
| fsuppsssupp 8291 | If the support of a functi... |
| fsuppxpfi 8292 | The cartesian product of t... |
| fczfsuppd 8293 | A constant function with v... |
| fsuppun 8294 | The union of two finitely ... |
| fsuppunfi 8295 | The union of the support o... |
| fsuppunbi 8296 | If the union of two classe... |
| 0fsupp 8297 | The empty set is a finitel... |
| snopfsupp 8298 | A singleton containing an ... |
| funsnfsupp 8299 | Finite support for a funct... |
| fsuppres 8300 | The restriction of a finit... |
| ressuppfi 8301 | If the support of the rest... |
| resfsupp 8302 | If the restriction of a fu... |
| resfifsupp 8303 | The restriction of a funct... |
| frnfsuppbi 8304 | Two ways of saying that a ... |
| fsuppmptif 8305 | A function mapping an argu... |
| fsuppcolem 8306 | Lemma for ~ fsuppco . For... |
| fsuppco 8307 | The composition of a 1-1 f... |
| fsuppco2 8308 | The composition of a funct... |
| fsuppcor 8309 | The composition of a funct... |
| mapfienlem1 8310 | Lemma 1 for ~ mapfien . (... |
| mapfienlem2 8311 | Lemma 2 for ~ mapfien . (... |
| mapfienlem3 8312 | Lemma 3 for ~ mapfien . (... |
| mapfien 8313 | A bijection of the base se... |
| mapfien2 8314 | Equinumerousity relation f... |
| sniffsupp 8315 | A function mapping all but... |
| fival 8318 | The set of all the finite ... |
| elfi 8319 | Specific properties of an ... |
| elfi2 8320 | The empty intersection nee... |
| elfir 8321 | Sufficient condition for a... |
| intrnfi 8322 | Sufficient condition for t... |
| iinfi 8323 | An indexed intersection of... |
| inelfi 8324 | The intersection of two se... |
| ssfii 8325 | Any element of a set ` A `... |
| fi0 8326 | The set of finite intersec... |
| fieq0 8327 | If ` A ` is not empty, the... |
| fiin 8328 | The elements of ` ( fi `` ... |
| dffi2 8329 | The set of finite intersec... |
| fiss 8330 | Subset relationship for fu... |
| inficl 8331 | A set which is closed unde... |
| fipwuni 8332 | The set of finite intersec... |
| fisn 8333 | A singleton is closed unde... |
| fiuni 8334 | The union of the finite in... |
| fipwss 8335 | If a set is a family of su... |
| elfiun 8336 | A finite intersection of e... |
| dffi3 8337 | The set of finite intersec... |
| fifo 8338 | Describe a surjection from... |
| marypha1lem 8339 | Core induction for Philip ... |
| marypha1 8340 | (Philip) Hall's marriage t... |
| marypha2lem1 8341 | Lemma for ~ marypha2 . Pr... |
| marypha2lem2 8342 | Lemma for ~ marypha2 . Pr... |
| marypha2lem3 8343 | Lemma for ~ marypha2 . Pr... |
| marypha2lem4 8344 | Lemma for ~ marypha2 . Pr... |
| marypha2 8345 | Version of ~ marypha1 usin... |
| dfsup2 8350 | Quantifier free definition... |
| supeq1 8351 | Equality theorem for supre... |
| supeq1d 8352 | Equality deduction for sup... |
| supeq1i 8353 | Equality inference for sup... |
| supeq2 8354 | Equality theorem for supre... |
| supeq3 8355 | Equality theorem for supre... |
| supeq123d 8356 | Equality deduction for sup... |
| nfsup 8357 | Hypothesis builder for sup... |
| supmo 8358 | Any class ` B ` has at mos... |
| supexd 8359 | A supremum is a set. (Con... |
| supeu 8360 | A supremum is unique. Sim... |
| supval2 8361 | Alternate expression for t... |
| eqsup 8362 | Sufficient condition for a... |
| eqsupd 8363 | Sufficient condition for a... |
| supcl 8364 | A supremum belongs to its ... |
| supub 8365 | A supremum is an upper bou... |
| suplub 8366 | A supremum is the least up... |
| suplub2 8367 | Bidirectional form of ~ su... |
| supnub 8368 | An upper bound is not less... |
| supex 8369 | A supremum is a set. (Con... |
| sup00 8370 | The supremum under an empt... |
| sup0riota 8371 | The supremum of an empty s... |
| sup0 8372 | The supremum of an empty s... |
| supmax 8373 | The greatest element of a ... |
| fisup2g 8374 | A finite set satisfies the... |
| fisupcl 8375 | A nonempty finite set cont... |
| supgtoreq 8376 | The supremum of a finite s... |
| suppr 8377 | The supremum of a pair. (... |
| supsn 8378 | The supremum of a singleto... |
| supisolem 8379 | Lemma for ~ supiso . (Con... |
| supisoex 8380 | Lemma for ~ supiso . (Con... |
| supiso 8381 | Image of a supremum under ... |
| infeq1 8382 | Equality theorem for infim... |
| infeq1d 8383 | Equality deduction for inf... |
| infeq1i 8384 | Equality inference for inf... |
| infeq2 8385 | Equality theorem for infim... |
| infeq3 8386 | Equality theorem for infim... |
| infeq123d 8387 | Equality deduction for inf... |
| nfinf 8388 | Hypothesis builder for inf... |
| infexd 8389 | An infimum is a set. (Con... |
| eqinf 8390 | Sufficient condition for a... |
| eqinfd 8391 | Sufficient condition for a... |
| infval 8392 | Alternate expression for t... |
| infcllem 8393 | Lemma for ~ infcl , ~ infl... |
| infcl 8394 | An infimum belongs to its ... |
| inflb 8395 | An infimum is a lower boun... |
| infglb 8396 | An infimum is the greatest... |
| infglbb 8397 | Bidirectional form of ~ in... |
| infnlb 8398 | A lower bound is not great... |
| infex 8399 | An infimum is a set. (Con... |
| infmin 8400 | The smallest element of a ... |
| infmo 8401 | Any class ` B ` has at mos... |
| infeu 8402 | An infimum is unique. (Co... |
| fimin2g 8403 | A finite set has a minimum... |
| fiming 8404 | A finite set has a minimum... |
| fiinfg 8405 | Lemma showing existence an... |
| fiinf2g 8406 | A finite set satisfies the... |
| fiinfcl 8407 | A nonempty finite set cont... |
| infltoreq 8408 | The infimum of a finite se... |
| infpr 8409 | The infimum of a pair. (C... |
| infsn 8410 | The infimum of a singleton... |
| inf00 8411 | The infimum regarding an e... |
| infempty 8412 | The infimum of an empty se... |
| infiso 8413 | Image of an infimum under ... |
| dfoi 8416 | Rewrite ~ df-oi with abbre... |
| oieq1 8417 | Equality theorem for ordin... |
| oieq2 8418 | Equality theorem for ordin... |
| nfoi 8419 | Hypothesis builder for ord... |
| ordiso2 8420 | Generalize ~ ordiso to pro... |
| ordiso 8421 | Order-isomorphic ordinal n... |
| ordtypecbv 8422 | Lemma for ~ ordtype . (Co... |
| ordtypelem1 8423 | Lemma for ~ ordtype . (Co... |
| ordtypelem2 8424 | Lemma for ~ ordtype . (Co... |
| ordtypelem3 8425 | Lemma for ~ ordtype . (Co... |
| ordtypelem4 8426 | Lemma for ~ ordtype . (Co... |
| ordtypelem5 8427 | Lemma for ~ ordtype . (Co... |
| ordtypelem6 8428 | Lemma for ~ ordtype . (Co... |
| ordtypelem7 8429 | Lemma for ~ ordtype . ` ra... |
| ordtypelem8 8430 | Lemma for ~ ordtype . (Co... |
| ordtypelem9 8431 | Lemma for ~ ordtype . Eit... |
| ordtypelem10 8432 | Lemma for ~ ordtype . Usi... |
| oi0 8433 | Definition of the ordinal ... |
| oicl 8434 | The order type of the well... |
| oif 8435 | The order isomorphism of t... |
| oiiso2 8436 | The order isomorphism of t... |
| ordtype 8437 | For any set-like well-orde... |
| oiiniseg 8438 | ` ran F ` is an initial se... |
| ordtype2 8439 | For any set-like well-orde... |
| oiexg 8440 | The order isomorphism on a... |
| oion 8441 | The order type of the well... |
| oiiso 8442 | The order isomorphism of t... |
| oien 8443 | The order type of a well-o... |
| oieu 8444 | Uniqueness of the unique o... |
| oismo 8445 | When ` A ` is a subclass o... |
| oiid 8446 | The order type of an ordin... |
| hartogslem1 8447 | Lemma for ~ hartogs . (Co... |
| hartogslem2 8448 | Lemma for ~ hartogs . (Co... |
| hartogs 8449 | Given any set, the Hartogs... |
| wofib 8450 | The only sets which are we... |
| wemaplem1 8451 | Value of the lexicographic... |
| wemaplem2 8452 | Lemma for ~ wemapso . Tra... |
| wemaplem3 8453 | Lemma for ~ wemapso . Tra... |
| wemappo 8454 | Construct lexicographic or... |
| wemapsolem 8455 | Lemma for ~ wemapso . (Co... |
| wemapso 8456 | Construct lexicographic or... |
| wemapso2lem 8457 | Lemma for ~ wemapso2 . (C... |
| wemapso2 8458 | An alternative to having a... |
| card2on 8459 | Proof that the alternate d... |
| card2inf 8460 | The definition ~ cardval2 ... |
| harf 8465 | Functionality of the Harto... |
| harcl 8466 | Closure of the Hartogs fun... |
| harval 8467 | Function value of the Hart... |
| elharval 8468 | The Hartogs number of a se... |
| harndom 8469 | The Hartogs number of a se... |
| harword 8470 | Weak ordering property of ... |
| relwdom 8471 | Weak dominance is a relati... |
| brwdom 8472 | Property of weak dominance... |
| brwdomi 8473 | Property of weak dominance... |
| brwdomn0 8474 | Weak dominance over nonemp... |
| 0wdom 8475 | Any set weakly dominates t... |
| fowdom 8476 | An onto function implies w... |
| wdomref 8477 | Reflexivity of weak domina... |
| brwdom2 8478 | Alternate characterization... |
| domwdom 8479 | Weak dominance is implied ... |
| wdomtr 8480 | Transitivity of weak domin... |
| wdomen1 8481 | Equality-like theorem for ... |
| wdomen2 8482 | Equality-like theorem for ... |
| wdompwdom 8483 | Weak dominance strengthens... |
| canthwdom 8484 | Cantor's Theorem, stated u... |
| wdom2d 8485 | Deduce weak dominance from... |
| wdomd 8486 | Deduce weak dominance from... |
| brwdom3 8487 | Condition for weak dominan... |
| brwdom3i 8488 | Weak dominance implies exi... |
| unwdomg 8489 | Weak dominance of a (disjo... |
| xpwdomg 8490 | Weak dominance of a Cartes... |
| wdomima2g 8491 | A set is weakly dominant o... |
| wdomimag 8492 | A set is weakly dominant o... |
| unxpwdom2 8493 | Lemma for ~ unxpwdom . (C... |
| unxpwdom 8494 | If a Cartesian product is ... |
| harwdom 8495 | The Hartogs function is we... |
| ixpiunwdom 8496 | Describe an onto function ... |
| axreg2 8498 | Axiom of Regularity expres... |
| zfregcl 8499 | The Axiom of Regularity wi... |
| zfreg 8500 | The Axiom of Regularity us... |
| zfregclOLD 8501 | Obsolete version of ~ zfre... |
| zfregOLD 8502 | Obsolete version of ~ zfre... |
| zfreg2OLD 8503 | Alternate version of ~ zfr... |
| elirrv 8504 | The membership relation is... |
| elirr 8505 | No class is a member of it... |
| sucprcreg 8506 | A class is equal to its su... |
| ruv 8507 | The Russell class is equal... |
| ruALT 8508 | Alternate proof of ~ ru , ... |
| zfregfr 8509 | The epsilon relation is we... |
| en2lp 8510 | No class has 2-cycle membe... |
| en3lplem1 8511 | Lemma for ~ en3lp . (Cont... |
| en3lplem2 8512 | Lemma for ~ en3lp . (Cont... |
| en3lp 8513 | No class has 3-cycle membe... |
| preleq 8514 | Equality of two unordered ... |
| opthreg 8515 | Theorem for alternate repr... |
| suc11reg 8516 | The successor operation be... |
| dford2 8517 | Assuming ~ ax-reg , an ord... |
| inf0 8518 | Our Axiom of Infinity deri... |
| inf1 8519 | Variation of Axiom of Infi... |
| inf2 8520 | Variation of Axiom of Infi... |
| inf3lema 8521 | Lemma for our Axiom of Inf... |
| inf3lemb 8522 | Lemma for our Axiom of Inf... |
| inf3lemc 8523 | Lemma for our Axiom of Inf... |
| inf3lemd 8524 | Lemma for our Axiom of Inf... |
| inf3lem1 8525 | Lemma for our Axiom of Inf... |
| inf3lem2 8526 | Lemma for our Axiom of Inf... |
| inf3lem3 8527 | Lemma for our Axiom of Inf... |
| inf3lem4 8528 | Lemma for our Axiom of Inf... |
| inf3lem5 8529 | Lemma for our Axiom of Inf... |
| inf3lem6 8530 | Lemma for our Axiom of Inf... |
| inf3lem7 8531 | Lemma for our Axiom of Inf... |
| inf3 8532 | Our Axiom of Infinity ~ ax... |
| infeq5i 8533 | Half of ~ infeq5 . (Contr... |
| infeq5 8534 | The statement "there exist... |
| zfinf 8536 | Axiom of Infinity expresse... |
| axinf2 8537 | A standard version of Axio... |
| zfinf2 8539 | A standard version of the ... |
| omex 8540 | The existence of omega (th... |
| axinf 8541 | The first version of the A... |
| inf5 8542 | The statement "there exist... |
| omelon 8543 | Omega is an ordinal number... |
| dfom3 8544 | The class of natural numbe... |
| elom3 8545 | A simplification of ~ elom... |
| dfom4 8546 | A simplification of ~ df-o... |
| dfom5 8547 | ` _om ` is the smallest li... |
| oancom 8548 | Ordinal addition is not co... |
| isfinite 8549 | A set is finite iff it is ... |
| fict 8550 | A finite set is countable ... |
| nnsdom 8551 | A natural number is strict... |
| omenps 8552 | Omega is equinumerous to a... |
| omensuc 8553 | The set of natural numbers... |
| infdifsn 8554 | Removing a singleton from ... |
| infdiffi 8555 | Removing a finite set from... |
| unbnn3 8556 | Any unbounded subset of na... |
| noinfep 8557 | Using the Axiom of Regular... |
| cantnffval 8560 | The value of the Cantor no... |
| cantnfdm 8561 | The domain of the Cantor n... |
| cantnfvalf 8562 | Lemma for ~ cantnf . The ... |
| cantnfs 8563 | Elementhood in the set of ... |
| cantnfcl 8564 | Basic properties of the or... |
| cantnfval 8565 | The value of the Cantor no... |
| cantnfval2 8566 | Alternate expression for t... |
| cantnfsuc 8567 | The value of the recursive... |
| cantnfle 8568 | A lower bound on the ` CNF... |
| cantnflt 8569 | An upper bound on the part... |
| cantnflt2 8570 | An upper bound on the ` CN... |
| cantnff 8571 | The ` CNF ` function is a ... |
| cantnf0 8572 | The value of the zero func... |
| cantnfrescl 8573 | A function is finitely sup... |
| cantnfres 8574 | The ` CNF ` function respe... |
| cantnfp1lem1 8575 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem2 8576 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem3 8577 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1 8578 | If ` F ` is created by add... |
| oemapso 8579 | The relation ` T ` is a st... |
| oemapval 8580 | Value of the relation ` T ... |
| oemapvali 8581 | If ` F < G ` , then there ... |
| cantnflem1a 8582 | Lemma for ~ cantnf . (Con... |
| cantnflem1b 8583 | Lemma for ~ cantnf . (Con... |
| cantnflem1c 8584 | Lemma for ~ cantnf . (Con... |
| cantnflem1d 8585 | Lemma for ~ cantnf . (Con... |
| cantnflem1 8586 | Lemma for ~ cantnf . This... |
| cantnflem2 8587 | Lemma for ~ cantnf . (Con... |
| cantnflem3 8588 | Lemma for ~ cantnf . Here... |
| cantnflem4 8589 | Lemma for ~ cantnf . Comp... |
| cantnf 8590 | The Cantor Normal Form the... |
| oemapwe 8591 | The lexicographic order on... |
| cantnffval2 8592 | An alternate definition of... |
| cantnff1o 8593 | Simplify the isomorphism o... |
| wemapwe 8594 | Construct lexicographic or... |
| oef1o 8595 | A bijection of the base se... |
| cnfcomlem 8596 | Lemma for ~ cnfcom . (Con... |
| cnfcom 8597 | Any ordinal ` B ` is equin... |
| cnfcom2lem 8598 | Lemma for ~ cnfcom2 . (Co... |
| cnfcom2 8599 | Any nonzero ordinal ` B ` ... |
| cnfcom3lem 8600 | Lemma for ~ cnfcom3 . (Co... |
| cnfcom3 8601 | Any infinite ordinal ` B `... |
| cnfcom3clem 8602 | Lemma for ~ cnfcom3c . (C... |
| cnfcom3c 8603 | Wrap the construction of ~... |
| trcl 8604 | For any set ` A ` , show t... |
| tz9.1 8605 | Every set has a transitive... |
| tz9.1c 8606 | Alternate expression for t... |
| epfrs 8607 | The strong form of the Axi... |
| zfregs 8608 | The strong form of the Axi... |
| zfregs2 8609 | Alternate strong form of t... |
| setind 8610 | Set (epsilon) induction. ... |
| setind2 8611 | Set (epsilon) induction, s... |
| tcvalg 8614 | Value of the transitive cl... |
| tcid 8615 | Defining property of the t... |
| tctr 8616 | Defining property of the t... |
| tcmin 8617 | Defining property of the t... |
| tc2 8618 | A variant of the definitio... |
| tcsni 8619 | The transitive closure of ... |
| tcss 8620 | The transitive closure fun... |
| tcel 8621 | The transitive closure fun... |
| tcidm 8622 | The transitive closure fun... |
| tc0 8623 | The transitive closure of ... |
| tc00 8624 | The transitive closure is ... |
| r1funlim 8629 | The cumulative hierarchy o... |
| r1fnon 8630 | The cumulative hierarchy o... |
| r10 8631 | Value of the cumulative hi... |
| r1sucg 8632 | Value of the cumulative hi... |
| r1suc 8633 | Value of the cumulative hi... |
| r1limg 8634 | Value of the cumulative hi... |
| r1lim 8635 | Value of the cumulative hi... |
| r1fin 8636 | The first ` _om ` levels o... |
| r1sdom 8637 | Each stage in the cumulati... |
| r111 8638 | The cumulative hierarchy i... |
| r1tr 8639 | The cumulative hierarchy o... |
| r1tr2 8640 | The union of a cumulative ... |
| r1ordg 8641 | Ordering relation for the ... |
| r1ord3g 8642 | Ordering relation for the ... |
| r1ord 8643 | Ordering relation for the ... |
| r1ord2 8644 | Ordering relation for the ... |
| r1ord3 8645 | Ordering relation for the ... |
| r1sssuc 8646 | The value of the cumulativ... |
| r1pwss 8647 | Each set of the cumulative... |
| r1sscl 8648 | Each set of the cumulative... |
| r1val1 8649 | The value of the cumulativ... |
| tz9.12lem1 8650 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem2 8651 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem3 8652 | Lemma for ~ tz9.12 . (Con... |
| tz9.12 8653 | A set is well-founded if a... |
| tz9.13 8654 | Every set is well-founded,... |
| tz9.13g 8655 | Every set is well-founded,... |
| rankwflemb 8656 | Two ways of saying a set i... |
| rankf 8657 | The domain and range of th... |
| rankon 8658 | The rank of a set is an or... |
| r1elwf 8659 | Any member of the cumulati... |
| rankvalb 8660 | Value of the rank function... |
| rankr1ai 8661 | One direction of ~ rankr1a... |
| rankvaln 8662 | Value of the rank function... |
| rankidb 8663 | Identity law for the rank ... |
| rankdmr1 8664 | A rank is a member of the ... |
| rankr1ag 8665 | A version of ~ rankr1a tha... |
| rankr1bg 8666 | A relationship between ran... |
| r1rankidb 8667 | Any set is a subset of the... |
| r1elssi 8668 | The range of the ` R1 ` fu... |
| r1elss 8669 | The range of the ` R1 ` fu... |
| pwwf 8670 | A power set is well-founde... |
| sswf 8671 | A subset of a well-founded... |
| snwf 8672 | A singleton is well-founde... |
| unwf 8673 | A binary union is well-fou... |
| prwf 8674 | An unordered pair is well-... |
| opwf 8675 | An ordered pair is well-fo... |
| unir1 8676 | The cumulative hierarchy o... |
| jech9.3 8677 | Every set belongs to some ... |
| rankwflem 8678 | Every set is well-founded,... |
| rankval 8679 | Value of the rank function... |
| rankvalg 8680 | Value of the rank function... |
| rankval2 8681 | Value of an alternate defi... |
| uniwf 8682 | A union is well-founded if... |
| rankr1clem 8683 | Lemma for ~ rankr1c . (Co... |
| rankr1c 8684 | A relationship between the... |
| rankidn 8685 | A relationship between the... |
| rankpwi 8686 | The rank of a power set. ... |
| rankelb 8687 | The membership relation is... |
| wfelirr 8688 | A well-founded set is not ... |
| rankval3b 8689 | The value of the rank func... |
| ranksnb 8690 | The rank of a singleton. ... |
| rankonidlem 8691 | Lemma for ~ rankonid . (C... |
| rankonid 8692 | The rank of an ordinal num... |
| onwf 8693 | The ordinals are all well-... |
| onssr1 8694 | Initial segments of the or... |
| rankr1g 8695 | A relationship between the... |
| rankid 8696 | Identity law for the rank ... |
| rankr1 8697 | A relationship between the... |
| ssrankr1 8698 | A relationship between an ... |
| rankr1a 8699 | A relationship between ran... |
| r1val2 8700 | The value of the cumulativ... |
| r1val3 8701 | The value of the cumulativ... |
| rankel 8702 | The membership relation is... |
| rankval3 8703 | The value of the rank func... |
| bndrank 8704 | Any class whose elements h... |
| unbndrank 8705 | The elements of a proper c... |
| rankpw 8706 | The rank of a power set. ... |
| ranklim 8707 | The rank of a set belongs ... |
| r1pw 8708 | A stronger property of ` R... |
| r1pwALT 8709 | Alternate shorter proof of... |
| r1pwcl 8710 | The cumulative hierarchy o... |
| rankssb 8711 | The subset relation is inh... |
| rankss 8712 | The subset relation is inh... |
| rankunb 8713 | The rank of the union of t... |
| rankprb 8714 | The rank of an unordered p... |
| rankopb 8715 | The rank of an ordered pai... |
| rankuni2b 8716 | The value of the rank func... |
| ranksn 8717 | The rank of a singleton. ... |
| rankuni2 8718 | The rank of a union. Part... |
| rankun 8719 | The rank of the union of t... |
| rankpr 8720 | The rank of an unordered p... |
| rankop 8721 | The rank of an ordered pai... |
| r1rankid 8722 | Any set is a subset of the... |
| rankeq0b 8723 | A set is empty iff its ran... |
| rankeq0 8724 | A set is empty iff its ran... |
| rankr1id 8725 | The rank of the hierarchy ... |
| rankuni 8726 | The rank of a union. Part... |
| rankr1b 8727 | A relationship between ran... |
| ranksuc 8728 | The rank of a successor. ... |
| rankuniss 8729 | Upper bound of the rank of... |
| rankval4 8730 | The rank of a set is the s... |
| rankbnd 8731 | The rank of a set is bound... |
| rankbnd2 8732 | The rank of a set is bound... |
| rankc1 8733 | A relationship that can be... |
| rankc2 8734 | A relationship that can be... |
| rankelun 8735 | Rank membership is inherit... |
| rankelpr 8736 | Rank membership is inherit... |
| rankelop 8737 | Rank membership is inherit... |
| rankxpl 8738 | A lower bound on the rank ... |
| rankxpu 8739 | An upper bound on the rank... |
| rankfu 8740 | An upper bound on the rank... |
| rankmapu 8741 | An upper bound on the rank... |
| rankxplim 8742 | The rank of a Cartesian pr... |
| rankxplim2 8743 | If the rank of a Cartesian... |
| rankxplim3 8744 | The rank of a Cartesian pr... |
| rankxpsuc 8745 | The rank of a Cartesian pr... |
| tcwf 8746 | The transitive closure fun... |
| tcrank 8747 | This theorem expresses two... |
| scottex 8748 | Scott's trick collects all... |
| scott0 8749 | Scott's trick collects all... |
| scottexs 8750 | Theorem scheme version of ... |
| scott0s 8751 | Theorem scheme version of ... |
| cplem1 8752 | Lemma for the Collection P... |
| cplem2 8753 | -Lemma for the Collection ... |
| cp 8754 | Collection Principle. Thi... |
| bnd 8755 | A very strong generalizati... |
| bnd2 8756 | A variant of the Boundedne... |
| kardex 8757 | The collection of all sets... |
| karden 8758 | If we allow the Axiom of R... |
| htalem 8759 | Lemma for defining an emul... |
| hta 8760 | A ZFC emulation of Hilbert... |
| cardf2 8769 | The cardinality function i... |
| cardon 8770 | The cardinal number of a s... |
| isnum2 8771 | A way to express well-orde... |
| isnumi 8772 | A set equinumerous to an o... |
| ennum 8773 | Equinumerous sets are equi... |
| finnum 8774 | Every finite set is numera... |
| onenon 8775 | Every ordinal number is nu... |
| tskwe 8776 | A Tarski set is well-order... |
| xpnum 8777 | The cartesian product of n... |
| cardval3 8778 | An alternate definition of... |
| cardid2 8779 | Any numerable set is equin... |
| isnum3 8780 | A set is numerable iff it ... |
| oncardval 8781 | The value of the cardinal ... |
| oncardid 8782 | Any ordinal number is equi... |
| cardonle 8783 | The cardinal of an ordinal... |
| card0 8784 | The cardinality of the emp... |
| cardidm 8785 | The cardinality function i... |
| oncard 8786 | A set is a cardinal number... |
| ficardom 8787 | The cardinal number of a f... |
| ficardid 8788 | A finite set is equinumero... |
| cardnn 8789 | The cardinality of a natur... |
| cardnueq0 8790 | The empty set is the only ... |
| cardne 8791 | No member of a cardinal nu... |
| carden2a 8792 | If two sets have equal non... |
| carden2b 8793 | If two sets are equinumero... |
| card1 8794 | A set has cardinality one ... |
| cardsn 8795 | A singleton has cardinalit... |
| carddomi2 8796 | Two sets have the dominanc... |
| sdomsdomcardi 8797 | A set strictly dominates i... |
| cardlim 8798 | An infinite cardinal is a ... |
| cardsdomelir 8799 | A cardinal strictly domina... |
| cardsdomel 8800 | A cardinal strictly domina... |
| iscard 8801 | Two ways to express the pr... |
| iscard2 8802 | Two ways to express the pr... |
| carddom2 8803 | Two numerable sets have th... |
| harcard 8804 | The class of ordinal numbe... |
| cardprclem 8805 | Lemma for ~ cardprc . (Co... |
| cardprc 8806 | The class of all cardinal ... |
| carduni 8807 | The union of a set of card... |
| cardiun 8808 | The indexed union of a set... |
| cardennn 8809 | If ` A ` is equinumerous t... |
| cardsucinf 8810 | The cardinality of the suc... |
| cardsucnn 8811 | The cardinality of the suc... |
| cardom 8812 | The set of natural numbers... |
| carden2 8813 | Two numerable sets are equ... |
| cardsdom2 8814 | A numerable set is strictl... |
| domtri2 8815 | Trichotomy of dominance fo... |
| nnsdomel 8816 | Strict dominance and eleme... |
| cardval2 8817 | An alternate version of th... |
| isinffi 8818 | An infinite set contains s... |
| fidomtri 8819 | Trichotomy of dominance wi... |
| fidomtri2 8820 | Trichotomy of dominance wi... |
| harsdom 8821 | The Hartogs number of a we... |
| onsdom 8822 | Any well-orderable set is ... |
| harval2 8823 | An alternate expression fo... |
| cardmin2 8824 | The smallest ordinal that ... |
| pm54.43lem 8825 | In Theorem *54.43 of [Whit... |
| pm54.43 8826 | Theorem *54.43 of [Whitehe... |
| pr2nelem 8827 | Lemma for ~ pr2ne . (Cont... |
| pr2ne 8828 | If an unordered pair has t... |
| prdom2 8829 | An unordered pair has at m... |
| en2eqpr 8830 | Building a set with two el... |
| en2eleq 8831 | Express a set of pair card... |
| en2other2 8832 | Taking the other element t... |
| dif1card 8833 | The cardinality of a nonem... |
| leweon 8834 | Lexicographical order is a... |
| r0weon 8835 | A set-like well-ordering o... |
| infxpenlem 8836 | Lemma for ~ infxpen . (Co... |
| infxpen 8837 | Every infinite ordinal is ... |
| xpomen 8838 | The Cartesian product of o... |
| xpct 8839 | The cartesian product of t... |
| infxpidm2 8840 | The Cartesian product of a... |
| infxpenc 8841 | A canonical version of ~ i... |
| infxpenc2lem1 8842 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem2 8843 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem3 8844 | Lemma for ~ infxpenc2 . (... |
| infxpenc2 8845 | Existence form of ~ infxpe... |
| iunmapdisj 8846 | The union ` U_ n e. C ( A ... |
| fseqenlem1 8847 | Lemma for ~ fseqen . (Con... |
| fseqenlem2 8848 | Lemma for ~ fseqen . (Con... |
| fseqdom 8849 | One half of ~ fseqen . (C... |
| fseqen 8850 | A set that is equinumerous... |
| infpwfidom 8851 | The collection of finite s... |
| dfac8alem 8852 | Lemma for ~ dfac8a . If t... |
| dfac8a 8853 | Numeration theorem: every ... |
| dfac8b 8854 | The well-ordering theorem:... |
| dfac8clem 8855 | Lemma for ~ dfac8c . (Con... |
| dfac8c 8856 | If the union of a set is w... |
| ac10ct 8857 | A proof of the Well orderi... |
| ween 8858 | A set is numerable iff it ... |
| ac5num 8859 | A version of ~ ac5b with t... |
| ondomen 8860 | If a set is dominated by a... |
| numdom 8861 | A set dominated by a numer... |
| ssnum 8862 | A subset of a numerable se... |
| onssnum 8863 | All subsets of the ordinal... |
| indcardi 8864 | Indirect strong induction ... |
| acnrcl 8865 | Reverse closure for the ch... |
| acneq 8866 | Equality theorem for the c... |
| isacn 8867 | The property of being a ch... |
| acni 8868 | The property of being a ch... |
| acni2 8869 | The property of being a ch... |
| acni3 8870 | The property of being a ch... |
| acnlem 8871 | Construct a mapping satisf... |
| numacn 8872 | A well-orderable set has c... |
| finacn 8873 | Every set has finite choic... |
| acndom 8874 | A set with long choice seq... |
| acnnum 8875 | A set ` X ` which has choi... |
| acnen 8876 | The class of choice sets o... |
| acndom2 8877 | A set smaller than one wit... |
| acnen2 8878 | The class of sets with cho... |
| fodomacn 8879 | A version of ~ fodom that ... |
| fodomnum 8880 | A version of ~ fodom that ... |
| fonum 8881 | A surjection maps numerabl... |
| numwdom 8882 | A surjection maps numerabl... |
| fodomfi2 8883 | Onto functions define domi... |
| wdomfil 8884 | Weak dominance agrees with... |
| infpwfien 8885 | Any infinite well-orderabl... |
| inffien 8886 | The set of finite intersec... |
| wdomnumr 8887 | Weak dominance agrees with... |
| alephfnon 8888 | The aleph function is a fu... |
| aleph0 8889 | The first infinite cardina... |
| alephlim 8890 | Value of the aleph functio... |
| alephsuc 8891 | Value of the aleph functio... |
| alephon 8892 | An aleph is an ordinal num... |
| alephcard 8893 | Every aleph is a cardinal ... |
| alephnbtwn 8894 | No cardinal can be sandwic... |
| alephnbtwn2 8895 | No set has equinumerosity ... |
| alephordilem1 8896 | Lemma for ~ alephordi . (... |
| alephordi 8897 | Strict ordering property o... |
| alephord 8898 | Ordering property of the a... |
| alephord2 8899 | Ordering property of the a... |
| alephord2i 8900 | Ordering property of the a... |
| alephord3 8901 | Ordering property of the a... |
| alephsucdom 8902 | A set dominated by an alep... |
| alephsuc2 8903 | An alternate representatio... |
| alephdom 8904 | Relationship between inclu... |
| alephgeom 8905 | Every aleph is greater tha... |
| alephislim 8906 | Every aleph is a limit ord... |
| aleph11 8907 | The aleph function is one-... |
| alephf1 8908 | The aleph function is a on... |
| alephsdom 8909 | If an ordinal is smaller t... |
| alephdom2 8910 | A dominated initial ordina... |
| alephle 8911 | The argument of the aleph ... |
| cardaleph 8912 | Given any transfinite card... |
| cardalephex 8913 | Every transfinite cardinal... |
| infenaleph 8914 | An infinite numerable set ... |
| isinfcard 8915 | Two ways to express the pr... |
| iscard3 8916 | Two ways to express the pr... |
| cardnum 8917 | Two ways to express the cl... |
| alephinit 8918 | An infinite initial ordina... |
| carduniima 8919 | The union of the image of ... |
| cardinfima 8920 | If a mapping to cardinals ... |
| alephiso 8921 | Aleph is an order isomorph... |
| alephprc 8922 | The class of all transfini... |
| alephsson 8923 | The class of transfinite c... |
| unialeph 8924 | The union of the class of ... |
| alephsmo 8925 | The aleph function is stri... |
| alephf1ALT 8926 | Alternate proof of ~ aleph... |
| alephfplem1 8927 | Lemma for ~ alephfp . (Co... |
| alephfplem2 8928 | Lemma for ~ alephfp . (Co... |
| alephfplem3 8929 | Lemma for ~ alephfp . (Co... |
| alephfplem4 8930 | Lemma for ~ alephfp . (Co... |
| alephfp 8931 | The aleph function has a f... |
| alephfp2 8932 | The aleph function has at ... |
| alephval3 8933 | An alternate way to expres... |
| alephsucpw2 8934 | The power set of an aleph ... |
| mappwen 8935 | Power rule for cardinal ar... |
| finnisoeu 8936 | A finite totally ordered s... |
| iunfictbso 8937 | Countability of a countabl... |
| aceq1 8940 | Equivalence of two version... |
| aceq0 8941 | Equivalence of two version... |
| aceq2 8942 | Equivalence of two version... |
| aceq3lem 8943 | Lemma for ~ dfac3 . (Cont... |
| dfac3 8944 | Equivalence of two version... |
| dfac4 8945 | Equivalence of two version... |
| dfac5lem1 8946 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem2 8947 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem3 8948 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4 8949 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem5 8950 | Lemma for ~ dfac5 . (Cont... |
| dfac5 8951 | Equivalence of two version... |
| dfac2a 8952 | Our Axiom of Choice (in th... |
| dfac2 8953 | Axiom of Choice (first for... |
| dfac7 8954 | Equivalence of the Axiom o... |
| dfac0 8955 | Equivalence of two version... |
| dfac1 8956 | Equivalence of two version... |
| dfac8 8957 | A proof of the equivalency... |
| dfac9 8958 | Equivalence of the axiom o... |
| dfac10 8959 | Axiom of Choice equivalent... |
| dfac10c 8960 | Axiom of Choice equivalent... |
| dfac10b 8961 | Axiom of Choice equivalent... |
| acacni 8962 | A choice equivalent: every... |
| dfacacn 8963 | A choice equivalent: every... |
| dfac13 8964 | The axiom of choice holds ... |
| dfac12lem1 8965 | Lemma for ~ dfac12 . (Con... |
| dfac12lem2 8966 | Lemma for ~ dfac12 . (Con... |
| dfac12lem3 8967 | Lemma for ~ dfac12 . (Con... |
| dfac12r 8968 | The axiom of choice holds ... |
| dfac12k 8969 | Equivalence of ~ dfac12 an... |
| dfac12a 8970 | The axiom of choice holds ... |
| dfac12 8971 | The axiom of choice holds ... |
| kmlem1 8972 | Lemma for 5-quantifier AC ... |
| kmlem2 8973 | Lemma for 5-quantifier AC ... |
| kmlem3 8974 | Lemma for 5-quantifier AC ... |
| kmlem4 8975 | Lemma for 5-quantifier AC ... |
| kmlem5 8976 | Lemma for 5-quantifier AC ... |
| kmlem6 8977 | Lemma for 5-quantifier AC ... |
| kmlem7 8978 | Lemma for 5-quantifier AC ... |
| kmlem8 8979 | Lemma for 5-quantifier AC ... |
| kmlem9 8980 | Lemma for 5-quantifier AC ... |
| kmlem10 8981 | Lemma for 5-quantifier AC ... |
| kmlem11 8982 | Lemma for 5-quantifier AC ... |
| kmlem12 8983 | Lemma for 5-quantifier AC ... |
| kmlem13 8984 | Lemma for 5-quantifier AC ... |
| kmlem14 8985 | Lemma for 5-quantifier AC ... |
| kmlem15 8986 | Lemma for 5-quantifier AC ... |
| kmlem16 8987 | Lemma for 5-quantifier AC ... |
| dfackm 8988 | Equivalence of the Axiom o... |
| cdafn 8991 | Cardinal number addition i... |
| cdaval 8992 | Value of cardinal addition... |
| uncdadom 8993 | Cardinal addition dominate... |
| cdaun 8994 | Cardinal addition is equin... |
| cdaen 8995 | Cardinal addition of equin... |
| cdaenun 8996 | Cardinal addition is equin... |
| cda1en 8997 | Cardinal addition with car... |
| cda1dif 8998 | Adding and subtracting one... |
| pm110.643 8999 | 1+1=2 for cardinal number ... |
| pm110.643ALT 9000 | Alternate proof of ~ pm110... |
| cda0en 9001 | Cardinal addition with car... |
| xp2cda 9002 | Two times a cardinal numbe... |
| cdacomen 9003 | Commutative law for cardin... |
| cdaassen 9004 | Associative law for cardin... |
| xpcdaen 9005 | Cardinal multiplication di... |
| mapcdaen 9006 | Sum of exponents law for c... |
| pwcdaen 9007 | Sum of exponents law for c... |
| cdadom1 9008 | Ordering law for cardinal ... |
| cdadom2 9009 | Ordering law for cardinal ... |
| cdadom3 9010 | A set is dominated by its ... |
| cdaxpdom 9011 | Cartesian product dominate... |
| cdafi 9012 | The cardinal sum of two fi... |
| cdainflem 9013 | Any partition of omega int... |
| cdainf 9014 | A set is infinite iff the ... |
| infcda1 9015 | An infinite set is equinum... |
| pwcda1 9016 | The sum of a powerset with... |
| pwcdaidm 9017 | If the natural numbers inj... |
| cdalepw 9018 | If ` A ` is idempotent und... |
| onacda 9019 | The cardinal and ordinal s... |
| cardacda 9020 | The cardinal sum is equinu... |
| cdanum 9021 | The cardinal sum of two nu... |
| unnum 9022 | The union of two numerable... |
| nnacda 9023 | The cardinal and ordinal s... |
| ficardun 9024 | The cardinality of the uni... |
| ficardun2 9025 | The cardinality of the uni... |
| pwsdompw 9026 | Lemma for ~ domtriom . Th... |
| unctb 9027 | The union of two countable... |
| infcdaabs 9028 | Absorption law for additio... |
| infunabs 9029 | An infinite set is equinum... |
| infcda 9030 | The sum of two cardinal nu... |
| infdif 9031 | The cardinality of an infi... |
| infdif2 9032 | Cardinality ordering for a... |
| infxpdom 9033 | Dominance law for multipli... |
| infxpabs 9034 | Absorption law for multipl... |
| infunsdom1 9035 | The union of two sets that... |
| infunsdom 9036 | The union of two sets that... |
| infxp 9037 | Absorption law for multipl... |
| pwcdadom 9038 | A property of dominance ov... |
| infpss 9039 | Every infinite set has an ... |
| infmap2 9040 | An exponentiation law for ... |
| ackbij2lem1 9041 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem1 9042 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem2 9043 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem3 9044 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem4 9045 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem5 9046 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem6 9047 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem7 9048 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem8 9049 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem9 9050 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem10 9051 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem11 9052 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem12 9053 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem13 9054 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem14 9055 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem15 9056 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem16 9057 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem17 9058 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem18 9059 | Lemma for ~ ackbij1 . (Co... |
| ackbij1 9060 | The Ackermann bijection, p... |
| ackbij1b 9061 | The Ackermann bijection, p... |
| ackbij2lem2 9062 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem3 9063 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem4 9064 | Lemma for ~ ackbij2 . (Co... |
| ackbij2 9065 | The Ackermann bijection, p... |
| r1om 9066 | The set of hereditarily fi... |
| fictb 9067 | A set is countable iff its... |
| cflem 9068 | A lemma used to simplify c... |
| cfval 9069 | Value of the cofinality fu... |
| cff 9070 | Cofinality is a function o... |
| cfub 9071 | An upper bound on cofinali... |
| cflm 9072 | Value of the cofinality fu... |
| cf0 9073 | Value of the cofinality fu... |
| cardcf 9074 | Cofinality is a cardinal n... |
| cflecard 9075 | Cofinality is bounded by t... |
| cfle 9076 | Cofinality is bounded by i... |
| cfon 9077 | The cofinality of any set ... |
| cfeq0 9078 | Only the ordinal zero has ... |
| cfsuc 9079 | Value of the cofinality fu... |
| cff1 9080 | There is always a map from... |
| cfflb 9081 | If there is a cofinal map ... |
| cfval2 9082 | Another expression for the... |
| coflim 9083 | A simpler expression for t... |
| cflim3 9084 | Another expression for the... |
| cflim2 9085 | The cofinality function is... |
| cfom 9086 | Value of the cofinality fu... |
| cfss 9087 | There is a cofinal subset ... |
| cfslb 9088 | Any cofinal subset of ` A ... |
| cfslbn 9089 | Any subset of ` A ` smalle... |
| cfslb2n 9090 | Any small collection of sm... |
| cofsmo 9091 | Any cofinal map implies th... |
| cfsmolem 9092 | Lemma for ~ cfsmo . (Cont... |
| cfsmo 9093 | The map in ~ cff1 can be a... |
| cfcoflem 9094 | Lemma for ~ cfcof , showin... |
| coftr 9095 | If there is a cofinal map ... |
| cfcof 9096 | If there is a cofinal map ... |
| cfidm 9097 | The cofinality function is... |
| alephsing 9098 | The cofinality of a limit ... |
| sornom 9099 | The range of a single-step... |
| isfin1a 9114 | Definition of a Ia-finite ... |
| fin1ai 9115 | Property of a Ia-finite se... |
| isfin2 9116 | Definition of a II-finite ... |
| fin2i 9117 | Property of a II-finite se... |
| isfin3 9118 | Definition of a III-finite... |
| isfin4 9119 | Definition of a IV-finite ... |
| fin4i 9120 | Infer that a set is IV-inf... |
| isfin5 9121 | Definition of a V-finite s... |
| isfin6 9122 | Definition of a VI-finite ... |
| isfin7 9123 | Definition of a VII-finite... |
| sdom2en01 9124 | A set with less than two e... |
| infpssrlem1 9125 | Lemma for ~ infpssr . (Co... |
| infpssrlem2 9126 | Lemma for ~ infpssr . (Co... |
| infpssrlem3 9127 | Lemma for ~ infpssr . (Co... |
| infpssrlem4 9128 | Lemma for ~ infpssr . (Co... |
| infpssrlem5 9129 | Lemma for ~ infpssr . (Co... |
| infpssr 9130 | Dedekind infinity implies ... |
| fin4en1 9131 | Dedekind finite is a cardi... |
| ssfin4 9132 | Dedekind finite sets have ... |
| domfin4 9133 | A set dominated by a Dedek... |
| ominf4 9134 | ` _om ` is Dedekind infini... |
| infpssALT 9135 | Alternate proof of ~ infps... |
| isfin4-2 9136 | Alternate definition of IV... |
| isfin4-3 9137 | Alternate definition of IV... |
| fin23lem7 9138 | Lemma for ~ isfin2-2 . Th... |
| fin23lem11 9139 | Lemma for ~ isfin2-2 . (C... |
| fin2i2 9140 | A II-finite set contains m... |
| isfin2-2 9141 | ` Fin2 ` expressed in term... |
| ssfin2 9142 | A subset of a II-finite se... |
| enfin2i 9143 | II-finiteness is a cardina... |
| fin23lem24 9144 | Lemma for ~ fin23 . In a ... |
| fincssdom 9145 | In a chain of finite sets,... |
| fin23lem25 9146 | Lemma for ~ fin23 . In a ... |
| fin23lem26 9147 | Lemma for ~ fin23lem22 . ... |
| fin23lem23 9148 | Lemma for ~ fin23lem22 . ... |
| fin23lem22 9149 | Lemma for ~ fin23 but coul... |
| fin23lem27 9150 | The mapping constructed in... |
| isfin3ds 9151 | Property of a III-finite s... |
| ssfin3ds 9152 | A subset of a III-finite s... |
| fin23lem12 9153 | The beginning of the proof... |
| fin23lem13 9154 | Lemma for ~ fin23 . Each ... |
| fin23lem14 9155 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem15 9156 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem16 9157 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem19 9158 | Lemma for ~ fin23 . The f... |
| fin23lem20 9159 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem17 9160 | Lemma for ~ fin23 . By ? ... |
| fin23lem21 9161 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem28 9162 | Lemma for ~ fin23 . The r... |
| fin23lem29 9163 | Lemma for ~ fin23 . The r... |
| fin23lem30 9164 | Lemma for ~ fin23 . The r... |
| fin23lem31 9165 | Lemma for ~ fin23 . The r... |
| fin23lem32 9166 | Lemma for ~ fin23 . Wrap ... |
| fin23lem33 9167 | Lemma for ~ fin23 . Disch... |
| fin23lem34 9168 | Lemma for ~ fin23 . Estab... |
| fin23lem35 9169 | Lemma for ~ fin23 . Stric... |
| fin23lem36 9170 | Lemma for ~ fin23 . Weak ... |
| fin23lem38 9171 | Lemma for ~ fin23 . The c... |
| fin23lem39 9172 | Lemma for ~ fin23 . Thus,... |
| fin23lem40 9173 | Lemma for ~ fin23 . ` Fin2... |
| fin23lem41 9174 | Lemma for ~ fin23 . A set... |
| isf32lem1 9175 | Lemma for ~ isfin3-2 . De... |
| isf32lem2 9176 | Lemma for ~ isfin3-2 . No... |
| isf32lem3 9177 | Lemma for ~ isfin3-2 . Be... |
| isf32lem4 9178 | Lemma for ~ isfin3-2 . Be... |
| isf32lem5 9179 | Lemma for ~ isfin3-2 . Th... |
| isf32lem6 9180 | Lemma for ~ isfin3-2 . Ea... |
| isf32lem7 9181 | Lemma for ~ isfin3-2 . Di... |
| isf32lem8 9182 | Lemma for ~ isfin3-2 . K ... |
| isf32lem9 9183 | Lemma for ~ isfin3-2 . Co... |
| isf32lem10 9184 | Lemma for isfin3-2 . Writ... |
| isf32lem11 9185 | Lemma for ~ isfin3-2 . Re... |
| isf32lem12 9186 | Lemma for ~ isfin3-2 . (C... |
| isfin32i 9187 | One half of ~ isfin3-2 . ... |
| isf33lem 9188 | Lemma for ~ isfin3-3 . (C... |
| isfin3-2 9189 | Weakly Dedekind-infinite s... |
| isfin3-3 9190 | Weakly Dedekind-infinite s... |
| fin33i 9191 | Inference from ~ isfin3-3 ... |
| compsscnvlem 9192 | Lemma for ~ compsscnv . (... |
| compsscnv 9193 | Complementation on a power... |
| isf34lem1 9194 | Lemma for ~ isfin3-4 . (C... |
| isf34lem2 9195 | Lemma for ~ isfin3-4 . (C... |
| compssiso 9196 | Complementation is an anti... |
| isf34lem3 9197 | Lemma for ~ isfin3-4 . (C... |
| compss 9198 | Express image under of the... |
| isf34lem4 9199 | Lemma for ~ isfin3-4 . (C... |
| isf34lem5 9200 | Lemma for ~ isfin3-4 . (C... |
| isf34lem7 9201 | Lemma for ~ isfin3-4 . (C... |
| isf34lem6 9202 | Lemma for ~ isfin3-4 . (C... |
| fin34i 9203 | Inference from ~ isfin3-4 ... |
| isfin3-4 9204 | Weakly Dedekind-infinite s... |
| fin11a 9205 | Every I-finite set is Ia-f... |
| enfin1ai 9206 | Ia-finiteness is a cardina... |
| isfin1-2 9207 | A set is finite in the usu... |
| isfin1-3 9208 | A set is I-finite iff ever... |
| isfin1-4 9209 | A set is I-finite iff ever... |
| dffin1-5 9210 | Compact quantifier-free ve... |
| fin23 9211 | Every II-finite set (every... |
| fin34 9212 | Every III-finite set is IV... |
| isfin5-2 9213 | Alternate definition of V-... |
| fin45 9214 | Every IV-finite set is V-f... |
| fin56 9215 | Every V-finite set is VI-f... |
| fin17 9216 | Every I-finite set is VII-... |
| fin67 9217 | Every VI-finite set is VII... |
| isfin7-2 9218 | A set is VII-finite iff it... |
| fin71num 9219 | A well-orderable set is VI... |
| dffin7-2 9220 | Class form of ~ isfin7-2 .... |
| dfacfin7 9221 | Axiom of Choice equivalent... |
| fin1a2lem1 9222 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem2 9223 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem3 9224 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem4 9225 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem5 9226 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem6 9227 | Lemma for ~ fin1a2 . Esta... |
| fin1a2lem7 9228 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem8 9229 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem9 9230 | Lemma for ~ fin1a2 . In a... |
| fin1a2lem10 9231 | Lemma for ~ fin1a2 . A no... |
| fin1a2lem11 9232 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem12 9233 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem13 9234 | Lemma for ~ fin1a2 . (Con... |
| fin12 9235 | Weak theorem which skips I... |
| fin1a2s 9236 | An II-infinite set can hav... |
| fin1a2 9237 | Every Ia-finite set is II-... |
| itunifval 9238 | Function value of iterated... |
| itunifn 9239 | Functionality of the itera... |
| ituni0 9240 | A zero-fold iterated union... |
| itunisuc 9241 | Successor iterated union. ... |
| itunitc1 9242 | Each union iterate is a me... |
| itunitc 9243 | The union of all union ite... |
| ituniiun 9244 | Unwrap an iterated union f... |
| hsmexlem7 9245 | Lemma for ~ hsmex . Prope... |
| hsmexlem8 9246 | Lemma for ~ hsmex . Prope... |
| hsmexlem9 9247 | Lemma for ~ hsmex . Prope... |
| hsmexlem1 9248 | Lemma for ~ hsmex . Bound... |
| hsmexlem2 9249 | Lemma for ~ hsmex . Bound... |
| hsmexlem3 9250 | Lemma for ~ hsmex . Clear... |
| hsmexlem4 9251 | Lemma for ~ hsmex . The c... |
| hsmexlem5 9252 | Lemma for ~ hsmex . Combi... |
| hsmexlem6 9253 | Lemma for ~ hsmex . (Cont... |
| hsmex 9254 | The collection of heredita... |
| hsmex2 9255 | The set of hereditary size... |
| hsmex3 9256 | The set of hereditary size... |
| axcc2lem 9258 | Lemma for ~ axcc2 . (Cont... |
| axcc2 9259 | A possibly more useful ver... |
| axcc3 9260 | A possibly more useful ver... |
| axcc4 9261 | A version of ~ axcc3 that ... |
| acncc 9262 | An ~ ax-cc equivalent: eve... |
| axcc4dom 9263 | Relax the constraint on ~ ... |
| domtriomlem 9264 | Lemma for ~ domtriom . (C... |
| domtriom 9265 | Trichotomy of equinumerosi... |
| fin41 9266 | Under countable choice, th... |
| dominf 9267 | A nonempty set that is a s... |
| dcomex 9269 | The Axiom of Dependent Cho... |
| axdc2lem 9270 | Lemma for ~ axdc2 . We co... |
| axdc2 9271 | An apparent strengthening ... |
| axdc3lem 9272 | The class ` S ` of finite ... |
| axdc3lem2 9273 | Lemma for ~ axdc3 . We ha... |
| axdc3lem3 9274 | Simple substitution lemma ... |
| axdc3lem4 9275 | Lemma for ~ axdc3 . We ha... |
| axdc3 9276 | Dependent Choice. Axiom D... |
| axdc4lem 9277 | Lemma for ~ axdc4 . (Cont... |
| axdc4 9278 | A more general version of ... |
| axcclem 9279 | Lemma for ~ axcc . (Contr... |
| axcc 9280 | Although CC can be proven ... |
| zfac 9282 | Axiom of Choice expressed ... |
| ac2 9283 | Axiom of Choice equivalent... |
| ac3 9284 | Axiom of Choice using abbr... |
| axac3 9286 | This theorem asserts that ... |
| ackm 9287 | A remarkable equivalent to... |
| axac2 9288 | Derive ~ ax-ac2 from ~ ax-... |
| axac 9289 | Derive ~ ax-ac from ~ ax-a... |
| axaci 9290 | Apply a choice equivalent.... |
| cardeqv 9291 | All sets are well-orderabl... |
| numth3 9292 | All sets are well-orderabl... |
| numth2 9293 | Numeration theorem: any se... |
| numth 9294 | Numeration theorem: every ... |
| ac7 9295 | An Axiom of Choice equival... |
| ac7g 9296 | An Axiom of Choice equival... |
| ac4 9297 | Equivalent of Axiom of Cho... |
| ac4c 9298 | Equivalent of Axiom of Cho... |
| ac5 9299 | An Axiom of Choice equival... |
| ac5b 9300 | Equivalent of Axiom of Cho... |
| ac6num 9301 | A version of ~ ac6 which t... |
| ac6 9302 | Equivalent of Axiom of Cho... |
| ac6c4 9303 | Equivalent of Axiom of Cho... |
| ac6c5 9304 | Equivalent of Axiom of Cho... |
| ac9 9305 | An Axiom of Choice equival... |
| ac6s 9306 | Equivalent of Axiom of Cho... |
| ac6n 9307 | Equivalent of Axiom of Cho... |
| ac6s2 9308 | Generalization of the Axio... |
| ac6s3 9309 | Generalization of the Axio... |
| ac6sg 9310 | ~ ac6s with sethood as ant... |
| ac6sf 9311 | Version of ~ ac6 with boun... |
| ac6s4 9312 | Generalization of the Axio... |
| ac6s5 9313 | Generalization of the Axio... |
| ac8 9314 | An Axiom of Choice equival... |
| ac9s 9315 | An Axiom of Choice equival... |
| numthcor 9316 | Any set is strictly domina... |
| weth 9317 | Well-ordering theorem: any... |
| zorn2lem1 9318 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem2 9319 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem3 9320 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem4 9321 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem5 9322 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem6 9323 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem7 9324 | Lemma for ~ zorn2 . (Cont... |
| zorn2g 9325 | Zorn's Lemma of [Monk1] p.... |
| zorng 9326 | Zorn's Lemma. If the unio... |
| zornn0g 9327 | Variant of Zorn's lemma ~ ... |
| zorn2 9328 | Zorn's Lemma of [Monk1] p.... |
| zorn 9329 | Zorn's Lemma. If the unio... |
| zornn0 9330 | Variant of Zorn's lemma ~ ... |
| ttukeylem1 9331 | Lemma for ~ ttukey . Expa... |
| ttukeylem2 9332 | Lemma for ~ ttukey . A pr... |
| ttukeylem3 9333 | Lemma for ~ ttukey . (Con... |
| ttukeylem4 9334 | Lemma for ~ ttukey . (Con... |
| ttukeylem5 9335 | Lemma for ~ ttukey . The ... |
| ttukeylem6 9336 | Lemma for ~ ttukey . (Con... |
| ttukeylem7 9337 | Lemma for ~ ttukey . (Con... |
| ttukey2g 9338 | The Teichmüller-Tukey... |
| ttukeyg 9339 | The Teichmüller-Tukey... |
| ttukey 9340 | The Teichmüller-Tukey... |
| axdclem 9341 | Lemma for ~ axdc . (Contr... |
| axdclem2 9342 | Lemma for ~ axdc . Using ... |
| axdc 9343 | This theorem derives ~ ax-... |
| fodom 9344 | An onto function implies d... |
| fodomg 9345 | An onto function implies d... |
| dmct 9346 | The domain of a countable ... |
| rnct 9347 | The range of a countable s... |
| fodomb 9348 | Equivalence of an onto map... |
| wdomac 9349 | When assuming AC, weak and... |
| brdom3 9350 | Equivalence to a dominance... |
| brdom5 9351 | An equivalence to a domina... |
| brdom4 9352 | An equivalence to a domina... |
| brdom7disj 9353 | An equivalence to a domina... |
| brdom6disj 9354 | An equivalence to a domina... |
| fin71ac 9355 | Once we allow AC, the "str... |
| imadomg 9356 | An image of a function und... |
| fimact 9357 | The image by a function of... |
| fnrndomg 9358 | The range of a function is... |
| fnct 9359 | If the domain of a functio... |
| mptct 9360 | A countable mapping set is... |
| iunfo 9361 | Existence of an onto funct... |
| iundom2g 9362 | An upper bound for the car... |
| iundomg 9363 | An upper bound for the car... |
| iundom 9364 | An upper bound for the car... |
| unidom 9365 | An upper bound for the car... |
| uniimadom 9366 | An upper bound for the car... |
| uniimadomf 9367 | An upper bound for the car... |
| cardval 9368 | The value of the cardinal ... |
| cardid 9369 | Any set is equinumerous to... |
| cardidg 9370 | Any set is equinumerous to... |
| cardidd 9371 | Any set is equinumerous to... |
| cardf 9372 | The cardinality function i... |
| carden 9373 | Two sets are equinumerous ... |
| cardeq0 9374 | Only the empty set has car... |
| unsnen 9375 | Equinumerosity of a set wi... |
| carddom 9376 | Two sets have the dominanc... |
| cardsdom 9377 | Two sets have the strict d... |
| domtri 9378 | Trichotomy law for dominan... |
| entric 9379 | Trichotomy of equinumerosi... |
| entri2 9380 | Trichotomy of dominance an... |
| entri3 9381 | Trichotomy of dominance. ... |
| sdomsdomcard 9382 | A set strictly dominates i... |
| canth3 9383 | Cantor's theorem in terms ... |
| infxpidm 9384 | The Cartesian product of a... |
| ondomon 9385 | The collection of ordinal ... |
| cardmin 9386 | The smallest ordinal that ... |
| ficard 9387 | A set is finite iff its ca... |
| infinf 9388 | Equivalence between two in... |
| unirnfdomd 9389 | The union of the range of ... |
| konigthlem 9390 | Lemma for ~ konigth . (Co... |
| konigth 9391 | Konig's Theorem. If ` m (... |
| alephsucpw 9392 | The power set of an aleph ... |
| aleph1 9393 | The set exponentiation of ... |
| alephval2 9394 | An alternate way to expres... |
| dominfac 9395 | A nonempty set that is a s... |
| iunctb 9396 | The countable union of cou... |
| unictb 9397 | The countable union of cou... |
| infmap 9398 | An exponentiation law for ... |
| alephadd 9399 | The sum of two alephs is t... |
| alephmul 9400 | The product of two alephs ... |
| alephexp1 9401 | An exponentiation law for ... |
| alephsuc3 9402 | An alternate representatio... |
| alephexp2 9403 | An expression equinumerous... |
| alephreg 9404 | A successor aleph is regul... |
| pwcfsdom 9405 | A corollary of Konig's The... |
| cfpwsdom 9406 | A corollary of Konig's The... |
| alephom 9407 | From ~ canth2 , we know th... |
| smobeth 9408 | The beth function is stric... |
| nd1 9409 | A lemma for proving condit... |
| nd2 9410 | A lemma for proving condit... |
| nd3 9411 | A lemma for proving condit... |
| nd4 9412 | A lemma for proving condit... |
| axextnd 9413 | A version of the Axiom of ... |
| axrepndlem1 9414 | Lemma for the Axiom of Rep... |
| axrepndlem2 9415 | Lemma for the Axiom of Rep... |
| axrepnd 9416 | A version of the Axiom of ... |
| axunndlem1 9417 | Lemma for the Axiom of Uni... |
| axunnd 9418 | A version of the Axiom of ... |
| axpowndlem1 9419 | Lemma for the Axiom of Pow... |
| axpowndlem2 9420 | Lemma for the Axiom of Pow... |
| axpowndlem3 9421 | Lemma for the Axiom of Pow... |
| axpowndlem4 9422 | Lemma for the Axiom of Pow... |
| axpownd 9423 | A version of the Axiom of ... |
| axregndlem1 9424 | Lemma for the Axiom of Reg... |
| axregndlem2 9425 | Lemma for the Axiom of Reg... |
| axregnd 9426 | A version of the Axiom of ... |
| axinfndlem1 9427 | Lemma for the Axiom of Inf... |
| axinfnd 9428 | A version of the Axiom of ... |
| axacndlem1 9429 | Lemma for the Axiom of Cho... |
| axacndlem2 9430 | Lemma for the Axiom of Cho... |
| axacndlem3 9431 | Lemma for the Axiom of Cho... |
| axacndlem4 9432 | Lemma for the Axiom of Cho... |
| axacndlem5 9433 | Lemma for the Axiom of Cho... |
| axacnd 9434 | A version of the Axiom of ... |
| zfcndext 9435 | Axiom of Extensionality ~ ... |
| zfcndrep 9436 | Axiom of Replacement ~ ax-... |
| zfcndun 9437 | Axiom of Union ~ ax-un , r... |
| zfcndpow 9438 | Axiom of Power Sets ~ ax-p... |
| zfcndreg 9439 | Axiom of Regularity ~ ax-r... |
| zfcndinf 9440 | Axiom of Infinity ~ ax-inf... |
| zfcndac 9441 | Axiom of Choice ~ ax-ac , ... |
| elgch 9444 | Elementhood in the collect... |
| fingch 9445 | A finite set is a GCH-set.... |
| gchi 9446 | The only GCH-sets which ha... |
| gchen1 9447 | If ` A <_ B < ~P A ` , and... |
| gchen2 9448 | If ` A < B <_ ~P A ` , and... |
| gchor 9449 | If ` A <_ B <_ ~P A ` , an... |
| engch 9450 | The property of being a GC... |
| gchdomtri 9451 | Under certain conditions, ... |
| fpwwe2cbv 9452 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem1 9453 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem2 9454 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem3 9455 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem5 9456 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem6 9457 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem7 9458 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem8 9459 | Lemma for ~ fpwwe2 . Show... |
| fpwwe2lem9 9460 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem10 9461 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem11 9462 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem12 9463 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem13 9464 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2 9465 | Given any function ` F ` f... |
| fpwwecbv 9466 | Lemma for ~ fpwwe . (Cont... |
| fpwwelem 9467 | Lemma for ~ fpwwe . (Cont... |
| fpwwe 9468 | Given any function ` F ` f... |
| canth4 9469 | An "effective" form of Can... |
| canthnumlem 9470 | Lemma for ~ canthnum . (C... |
| canthnum 9471 | The set of well-orderable ... |
| canthwelem 9472 | Lemma for ~ canthwe . (Co... |
| canthwe 9473 | The set of well-orders of ... |
| canthp1lem1 9474 | Lemma for ~ canthp1 . (Co... |
| canthp1lem2 9475 | Lemma for ~ canthp1 . (Co... |
| canthp1 9476 | A slightly stronger form o... |
| finngch 9477 | The exclusion of finite se... |
| gchcda1 9478 | An infinite GCH-set is ide... |
| gchinf 9479 | An infinite GCH-set is Ded... |
| pwfseqlem1 9480 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem2 9481 | Lemma for ~ pwfseq . (Con... |
| pwfseqlem3 9482 | Lemma for ~ pwfseq . Usin... |
| pwfseqlem4a 9483 | Lemma for ~ pwfseqlem4 . ... |
| pwfseqlem4 9484 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem5 9485 | Lemma for ~ pwfseq . Alth... |
| pwfseq 9486 | The powerset of a Dedekind... |
| pwxpndom2 9487 | The powerset of a Dedekind... |
| pwxpndom 9488 | The powerset of a Dedekind... |
| pwcdandom 9489 | The powerset of a Dedekind... |
| gchcdaidm 9490 | An infinite GCH-set is ide... |
| gchxpidm 9491 | An infinite GCH-set is ide... |
| gchpwdom 9492 | A relationship between dom... |
| gchaleph 9493 | If ` ( aleph `` A ) ` is a... |
| gchaleph2 9494 | If ` ( aleph `` A ) ` and ... |
| hargch 9495 | If ` A + ~~ ~P A ` , then ... |
| alephgch 9496 | If ` ( aleph `` suc A ) ` ... |
| gch2 9497 | It is sufficient to requir... |
| gch3 9498 | An equivalent formulation ... |
| gch-kn 9499 | The equivalence of two ver... |
| gchaclem 9500 | Lemma for ~ gchac (obsolet... |
| gchhar 9501 | A "local" form of ~ gchac ... |
| gchacg 9502 | A "local" form of ~ gchac ... |
| gchac 9503 | The Generalized Continuum ... |
| elwina 9508 | Conditions of weak inacces... |
| elina 9509 | Conditions of strong inacc... |
| winaon 9510 | A weakly inaccessible card... |
| inawinalem 9511 | Lemma for ~ inawina . (Co... |
| inawina 9512 | Every strongly inaccessibl... |
| omina 9513 | ` _om ` is a strongly inac... |
| winacard 9514 | A weakly inaccessible card... |
| winainflem 9515 | A weakly inaccessible card... |
| winainf 9516 | A weakly inaccessible card... |
| winalim 9517 | A weakly inaccessible card... |
| winalim2 9518 | A nontrivial weakly inacce... |
| winafp 9519 | A nontrivial weakly inacce... |
| winafpi 9520 | This theorem, which states... |
| gchina 9521 | Assuming the GCH, weakly a... |
| iswun 9526 | Properties of a weak unive... |
| wuntr 9527 | A weak universe is transit... |
| wununi 9528 | A weak universe is closed ... |
| wunpw 9529 | A weak universe is closed ... |
| wunelss 9530 | The elements of a weak uni... |
| wunpr 9531 | A weak universe is closed ... |
| wunun 9532 | A weak universe is closed ... |
| wuntp 9533 | A weak universe is closed ... |
| wunss 9534 | A weak universe is closed ... |
| wunin 9535 | A weak universe is closed ... |
| wundif 9536 | A weak universe is closed ... |
| wunint 9537 | A weak universe is closed ... |
| wunsn 9538 | A weak universe is closed ... |
| wunsuc 9539 | A weak universe is closed ... |
| wun0 9540 | A weak universe contains t... |
| wunr1om 9541 | A weak universe is infinit... |
| wunom 9542 | A weak universe contains a... |
| wunfi 9543 | A weak universe contains a... |
| wunop 9544 | A weak universe is closed ... |
| wunot 9545 | A weak universe is closed ... |
| wunxp 9546 | A weak universe is closed ... |
| wunpm 9547 | A weak universe is closed ... |
| wunmap 9548 | A weak universe is closed ... |
| wunf 9549 | A weak universe is closed ... |
| wundm 9550 | A weak universe is closed ... |
| wunrn 9551 | A weak universe is closed ... |
| wuncnv 9552 | A weak universe is closed ... |
| wunres 9553 | A weak universe is closed ... |
| wunfv 9554 | A weak universe is closed ... |
| wunco 9555 | A weak universe is closed ... |
| wuntpos 9556 | A weak universe is closed ... |
| intwun 9557 | The intersection of a coll... |
| r1limwun 9558 | Each limit stage in the cu... |
| r1wunlim 9559 | The weak universes in the ... |
| wunex2 9560 | Construct a weak universe ... |
| wunex 9561 | Construct a weak universe ... |
| uniwun 9562 | Every set is contained in ... |
| wunex3 9563 | Construct a weak universe ... |
| wuncval 9564 | Value of the weak universe... |
| wuncid 9565 | The weak universe closure ... |
| wunccl 9566 | The weak universe closure ... |
| wuncss 9567 | The weak universe closure ... |
| wuncidm 9568 | The weak universe closure ... |
| wuncval2 9569 | Our earlier expression for... |
| eltskg 9572 | Properties of a Tarski cla... |
| eltsk2g 9573 | Properties of a Tarski cla... |
| tskpwss 9574 | First axiom of a Tarski cl... |
| tskpw 9575 | Second axiom of a Tarski c... |
| tsken 9576 | Third axiom of a Tarski cl... |
| 0tsk 9577 | The empty set is a (transi... |
| tsksdom 9578 | An element of a Tarski cla... |
| tskssel 9579 | A part of a Tarski class s... |
| tskss 9580 | The subsets of an element ... |
| tskin 9581 | The intersection of two el... |
| tsksn 9582 | A singleton of an element ... |
| tsktrss 9583 | A transitive element of a ... |
| tsksuc 9584 | If an element of a Tarski ... |
| tsk0 9585 | A nonempty Tarski class co... |
| tsk1 9586 | One is an element of a non... |
| tsk2 9587 | Two is an element of a non... |
| 2domtsk 9588 | If a Tarski class is not e... |
| tskr1om 9589 | A nonempty Tarski class is... |
| tskr1om2 9590 | A nonempty Tarski class co... |
| tskinf 9591 | A nonempty Tarski class is... |
| tskpr 9592 | If ` A ` and ` B ` are mem... |
| tskop 9593 | If ` A ` and ` B ` are mem... |
| tskxpss 9594 | A Cartesian product of two... |
| tskwe2 9595 | A Tarski class is well-ord... |
| inttsk 9596 | The intersection of a coll... |
| inar1 9597 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omALT 9598 | Alternate proof of ~ r1om ... |
| rankcf 9599 | Any set must be at least a... |
| inatsk 9600 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omtsk 9601 | The set of hereditarily fi... |
| tskord 9602 | A Tarski class contains al... |
| tskcard 9603 | An even more direct relati... |
| r1tskina 9604 | There is a direct relation... |
| tskuni 9605 | The union of an element of... |
| tskwun 9606 | A nonempty transitive Tars... |
| tskint 9607 | The intersection of an ele... |
| tskun 9608 | The union of two elements ... |
| tskxp 9609 | The Cartesian product of t... |
| tskmap 9610 | Set exponentiation is an e... |
| tskurn 9611 | A transitive Tarski class ... |
| elgrug 9614 | Properties of a Grothendie... |
| grutr 9615 | A Grothendieck universe is... |
| gruelss 9616 | A Grothendieck universe is... |
| grupw 9617 | A Grothendieck universe co... |
| gruss 9618 | Any subset of an element o... |
| grupr 9619 | A Grothendieck universe co... |
| gruurn 9620 | A Grothendieck universe co... |
| gruiun 9621 | If ` B ( x ) ` is a family... |
| gruuni 9622 | A Grothendieck universe co... |
| grurn 9623 | A Grothendieck universe co... |
| gruima 9624 | A Grothendieck universe co... |
| gruel 9625 | Any element of an element ... |
| grusn 9626 | A Grothendieck universe co... |
| gruop 9627 | A Grothendieck universe co... |
| gruun 9628 | A Grothendieck universe co... |
| gruxp 9629 | A Grothendieck universe co... |
| grumap 9630 | A Grothendieck universe co... |
| gruixp 9631 | A Grothendieck universe co... |
| gruiin 9632 | A Grothendieck universe co... |
| gruf 9633 | A Grothendieck universe co... |
| gruen 9634 | A Grothendieck universe co... |
| gruwun 9635 | A nonempty Grothendieck un... |
| intgru 9636 | The intersection of a fami... |
| ingru 9637 | The intersection of a univ... |
| wfgru 9638 | The wellfounded part of a ... |
| grudomon 9639 | Each ordinal that is compa... |
| gruina 9640 | If a Grothendieck universe... |
| grur1a 9641 | A characterization of Grot... |
| grur1 9642 | A characterization of Grot... |
| grutsk1 9643 | Grothendieck universes are... |
| grutsk 9644 | Grothendieck universes are... |
| axgroth5 9646 | The Tarski-Grothendieck ax... |
| axgroth2 9647 | Alternate version of the T... |
| grothpw 9648 | Derive the Axiom of Power ... |
| grothpwex 9649 | Derive the Axiom of Power ... |
| axgroth6 9650 | The Tarski-Grothendieck ax... |
| grothomex 9651 | The Tarski-Grothendieck Ax... |
| grothac 9652 | The Tarski-Grothendieck Ax... |
| axgroth3 9653 | Alternate version of the T... |
| axgroth4 9654 | Alternate version of the T... |
| grothprimlem 9655 | Lemma for ~ grothprim . E... |
| grothprim 9656 | The Tarski-Grothendieck Ax... |
| grothtsk 9657 | The Tarski-Grothendieck Ax... |
| inaprc 9658 | An equivalent to the Tarsk... |
| tskmval 9661 | Value of our tarski map. ... |
| tskmid 9662 | The set ` A ` is an elemen... |
| tskmcl 9663 | A Tarski class that contai... |
| sstskm 9664 | Being a part of ` ( tarski... |
| eltskm 9665 | Belonging to ` ( tarskiMap... |
| elni 9698 | Membership in the class of... |
| elni2 9699 | Membership in the class of... |
| pinn 9700 | A positive integer is a na... |
| pion 9701 | A positive integer is an o... |
| piord 9702 | A positive integer is ordi... |
| niex 9703 | The class of positive inte... |
| 0npi 9704 | The empty set is not a pos... |
| 1pi 9705 | Ordinal 'one' is a positiv... |
| addpiord 9706 | Positive integer addition ... |
| mulpiord 9707 | Positive integer multiplic... |
| mulidpi 9708 | 1 is an identity element f... |
| ltpiord 9709 | Positive integer 'less tha... |
| ltsopi 9710 | Positive integer 'less tha... |
| ltrelpi 9711 | Positive integer 'less tha... |
| dmaddpi 9712 | Domain of addition on posi... |
| dmmulpi 9713 | Domain of multiplication o... |
| addclpi 9714 | Closure of addition of pos... |
| mulclpi 9715 | Closure of multiplication ... |
| addcompi 9716 | Addition of positive integ... |
| addasspi 9717 | Addition of positive integ... |
| mulcompi 9718 | Multiplication of positive... |
| mulasspi 9719 | Multiplication of positive... |
| distrpi 9720 | Multiplication of positive... |
| addcanpi 9721 | Addition cancellation law ... |
| mulcanpi 9722 | Multiplication cancellatio... |
| addnidpi 9723 | There is no identity eleme... |
| ltexpi 9724 | Ordering on positive integ... |
| ltapi 9725 | Ordering property of addit... |
| ltmpi 9726 | Ordering property of multi... |
| 1lt2pi 9727 | One is less than two (one ... |
| nlt1pi 9728 | No positive integer is les... |
| indpi 9729 | Principle of Finite Induct... |
| enqbreq 9741 | Equivalence relation for p... |
| enqbreq2 9742 | Equivalence relation for p... |
| enqer 9743 | The equivalence relation f... |
| enqex 9744 | The equivalence relation f... |
| nqex 9745 | The class of positive frac... |
| 0nnq 9746 | The empty set is not a pos... |
| elpqn 9747 | Each positive fraction is ... |
| ltrelnq 9748 | Positive fraction 'less th... |
| pinq 9749 | The representatives of pos... |
| 1nq 9750 | The positive fraction 'one... |
| nqereu 9751 | There is a unique element ... |
| nqerf 9752 | Corollary of ~ nqereu : th... |
| nqercl 9753 | Corollary of ~ nqereu : cl... |
| nqerrel 9754 | Any member of ` ( N. X. N.... |
| nqerid 9755 | Corollary of ~ nqereu : th... |
| enqeq 9756 | Corollary of ~ nqereu : if... |
| nqereq 9757 | The function ` /Q ` acts a... |
| addpipq2 9758 | Addition of positive fract... |
| addpipq 9759 | Addition of positive fract... |
| addpqnq 9760 | Addition of positive fract... |
| mulpipq2 9761 | Multiplication of positive... |
| mulpipq 9762 | Multiplication of positive... |
| mulpqnq 9763 | Multiplication of positive... |
| ordpipq 9764 | Ordering of positive fract... |
| ordpinq 9765 | Ordering of positive fract... |
| addpqf 9766 | Closure of addition on pos... |
| addclnq 9767 | Closure of addition on pos... |
| mulpqf 9768 | Closure of multiplication ... |
| mulclnq 9769 | Closure of multiplication ... |
| addnqf 9770 | Domain of addition on posi... |
| mulnqf 9771 | Domain of multiplication o... |
| addcompq 9772 | Addition of positive fract... |
| addcomnq 9773 | Addition of positive fract... |
| mulcompq 9774 | Multiplication of positive... |
| mulcomnq 9775 | Multiplication of positive... |
| adderpqlem 9776 | Lemma for ~ adderpq . (Co... |
| mulerpqlem 9777 | Lemma for ~ mulerpq . (Co... |
| adderpq 9778 | Addition is compatible wit... |
| mulerpq 9779 | Multiplication is compatib... |
| addassnq 9780 | Addition of positive fract... |
| mulassnq 9781 | Multiplication of positive... |
| mulcanenq 9782 | Lemma for distributive law... |
| distrnq 9783 | Multiplication of positive... |
| 1nqenq 9784 | The equivalence class of r... |
| mulidnq 9785 | Multiplication identity el... |
| recmulnq 9786 | Relationship between recip... |
| recidnq 9787 | A positive fraction times ... |
| recclnq 9788 | Closure law for positive f... |
| recrecnq 9789 | Reciprocal of reciprocal o... |
| dmrecnq 9790 | Domain of reciprocal on po... |
| ltsonq 9791 | 'Less than' is a strict or... |
| lterpq 9792 | Compatibility of ordering ... |
| ltanq 9793 | Ordering property of addit... |
| ltmnq 9794 | Ordering property of multi... |
| 1lt2nq 9795 | One is less than two (one ... |
| ltaddnq 9796 | The sum of two fractions i... |
| ltexnq 9797 | Ordering on positive fract... |
| halfnq 9798 | One-half of any positive f... |
| nsmallnq 9799 | The is no smallest positiv... |
| ltbtwnnq 9800 | There exists a number betw... |
| ltrnq 9801 | Ordering property of recip... |
| archnq 9802 | For any fraction, there is... |
| npex 9808 | The class of positive real... |
| elnp 9809 | Membership in positive rea... |
| elnpi 9810 | Membership in positive rea... |
| prn0 9811 | A positive real is not emp... |
| prpssnq 9812 | A positive real is a subse... |
| elprnq 9813 | A positive real is a set o... |
| 0npr 9814 | The empty set is not a pos... |
| prcdnq 9815 | A positive real is closed ... |
| prub 9816 | A positive fraction not in... |
| prnmax 9817 | A positive real has no lar... |
| npomex 9818 | A simplifying observation,... |
| prnmadd 9819 | A positive real has no lar... |
| ltrelpr 9820 | Positive real 'less than' ... |
| genpv 9821 | Value of general operation... |
| genpelv 9822 | Membership in value of gen... |
| genpprecl 9823 | Pre-closure law for genera... |
| genpdm 9824 | Domain of general operatio... |
| genpn0 9825 | The result of an operation... |
| genpss 9826 | The result of an operation... |
| genpnnp 9827 | The result of an operation... |
| genpcd 9828 | Downward closure of an ope... |
| genpnmax 9829 | An operation on positive r... |
| genpcl 9830 | Closure of an operation on... |
| genpass 9831 | Associativity of an operat... |
| plpv 9832 | Value of addition on posit... |
| mpv 9833 | Value of multiplication on... |
| dmplp 9834 | Domain of addition on posi... |
| dmmp 9835 | Domain of multiplication o... |
| nqpr 9836 | The canonical embedding of... |
| 1pr 9837 | The positive real number '... |
| addclprlem1 9838 | Lemma to prove downward cl... |
| addclprlem2 9839 | Lemma to prove downward cl... |
| addclpr 9840 | Closure of addition on pos... |
| mulclprlem 9841 | Lemma to prove downward cl... |
| mulclpr 9842 | Closure of multiplication ... |
| addcompr 9843 | Addition of positive reals... |
| addasspr 9844 | Addition of positive reals... |
| mulcompr 9845 | Multiplication of positive... |
| mulasspr 9846 | Multiplication of positive... |
| distrlem1pr 9847 | Lemma for distributive law... |
| distrlem4pr 9848 | Lemma for distributive law... |
| distrlem5pr 9849 | Lemma for distributive law... |
| distrpr 9850 | Multiplication of positive... |
| 1idpr 9851 | 1 is an identity element f... |
| ltprord 9852 | Positive real 'less than' ... |
| psslinpr 9853 | Proper subset is a linear ... |
| ltsopr 9854 | Positive real 'less than' ... |
| prlem934 9855 | Lemma 9-3.4 of [Gleason] p... |
| ltaddpr 9856 | The sum of two positive re... |
| ltaddpr2 9857 | The sum of two positive re... |
| ltexprlem1 9858 | Lemma for Proposition 9-3.... |
| ltexprlem2 9859 | Lemma for Proposition 9-3.... |
| ltexprlem3 9860 | Lemma for Proposition 9-3.... |
| ltexprlem4 9861 | Lemma for Proposition 9-3.... |
| ltexprlem5 9862 | Lemma for Proposition 9-3.... |
| ltexprlem6 9863 | Lemma for Proposition 9-3.... |
| ltexprlem7 9864 | Lemma for Proposition 9-3.... |
| ltexpri 9865 | Proposition 9-3.5(iv) of [... |
| ltaprlem 9866 | Lemma for Proposition 9-3.... |
| ltapr 9867 | Ordering property of addit... |
| addcanpr 9868 | Addition cancellation law ... |
| prlem936 9869 | Lemma 9-3.6 of [Gleason] p... |
| reclem2pr 9870 | Lemma for Proposition 9-3.... |
| reclem3pr 9871 | Lemma for Proposition 9-3.... |
| reclem4pr 9872 | Lemma for Proposition 9-3.... |
| recexpr 9873 | The reciprocal of a positi... |
| suplem1pr 9874 | The union of a nonempty, b... |
| suplem2pr 9875 | The union of a set of posi... |
| supexpr 9876 | The union of a nonempty, b... |
| enrbreq 9885 | Equivalence relation for s... |
| enrer 9886 | The equivalence relation f... |
| enreceq 9887 | Equivalence class equality... |
| enrex 9888 | The equivalence relation f... |
| ltrelsr 9889 | Signed real 'less than' is... |
| addcmpblnr 9890 | Lemma showing compatibilit... |
| mulcmpblnrlem 9891 | Lemma used in lemma showin... |
| mulcmpblnr 9892 | Lemma showing compatibilit... |
| prsrlem1 9893 | Decomposing signed reals i... |
| addsrmo 9894 | There is at most one resul... |
| mulsrmo 9895 | There is at most one resul... |
| addsrpr 9896 | Addition of signed reals i... |
| mulsrpr 9897 | Multiplication of signed r... |
| ltsrpr 9898 | Ordering of signed reals i... |
| gt0srpr 9899 | Greater than zero in terms... |
| 0nsr 9900 | The empty set is not a sig... |
| 0r 9901 | The constant ` 0R ` is a s... |
| 1sr 9902 | The constant ` 1R ` is a s... |
| m1r 9903 | The constant ` -1R ` is a ... |
| addclsr 9904 | Closure of addition on sig... |
| mulclsr 9905 | Closure of multiplication ... |
| dmaddsr 9906 | Domain of addition on sign... |
| dmmulsr 9907 | Domain of multiplication o... |
| addcomsr 9908 | Addition of signed reals i... |
| addasssr 9909 | Addition of signed reals i... |
| mulcomsr 9910 | Multiplication of signed r... |
| mulasssr 9911 | Multiplication of signed r... |
| distrsr 9912 | Multiplication of signed r... |
| m1p1sr 9913 | Minus one plus one is zero... |
| m1m1sr 9914 | Minus one times minus one ... |
| ltsosr 9915 | Signed real 'less than' is... |
| 0lt1sr 9916 | 0 is less than 1 for signe... |
| 1ne0sr 9917 | 1 and 0 are distinct for s... |
| 0idsr 9918 | The signed real number 0 i... |
| 1idsr 9919 | 1 is an identity element f... |
| 00sr 9920 | A signed real times 0 is 0... |
| ltasr 9921 | Ordering property of addit... |
| pn0sr 9922 | A signed real plus its neg... |
| negexsr 9923 | Existence of negative sign... |
| recexsrlem 9924 | The reciprocal of a positi... |
| addgt0sr 9925 | The sum of two positive si... |
| mulgt0sr 9926 | The product of two positiv... |
| sqgt0sr 9927 | The square of a nonzero si... |
| recexsr 9928 | The reciprocal of a nonzer... |
| mappsrpr 9929 | Mapping from positive sign... |
| ltpsrpr 9930 | Mapping of order from posi... |
| map2psrpr 9931 | Equivalence for positive s... |
| supsrlem 9932 | Lemma for supremum theorem... |
| supsr 9933 | A nonempty, bounded set of... |
| opelcn 9950 | Ordered pair membership in... |
| opelreal 9951 | Ordered pair membership in... |
| elreal 9952 | Membership in class of rea... |
| elreal2 9953 | Ordered pair membership in... |
| 0ncn 9954 | The empty set is not a com... |
| ltrelre 9955 | 'Less than' is a relation ... |
| addcnsr 9956 | Addition of complex number... |
| mulcnsr 9957 | Multiplication of complex ... |
| eqresr 9958 | Equality of real numbers i... |
| addresr 9959 | Addition of real numbers i... |
| mulresr 9960 | Multiplication of real num... |
| ltresr 9961 | Ordering of real subset of... |
| ltresr2 9962 | Ordering of real subset of... |
| dfcnqs 9963 | Technical trick to permit ... |
| addcnsrec 9964 | Technical trick to permit ... |
| mulcnsrec 9965 | Technical trick to permit ... |
| axaddf 9966 | Addition is an operation o... |
| axmulf 9967 | Multiplication is an opera... |
| axcnex 9968 | The complex numbers form a... |
| axresscn 9969 | The real numbers are a sub... |
| ax1cn 9970 | 1 is a complex number. Ax... |
| axicn 9971 | ` _i ` is a complex number... |
| axaddcl 9972 | Closure law for addition o... |
| axaddrcl 9973 | Closure law for addition i... |
| axmulcl 9974 | Closure law for multiplica... |
| axmulrcl 9975 | Closure law for multiplica... |
| axmulcom 9976 | Multiplication of complex ... |
| axaddass 9977 | Addition of complex number... |
| axmulass 9978 | Multiplication of complex ... |
| axdistr 9979 | Distributive law for compl... |
| axi2m1 9980 | i-squared equals -1 (expre... |
| ax1ne0 9981 | 1 and 0 are distinct. Axi... |
| ax1rid 9982 | ` 1 ` is an identity eleme... |
| axrnegex 9983 | Existence of negative of r... |
| axrrecex 9984 | Existence of reciprocal of... |
| axcnre 9985 | A complex number can be ex... |
| axpre-lttri 9986 | Ordering on reals satisfie... |
| axpre-lttrn 9987 | Ordering on reals is trans... |
| axpre-ltadd 9988 | Ordering property of addit... |
| axpre-mulgt0 9989 | The product of two positiv... |
| axpre-sup 9990 | A nonempty, bounded-above ... |
| wuncn 9991 | A weak universe containing... |
| cnex 10017 | Alias for ~ ax-cnex . See... |
| addcl 10018 | Alias for ~ ax-addcl , for... |
| readdcl 10019 | Alias for ~ ax-addrcl , fo... |
| mulcl 10020 | Alias for ~ ax-mulcl , for... |
| remulcl 10021 | Alias for ~ ax-mulrcl , fo... |
| mulcom 10022 | Alias for ~ ax-mulcom , fo... |
| addass 10023 | Alias for ~ ax-addass , fo... |
| mulass 10024 | Alias for ~ ax-mulass , fo... |
| adddi 10025 | Alias for ~ ax-distr , for... |
| recn 10026 | A real number is a complex... |
| reex 10027 | The real numbers form a se... |
| reelprrecn 10028 | Reals are a subset of the ... |
| cnelprrecn 10029 | Complex numbers are a subs... |
| elimne0 10030 | Hypothesis for weak deduct... |
| adddir 10031 | Distributive law for compl... |
| 0cn 10032 | 0 is a complex number. Se... |
| 0cnd 10033 | 0 is a complex number, ded... |
| c0ex 10034 | 0 is a set (common case). ... |
| 1ex 10035 | 1 is a set. Common specia... |
| cnre 10036 | Alias for ~ ax-cnre , for ... |
| mulid1 10037 | ` 1 ` is an identity eleme... |
| mulid2 10038 | Identity law for multiplic... |
| 1re 10039 | ` 1 ` is a real number. T... |
| 0re 10040 | ` 0 ` is a real number. S... |
| 0red 10041 | ` 0 ` is a real number, de... |
| mulid1i 10042 | Identity law for multiplic... |
| mulid2i 10043 | Identity law for multiplic... |
| addcli 10044 | Closure law for addition. ... |
| mulcli 10045 | Closure law for multiplica... |
| mulcomi 10046 | Commutative law for multip... |
| mulcomli 10047 | Commutative law for multip... |
| addassi 10048 | Associative law for additi... |
| mulassi 10049 | Associative law for multip... |
| adddii 10050 | Distributive law (left-dis... |
| adddiri 10051 | Distributive law (right-di... |
| recni 10052 | A real number is a complex... |
| readdcli 10053 | Closure law for addition o... |
| remulcli 10054 | Closure law for multiplica... |
| 1red 10055 | 1 is an real number, deduc... |
| 1cnd 10056 | 1 is a complex number, ded... |
| mulid1d 10057 | Identity law for multiplic... |
| mulid2d 10058 | Identity law for multiplic... |
| addcld 10059 | Closure law for addition. ... |
| mulcld 10060 | Closure law for multiplica... |
| mulcomd 10061 | Commutative law for multip... |
| addassd 10062 | Associative law for additi... |
| mulassd 10063 | Associative law for multip... |
| adddid 10064 | Distributive law (left-dis... |
| adddird 10065 | Distributive law (right-di... |
| adddirp1d 10066 | Distributive law, plus 1 v... |
| joinlmuladdmuld 10067 | Join AB+CB into (A+C) on L... |
| recnd 10068 | Deduction from real number... |
| readdcld 10069 | Closure law for addition o... |
| remulcld 10070 | Closure law for multiplica... |
| pnfnre 10081 | Plus infinity is not a rea... |
| mnfnre 10082 | Minus infinity is not a re... |
| ressxr 10083 | The standard reals are a s... |
| rexpssxrxp 10084 | The Cartesian product of s... |
| rexr 10085 | A standard real is an exte... |
| 0xr 10086 | Zero is an extended real. ... |
| renepnf 10087 | No (finite) real equals pl... |
| renemnf 10088 | No real equals minus infin... |
| rexrd 10089 | A standard real is an exte... |
| renepnfd 10090 | No (finite) real equals pl... |
| renemnfd 10091 | No real equals minus infin... |
| pnfxr 10092 | Plus infinity belongs to t... |
| pnfex 10093 | Plus infinity exists (comm... |
| pnfnemnf 10094 | Plus and minus infinity ar... |
| mnfnepnf 10095 | Minus and plus infinity ar... |
| mnfxr 10096 | Minus infinity belongs to ... |
| rexri 10097 | A standard real is an exte... |
| renfdisj 10098 | The reals and the infiniti... |
| ltrelxr 10099 | 'Less than' is a relation ... |
| ltrel 10100 | 'Less than' is a relation.... |
| lerelxr 10101 | 'Less than or equal' is a ... |
| lerel 10102 | 'Less or equal to' is a re... |
| xrlenlt 10103 | 'Less than or equal to' ex... |
| xrlenltd 10104 | 'Less than or equal to' ex... |
| xrltnle 10105 | 'Less than' expressed in t... |
| xrnltled 10106 | 'Not less than ' implies '... |
| ssxr 10107 | The three (non-exclusive) ... |
| ltxrlt 10108 | The standard less-than ` <... |
| axlttri 10109 | Ordering on reals satisfie... |
| axlttrn 10110 | Ordering on reals is trans... |
| axltadd 10111 | Ordering property of addit... |
| axmulgt0 10112 | The product of two positiv... |
| axsup 10113 | A nonempty, bounded-above ... |
| lttr 10114 | Alias for ~ axlttrn , for ... |
| mulgt0 10115 | The product of two positiv... |
| lenlt 10116 | 'Less than or equal to' ex... |
| ltnle 10117 | 'Less than' expressed in t... |
| ltso 10118 | 'Less than' is a strict or... |
| gtso 10119 | 'Greater than' is a strict... |
| lttri2 10120 | Consequence of trichotomy.... |
| lttri3 10121 | Trichotomy law for 'less t... |
| lttri4 10122 | Trichotomy law for 'less t... |
| letri3 10123 | Trichotomy law. (Contribu... |
| leloe 10124 | 'Less than or equal to' ex... |
| eqlelt 10125 | Equality in terms of 'less... |
| ltle 10126 | 'Less than' implies 'less ... |
| leltne 10127 | 'Less than or equal to' im... |
| lelttr 10128 | Transitive law. (Contribu... |
| ltletr 10129 | Transitive law. (Contribu... |
| ltleletr 10130 | Transitive law, weaker for... |
| letr 10131 | Transitive law. (Contribu... |
| ltnr 10132 | 'Less than' is irreflexive... |
| leid 10133 | 'Less than or equal to' is... |
| ltne 10134 | 'Less than' implies not eq... |
| ltnsym 10135 | 'Less than' is not symmetr... |
| ltnsym2 10136 | 'Less than' is antisymmetr... |
| letric 10137 | Trichotomy law. (Contribu... |
| ltlen 10138 | 'Less than' expressed in t... |
| eqle 10139 | Equality implies 'less tha... |
| eqled 10140 | Equality implies 'less tha... |
| ltadd2 10141 | Addition to both sides of ... |
| ne0gt0 10142 | A nonzero nonnegative numb... |
| lecasei 10143 | Ordering elimination by ca... |
| lelttric 10144 | Trichotomy law. (Contribu... |
| ltlecasei 10145 | Ordering elimination by ca... |
| ltnri 10146 | 'Less than' is irreflexive... |
| eqlei 10147 | Equality implies 'less tha... |
| eqlei2 10148 | Equality implies 'less tha... |
| gtneii 10149 | 'Less than' implies not eq... |
| ltneii 10150 | 'Greater than' implies not... |
| lttri2i 10151 | Consequence of trichotomy.... |
| lttri3i 10152 | Consequence of trichotomy.... |
| letri3i 10153 | Consequence of trichotomy.... |
| leloei 10154 | 'Less than or equal to' in... |
| ltleni 10155 | 'Less than' expressed in t... |
| ltnsymi 10156 | 'Less than' is not symmetr... |
| lenlti 10157 | 'Less than or equal to' in... |
| ltnlei 10158 | 'Less than' in terms of 'l... |
| ltlei 10159 | 'Less than' implies 'less ... |
| ltleii 10160 | 'Less than' implies 'less ... |
| ltnei 10161 | 'Less than' implies not eq... |
| letrii 10162 | Trichotomy law for 'less t... |
| lttri 10163 | 'Less than' is transitive.... |
| lelttri 10164 | 'Less than or equal to', '... |
| ltletri 10165 | 'Less than', 'less than or... |
| letri 10166 | 'Less than or equal to' is... |
| le2tri3i 10167 | Extended trichotomy law fo... |
| ltadd2i 10168 | Addition to both sides of ... |
| mulgt0i 10169 | The product of two positiv... |
| mulgt0ii 10170 | The product of two positiv... |
| ltnrd 10171 | 'Less than' is irreflexive... |
| gtned 10172 | 'Less than' implies not eq... |
| ltned 10173 | 'Greater than' implies not... |
| ne0gt0d 10174 | A nonzero nonnegative numb... |
| lttrid 10175 | Ordering on reals satisfie... |
| lttri2d 10176 | Consequence of trichotomy.... |
| lttri3d 10177 | Consequence of trichotomy.... |
| lttri4d 10178 | Trichotomy law for 'less t... |
| letri3d 10179 | Consequence of trichotomy.... |
| leloed 10180 | 'Less than or equal to' in... |
| eqleltd 10181 | Equality in terms of 'less... |
| ltlend 10182 | 'Less than' expressed in t... |
| lenltd 10183 | 'Less than or equal to' in... |
| ltnled 10184 | 'Less than' in terms of 'l... |
| ltled 10185 | 'Less than' implies 'less ... |
| ltnsymd 10186 | 'Less than' implies 'less ... |
| nltled 10187 | 'Not less than ' implies '... |
| lensymd 10188 | 'Less than or equal to' im... |
| letrid 10189 | Trichotomy law for 'less t... |
| leltned 10190 | 'Less than or equal to' im... |
| leneltd 10191 | 'Less than or equal to' an... |
| mulgt0d 10192 | The product of two positiv... |
| ltadd2d 10193 | Addition to both sides of ... |
| letrd 10194 | Transitive law deduction f... |
| lelttrd 10195 | Transitive law deduction f... |
| ltadd2dd 10196 | Addition to both sides of ... |
| ltletrd 10197 | Transitive law deduction f... |
| lttrd 10198 | Transitive law deduction f... |
| lelttrdi 10199 | If a number is less than a... |
| dedekind 10200 | The Dedekind cut theorem. ... |
| dedekindle 10201 | The Dedekind cut theorem, ... |
| mul12 10202 | Commutative/associative la... |
| mul32 10203 | Commutative/associative la... |
| mul31 10204 | Commutative/associative la... |
| mul4 10205 | Rearrangement of 4 factors... |
| muladd11 10206 | A simple product of sums e... |
| 1p1times 10207 | Two times a number. (Cont... |
| peano2cn 10208 | A theorem for complex numb... |
| peano2re 10209 | A theorem for reals analog... |
| readdcan 10210 | Cancellation law for addit... |
| 00id 10211 | ` 0 ` is its own additive ... |
| mul02lem1 10212 | Lemma for ~ mul02 . If an... |
| mul02lem2 10213 | Lemma for ~ mul02 . Zero ... |
| mul02 10214 | Multiplication by ` 0 ` . ... |
| mul01 10215 | Multiplication by ` 0 ` . ... |
| addid1 10216 | ` 0 ` is an additive ident... |
| cnegex 10217 | Existence of the negative ... |
| cnegex2 10218 | Existence of a left invers... |
| addid2 10219 | ` 0 ` is a left identity f... |
| addcan 10220 | Cancellation law for addit... |
| addcan2 10221 | Cancellation law for addit... |
| addcom 10222 | Addition commutes. This u... |
| addid1i 10223 | ` 0 ` is an additive ident... |
| addid2i 10224 | ` 0 ` is a left identity f... |
| mul02i 10225 | Multiplication by 0. Theo... |
| mul01i 10226 | Multiplication by ` 0 ` . ... |
| addcomi 10227 | Addition commutes. Based ... |
| addcomli 10228 | Addition commutes. (Contr... |
| addcani 10229 | Cancellation law for addit... |
| addcan2i 10230 | Cancellation law for addit... |
| mul12i 10231 | Commutative/associative la... |
| mul32i 10232 | Commutative/associative la... |
| mul4i 10233 | Rearrangement of 4 factors... |
| mul02d 10234 | Multiplication by 0. Theo... |
| mul01d 10235 | Multiplication by ` 0 ` . ... |
| addid1d 10236 | ` 0 ` is an additive ident... |
| addid2d 10237 | ` 0 ` is a left identity f... |
| addcomd 10238 | Addition commutes. Based ... |
| addcand 10239 | Cancellation law for addit... |
| addcan2d 10240 | Cancellation law for addit... |
| addcanad 10241 | Cancelling a term on the l... |
| addcan2ad 10242 | Cancelling a term on the r... |
| addneintrd 10243 | Introducing a term on the ... |
| addneintr2d 10244 | Introducing a term on the ... |
| mul12d 10245 | Commutative/associative la... |
| mul32d 10246 | Commutative/associative la... |
| mul31d 10247 | Commutative/associative la... |
| mul4d 10248 | Rearrangement of 4 factors... |
| muladd11r 10249 | A simple product of sums e... |
| comraddd 10250 | Commute RHS addition, in d... |
| ltaddneg 10251 | Adding a negative number t... |
| ltaddnegr 10252 | Adding a negative number t... |
| add12 10253 | Commutative/associative la... |
| add32 10254 | Commutative/associative la... |
| add32r 10255 | Commutative/associative la... |
| add4 10256 | Rearrangement of 4 terms i... |
| add42 10257 | Rearrangement of 4 terms i... |
| add12i 10258 | Commutative/associative la... |
| add32i 10259 | Commutative/associative la... |
| add4i 10260 | Rearrangement of 4 terms i... |
| add42i 10261 | Rearrangement of 4 terms i... |
| add12d 10262 | Commutative/associative la... |
| add32d 10263 | Commutative/associative la... |
| add4d 10264 | Rearrangement of 4 terms i... |
| add42d 10265 | Rearrangement of 4 terms i... |
| 0cnALT 10270 | Alternate proof of ~ 0cn w... |
| negeu 10271 | Existential uniqueness of ... |
| subval 10272 | Value of subtraction, whic... |
| negeq 10273 | Equality theorem for negat... |
| negeqi 10274 | Equality inference for neg... |
| negeqd 10275 | Equality deduction for neg... |
| nfnegd 10276 | Deduction version of ~ nfn... |
| nfneg 10277 | Bound-variable hypothesis ... |
| csbnegg 10278 | Move class substitution in... |
| negex 10279 | A negative is a set. (Con... |
| subcl 10280 | Closure law for subtractio... |
| negcl 10281 | Closure law for negative. ... |
| negicn 10282 | ` -u _i ` is a complex num... |
| subf 10283 | Subtraction is an operatio... |
| subadd 10284 | Relationship between subtr... |
| subadd2 10285 | Relationship between subtr... |
| subsub23 10286 | Swap subtrahend and result... |
| pncan 10287 | Cancellation law for subtr... |
| pncan2 10288 | Cancellation law for subtr... |
| pncan3 10289 | Subtraction and addition o... |
| npcan 10290 | Cancellation law for subtr... |
| addsubass 10291 | Associative-type law for a... |
| addsub 10292 | Law for addition and subtr... |
| subadd23 10293 | Commutative/associative la... |
| addsub12 10294 | Commutative/associative la... |
| 2addsub 10295 | Law for subtraction and ad... |
| addsubeq4 10296 | Relation between sums and ... |
| pncan3oi 10297 | Subtraction and addition o... |
| mvrraddi 10298 | Move RHS right addition to... |
| mvlladdi 10299 | Move LHS left addition to ... |
| subid 10300 | Subtraction of a number fr... |
| subid1 10301 | Identity law for subtracti... |
| npncan 10302 | Cancellation law for subtr... |
| nppcan 10303 | Cancellation law for subtr... |
| nnpcan 10304 | Cancellation law for subtr... |
| nppcan3 10305 | Cancellation law for subtr... |
| subcan2 10306 | Cancellation law for subtr... |
| subeq0 10307 | If the difference between ... |
| npncan2 10308 | Cancellation law for subtr... |
| subsub2 10309 | Law for double subtraction... |
| nncan 10310 | Cancellation law for subtr... |
| subsub 10311 | Law for double subtraction... |
| nppcan2 10312 | Cancellation law for subtr... |
| subsub3 10313 | Law for double subtraction... |
| subsub4 10314 | Law for double subtraction... |
| sub32 10315 | Swap the second and third ... |
| nnncan 10316 | Cancellation law for subtr... |
| nnncan1 10317 | Cancellation law for subtr... |
| nnncan2 10318 | Cancellation law for subtr... |
| npncan3 10319 | Cancellation law for subtr... |
| pnpcan 10320 | Cancellation law for mixed... |
| pnpcan2 10321 | Cancellation law for mixed... |
| pnncan 10322 | Cancellation law for mixed... |
| ppncan 10323 | Cancellation law for mixed... |
| addsub4 10324 | Rearrangement of 4 terms i... |
| subadd4 10325 | Rearrangement of 4 terms i... |
| sub4 10326 | Rearrangement of 4 terms i... |
| neg0 10327 | Minus 0 equals 0. (Contri... |
| negid 10328 | Addition of a number and i... |
| negsub 10329 | Relationship between subtr... |
| subneg 10330 | Relationship between subtr... |
| negneg 10331 | A number is equal to the n... |
| neg11 10332 | Negative is one-to-one. (... |
| negcon1 10333 | Negative contraposition la... |
| negcon2 10334 | Negative contraposition la... |
| negeq0 10335 | A number is zero iff its n... |
| subcan 10336 | Cancellation law for subtr... |
| negsubdi 10337 | Distribution of negative o... |
| negdi 10338 | Distribution of negative o... |
| negdi2 10339 | Distribution of negative o... |
| negsubdi2 10340 | Distribution of negative o... |
| neg2sub 10341 | Relationship between subtr... |
| renegcli 10342 | Closure law for negative o... |
| resubcli 10343 | Closure law for subtractio... |
| renegcl 10344 | Closure law for negative o... |
| resubcl 10345 | Closure law for subtractio... |
| negreb 10346 | The negative of a real is ... |
| peano2cnm 10347 | "Reverse" second Peano pos... |
| peano2rem 10348 | "Reverse" second Peano pos... |
| negcli 10349 | Closure law for negative. ... |
| negidi 10350 | Addition of a number and i... |
| negnegi 10351 | A number is equal to the n... |
| subidi 10352 | Subtraction of a number fr... |
| subid1i 10353 | Identity law for subtracti... |
| negne0bi 10354 | A number is nonzero iff it... |
| negrebi 10355 | The negative of a real is ... |
| negne0i 10356 | The negative of a nonzero ... |
| subcli 10357 | Closure law for subtractio... |
| pncan3i 10358 | Subtraction and addition o... |
| negsubi 10359 | Relationship between subtr... |
| subnegi 10360 | Relationship between subtr... |
| subeq0i 10361 | If the difference between ... |
| neg11i 10362 | Negative is one-to-one. (... |
| negcon1i 10363 | Negative contraposition la... |
| negcon2i 10364 | Negative contraposition la... |
| negdii 10365 | Distribution of negative o... |
| negsubdii 10366 | Distribution of negative o... |
| negsubdi2i 10367 | Distribution of negative o... |
| subaddi 10368 | Relationship between subtr... |
| subadd2i 10369 | Relationship between subtr... |
| subaddrii 10370 | Relationship between subtr... |
| subsub23i 10371 | Swap subtrahend and result... |
| addsubassi 10372 | Associative-type law for s... |
| addsubi 10373 | Law for subtraction and ad... |
| subcani 10374 | Cancellation law for subtr... |
| subcan2i 10375 | Cancellation law for subtr... |
| pnncani 10376 | Cancellation law for mixed... |
| addsub4i 10377 | Rearrangement of 4 terms i... |
| 0reALT 10378 | Alternate proof of ~ 0re .... |
| negcld 10379 | Closure law for negative. ... |
| subidd 10380 | Subtraction of a number fr... |
| subid1d 10381 | Identity law for subtracti... |
| negidd 10382 | Addition of a number and i... |
| negnegd 10383 | A number is equal to the n... |
| negeq0d 10384 | A number is zero iff its n... |
| negne0bd 10385 | A number is nonzero iff it... |
| negcon1d 10386 | Contraposition law for una... |
| negcon1ad 10387 | Contraposition law for una... |
| neg11ad 10388 | The negatives of two compl... |
| negned 10389 | If two complex numbers are... |
| negne0d 10390 | The negative of a nonzero ... |
| negrebd 10391 | The negative of a real is ... |
| subcld 10392 | Closure law for subtractio... |
| pncand 10393 | Cancellation law for subtr... |
| pncan2d 10394 | Cancellation law for subtr... |
| pncan3d 10395 | Subtraction and addition o... |
| npcand 10396 | Cancellation law for subtr... |
| nncand 10397 | Cancellation law for subtr... |
| negsubd 10398 | Relationship between subtr... |
| subnegd 10399 | Relationship between subtr... |
| subeq0d 10400 | If the difference between ... |
| subne0d 10401 | Two unequal numbers have n... |
| subeq0ad 10402 | The difference of two comp... |
| subne0ad 10403 | If the difference of two c... |
| neg11d 10404 | If the difference between ... |
| negdid 10405 | Distribution of negative o... |
| negdi2d 10406 | Distribution of negative o... |
| negsubdid 10407 | Distribution of negative o... |
| negsubdi2d 10408 | Distribution of negative o... |
| neg2subd 10409 | Relationship between subtr... |
| subaddd 10410 | Relationship between subtr... |
| subadd2d 10411 | Relationship between subtr... |
| addsubassd 10412 | Associative-type law for s... |
| addsubd 10413 | Law for subtraction and ad... |
| subadd23d 10414 | Commutative/associative la... |
| addsub12d 10415 | Commutative/associative la... |
| npncand 10416 | Cancellation law for subtr... |
| nppcand 10417 | Cancellation law for subtr... |
| nppcan2d 10418 | Cancellation law for subtr... |
| nppcan3d 10419 | Cancellation law for subtr... |
| subsubd 10420 | Law for double subtraction... |
| subsub2d 10421 | Law for double subtraction... |
| subsub3d 10422 | Law for double subtraction... |
| subsub4d 10423 | Law for double subtraction... |
| sub32d 10424 | Swap the second and third ... |
| nnncand 10425 | Cancellation law for subtr... |
| nnncan1d 10426 | Cancellation law for subtr... |
| nnncan2d 10427 | Cancellation law for subtr... |
| npncan3d 10428 | Cancellation law for subtr... |
| pnpcand 10429 | Cancellation law for mixed... |
| pnpcan2d 10430 | Cancellation law for mixed... |
| pnncand 10431 | Cancellation law for mixed... |
| ppncand 10432 | Cancellation law for mixed... |
| subcand 10433 | Cancellation law for subtr... |
| subcan2d 10434 | Cancellation law for subtr... |
| subcanad 10435 | Cancellation law for subtr... |
| subneintrd 10436 | Introducing subtraction on... |
| subcan2ad 10437 | Cancellation law for subtr... |
| subneintr2d 10438 | Introducing subtraction on... |
| addsub4d 10439 | Rearrangement of 4 terms i... |
| subadd4d 10440 | Rearrangement of 4 terms i... |
| sub4d 10441 | Rearrangement of 4 terms i... |
| 2addsubd 10442 | Law for subtraction and ad... |
| addsubeq4d 10443 | Relation between sums and ... |
| mvlraddd 10444 | Move LHS right addition to... |
| mvrraddd 10445 | Move RHS right addition to... |
| subaddeqd 10446 | Transfer two terms of a su... |
| addlsub 10447 | Left-subtraction: Subtrac... |
| addrsub 10448 | Right-subtraction: Subtra... |
| subexsub 10449 | A subtraction law: Exchan... |
| addid0 10450 | If adding a number to a an... |
| addn0nid 10451 | Adding a nonzero number to... |
| pnpncand 10452 | Addition/subtraction cance... |
| subeqrev 10453 | Reverse the order of subtr... |
| pncan1 10454 | Cancellation law for addit... |
| npcan1 10455 | Cancellation law for subtr... |
| subeq0bd 10456 | If two complex numbers are... |
| renegcld 10457 | Closure law for negative o... |
| resubcld 10458 | Closure law for subtractio... |
| negn0 10459 | The image under negation o... |
| negf1o 10460 | Negation is an isomorphism... |
| kcnktkm1cn 10461 | k times k minus 1 is a com... |
| muladd 10462 | Product of two sums. (Con... |
| subdi 10463 | Distribution of multiplica... |
| subdir 10464 | Distribution of multiplica... |
| ine0 10465 | The imaginary unit ` _i ` ... |
| mulneg1 10466 | Product with negative is n... |
| mulneg2 10467 | The product with a negativ... |
| mulneg12 10468 | Swap the negative sign in ... |
| mul2neg 10469 | Product of two negatives. ... |
| submul2 10470 | Convert a subtraction to a... |
| mulm1 10471 | Product with minus one is ... |
| addneg1mul 10472 | Addition with product with... |
| mulsub 10473 | Product of two differences... |
| mulsub2 10474 | Swap the order of subtract... |
| mulm1i 10475 | Product with minus one is ... |
| mulneg1i 10476 | Product with negative is n... |
| mulneg2i 10477 | Product with negative is n... |
| mul2negi 10478 | Product of two negatives. ... |
| subdii 10479 | Distribution of multiplica... |
| subdiri 10480 | Distribution of multiplica... |
| muladdi 10481 | Product of two sums. (Con... |
| mulm1d 10482 | Product with minus one is ... |
| mulneg1d 10483 | Product with negative is n... |
| mulneg2d 10484 | Product with negative is n... |
| mul2negd 10485 | Product of two negatives. ... |
| subdid 10486 | Distribution of multiplica... |
| subdird 10487 | Distribution of multiplica... |
| subdir2d 10488 | Distribution of multiplica... |
| muladdd 10489 | Product of two sums. (Con... |
| mulsubd 10490 | Product of two differences... |
| muls1d 10491 | Multiplication by one minu... |
| mulsubfacd 10492 | Multiplication followed by... |
| gt0ne0 10493 | Positive implies nonzero. ... |
| lt0ne0 10494 | A number which is less tha... |
| ltadd1 10495 | Addition to both sides of ... |
| leadd1 10496 | Addition to both sides of ... |
| leadd2 10497 | Addition to both sides of ... |
| ltsubadd 10498 | 'Less than' relationship b... |
| ltsubadd2 10499 | 'Less than' relationship b... |
| lesubadd 10500 | 'Less than or equal to' re... |
| lesubadd2 10501 | 'Less than or equal to' re... |
| ltaddsub 10502 | 'Less than' relationship b... |
| ltaddsub2 10503 | 'Less than' relationship b... |
| leaddsub 10504 | 'Less than or equal to' re... |
| leaddsub2 10505 | 'Less than or equal to' re... |
| suble 10506 | Swap subtrahends in an ine... |
| lesub 10507 | Swap subtrahends in an ine... |
| ltsub23 10508 | 'Less than' relationship b... |
| ltsub13 10509 | 'Less than' relationship b... |
| le2add 10510 | Adding both sides of two '... |
| ltleadd 10511 | Adding both sides of two o... |
| leltadd 10512 | Adding both sides of two o... |
| lt2add 10513 | Adding both sides of two '... |
| addgt0 10514 | The sum of 2 positive numb... |
| addgegt0 10515 | The sum of nonnegative and... |
| addgtge0 10516 | The sum of nonnegative and... |
| addge0 10517 | The sum of 2 nonnegative n... |
| ltaddpos 10518 | Adding a positive number t... |
| ltaddpos2 10519 | Adding a positive number t... |
| ltsubpos 10520 | Subtracting a positive num... |
| posdif 10521 | Comparison of two numbers ... |
| lesub1 10522 | Subtraction from both side... |
| lesub2 10523 | Subtraction of both sides ... |
| ltsub1 10524 | Subtraction from both side... |
| ltsub2 10525 | Subtraction of both sides ... |
| lt2sub 10526 | Subtracting both sides of ... |
| le2sub 10527 | Subtracting both sides of ... |
| ltneg 10528 | Negative of both sides of ... |
| ltnegcon1 10529 | Contraposition of negative... |
| ltnegcon2 10530 | Contraposition of negative... |
| leneg 10531 | Negative of both sides of ... |
| lenegcon1 10532 | Contraposition of negative... |
| lenegcon2 10533 | Contraposition of negative... |
| lt0neg1 10534 | Comparison of a number and... |
| lt0neg2 10535 | Comparison of a number and... |
| le0neg1 10536 | Comparison of a number and... |
| le0neg2 10537 | Comparison of a number and... |
| addge01 10538 | A number is less than or e... |
| addge02 10539 | A number is less than or e... |
| add20 10540 | Two nonnegative numbers ar... |
| subge0 10541 | Nonnegative subtraction. ... |
| suble0 10542 | Nonpositive subtraction. ... |
| leaddle0 10543 | The sum of a real number a... |
| subge02 10544 | Nonnegative subtraction. ... |
| lesub0 10545 | Lemma to show a nonnegativ... |
| mulge0 10546 | The product of two nonnega... |
| mullt0 10547 | The product of two negativ... |
| msqgt0 10548 | A nonzero square is positi... |
| msqge0 10549 | A square is nonnegative. ... |
| 0lt1 10550 | 0 is less than 1. Theorem... |
| 0le1 10551 | 0 is less than or equal to... |
| relin01 10552 | An interval law for less t... |
| ltordlem 10553 | Lemma for ~ ltord1 . (Con... |
| ltord1 10554 | Infer an ordering relation... |
| leord1 10555 | Infer an ordering relation... |
| eqord1 10556 | Infer an ordering relation... |
| ltord2 10557 | Infer an ordering relation... |
| leord2 10558 | Infer an ordering relation... |
| eqord2 10559 | Infer an ordering relation... |
| wloglei 10560 | Form of ~ wlogle where bot... |
| wlogle 10561 | If the predicate ` ch ( x ... |
| leidi 10562 | 'Less than or equal to' is... |
| gt0ne0i 10563 | Positive means nonzero (us... |
| gt0ne0ii 10564 | Positive implies nonzero. ... |
| msqgt0i 10565 | A nonzero square is positi... |
| msqge0i 10566 | A square is nonnegative. ... |
| addgt0i 10567 | Addition of 2 positive num... |
| addge0i 10568 | Addition of 2 nonnegative ... |
| addgegt0i 10569 | Addition of nonnegative an... |
| addgt0ii 10570 | Addition of 2 positive num... |
| add20i 10571 | Two nonnegative numbers ar... |
| ltnegi 10572 | Negative of both sides of ... |
| lenegi 10573 | Negative of both sides of ... |
| ltnegcon2i 10574 | Contraposition of negative... |
| mulge0i 10575 | The product of two nonnega... |
| lesub0i 10576 | Lemma to show a nonnegativ... |
| ltaddposi 10577 | Adding a positive number t... |
| posdifi 10578 | Comparison of two numbers ... |
| ltnegcon1i 10579 | Contraposition of negative... |
| lenegcon1i 10580 | Contraposition of negative... |
| subge0i 10581 | Nonnegative subtraction. ... |
| ltadd1i 10582 | Addition to both sides of ... |
| leadd1i 10583 | Addition to both sides of ... |
| leadd2i 10584 | Addition to both sides of ... |
| ltsubaddi 10585 | 'Less than' relationship b... |
| lesubaddi 10586 | 'Less than or equal to' re... |
| ltsubadd2i 10587 | 'Less than' relationship b... |
| lesubadd2i 10588 | 'Less than or equal to' re... |
| ltaddsubi 10589 | 'Less than' relationship b... |
| lt2addi 10590 | Adding both side of two in... |
| le2addi 10591 | Adding both side of two in... |
| gt0ne0d 10592 | Positive implies nonzero. ... |
| lt0ne0d 10593 | Something less than zero i... |
| leidd 10594 | 'Less than or equal to' is... |
| msqgt0d 10595 | A nonzero square is positi... |
| msqge0d 10596 | A square is nonnegative. ... |
| lt0neg1d 10597 | Comparison of a number and... |
| lt0neg2d 10598 | Comparison of a number and... |
| le0neg1d 10599 | Comparison of a number and... |
| le0neg2d 10600 | Comparison of a number and... |
| addgegt0d 10601 | Addition of nonnegative an... |
| addgt0d 10602 | Addition of 2 positive num... |
| addge0d 10603 | Addition of 2 nonnegative ... |
| mulge0d 10604 | The product of two nonnega... |
| ltnegd 10605 | Negative of both sides of ... |
| lenegd 10606 | Negative of both sides of ... |
| ltnegcon1d 10607 | Contraposition of negative... |
| ltnegcon2d 10608 | Contraposition of negative... |
| lenegcon1d 10609 | Contraposition of negative... |
| lenegcon2d 10610 | Contraposition of negative... |
| ltaddposd 10611 | Adding a positive number t... |
| ltaddpos2d 10612 | Adding a positive number t... |
| ltsubposd 10613 | Subtracting a positive num... |
| posdifd 10614 | Comparison of two numbers ... |
| addge01d 10615 | A number is less than or e... |
| addge02d 10616 | A number is less than or e... |
| subge0d 10617 | Nonnegative subtraction. ... |
| suble0d 10618 | Nonpositive subtraction. ... |
| subge02d 10619 | Nonnegative subtraction. ... |
| ltadd1d 10620 | Addition to both sides of ... |
| leadd1d 10621 | Addition to both sides of ... |
| leadd2d 10622 | Addition to both sides of ... |
| ltsubaddd 10623 | 'Less than' relationship b... |
| lesubaddd 10624 | 'Less than or equal to' re... |
| ltsubadd2d 10625 | 'Less than' relationship b... |
| lesubadd2d 10626 | 'Less than or equal to' re... |
| ltaddsubd 10627 | 'Less than' relationship b... |
| ltaddsub2d 10628 | 'Less than' relationship b... |
| leaddsub2d 10629 | 'Less than or equal to' re... |
| subled 10630 | Swap subtrahends in an ine... |
| lesubd 10631 | Swap subtrahends in an ine... |
| ltsub23d 10632 | 'Less than' relationship b... |
| ltsub13d 10633 | 'Less than' relationship b... |
| lesub1d 10634 | Subtraction from both side... |
| lesub2d 10635 | Subtraction of both sides ... |
| ltsub1d 10636 | Subtraction from both side... |
| ltsub2d 10637 | Subtraction of both sides ... |
| ltadd1dd 10638 | Addition to both sides of ... |
| ltsub1dd 10639 | Subtraction from both side... |
| ltsub2dd 10640 | Subtraction of both sides ... |
| leadd1dd 10641 | Addition to both sides of ... |
| leadd2dd 10642 | Addition to both sides of ... |
| lesub1dd 10643 | Subtraction from both side... |
| lesub2dd 10644 | Subtraction of both sides ... |
| lesub3d 10645 | The result of subtracting ... |
| le2addd 10646 | Adding both side of two in... |
| le2subd 10647 | Subtracting both sides of ... |
| ltleaddd 10648 | Adding both sides of two o... |
| leltaddd 10649 | Adding both sides of two o... |
| lt2addd 10650 | Adding both side of two in... |
| lt2subd 10651 | Subtracting both sides of ... |
| possumd 10652 | Condition for a positive s... |
| sublt0d 10653 | When a subtraction gives a... |
| ltaddsublt 10654 | Addition and subtraction o... |
| 1le1 10655 | ` 1 <_ 1 ` . Common speci... |
| ixi 10656 | ` _i ` times itself is min... |
| recextlem1 10657 | Lemma for ~ recex . (Cont... |
| recextlem2 10658 | Lemma for ~ recex . (Cont... |
| recex 10659 | Existence of reciprocal of... |
| mulcand 10660 | Cancellation law for multi... |
| mulcan2d 10661 | Cancellation law for multi... |
| mulcanad 10662 | Cancellation of a nonzero ... |
| mulcan2ad 10663 | Cancellation of a nonzero ... |
| mulcan 10664 | Cancellation law for multi... |
| mulcan2 10665 | Cancellation law for multi... |
| mulcani 10666 | Cancellation law for multi... |
| mul0or 10667 | If a product is zero, one ... |
| mulne0b 10668 | The product of two nonzero... |
| mulne0 10669 | The product of two nonzero... |
| mulne0i 10670 | The product of two nonzero... |
| muleqadd 10671 | Property of numbers whose ... |
| receu 10672 | Existential uniqueness of ... |
| mulnzcnopr 10673 | Multiplication maps nonzer... |
| msq0i 10674 | A number is zero iff its s... |
| mul0ori 10675 | If a product is zero, one ... |
| msq0d 10676 | A number is zero iff its s... |
| mul0ord 10677 | If a product is zero, one ... |
| mulne0bd 10678 | The product of two nonzero... |
| mulne0d 10679 | The product of two nonzero... |
| mulcan1g 10680 | A generalized form of the ... |
| mulcan2g 10681 | A generalized form of the ... |
| mulne0bad 10682 | A factor of a nonzero comp... |
| mulne0bbd 10683 | A factor of a nonzero comp... |
| 1div0 10686 | You can't divide by zero, ... |
| divval 10687 | Value of division: if ` A ... |
| divmul 10688 | Relationship between divis... |
| divmul2 10689 | Relationship between divis... |
| divmul3 10690 | Relationship between divis... |
| divcl 10691 | Closure law for division. ... |
| reccl 10692 | Closure law for reciprocal... |
| divcan2 10693 | A cancellation law for div... |
| divcan1 10694 | A cancellation law for div... |
| diveq0 10695 | A ratio is zero iff the nu... |
| divne0b 10696 | The ratio of nonzero numbe... |
| divne0 10697 | The ratio of nonzero numbe... |
| recne0 10698 | The reciprocal of a nonzer... |
| recid 10699 | Multiplication of a number... |
| recid2 10700 | Multiplication of a number... |
| divrec 10701 | Relationship between divis... |
| divrec2 10702 | Relationship between divis... |
| divass 10703 | An associative law for div... |
| div23 10704 | A commutative/associative ... |
| div32 10705 | A commutative/associative ... |
| div13 10706 | A commutative/associative ... |
| div12 10707 | A commutative/associative ... |
| divmulass 10708 | An associative law for div... |
| divmulasscom 10709 | An associative/commutative... |
| divdir 10710 | Distribution of division o... |
| divcan3 10711 | A cancellation law for div... |
| divcan4 10712 | A cancellation law for div... |
| div11 10713 | One-to-one relationship fo... |
| divid 10714 | A number divided by itself... |
| div0 10715 | Division into zero is zero... |
| div1 10716 | A number divided by 1 is i... |
| 1div1e1 10717 | 1 divided by 1 is 1 (commo... |
| diveq1 10718 | Equality in terms of unit ... |
| divneg 10719 | Move negative sign inside ... |
| muldivdir 10720 | Distribution of division o... |
| divsubdir 10721 | Distribution of division o... |
| recrec 10722 | A number is equal to the r... |
| rec11 10723 | Reciprocal is one-to-one. ... |
| rec11r 10724 | Mutual reciprocals. (Cont... |
| divmuldiv 10725 | Multiplication of two rati... |
| divdivdiv 10726 | Division of two ratios. T... |
| divcan5 10727 | Cancellation of common fac... |
| divmul13 10728 | Swap the denominators in t... |
| divmul24 10729 | Swap the numerators in the... |
| divmuleq 10730 | Cross-multiply in an equal... |
| recdiv 10731 | The reciprocal of a ratio.... |
| divcan6 10732 | Cancellation of inverted f... |
| divdiv32 10733 | Swap denominators in a div... |
| divcan7 10734 | Cancel equal divisors in a... |
| dmdcan 10735 | Cancellation law for divis... |
| divdiv1 10736 | Division into a fraction. ... |
| divdiv2 10737 | Division by a fraction. (... |
| recdiv2 10738 | Division into a reciprocal... |
| ddcan 10739 | Cancellation in a double d... |
| divadddiv 10740 | Addition of two ratios. T... |
| divsubdiv 10741 | Subtraction of two ratios.... |
| conjmul 10742 | Two numbers whose reciproc... |
| rereccl 10743 | Closure law for reciprocal... |
| redivcl 10744 | Closure law for division o... |
| eqneg 10745 | A number equal to its nega... |
| eqnegd 10746 | A complex number equals it... |
| eqnegad 10747 | If a complex number equals... |
| div2neg 10748 | Quotient of two negatives.... |
| divneg2 10749 | Move negative sign inside ... |
| recclzi 10750 | Closure law for reciprocal... |
| recne0zi 10751 | The reciprocal of a nonzer... |
| recidzi 10752 | Multiplication of a number... |
| div1i 10753 | A number divided by 1 is i... |
| eqnegi 10754 | A number equal to its nega... |
| reccli 10755 | Closure law for reciprocal... |
| recidi 10756 | Multiplication of a number... |
| recreci 10757 | A number is equal to the r... |
| dividi 10758 | A number divided by itself... |
| div0i 10759 | Division into zero is zero... |
| divclzi 10760 | Closure law for division. ... |
| divcan1zi 10761 | A cancellation law for div... |
| divcan2zi 10762 | A cancellation law for div... |
| divreczi 10763 | Relationship between divis... |
| divcan3zi 10764 | A cancellation law for div... |
| divcan4zi 10765 | A cancellation law for div... |
| rec11i 10766 | Reciprocal is one-to-one. ... |
| divcli 10767 | Closure law for division. ... |
| divcan2i 10768 | A cancellation law for div... |
| divcan1i 10769 | A cancellation law for div... |
| divreci 10770 | Relationship between divis... |
| divcan3i 10771 | A cancellation law for div... |
| divcan4i 10772 | A cancellation law for div... |
| divne0i 10773 | The ratio of nonzero numbe... |
| rec11ii 10774 | Reciprocal is one-to-one. ... |
| divasszi 10775 | An associative law for div... |
| divmulzi 10776 | Relationship between divis... |
| divdirzi 10777 | Distribution of division o... |
| divdiv23zi 10778 | Swap denominators in a div... |
| divmuli 10779 | Relationship between divis... |
| divdiv32i 10780 | Swap denominators in a div... |
| divassi 10781 | An associative law for div... |
| divdiri 10782 | Distribution of division o... |
| div23i 10783 | A commutative/associative ... |
| div11i 10784 | One-to-one relationship fo... |
| divmuldivi 10785 | Multiplication of two rati... |
| divmul13i 10786 | Swap denominators of two r... |
| divadddivi 10787 | Addition of two ratios. T... |
| divdivdivi 10788 | Division of two ratios. T... |
| rerecclzi 10789 | Closure law for reciprocal... |
| rereccli 10790 | Closure law for reciprocal... |
| redivclzi 10791 | Closure law for division o... |
| redivcli 10792 | Closure law for division o... |
| div1d 10793 | A number divided by 1 is i... |
| reccld 10794 | Closure law for reciprocal... |
| recne0d 10795 | The reciprocal of a nonzer... |
| recidd 10796 | Multiplication of a number... |
| recid2d 10797 | Multiplication of a number... |
| recrecd 10798 | A number is equal to the r... |
| dividd 10799 | A number divided by itself... |
| div0d 10800 | Division into zero is zero... |
| divcld 10801 | Closure law for division. ... |
| divcan1d 10802 | A cancellation law for div... |
| divcan2d 10803 | A cancellation law for div... |
| divrecd 10804 | Relationship between divis... |
| divrec2d 10805 | Relationship between divis... |
| divcan3d 10806 | A cancellation law for div... |
| divcan4d 10807 | A cancellation law for div... |
| diveq0d 10808 | A ratio is zero iff the nu... |
| diveq1d 10809 | Equality in terms of unit ... |
| diveq1ad 10810 | The quotient of two comple... |
| diveq0ad 10811 | A fraction of complex numb... |
| divne1d 10812 | If two complex numbers are... |
| divne0bd 10813 | A ratio is zero iff the nu... |
| divnegd 10814 | Move negative sign inside ... |
| divneg2d 10815 | Move negative sign inside ... |
| div2negd 10816 | Quotient of two negatives.... |
| divne0d 10817 | The ratio of nonzero numbe... |
| recdivd 10818 | The reciprocal of a ratio.... |
| recdiv2d 10819 | Division into a reciprocal... |
| divcan6d 10820 | Cancellation of inverted f... |
| ddcand 10821 | Cancellation in a double d... |
| rec11d 10822 | Reciprocal is one-to-one. ... |
| divmuld 10823 | Relationship between divis... |
| div32d 10824 | A commutative/associative ... |
| div13d 10825 | A commutative/associative ... |
| divdiv32d 10826 | Swap denominators in a div... |
| divcan5d 10827 | Cancellation of common fac... |
| divcan5rd 10828 | Cancellation of common fac... |
| divcan7d 10829 | Cancel equal divisors in a... |
| dmdcand 10830 | Cancellation law for divis... |
| dmdcan2d 10831 | Cancellation law for divis... |
| divdiv1d 10832 | Division into a fraction. ... |
| divdiv2d 10833 | Division by a fraction. (... |
| divmul2d 10834 | Relationship between divis... |
| divmul3d 10835 | Relationship between divis... |
| divassd 10836 | An associative law for div... |
| div12d 10837 | A commutative/associative ... |
| div23d 10838 | A commutative/associative ... |
| divdird 10839 | Distribution of division o... |
| divsubdird 10840 | Distribution of division o... |
| div11d 10841 | One-to-one relationship fo... |
| divmuldivd 10842 | Multiplication of two rati... |
| divmul13d 10843 | Swap denominators of two r... |
| divmul24d 10844 | Swap the numerators in the... |
| divadddivd 10845 | Addition of two ratios. T... |
| divsubdivd 10846 | Subtraction of two ratios.... |
| divmuleqd 10847 | Cross-multiply in an equal... |
| divdivdivd 10848 | Division of two ratios. T... |
| diveq1bd 10849 | If two complex numbers are... |
| div2sub 10850 | Swap the order of subtract... |
| div2subd 10851 | Swap subtrahend and minuen... |
| rereccld 10852 | Closure law for reciprocal... |
| redivcld 10853 | Closure law for division o... |
| subrec 10854 | Subtraction of reciprocals... |
| subreci 10855 | Subtraction of reciprocals... |
| subrecd 10856 | Subtraction of reciprocals... |
| mvllmuld 10857 | Move LHS left multiplicati... |
| mvllmuli 10858 | Move LHS left multiplicati... |
| elimgt0 10859 | Hypothesis for weak deduct... |
| elimge0 10860 | Hypothesis for weak deduct... |
| ltp1 10861 | A number is less than itse... |
| lep1 10862 | A number is less than or e... |
| ltm1 10863 | A number minus 1 is less t... |
| lem1 10864 | A number minus 1 is less t... |
| letrp1 10865 | A transitive property of '... |
| p1le 10866 | A transitive property of p... |
| recgt0 10867 | The reciprocal of a positi... |
| prodgt0 10868 | Infer that a multiplicand ... |
| prodgt02 10869 | Infer that a multiplier is... |
| prodge0 10870 | Infer that a multiplicand ... |
| prodge02 10871 | Infer that a multiplier is... |
| ltmul1a 10872 | Lemma for ~ ltmul1 . Mult... |
| ltmul1 10873 | Multiplication of both sid... |
| ltmul2 10874 | Multiplication of both sid... |
| lemul1 10875 | Multiplication of both sid... |
| lemul2 10876 | Multiplication of both sid... |
| lemul1a 10877 | Multiplication of both sid... |
| lemul2a 10878 | Multiplication of both sid... |
| ltmul12a 10879 | Comparison of product of t... |
| lemul12b 10880 | Comparison of product of t... |
| lemul12a 10881 | Comparison of product of t... |
| mulgt1 10882 | The product of two numbers... |
| ltmulgt11 10883 | Multiplication by a number... |
| ltmulgt12 10884 | Multiplication by a number... |
| lemulge11 10885 | Multiplication by a number... |
| lemulge12 10886 | Multiplication by a number... |
| ltdiv1 10887 | Division of both sides of ... |
| lediv1 10888 | Division of both sides of ... |
| gt0div 10889 | Division of a positive num... |
| ge0div 10890 | Division of a nonnegative ... |
| divgt0 10891 | The ratio of two positive ... |
| divge0 10892 | The ratio of nonnegative a... |
| mulge0b 10893 | A condition for multiplica... |
| mulle0b 10894 | A condition for multiplica... |
| mulsuble0b 10895 | A condition for multiplica... |
| ltmuldiv 10896 | 'Less than' relationship b... |
| ltmuldiv2 10897 | 'Less than' relationship b... |
| ltdivmul 10898 | 'Less than' relationship b... |
| ledivmul 10899 | 'Less than or equal to' re... |
| ltdivmul2 10900 | 'Less than' relationship b... |
| lt2mul2div 10901 | 'Less than' relationship b... |
| ledivmul2 10902 | 'Less than or equal to' re... |
| lemuldiv 10903 | 'Less than or equal' relat... |
| lemuldiv2 10904 | 'Less than or equal' relat... |
| ltrec 10905 | The reciprocal of both sid... |
| lerec 10906 | The reciprocal of both sid... |
| lt2msq1 10907 | Lemma for ~ lt2msq . (Con... |
| lt2msq 10908 | Two nonnegative numbers co... |
| ltdiv2 10909 | Division of a positive num... |
| ltrec1 10910 | Reciprocal swap in a 'less... |
| lerec2 10911 | Reciprocal swap in a 'less... |
| ledivdiv 10912 | Invert ratios of positive ... |
| lediv2 10913 | Division of a positive num... |
| ltdiv23 10914 | Swap denominator with othe... |
| lediv23 10915 | Swap denominator with othe... |
| lediv12a 10916 | Comparison of ratio of two... |
| lediv2a 10917 | Division of both sides of ... |
| reclt1 10918 | The reciprocal of a positi... |
| recgt1 10919 | The reciprocal of a positi... |
| recgt1i 10920 | The reciprocal of a number... |
| recp1lt1 10921 | Construct a number less th... |
| recreclt 10922 | Given a positive number ` ... |
| le2msq 10923 | The square function on non... |
| msq11 10924 | The square of a nonnegativ... |
| ledivp1 10925 | Less-than-or-equal-to and ... |
| squeeze0 10926 | If a nonnegative number is... |
| ltp1i 10927 | A number is less than itse... |
| recgt0i 10928 | The reciprocal of a positi... |
| recgt0ii 10929 | The reciprocal of a positi... |
| prodgt0i 10930 | Infer that a multiplicand ... |
| prodge0i 10931 | Infer that a multiplicand ... |
| divgt0i 10932 | The ratio of two positive ... |
| divge0i 10933 | The ratio of nonnegative a... |
| ltreci 10934 | The reciprocal of both sid... |
| lereci 10935 | The reciprocal of both sid... |
| lt2msqi 10936 | The square function on non... |
| le2msqi 10937 | The square function on non... |
| msq11i 10938 | The square of a nonnegativ... |
| divgt0i2i 10939 | The ratio of two positive ... |
| ltrecii 10940 | The reciprocal of both sid... |
| divgt0ii 10941 | The ratio of two positive ... |
| ltmul1i 10942 | Multiplication of both sid... |
| ltdiv1i 10943 | Division of both sides of ... |
| ltmuldivi 10944 | 'Less than' relationship b... |
| ltmul2i 10945 | Multiplication of both sid... |
| lemul1i 10946 | Multiplication of both sid... |
| lemul2i 10947 | Multiplication of both sid... |
| ltdiv23i 10948 | Swap denominator with othe... |
| ledivp1i 10949 | Less-than-or-equal-to and ... |
| ltdivp1i 10950 | Less-than and division rel... |
| ltdiv23ii 10951 | Swap denominator with othe... |
| ltmul1ii 10952 | Multiplication of both sid... |
| ltdiv1ii 10953 | Division of both sides of ... |
| ltp1d 10954 | A number is less than itse... |
| lep1d 10955 | A number is less than or e... |
| ltm1d 10956 | A number minus 1 is less t... |
| lem1d 10957 | A number minus 1 is less t... |
| recgt0d 10958 | The reciprocal of a positi... |
| divgt0d 10959 | The ratio of two positive ... |
| mulgt1d 10960 | The product of two numbers... |
| lemulge11d 10961 | Multiplication by a number... |
| lemulge12d 10962 | Multiplication by a number... |
| lemul1ad 10963 | Multiplication of both sid... |
| lemul2ad 10964 | Multiplication of both sid... |
| ltmul12ad 10965 | Comparison of product of t... |
| lemul12ad 10966 | Comparison of product of t... |
| lemul12bd 10967 | Comparison of product of t... |
| fimaxre 10968 | A finite set of real numbe... |
| fimaxre2 10969 | A nonempty finite set of r... |
| fimaxre3 10970 | A nonempty finite set of r... |
| negfi 10971 | The negation of a finite s... |
| fiminre 10972 | A nonempty finite set of r... |
| lbreu 10973 | If a set of reals contains... |
| lbcl 10974 | If a set of reals contains... |
| lble 10975 | If a set of reals contains... |
| lbinf 10976 | If a set of reals contains... |
| lbinfcl 10977 | If a set of reals contains... |
| lbinfle 10978 | If a set of reals contains... |
| sup2 10979 | A nonempty, bounded-above ... |
| sup3 10980 | A version of the completen... |
| infm3lem 10981 | Lemma for ~ infm3 . (Cont... |
| infm3 10982 | The completeness axiom for... |
| suprcl 10983 | Closure of supremum of a n... |
| suprub 10984 | A member of a nonempty bou... |
| suprubd 10985 | Natural deduction form of ... |
| suprcld 10986 | Natural deduction form of ... |
| suprlub 10987 | The supremum of a nonempty... |
| suprnub 10988 | An upper bound is not less... |
| suprleub 10989 | The supremum of a nonempty... |
| supaddc 10990 | The supremum function dist... |
| supadd 10991 | The supremum function dist... |
| supmul1 10992 | The supremum function dist... |
| supmullem1 10993 | Lemma for ~ supmul . (Con... |
| supmullem2 10994 | Lemma for ~ supmul . (Con... |
| supmul 10995 | The supremum function dist... |
| sup3ii 10996 | A version of the completen... |
| suprclii 10997 | Closure of supremum of a n... |
| suprubii 10998 | A member of a nonempty bou... |
| suprlubii 10999 | The supremum of a nonempty... |
| suprnubii 11000 | An upper bound is not less... |
| suprleubii 11001 | The supremum of a nonempty... |
| riotaneg 11002 | The negative of the unique... |
| negiso 11003 | Negation is an order anti-... |
| dfinfre 11004 | The infimum of a set of re... |
| infrecl 11005 | Closure of infimum of a no... |
| infrenegsup 11006 | The infimum of a set of re... |
| infregelb 11007 | Any lower bound of a nonem... |
| infrelb 11008 | If a nonempty set of real ... |
| supfirege 11009 | The supremum of a finite s... |
| inelr 11010 | The imaginary unit ` _i ` ... |
| rimul 11011 | A real number times the im... |
| cru 11012 | The representation of comp... |
| crne0 11013 | The real representation of... |
| creur 11014 | The real part of a complex... |
| creui 11015 | The imaginary part of a co... |
| cju 11016 | The complex conjugate of a... |
| ofsubeq0 11017 | Function analogue of ~ sub... |
| ofnegsub 11018 | Function analogue of ~ neg... |
| ofsubge0 11019 | Function analogue of ~ sub... |
| nnexALT 11022 | Alternate proof of ~ nnex ... |
| peano5nni 11023 | Peano's inductive postulat... |
| nnssre 11024 | The positive integers are ... |
| nnsscn 11025 | The positive integers are ... |
| nnex 11026 | The set of positive intege... |
| nnre 11027 | A positive integer is a re... |
| nncn 11028 | A positive integer is a co... |
| nnrei 11029 | A positive integer is a re... |
| nncni 11030 | A positive integer is a co... |
| 1nn 11031 | Peano postulate: 1 is a po... |
| peano2nn 11032 | Peano postulate: a success... |
| dfnn2 11033 | Alternate definition of th... |
| dfnn3 11034 | Alternate definition of th... |
| nnred 11035 | A positive integer is a re... |
| nncnd 11036 | A positive integer is a co... |
| peano2nnd 11037 | Peano postulate: a success... |
| nnind 11038 | Principle of Mathematical ... |
| nnindALT 11039 | Principle of Mathematical ... |
| nn1m1nn 11040 | Every positive integer is ... |
| nn1suc 11041 | If a statement holds for 1... |
| nnaddcl 11042 | Closure of addition of pos... |
| nnmulcl 11043 | Closure of multiplication ... |
| nnmulcli 11044 | Closure of multiplication ... |
| nn2ge 11045 | There exists a positive in... |
| nnge1 11046 | A positive integer is one ... |
| nngt1ne1 11047 | A positive integer is grea... |
| nnle1eq1 11048 | A positive integer is less... |
| nngt0 11049 | A positive integer is posi... |
| nnnlt1 11050 | A positive integer is not ... |
| nnnle0 11051 | A positive integer is not ... |
| 0nnn 11052 | Zero is not a positive int... |
| nnne0 11053 | A positive integer is nonz... |
| nngt0i 11054 | A positive integer is posi... |
| nnne0i 11055 | A positive integer is nonz... |
| nndivre 11056 | The quotient of a real and... |
| nnrecre 11057 | The reciprocal of a positi... |
| nnrecgt0 11058 | The reciprocal of a positi... |
| nnsub 11059 | Subtraction of positive in... |
| nnsubi 11060 | Subtraction of positive in... |
| nndiv 11061 | Two ways to express " ` A ... |
| nndivtr 11062 | Transitive property of div... |
| nnge1d 11063 | A positive integer is one ... |
| nngt0d 11064 | A positive integer is posi... |
| nnne0d 11065 | A positive integer is nonz... |
| nnrecred 11066 | The reciprocal of a positi... |
| nnaddcld 11067 | Closure of addition of pos... |
| nnmulcld 11068 | Closure of multiplication ... |
| nndivred 11069 | A positive integer is one ... |
| 0ne1 11088 | ` 0 =/= 1 ` (common case);... |
| 1m1e0 11089 | ` ( 1 - 1 ) = 0 ` (common ... |
| 2re 11090 | The number 2 is real. (Co... |
| 2cn 11091 | The number 2 is a complex ... |
| 2ex 11092 | 2 is a set (common case). ... |
| 2cnd 11093 | 2 is a complex number, ded... |
| 3re 11094 | The number 3 is real. (Co... |
| 3cn 11095 | The number 3 is a complex ... |
| 3ex 11096 | 3 is a set (common case). ... |
| 4re 11097 | The number 4 is real. (Co... |
| 4cn 11098 | The number 4 is a complex ... |
| 5re 11099 | The number 5 is real. (Co... |
| 5cn 11100 | The number 5 is complex. ... |
| 6re 11101 | The number 6 is real. (Co... |
| 6cn 11102 | The number 6 is complex. ... |
| 7re 11103 | The number 7 is real. (Co... |
| 7cn 11104 | The number 7 is complex. ... |
| 8re 11105 | The number 8 is real. (Co... |
| 8cn 11106 | The number 8 is complex. ... |
| 9re 11107 | The number 9 is real. (Co... |
| 9cn 11108 | The number 9 is complex. ... |
| 10reOLD 11109 | Obsolete version of ~ 10re... |
| 0le0 11110 | Zero is nonnegative. (Con... |
| 0le2 11111 | 0 is less than or equal to... |
| 2pos 11112 | The number 2 is positive. ... |
| 2ne0 11113 | The number 2 is nonzero. ... |
| 3pos 11114 | The number 3 is positive. ... |
| 3ne0 11115 | The number 3 is nonzero. ... |
| 4pos 11116 | The number 4 is positive. ... |
| 4ne0 11117 | The number 4 is nonzero. ... |
| 5pos 11118 | The number 5 is positive. ... |
| 6pos 11119 | The number 6 is positive. ... |
| 7pos 11120 | The number 7 is positive. ... |
| 8pos 11121 | The number 8 is positive. ... |
| 9pos 11122 | The number 9 is positive. ... |
| 10posOLD 11123 | The number 10 is positive.... |
| neg1cn 11124 | -1 is a complex number (co... |
| neg1rr 11125 | -1 is a real number (commo... |
| neg1ne0 11126 | -1 is nonzero (common case... |
| neg1lt0 11127 | -1 is less than 0 (common ... |
| negneg1e1 11128 | ` -u -u 1 ` is 1 (common c... |
| 1pneg1e0 11129 | ` 1 + -u 1 ` is 0 (common ... |
| 0m0e0 11130 | 0 minus 0 equals 0 (common... |
| 1m0e1 11131 | 1 - 0 = 1 (common case). ... |
| 0p1e1 11132 | 0 + 1 = 1. (Contributed b... |
| 1p0e1 11133 | 1 + 0 = 1. (Contributed b... |
| 1p1e2 11134 | 1 + 1 = 2. (Contributed b... |
| 2m1e1 11135 | 2 - 1 = 1. The result is ... |
| 1e2m1 11136 | 1 = 2 - 1 (common case). ... |
| 3m1e2 11137 | 3 - 1 = 2. (Contributed b... |
| 4m1e3 11138 | 4 - 1 = 3. (Contributed b... |
| 5m1e4 11139 | 5 - 1 = 4. (Contributed b... |
| 6m1e5 11140 | 6 - 1 = 5. (Contributed b... |
| 7m1e6 11141 | 7 - 1 = 6. (Contributed b... |
| 8m1e7 11142 | 8 - 1 = 7. (Contributed b... |
| 9m1e8 11143 | 9 - 1 = 8. (Contributed b... |
| 2p2e4 11144 | Two plus two equals four. ... |
| 2times 11145 | Two times a number. (Cont... |
| times2 11146 | A number times 2. (Contri... |
| 2timesi 11147 | Two times a number. (Cont... |
| times2i 11148 | A number times 2. (Contri... |
| 2txmxeqx 11149 | Two times a complex number... |
| 2div2e1 11150 | 2 divided by 2 is 1 (commo... |
| 2p1e3 11151 | 2 + 1 = 3. (Contributed b... |
| 1p2e3 11152 | 1 + 2 = 3 (common case). ... |
| 3p1e4 11153 | 3 + 1 = 4. (Contributed b... |
| 4p1e5 11154 | 4 + 1 = 5. (Contributed b... |
| 5p1e6 11155 | 5 + 1 = 6. (Contributed b... |
| 6p1e7 11156 | 6 + 1 = 7. (Contributed b... |
| 7p1e8 11157 | 7 + 1 = 8. (Contributed b... |
| 8p1e9 11158 | 8 + 1 = 9. (Contributed b... |
| 9p1e10OLD 11159 | 9 + 1 = 10. (Contributed ... |
| 3p2e5 11160 | 3 + 2 = 5. (Contributed b... |
| 3p3e6 11161 | 3 + 3 = 6. (Contributed b... |
| 4p2e6 11162 | 4 + 2 = 6. (Contributed b... |
| 4p3e7 11163 | 4 + 3 = 7. (Contributed b... |
| 4p4e8 11164 | 4 + 4 = 8. (Contributed b... |
| 5p2e7 11165 | 5 + 2 = 7. (Contributed b... |
| 5p3e8 11166 | 5 + 3 = 8. (Contributed b... |
| 5p4e9 11167 | 5 + 4 = 9. (Contributed b... |
| 5p5e10OLD 11168 | 5 + 5 = 10. (Contributed ... |
| 6p2e8 11169 | 6 + 2 = 8. (Contributed b... |
| 6p3e9 11170 | 6 + 3 = 9. (Contributed b... |
| 6p4e10OLD 11171 | 6 + 4 = 10. (Contributed ... |
| 7p2e9 11172 | 7 + 2 = 9. (Contributed b... |
| 7p3e10OLD 11173 | 7 + 3 = 10. (Contributed ... |
| 8p2e10OLD 11174 | 8 + 2 = 10. (Contributed ... |
| 1t1e1 11175 | 1 times 1 equals 1. (Cont... |
| 2t1e2 11176 | 2 times 1 equals 2. (Cont... |
| 2t2e4 11177 | 2 times 2 equals 4. (Cont... |
| 3t1e3 11178 | 3 times 1 equals 3. (Cont... |
| 3t2e6 11179 | 3 times 2 equals 6. (Cont... |
| 3t3e9 11180 | 3 times 3 equals 9. (Cont... |
| 4t2e8 11181 | 4 times 2 equals 8. (Cont... |
| 5t2e10OLD 11182 | 5 times 2 equals 10. (Con... |
| 2t0e0 11183 | 2 times 0 equals 0. (Cont... |
| 4d2e2 11184 | One half of four is two. ... |
| 2nn 11185 | 2 is a positive integer. ... |
| 3nn 11186 | 3 is a positive integer. ... |
| 4nn 11187 | 4 is a positive integer. ... |
| 5nn 11188 | 5 is a positive integer. ... |
| 6nn 11189 | 6 is a positive integer. ... |
| 7nn 11190 | 7 is a positive integer. ... |
| 8nn 11191 | 8 is a positive integer. ... |
| 9nn 11192 | 9 is a positive integer. ... |
| 10nnOLD 11193 | Obsolete version of ~ 10nn... |
| 1lt2 11194 | 1 is less than 2. (Contri... |
| 2lt3 11195 | 2 is less than 3. (Contri... |
| 1lt3 11196 | 1 is less than 3. (Contri... |
| 3lt4 11197 | 3 is less than 4. (Contri... |
| 2lt4 11198 | 2 is less than 4. (Contri... |
| 1lt4 11199 | 1 is less than 4. (Contri... |
| 4lt5 11200 | 4 is less than 5. (Contri... |
| 3lt5 11201 | 3 is less than 5. (Contri... |
| 2lt5 11202 | 2 is less than 5. (Contri... |
| 1lt5 11203 | 1 is less than 5. (Contri... |
| 5lt6 11204 | 5 is less than 6. (Contri... |
| 4lt6 11205 | 4 is less than 6. (Contri... |
| 3lt6 11206 | 3 is less than 6. (Contri... |
| 2lt6 11207 | 2 is less than 6. (Contri... |
| 1lt6 11208 | 1 is less than 6. (Contri... |
| 6lt7 11209 | 6 is less than 7. (Contri... |
| 5lt7 11210 | 5 is less than 7. (Contri... |
| 4lt7 11211 | 4 is less than 7. (Contri... |
| 3lt7 11212 | 3 is less than 7. (Contri... |
| 2lt7 11213 | 2 is less than 7. (Contri... |
| 1lt7 11214 | 1 is less than 7. (Contri... |
| 7lt8 11215 | 7 is less than 8. (Contri... |
| 6lt8 11216 | 6 is less than 8. (Contri... |
| 5lt8 11217 | 5 is less than 8. (Contri... |
| 4lt8 11218 | 4 is less than 8. (Contri... |
| 3lt8 11219 | 3 is less than 8. (Contri... |
| 2lt8 11220 | 2 is less than 8. (Contri... |
| 1lt8 11221 | 1 is less than 8. (Contri... |
| 8lt9 11222 | 8 is less than 9. (Contri... |
| 7lt9 11223 | 7 is less than 9. (Contri... |
| 6lt9 11224 | 6 is less than 9. (Contri... |
| 5lt9 11225 | 5 is less than 9. (Contri... |
| 4lt9 11226 | 4 is less than 9. (Contri... |
| 3lt9 11227 | 3 is less than 9. (Contri... |
| 2lt9 11228 | 2 is less than 9. (Contri... |
| 1lt9 11229 | 1 is less than 9. (Contri... |
| 9lt10OLD 11230 | 9 is less than 10. (Contr... |
| 8lt10OLD 11231 | 8 is less than 10. (Contr... |
| 7lt10OLD 11232 | 7 is less than 10. (Contr... |
| 6lt10OLD 11233 | 6 is less than 10. (Contr... |
| 5lt10OLD 11234 | 5 is less than 10. (Contr... |
| 4lt10OLD 11235 | 4 is less than 10. (Contr... |
| 3lt10OLD 11236 | 3 is less than 10. (Contr... |
| 2lt10OLD 11237 | 2 is less than 10. (Contr... |
| 1lt10OLD 11238 | 1 is less than 10. (Contr... |
| 0ne2 11239 | 0 is not equal to 2. (Con... |
| 1ne2 11240 | 1 is not equal to 2. (Con... |
| 1le2 11241 | 1 is less than or equal to... |
| 2cnne0 11242 | 2 is a nonzero complex num... |
| 2rene0 11243 | 2 is a nonzero real number... |
| 1le3 11244 | 1 is less than or equal to... |
| neg1mulneg1e1 11245 | ` -u 1 x. -u 1 ` is 1 (com... |
| halfre 11246 | One-half is real. (Contri... |
| halfcn 11247 | One-half is complex. (Con... |
| halfgt0 11248 | One-half is greater than z... |
| halfge0 11249 | One-half is not negative. ... |
| halflt1 11250 | One-half is less than one.... |
| 1mhlfehlf 11251 | Prove that 1 - 1/2 = 1/2. ... |
| 8th4div3 11252 | An eighth of four thirds i... |
| halfpm6th 11253 | One half plus or minus one... |
| it0e0 11254 | i times 0 equals 0 (common... |
| 2mulicn 11255 | ` ( 2 x. _i ) e. CC ` (com... |
| 2muline0 11256 | ` ( 2 x. _i ) =/= 0 ` (com... |
| halfcl 11257 | Closure of half of a numbe... |
| rehalfcl 11258 | Real closure of half. (Co... |
| half0 11259 | Half of a number is zero i... |
| 2halves 11260 | Two halves make a whole. ... |
| halfpos2 11261 | A number is positive iff i... |
| halfpos 11262 | A positive number is great... |
| halfnneg2 11263 | A number is nonnegative if... |
| halfaddsubcl 11264 | Closure of half-sum and ha... |
| halfaddsub 11265 | Sum and difference of half... |
| subhalfhalf 11266 | Subtracting the half of a ... |
| lt2halves 11267 | A sum is less than the who... |
| addltmul 11268 | Sum is less than product f... |
| nominpos 11269 | There is no smallest posit... |
| avglt1 11270 | Ordering property for aver... |
| avglt2 11271 | Ordering property for aver... |
| avgle1 11272 | Ordering property for aver... |
| avgle2 11273 | Ordering property for aver... |
| avgle 11274 | The average of two numbers... |
| 2timesd 11275 | Two times a number. (Cont... |
| times2d 11276 | A number times 2. (Contri... |
| halfcld 11277 | Closure of half of a numbe... |
| 2halvesd 11278 | Two halves make a whole. ... |
| rehalfcld 11279 | Real closure of half. (Co... |
| lt2halvesd 11280 | A sum is less than the who... |
| rehalfcli 11281 | Half a real number is real... |
| lt2addmuld 11282 | If two real numbers are le... |
| add1p1 11283 | Adding two times 1 to a nu... |
| sub1m1 11284 | Subtracting two times 1 fr... |
| cnm2m1cnm3 11285 | Subtracting 2 and afterwar... |
| xp1d2m1eqxm1d2 11286 | A complex number increased... |
| div4p1lem1div2 11287 | An integer greater than 5,... |
| nnunb 11288 | The set of positive intege... |
| arch 11289 | Archimedean property of re... |
| nnrecl 11290 | There exists a positive in... |
| bndndx 11291 | A bounded real sequence ` ... |
| elnn0 11294 | Nonnegative integers expre... |
| nnssnn0 11295 | Positive naturals are a su... |
| nn0ssre 11296 | Nonnegative integers are a... |
| nn0sscn 11297 | Nonnegative integers are a... |
| nn0ex 11298 | The set of nonnegative int... |
| nnnn0 11299 | A positive integer is a no... |
| nnnn0i 11300 | A positive integer is a no... |
| nn0re 11301 | A nonnegative integer is a... |
| nn0cn 11302 | A nonnegative integer is a... |
| nn0rei 11303 | A nonnegative integer is a... |
| nn0cni 11304 | A nonnegative integer is a... |
| dfn2 11305 | The set of positive intege... |
| elnnne0 11306 | The positive integer prope... |
| 0nn0 11307 | 0 is a nonnegative integer... |
| 1nn0 11308 | 1 is a nonnegative integer... |
| 2nn0 11309 | 2 is a nonnegative integer... |
| 3nn0 11310 | 3 is a nonnegative integer... |
| 4nn0 11311 | 4 is a nonnegative integer... |
| 5nn0 11312 | 5 is a nonnegative integer... |
| 6nn0 11313 | 6 is a nonnegative integer... |
| 7nn0 11314 | 7 is a nonnegative integer... |
| 8nn0 11315 | 8 is a nonnegative integer... |
| 9nn0 11316 | 9 is a nonnegative integer... |
| 10nn0OLD 11317 | Obsolete version of ~ 10nn... |
| nn0ge0 11318 | A nonnegative integer is g... |
| nn0nlt0 11319 | A nonnegative integer is n... |
| nn0ge0i 11320 | Nonnegative integers are n... |
| nn0le0eq0 11321 | A nonnegative integer is l... |
| nn0p1gt0 11322 | A nonnegative integer incr... |
| nnnn0addcl 11323 | A positive integer plus a ... |
| nn0nnaddcl 11324 | A nonnegative integer plus... |
| 0mnnnnn0 11325 | The result of subtracting ... |
| un0addcl 11326 | If ` S ` is closed under a... |
| un0mulcl 11327 | If ` S ` is closed under m... |
| nn0addcl 11328 | Closure of addition of non... |
| nn0mulcl 11329 | Closure of multiplication ... |
| nn0addcli 11330 | Closure of addition of non... |
| nn0mulcli 11331 | Closure of multiplication ... |
| nn0p1nn 11332 | A nonnegative integer plus... |
| peano2nn0 11333 | Second Peano postulate for... |
| nnm1nn0 11334 | A positive integer minus 1... |
| elnn0nn 11335 | The nonnegative integer pr... |
| elnnnn0 11336 | The positive integer prope... |
| elnnnn0b 11337 | The positive integer prope... |
| elnnnn0c 11338 | The positive integer prope... |
| nn0addge1 11339 | A number is less than or e... |
| nn0addge2 11340 | A number is less than or e... |
| nn0addge1i 11341 | A number is less than or e... |
| nn0addge2i 11342 | A number is less than or e... |
| nn0sub 11343 | Subtraction of nonnegative... |
| ltsubnn0 11344 | Subtracting a nonnegative ... |
| nn0negleid 11345 | A nonnegative integer is g... |
| difgtsumgt 11346 | If the difference of a rea... |
| nn0le2xi 11347 | A nonnegative integer is l... |
| nn0lele2xi 11348 | 'Less than or equal to' im... |
| frnnn0supp 11349 | Two ways to write the supp... |
| frnnn0fsupp 11350 | A function on ` NN0 ` is f... |
| nnnn0d 11351 | A positive integer is a no... |
| nn0red 11352 | A nonnegative integer is a... |
| nn0cnd 11353 | A nonnegative integer is a... |
| nn0ge0d 11354 | A nonnegative integer is g... |
| nn0addcld 11355 | Closure of addition of non... |
| nn0mulcld 11356 | Closure of multiplication ... |
| nn0readdcl 11357 | Closure law for addition o... |
| nn0n0n1ge2 11358 | A nonnegative integer whic... |
| nn0n0n1ge2b 11359 | A nonnegative integer is n... |
| nn0ge2m1nn 11360 | If a nonnegative integer i... |
| nn0ge2m1nn0 11361 | If a nonnegative integer i... |
| nn0nndivcl 11362 | Closure law for dividing o... |
| elxnn0 11365 | An extended nonnegative in... |
| nn0ssxnn0 11366 | The standard nonnegative i... |
| nn0xnn0 11367 | A standard nonnegative int... |
| xnn0xr 11368 | An extended nonnegative in... |
| 0xnn0 11369 | Zero is an extended nonneg... |
| pnf0xnn0 11370 | Positive infinity is an ex... |
| nn0nepnf 11371 | No standard nonnegative in... |
| nn0xnn0d 11372 | A standard nonnegative int... |
| nn0nepnfd 11373 | No standard nonnegative in... |
| xnn0nemnf 11374 | No extended nonnegative in... |
| xnn0xrnemnf 11375 | The extended nonnegative i... |
| xnn0nnn0pnf 11376 | An extended nonnegative in... |
| elz 11379 | Membership in the set of i... |
| nnnegz 11380 | The negative of a positive... |
| zre 11381 | An integer is a real. (Co... |
| zcn 11382 | An integer is a complex nu... |
| zrei 11383 | An integer is a real numbe... |
| zssre 11384 | The integers are a subset ... |
| zsscn 11385 | The integers are a subset ... |
| zex 11386 | The set of integers exists... |
| elnnz 11387 | Positive integer property ... |
| 0z 11388 | Zero is an integer. (Cont... |
| 0zd 11389 | Zero is an integer, deduct... |
| elnn0z 11390 | Nonnegative integer proper... |
| elznn0nn 11391 | Integer property expressed... |
| elznn0 11392 | Integer property expressed... |
| elznn 11393 | Integer property expressed... |
| elz2 11394 | Membership in the set of i... |
| dfz2 11395 | Alternative definition of ... |
| zexALT 11396 | Alternate proof of ~ zex .... |
| nnssz 11397 | Positive integers are a su... |
| nn0ssz 11398 | Nonnegative integers are a... |
| nnz 11399 | A positive integer is an i... |
| nn0z 11400 | A nonnegative integer is a... |
| nnzi 11401 | A positive integer is an i... |
| nn0zi 11402 | A nonnegative integer is a... |
| elnnz1 11403 | Positive integer property ... |
| znnnlt1 11404 | An integer is not a positi... |
| nnzrab 11405 | Positive integers expresse... |
| nn0zrab 11406 | Nonnegative integers expre... |
| 1z 11407 | One is an integer. (Contr... |
| 1zzd 11408 | 1 is an integer, deductive... |
| 2z 11409 | 2 is an integer. (Contrib... |
| 3z 11410 | 3 is an integer. (Contrib... |
| 4z 11411 | 4 is an integer. (Contrib... |
| znegcl 11412 | Closure law for negative i... |
| neg1z 11413 | -1 is an integer (common c... |
| znegclb 11414 | A complex number is an int... |
| nn0negz 11415 | The negative of a nonnegat... |
| nn0negzi 11416 | The negative of a nonnegat... |
| zaddcl 11417 | Closure of addition of int... |
| peano2z 11418 | Second Peano postulate gen... |
| zsubcl 11419 | Closure of subtraction of ... |
| peano2zm 11420 | "Reverse" second Peano pos... |
| zletr 11421 | Transitive law of ordering... |
| zrevaddcl 11422 | Reverse closure law for ad... |
| znnsub 11423 | The positive difference of... |
| znn0sub 11424 | The nonnegative difference... |
| nzadd 11425 | The sum of a real number n... |
| zmulcl 11426 | Closure of multiplication ... |
| zltp1le 11427 | Integer ordering relation.... |
| zleltp1 11428 | Integer ordering relation.... |
| zlem1lt 11429 | Integer ordering relation.... |
| zltlem1 11430 | Integer ordering relation.... |
| zgt0ge1 11431 | An integer greater than ` ... |
| nnleltp1 11432 | Positive integer ordering ... |
| nnltp1le 11433 | Positive integer ordering ... |
| nnaddm1cl 11434 | Closure of addition of pos... |
| nn0ltp1le 11435 | Nonnegative integer orderi... |
| nn0leltp1 11436 | Nonnegative integer orderi... |
| nn0ltlem1 11437 | Nonnegative integer orderi... |
| nn0sub2 11438 | Subtraction of nonnegative... |
| nn0lt10b 11439 | A nonnegative integer less... |
| nn0lt2 11440 | A nonnegative integer less... |
| nn0le2is012 11441 | A nonnegative integer whic... |
| nn0lem1lt 11442 | Nonnegative integer orderi... |
| nnlem1lt 11443 | Positive integer ordering ... |
| nnltlem1 11444 | Positive integer ordering ... |
| nnm1ge0 11445 | A positive integer decreas... |
| nn0ge0div 11446 | Division of a nonnegative ... |
| zdiv 11447 | Two ways to express " ` M ... |
| zdivadd 11448 | Property of divisibility: ... |
| zdivmul 11449 | Property of divisibility: ... |
| zextle 11450 | An extensionality-like pro... |
| zextlt 11451 | An extensionality-like pro... |
| recnz 11452 | The reciprocal of a number... |
| btwnnz 11453 | A number between an intege... |
| gtndiv 11454 | A larger number does not d... |
| halfnz 11455 | One-half is not an integer... |
| 3halfnz 11456 | Three halves is not an int... |
| suprzcl 11457 | The supremum of a bounded-... |
| prime 11458 | Two ways to express " ` A ... |
| msqznn 11459 | The square of a nonzero in... |
| zneo 11460 | No even integer equals an ... |
| nneo 11461 | A positive integer is even... |
| nneoi 11462 | A positive integer is even... |
| zeo 11463 | An integer is even or odd.... |
| zeo2 11464 | An integer is even or odd ... |
| peano2uz2 11465 | Second Peano postulate for... |
| peano5uzi 11466 | Peano's inductive postulat... |
| peano5uzti 11467 | Peano's inductive postulat... |
| dfuzi 11468 | An expression for the uppe... |
| uzind 11469 | Induction on the upper int... |
| uzind2 11470 | Induction on the upper int... |
| uzind3 11471 | Induction on the upper int... |
| nn0ind 11472 | Principle of Mathematical ... |
| nn0indALT 11473 | Principle of Mathematical ... |
| nn0indd 11474 | Principle of Mathematical ... |
| fzind 11475 | Induction on the integers ... |
| fnn0ind 11476 | Induction on the integers ... |
| nn0ind-raph 11477 | Principle of Mathematical ... |
| zindd 11478 | Principle of Mathematical ... |
| btwnz 11479 | Any real number can be san... |
| nn0zd 11480 | A positive integer is an i... |
| nnzd 11481 | A nonnegative integer is a... |
| zred 11482 | An integer is a real numbe... |
| zcnd 11483 | An integer is a complex nu... |
| znegcld 11484 | Closure law for negative i... |
| peano2zd 11485 | Deduction from second Pean... |
| zaddcld 11486 | Closure of addition of int... |
| zsubcld 11487 | Closure of subtraction of ... |
| zmulcld 11488 | Closure of multiplication ... |
| znnn0nn 11489 | The negative of a negative... |
| zadd2cl 11490 | Increasing an integer by 2... |
| zriotaneg 11491 | The negative of the unique... |
| suprfinzcl 11492 | The supremum of a nonempty... |
| dfdecOLD 11495 | Define the "decimal constr... |
| 9p1e10 11496 | 9 + 1 = 10. (Contributed ... |
| dfdec10 11497 | Version of the definition ... |
| decex 11498 | A decimal number is a set.... |
| decexOLD 11499 | Obsolete proof of ~ decex ... |
| deceq1 11500 | Equality theorem for the d... |
| deceq1OLD 11501 | Obsolete proof of ~ deceq1... |
| deceq2 11502 | Equality theorem for the d... |
| deceq2OLD 11503 | Obsolete proof of ~ deceq1... |
| deceq1i 11504 | Equality theorem for the d... |
| deceq2i 11505 | Equality theorem for the d... |
| deceq12i 11506 | Equality theorem for the d... |
| numnncl 11507 | Closure for a numeral (wit... |
| num0u 11508 | Add a zero in the units pl... |
| num0h 11509 | Add a zero in the higher p... |
| numcl 11510 | Closure for a decimal inte... |
| numsuc 11511 | The successor of a decimal... |
| deccl 11512 | Closure for a numeral. (C... |
| decclOLD 11513 | Obsolete proof of ~ deccl ... |
| 10nn 11514 | 10 is a positive integer. ... |
| 10pos 11515 | The number 10 is positive.... |
| 10nn0 11516 | 10 is a nonnegative intege... |
| 10re 11517 | The number 10 is real. (C... |
| decnncl 11518 | Closure for a numeral. (C... |
| decnnclOLD 11519 | Obsolete proof of ~ decnnc... |
| dec0u 11520 | Add a zero in the units pl... |
| dec0uOLD 11521 | Obsolete version of ~ dec0... |
| dec0h 11522 | Add a zero in the higher p... |
| dec0hOLD 11523 | Obsolete proof of ~ dec0h ... |
| numnncl2 11524 | Closure for a decimal inte... |
| decnncl2 11525 | Closure for a decimal inte... |
| decnncl2OLD 11526 | Obsolete proof of ~ decnnc... |
| numlt 11527 | Comparing two decimal inte... |
| numltc 11528 | Comparing two decimal inte... |
| le9lt10 11529 | A "decimal digit" (i.e. a ... |
| declt 11530 | Comparing two decimal inte... |
| decltOLD 11531 | Obsolete proof of ~ declt ... |
| decltc 11532 | Comparing two decimal inte... |
| decltcOLD 11533 | Obsolete version of ~ decl... |
| declth 11534 | Comparing two decimal inte... |
| decsuc 11535 | The successor of a decimal... |
| decsucOLD 11536 | Obsolete proof of ~ decsuc... |
| 3declth 11537 | Comparing two decimal inte... |
| 3decltc 11538 | Comparing two decimal inte... |
| 3decltcOLD 11539 | Obsolete version of ~ 3dec... |
| decle 11540 | Comparing two decimal inte... |
| decleh 11541 | Comparing two decimal inte... |
| declei 11542 | Comparing a digit to a dec... |
| decleOLD 11543 | Obsolete version of ~ decl... |
| declecOLD 11544 | Obsolete version of ~ decl... |
| numlti 11545 | Comparing a digit to a dec... |
| declti 11546 | Comparing a digit to a dec... |
| decltdi 11547 | Comparing a digit to a dec... |
| decltiOLD 11548 | Obsolete version of ~ decl... |
| numsucc 11549 | The successor of a decimal... |
| decsucc 11550 | The successor of a decimal... |
| decsuccOLD 11551 | Obsolete version of ~ decs... |
| 1e0p1 11552 | The successor of zero. (C... |
| dec10p 11553 | Ten plus an integer. (Con... |
| dec10pOLD 11554 | Obsolete version of ~ dec1... |
| dec10OLD 11555 | The decimal form of 10. N... |
| 9p1e10bOLD 11556 | Obsolete proof of ~ 9p1e10... |
| numma 11557 | Perform a multiply-add of ... |
| nummac 11558 | Perform a multiply-add of ... |
| numma2c 11559 | Perform a multiply-add of ... |
| numadd 11560 | Add two decimal integers `... |
| numaddc 11561 | Add two decimal integers `... |
| nummul1c 11562 | The product of a decimal i... |
| nummul2c 11563 | The product of a decimal i... |
| decma 11564 | Perform a multiply-add of ... |
| decmaOLD 11565 | Obsolete proof of ~ decma ... |
| decmac 11566 | Perform a multiply-add of ... |
| decmacOLD 11567 | Obsolete proof of ~ decmac... |
| decma2c 11568 | Perform a multiply-add of ... |
| decma2cOLD 11569 | Obsolete proof of ~ decma2... |
| decadd 11570 | Add two numerals ` M ` and... |
| decaddOLD 11571 | Obsolete proof of ~ decadd... |
| decaddc 11572 | Add two numerals ` M ` and... |
| decaddcOLD 11573 | Obsolete proof of ~ decadd... |
| decaddc2OLD 11574 | Obsolete version of ~ deca... |
| decaddc2 11575 | Add two numerals ` M ` and... |
| decrmanc 11576 | Perform a multiply-add of ... |
| decrmac 11577 | Perform a multiply-add of ... |
| decaddm10 11578 | The sum of two multiples o... |
| decaddi 11579 | Add two numerals ` M ` and... |
| decaddci 11580 | Add two numerals ` M ` and... |
| decaddci2 11581 | Add two numerals ` M ` and... |
| decaddci2OLD 11582 | Obsolete version of ~ deca... |
| decsubi 11583 | Difference between a numer... |
| decsubiOLD 11584 | Obsolete proof of ~ decsub... |
| decmul1 11585 | The product of a numeral w... |
| decmul1OLD 11586 | Obsolete proof of ~ decmul... |
| decmul1c 11587 | The product of a numeral w... |
| decmul1cOLD 11588 | Obsolete proof of ~ decmul... |
| decmul2c 11589 | The product of a numeral w... |
| decmul2cOLD 11590 | Obsolete proof of ~ decmul... |
| decmulnc 11591 | The product of a numeral w... |
| 11multnc 11592 | The product of 11 (as nume... |
| decmul10add 11593 | A multiplication of a numb... |
| decmul10addOLD 11594 | Obsolete proof of ~ decmul... |
| 6p5lem 11595 | Lemma for ~ 6p5e11 and rel... |
| 5p5e10 11596 | 5 + 5 = 10. (Contributed ... |
| 5p5e10bOLD 11597 | Obsolete proof of ~ 5p5e10... |
| 6p4e10 11598 | 6 + 4 = 10. (Contributed ... |
| 6p4e10bOLD 11599 | Obsolete proof of ~ 6p4e10... |
| 6p5e11 11600 | 6 + 5 = 11. (Contributed ... |
| 6p5e11OLD 11601 | Obsolete proof of ~ 6p5e11... |
| 6p6e12 11602 | 6 + 6 = 12. (Contributed ... |
| 7p3e10 11603 | 7 + 3 = 10. (Contributed ... |
| 7p3e10bOLD 11604 | Obsolete proof of ~ 7p3e10... |
| 7p4e11 11605 | 7 + 4 = 11. (Contributed ... |
| 7p4e11OLD 11606 | Obsolete proof of ~ 7p4e11... |
| 7p5e12 11607 | 7 + 5 = 12. (Contributed ... |
| 7p6e13 11608 | 7 + 6 = 13. (Contributed ... |
| 7p7e14 11609 | 7 + 7 = 14. (Contributed ... |
| 8p2e10 11610 | 8 + 2 = 10. (Contributed ... |
| 8p2e10bOLD 11611 | Obsolete proof of ~ 8p2e10... |
| 8p3e11 11612 | 8 + 3 = 11. (Contributed ... |
| 8p3e11OLD 11613 | Obsolete proof of ~ 8p3e11... |
| 8p4e12 11614 | 8 + 4 = 12. (Contributed ... |
| 8p5e13 11615 | 8 + 5 = 13. (Contributed ... |
| 8p6e14 11616 | 8 + 6 = 14. (Contributed ... |
| 8p7e15 11617 | 8 + 7 = 15. (Contributed ... |
| 8p8e16 11618 | 8 + 8 = 16. (Contributed ... |
| 9p2e11 11619 | 9 + 2 = 11. (Contributed ... |
| 9p2e11OLD 11620 | Obsolete proof of ~ 9p2e11... |
| 9p3e12 11621 | 9 + 3 = 12. (Contributed ... |
| 9p4e13 11622 | 9 + 4 = 13. (Contributed ... |
| 9p5e14 11623 | 9 + 5 = 14. (Contributed ... |
| 9p6e15 11624 | 9 + 6 = 15. (Contributed ... |
| 9p7e16 11625 | 9 + 7 = 16. (Contributed ... |
| 9p8e17 11626 | 9 + 8 = 17. (Contributed ... |
| 9p9e18 11627 | 9 + 9 = 18. (Contributed ... |
| 10p10e20 11628 | 10 + 10 = 20. (Contribute... |
| 10p10e20OLD 11629 | Obsolete version of ~ 10p1... |
| 10m1e9 11630 | 10 - 1 = 9. (Contributed ... |
| 4t3lem 11631 | Lemma for ~ 4t3e12 and rel... |
| 4t3e12 11632 | 4 times 3 equals 12. (Con... |
| 4t4e16 11633 | 4 times 4 equals 16. (Con... |
| 5t2e10 11634 | 5 times 2 equals 10. (Con... |
| 5t3e15 11635 | 5 times 3 equals 15. (Con... |
| 5t3e15OLD 11636 | Obsolete proof of ~ 5t3e15... |
| 5t4e20 11637 | 5 times 4 equals 20. (Con... |
| 5t4e20OLD 11638 | Obsolete proof of ~ 5t4e20... |
| 5t5e25 11639 | 5 times 5 equals 25. (Con... |
| 5t5e25OLD 11640 | Obsolete proof of ~ 5t5e25... |
| 6t2e12 11641 | 6 times 2 equals 12. (Con... |
| 6t3e18 11642 | 6 times 3 equals 18. (Con... |
| 6t4e24 11643 | 6 times 4 equals 24. (Con... |
| 6t5e30 11644 | 6 times 5 equals 30. (Con... |
| 6t5e30OLD 11645 | Obsolete proof of ~ 6t5e30... |
| 6t6e36 11646 | 6 times 6 equals 36. (Con... |
| 6t6e36OLD 11647 | Obsolete proof of ~ 6t6e36... |
| 7t2e14 11648 | 7 times 2 equals 14. (Con... |
| 7t3e21 11649 | 7 times 3 equals 21. (Con... |
| 7t4e28 11650 | 7 times 4 equals 28. (Con... |
| 7t5e35 11651 | 7 times 5 equals 35. (Con... |
| 7t6e42 11652 | 7 times 6 equals 42. (Con... |
| 7t7e49 11653 | 7 times 7 equals 49. (Con... |
| 8t2e16 11654 | 8 times 2 equals 16. (Con... |
| 8t3e24 11655 | 8 times 3 equals 24. (Con... |
| 8t4e32 11656 | 8 times 4 equals 32. (Con... |
| 8t5e40 11657 | 8 times 5 equals 40. (Con... |
| 8t5e40OLD 11658 | Obsolete proof of ~ 8t5e40... |
| 8t6e48 11659 | 8 times 6 equals 48. (Con... |
| 8t6e48OLD 11660 | Obsolete proof of ~ 8t6e48... |
| 8t7e56 11661 | 8 times 7 equals 56. (Con... |
| 8t8e64 11662 | 8 times 8 equals 64. (Con... |
| 9t2e18 11663 | 9 times 2 equals 18. (Con... |
| 9t3e27 11664 | 9 times 3 equals 27. (Con... |
| 9t4e36 11665 | 9 times 4 equals 36. (Con... |
| 9t5e45 11666 | 9 times 5 equals 45. (Con... |
| 9t6e54 11667 | 9 times 6 equals 54. (Con... |
| 9t7e63 11668 | 9 times 7 equals 63. (Con... |
| 9t8e72 11669 | 9 times 8 equals 72. (Con... |
| 9t9e81 11670 | 9 times 9 equals 81. (Con... |
| 9t11e99 11671 | 9 times 11 equals 99. (Co... |
| 9t11e99OLD 11672 | Obsolete proof of ~ 9t11e9... |
| 9lt10 11673 | 9 is less than 10. (Contr... |
| 8lt10 11674 | 8 is less than 10. (Contr... |
| 7lt10 11675 | 7 is less than 10. (Contr... |
| 6lt10 11676 | 6 is less than 10. (Contr... |
| 5lt10 11677 | 5 is less than 10. (Contr... |
| 4lt10 11678 | 4 is less than 10. (Contr... |
| 3lt10 11679 | 3 is less than 10. (Contr... |
| 2lt10 11680 | 2 is less than 10. (Contr... |
| 1lt10 11681 | 1 is less than 10. (Contr... |
| decbin0 11682 | Decompose base 4 into base... |
| decbin2 11683 | Decompose base 4 into base... |
| decbin3 11684 | Decompose base 4 into base... |
| halfthird 11685 | Half minus a third. (Cont... |
| 5recm6rec 11686 | One fifth minus one sixth.... |
| uzval 11689 | The value of the upper int... |
| uzf 11690 | The domain and range of th... |
| eluz1 11691 | Membership in the upper se... |
| eluzel2 11692 | Implication of membership ... |
| eluz2 11693 | Membership in an upper set... |
| eluzmn 11694 | Membership in an earlier u... |
| eluz1i 11695 | Membership in an upper set... |
| eluzuzle 11696 | An integer in an upper set... |
| eluzelz 11697 | A member of an upper set o... |
| eluzelre 11698 | A member of an upper set o... |
| eluzelcn 11699 | A member of an upper set o... |
| eluzle 11700 | Implication of membership ... |
| eluz 11701 | Membership in an upper set... |
| uzid 11702 | Membership of the least me... |
| uzn0 11703 | The upper integers are all... |
| uztrn 11704 | Transitive law for sets of... |
| uztrn2 11705 | Transitive law for sets of... |
| uzneg 11706 | Contraposition law for upp... |
| uzssz 11707 | An upper set of integers i... |
| uzss 11708 | Subset relationship for tw... |
| uztric 11709 | Totality of the ordering r... |
| uz11 11710 | The upper integers functio... |
| eluzp1m1 11711 | Membership in the next upp... |
| eluzp1l 11712 | Strict ordering implied by... |
| eluzp1p1 11713 | Membership in the next upp... |
| eluzaddi 11714 | Membership in a later uppe... |
| eluzsubi 11715 | Membership in an earlier u... |
| eluzadd 11716 | Membership in a later uppe... |
| eluzsub 11717 | Membership in an earlier u... |
| uzm1 11718 | Choices for an element of ... |
| uznn0sub 11719 | The nonnegative difference... |
| uzin 11720 | Intersection of two upper ... |
| uzp1 11721 | Choices for an element of ... |
| nn0uz 11722 | Nonnegative integers expre... |
| nnuz 11723 | Positive integers expresse... |
| elnnuz 11724 | A positive integer express... |
| elnn0uz 11725 | A nonnegative integer expr... |
| eluz2nn 11726 | An integer is greater than... |
| eluzge2nn0 11727 | If an integer is greater t... |
| eluz2n0 11728 | An integer greater than or... |
| uzuzle23 11729 | An integer in the upper se... |
| eluzge3nn 11730 | If an integer is greater t... |
| uz3m2nn 11731 | An integer greater than or... |
| 1eluzge0 11732 | 1 is an integer greater th... |
| 2eluzge0 11733 | 2 is an integer greater th... |
| 2eluzge1 11734 | 2 is an integer greater th... |
| uznnssnn 11735 | The upper integers startin... |
| raluz 11736 | Restricted universal quant... |
| raluz2 11737 | Restricted universal quant... |
| rexuz 11738 | Restricted existential qua... |
| rexuz2 11739 | Restricted existential qua... |
| 2rexuz 11740 | Double existential quantif... |
| peano2uz 11741 | Second Peano postulate for... |
| peano2uzs 11742 | Second Peano postulate for... |
| peano2uzr 11743 | Reversed second Peano axio... |
| uzaddcl 11744 | Addition closure law for a... |
| nn0pzuz 11745 | The sum of a nonnegative i... |
| uzind4 11746 | Induction on the upper set... |
| uzind4ALT 11747 | Induction on the upper set... |
| uzind4s 11748 | Induction on the upper set... |
| uzind4s2 11749 | Induction on the upper set... |
| uzind4i 11750 | Induction on the upper int... |
| uzwo 11751 | Well-ordering principle: a... |
| uzwo2 11752 | Well-ordering principle: a... |
| nnwo 11753 | Well-ordering principle: a... |
| nnwof 11754 | Well-ordering principle: a... |
| nnwos 11755 | Well-ordering principle: a... |
| indstr 11756 | Strong Mathematical Induct... |
| eluznn0 11757 | Membership in a nonnegativ... |
| eluznn 11758 | Membership in a positive u... |
| eluz2b1 11759 | Two ways to say "an intege... |
| eluz2gt1 11760 | An integer greater than or... |
| eluz2b2 11761 | Two ways to say "an intege... |
| eluz2b3 11762 | Two ways to say "an intege... |
| uz2m1nn 11763 | One less than an integer g... |
| 1nuz2 11764 | 1 is not in ` ( ZZ>= `` 2 ... |
| elnn1uz2 11765 | A positive integer is eith... |
| uz2mulcl 11766 | Closure of multiplication ... |
| indstr2 11767 | Strong Mathematical Induct... |
| uzinfi 11768 | Extract the lower bound of... |
| nninf 11769 | The infimum of the set of ... |
| nn0inf 11770 | The infimum of the set of ... |
| infssuzle 11771 | The infimum of a subset of... |
| infssuzcl 11772 | The infimum of a subset of... |
| ublbneg 11773 | The image under negation o... |
| eqreznegel 11774 | Two ways to express the im... |
| supminf 11775 | The supremum of a bounded-... |
| lbzbi 11776 | If a set of reals is bound... |
| zsupss 11777 | Any nonempty bounded subse... |
| suprzcl2 11778 | The supremum of a bounded-... |
| suprzub 11779 | The supremum of a bounded-... |
| uzsupss 11780 | Any bounded subset of an u... |
| nn01to3 11781 | A (nonnegative) integer be... |
| nn0ge2m1nnALT 11782 | Alternate proof of ~ nn0ge... |
| uzwo3 11783 | Well-ordering principle: a... |
| zmin 11784 | There is a unique smallest... |
| zmax 11785 | There is a unique largest ... |
| zbtwnre 11786 | There is a unique integer ... |
| rebtwnz 11787 | There is a unique greatest... |
| elq 11790 | Membership in the set of r... |
| qmulz 11791 | If ` A ` is rational, then... |
| znq 11792 | The ratio of an integer an... |
| qre 11793 | A rational number is a rea... |
| zq 11794 | An integer is a rational n... |
| zssq 11795 | The integers are a subset ... |
| nn0ssq 11796 | The nonnegative integers a... |
| nnssq 11797 | The positive integers are ... |
| qssre 11798 | The rationals are a subset... |
| qsscn 11799 | The rationals are a subset... |
| qex 11800 | The set of rational number... |
| nnq 11801 | A positive integer is rati... |
| qcn 11802 | A rational number is a com... |
| qexALT 11803 | Alternate proof of ~ qex .... |
| qaddcl 11804 | Closure of addition of rat... |
| qnegcl 11805 | Closure law for the negati... |
| qmulcl 11806 | Closure of multiplication ... |
| qsubcl 11807 | Closure of subtraction of ... |
| qreccl 11808 | Closure of reciprocal of r... |
| qdivcl 11809 | Closure of division of rat... |
| qrevaddcl 11810 | Reverse closure law for ad... |
| nnrecq 11811 | The reciprocal of a positi... |
| irradd 11812 | The sum of an irrational n... |
| irrmul 11813 | The product of an irration... |
| rpnnen1lem2 11814 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem1 11815 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem3 11816 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem4 11817 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem5 11818 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem6 11819 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1 11820 | One half of ~ rpnnen , whe... |
| rpnnen1lem1OLD 11821 | Lemma for ~ rpnnen1OLD . ... |
| rpnnen1lem3OLD 11822 | Lemma for ~ rpnnen1OLD . ... |
| rpnnen1lem4OLD 11823 | Lemma for ~ rpnnen1OLD . ... |
| rpnnen1lem5OLD 11824 | Lemma for ~ rpnnen1OLD . ... |
| rpnnen1OLD 11825 | One half of ~ rpnnen , whe... |
| reexALT 11826 | Alternate proof of ~ reex ... |
| cnref1o 11827 | There is a natural one-to-... |
| cnexALT 11828 | The set of complex numbers... |
| xrex 11829 | The set of extended reals ... |
| addex 11830 | The addition operation is ... |
| mulex 11831 | The multiplication operati... |
| elrp 11834 | Membership in the set of p... |
| elrpii 11835 | Membership in the set of p... |
| 1rp 11836 | 1 is a positive real. (Co... |
| 2rp 11837 | 2 is a positive real. (Co... |
| 3rp 11838 | 3 is a positive real. (Co... |
| rpre 11839 | A positive real is a real.... |
| rpxr 11840 | A positive real is an exte... |
| rpcn 11841 | A positive real is a compl... |
| nnrp 11842 | A positive integer is a po... |
| rpssre 11843 | The positive reals are a s... |
| rpgt0 11844 | A positive real is greater... |
| rpge0 11845 | A positive real is greater... |
| rpregt0 11846 | A positive real is a posit... |
| rprege0 11847 | A positive real is a nonne... |
| rpne0 11848 | A positive real is nonzero... |
| rprene0 11849 | A positive real is a nonze... |
| rpcnne0 11850 | A positive real is a nonze... |
| rpcndif0 11851 | A positive real number is ... |
| ralrp 11852 | Quantification over positi... |
| rexrp 11853 | Quantification over positi... |
| rpaddcl 11854 | Closure law for addition o... |
| rpmulcl 11855 | Closure law for multiplica... |
| rpdivcl 11856 | Closure law for division o... |
| rpreccl 11857 | Closure law for reciprocat... |
| rphalfcl 11858 | Closure law for half of a ... |
| rpgecl 11859 | A number greater or equal ... |
| rphalflt 11860 | Half of a positive real is... |
| rerpdivcl 11861 | Closure law for division o... |
| ge0p1rp 11862 | A nonnegative number plus ... |
| rpneg 11863 | Either a nonzero real or i... |
| negelrp 11864 | Elementhood of a negation ... |
| 0nrp 11865 | Zero is not a positive rea... |
| ltsubrp 11866 | Subtracting a positive rea... |
| ltaddrp 11867 | Adding a positive number t... |
| difrp 11868 | Two ways to say one number... |
| elrpd 11869 | Membership in the set of p... |
| nnrpd 11870 | A positive integer is a po... |
| zgt1rpn0n1 11871 | An integer greater than 1 ... |
| rpred 11872 | A positive real is a real.... |
| rpxrd 11873 | A positive real is an exte... |
| rpcnd 11874 | A positive real is a compl... |
| rpgt0d 11875 | A positive real is greater... |
| rpge0d 11876 | A positive real is greater... |
| rpne0d 11877 | A positive real is nonzero... |
| rpregt0d 11878 | A positive real is real an... |
| rprege0d 11879 | A positive real is real an... |
| rprene0d 11880 | A positive real is a nonze... |
| rpcnne0d 11881 | A positive real is a nonze... |
| rpreccld 11882 | Closure law for reciprocat... |
| rprecred 11883 | Closure law for reciprocat... |
| rphalfcld 11884 | Closure law for half of a ... |
| reclt1d 11885 | The reciprocal of a positi... |
| recgt1d 11886 | The reciprocal of a positi... |
| rpaddcld 11887 | Closure law for addition o... |
| rpmulcld 11888 | Closure law for multiplica... |
| rpdivcld 11889 | Closure law for division o... |
| ltrecd 11890 | The reciprocal of both sid... |
| lerecd 11891 | The reciprocal of both sid... |
| ltrec1d 11892 | Reciprocal swap in a 'less... |
| lerec2d 11893 | Reciprocal swap in a 'less... |
| lediv2ad 11894 | Division of both sides of ... |
| ltdiv2d 11895 | Division of a positive num... |
| lediv2d 11896 | Division of a positive num... |
| ledivdivd 11897 | Invert ratios of positive ... |
| divge1 11898 | The ratio of a number over... |
| divlt1lt 11899 | A real number divided by a... |
| divle1le 11900 | A real number divided by a... |
| ledivge1le 11901 | If a number is less than o... |
| ge0p1rpd 11902 | A nonnegative number plus ... |
| rerpdivcld 11903 | Closure law for division o... |
| ltsubrpd 11904 | Subtracting a positive rea... |
| ltaddrpd 11905 | Adding a positive number t... |
| ltaddrp2d 11906 | Adding a positive number t... |
| ltmulgt11d 11907 | Multiplication by a number... |
| ltmulgt12d 11908 | Multiplication by a number... |
| gt0divd 11909 | Division of a positive num... |
| ge0divd 11910 | Division of a nonnegative ... |
| rpgecld 11911 | A number greater or equal ... |
| divge0d 11912 | The ratio of nonnegative a... |
| ltmul1d 11913 | The ratio of nonnegative a... |
| ltmul2d 11914 | Multiplication of both sid... |
| lemul1d 11915 | Multiplication of both sid... |
| lemul2d 11916 | Multiplication of both sid... |
| ltdiv1d 11917 | Division of both sides of ... |
| lediv1d 11918 | Division of both sides of ... |
| ltmuldivd 11919 | 'Less than' relationship b... |
| ltmuldiv2d 11920 | 'Less than' relationship b... |
| lemuldivd 11921 | 'Less than or equal to' re... |
| lemuldiv2d 11922 | 'Less than or equal to' re... |
| ltdivmuld 11923 | 'Less than' relationship b... |
| ltdivmul2d 11924 | 'Less than' relationship b... |
| ledivmuld 11925 | 'Less than or equal to' re... |
| ledivmul2d 11926 | 'Less than or equal to' re... |
| ltmul1dd 11927 | The ratio of nonnegative a... |
| ltmul2dd 11928 | Multiplication of both sid... |
| ltdiv1dd 11929 | Division of both sides of ... |
| lediv1dd 11930 | Division of both sides of ... |
| lediv12ad 11931 | Comparison of ratio of two... |
| mul2lt0rlt0 11932 | If the result of a multipl... |
| mul2lt0rgt0 11933 | If the result of a multipl... |
| mul2lt0llt0 11934 | If the result of a multipl... |
| mul2lt0lgt0 11935 | If the result of a multipl... |
| mul2lt0bi 11936 | If the result of a multipl... |
| ltdiv23d 11937 | Swap denominator with othe... |
| lediv23d 11938 | Swap denominator with othe... |
| lt2mul2divd 11939 | The ratio of nonnegative a... |
| nnledivrp 11940 | Division of a positive int... |
| nn0ledivnn 11941 | Division of a nonnegative ... |
| addlelt 11942 | If the sum of a real numbe... |
| ltxr 11949 | The 'less than' binary rel... |
| elxr 11950 | Membership in the set of e... |
| xrnemnf 11951 | An extended real other tha... |
| xrnepnf 11952 | An extended real other tha... |
| xrltnr 11953 | The extended real 'less th... |
| ltpnf 11954 | Any (finite) real is less ... |
| ltpnfd 11955 | Any (finite) real is less ... |
| 0ltpnf 11956 | Zero is less than plus inf... |
| mnflt 11957 | Minus infinity is less tha... |
| mnfltd 11958 | Minus infinity is less tha... |
| mnflt0 11959 | Minus infinity is less tha... |
| mnfltpnf 11960 | Minus infinity is less tha... |
| mnfltxr 11961 | Minus infinity is less tha... |
| pnfnlt 11962 | No extended real is greate... |
| nltmnf 11963 | No extended real is less t... |
| pnfge 11964 | Plus infinity is an upper ... |
| xnn0n0n1ge2b 11965 | An extended nonnegative in... |
| 0lepnf 11966 | 0 less than or equal to po... |
| xnn0ge0 11967 | An extended nonnegative in... |
| nn0pnfge0OLD 11968 | Obsolete version of ~ xnn0... |
| mnfle 11969 | Minus infinity is less tha... |
| xrltnsym 11970 | Ordering on the extended r... |
| xrltnsym2 11971 | 'Less than' is antisymmetr... |
| xrlttri 11972 | Ordering on the extended r... |
| xrlttr 11973 | Ordering on the extended r... |
| xrltso 11974 | 'Less than' is a strict or... |
| xrlttri2 11975 | Trichotomy law for 'less t... |
| xrlttri3 11976 | Trichotomy law for 'less t... |
| xrleloe 11977 | 'Less than or equal' expre... |
| xrleltne 11978 | 'Less than or equal to' im... |
| xrltlen 11979 | 'Less than' expressed in t... |
| dfle2 11980 | Alternative definition of ... |
| dflt2 11981 | Alternative definition of ... |
| xrltle 11982 | 'Less than' implies 'less ... |
| xrleid 11983 | 'Less than or equal to' is... |
| xrletri 11984 | Trichotomy law for extende... |
| xrletri3 11985 | Trichotomy law for extende... |
| xrletrid 11986 | Trichotomy law for extende... |
| xrlelttr 11987 | Transitive law for orderin... |
| xrltletr 11988 | Transitive law for orderin... |
| xrletr 11989 | Transitive law for orderin... |
| xrlttrd 11990 | Transitive law for orderin... |
| xrlelttrd 11991 | Transitive law for orderin... |
| xrltletrd 11992 | Transitive law for orderin... |
| xrletrd 11993 | Transitive law for orderin... |
| xrltne 11994 | 'Less than' implies not eq... |
| nltpnft 11995 | An extended real is not le... |
| xgepnf 11996 | An extended real which is ... |
| ngtmnft 11997 | An extended real is not gr... |
| xlemnf 11998 | An extended real which is ... |
| xrrebnd 11999 | An extended real is real i... |
| xrre 12000 | A way of proving that an e... |
| xrre2 12001 | An extended real between t... |
| xrre3 12002 | A way of proving that an e... |
| ge0gtmnf 12003 | A nonnegative extended rea... |
| ge0nemnf 12004 | A nonnegative extended rea... |
| xrrege0 12005 | A nonnegative extended rea... |
| xrmax1 12006 | An extended real is less t... |
| xrmax2 12007 | An extended real is less t... |
| xrmin1 12008 | The minimum of two extende... |
| xrmin2 12009 | The minimum of two extende... |
| xrmaxeq 12010 | The maximum of two extende... |
| xrmineq 12011 | The minimum of two extende... |
| xrmaxlt 12012 | Two ways of saying the max... |
| xrltmin 12013 | Two ways of saying an exte... |
| xrmaxle 12014 | Two ways of saying the max... |
| xrlemin 12015 | Two ways of saying a numbe... |
| max1 12016 | A number is less than or e... |
| max1ALT 12017 | A number is less than or e... |
| max2 12018 | A number is less than or e... |
| 2resupmax 12019 | The supremum of two real n... |
| min1 12020 | The minimum of two numbers... |
| min2 12021 | The minimum of two numbers... |
| maxle 12022 | Two ways of saying the max... |
| lemin 12023 | Two ways of saying a numbe... |
| maxlt 12024 | Two ways of saying the max... |
| ltmin 12025 | Two ways of saying a numbe... |
| lemaxle 12026 | A real number which is les... |
| max0sub 12027 | Decompose a real number in... |
| ifle 12028 | An if statement transforms... |
| z2ge 12029 | There exists an integer gr... |
| qbtwnre 12030 | The rational numbers are d... |
| qbtwnxr 12031 | The rational numbers are d... |
| qsqueeze 12032 | If a nonnegative real is l... |
| qextltlem 12033 | Lemma for ~ qextlt and qex... |
| qextlt 12034 | An extensionality-like pro... |
| qextle 12035 | An extensionality-like pro... |
| xralrple 12036 | Show that ` A ` is less th... |
| alrple 12037 | Show that ` A ` is less th... |
| xnegeq 12038 | Equality of two extended n... |
| xnegex 12039 | A negative extended real e... |
| xnegpnf 12040 | Minus ` +oo ` . Remark of... |
| xnegmnf 12041 | Minus ` -oo ` . Remark of... |
| rexneg 12042 | Minus a real number. Rema... |
| xneg0 12043 | The negative of zero. (Co... |
| xnegcl 12044 | Closure of extended real n... |
| xnegneg 12045 | Extended real version of ~... |
| xneg11 12046 | Extended real version of ~... |
| xltnegi 12047 | Forward direction of ~ xlt... |
| xltneg 12048 | Extended real version of ~... |
| xleneg 12049 | Extended real version of ~... |
| xlt0neg1 12050 | Extended real version of ~... |
| xlt0neg2 12051 | Extended real version of ~... |
| xle0neg1 12052 | Extended real version of ~... |
| xle0neg2 12053 | Extended real version of ~... |
| xaddval 12054 | Value of the extended real... |
| xaddf 12055 | The extended real addition... |
| xmulval 12056 | Value of the extended real... |
| xaddpnf1 12057 | Addition of positive infin... |
| xaddpnf2 12058 | Addition of positive infin... |
| xaddmnf1 12059 | Addition of negative infin... |
| xaddmnf2 12060 | Addition of negative infin... |
| pnfaddmnf 12061 | Addition of positive and n... |
| mnfaddpnf 12062 | Addition of negative and p... |
| rexadd 12063 | The extended real addition... |
| rexsub 12064 | Extended real subtraction ... |
| rexaddd 12065 | The extended real addition... |
| xnn0xaddcl 12066 | The extended nonnegative i... |
| xaddnemnf 12067 | Closure of extended real a... |
| xaddnepnf 12068 | Closure of extended real a... |
| xnegid 12069 | Extended real version of ~... |
| xaddcl 12070 | The extended real addition... |
| xaddcom 12071 | The extended real addition... |
| xaddid1 12072 | Extended real version of ~... |
| xaddid2 12073 | Extended real version of ~... |
| xaddid1d 12074 | ` 0 ` is a right identity ... |
| xnn0lenn0nn0 12075 | An extended nonnegative in... |
| xnn0le2is012 12076 | An extended nonnegative in... |
| xnn0xadd0 12077 | The sum of two extended no... |
| xnegdi 12078 | Extended real version of ~... |
| xaddass 12079 | Associativity of extended ... |
| xaddass2 12080 | Associativity of extended ... |
| xpncan 12081 | Extended real version of ~... |
| xnpcan 12082 | Extended real version of ~... |
| xleadd1a 12083 | Extended real version of ~... |
| xleadd2a 12084 | Commuted form of ~ xleadd1... |
| xleadd1 12085 | Weakened version of ~ xlea... |
| xltadd1 12086 | Extended real version of ~... |
| xltadd2 12087 | Extended real version of ~... |
| xaddge0 12088 | The sum of nonnegative ext... |
| xle2add 12089 | Extended real version of ~... |
| xlt2add 12090 | Extended real version of ~... |
| xsubge0 12091 | Extended real version of ~... |
| xposdif 12092 | Extended real version of ~... |
| xlesubadd 12093 | Under certain conditions, ... |
| xmullem 12094 | Lemma for ~ rexmul . (Con... |
| xmullem2 12095 | Lemma for ~ xmulneg1 . (C... |
| xmulcom 12096 | Extended real multiplicati... |
| xmul01 12097 | Extended real version of ~... |
| xmul02 12098 | Extended real version of ~... |
| xmulneg1 12099 | Extended real version of ~... |
| xmulneg2 12100 | Extended real version of ~... |
| rexmul 12101 | The extended real multipli... |
| xmulf 12102 | The extended real multipli... |
| xmulcl 12103 | Closure of extended real m... |
| xmulpnf1 12104 | Multiplication by plus inf... |
| xmulpnf2 12105 | Multiplication by plus inf... |
| xmulmnf1 12106 | Multiplication by minus in... |
| xmulmnf2 12107 | Multiplication by minus in... |
| xmulpnf1n 12108 | Multiplication by plus inf... |
| xmulid1 12109 | Extended real version of ~... |
| xmulid2 12110 | Extended real version of ~... |
| xmulm1 12111 | Extended real version of ~... |
| xmulasslem2 12112 | Lemma for ~ xmulass . (Co... |
| xmulgt0 12113 | Extended real version of ~... |
| xmulge0 12114 | Extended real version of ~... |
| xmulasslem 12115 | Lemma for ~ xmulass . (Co... |
| xmulasslem3 12116 | Lemma for ~ xmulass . (Co... |
| xmulass 12117 | Associativity of the exten... |
| xlemul1a 12118 | Extended real version of ~... |
| xlemul2a 12119 | Extended real version of ~... |
| xlemul1 12120 | Extended real version of ~... |
| xlemul2 12121 | Extended real version of ~... |
| xltmul1 12122 | Extended real version of ~... |
| xltmul2 12123 | Extended real version of ~... |
| xadddilem 12124 | Lemma for ~ xadddi . (Con... |
| xadddi 12125 | Distributive property for ... |
| xadddir 12126 | Commuted version of ~ xadd... |
| xadddi2 12127 | The assumption that the mu... |
| xadddi2r 12128 | Commuted version of ~ xadd... |
| x2times 12129 | Extended real version of ~... |
| xnegcld 12130 | Closure of extended real n... |
| xaddcld 12131 | The extended real addition... |
| xmulcld 12132 | Closure of extended real m... |
| xadd4d 12133 | Rearrangement of 4 terms i... |
| xnn0add4d 12134 | Rearrangement of 4 terms i... |
| xrsupexmnf 12135 | Adding minus infinity to a... |
| xrinfmexpnf 12136 | Adding plus infinity to a ... |
| xrsupsslem 12137 | Lemma for ~ xrsupss . (Co... |
| xrinfmsslem 12138 | Lemma for ~ xrinfmss . (C... |
| xrsupss 12139 | Any subset of extended rea... |
| xrinfmss 12140 | Any subset of extended rea... |
| xrinfmss2 12141 | Any subset of extended rea... |
| xrub 12142 | By quantifying only over r... |
| supxr 12143 | The supremum of a set of e... |
| supxr2 12144 | The supremum of a set of e... |
| supxrcl 12145 | The supremum of an arbitra... |
| supxrun 12146 | The supremum of the union ... |
| supxrmnf 12147 | Adding minus infinity to a... |
| supxrpnf 12148 | The supremum of a set of e... |
| supxrunb1 12149 | The supremum of an unbound... |
| supxrunb2 12150 | The supremum of an unbound... |
| supxrbnd1 12151 | The supremum of a bounded-... |
| supxrbnd2 12152 | The supremum of a bounded-... |
| xrsup0 12153 | The supremum of an empty s... |
| supxrub 12154 | A member of a set of exten... |
| supxrlub 12155 | The supremum of a set of e... |
| supxrleub 12156 | The supremum of a set of e... |
| supxrre 12157 | The real and extended real... |
| supxrbnd 12158 | The supremum of a bounded-... |
| supxrgtmnf 12159 | The supremum of a nonempty... |
| supxrre1 12160 | The supremum of a nonempty... |
| supxrre2 12161 | The supremum of a nonempty... |
| supxrss 12162 | Smaller sets of extended r... |
| infxrcl 12163 | The infimum of an arbitrar... |
| infxrlb 12164 | A member of a set of exten... |
| infxrgelb 12165 | The infimum of a set of ex... |
| infxrre 12166 | The real and extended real... |
| infxrmnf 12167 | The infinimum of a set of ... |
| xrinf0 12168 | The infimum of the empty s... |
| infxrss 12169 | Larger sets of extended re... |
| reltre 12170 | For all real numbers there... |
| rpltrp 12171 | For all positive real numb... |
| reltxrnmnf 12172 | For all extended real numb... |
| infmremnf 12173 | The infimum of the reals i... |
| infmrp1 12174 | The infimum of the positiv... |
| ixxval 12183 | Value of the interval func... |
| elixx1 12184 | Membership in an interval ... |
| ixxf 12185 | The set of intervals of ex... |
| ixxex 12186 | The set of intervals of ex... |
| ixxssxr 12187 | The set of intervals of ex... |
| elixx3g 12188 | Membership in a set of ope... |
| ixxssixx 12189 | An interval is a subset of... |
| ixxdisj 12190 | Split an interval into dis... |
| ixxun 12191 | Split an interval into two... |
| ixxin 12192 | Intersection of two interv... |
| ixxss1 12193 | Subset relationship for in... |
| ixxss2 12194 | Subset relationship for in... |
| ixxss12 12195 | Subset relationship for in... |
| ixxub 12196 | Extract the upper bound of... |
| ixxlb 12197 | Extract the lower bound of... |
| iooex 12198 | The set of open intervals ... |
| iooval 12199 | Value of the open interval... |
| ioo0 12200 | An empty open interval of ... |
| ioon0 12201 | An open interval of extend... |
| ndmioo 12202 | The open interval function... |
| iooid 12203 | An open interval with iden... |
| elioo3g 12204 | Membership in a set of ope... |
| elioore 12205 | A member of an open interv... |
| lbioo 12206 | An open interval does not ... |
| ubioo 12207 | An open interval does not ... |
| iooval2 12208 | Value of the open interval... |
| iooin 12209 | Intersection of two open i... |
| iooss1 12210 | Subset relationship for op... |
| iooss2 12211 | Subset relationship for op... |
| iocval 12212 | Value of the open-below, c... |
| icoval 12213 | Value of the closed-below,... |
| iccval 12214 | Value of the closed interv... |
| elioo1 12215 | Membership in an open inte... |
| elioo2 12216 | Membership in an open inte... |
| elioc1 12217 | Membership in an open-belo... |
| elico1 12218 | Membership in a closed-bel... |
| elicc1 12219 | Membership in a closed int... |
| iccid 12220 | A closed interval with ide... |
| ico0 12221 | An empty open interval of ... |
| ioc0 12222 | An empty open interval of ... |
| icc0 12223 | An empty closed interval o... |
| elicod 12224 | Membership in a left close... |
| icogelb 12225 | An element of a left close... |
| elicore 12226 | A member of a left closed,... |
| ubioc1 12227 | The upper bound belongs to... |
| lbico1 12228 | The lower bound belongs to... |
| iccleub 12229 | An element of a closed int... |
| iccgelb 12230 | An element of a closed int... |
| elioo5 12231 | Membership in an open inte... |
| eliooxr 12232 | A nonempty open interval s... |
| eliooord 12233 | Ordering implied by a memb... |
| elioo4g 12234 | Membership in an open inte... |
| ioossre 12235 | An open interval is a set ... |
| elioc2 12236 | Membership in an open-belo... |
| elico2 12237 | Membership in a closed-bel... |
| elicc2 12238 | Membership in a closed rea... |
| elicc2i 12239 | Inference for membership i... |
| elicc4 12240 | Membership in a closed rea... |
| iccss 12241 | Condition for a closed int... |
| iccssioo 12242 | Condition for a closed int... |
| icossico 12243 | Condition for a closed-bel... |
| iccss2 12244 | Condition for a closed int... |
| iccssico 12245 | Condition for a closed int... |
| iccssioo2 12246 | Condition for a closed int... |
| iccssico2 12247 | Condition for a closed int... |
| ioomax 12248 | The open interval from min... |
| iccmax 12249 | The closed interval from m... |
| ioopos 12250 | The set of positive reals ... |
| ioorp 12251 | The set of positive reals ... |
| iooshf 12252 | Shift the arguments of the... |
| iocssre 12253 | A closed-above interval wi... |
| icossre 12254 | A closed-below interval wi... |
| iccssre 12255 | A closed real interval is ... |
| iccssxr 12256 | A closed interval is a set... |
| iocssxr 12257 | An open-below, closed-abov... |
| icossxr 12258 | A closed-below, open-above... |
| ioossicc 12259 | An open interval is a subs... |
| icossicc 12260 | A closed-below, open-above... |
| iocssicc 12261 | A closed-above, open-below... |
| ioossico 12262 | An open interval is a subs... |
| iocssioo 12263 | Condition for a closed int... |
| icossioo 12264 | Condition for a closed int... |
| ioossioo 12265 | Condition for an open inte... |
| iccsupr 12266 | A nonempty subset of a clo... |
| elioopnf 12267 | Membership in an unbounded... |
| elioomnf 12268 | Membership in an unbounded... |
| elicopnf 12269 | Membership in a closed unb... |
| repos 12270 | Two ways of saying that a ... |
| ioof 12271 | The set of open intervals ... |
| iccf 12272 | The set of closed interval... |
| unirnioo 12273 | The union of the range of ... |
| dfioo2 12274 | Alternate definition of th... |
| ioorebas 12275 | Open intervals are element... |
| xrge0neqmnf 12276 | An extended nonnegative re... |
| xrge0nre 12277 | An extended real which is ... |
| elrege0 12278 | The predicate "is a nonneg... |
| nn0rp0 12279 | A nonnegative integer is a... |
| rge0ssre 12280 | Nonnegative real numbers a... |
| elxrge0 12281 | Elementhood in the set of ... |
| 0e0icopnf 12282 | 0 is a member of ` ( 0 [,)... |
| 0e0iccpnf 12283 | 0 is a member of ` ( 0 [,]... |
| ge0addcl 12284 | The nonnegative reals are ... |
| ge0mulcl 12285 | The nonnegative reals are ... |
| ge0xaddcl 12286 | The nonnegative reals are ... |
| ge0xmulcl 12287 | The nonnegative extended r... |
| lbicc2 12288 | The lower bound of a close... |
| ubicc2 12289 | The upper bound of a close... |
| 0elunit 12290 | Zero is an element of the ... |
| 1elunit 12291 | One is an element of the c... |
| iooneg 12292 | Membership in a negated op... |
| iccneg 12293 | Membership in a negated cl... |
| icoshft 12294 | A shifted real is a member... |
| icoshftf1o 12295 | Shifting a closed-below, o... |
| icoun 12296 | The union of end-to-end cl... |
| icodisj 12297 | End-to-end closed-below, o... |
| snunioo 12298 | The closure of one end of ... |
| snunico 12299 | The closure of the open en... |
| snunioc 12300 | The closure of the open en... |
| prunioo 12301 | The closure of an open rea... |
| ioodisj 12302 | If the upper bound of one ... |
| ioojoin 12303 | Join two open intervals to... |
| difreicc 12304 | The class difference of ` ... |
| iccsplit 12305 | Split a closed interval in... |
| iccshftr 12306 | Membership in a shifted in... |
| iccshftri 12307 | Membership in a shifted in... |
| iccshftl 12308 | Membership in a shifted in... |
| iccshftli 12309 | Membership in a shifted in... |
| iccdil 12310 | Membership in a dilated in... |
| iccdili 12311 | Membership in a dilated in... |
| icccntr 12312 | Membership in a contracted... |
| icccntri 12313 | Membership in a contracted... |
| divelunit 12314 | A condition for a ratio to... |
| lincmb01cmp 12315 | A linear combination of tw... |
| iccf1o 12316 | Describe a bijection from ... |
| iccen 12317 | Any nontrivial closed inte... |
| xov1plusxeqvd 12318 | A complex number ` X ` is ... |
| unitssre 12319 | ` ( 0 [,] 1 ) ` is a subse... |
| supicc 12320 | Supremum of a bounded set ... |
| supiccub 12321 | The supremum of a bounded ... |
| supicclub 12322 | The supremum of a bounded ... |
| supicclub2 12323 | The supremum of a bounded ... |
| zltaddlt1le 12324 | The sum of an integer and ... |
| xnn0xrge0 12325 | An extended nonnegative in... |
| fzval 12328 | The value of a finite set ... |
| fzval2 12329 | An alternative way of expr... |
| fzf 12330 | Establish the domain and c... |
| elfz1 12331 | Membership in a finite set... |
| elfz 12332 | Membership in a finite set... |
| elfz2 12333 | Membership in a finite set... |
| elfz5 12334 | Membership in a finite set... |
| elfz4 12335 | Membership in a finite set... |
| elfzuzb 12336 | Membership in a finite set... |
| eluzfz 12337 | Membership in a finite set... |
| elfzuz 12338 | A member of a finite set o... |
| elfzuz3 12339 | Membership in a finite set... |
| elfzel2 12340 | Membership in a finite set... |
| elfzel1 12341 | Membership in a finite set... |
| elfzelz 12342 | A member of a finite set o... |
| fzssz 12343 | A finite sequence of integ... |
| elfzle1 12344 | A member of a finite set o... |
| elfzle2 12345 | A member of a finite set o... |
| elfzuz2 12346 | Implication of membership ... |
| elfzle3 12347 | Membership in a finite set... |
| eluzfz1 12348 | Membership in a finite set... |
| eluzfz2 12349 | Membership in a finite set... |
| eluzfz2b 12350 | Membership in a finite set... |
| elfz3 12351 | Membership in a finite set... |
| elfz1eq 12352 | Membership in a finite set... |
| elfzubelfz 12353 | If there is a member in a ... |
| peano2fzr 12354 | A Peano-postulate-like the... |
| fzn0 12355 | Properties of a finite int... |
| fz0 12356 | A finite set of sequential... |
| fzn 12357 | A finite set of sequential... |
| fzen 12358 | A shifted finite set of se... |
| fz1n 12359 | A 1-based finite set of se... |
| 0nelfz1 12360 | 0 is not an element of a f... |
| 0fz1 12361 | Two ways to say a finite 1... |
| fz10 12362 | There are no integers betw... |
| uzsubsubfz 12363 | Membership of an integer g... |
| uzsubsubfz1 12364 | Membership of an integer g... |
| ige3m2fz 12365 | Membership of an integer g... |
| fzsplit2 12366 | Split a finite interval of... |
| fzsplit 12367 | Split a finite interval of... |
| fzdisj 12368 | Condition for two finite i... |
| fz01en 12369 | 0-based and 1-based finite... |
| elfznn 12370 | A member of a finite set o... |
| elfz1end 12371 | A nonempty finite range of... |
| fz1ssnn 12372 | A finite set of positive i... |
| fznn0sub 12373 | Subtraction closure for a ... |
| fzmmmeqm 12374 | Subtracting the difference... |
| fzaddel 12375 | Membership of a sum in a f... |
| fzadd2 12376 | Membership of a sum in a f... |
| fzsubel 12377 | Membership of a difference... |
| fzopth 12378 | A finite set of sequential... |
| fzass4 12379 | Two ways to express a nond... |
| fzss1 12380 | Subset relationship for fi... |
| fzss2 12381 | Subset relationship for fi... |
| fzssuz 12382 | A finite set of sequential... |
| fzsn 12383 | A finite interval of integ... |
| fzssp1 12384 | Subset relationship for fi... |
| fzssnn 12385 | Finite sets of sequential ... |
| ssfzunsnext 12386 | A subset of a finite seque... |
| ssfzunsn 12387 | A subset of a finite seque... |
| fzsuc 12388 | Join a successor to the en... |
| fzpred 12389 | Join a predecessor to the ... |
| fzpreddisj 12390 | A finite set of sequential... |
| elfzp1 12391 | Append an element to a fin... |
| fzp1ss 12392 | Subset relationship for fi... |
| fzelp1 12393 | Membership in a set of seq... |
| fzp1elp1 12394 | Add one to an element of a... |
| fznatpl1 12395 | Shift membership in a fini... |
| fzpr 12396 | A finite interval of integ... |
| fztp 12397 | A finite interval of integ... |
| fzsuc2 12398 | Join a successor to the en... |
| fzp1disj 12399 | ` ( M ... ( N + 1 ) ) ` is... |
| fzdifsuc 12400 | Remove a successor from th... |
| fzprval 12401 | Two ways of defining the f... |
| fztpval 12402 | Two ways of defining the f... |
| fzrev 12403 | Reversal of start and end ... |
| fzrev2 12404 | Reversal of start and end ... |
| fzrev2i 12405 | Reversal of start and end ... |
| fzrev3 12406 | The "complement" of a memb... |
| fzrev3i 12407 | The "complement" of a memb... |
| fznn 12408 | Finite set of sequential i... |
| elfz1b 12409 | Membership in a 1 based fi... |
| elfz1uz 12410 | Membership in a 1 based fi... |
| elfzm11 12411 | Membership in a finite set... |
| uzsplit 12412 | Express an upper integer s... |
| uzdisj 12413 | The first ` N ` elements o... |
| fseq1p1m1 12414 | Add/remove an item to/from... |
| fseq1m1p1 12415 | Add/remove an item to/from... |
| fz1sbc 12416 | Quantification over a one-... |
| elfzp1b 12417 | An integer is a member of ... |
| elfzm1b 12418 | An integer is a member of ... |
| elfzp12 12419 | Options for membership in ... |
| fzm1 12420 | Choices for an element of ... |
| fzneuz 12421 | No finite set of sequentia... |
| fznuz 12422 | Disjointness of the upper ... |
| uznfz 12423 | Disjointness of the upper ... |
| fzp1nel 12424 | One plus the upper bound o... |
| fzrevral 12425 | Reversal of scanning order... |
| fzrevral2 12426 | Reversal of scanning order... |
| fzrevral3 12427 | Reversal of scanning order... |
| fzshftral 12428 | Shift the scanning order i... |
| ige2m1fz1 12429 | Membership of an integer g... |
| ige2m1fz 12430 | Membership in a 0 based fi... |
| elfz2nn0 12431 | Membership in a finite set... |
| fznn0 12432 | Characterization of a fini... |
| elfznn0 12433 | A member of a finite set o... |
| elfz3nn0 12434 | The upper bound of a nonem... |
| fz0ssnn0 12435 | Finite sets of sequential ... |
| 0elfz 12436 | 0 is an element of a finit... |
| nn0fz0 12437 | A nonnegative integer is a... |
| elfz0add 12438 | An element of a finite set... |
| fz0sn 12439 | An integer range from 0 to... |
| fz0tp 12440 | An integer range from 0 to... |
| fz0to3un2pr 12441 | An integer range from 0 to... |
| fz0to4untppr 12442 | An integer range from 0 to... |
| elfz0ubfz0 12443 | An element of a finite set... |
| elfz0fzfz0 12444 | A member of a finite set o... |
| fz0fzelfz0 12445 | If a member of a finite se... |
| fznn0sub2 12446 | Subtraction closure for a ... |
| uzsubfz0 12447 | Membership of an integer g... |
| fz0fzdiffz0 12448 | The difference of an integ... |
| elfzmlbm 12449 | Subtracting the lower boun... |
| elfzmlbp 12450 | Subtracting the lower boun... |
| fzctr 12451 | Lemma for theorems about t... |
| difelfzle 12452 | The difference of two inte... |
| difelfznle 12453 | The difference of two inte... |
| nn0split 12454 | Express the set of nonnega... |
| nn0disj 12455 | The first ` N + 1 ` elemen... |
| fz0sn0fz1 12456 | A finite set of sequential... |
| fvffz0 12457 | The function value of a fu... |
| 1fv 12458 | A one value function. (Co... |
| 4fvwrd4 12459 | The first four function va... |
| 2ffzeq 12460 | Two functions over 0 based... |
| preduz 12461 | The value of the predecess... |
| prednn 12462 | The value of the predecess... |
| prednn0 12463 | The value of the predecess... |
| predfz 12464 | Calculate the predecessor ... |
| fzof 12467 | Functionality of the half-... |
| elfzoel1 12468 | Reverse closure for half-o... |
| elfzoel2 12469 | Reverse closure for half-o... |
| elfzoelz 12470 | Reverse closure for half-o... |
| fzoval 12471 | Value of the half-open int... |
| elfzo 12472 | Membership in a half-open ... |
| elfzo2 12473 | Membership in a half-open ... |
| elfzouz 12474 | Membership in a half-open ... |
| nelfzo 12475 | An integer not being a mem... |
| fzolb 12476 | The left endpoint of a hal... |
| fzolb2 12477 | The left endpoint of a hal... |
| elfzole1 12478 | A member in a half-open in... |
| elfzolt2 12479 | A member in a half-open in... |
| elfzolt3 12480 | Membership in a half-open ... |
| elfzolt2b 12481 | A member in a half-open in... |
| elfzolt3b 12482 | Membership in a half-open ... |
| fzonel 12483 | A half-open range does not... |
| elfzouz2 12484 | The upper bound of a half-... |
| elfzofz 12485 | A half-open range is conta... |
| elfzo3 12486 | Express membership in a ha... |
| fzon0 12487 | A half-open integer interv... |
| fzossfz 12488 | A half-open range is conta... |
| fzon 12489 | A half-open set of sequent... |
| fzo0n 12490 | A half-open range of nonne... |
| fzonlt0 12491 | A half-open integer range ... |
| fzo0 12492 | Half-open sets with equal ... |
| fzonnsub 12493 | If ` K < N ` then ` N - K ... |
| fzonnsub2 12494 | If ` M < N ` then ` N - M ... |
| fzoss1 12495 | Subset relationship for ha... |
| fzoss2 12496 | Subset relationship for ha... |
| fzossrbm1 12497 | Subset of a half open rang... |
| fzo0ss1 12498 | Subset relationship for ha... |
| fzossnn0 12499 | A half-open integer range ... |
| fzospliti 12500 | One direction of splitting... |
| fzosplit 12501 | Split a half-open integer ... |
| fzodisj 12502 | Abutting half-open integer... |
| fzouzsplit 12503 | Split an upper integer set... |
| fzouzdisj 12504 | A half-open integer range ... |
| fzodisjsn 12505 | A half-open integer range ... |
| prinfzo0 12506 | The intersection of a half... |
| lbfzo0 12507 | An integer is strictly gre... |
| elfzo0 12508 | Membership in a half-open ... |
| elfzo0z 12509 | Membership in a half-open ... |
| nn0p1elfzo 12510 | A nonnegative integer incr... |
| elfzo0le 12511 | A member in a half-open ra... |
| elfzonn0 12512 | A member of a half-open ra... |
| fzonmapblen 12513 | The result of subtracting ... |
| fzofzim 12514 | If a nonnegative integer i... |
| fz1fzo0m1 12515 | Translation of one between... |
| fzossnn 12516 | Half-open integer ranges s... |
| elfzo1 12517 | Membership in a half-open ... |
| fzo1fzo0n0 12518 | An integer between 1 and a... |
| fzo0n0 12519 | A half-open integer range ... |
| fzoaddel 12520 | Translate membership in a ... |
| fzo0addel 12521 | Translate membership in a ... |
| fzo0addelr 12522 | Translate membership in a ... |
| fzoaddel2 12523 | Translate membership in a ... |
| elfzoext 12524 | Membership of an integer i... |
| elincfzoext 12525 | Membership of an increased... |
| fzosubel 12526 | Translate membership in a ... |
| fzosubel2 12527 | Membership in a translated... |
| fzosubel3 12528 | Membership in a translated... |
| eluzgtdifelfzo 12529 | Membership of the differen... |
| ige2m2fzo 12530 | Membership of an integer g... |
| fzocatel 12531 | Translate membership in a ... |
| ubmelfzo 12532 | If an integer in a 1 based... |
| elfzodifsumelfzo 12533 | If an integer is in a half... |
| elfzom1elp1fzo 12534 | Membership of an integer i... |
| elfzom1elfzo 12535 | Membership in a half-open ... |
| fzval3 12536 | Expressing a closed intege... |
| fz0add1fz1 12537 | Translate membership in a ... |
| fzosn 12538 | Expressing a singleton as ... |
| elfzomin 12539 | Membership of an integer i... |
| zpnn0elfzo 12540 | Membership of an integer i... |
| zpnn0elfzo1 12541 | Membership of an integer i... |
| fzosplitsnm1 12542 | Removing a singleton from ... |
| elfzonlteqm1 12543 | If an element of a half-op... |
| fzonn0p1 12544 | A nonnegative integer is e... |
| fzossfzop1 12545 | A half-open range of nonne... |
| fzonn0p1p1 12546 | If a nonnegative integer i... |
| elfzom1p1elfzo 12547 | Increasing an element of a... |
| fzo0ssnn0 12548 | Half-open integer ranges s... |
| fzo0ssnn0OLD 12549 | Obsolete proof of ~ fzo0ss... |
| fzo01 12550 | Expressing the singleton o... |
| fzo12sn 12551 | A 1-based half-open intege... |
| fzo13pr 12552 | A 1-based half-open intege... |
| fzo0to2pr 12553 | A half-open integer range ... |
| fzo0to3tp 12554 | A half-open integer range ... |
| fzo0to42pr 12555 | A half-open integer range ... |
| fzo1to4tp 12556 | A half-open integer range ... |
| fzo0sn0fzo1 12557 | A half-open range of nonne... |
| elfzo0l 12558 | A member of a half-open ra... |
| fzoend 12559 | The endpoint of a half-ope... |
| fzo0end 12560 | The endpoint of a zero-bas... |
| ssfzo12 12561 | Subset relationship for ha... |
| ssfzoulel 12562 | If a half-open integer ran... |
| ssfzo12bi 12563 | Subset relationship for ha... |
| ubmelm1fzo 12564 | The result of subtracting ... |
| fzofzp1 12565 | If a point is in a half-op... |
| fzofzp1b 12566 | If a point is in a half-op... |
| elfzom1b 12567 | An integer is a member of ... |
| elfzom1elp1fzo1 12568 | Membership of a nonnegativ... |
| elfzo1elm1fzo0 12569 | Membership of a positive i... |
| elfzonelfzo 12570 | If an element of a half-op... |
| fzonfzoufzol 12571 | If an element of a half-op... |
| elfzomelpfzo 12572 | An integer increased by an... |
| elfznelfzo 12573 | A value in a finite set of... |
| elfznelfzob 12574 | A value in a finite set of... |
| peano2fzor 12575 | A Peano-postulate-like the... |
| fzosplitsn 12576 | Extending a half-open rang... |
| fzosplitpr 12577 | Extending a half-open inte... |
| fzosplitprm1 12578 | Extending a half-open inte... |
| fzosplitsni 12579 | Membership in a half-open ... |
| fzisfzounsn 12580 | A finite interval of integ... |
| elfzr 12581 | A member of a finite inter... |
| elfzlmr 12582 | A member of a finite inter... |
| elfz0lmr 12583 | A member of a finite inter... |
| fzostep1 12584 | Two possibilities for a nu... |
| fzoshftral 12585 | Shift the scanning order i... |
| fzind2 12586 | Induction on the integers ... |
| fvinim0ffz 12587 | The function values for th... |
| injresinjlem 12588 | Lemma for ~ injresinj . (... |
| injresinj 12589 | A function whose restricti... |
| subfzo0 12590 | The difference between two... |
| flval 12595 | Value of the floor (greate... |
| flcl 12596 | The floor (greatest intege... |
| reflcl 12597 | The floor (greatest intege... |
| fllelt 12598 | A basic property of the fl... |
| flcld 12599 | The floor (greatest intege... |
| flle 12600 | A basic property of the fl... |
| flltp1 12601 | A basic property of the fl... |
| fllep1 12602 | A basic property of the fl... |
| fraclt1 12603 | The fractional part of a r... |
| fracle1 12604 | The fractional part of a r... |
| fracge0 12605 | The fractional part of a r... |
| flge 12606 | The floor function value i... |
| fllt 12607 | The floor function value i... |
| flflp1 12608 | Move floor function betwee... |
| flid 12609 | An integer is its own floo... |
| flidm 12610 | The floor function is idem... |
| flidz 12611 | A real number equals its f... |
| flltnz 12612 | If A is not an integer, th... |
| flwordi 12613 | Ordering relationship for ... |
| flword2 12614 | Ordering relationship for ... |
| flval2 12615 | An alternate way to define... |
| flval3 12616 | An alternate way to define... |
| flbi 12617 | A condition equivalent to ... |
| flbi2 12618 | A condition equivalent to ... |
| adddivflid 12619 | The floor of a sum of an i... |
| ico01fl0 12620 | The floor of a real number... |
| flge0nn0 12621 | The floor of a number grea... |
| flge1nn 12622 | The floor of a number grea... |
| fldivnn0 12623 | The floor function of a di... |
| refldivcl 12624 | The floor function of a di... |
| divfl0 12625 | The floor of a fraction is... |
| fladdz 12626 | An integer can be moved in... |
| flzadd 12627 | An integer can be moved in... |
| flmulnn0 12628 | Move a nonnegative integer... |
| btwnzge0 12629 | A real bounded between an ... |
| 2tnp1ge0ge0 12630 | Two times an integer plus ... |
| flhalf 12631 | Ordering relation for the ... |
| fldivle 12632 | The floor function of a di... |
| fldivnn0le 12633 | The floor function of a di... |
| flltdivnn0lt 12634 | The floor function of a di... |
| ltdifltdiv 12635 | If the dividend of a divis... |
| fldiv4p1lem1div2 12636 | The floor of an integer eq... |
| fldiv4lem1div2uz2 12637 | The floor of an integer gr... |
| fldiv4lem1div2 12638 | The floor of a positive in... |
| ceilval 12639 | The value of the ceiling f... |
| dfceil2 12640 | Alternative definition of ... |
| ceilval2 12641 | The value of the ceiling f... |
| ceicl 12642 | The ceiling function retur... |
| ceilcl 12643 | Closure of the ceiling fun... |
| ceige 12644 | The ceiling of a real numb... |
| ceilge 12645 | The ceiling of a real numb... |
| ceim1l 12646 | One less than the ceiling ... |
| ceilm1lt 12647 | One less than the ceiling ... |
| ceile 12648 | The ceiling of a real numb... |
| ceille 12649 | The ceiling of a real numb... |
| ceilid 12650 | An integer is its own ceil... |
| ceilidz 12651 | A real number equals its c... |
| flleceil 12652 | The floor of a real number... |
| fleqceilz 12653 | A real number is an intege... |
| quoremz 12654 | Quotient and remainder of ... |
| quoremnn0 12655 | Quotient and remainder of ... |
| quoremnn0ALT 12656 | Alternate proof of ~ quore... |
| intfrac2 12657 | Decompose a real into inte... |
| intfracq 12658 | Decompose a rational numbe... |
| fldiv 12659 | Cancellation of the embedd... |
| fldiv2 12660 | Cancellation of an embedde... |
| fznnfl 12661 | Finite set of sequential i... |
| uzsup 12662 | An upper set of integers i... |
| ioopnfsup 12663 | An upper set of reals is u... |
| icopnfsup 12664 | An upper set of reals is u... |
| rpsup 12665 | The positive reals are unb... |
| resup 12666 | The real numbers are unbou... |
| xrsup 12667 | The extended real numbers ... |
| modval 12670 | The value of the modulo op... |
| modvalr 12671 | The value of the modulo op... |
| modcl 12672 | Closure law for the modulo... |
| flpmodeq 12673 | Partition of a division in... |
| modcld 12674 | Closure law for the modulo... |
| mod0 12675 | ` A mod B ` is zero iff ` ... |
| mulmod0 12676 | The product of an integer ... |
| negmod0 12677 | ` A ` is divisible by ` B ... |
| modge0 12678 | The modulo operation is no... |
| modlt 12679 | The modulo operation is le... |
| modelico 12680 | Modular reduction produces... |
| moddiffl 12681 | The modulo operation diffe... |
| moddifz 12682 | The modulo operation diffe... |
| modfrac 12683 | The fractional part of a n... |
| flmod 12684 | The floor function express... |
| intfrac 12685 | Break a number into its in... |
| zmod10 12686 | An integer modulo 1 is 0. ... |
| zmod1congr 12687 | Two arbitrary integers are... |
| modmulnn 12688 | Move a positive integer in... |
| modvalp1 12689 | The value of the modulo op... |
| zmodcl 12690 | Closure law for the modulo... |
| zmodcld 12691 | Closure law for the modulo... |
| zmodfz 12692 | An integer mod ` B ` lies ... |
| zmodfzo 12693 | An integer mod ` B ` lies ... |
| zmodfzp1 12694 | An integer mod ` B ` lies ... |
| modid 12695 | Identity law for modulo. ... |
| modid0 12696 | A positive real number mod... |
| modid2 12697 | Identity law for modulo. ... |
| zmodid2 12698 | Identity law for modulo re... |
| zmodidfzo 12699 | Identity law for modulo re... |
| zmodidfzoimp 12700 | Identity law for modulo re... |
| 0mod 12701 | Special case: 0 modulo a p... |
| 1mod 12702 | Special case: 1 modulo a r... |
| modabs 12703 | Absorption law for modulo.... |
| modabs2 12704 | Absorption law for modulo.... |
| modcyc 12705 | The modulo operation is pe... |
| modcyc2 12706 | The modulo operation is pe... |
| modadd1 12707 | Addition property of the m... |
| modaddabs 12708 | Absorption law for modulo.... |
| modaddmod 12709 | The sum of a real number m... |
| muladdmodid 12710 | The sum of a positive real... |
| mulp1mod1 12711 | The product of an integer ... |
| modmuladd 12712 | Decomposition of an intege... |
| modmuladdim 12713 | Implication of a decomposi... |
| modmuladdnn0 12714 | Implication of a decomposi... |
| negmod 12715 | The negation of a number m... |
| m1modnnsub1 12716 | Minus one modulo a positiv... |
| m1modge3gt1 12717 | Minus one modulo an intege... |
| addmodid 12718 | The sum of a positive inte... |
| addmodidr 12719 | The sum of a positive inte... |
| modadd2mod 12720 | The sum of a real number m... |
| modm1p1mod0 12721 | If an real number modulo a... |
| modltm1p1mod 12722 | If a real number modulo a ... |
| modmul1 12723 | Multiplication property of... |
| modmul12d 12724 | Multiplication property of... |
| modnegd 12725 | Negation property of the m... |
| modadd12d 12726 | Additive property of the m... |
| modsub12d 12727 | Subtraction property of th... |
| modsubmod 12728 | The difference of a real n... |
| modsubmodmod 12729 | The difference of a real n... |
| 2txmodxeq0 12730 | Two times a positive real ... |
| 2submod 12731 | If a real number is betwee... |
| modifeq2int 12732 | If a nonnegative integer i... |
| modaddmodup 12733 | The sum of an integer modu... |
| modaddmodlo 12734 | The sum of an integer modu... |
| modmulmod 12735 | The product of a real numb... |
| modmulmodr 12736 | The product of an integer ... |
| modaddmulmod 12737 | The sum of a real number a... |
| moddi 12738 | Distribute multiplication ... |
| modsubdir 12739 | Distribute the modulo oper... |
| modeqmodmin 12740 | A real number equals the d... |
| modirr 12741 | A number modulo an irratio... |
| modfzo0difsn 12742 | For a number within a half... |
| modsumfzodifsn 12743 | The sum of a number within... |
| modlteq 12744 | Two nonnegative integers l... |
| addmodlteq 12745 | Two nonnegative integers l... |
| om2uz0i 12746 | The mapping ` G ` is a one... |
| om2uzsuci 12747 | The value of ` G ` (see ~ ... |
| om2uzuzi 12748 | The value ` G ` (see ~ om2... |
| om2uzlti 12749 | Less-than relation for ` G... |
| om2uzlt2i 12750 | The mapping ` G ` (see ~ o... |
| om2uzrani 12751 | Range of ` G ` (see ~ om2u... |
| om2uzf1oi 12752 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzisoi 12753 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzoi 12754 | An alternative definition ... |
| om2uzrdg 12755 | A helper lemma for the val... |
| uzrdglem 12756 | A helper lemma for the val... |
| uzrdgfni 12757 | The recursive definition g... |
| uzrdg0i 12758 | Initial value of a recursi... |
| uzrdgsuci 12759 | Successor value of a recur... |
| ltweuz 12760 | ` < ` is a well-founded re... |
| ltwenn 12761 | Less than well-orders the ... |
| ltwefz 12762 | Less than well-orders a se... |
| uzenom 12763 | An upper integer set is de... |
| uzinf 12764 | An upper integer set is in... |
| nnnfi 12765 | The set of positive intege... |
| uzrdgxfr 12766 | Transfer the value of the ... |
| fzennn 12767 | The cardinality of a finit... |
| fzen2 12768 | The cardinality of a finit... |
| cardfz 12769 | The cardinality of a finit... |
| hashgf1o 12770 | ` G ` maps ` _om ` one-to-... |
| fzfi 12771 | A finite interval of integ... |
| fzfid 12772 | Commonly used special case... |
| fzofi 12773 | Half-open integer sets are... |
| fsequb 12774 | The values of a finite rea... |
| fsequb2 12775 | The values of a finite rea... |
| fseqsupcl 12776 | The values of a finite rea... |
| fseqsupubi 12777 | The values of a finite rea... |
| nn0ennn 12778 | The nonnegative integers a... |
| nnenom 12779 | The set of positive intege... |
| nnct 12780 | ` NN ` is countable. (Con... |
| uzindi 12781 | Indirect strong induction ... |
| axdc4uzlem 12782 | Lemma for ~ axdc4uz . (Co... |
| axdc4uz 12783 | A version of ~ axdc4 that ... |
| ssnn0fi 12784 | A subset of the nonnegativ... |
| rabssnn0fi 12785 | A subset of the nonnegativ... |
| uzsinds 12786 | Strong (or "total") induct... |
| nnsinds 12787 | Strong (or "total") induct... |
| nn0sinds 12788 | Strong (or "total") induct... |
| fsuppmapnn0fiublem 12789 | Lemma for ~ fsuppmapnn0fiu... |
| fsuppmapnn0fiub 12790 | If all functions of a fini... |
| fsuppmapnn0fiubOLD 12791 | Obsolete proof of ~ fsuppm... |
| fsuppmapnn0fiubex 12792 | If all functions of a fini... |
| fsuppmapnn0fiub0 12793 | If all functions of a fini... |
| suppssfz 12794 | Condition for a function o... |
| fsuppmapnn0ub 12795 | If a function over the non... |
| fsuppmapnn0fz 12796 | If a function over the non... |
| mptnn0fsupp 12797 | A mapping from the nonnega... |
| mptnn0fsuppd 12798 | A mapping from the nonnega... |
| mptnn0fsuppr 12799 | A finitely supported mappi... |
| f13idfv 12800 | A one-to-one function with... |
| seqex 12803 | Existence of the sequence ... |
| seqeq1 12804 | Equality theorem for the s... |
| seqeq2 12805 | Equality theorem for the s... |
| seqeq3 12806 | Equality theorem for the s... |
| seqeq1d 12807 | Equality deduction for the... |
| seqeq2d 12808 | Equality deduction for the... |
| seqeq3d 12809 | Equality deduction for the... |
| seqeq123d 12810 | Equality deduction for the... |
| nfseq 12811 | Hypothesis builder for the... |
| seqval 12812 | Value of the sequence buil... |
| seqfn 12813 | The sequence builder funct... |
| seq1 12814 | Value of the sequence buil... |
| seq1i 12815 | Value of the sequence buil... |
| seqp1 12816 | Value of the sequence buil... |
| seqp1i 12817 | Value of the sequence buil... |
| seqm1 12818 | Value of the sequence buil... |
| seqcl2 12819 | Closure properties of the ... |
| seqf2 12820 | Range of the recursive seq... |
| seqcl 12821 | Closure properties of the ... |
| seqf 12822 | Range of the recursive seq... |
| seqfveq2 12823 | Equality of sequences. (C... |
| seqfeq2 12824 | Equality of sequences. (C... |
| seqfveq 12825 | Equality of sequences. (C... |
| seqfeq 12826 | Equality of sequences. (C... |
| seqshft2 12827 | Shifting the index set of ... |
| seqres 12828 | Restricting its characteri... |
| serf 12829 | An infinite series of comp... |
| serfre 12830 | An infinite series of real... |
| monoord 12831 | Ordering relation for a mo... |
| monoord2 12832 | Ordering relation for a mo... |
| sermono 12833 | The partial sums in an inf... |
| seqsplit 12834 | Split a sequence into two ... |
| seq1p 12835 | Removing the first term fr... |
| seqcaopr3 12836 | Lemma for ~ seqcaopr2 . (... |
| seqcaopr2 12837 | The sum of two infinite se... |
| seqcaopr 12838 | The sum of two infinite se... |
| seqf1olem2a 12839 | Lemma for ~ seqf1o . (Con... |
| seqf1olem1 12840 | Lemma for ~ seqf1o . (Con... |
| seqf1olem2 12841 | Lemma for ~ seqf1o . (Con... |
| seqf1o 12842 | Rearrange a sum via an arb... |
| seradd 12843 | The sum of two infinite se... |
| sersub 12844 | The difference of two infi... |
| seqid3 12845 | A sequence that consists e... |
| seqid 12846 | Discard the first few term... |
| seqid2 12847 | The last few terms of a se... |
| seqhomo 12848 | Apply a homomorphism to a ... |
| seqz 12849 | If the operation ` .+ ` ha... |
| seqfeq4 12850 | Equality of series under d... |
| seqfeq3 12851 | Equality of series under d... |
| seqdistr 12852 | The distributive property ... |
| ser0 12853 | The value of the partial s... |
| ser0f 12854 | A zero-valued infinite ser... |
| serge0 12855 | A finite sum of nonnegativ... |
| serle 12856 | Comparison of partial sums... |
| ser1const 12857 | Value of the partial serie... |
| seqof 12858 | Distribute function operat... |
| seqof2 12859 | Distribute function operat... |
| expval 12862 | Value of exponentiation to... |
| expnnval 12863 | Value of exponentiation to... |
| exp0 12864 | Value of a complex number ... |
| 0exp0e1 12865 | ` 0 ^ 0 = 1 ` (common case... |
| exp1 12866 | Value of a complex number ... |
| expp1 12867 | Value of a complex number ... |
| expneg 12868 | Value of a complex number ... |
| expneg2 12869 | Value of a complex number ... |
| expn1 12870 | A number to the negative o... |
| expcllem 12871 | Lemma for proving nonnegat... |
| expcl2lem 12872 | Lemma for proving integer ... |
| nnexpcl 12873 | Closure of exponentiation ... |
| nn0expcl 12874 | Closure of exponentiation ... |
| zexpcl 12875 | Closure of exponentiation ... |
| qexpcl 12876 | Closure of exponentiation ... |
| reexpcl 12877 | Closure of exponentiation ... |
| expcl 12878 | Closure law for nonnegativ... |
| rpexpcl 12879 | Closure law for exponentia... |
| reexpclz 12880 | Closure of exponentiation ... |
| qexpclz 12881 | Closure of exponentiation ... |
| m1expcl2 12882 | Closure of exponentiation ... |
| m1expcl 12883 | Closure of exponentiation ... |
| expclzlem 12884 | Closure law for integer ex... |
| expclz 12885 | Closure law for integer ex... |
| nn0expcli 12886 | Closure of exponentiation ... |
| nn0sqcl 12887 | The square of a nonnegativ... |
| expm1t 12888 | Exponentiation in terms of... |
| 1exp 12889 | Value of one raised to a n... |
| expeq0 12890 | Positive integer exponenti... |
| expne0 12891 | Positive integer exponenti... |
| expne0i 12892 | Nonnegative integer expone... |
| expgt0 12893 | Nonnegative integer expone... |
| expnegz 12894 | Value of a complex number ... |
| 0exp 12895 | Value of zero raised to a ... |
| expge0 12896 | Nonnegative integer expone... |
| expge1 12897 | Nonnegative integer expone... |
| expgt1 12898 | Positive integer exponenti... |
| mulexp 12899 | Positive integer exponenti... |
| mulexpz 12900 | Integer exponentiation of ... |
| exprec 12901 | Nonnegative integer expone... |
| expadd 12902 | Sum of exponents law for n... |
| expaddzlem 12903 | Lemma for ~ expaddz . (Co... |
| expaddz 12904 | Sum of exponents law for i... |
| expmul 12905 | Product of exponents law f... |
| expmulz 12906 | Product of exponents law f... |
| m1expeven 12907 | Exponentiation of negative... |
| expsub 12908 | Exponent subtraction law f... |
| expp1z 12909 | Value of a nonzero complex... |
| expm1 12910 | Value of a complex number ... |
| expdiv 12911 | Nonnegative integer expone... |
| ltexp2a 12912 | Ordering relationship for ... |
| expcan 12913 | Cancellation law for expon... |
| ltexp2 12914 | Ordering law for exponenti... |
| leexp2 12915 | Ordering law for exponenti... |
| leexp2a 12916 | Weak ordering relationship... |
| ltexp2r 12917 | The power of a positive nu... |
| leexp2r 12918 | Weak ordering relationship... |
| leexp1a 12919 | Weak mantissa ordering rel... |
| exple1 12920 | Nonnegative integer expone... |
| expubnd 12921 | An upper bound on ` A ^ N ... |
| sqval 12922 | Value of the square of a c... |
| sqneg 12923 | The square of the negative... |
| sqsubswap 12924 | Swap the order of subtract... |
| sqcl 12925 | Closure of square. (Contr... |
| sqmul 12926 | Distribution of square ove... |
| sqeq0 12927 | A number is zero iff its s... |
| sqdiv 12928 | Distribution of square ove... |
| sqdivid 12929 | The square of a nonzero nu... |
| sqne0 12930 | A number is nonzero iff it... |
| resqcl 12931 | Closure of the square of a... |
| sqgt0 12932 | The square of a nonzero re... |
| nnsqcl 12933 | The naturals are closed un... |
| zsqcl 12934 | Integers are closed under ... |
| qsqcl 12935 | The square of a rational i... |
| sq11 12936 | The square function is one... |
| lt2sq 12937 | The square function on non... |
| le2sq 12938 | The square function on non... |
| le2sq2 12939 | The square of a 'less than... |
| sqge0 12940 | A square of a real is nonn... |
| zsqcl2 12941 | The square of an integer i... |
| sumsqeq0 12942 | Two real numbers are equal... |
| sqvali 12943 | Value of square. Inferenc... |
| sqcli 12944 | Closure of square. (Contr... |
| sqeq0i 12945 | A number is zero iff its s... |
| sqrecii 12946 | Square of reciprocal. (Co... |
| sqmuli 12947 | Distribution of square ove... |
| sqdivi 12948 | Distribution of square ove... |
| resqcli 12949 | Closure of square in reals... |
| sqgt0i 12950 | The square of a nonzero re... |
| sqge0i 12951 | A square of a real is nonn... |
| lt2sqi 12952 | The square function on non... |
| le2sqi 12953 | The square function on non... |
| sq11i 12954 | The square function is one... |
| sq0 12955 | The square of 0 is 0. (Co... |
| sq0i 12956 | If a number is zero, its s... |
| sq0id 12957 | If a number is zero, its s... |
| sq1 12958 | The square of 1 is 1. (Co... |
| neg1sqe1 12959 | ` -u 1 ` squared is 1 (com... |
| sq2 12960 | The square of 2 is 4. (Co... |
| sq3 12961 | The square of 3 is 9. (Co... |
| sq4e2t8 12962 | The square of 4 is 2 times... |
| cu2 12963 | The cube of 2 is 8. (Cont... |
| irec 12964 | The reciprocal of ` _i ` .... |
| i2 12965 | ` _i ` squared. (Contribu... |
| i3 12966 | ` _i ` cubed. (Contribute... |
| i4 12967 | ` _i ` to the fourth power... |
| nnlesq 12968 | A positive integer is less... |
| iexpcyc 12969 | Taking ` _i ` to the ` K `... |
| expnass 12970 | A counterexample showing t... |
| sqlecan 12971 | Cancel one factor of a squ... |
| subsq 12972 | Factor the difference of t... |
| subsq2 12973 | Express the difference of ... |
| binom2i 12974 | The square of a binomial. ... |
| subsqi 12975 | Factor the difference of t... |
| sqeqori 12976 | The squares of two complex... |
| subsq0i 12977 | The two solutions to the d... |
| sqeqor 12978 | The squares of two complex... |
| binom2 12979 | The square of a binomial. ... |
| binom21 12980 | Special case of ~ binom2 w... |
| binom2sub 12981 | Expand the square of a sub... |
| binom2sub1 12982 | Special case of ~ binom2su... |
| binom2subi 12983 | Expand the square of a sub... |
| mulbinom2 12984 | The square of a binomial w... |
| binom3 12985 | The cube of a binomial. (... |
| sq01 12986 | If a complex number equals... |
| zesq 12987 | An integer is even iff its... |
| nnesq 12988 | A positive integer is even... |
| crreczi 12989 | Reciprocal of a complex nu... |
| bernneq 12990 | Bernoulli's inequality, du... |
| bernneq2 12991 | Variation of Bernoulli's i... |
| bernneq3 12992 | A corollary of ~ bernneq .... |
| expnbnd 12993 | Exponentiation with a mant... |
| expnlbnd 12994 | The reciprocal of exponent... |
| expnlbnd2 12995 | The reciprocal of exponent... |
| expmulnbnd 12996 | Exponentiation with a mant... |
| digit2 12997 | Two ways to express the ` ... |
| digit1 12998 | Two ways to express the ` ... |
| modexp 12999 | Exponentiation property of... |
| discr1 13000 | A nonnegative quadratic fo... |
| discr 13001 | If a quadratic polynomial ... |
| exp0d 13002 | Value of a complex number ... |
| exp1d 13003 | Value of a complex number ... |
| expeq0d 13004 | Positive integer exponenti... |
| sqvald 13005 | Value of square. Inferenc... |
| sqcld 13006 | Closure of square. (Contr... |
| sqeq0d 13007 | A number is zero iff its s... |
| expcld 13008 | Closure law for nonnegativ... |
| expp1d 13009 | Value of a complex number ... |
| expaddd 13010 | Sum of exponents law for n... |
| expmuld 13011 | Product of exponents law f... |
| sqrecd 13012 | Square of reciprocal. (Co... |
| expclzd 13013 | Closure law for integer ex... |
| expne0d 13014 | Nonnegative integer expone... |
| expnegd 13015 | Value of a complex number ... |
| exprecd 13016 | Nonnegative integer expone... |
| expp1zd 13017 | Value of a nonzero complex... |
| expm1d 13018 | Value of a complex number ... |
| expsubd 13019 | Exponent subtraction law f... |
| sqmuld 13020 | Distribution of square ove... |
| sqdivd 13021 | Distribution of square ove... |
| expdivd 13022 | Nonnegative integer expone... |
| mulexpd 13023 | Positive integer exponenti... |
| 0expd 13024 | Value of zero raised to a ... |
| reexpcld 13025 | Closure of exponentiation ... |
| expge0d 13026 | Nonnegative integer expone... |
| expge1d 13027 | Nonnegative integer expone... |
| sqoddm1div8 13028 | A squared odd number minus... |
| nnsqcld 13029 | The naturals are closed un... |
| nnexpcld 13030 | Closure of exponentiation ... |
| nn0expcld 13031 | Closure of exponentiation ... |
| rpexpcld 13032 | Closure law for exponentia... |
| ltexp2rd 13033 | The power of a positive nu... |
| reexpclzd 13034 | Closure of exponentiation ... |
| resqcld 13035 | Closure of square in reals... |
| sqge0d 13036 | A square of a real is nonn... |
| sqgt0d 13037 | The square of a nonzero re... |
| ltexp2d 13038 | Ordering relationship for ... |
| leexp2d 13039 | Ordering law for exponenti... |
| expcand 13040 | Ordering relationship for ... |
| leexp2ad 13041 | Ordering relationship for ... |
| leexp2rd 13042 | Ordering relationship for ... |
| lt2sqd 13043 | The square function on non... |
| le2sqd 13044 | The square function on non... |
| sq11d 13045 | The square function is one... |
| mulsubdivbinom2 13046 | The square of a binomial w... |
| muldivbinom2 13047 | The square of a binomial w... |
| sq10 13048 | The square of 10 is 100. ... |
| sq10e99m1 13049 | The square of 10 is 99 plu... |
| 3dec 13050 | A "decimal constructor" wh... |
| sq10OLD 13051 | Old version of ~ sq10 . O... |
| sq10e99m1OLD 13052 | Old version of ~ sq10e99m1... |
| 3decOLD 13053 | Old version of ~ 3dec . O... |
| nn0le2msqi 13054 | The square function on non... |
| nn0opthlem1 13055 | A rather pretty lemma for ... |
| nn0opthlem2 13056 | Lemma for ~ nn0opthi . (C... |
| nn0opthi 13057 | An ordered pair theorem fo... |
| nn0opth2i 13058 | An ordered pair theorem fo... |
| nn0opth2 13059 | An ordered pair theorem fo... |
| facnn 13062 | Value of the factorial fun... |
| fac0 13063 | The factorial of 0. (Cont... |
| fac1 13064 | The factorial of 1. (Cont... |
| facp1 13065 | The factorial of a success... |
| fac2 13066 | The factorial of 2. (Cont... |
| fac3 13067 | The factorial of 3. (Cont... |
| fac4 13068 | The factorial of 4. (Cont... |
| facnn2 13069 | Value of the factorial fun... |
| faccl 13070 | Closure of the factorial f... |
| faccld 13071 | Closure of the factorial f... |
| facmapnn 13072 | The factorial function res... |
| facne0 13073 | The factorial function is ... |
| facdiv 13074 | A positive integer divides... |
| facndiv 13075 | No positive integer (great... |
| facwordi 13076 | Ordering property of facto... |
| faclbnd 13077 | A lower bound for the fact... |
| faclbnd2 13078 | A lower bound for the fact... |
| faclbnd3 13079 | A lower bound for the fact... |
| faclbnd4lem1 13080 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4lem2 13081 | Lemma for ~ faclbnd4 . Us... |
| faclbnd4lem3 13082 | Lemma for ~ faclbnd4 . Th... |
| faclbnd4lem4 13083 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4 13084 | Variant of ~ faclbnd5 prov... |
| faclbnd5 13085 | The factorial function gro... |
| faclbnd6 13086 | Geometric lower bound for ... |
| facubnd 13087 | An upper bound for the fac... |
| facavg 13088 | The product of two factori... |
| bcval 13091 | Value of the binomial coef... |
| bcval2 13092 | Value of the binomial coef... |
| bcval3 13093 | Value of the binomial coef... |
| bcval4 13094 | Value of the binomial coef... |
| bcrpcl 13095 | Closure of the binomial co... |
| bccmpl 13096 | "Complementing" its second... |
| bcn0 13097 | ` N ` choose 0 is 1. Rema... |
| bc0k 13098 | The binomial coefficient "... |
| bcnn 13099 | ` N ` choose ` N ` is 1. ... |
| bcn1 13100 | Binomial coefficient: ` N ... |
| bcnp1n 13101 | Binomial coefficient: ` N ... |
| bcm1k 13102 | The proportion of one bino... |
| bcp1n 13103 | The proportion of one bino... |
| bcp1nk 13104 | The proportion of one bino... |
| bcval5 13105 | Write out the top and bott... |
| bcn2 13106 | Binomial coefficient: ` N ... |
| bcp1m1 13107 | Compute the binomial coeff... |
| bcpasc 13108 | Pascal's rule for the bino... |
| bccl 13109 | A binomial coefficient, in... |
| bccl2 13110 | A binomial coefficient, in... |
| bcn2m1 13111 | Compute the binomial coeff... |
| bcn2p1 13112 | Compute the binomial coeff... |
| permnn 13113 | The number of permutations... |
| bcnm1 13114 | The binomial coefficent of... |
| 4bc3eq4 13115 | The value of four choose t... |
| 4bc2eq6 13116 | The value of four choose t... |
| hashkf 13119 | The finite part of the siz... |
| hashgval 13120 | The value of the ` # ` fun... |
| hashginv 13121 | ` ``' G ` maps the size fu... |
| hashinf 13122 | The value of the ` # ` fun... |
| hashbnd 13123 | If ` A ` has size bounded ... |
| hashfxnn0 13124 | The size function is a fun... |
| hashf 13125 | The size function maps all... |
| hashfOLD 13126 | Obsolete version of ~ hash... |
| hashxnn0 13127 | The value of the hash func... |
| hashresfn 13128 | Restriction of the domain ... |
| dmhashres 13129 | Restriction of the domain ... |
| hashnn0pnf 13130 | The value of the hash func... |
| hashnnn0genn0 13131 | If the size of a set is no... |
| hashnemnf 13132 | The size of a set is never... |
| hashv01gt1 13133 | The size of a set is eithe... |
| hashfz1 13134 | The set ` ( 1 ... N ) ` ha... |
| hashen 13135 | Two finite sets have the s... |
| hasheni 13136 | Equinumerous sets have the... |
| hasheqf1o 13137 | The size of two finite set... |
| fiinfnf1o 13138 | There is no bijection betw... |
| focdmex 13139 | The codomain of an onto fu... |
| hasheqf1oi 13140 | The size of two sets is eq... |
| hasheqf1oiOLD 13141 | Obsolete version of ~ hash... |
| hashf1rn 13142 | The size of a finite set w... |
| hashf1rnOLD 13143 | Obsolete version of ~ hash... |
| hasheqf1od 13144 | The size of two sets is eq... |
| fz1eqb 13145 | Two possibly-empty 1-based... |
| hashcard 13146 | The size function of the c... |
| hashcl 13147 | Closure of the ` # ` funct... |
| hashxrcl 13148 | Extended real closure of t... |
| hashclb 13149 | Reverse closure of the ` #... |
| nfile 13150 | The size of any infinite s... |
| hashvnfin 13151 | A set of finite size is a ... |
| hashnfinnn0 13152 | The size of an infinite se... |
| isfinite4 13153 | A finite set is equinumero... |
| hasheq0 13154 | Two ways of saying a finit... |
| hashneq0 13155 | Two ways of saying a set i... |
| hashgt0n0 13156 | If the size of a set is gr... |
| hashnncl 13157 | Positive natural closure o... |
| hash0 13158 | The empty set has size zer... |
| hashsng 13159 | The size of a singleton. ... |
| hashen1 13160 | A set with only one elemen... |
| hashrabrsn 13161 | The size of a restricted c... |
| hashrabsn01 13162 | The size of a restricted c... |
| hashrabsn1 13163 | If the size of a restricte... |
| hashfn 13164 | A function is equinumerous... |
| fseq1hash 13165 | The value of the size func... |
| hashgadd 13166 | ` G ` maps ordinal additio... |
| hashgval2 13167 | A short expression for the... |
| hashdom 13168 | Dominance relation for the... |
| hashdomi 13169 | Non-strict order relation ... |
| hashsdom 13170 | Strict dominance relation ... |
| hashun 13171 | The size of the union of d... |
| hashun2 13172 | The size of the union of f... |
| hashun3 13173 | The size of the union of f... |
| hashinfxadd 13174 | The extended real addition... |
| hashunx 13175 | The size of the union of d... |
| hashge0 13176 | The cardinality of a set i... |
| hashgt0 13177 | The cardinality of a nonem... |
| hashge1 13178 | The cardinality of a nonem... |
| 1elfz0hash 13179 | 1 is an element of the fin... |
| hashnn0n0nn 13180 | If a nonnegative integer i... |
| hashunsng 13181 | The size of the union of a... |
| hashprg 13182 | The size of an unordered p... |
| hashprgOLD 13183 | Obsolete version of ~ hash... |
| elprchashprn2 13184 | If one element of an unord... |
| hashprb 13185 | The size of an unordered p... |
| hashprdifel 13186 | The elements of an unorder... |
| prhash2ex 13187 | There is (at least) one se... |
| hashle00 13188 | If the size of a set is le... |
| hashgt0elex 13189 | If the size of a set is gr... |
| hashgt0elexb 13190 | The size of a set is great... |
| hashp1i 13191 | Size of a finite ordinal. ... |
| hash1 13192 | Size of a finite ordinal. ... |
| hash2 13193 | Size of a finite ordinal. ... |
| hash3 13194 | Size of a finite ordinal. ... |
| hash4 13195 | Size of a finite ordinal. ... |
| pr0hash2ex 13196 | There is (at least) one se... |
| hashss 13197 | The size of a subset is le... |
| prsshashgt1 13198 | The size of a superset of ... |
| hashin 13199 | The size of the intersecti... |
| hashssdif 13200 | The size of the difference... |
| hashdif 13201 | The size of the difference... |
| hashdifsn 13202 | The size of the difference... |
| hashdifpr 13203 | The size of the difference... |
| hashsn01 13204 | The size of a singleton is... |
| hashsnle1 13205 | The size of a singleton is... |
| hashsnlei 13206 | Get an upper bound on a co... |
| hash1snb 13207 | The size of a set is 1 if ... |
| euhash1 13208 | The size of a set is 1 in ... |
| hash1n0 13209 | If the size of a set is 1 ... |
| hashgt12el 13210 | In a set with more than on... |
| hashgt12el2 13211 | In a set with more than on... |
| hashunlei 13212 | Get an upper bound on a co... |
| hashsslei 13213 | Get an upper bound on a co... |
| hashfz 13214 | Value of the numeric cardi... |
| fzsdom2 13215 | Condition for finite range... |
| hashfzo 13216 | Cardinality of a half-open... |
| hashfzo0 13217 | Cardinality of a half-open... |
| hashfzp1 13218 | Value of the numeric cardi... |
| hashfz0 13219 | Value of the numeric cardi... |
| hashxplem 13220 | Lemma for ~ hashxp . (Con... |
| hashxp 13221 | The size of the Cartesian ... |
| hashmap 13222 | The size of the set expone... |
| hashpw 13223 | The size of the power set ... |
| hashfun 13224 | A finite set is a function... |
| hashres 13225 | The number of elements of ... |
| hashreshashfun 13226 | The number of elements of ... |
| hashimarn 13227 | The size of the image of a... |
| hashimarni 13228 | If the size of the image o... |
| resunimafz0 13229 | TODO-AV: Revise using ` F... |
| fnfz0hash 13230 | The size of a function on ... |
| ffz0hash 13231 | The size of a function on ... |
| fnfz0hashnn0 13232 | The size of a function on ... |
| ffzo0hash 13233 | The size of a function on ... |
| fnfzo0hash 13234 | The size of a function on ... |
| fnfzo0hashnn0 13235 | The value of the size func... |
| hashbclem 13236 | Lemma for ~ hashbc : induc... |
| hashbc 13237 | The binomial coefficient c... |
| hashfacen 13238 | The number of bijections b... |
| hashf1lem1 13239 | Lemma for ~ hashf1 . (Con... |
| hashf1lem2 13240 | Lemma for ~ hashf1 . (Con... |
| hashf1 13241 | The permutation number ` |... |
| hashfac 13242 | A factorial counts the num... |
| leiso 13243 | Two ways to write a strict... |
| leisorel 13244 | Version of ~ isorel for st... |
| fz1isolem 13245 | Lemma for ~ fz1iso . (Con... |
| fz1iso 13246 | Any finite ordered set has... |
| ishashinf 13247 | Any set that is not finite... |
| seqcoll 13248 | The function ` F ` contain... |
| seqcoll2 13249 | The function ` F ` contain... |
| hashprlei 13250 | An unordered pair has at m... |
| hash2pr 13251 | A set of size two is an un... |
| hash2prde 13252 | A set of size two is an un... |
| hash2exprb 13253 | A set of size two is an un... |
| hash2prb 13254 | A set of size two is a pro... |
| prprrab 13255 | The set of proper pairs of... |
| nehash2 13256 | The cardinality of a set w... |
| hash2prd 13257 | A set of size two is an un... |
| hash2pwpr 13258 | If the size of a subset of... |
| hashle2pr 13259 | A nonempty set of size les... |
| hashle2prv 13260 | A nonempty subset of a pow... |
| pr2pwpr 13261 | The set of subsets of a pa... |
| hashge2el2dif 13262 | A set with size at least 2... |
| hashge2el2difr 13263 | A set with at least 2 diff... |
| hashge2el2difb 13264 | A set has size at least 2 ... |
| hashdmpropge2 13265 | The size of the domain of ... |
| hashtplei 13266 | An unordered triple has at... |
| hashtpg 13267 | The size of an unordered t... |
| hashge3el3dif 13268 | A set with size at least 3... |
| elss2prb 13269 | An element of the set of s... |
| hash2sspr 13270 | A subset of size two is an... |
| exprelprel 13271 | If there is an element of ... |
| hash3tr 13272 | A set of size three is an ... |
| hash1to3 13273 | If the size of a set is be... |
| fundmge2nop0 13274 | A function with a domain c... |
| fundmge2nop 13275 | A function with a domain c... |
| fun2dmnop0 13276 | A function with a domain c... |
| fun2dmnop 13277 | A function with a domain c... |
| brfi1indlem 13278 | TODO-AV1: no lemma, but se... |
| fi1uzind 13279 | Properties of an ordered p... |
| brfi1uzind 13280 | Properties of a binary rel... |
| brfi1ind 13281 | Properties of a binary rel... |
| brfi1indALT 13282 | Alternate proof of ~ brfi1... |
| opfi1uzind 13283 | Properties of an ordered p... |
| opfi1ind 13284 | Properties of an ordered p... |
| fi1uzindOLD 13285 | Obsolete version of ~ fi1u... |
| brfi1uzindOLD 13286 | Obsolete version of ~ brfi... |
| brfi1indOLD 13287 | Obsolete version of ~ brfi... |
| brfi1indALTOLD 13288 | Obsolete version of ~ brfi... |
| opfi1uzindOLD 13289 | Obsolete version of ~ opfi... |
| opfi1indOLD 13290 | Obsolete version of ~ opfi... |
| iswrd 13307 | Property of being a word o... |
| wrdval 13308 | Value of the set of words ... |
| iswrdi 13309 | A zero-based sequence is a... |
| wrdf 13310 | A word is a zero-based seq... |
| iswrdb 13311 | A word over an alphabet is... |
| wrddm 13312 | The indices of a word (i.e... |
| sswrd 13313 | The set of words respects ... |
| snopiswrd 13314 | A singleton of an ordered ... |
| wrdexg 13315 | The set of words over a se... |
| wrdexb 13316 | The set of words over a se... |
| wrdexi 13317 | The set of words over a se... |
| wrdsymbcl 13318 | A symbol within a word ove... |
| wrdfn 13319 | A word is a function with ... |
| wrdv 13320 | A word over an alphabet is... |
| wrdlndm 13321 | The length of a word is no... |
| iswrdsymb 13322 | An arbitrary word is a wor... |
| wrdfin 13323 | A word is a finite set. (... |
| lencl 13324 | The length of a word is a ... |
| lennncl 13325 | The length of a nonempty w... |
| wrdffz 13326 | A word is a function from ... |
| wrdeq 13327 | Equality theorem for the s... |
| wrdeqi 13328 | Equality theorem for the s... |
| iswrddm0 13329 | A function with empty doma... |
| wrd0 13330 | The empty set is a word (t... |
| 0wrd0 13331 | The empty word is the only... |
| ffz0iswrd 13332 | A sequence with zero-based... |
| nfwrd 13333 | Hypothesis builder for ` W... |
| csbwrdg 13334 | Class substitution for the... |
| wrdnval 13335 | Words of a fixed length ar... |
| wrdmap 13336 | Words as a mapping. (Cont... |
| hashwrdn 13337 | If there is only a finite ... |
| wrdnfi 13338 | If there is only a finite ... |
| wrdsymb0 13339 | A symbol at a position "ou... |
| wrdlenge1n0 13340 | A word with length at leas... |
| wrdlenge2n0 13341 | A word with length at leas... |
| wrdsymb1 13342 | The first symbol of a none... |
| wrdlen1 13343 | A word of length 1 starts ... |
| fstwrdne 13344 | The first symbol of a none... |
| fstwrdne0 13345 | The first symbol of a none... |
| eqwrd 13346 | Two words are equal iff th... |
| elovmpt2wrd 13347 | Implications for the value... |
| elovmptnn0wrd 13348 | Implications for the value... |
| wrdred1 13349 | A word truncated by a symb... |
| wrdred1hash 13350 | The length of a word trunc... |
| lsw 13351 | Extract the last symbol of... |
| lsw0 13352 | The last symbol of an empt... |
| lsw0g 13353 | The last symbol of an empt... |
| lsw1 13354 | The last symbol of a word ... |
| lswcl 13355 | Closure of the last symbol... |
| lswlgt0cl 13356 | The last symbol of a nonem... |
| ccatfn 13357 | The concatenation operator... |
| ccatfval 13358 | Value of the concatenation... |
| ccatcl 13359 | The concatenation of two w... |
| ccatlen 13360 | The length of a concatenat... |
| ccatval1 13361 | Value of a symbol in the l... |
| ccatval2 13362 | Value of a symbol in the r... |
| ccatval3 13363 | Value of a symbol in the r... |
| elfzelfzccat 13364 | An element of a finite set... |
| ccatvalfn 13365 | The concatenation of two w... |
| ccatsymb 13366 | The symbol at a given posi... |
| ccatfv0 13367 | The first symbol of a conc... |
| ccatval1lsw 13368 | The last symbol of the lef... |
| ccatlid 13369 | Concatenation of a word by... |
| ccatrid 13370 | Concatenation of a word by... |
| ccatass 13371 | Associative law for concat... |
| ccatrn 13372 | The range of a concatenate... |
| lswccatn0lsw 13373 | The last symbol of a word ... |
| lswccat0lsw 13374 | The last symbol of a word ... |
| ccatalpha 13375 | A concatenation of two arb... |
| ccatrcl1 13376 | Reverse closure of a conca... |
| ids1 13377 | Identity function protecti... |
| s1val 13378 | Value of a single-symbol w... |
| s1rn 13379 | The range of a single-symb... |
| s1eq 13380 | Equality theorem for a sin... |
| s1eqd 13381 | Equality theorem for a sin... |
| s1cl 13382 | A singleton word is a word... |
| s1cld 13383 | A singleton word is a word... |
| s1cli 13384 | A singleton word is a word... |
| s1len 13385 | Length of a singleton word... |
| s1nz 13386 | A singleton word is not th... |
| s1nzOLD 13387 | Obsolete proof of ~ s1nz a... |
| s1dm 13388 | The domain of a singleton ... |
| s1dmALT 13389 | Alternate version of ~ s1d... |
| s1fv 13390 | Sole symbol of a singleton... |
| lsws1 13391 | The last symbol of a singl... |
| eqs1 13392 | A word of length 1 is a si... |
| wrdl1exs1 13393 | A word of length 1 is a si... |
| wrdl1s1 13394 | A word of length 1 is a si... |
| s111 13395 | The singleton word functio... |
| ccatws1cl 13396 | The concatenation of a wor... |
| ccat2s1cl 13397 | The concatenation of two s... |
| ccatws1len 13398 | The length of the concaten... |
| wrdlenccats1lenm1 13399 | The length of a word is th... |
| ccat2s1len 13400 | The length of the concaten... |
| ccatw2s1cl 13401 | The concatenation of a wor... |
| ccatw2s1len 13402 | The length of the concaten... |
| ccats1val1 13403 | Value of a symbol in the l... |
| ccats1val2 13404 | Value of the symbol concat... |
| ccat2s1p1 13405 | Extract the first of two c... |
| ccat2s1p2 13406 | Extract the second of two ... |
| ccatw2s1ass 13407 | Associative law for a conc... |
| ccatws1lenrevOLD 13408 | Obsolete theorem as of 24-... |
| ccatws1n0 13409 | The concatenation of a wor... |
| ccatws1ls 13410 | The last symbol of the con... |
| lswccats1 13411 | The last symbol of a word ... |
| lswccats1fst 13412 | The last symbol of a nonem... |
| ccatw2s1p1 13413 | Extract the symbol of the ... |
| ccatw2s1p2 13414 | Extract the second of two ... |
| ccat2s1fvw 13415 | Extract a symbol of a word... |
| ccat2s1fst 13416 | The first symbol of the co... |
| swrdval 13417 | Value of a subword. (Cont... |
| swrd00 13418 | A zero length substring. ... |
| swrdcl 13419 | Closure of the subword ext... |
| swrdval2 13420 | Value of the subword extra... |
| swrd0val 13421 | Value of the subword extra... |
| swrd0len 13422 | Length of a left-anchored ... |
| swrdlen 13423 | Length of an extracted sub... |
| swrdfv 13424 | A symbol in an extracted s... |
| swrdf 13425 | A subword of a word is a f... |
| swrdvalfn 13426 | Value of the subword extra... |
| swrd0f 13427 | A left-anchored subword of... |
| swrdid 13428 | A word is a subword of its... |
| swrdrn 13429 | The range of a subword of ... |
| swrdn0 13430 | A prefixing subword consis... |
| swrdlend 13431 | The value of the subword e... |
| swrdnd 13432 | The value of the subword e... |
| swrdnd2 13433 | Value of the subword extra... |
| swrd0 13434 | A subword of an empty set ... |
| swrdrlen 13435 | Length of a right-anchored... |
| swrd0len0 13436 | Length of a prefix of a wo... |
| addlenrevswrd 13437 | The sum of the lengths of ... |
| addlenswrd 13438 | The sum of the lengths of ... |
| swrd0fv 13439 | A symbol in an left-anchor... |
| swrd0fv0 13440 | The first symbol in a left... |
| swrdtrcfv 13441 | A symbol in a word truncat... |
| swrdtrcfv0 13442 | The first symbol in a word... |
| swrd0fvlsw 13443 | The last symbol in a left-... |
| swrdeq 13444 | Two subwords of words are ... |
| swrdlen2 13445 | Length of an extracted sub... |
| swrdfv2 13446 | A symbol in an extracted s... |
| swrdsb0eq 13447 | Two subwords with the same... |
| swrdsbslen 13448 | Two subwords with the same... |
| swrdspsleq 13449 | Two words have a common su... |
| swrdtrcfvl 13450 | The last symbol in a word ... |
| swrds1 13451 | Extract a single symbol fr... |
| swrdlsw 13452 | Extract the last single sy... |
| 2swrdeqwrdeq 13453 | Two words are equal if and... |
| 2swrd1eqwrdeq 13454 | Two (nonempty) words are e... |
| disjxwrd 13455 | Sets of words are disjoint... |
| ccatswrd 13456 | Joining two adjacent subwo... |
| swrdccat1 13457 | Recover the left half of a... |
| swrdccat2 13458 | Recover the right half of ... |
| swrdswrdlem 13459 | Lemma for ~ swrdswrd . (C... |
| swrdswrd 13460 | A subword of a subword. (... |
| swrd0swrd 13461 | A prefix of a subword. (C... |
| swrdswrd0 13462 | A subword of a prefix. (C... |
| swrd0swrd0 13463 | A prefix of a prefix. (Co... |
| swrd0swrdid 13464 | A prefix of a prefix with ... |
| wrdcctswrd 13465 | The concatenation of two p... |
| lencctswrd 13466 | The length of two concaten... |
| lenrevcctswrd 13467 | The length of two reversel... |
| swrdccatwrd 13468 | Reconstruct a nonempty wor... |
| ccats1swrdeq 13469 | The last symbol of a word ... |
| ccatopth 13470 | An ~ opth -like theorem fo... |
| ccatopth2 13471 | An ~ opth -like theorem fo... |
| ccatlcan 13472 | Concatenation of words is ... |
| ccatrcan 13473 | Concatenation of words is ... |
| wrdeqs1cat 13474 | Decompose a nonempty word ... |
| cats1un 13475 | Express a word with an ext... |
| wrdind 13476 | Perform induction over the... |
| wrd2ind 13477 | Perform induction over the... |
| ccats1swrdeqrex 13478 | There exists a symbol such... |
| reuccats1lem 13479 | Lemma for ~ reuccats1 . (... |
| reuccats1 13480 | A set of words having the ... |
| reuccats1v 13481 | A set of words having the ... |
| swrdccatfn 13482 | The subword of a concatena... |
| swrdccatin1 13483 | The subword of a concatena... |
| swrdccatin12lem1 13484 | Lemma 1 for ~ swrdccatin12... |
| swrdccatin12lem2a 13485 | Lemma 1 for ~ swrdccatin12... |
| swrdccatin12lem2b 13486 | Lemma 2 for ~ swrdccatin12... |
| swrdccatin2 13487 | The subword of a concatena... |
| swrdccatin12lem2c 13488 | Lemma for ~ swrdccatin12le... |
| swrdccatin12lem2 13489 | Lemma 2 for ~ swrdccatin12... |
| swrdccatin12lem3 13490 | Lemma 3 for ~ swrdccatin12... |
| swrdccatin12 13491 | The subword of a concatena... |
| swrdccat3 13492 | The subword of a concatena... |
| swrdccat 13493 | The subword of a concatena... |
| swrdccat3a 13494 | A prefix of a concatenatio... |
| swrdccat3blem 13495 | Lemma for ~ swrdccat3b . ... |
| swrdccat3b 13496 | A suffix of a concatenatio... |
| swrdccatid 13497 | A prefix of a concatenatio... |
| ccats1swrdeqbi 13498 | A word is a prefix of a wo... |
| swrdccatin1d 13499 | The subword of a concatena... |
| swrdccatin2d 13500 | The subword of a concatena... |
| swrdccatin12d 13501 | The subword of a concatena... |
| splval 13502 | Value of the substring rep... |
| splcl 13503 | Closure of the substring r... |
| splid 13504 | Splicing a subword for the... |
| spllen 13505 | The length of a splice. (... |
| splfv1 13506 | Symbols to the left of a s... |
| splfv2a 13507 | Symbols within the replace... |
| splval2 13508 | Value of a splice, assumin... |
| revval 13509 | Value of the word reversin... |
| revcl 13510 | The reverse of a word is a... |
| revlen 13511 | The reverse of a word has ... |
| revfv 13512 | Reverse of a word at a poi... |
| rev0 13513 | The empty word is its own ... |
| revs1 13514 | Singleton words are their ... |
| revccat 13515 | Antiautomorphic property o... |
| revrev 13516 | Reversion is an involution... |
| reps 13517 | Construct a function mappi... |
| repsundef 13518 | A function mapping a half-... |
| repsconst 13519 | Construct a function mappi... |
| repsf 13520 | The constructed function m... |
| repswsymb 13521 | The symbols of a "repeated... |
| repsw 13522 | A function mapping a half-... |
| repswlen 13523 | The length of a "repeated ... |
| repsw0 13524 | The "repeated symbol word"... |
| repsdf2 13525 | Alternative definition of ... |
| repswsymball 13526 | All the symbols of a "repe... |
| repswsymballbi 13527 | A word is a "repeated symb... |
| repswfsts 13528 | The first symbol of a none... |
| repswlsw 13529 | The last symbol of a nonem... |
| repsw1 13530 | The "repeated symbol word"... |
| repswswrd 13531 | A subword of a "repeated s... |
| repswccat 13532 | The concatenation of two "... |
| repswrevw 13533 | The reverse of a "repeated... |
| cshfn 13536 | Perform a cyclical shift f... |
| cshword 13537 | Perform a cyclical shift f... |
| cshnz 13538 | A cyclical shift is the em... |
| 0csh0 13539 | Cyclically shifting an emp... |
| cshw0 13540 | A word cyclically shifted ... |
| cshwmodn 13541 | Cyclically shifting a word... |
| cshwsublen 13542 | Cyclically shifting a word... |
| cshwn 13543 | A word cyclically shifted ... |
| cshwcl 13544 | A cyclically shifted word ... |
| cshwlen 13545 | The length of a cyclically... |
| cshwf 13546 | A cyclically shifted word ... |
| cshwfn 13547 | A cyclically shifted word ... |
| cshwrn 13548 | The range of a cyclically ... |
| cshwidxmod 13549 | The symbol at a given inde... |
| cshwidxmodr 13550 | The symbol at a given inde... |
| cshwidx0mod 13551 | The symbol at index 0 of a... |
| cshwidx0 13552 | The symbol at index 0 of a... |
| cshwidxm1 13553 | The symbol at index ((n-N)... |
| cshwidxm 13554 | The symbol at index (n-N) ... |
| cshwidxn 13555 | The symbol at index (n-1) ... |
| cshf1 13556 | Cyclically shifting a word... |
| cshinj 13557 | If a word is injectiv (reg... |
| repswcshw 13558 | A cyclically shifted "repe... |
| 2cshw 13559 | Cyclically shifting a word... |
| 2cshwid 13560 | Cyclically shifting a word... |
| lswcshw 13561 | The last symbol of a word ... |
| 2cshwcom 13562 | Cyclically shifting a word... |
| cshwleneq 13563 | If the results of cyclical... |
| 3cshw 13564 | Cyclically shifting a word... |
| cshweqdif2 13565 | If cyclically shifting two... |
| cshweqdifid 13566 | If cyclically shifting a w... |
| cshweqrep 13567 | If cyclically shifting a w... |
| cshw1 13568 | If cyclically shifting a w... |
| cshw1repsw 13569 | If cyclically shifting a w... |
| cshwsexa 13570 | The class of (different!) ... |
| 2cshwcshw 13571 | If a word is a cyclically ... |
| scshwfzeqfzo 13572 | For a nonempty word the se... |
| cshwcshid 13573 | A cyclically shifted word ... |
| cshwcsh2id 13574 | A cyclically shifted word ... |
| cshimadifsn 13575 | The image of a cyclically ... |
| cshimadifsn0 13576 | The image of a cyclically ... |
| wrdco 13577 | Mapping a word by a functi... |
| lenco 13578 | Length of a mapped word is... |
| s1co 13579 | Mapping of a singleton wor... |
| revco 13580 | Mapping of words commutes ... |
| ccatco 13581 | Mapping of words commutes ... |
| cshco 13582 | Mapping of words commutes ... |
| swrdco 13583 | Mapping of words commutes ... |
| lswco 13584 | Mapping of (nonempty) word... |
| repsco 13585 | Mapping of words commutes ... |
| cats1cld 13600 | Closure of concatenation w... |
| cats1co 13601 | Closure of concatenation w... |
| cats1cli 13602 | Closure of concatenation w... |
| cats1fvn 13603 | The last symbol of a conca... |
| cats1fv 13604 | A symbol other than the la... |
| cats1len 13605 | The length of concatenatio... |
| cats1cat 13606 | Closure of concatenation w... |
| cats2cat 13607 | Closure of concatenation o... |
| s2eqd 13608 | Equality theorem for a dou... |
| s3eqd 13609 | Equality theorem for a len... |
| s4eqd 13610 | Equality theorem for a len... |
| s5eqd 13611 | Equality theorem for a len... |
| s6eqd 13612 | Equality theorem for a len... |
| s7eqd 13613 | Equality theorem for a len... |
| s8eqd 13614 | Equality theorem for a len... |
| s3eq2 13615 | Equality theorem for a len... |
| s2cld 13616 | A doubleton word is a word... |
| s3cld 13617 | A length 3 string is a wor... |
| s4cld 13618 | A length 4 string is a wor... |
| s5cld 13619 | A length 5 string is a wor... |
| s6cld 13620 | A length 6 string is a wor... |
| s7cld 13621 | A length 7 string is a wor... |
| s8cld 13622 | A length 7 string is a wor... |
| s2cl 13623 | A doubleton word is a word... |
| s3cl 13624 | A length 3 string is a wor... |
| s2cli 13625 | A doubleton word is a word... |
| s3cli 13626 | A length 3 string is a wor... |
| s4cli 13627 | A length 4 string is a wor... |
| s5cli 13628 | A length 5 string is a wor... |
| s6cli 13629 | A length 6 string is a wor... |
| s7cli 13630 | A length 7 string is a wor... |
| s8cli 13631 | A length 8 string is a wor... |
| s2fv0 13632 | Extract the first symbol f... |
| s2fv1 13633 | Extract the second symbol ... |
| s2len 13634 | The length of a doubleton ... |
| s2dm 13635 | The domain of a doubleton ... |
| s3fv0 13636 | Extract the first symbol f... |
| s3fv1 13637 | Extract the second symbol ... |
| s3fv2 13638 | Extract the third symbol f... |
| s3len 13639 | The length of a length 3 s... |
| s4fv0 13640 | Extract the first symbol f... |
| s4fv1 13641 | Extract the second symbol ... |
| s4fv2 13642 | Extract the third symbol f... |
| s4fv3 13643 | Extract the fourth symbol ... |
| s4len 13644 | The length of a length 4 s... |
| s5len 13645 | The length of a length 5 s... |
| s6len 13646 | The length of a length 6 s... |
| s7len 13647 | The length of a length 7 s... |
| s8len 13648 | The length of a length 8 s... |
| lsws2 13649 | The last symbol of a doubl... |
| lsws3 13650 | The last symbol of a 3 let... |
| lsws4 13651 | The last symbol of a 4 let... |
| s2prop 13652 | A length 2 word is an unor... |
| s2dmALT 13653 | Alternate version of ~ s2d... |
| s3tpop 13654 | A length 3 word is an unor... |
| s4prop 13655 | A length 4 word is a union... |
| s3fn 13656 | A length 3 word is a funct... |
| funcnvs1 13657 | The converse of a singleto... |
| funcnvs2 13658 | The converse of a length 2... |
| funcnvs3 13659 | The converse of a length 3... |
| funcnvs4 13660 | The converse of a length 4... |
| s2f1o 13661 | A length 2 word with mutua... |
| f1oun2prg 13662 | A union of unordered pairs... |
| s4f1o 13663 | A length 4 word with mutua... |
| s4dom 13664 | The domain of a length 4 w... |
| s2co 13665 | Mapping a doubleton word b... |
| s3co 13666 | Mapping a length 3 string ... |
| s0s1 13667 | Concatenation of fixed len... |
| s1s2 13668 | Concatenation of fixed len... |
| s1s3 13669 | Concatenation of fixed len... |
| s1s4 13670 | Concatenation of fixed len... |
| s1s5 13671 | Concatenation of fixed len... |
| s1s6 13672 | Concatenation of fixed len... |
| s1s7 13673 | Concatenation of fixed len... |
| s2s2 13674 | Concatenation of fixed len... |
| s4s2 13675 | Concatenation of fixed len... |
| s4s3 13676 | Concatenation of fixed len... |
| s4s4 13677 | Concatenation of fixed len... |
| s3s4 13678 | Concatenation of fixed len... |
| s2s5 13679 | Concatenation of fixed len... |
| s5s2 13680 | Concatenation of fixed len... |
| s2eq2s1eq 13681 | Two length 2 words are equ... |
| s2eq2seq 13682 | Two length 2 words are equ... |
| s3eqs2s1eq 13683 | Two length 3 words are equ... |
| s3eq3seq 13684 | Two length 3 words are equ... |
| swrds2 13685 | Extract two adjacent symbo... |
| wrdlen2i 13686 | Implications of a word of ... |
| wrd2pr2op 13687 | A word of length 2 represe... |
| wrdlen2 13688 | A word of length 2. (Cont... |
| wrdlen2s2 13689 | A word of length 2 as doub... |
| wrdl2exs2 13690 | A word of length 2 is a do... |
| wrd3tpop 13691 | A word of length 3 represe... |
| wrdlen3s3 13692 | A word of length 3 as leng... |
| repsw2 13693 | The "repeated symbol word"... |
| repsw3 13694 | The "repeated symbol word"... |
| swrd2lsw 13695 | Extract the last two symbo... |
| 2swrd2eqwrdeq 13696 | Two words of length at lea... |
| ccatw2s1ccatws2 13697 | The concatenation of a wor... |
| ccat2s1fvwALT 13698 | Alternate proof of ~ ccat2... |
| wwlktovf 13699 | Lemma 1 for ~ wrd2f1tovbij... |
| wwlktovf1 13700 | Lemma 2 for ~ wrd2f1tovbij... |
| wwlktovfo 13701 | Lemma 3 for ~ wrd2f1tovbij... |
| wwlktovf1o 13702 | Lemma 4 for ~ wrd2f1tovbij... |
| wrd2f1tovbij 13703 | There is a bijection betwe... |
| eqwrds3 13704 | A word is equal with a len... |
| wrdl3s3 13705 | A word of length 3 is a le... |
| s3sndisj 13706 | The singletons consisting ... |
| s3iunsndisj 13707 | The union of singletons co... |
| ofccat 13708 | Letterwise operations on w... |
| ofs1 13709 | Letterwise operations on a... |
| ofs2 13710 | Letterwise operations on a... |
| coss12d 13711 | Subset deduction for compo... |
| trrelssd 13712 | The composition of subclas... |
| xpcogend 13713 | The most interesting case ... |
| xpcoidgend 13714 | If two classes are not dis... |
| cotr2g 13715 | Two ways of saying that th... |
| cotr2 13716 | Two ways of saying a relat... |
| cotr3 13717 | Two ways of saying a relat... |
| coemptyd 13718 | Deduction about compositio... |
| xptrrel 13719 | The cross product is alway... |
| 0trrel 13720 | The empty class is a trans... |
| cleq1lem 13721 | Equality implies bijection... |
| cleq1 13722 | Equality of relations impl... |
| clsslem 13723 | The closure of a subclass ... |
| trcleq1 13728 | Equality of relations impl... |
| trclsslem 13729 | The transitive closure (as... |
| trcleq2lem 13730 | Equality implies bijection... |
| cvbtrcl 13731 | Change of bound variable i... |
| trcleq12lem 13732 | Equality implies bijection... |
| trclexlem 13733 | Existence of relation impl... |
| trclublem 13734 | If a relation exists then ... |
| trclubi 13735 | The Cartesian product of t... |
| trclubiOLD 13736 | Obsolete version of ~ trcl... |
| trclubgi 13737 | The union with the Cartesi... |
| trclubgiOLD 13738 | Obsolete version of ~ trcl... |
| trclub 13739 | The Cartesian product of t... |
| trclubg 13740 | The union with the Cartesi... |
| trclfv 13741 | The transitive closure of ... |
| brintclab 13742 | Two ways to express a bina... |
| brtrclfv 13743 | Two ways of expressing the... |
| brcnvtrclfv 13744 | Two ways of expressing the... |
| brtrclfvcnv 13745 | Two ways of expressing the... |
| brcnvtrclfvcnv 13746 | Two ways of expressing the... |
| trclfvss 13747 | The transitive closure (as... |
| trclfvub 13748 | The transitive closure of ... |
| trclfvlb 13749 | The transitive closure of ... |
| trclfvcotr 13750 | The transitive closure of ... |
| trclfvlb2 13751 | The transitive closure of ... |
| trclfvlb3 13752 | The transitive closure of ... |
| cotrtrclfv 13753 | The transitive closure of ... |
| trclidm 13754 | The transitive closure of ... |
| trclun 13755 | Transitive closure of a un... |
| trclfvg 13756 | The value of the transitiv... |
| trclfvcotrg 13757 | The value of the transitiv... |
| reltrclfv 13758 | The transitive closure of ... |
| dmtrclfv 13759 | The domain of the transiti... |
| relexp0g 13762 | A relation composed zero t... |
| relexp0 13763 | A relation composed zero t... |
| relexp0d 13764 | A relation composed zero t... |
| relexpsucnnr 13765 | A reduction for relation e... |
| relexp1g 13766 | A relation composed once i... |
| dfid5 13767 | Identity relation is equal... |
| dfid6 13768 | Identity relation expresse... |
| relexpsucr 13769 | A reduction for relation e... |
| relexpsucrd 13770 | A reduction for relation e... |
| relexp1d 13771 | A relation composed once i... |
| relexpsucnnl 13772 | A reduction for relation e... |
| relexpsucl 13773 | A reduction for relation e... |
| relexpsucld 13774 | A reduction for relation e... |
| relexpcnv 13775 | Commutation of converse an... |
| relexpcnvd 13776 | Commutation of converse an... |
| relexp0rel 13777 | The exponentiation of a cl... |
| relexprelg 13778 | The exponentiation of a cl... |
| relexprel 13779 | The exponentiation of a re... |
| relexpreld 13780 | The exponentiation of a re... |
| relexpnndm 13781 | The domain of an exponenti... |
| relexpdmg 13782 | The domain of an exponenti... |
| relexpdm 13783 | The domain of an exponenti... |
| relexpdmd 13784 | The domain of an exponenti... |
| relexpnnrn 13785 | The range of an exponentia... |
| relexprng 13786 | The range of an exponentia... |
| relexprn 13787 | The range of an exponentia... |
| relexprnd 13788 | The range of an exponentia... |
| relexpfld 13789 | The field of an exponentia... |
| relexpfldd 13790 | The field of an exponentia... |
| relexpaddnn 13791 | Relation composition becom... |
| relexpuzrel 13792 | The exponentiation of a cl... |
| relexpaddg 13793 | Relation composition becom... |
| relexpaddd 13794 | Relation composition becom... |
| dfrtrclrec2 13797 | If two elements are connec... |
| rtrclreclem1 13798 | The reflexive, transitive ... |
| rtrclreclem2 13799 | The reflexive, transitive ... |
| rtrclreclem3 13800 | The reflexive, transitive ... |
| rtrclreclem4 13801 | The reflexive, transitive ... |
| dfrtrcl2 13802 | The two definitions ` t* `... |
| relexpindlem 13803 | Principle of transitive in... |
| relexpind 13804 | Principle of transitive in... |
| rtrclind 13805 | Principle of transitive in... |
| shftlem 13808 | Two ways to write a shifte... |
| shftuz 13809 | A shift of the upper integ... |
| shftfval 13810 | The value of the sequence ... |
| shftdm 13811 | Domain of a relation shift... |
| shftfib 13812 | Value of a fiber of the re... |
| shftfn 13813 | Functionality and domain o... |
| shftval 13814 | Value of a sequence shifte... |
| shftval2 13815 | Value of a sequence shifte... |
| shftval3 13816 | Value of a sequence shifte... |
| shftval4 13817 | Value of a sequence shifte... |
| shftval5 13818 | Value of a shifted sequenc... |
| shftf 13819 | Functionality of a shifted... |
| 2shfti 13820 | Composite shift operations... |
| shftidt2 13821 | Identity law for the shift... |
| shftidt 13822 | Identity law for the shift... |
| shftcan1 13823 | Cancellation law for the s... |
| shftcan2 13824 | Cancellation law for the s... |
| seqshft 13825 | Shifting the index set of ... |
| sgnval 13828 | Value of Signum function. ... |
| sgn0 13829 | Proof that signum of 0 is ... |
| sgnp 13830 | Proof that signum of posit... |
| sgnrrp 13831 | Proof that signum of posit... |
| sgn1 13832 | Proof that the signum of 1... |
| sgnpnf 13833 | Proof that the signum of `... |
| sgnn 13834 | Proof that signum of negat... |
| sgnmnf 13835 | Proof that the signum of `... |
| cjval 13842 | The value of the conjugate... |
| cjth 13843 | The defining property of t... |
| cjf 13844 | Domain and codomain of the... |
| cjcl 13845 | The conjugate of a complex... |
| reval 13846 | The value of the real part... |
| imval 13847 | The value of the imaginary... |
| imre 13848 | The imaginary part of a co... |
| reim 13849 | The real part of a complex... |
| recl 13850 | The real part of a complex... |
| imcl 13851 | The imaginary part of a co... |
| ref 13852 | Domain and codomain of the... |
| imf 13853 | Domain and codomain of the... |
| crre 13854 | The real part of a complex... |
| crim 13855 | The real part of a complex... |
| replim 13856 | Reconstruct a complex numb... |
| remim 13857 | Value of the conjugate of ... |
| reim0 13858 | The imaginary part of a re... |
| reim0b 13859 | A number is real iff its i... |
| rereb 13860 | A number is real iff it eq... |
| mulre 13861 | A product with a nonzero r... |
| rere 13862 | A real number equals its r... |
| cjreb 13863 | A number is real iff it eq... |
| recj 13864 | Real part of a complex con... |
| reneg 13865 | Real part of negative. (C... |
| readd 13866 | Real part distributes over... |
| resub 13867 | Real part distributes over... |
| remullem 13868 | Lemma for ~ remul , ~ immu... |
| remul 13869 | Real part of a product. (... |
| remul2 13870 | Real part of a product. (... |
| rediv 13871 | Real part of a division. ... |
| imcj 13872 | Imaginary part of a comple... |
| imneg 13873 | The imaginary part of a ne... |
| imadd 13874 | Imaginary part distributes... |
| imsub 13875 | Imaginary part distributes... |
| immul 13876 | Imaginary part of a produc... |
| immul2 13877 | Imaginary part of a produc... |
| imdiv 13878 | Imaginary part of a divisi... |
| cjre 13879 | A real number equals its c... |
| cjcj 13880 | The conjugate of the conju... |
| cjadd 13881 | Complex conjugate distribu... |
| cjmul 13882 | Complex conjugate distribu... |
| ipcnval 13883 | Standard inner product on ... |
| cjmulrcl 13884 | A complex number times its... |
| cjmulval 13885 | A complex number times its... |
| cjmulge0 13886 | A complex number times its... |
| cjneg 13887 | Complex conjugate of negat... |
| addcj 13888 | A number plus its conjugat... |
| cjsub 13889 | Complex conjugate distribu... |
| cjexp 13890 | Complex conjugate of posit... |
| imval2 13891 | The imaginary part of a nu... |
| re0 13892 | The real part of zero. (C... |
| im0 13893 | The imaginary part of zero... |
| re1 13894 | The real part of one. (Co... |
| im1 13895 | The imaginary part of one.... |
| rei 13896 | The real part of ` _i ` . ... |
| imi 13897 | The imaginary part of ` _i... |
| cj0 13898 | The conjugate of zero. (C... |
| cji 13899 | The complex conjugate of t... |
| cjreim 13900 | The conjugate of a represe... |
| cjreim2 13901 | The conjugate of the repre... |
| cj11 13902 | Complex conjugate is a one... |
| cjne0 13903 | A number is nonzero iff it... |
| cjdiv 13904 | Complex conjugate distribu... |
| cnrecnv 13905 | The inverse to the canonic... |
| sqeqd 13906 | A deduction for showing tw... |
| recli 13907 | The real part of a complex... |
| imcli 13908 | The imaginary part of a co... |
| cjcli 13909 | Closure law for complex co... |
| replimi 13910 | Construct a complex number... |
| cjcji 13911 | The conjugate of the conju... |
| reim0bi 13912 | A number is real iff its i... |
| rerebi 13913 | A real number equals its r... |
| cjrebi 13914 | A number is real iff it eq... |
| recji 13915 | Real part of a complex con... |
| imcji 13916 | Imaginary part of a comple... |
| cjmulrcli 13917 | A complex number times its... |
| cjmulvali 13918 | A complex number times its... |
| cjmulge0i 13919 | A complex number times its... |
| renegi 13920 | Real part of negative. (C... |
| imnegi 13921 | Imaginary part of negative... |
| cjnegi 13922 | Complex conjugate of negat... |
| addcji 13923 | A number plus its conjugat... |
| readdi 13924 | Real part distributes over... |
| imaddi 13925 | Imaginary part distributes... |
| remuli 13926 | Real part of a product. (... |
| immuli 13927 | Imaginary part of a produc... |
| cjaddi 13928 | Complex conjugate distribu... |
| cjmuli 13929 | Complex conjugate distribu... |
| ipcni 13930 | Standard inner product on ... |
| cjdivi 13931 | Complex conjugate distribu... |
| crrei 13932 | The real part of a complex... |
| crimi 13933 | The imaginary part of a co... |
| recld 13934 | The real part of a complex... |
| imcld 13935 | The imaginary part of a co... |
| cjcld 13936 | Closure law for complex co... |
| replimd 13937 | Construct a complex number... |
| remimd 13938 | Value of the conjugate of ... |
| cjcjd 13939 | The conjugate of the conju... |
| reim0bd 13940 | A number is real iff its i... |
| rerebd 13941 | A real number equals its r... |
| cjrebd 13942 | A number is real iff it eq... |
| cjne0d 13943 | A number is nonzero iff it... |
| recjd 13944 | Real part of a complex con... |
| imcjd 13945 | Imaginary part of a comple... |
| cjmulrcld 13946 | A complex number times its... |
| cjmulvald 13947 | A complex number times its... |
| cjmulge0d 13948 | A complex number times its... |
| renegd 13949 | Real part of negative. (C... |
| imnegd 13950 | Imaginary part of negative... |
| cjnegd 13951 | Complex conjugate of negat... |
| addcjd 13952 | A number plus its conjugat... |
| cjexpd 13953 | Complex conjugate of posit... |
| readdd 13954 | Real part distributes over... |
| imaddd 13955 | Imaginary part distributes... |
| resubd 13956 | Real part distributes over... |
| imsubd 13957 | Imaginary part distributes... |
| remuld 13958 | Real part of a product. (... |
| immuld 13959 | Imaginary part of a produc... |
| cjaddd 13960 | Complex conjugate distribu... |
| cjmuld 13961 | Complex conjugate distribu... |
| ipcnd 13962 | Standard inner product on ... |
| cjdivd 13963 | Complex conjugate distribu... |
| rered 13964 | A real number equals its r... |
| reim0d 13965 | The imaginary part of a re... |
| cjred 13966 | A real number equals its c... |
| remul2d 13967 | Real part of a product. (... |
| immul2d 13968 | Imaginary part of a produc... |
| redivd 13969 | Real part of a division. ... |
| imdivd 13970 | Imaginary part of a divisi... |
| crred 13971 | The real part of a complex... |
| crimd 13972 | The imaginary part of a co... |
| sqrtval 13977 | Value of square root funct... |
| absval 13978 | The absolute value (modulu... |
| rennim 13979 | A real number does not lie... |
| cnpart 13980 | The specification of restr... |
| sqr0lem 13981 | Square root of zero. (Con... |
| sqrt0 13982 | Square root of zero. (Con... |
| sqrlem1 13983 | Lemma for ~ 01sqrex . (Co... |
| sqrlem2 13984 | Lemma for ~ 01sqrex . (Co... |
| sqrlem3 13985 | Lemma for ~ 01sqrex . (Co... |
| sqrlem4 13986 | Lemma for ~ 01sqrex . (Co... |
| sqrlem5 13987 | Lemma for ~ 01sqrex . (Co... |
| sqrlem6 13988 | Lemma for ~ 01sqrex . (Co... |
| sqrlem7 13989 | Lemma for ~ 01sqrex . (Co... |
| 01sqrex 13990 | Existence of a square root... |
| resqrex 13991 | Existence of a square root... |
| sqrmo 13992 | Uniqueness for the square ... |
| resqreu 13993 | Existence and uniqueness f... |
| resqrtcl 13994 | Closure of the square root... |
| resqrtthlem 13995 | Lemma for ~ resqrtth . (C... |
| resqrtth 13996 | Square root theorem over t... |
| remsqsqrt 13997 | Square of square root. (C... |
| sqrtge0 13998 | The square root function i... |
| sqrtgt0 13999 | The square root function i... |
| sqrtmul 14000 | Square root distributes ov... |
| sqrtle 14001 | Square root is monotonic. ... |
| sqrtlt 14002 | Square root is strictly mo... |
| sqrt11 14003 | The square root function i... |
| sqrt00 14004 | A square root is zero iff ... |
| rpsqrtcl 14005 | The square root of a posit... |
| sqrtdiv 14006 | Square root distributes ov... |
| sqrtneglem 14007 | The square root of a negat... |
| sqrtneg 14008 | The square root of a negat... |
| sqrtsq2 14009 | Relationship between squar... |
| sqrtsq 14010 | Square root of square. (C... |
| sqrtmsq 14011 | Square root of square. (C... |
| sqrt1 14012 | The square root of 1 is 1.... |
| sqrt4 14013 | The square root of 4 is 2.... |
| sqrt9 14014 | The square root of 9 is 3.... |
| sqrt2gt1lt2 14015 | The square root of 2 is bo... |
| sqrtm1 14016 | The imaginary unit is the ... |
| absneg 14017 | Absolute value of negative... |
| abscl 14018 | Real closure of absolute v... |
| abscj 14019 | The absolute value of a nu... |
| absvalsq 14020 | Square of value of absolut... |
| absvalsq2 14021 | Square of value of absolut... |
| sqabsadd 14022 | Square of absolute value o... |
| sqabssub 14023 | Square of absolute value o... |
| absval2 14024 | Value of absolute value fu... |
| abs0 14025 | The absolute value of 0. ... |
| absi 14026 | The absolute value of the ... |
| absge0 14027 | Absolute value is nonnegat... |
| absrpcl 14028 | The absolute value of a no... |
| abs00 14029 | The absolute value of a nu... |
| abs00ad 14030 | A complex number is zero i... |
| abs00bd 14031 | If a complex number is zer... |
| absreimsq 14032 | Square of the absolute val... |
| absreim 14033 | Absolute value of a number... |
| absmul 14034 | Absolute value distributes... |
| absdiv 14035 | Absolute value distributes... |
| absid 14036 | A nonnegative number is it... |
| abs1 14037 | The absolute value of 1. ... |
| absnid 14038 | A negative number is the n... |
| leabs 14039 | A real number is less than... |
| absor 14040 | The absolute value of a re... |
| absre 14041 | Absolute value of a real n... |
| absresq 14042 | Square of the absolute val... |
| absmod0 14043 | ` A ` is divisible by ` B ... |
| absexp 14044 | Absolute value of positive... |
| absexpz 14045 | Absolute value of integer ... |
| abssq 14046 | Square can be moved in and... |
| sqabs 14047 | The squares of two reals a... |
| absrele 14048 | The absolute value of a co... |
| absimle 14049 | The absolute value of a co... |
| max0add 14050 | The sum of the positive an... |
| absz 14051 | A real number is an intege... |
| nn0abscl 14052 | The absolute value of an i... |
| zabscl 14053 | The absolute value of an i... |
| abslt 14054 | Absolute value and 'less t... |
| absle 14055 | Absolute value and 'less t... |
| abssubne0 14056 | If the absolute value of a... |
| absdiflt 14057 | The absolute value of a di... |
| absdifle 14058 | The absolute value of a di... |
| elicc4abs 14059 | Membership in a symmetric ... |
| lenegsq 14060 | Comparison to a nonnegativ... |
| releabs 14061 | The real part of a number ... |
| recval 14062 | Reciprocal expressed with ... |
| absidm 14063 | The absolute value functio... |
| absgt0 14064 | The absolute value of a no... |
| nnabscl 14065 | The absolute value of a no... |
| abssub 14066 | Swapping order of subtract... |
| abssubge0 14067 | Absolute value of a nonneg... |
| abssuble0 14068 | Absolute value of a nonpos... |
| absmax 14069 | The maximum of two numbers... |
| abstri 14070 | Triangle inequality for ab... |
| abs3dif 14071 | Absolute value of differen... |
| abs2dif 14072 | Difference of absolute val... |
| abs2dif2 14073 | Difference of absolute val... |
| abs2difabs 14074 | Absolute value of differen... |
| abs1m 14075 | For any complex number, th... |
| recan 14076 | Cancellation law involving... |
| absf 14077 | Mapping domain and codomai... |
| abs3lem 14078 | Lemma involving absolute v... |
| abslem2 14079 | Lemma involving absolute v... |
| rddif 14080 | The difference between a r... |
| absrdbnd 14081 | Bound on the absolute valu... |
| fzomaxdiflem 14082 | Lemma for ~ fzomaxdif . (... |
| fzomaxdif 14083 | A bound on the separation ... |
| uzin2 14084 | The upper integers are clo... |
| rexanuz 14085 | Combine two different uppe... |
| rexanre 14086 | Combine two different uppe... |
| rexfiuz 14087 | Combine finitely many diff... |
| rexuz3 14088 | Restrict the base of the u... |
| rexanuz2 14089 | Combine two different uppe... |
| r19.29uz 14090 | A version of ~ 19.29 for u... |
| r19.2uz 14091 | A version of ~ r19.2z for ... |
| rexuzre 14092 | Convert an upper real quan... |
| rexico 14093 | Restrict the base of an up... |
| cau3lem 14094 | Lemma for ~ cau3 . (Contr... |
| cau3 14095 | Convert between three-quan... |
| cau4 14096 | Change the base of a Cauch... |
| caubnd2 14097 | A Cauchy sequence of compl... |
| caubnd 14098 | A Cauchy sequence of compl... |
| sqreulem 14099 | Lemma for ~ sqreu : write ... |
| sqreu 14100 | Existence and uniqueness f... |
| sqrtcl 14101 | Closure of the square root... |
| sqrtthlem 14102 | Lemma for ~ sqrtth . (Con... |
| sqrtf 14103 | Mapping domain and codomai... |
| sqrtth 14104 | Square root theorem over t... |
| sqrtrege0 14105 | The square root function m... |
| eqsqrtor 14106 | Solve an equation containi... |
| eqsqrtd 14107 | A deduction for showing th... |
| eqsqrt2d 14108 | A deduction for showing th... |
| amgm2 14109 | Arithmetic-geometric mean ... |
| sqrtthi 14110 | Square root theorem. Theo... |
| sqrtcli 14111 | The square root of a nonne... |
| sqrtgt0i 14112 | The square root of a posit... |
| sqrtmsqi 14113 | Square root of square. (C... |
| sqrtsqi 14114 | Square root of square. (C... |
| sqsqrti 14115 | Square of square root. (C... |
| sqrtge0i 14116 | The square root of a nonne... |
| absidi 14117 | A nonnegative number is it... |
| absnidi 14118 | A negative number is the n... |
| leabsi 14119 | A real number is less than... |
| absori 14120 | The absolute value of a re... |
| absrei 14121 | Absolute value of a real n... |
| sqrtpclii 14122 | The square root of a posit... |
| sqrtgt0ii 14123 | The square root of a posit... |
| sqrt11i 14124 | The square root function i... |
| sqrtmuli 14125 | Square root distributes ov... |
| sqrtmulii 14126 | Square root distributes ov... |
| sqrtmsq2i 14127 | Relationship between squar... |
| sqrtlei 14128 | Square root is monotonic. ... |
| sqrtlti 14129 | Square root is strictly mo... |
| abslti 14130 | Absolute value and 'less t... |
| abslei 14131 | Absolute value and 'less t... |
| absvalsqi 14132 | Square of value of absolut... |
| absvalsq2i 14133 | Square of value of absolut... |
| abscli 14134 | Real closure of absolute v... |
| absge0i 14135 | Absolute value is nonnegat... |
| absval2i 14136 | Value of absolute value fu... |
| abs00i 14137 | The absolute value of a nu... |
| absgt0i 14138 | The absolute value of a no... |
| absnegi 14139 | Absolute value of negative... |
| abscji 14140 | The absolute value of a nu... |
| releabsi 14141 | The real part of a number ... |
| abssubi 14142 | Swapping order of subtract... |
| absmuli 14143 | Absolute value distributes... |
| sqabsaddi 14144 | Square of absolute value o... |
| sqabssubi 14145 | Square of absolute value o... |
| absdivzi 14146 | Absolute value distributes... |
| abstrii 14147 | Triangle inequality for ab... |
| abs3difi 14148 | Absolute value of differen... |
| abs3lemi 14149 | Lemma involving absolute v... |
| rpsqrtcld 14150 | The square root of a posit... |
| sqrtgt0d 14151 | The square root of a posit... |
| absnidd 14152 | A negative number is the n... |
| leabsd 14153 | A real number is less than... |
| absord 14154 | The absolute value of a re... |
| absred 14155 | Absolute value of a real n... |
| resqrtcld 14156 | The square root of a nonne... |
| sqrtmsqd 14157 | Square root of square. (C... |
| sqrtsqd 14158 | Square root of square. (C... |
| sqrtge0d 14159 | The square root of a nonne... |
| sqrtnegd 14160 | The square root of a negat... |
| absidd 14161 | A nonnegative number is it... |
| sqrtdivd 14162 | Square root distributes ov... |
| sqrtmuld 14163 | Square root distributes ov... |
| sqrtsq2d 14164 | Relationship between squar... |
| sqrtled 14165 | Square root is monotonic. ... |
| sqrtltd 14166 | Square root is strictly mo... |
| sqr11d 14167 | The square root function i... |
| absltd 14168 | Absolute value and 'less t... |
| absled 14169 | Absolute value and 'less t... |
| abssubge0d 14170 | Absolute value of a nonneg... |
| abssuble0d 14171 | Absolute value of a nonpos... |
| absdifltd 14172 | The absolute value of a di... |
| absdifled 14173 | The absolute value of a di... |
| icodiamlt 14174 | Two elements in a half-ope... |
| abscld 14175 | Real closure of absolute v... |
| sqrtcld 14176 | Closure of the square root... |
| sqrtrege0d 14177 | The real part of the squar... |
| sqsqrtd 14178 | Square root theorem. Theo... |
| msqsqrtd 14179 | Square root theorem. Theo... |
| sqr00d 14180 | A square root is zero iff ... |
| absvalsqd 14181 | Square of value of absolut... |
| absvalsq2d 14182 | Square of value of absolut... |
| absge0d 14183 | Absolute value is nonnegat... |
| absval2d 14184 | Value of absolute value fu... |
| abs00d 14185 | The absolute value of a nu... |
| absne0d 14186 | The absolute value of a nu... |
| absrpcld 14187 | The absolute value of a no... |
| absnegd 14188 | Absolute value of negative... |
| abscjd 14189 | The absolute value of a nu... |
| releabsd 14190 | The real part of a number ... |
| absexpd 14191 | Absolute value of positive... |
| abssubd 14192 | Swapping order of subtract... |
| absmuld 14193 | Absolute value distributes... |
| absdivd 14194 | Absolute value distributes... |
| abstrid 14195 | Triangle inequality for ab... |
| abs2difd 14196 | Difference of absolute val... |
| abs2dif2d 14197 | Difference of absolute val... |
| abs2difabsd 14198 | Absolute value of differen... |
| abs3difd 14199 | Absolute value of differen... |
| abs3lemd 14200 | Lemma involving absolute v... |
| limsupgord 14203 | Ordering property of the s... |
| limsupcl 14204 | Closure of the superior li... |
| limsupval 14205 | The superior limit of an i... |
| limsupgf 14206 | Closure of the superior li... |
| limsupgval 14207 | Value of the superior limi... |
| limsupgle 14208 | The defining property of t... |
| limsuple 14209 | The defining property of t... |
| limsuplt 14210 | The defining property of t... |
| limsupval2 14211 | The superior limit, relati... |
| limsupgre 14212 | If a sequence of real numb... |
| limsupbnd1 14213 | If a sequence is eventuall... |
| limsupbnd2 14214 | If a sequence is eventuall... |
| climrel 14223 | The limit relation is a re... |
| rlimrel 14224 | The limit relation is a re... |
| clim 14225 | Express the predicate: Th... |
| rlim 14226 | Express the predicate: Th... |
| rlim2 14227 | Rewrite ~ rlim for a mappi... |
| rlim2lt 14228 | Use strictly less-than in ... |
| rlim3 14229 | Restrict the range of the ... |
| climcl 14230 | Closure of the limit of a ... |
| rlimpm 14231 | Closure of a function with... |
| rlimf 14232 | Closure of a function with... |
| rlimss 14233 | Domain closure of a functi... |
| rlimcl 14234 | Closure of the limit of a ... |
| clim2 14235 | Express the predicate: Th... |
| clim2c 14236 | Express the predicate ` F ... |
| clim0 14237 | Express the predicate ` F ... |
| clim0c 14238 | Express the predicate ` F ... |
| rlim0 14239 | Express the predicate ` B ... |
| rlim0lt 14240 | Use strictly less-than in ... |
| climi 14241 | Convergence of a sequence ... |
| climi2 14242 | Convergence of a sequence ... |
| climi0 14243 | Convergence of a sequence ... |
| rlimi 14244 | Convergence at infinity of... |
| rlimi2 14245 | Convergence at infinity of... |
| ello1 14246 | Elementhood in the set of ... |
| ello12 14247 | Elementhood in the set of ... |
| ello12r 14248 | Sufficient condition for e... |
| lo1f 14249 | An eventually upper bounde... |
| lo1dm 14250 | An eventually upper bounde... |
| lo1bdd 14251 | The defining property of a... |
| ello1mpt 14252 | Elementhood in the set of ... |
| ello1mpt2 14253 | Elementhood in the set of ... |
| ello1d 14254 | Sufficient condition for e... |
| lo1bdd2 14255 | If an eventually bounded f... |
| lo1bddrp 14256 | Refine ~ o1bdd2 to give a ... |
| elo1 14257 | Elementhood in the set of ... |
| elo12 14258 | Elementhood in the set of ... |
| elo12r 14259 | Sufficient condition for e... |
| o1f 14260 | An eventually bounded func... |
| o1dm 14261 | An eventually bounded func... |
| o1bdd 14262 | The defining property of a... |
| lo1o1 14263 | A function is eventually b... |
| lo1o12 14264 | A function is eventually b... |
| elo1mpt 14265 | Elementhood in the set of ... |
| elo1mpt2 14266 | Elementhood in the set of ... |
| elo1d 14267 | Sufficient condition for e... |
| o1lo1 14268 | A real function is eventua... |
| o1lo12 14269 | A lower bounded real funct... |
| o1lo1d 14270 | A real eventually bounded ... |
| icco1 14271 | Derive eventual boundednes... |
| o1bdd2 14272 | If an eventually bounded f... |
| o1bddrp 14273 | Refine ~ o1bdd2 to give a ... |
| climconst 14274 | An (eventually) constant s... |
| rlimconst 14275 | A constant sequence conver... |
| rlimclim1 14276 | Forward direction of ~ rli... |
| rlimclim 14277 | A sequence on an upper int... |
| climrlim2 14278 | Produce a real limit from ... |
| climconst2 14279 | A constant sequence conver... |
| climz 14280 | The zero sequence converge... |
| rlimuni 14281 | A real function whose doma... |
| rlimdm 14282 | Two ways to express that a... |
| climuni 14283 | An infinite sequence of co... |
| fclim 14284 | The limit relation is func... |
| climdm 14285 | Two ways to express that a... |
| climeu 14286 | An infinite sequence of co... |
| climreu 14287 | An infinite sequence of co... |
| climmo 14288 | An infinite sequence of co... |
| rlimres 14289 | The restriction of a funct... |
| lo1res 14290 | The restriction of an even... |
| o1res 14291 | The restriction of an even... |
| rlimres2 14292 | The restriction of a funct... |
| lo1res2 14293 | The restriction of a funct... |
| o1res2 14294 | The restriction of a funct... |
| lo1resb 14295 | The restriction of a funct... |
| rlimresb 14296 | The restriction of a funct... |
| o1resb 14297 | The restriction of a funct... |
| climeq 14298 | Two functions that are eve... |
| lo1eq 14299 | Two functions that are eve... |
| rlimeq 14300 | Two functions that are eve... |
| o1eq 14301 | Two functions that are eve... |
| climmpt 14302 | Exhibit a function ` G ` w... |
| 2clim 14303 | If two sequences converge ... |
| climmpt2 14304 | Relate an integer limit on... |
| climshftlem 14305 | A shifted function converg... |
| climres 14306 | A function restricted to u... |
| climshft 14307 | A shifted function converg... |
| serclim0 14308 | The zero series converges ... |
| rlimcld2 14309 | If ` D ` is a closed set i... |
| rlimrege0 14310 | The limit of a sequence of... |
| rlimrecl 14311 | The limit of a real sequen... |
| rlimge0 14312 | The limit of a sequence of... |
| climshft2 14313 | A shifted function converg... |
| climrecl 14314 | The limit of a convergent ... |
| climge0 14315 | A nonnegative sequence con... |
| climabs0 14316 | Convergence to zero of the... |
| o1co 14317 | Sufficient condition for t... |
| o1compt 14318 | Sufficient condition for t... |
| rlimcn1 14319 | Image of a limit under a c... |
| rlimcn1b 14320 | Image of a limit under a c... |
| rlimcn2 14321 | Image of a limit under a c... |
| climcn1 14322 | Image of a limit under a c... |
| climcn2 14323 | Image of a limit under a c... |
| addcn2 14324 | Complex number addition is... |
| subcn2 14325 | Complex number subtraction... |
| mulcn2 14326 | Complex number multiplicat... |
| reccn2 14327 | The reciprocal function is... |
| cn1lem 14328 | A sufficient condition for... |
| abscn2 14329 | The absolute value functio... |
| cjcn2 14330 | The complex conjugate func... |
| recn2 14331 | The real part function is ... |
| imcn2 14332 | The imaginary part functio... |
| climcn1lem 14333 | The limit of a continuous ... |
| climabs 14334 | Limit of the absolute valu... |
| climcj 14335 | Limit of the complex conju... |
| climre 14336 | Limit of the real part of ... |
| climim 14337 | Limit of the imaginary par... |
| rlimmptrcl 14338 | Reverse closure for a real... |
| rlimabs 14339 | Limit of the absolute valu... |
| rlimcj 14340 | Limit of the complex conju... |
| rlimre 14341 | Limit of the real part of ... |
| rlimim 14342 | Limit of the imaginary par... |
| o1of2 14343 | Show that a binary operati... |
| o1add 14344 | The sum of two eventually ... |
| o1mul 14345 | The product of two eventua... |
| o1sub 14346 | The difference of two even... |
| rlimo1 14347 | Any function with a finite... |
| rlimdmo1 14348 | A convergent function is e... |
| o1rlimmul 14349 | The product of an eventual... |
| o1const 14350 | A constant function is eve... |
| lo1const 14351 | A constant function is eve... |
| lo1mptrcl 14352 | Reverse closure for an eve... |
| o1mptrcl 14353 | Reverse closure for an eve... |
| o1add2 14354 | The sum of two eventually ... |
| o1mul2 14355 | The product of two eventua... |
| o1sub2 14356 | The product of two eventua... |
| lo1add 14357 | The sum of two eventually ... |
| lo1mul 14358 | The product of an eventual... |
| lo1mul2 14359 | The product of an eventual... |
| o1dif 14360 | If the difference of two f... |
| lo1sub 14361 | The difference of an event... |
| climadd 14362 | Limit of the sum of two co... |
| climmul 14363 | Limit of the product of tw... |
| climsub 14364 | Limit of the difference of... |
| climaddc1 14365 | Limit of a constant ` C ` ... |
| climaddc2 14366 | Limit of a constant ` C ` ... |
| climmulc2 14367 | Limit of a sequence multip... |
| climsubc1 14368 | Limit of a constant ` C ` ... |
| climsubc2 14369 | Limit of a constant ` C ` ... |
| climle 14370 | Comparison of the limits o... |
| climsqz 14371 | Convergence of a sequence ... |
| climsqz2 14372 | Convergence of a sequence ... |
| rlimadd 14373 | Limit of the sum of two co... |
| rlimsub 14374 | Limit of the difference of... |
| rlimmul 14375 | Limit of the product of tw... |
| rlimdiv 14376 | Limit of the quotient of t... |
| rlimneg 14377 | Limit of the negative of a... |
| rlimle 14378 | Comparison of the limits o... |
| rlimsqzlem 14379 | Lemma for ~ rlimsqz and ~ ... |
| rlimsqz 14380 | Convergence of a sequence ... |
| rlimsqz2 14381 | Convergence of a sequence ... |
| lo1le 14382 | Transfer eventual upper bo... |
| o1le 14383 | Transfer eventual boundedn... |
| rlimno1 14384 | A function whose inverse c... |
| clim2ser 14385 | The limit of an infinite s... |
| clim2ser2 14386 | The limit of an infinite s... |
| iserex 14387 | An infinite series converg... |
| isermulc2 14388 | Multiplication of an infin... |
| climlec2 14389 | Comparison of a constant t... |
| iserle 14390 | Comparison of the limits o... |
| iserge0 14391 | The limit of an infinite s... |
| climub 14392 | The limit of a monotonic s... |
| climserle 14393 | The partial sums of a conv... |
| isershft 14394 | Index shift of the limit o... |
| isercolllem1 14395 | Lemma for ~ isercoll . (C... |
| isercolllem2 14396 | Lemma for ~ isercoll . (C... |
| isercolllem3 14397 | Lemma for ~ isercoll . (C... |
| isercoll 14398 | Rearrange an infinite seri... |
| isercoll2 14399 | Generalize ~ isercoll so t... |
| climsup 14400 | A bounded monotonic sequen... |
| climcau 14401 | A converging sequence of c... |
| climbdd 14402 | A converging sequence of c... |
| caucvgrlem 14403 | Lemma for ~ caurcvgr . (C... |
| caurcvgr 14404 | A Cauchy sequence of real ... |
| caucvgrlem2 14405 | Lemma for ~ caucvgr . (Co... |
| caucvgr 14406 | A Cauchy sequence of compl... |
| caurcvg 14407 | A Cauchy sequence of real ... |
| caurcvg2 14408 | A Cauchy sequence of real ... |
| caucvg 14409 | A Cauchy sequence of compl... |
| caucvgb 14410 | A function is convergent i... |
| serf0 14411 | If an infinite series conv... |
| iseraltlem1 14412 | Lemma for ~ iseralt . A d... |
| iseraltlem2 14413 | Lemma for ~ iseralt . The... |
| iseraltlem3 14414 | Lemma for ~ iseralt . Fro... |
| iseralt 14415 | The alternating series tes... |
| sumex 14418 | A sum is a set. (Contribu... |
| sumeq1 14419 | Equality theorem for a sum... |
| nfsum1 14420 | Bound-variable hypothesis ... |
| nfsum 14421 | Bound-variable hypothesis ... |
| sumeq2w 14422 | Equality theorem for sum, ... |
| sumeq2ii 14423 | Equality theorem for sum, ... |
| sumeq2 14424 | Equality theorem for sum. ... |
| cbvsum 14425 | Change bound variable in a... |
| cbvsumv 14426 | Change bound variable in a... |
| cbvsumi 14427 | Change bound variable in a... |
| sumeq1i 14428 | Equality inference for sum... |
| sumeq2i 14429 | Equality inference for sum... |
| sumeq12i 14430 | Equality inference for sum... |
| sumeq1d 14431 | Equality deduction for sum... |
| sumeq2d 14432 | Equality deduction for sum... |
| sumeq2dv 14433 | Equality deduction for sum... |
| sumeq2ad 14434 | Equality deduction for sum... |
| sumeq2sdv 14435 | Equality deduction for sum... |
| 2sumeq2dv 14436 | Equality deduction for dou... |
| sumeq12dv 14437 | Equality deduction for sum... |
| sumeq12rdv 14438 | Equality deduction for sum... |
| sum2id 14439 | The second class argument ... |
| sumfc 14440 | A lemma to facilitate conv... |
| fz1f1o 14441 | A lemma for working with f... |
| sumrblem 14442 | Lemma for ~ sumrb . (Cont... |
| fsumcvg 14443 | The sequence of partial su... |
| sumrb 14444 | Rebase the starting point ... |
| summolem3 14445 | Lemma for ~ summo . (Cont... |
| summolem2a 14446 | Lemma for ~ summo . (Cont... |
| summolem2 14447 | Lemma for ~ summo . (Cont... |
| summo 14448 | A sum has at most one limi... |
| zsum 14449 | Series sum with index set ... |
| isum 14450 | Series sum with an upper i... |
| fsum 14451 | The value of a sum over a ... |
| sum0 14452 | Any sum over the empty set... |
| sumz 14453 | Any sum of zero over a sum... |
| fsumf1o 14454 | Re-index a finite sum usin... |
| sumss 14455 | Change the index set to a ... |
| fsumss 14456 | Change the index set to a ... |
| sumss2 14457 | Change the index set of a ... |
| fsumcvg2 14458 | The sequence of partial su... |
| fsumsers 14459 | Special case of series sum... |
| fsumcvg3 14460 | A finite sum is convergent... |
| fsumser 14461 | A finite sum expressed in ... |
| fsumcl2lem 14462 | - Lemma for finite sum clo... |
| fsumcllem 14463 | - Lemma for finite sum clo... |
| fsumcl 14464 | Closure of a finite sum of... |
| fsumrecl 14465 | Closure of a finite sum of... |
| fsumzcl 14466 | Closure of a finite sum of... |
| fsumnn0cl 14467 | Closure of a finite sum of... |
| fsumrpcl 14468 | Closure of a finite sum of... |
| fsumzcl2 14469 | A finite sum with integer ... |
| fsumadd 14470 | The sum of two finite sums... |
| fsumsplit 14471 | Split a sum into two parts... |
| fsumsplitf 14472 | Split a sum into two parts... |
| sumsnf 14473 | A sum of a singleton is th... |
| fsumsplitsn 14474 | Separate out a term in a f... |
| sumsn 14475 | A sum of a singleton is th... |
| fsum1 14476 | The finite sum of ` A ( k ... |
| sumpr 14477 | A sum over a pair is the s... |
| sumtp 14478 | A sum over a triple is the... |
| sumsns 14479 | A sum of a singleton is th... |
| fsumm1 14480 | Separate out the last term... |
| fzosump1 14481 | Separate out the last term... |
| fsum1p 14482 | Separate out the first ter... |
| fsummsnunz 14483 | A finite sum all of whose ... |
| fsumsplitsnun 14484 | Separate out a term in a f... |
| fsummsnunzOLD 14485 | Obsolete version of ~ fsum... |
| fsumsplitsnunOLD 14486 | Obsolete version of ~ fsum... |
| fsump1 14487 | The addition of the next t... |
| isumclim 14488 | An infinite sum equals the... |
| isumclim2 14489 | A converging series conver... |
| isumclim3 14490 | The sequence of partial fi... |
| sumnul 14491 | The sum of a non-convergen... |
| isumcl 14492 | The sum of a converging in... |
| isummulc2 14493 | An infinite sum multiplied... |
| isummulc1 14494 | An infinite sum multiplied... |
| isumdivc 14495 | An infinite sum divided by... |
| isumrecl 14496 | The sum of a converging in... |
| isumge0 14497 | An infinite sum of nonnega... |
| isumadd 14498 | Addition of infinite sums.... |
| sumsplit 14499 | Split a sum into two parts... |
| fsump1i 14500 | Optimized version of ~ fsu... |
| fsum2dlem 14501 | Lemma for ~ fsum2d - induc... |
| fsum2d 14502 | Write a double sum as a su... |
| fsumxp 14503 | Combine two sums into a si... |
| fsumcnv 14504 | Transform a region of summ... |
| fsumcom2 14505 | Interchange order of summa... |
| fsumcom2OLD 14506 | Obsolete proof of ~ fsumco... |
| fsumcom 14507 | Interchange order of summa... |
| fsum0diaglem 14508 | Lemma for ~ fsum0diag . (... |
| fsum0diag 14509 | Two ways to express "the s... |
| mptfzshft 14510 | 1-1 onto function in maps-... |
| fsumrev 14511 | Reversal of a finite sum. ... |
| fsumshft 14512 | Index shift of a finite su... |
| fsumshftm 14513 | Negative index shift of a ... |
| fsumrev2 14514 | Reversal of a finite sum. ... |
| fsum0diag2 14515 | Two ways to express "the s... |
| fsummulc2 14516 | A finite sum multiplied by... |
| fsummulc1 14517 | A finite sum multiplied by... |
| fsumdivc 14518 | A finite sum divided by a ... |
| fsumneg 14519 | Negation of a finite sum. ... |
| fsumsub 14520 | Split a finite sum over a ... |
| fsum2mul 14521 | Separate the nested sum of... |
| fsumconst 14522 | The sum of constant terms ... |
| fsumdifsnconst 14523 | The sum of constant terms ... |
| modfsummodslem1 14524 | Lemma 1 for ~ modfsummods ... |
| modfsummods 14525 | Induction step for ~ modfs... |
| modfsummod 14526 | A finite sum modulo a posi... |
| fsumge0 14527 | If all of the terms of a f... |
| fsumless 14528 | A shorter sum of nonnegati... |
| fsumge1 14529 | A sum of nonnegative numbe... |
| fsum00 14530 | A sum of nonnegative numbe... |
| fsumle 14531 | If all of the terms of fin... |
| fsumlt 14532 | If every term in one finit... |
| fsumabs 14533 | Generalized triangle inequ... |
| telfsumo 14534 | Sum of a telescoping serie... |
| telfsumo2 14535 | Sum of a telescoping serie... |
| telfsum 14536 | Sum of a telescoping serie... |
| telfsum2 14537 | Sum of a telescoping serie... |
| fsumparts 14538 | Summation by parts. (Cont... |
| fsumrelem 14539 | Lemma for ~ fsumre , ~ fsu... |
| fsumre 14540 | The real part of a sum. (... |
| fsumim 14541 | The imaginary part of a su... |
| fsumcj 14542 | The complex conjugate of a... |
| fsumrlim 14543 | Limit of a finite sum of c... |
| fsumo1 14544 | The finite sum of eventual... |
| o1fsum 14545 | If ` A ( k ) ` is O(1), th... |
| seqabs 14546 | Generalized triangle inequ... |
| iserabs 14547 | Generalized triangle inequ... |
| cvgcmp 14548 | A comparison test for conv... |
| cvgcmpub 14549 | An upper bound for the lim... |
| cvgcmpce 14550 | A comparison test for conv... |
| abscvgcvg 14551 | An absolutely convergent s... |
| climfsum 14552 | Limit of a finite sum of c... |
| fsumiun 14553 | Sum over a disjoint indexe... |
| hashiun 14554 | The cardinality of a disjo... |
| hash2iun 14555 | The cardinality of a neste... |
| hash2iun1dif1 14556 | The cardinality of a neste... |
| hashrabrex 14557 | The number of elements in ... |
| hashuni 14558 | The cardinality of a disjo... |
| qshash 14559 | The cardinality of a set w... |
| ackbijnn 14560 | Translate the Ackermann bi... |
| binomlem 14561 | Lemma for ~ binom (binomia... |
| binom 14562 | The binomial theorem: ` ( ... |
| binom1p 14563 | Special case of the binomi... |
| binom11 14564 | Special case of the binomi... |
| binom1dif 14565 | A summation for the differ... |
| bcxmaslem1 14566 | Lemma for ~ bcxmas . (Con... |
| bcxmas 14567 | Parallel summation (Christ... |
| incexclem 14568 | Lemma for ~ incexc . (Con... |
| incexc 14569 | The inclusion/exclusion pr... |
| incexc2 14570 | The inclusion/exclusion pr... |
| isumshft 14571 | Index shift of an infinite... |
| isumsplit 14572 | Split off the first ` N ` ... |
| isum1p 14573 | The infinite sum of a conv... |
| isumnn0nn 14574 | Sum from 0 to infinity in ... |
| isumrpcl 14575 | The infinite sum of positi... |
| isumle 14576 | Comparison of two infinite... |
| isumless 14577 | A finite sum of nonnegativ... |
| isumsup2 14578 | An infinite sum of nonnega... |
| isumsup 14579 | An infinite sum of nonnega... |
| isumltss 14580 | A partial sum of a series ... |
| climcndslem1 14581 | Lemma for ~ climcnds : bou... |
| climcndslem2 14582 | Lemma for ~ climcnds : bou... |
| climcnds 14583 | The Cauchy condensation te... |
| divrcnv 14584 | The sequence of reciprocal... |
| divcnv 14585 | The sequence of reciprocal... |
| flo1 14586 | The floor function satisfi... |
| divcnvshft 14587 | Limit of a ratio function.... |
| supcvg 14588 | Extract a sequence ` f ` i... |
| infcvgaux1i 14589 | Auxiliary theorem for appl... |
| infcvgaux2i 14590 | Auxiliary theorem for appl... |
| harmonic 14591 | The harmonic series ` H ` ... |
| arisum 14592 | Arithmetic series sum of t... |
| arisum2 14593 | Arithmetic series sum of t... |
| trireciplem 14594 | Lemma for ~ trirecip . Sh... |
| trirecip 14595 | The sum of the reciprocals... |
| expcnv 14596 | A sequence of powers of a ... |
| explecnv 14597 | A sequence of terms conver... |
| geoserg 14598 | The value of the finite ge... |
| geoser 14599 | The value of the finite ge... |
| pwm1geoser 14600 | The n-th power of a number... |
| geolim 14601 | The partial sums in the in... |
| geolim2 14602 | The partial sums in the ge... |
| georeclim 14603 | The limit of a geometric s... |
| geo2sum 14604 | The value of the finite ge... |
| geo2sum2 14605 | The value of the finite ge... |
| geo2lim 14606 | The value of the infinite ... |
| geomulcvg 14607 | The geometric series conve... |
| geoisum 14608 | The infinite sum of ` 1 + ... |
| geoisumr 14609 | The infinite sum of recipr... |
| geoisum1 14610 | The infinite sum of ` A ^ ... |
| geoisum1c 14611 | The infinite sum of ` A x.... |
| 0.999... 14612 | The recurring decimal 0.99... |
| 0.999...OLD 14613 | Obsolete version of ~ 0.99... |
| geoihalfsum 14614 | Prove that the infinite ge... |
| cvgrat 14615 | Ratio test for convergence... |
| mertenslem1 14616 | Lemma for ~ mertens . (Co... |
| mertenslem2 14617 | Lemma for ~ mertens . (Co... |
| mertens 14618 | Mertens' theorem. If ` A ... |
| prodf 14619 | An infinite product of com... |
| clim2prod 14620 | The limit of an infinite p... |
| clim2div 14621 | The limit of an infinite p... |
| prodfmul 14622 | The product of two infinit... |
| prodf1 14623 | The value of the partial p... |
| prodf1f 14624 | A one-valued infinite prod... |
| prodfclim1 14625 | The constant one product c... |
| prodfn0 14626 | No term of a nonzero infin... |
| prodfrec 14627 | The reciprocal of an infin... |
| prodfdiv 14628 | The quotient of two infini... |
| ntrivcvg 14629 | A non-trivially converging... |
| ntrivcvgn0 14630 | A product that converges t... |
| ntrivcvgfvn0 14631 | Any value of a product seq... |
| ntrivcvgtail 14632 | A tail of a non-trivially ... |
| ntrivcvgmullem 14633 | Lemma for ~ ntrivcvgmul . ... |
| ntrivcvgmul 14634 | The product of two non-tri... |
| prodex 14637 | A product is a set. (Cont... |
| prodeq1f 14638 | Equality theorem for a pro... |
| prodeq1 14639 | Equality theorem for a pro... |
| nfcprod1 14640 | Bound-variable hypothesis ... |
| nfcprod 14641 | Bound-variable hypothesis ... |
| prodeq2w 14642 | Equality theorem for produ... |
| prodeq2ii 14643 | Equality theorem for produ... |
| prodeq2 14644 | Equality theorem for produ... |
| cbvprod 14645 | Change bound variable in a... |
| cbvprodv 14646 | Change bound variable in a... |
| cbvprodi 14647 | Change bound variable in a... |
| prodeq1i 14648 | Equality inference for pro... |
| prodeq2i 14649 | Equality inference for pro... |
| prodeq12i 14650 | Equality inference for pro... |
| prodeq1d 14651 | Equality deduction for pro... |
| prodeq2d 14652 | Equality deduction for pro... |
| prodeq2dv 14653 | Equality deduction for pro... |
| prodeq2sdv 14654 | Equality deduction for pro... |
| 2cprodeq2dv 14655 | Equality deduction for dou... |
| prodeq12dv 14656 | Equality deduction for pro... |
| prodeq12rdv 14657 | Equality deduction for pro... |
| prod2id 14658 | The second class argument ... |
| prodrblem 14659 | Lemma for ~ prodrb . (Con... |
| fprodcvg 14660 | The sequence of partial pr... |
| prodrblem2 14661 | Lemma for ~ prodrb . (Con... |
| prodrb 14662 | Rebase the starting point ... |
| prodmolem3 14663 | Lemma for ~ prodmo . (Con... |
| prodmolem2a 14664 | Lemma for ~ prodmo . (Con... |
| prodmolem2 14665 | Lemma for ~ prodmo . (Con... |
| prodmo 14666 | A product has at most one ... |
| zprod 14667 | Series product with index ... |
| iprod 14668 | Series product with an upp... |
| zprodn0 14669 | Nonzero series product wit... |
| iprodn0 14670 | Nonzero series product wit... |
| fprod 14671 | The value of a product ove... |
| fprodntriv 14672 | A non-triviality lemma for... |
| prod0 14673 | A product over the empty s... |
| prod1 14674 | Any product of one over a ... |
| prodfc 14675 | A lemma to facilitate conv... |
| fprodf1o 14676 | Re-index a finite product ... |
| prodss 14677 | Change the index set to a ... |
| fprodss 14678 | Change the index set to a ... |
| fprodser 14679 | A finite product expressed... |
| fprodcl2lem 14680 | Finite product closure lem... |
| fprodcllem 14681 | Finite product closure lem... |
| fprodcl 14682 | Closure of a finite produc... |
| fprodrecl 14683 | Closure of a finite produc... |
| fprodzcl 14684 | Closure of a finite produc... |
| fprodnncl 14685 | Closure of a finite produc... |
| fprodrpcl 14686 | Closure of a finite produc... |
| fprodnn0cl 14687 | Closure of a finite produc... |
| fprodcllemf 14688 | Finite product closure lem... |
| fprodreclf 14689 | Closure of a finite produc... |
| fprodmul 14690 | The product of two finite ... |
| fproddiv 14691 | The quotient of two finite... |
| prodsn 14692 | A product of a singleton i... |
| fprod1 14693 | A finite product of only o... |
| prodsnf 14694 | A product of a singleton i... |
| climprod1 14695 | The limit of a product ove... |
| fprodsplit 14696 | Split a finite product int... |
| fprodm1 14697 | Separate out the last term... |
| fprod1p 14698 | Separate out the first ter... |
| fprodp1 14699 | Multiply in the last term ... |
| fprodm1s 14700 | Separate out the last term... |
| fprodp1s 14701 | Multiply in the last term ... |
| prodsns 14702 | A product of the singleton... |
| fprodfac 14703 | Factorial using product no... |
| fprodabs 14704 | The absolute value of a fi... |
| fprodeq0 14705 | Anything finite product co... |
| fprodshft 14706 | Shift the index of a finit... |
| fprodrev 14707 | Reversal of a finite produ... |
| fprodconst 14708 | The product of constant te... |
| fprodn0 14709 | A finite product of nonzer... |
| fprod2dlem 14710 | Lemma for ~ fprod2d - indu... |
| fprod2d 14711 | Write a double product as ... |
| fprodxp 14712 | Combine two products into ... |
| fprodcnv 14713 | Transform a product region... |
| fprodcom2 14714 | Interchange order of multi... |
| fprodcom2OLD 14715 | Obsolete proof of ~ fprodc... |
| fprodcom 14716 | Interchange product order.... |
| fprod0diag 14717 | Two ways to express "the p... |
| fproddivf 14718 | The quotient of two finite... |
| fprodsplitf 14719 | Split a finite product int... |
| fprodsplitsn 14720 | Separate out a term in a f... |
| fprodsplit1f 14721 | Separate out a term in a f... |
| fprodn0f 14722 | A finite product of nonzer... |
| fprodclf 14723 | Closure of a finite produc... |
| fprodge0 14724 | If all the terms of a fini... |
| fprodeq0g 14725 | Any finite product contain... |
| fprodge1 14726 | If all of the terms of a f... |
| fprodle 14727 | If all the terms of two fi... |
| fprodmodd 14728 | If all factors of two fini... |
| iprodclim 14729 | An infinite product equals... |
| iprodclim2 14730 | A converging product conve... |
| iprodclim3 14731 | The sequence of partial fi... |
| iprodcl 14732 | The product of a non-trivi... |
| iprodrecl 14733 | The product of a non-trivi... |
| iprodmul 14734 | Multiplication of infinite... |
| risefacval 14739 | The value of the rising fa... |
| fallfacval 14740 | The value of the falling f... |
| risefacval2 14741 | One-based value of rising ... |
| fallfacval2 14742 | One-based value of falling... |
| fallfacval3 14743 | A product representation o... |
| risefaccllem 14744 | Lemma for rising factorial... |
| fallfaccllem 14745 | Lemma for falling factoria... |
| risefaccl 14746 | Closure law for rising fac... |
| fallfaccl 14747 | Closure law for falling fa... |
| rerisefaccl 14748 | Closure law for rising fac... |
| refallfaccl 14749 | Closure law for falling fa... |
| nnrisefaccl 14750 | Closure law for rising fac... |
| zrisefaccl 14751 | Closure law for rising fac... |
| zfallfaccl 14752 | Closure law for falling fa... |
| nn0risefaccl 14753 | Closure law for rising fac... |
| rprisefaccl 14754 | Closure law for rising fac... |
| risefallfac 14755 | A relationship between ris... |
| fallrisefac 14756 | A relationship between fal... |
| risefall0lem 14757 | Lemma for ~ risefac0 and ~... |
| risefac0 14758 | The value of the rising fa... |
| fallfac0 14759 | The value of the falling f... |
| risefacp1 14760 | The value of the rising fa... |
| fallfacp1 14761 | The value of the falling f... |
| risefacp1d 14762 | The value of the rising fa... |
| fallfacp1d 14763 | The value of the falling f... |
| risefac1 14764 | The value of rising factor... |
| fallfac1 14765 | The value of falling facto... |
| risefacfac 14766 | Relate rising factorial to... |
| fallfacfwd 14767 | The forward difference of ... |
| 0fallfac 14768 | The value of the zero fall... |
| 0risefac 14769 | The value of the zero risi... |
| binomfallfaclem1 14770 | Lemma for ~ binomfallfac .... |
| binomfallfaclem2 14771 | Lemma for ~ binomfallfac .... |
| binomfallfac 14772 | A version of the binomial ... |
| binomrisefac 14773 | A version of the binomial ... |
| fallfacval4 14774 | Represent the falling fact... |
| bcfallfac 14775 | Binomial coefficient in te... |
| fallfacfac 14776 | Relate falling factorial t... |
| bpolylem 14779 | Lemma for ~ bpolyval . (C... |
| bpolyval 14780 | The value of the Bernoulli... |
| bpoly0 14781 | The value of the Bernoulli... |
| bpoly1 14782 | The value of the Bernoulli... |
| bpolycl 14783 | Closure law for Bernoulli ... |
| bpolysum 14784 | A sum for Bernoulli polyno... |
| bpolydiflem 14785 | Lemma for ~ bpolydif . (C... |
| bpolydif 14786 | Calculate the difference b... |
| fsumkthpow 14787 | A closed-form expression f... |
| bpoly2 14788 | The Bernoulli polynomials ... |
| bpoly3 14789 | The Bernoulli polynomials ... |
| bpoly4 14790 | The Bernoulli polynomials ... |
| fsumcube 14791 | Express the sum of cubes i... |
| eftcl 14804 | Closure of a term in the s... |
| reeftcl 14805 | The terms of the series ex... |
| eftabs 14806 | The absolute value of a te... |
| eftval 14807 | The value of a term in the... |
| efcllem 14808 | Lemma for ~ efcl . The se... |
| ef0lem 14809 | The series defining the ex... |
| efval 14810 | Value of the exponential f... |
| esum 14811 | Value of Euler's constant ... |
| eff 14812 | Domain and codomain of the... |
| efcl 14813 | Closure law for the expone... |
| efval2 14814 | Value of the exponential f... |
| efcvg 14815 | The series that defines th... |
| efcvgfsum 14816 | Exponential function conve... |
| reefcl 14817 | The exponential function i... |
| reefcld 14818 | The exponential function i... |
| ere 14819 | Euler's constant ` _e ` = ... |
| ege2le3 14820 | Lemma for ~ egt2lt3 . (Co... |
| ef0 14821 | Value of the exponential f... |
| efcj 14822 | Exponential function of a ... |
| efaddlem 14823 | Lemma for ~ efadd (exponen... |
| efadd 14824 | Sum of exponents law for e... |
| fprodefsum 14825 | Move the exponential funct... |
| efcan 14826 | Cancellation of law for ex... |
| efne0 14827 | The exponential function n... |
| efneg 14828 | Exponent of a negative num... |
| eff2 14829 | The exponential function m... |
| efsub 14830 | Difference of exponents la... |
| efexp 14831 | Exponential function to an... |
| efzval 14832 | Value of the exponential f... |
| efgt0 14833 | The exponential function o... |
| rpefcl 14834 | The exponential function o... |
| rpefcld 14835 | The exponential function o... |
| eftlcvg 14836 | The tail series of the exp... |
| eftlcl 14837 | Closure of the sum of an i... |
| reeftlcl 14838 | Closure of the sum of an i... |
| eftlub 14839 | An upper bound on the abso... |
| efsep 14840 | Separate out the next term... |
| effsumlt 14841 | The partial sums of the se... |
| eft0val 14842 | The value of the first ter... |
| ef4p 14843 | Separate out the first fou... |
| efgt1p2 14844 | The exponential function o... |
| efgt1p 14845 | The exponential function o... |
| efgt1 14846 | The exponential function o... |
| eflt 14847 | The exponential function o... |
| efle 14848 | The exponential function o... |
| reef11 14849 | The exponential function o... |
| reeff1 14850 | The exponential function m... |
| eflegeo 14851 | The exponential function o... |
| sinval 14852 | Value of the sine function... |
| cosval 14853 | Value of the cosine functi... |
| sinf 14854 | Domain and codomain of the... |
| cosf 14855 | Domain and codomain of the... |
| sincl 14856 | Closure of the sine functi... |
| coscl 14857 | Closure of the cosine func... |
| tanval 14858 | Value of the tangent funct... |
| tancl 14859 | The closure of the tangent... |
| sincld 14860 | Closure of the sine functi... |
| coscld 14861 | Closure of the cosine func... |
| tancld 14862 | Closure of the tangent fun... |
| tanval2 14863 | Express the tangent functi... |
| tanval3 14864 | Express the tangent functi... |
| resinval 14865 | The sine of a real number ... |
| recosval 14866 | The cosine of a real numbe... |
| efi4p 14867 | Separate out the first fou... |
| resin4p 14868 | Separate out the first fou... |
| recos4p 14869 | Separate out the first fou... |
| resincl 14870 | The sine of a real number ... |
| recoscl 14871 | The cosine of a real numbe... |
| retancl 14872 | The closure of the tangent... |
| resincld 14873 | Closure of the sine functi... |
| recoscld 14874 | Closure of the cosine func... |
| retancld 14875 | Closure of the tangent fun... |
| sinneg 14876 | The sine of a negative is ... |
| cosneg 14877 | The cosines of a number an... |
| tanneg 14878 | The tangent of a negative ... |
| sin0 14879 | Value of the sine function... |
| cos0 14880 | Value of the cosine functi... |
| tan0 14881 | The value of the tangent f... |
| efival 14882 | The exponential function i... |
| efmival 14883 | The exponential function i... |
| sinhval 14884 | Value of the hyperbolic si... |
| coshval 14885 | Value of the hyperbolic co... |
| resinhcl 14886 | The hyperbolic sine of a r... |
| rpcoshcl 14887 | The hyperbolic cosine of a... |
| recoshcl 14888 | The hyperbolic cosine of a... |
| retanhcl 14889 | The hyperbolic tangent of ... |
| tanhlt1 14890 | The hyperbolic tangent of ... |
| tanhbnd 14891 | The hyperbolic tangent of ... |
| efeul 14892 | Eulerian representation of... |
| efieq 14893 | The exponentials of two im... |
| sinadd 14894 | Addition formula for sine.... |
| cosadd 14895 | Addition formula for cosin... |
| tanaddlem 14896 | A useful intermediate step... |
| tanadd 14897 | Addition formula for tange... |
| sinsub 14898 | Sine of difference. (Cont... |
| cossub 14899 | Cosine of difference. (Co... |
| addsin 14900 | Sum of sines. (Contribute... |
| subsin 14901 | Difference of sines. (Con... |
| sinmul 14902 | Product of sines can be re... |
| cosmul 14903 | Product of cosines can be ... |
| addcos 14904 | Sum of cosines. (Contribu... |
| subcos 14905 | Difference of cosines. (C... |
| sincossq 14906 | Sine squared plus cosine s... |
| sin2t 14907 | Double-angle formula for s... |
| cos2t 14908 | Double-angle formula for c... |
| cos2tsin 14909 | Double-angle formula for c... |
| sinbnd 14910 | The sine of a real number ... |
| cosbnd 14911 | The cosine of a real numbe... |
| sinbnd2 14912 | The sine of a real number ... |
| cosbnd2 14913 | The cosine of a real numbe... |
| ef01bndlem 14914 | Lemma for ~ sin01bnd and ~... |
| sin01bnd 14915 | Bounds on the sine of a po... |
| cos01bnd 14916 | Bounds on the cosine of a ... |
| cos1bnd 14917 | Bounds on the cosine of 1.... |
| cos2bnd 14918 | Bounds on the cosine of 2.... |
| sinltx 14919 | The sine of a positive rea... |
| sin01gt0 14920 | The sine of a positive rea... |
| cos01gt0 14921 | The cosine of a positive r... |
| sin02gt0 14922 | The sine of a positive rea... |
| sincos1sgn 14923 | The signs of the sine and ... |
| sincos2sgn 14924 | The signs of the sine and ... |
| sin4lt0 14925 | The sine of 4 is negative.... |
| absefi 14926 | The absolute value of the ... |
| absef 14927 | The absolute value of the ... |
| absefib 14928 | A number is real iff its i... |
| efieq1re 14929 | A number whose imaginary e... |
| demoivre 14930 | De Moivre's Formula. Proo... |
| demoivreALT 14931 | Alternate proof of ~ demoi... |
| eirrlem 14932 | Lemma for ~ eirr . (Contr... |
| eirr 14933 | ` _e ` is irrational. (Co... |
| egt2lt3 14934 | Euler's constant ` _e ` = ... |
| epos 14935 | Euler's constant ` _e ` is... |
| epr 14936 | Euler's constant ` _e ` is... |
| ene0 14937 | ` _e ` is not 0. (Contrib... |
| ene1 14938 | ` _e ` is not 1. (Contrib... |
| xpnnen 14939 | The Cartesian product of t... |
| znnenlem 14940 | Lemma for ~ znnen . (Cont... |
| znnen 14941 | The set of integers and th... |
| qnnen 14942 | The rational numbers are c... |
| rpnnen2lem1 14943 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem2 14944 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem3 14945 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem4 14946 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem5 14947 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem6 14948 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem7 14949 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem8 14950 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem9 14951 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem10 14952 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem11 14953 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem12 14954 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2 14955 | The other half of ~ rpnnen... |
| rpnnen 14956 | The cardinality of the con... |
| rexpen 14957 | The real numbers are equin... |
| cpnnen 14958 | The complex numbers are eq... |
| rucALT 14959 | Alternate proof of ~ ruc .... |
| ruclem1 14960 | Lemma for ~ ruc (the reals... |
| ruclem2 14961 | Lemma for ~ ruc . Orderin... |
| ruclem3 14962 | Lemma for ~ ruc . The con... |
| ruclem4 14963 | Lemma for ~ ruc . Initial... |
| ruclem6 14964 | Lemma for ~ ruc . Domain ... |
| ruclem7 14965 | Lemma for ~ ruc . Success... |
| ruclem8 14966 | Lemma for ~ ruc . The int... |
| ruclem9 14967 | Lemma for ~ ruc . The fir... |
| ruclem10 14968 | Lemma for ~ ruc . Every f... |
| ruclem11 14969 | Lemma for ~ ruc . Closure... |
| ruclem12 14970 | Lemma for ~ ruc . The sup... |
| ruclem13 14971 | Lemma for ~ ruc . There i... |
| ruc 14972 | The set of positive intege... |
| resdomq 14973 | The set of rationals is st... |
| aleph1re 14974 | There are at least aleph-o... |
| aleph1irr 14975 | There are at least aleph-o... |
| cnso 14976 | The complex numbers can be... |
| sqrt2irrlem 14977 | Lemma for ~ sqrt2irr . Th... |
| sqrt2irrlemOLD 14978 | Obsolete proof of ~ sqrt2i... |
| sqrt2irr 14979 | The square root of 2 is ir... |
| sqrt2re 14980 | The square root of 2 exist... |
| nthruc 14981 | The sequence ` NN ` , ` ZZ... |
| nthruz 14982 | The sequence ` NN ` , ` NN... |
| divides 14985 | Define the divides relatio... |
| dvdsval2 14986 | One nonzero integer divide... |
| dvdsval3 14987 | One nonzero integer divide... |
| dvdszrcl 14988 | Reverse closure for the di... |
| nndivdvds 14989 | Strong form of ~ dvdsval2 ... |
| nndivides 14990 | Definition of the divides ... |
| moddvds 14991 | Two ways to say ` A == B `... |
| dvds0lem 14992 | A lemma to assist theorems... |
| dvds1lem 14993 | A lemma to assist theorems... |
| dvds2lem 14994 | A lemma to assist theorems... |
| iddvds 14995 | An integer divides itself.... |
| 1dvds 14996 | 1 divides any integer. Th... |
| dvds0 14997 | Any integer divides 0. Th... |
| negdvdsb 14998 | An integer divides another... |
| dvdsnegb 14999 | An integer divides another... |
| absdvdsb 15000 | An integer divides another... |
| dvdsabsb 15001 | An integer divides another... |
| 0dvds 15002 | Only 0 is divisible by 0. ... |
| dvdsmul1 15003 | An integer divides a multi... |
| dvdsmul2 15004 | An integer divides a multi... |
| iddvdsexp 15005 | An integer divides a posit... |
| muldvds1 15006 | If a product divides an in... |
| muldvds2 15007 | If a product divides an in... |
| dvdscmul 15008 | Multiplication by a consta... |
| dvdsmulc 15009 | Multiplication by a consta... |
| dvdscmulr 15010 | Cancellation law for the d... |
| dvdsmulcr 15011 | Cancellation law for the d... |
| summodnegmod 15012 | The sum of two integers mo... |
| modmulconst 15013 | Constant multiplication in... |
| dvds2ln 15014 | If an integer divides each... |
| dvds2add 15015 | If an integer divides each... |
| dvds2sub 15016 | If an integer divides each... |
| dvds2subd 15017 | Natural deduction form of ... |
| dvdstr 15018 | The divides relation is tr... |
| dvdsmultr1 15019 | If an integer divides anot... |
| dvdsmultr1d 15020 | Natural deduction form of ... |
| dvdsmultr2 15021 | If an integer divides anot... |
| ordvdsmul 15022 | If an integer divides eith... |
| dvdssub2 15023 | If an integer divides a di... |
| dvdsadd 15024 | An integer divides another... |
| dvdsaddr 15025 | An integer divides another... |
| dvdssub 15026 | An integer divides another... |
| dvdssubr 15027 | An integer divides another... |
| dvdsadd2b 15028 | Adding a multiple of the b... |
| dvdsaddre2b 15029 | Adding a multiple of the b... |
| fsumdvds 15030 | If every term in a sum is ... |
| dvdslelem 15031 | Lemma for ~ dvdsle . (Con... |
| dvdsle 15032 | The divisors of a positive... |
| dvdsleabs 15033 | The divisors of a nonzero ... |
| dvdsleabs2 15034 | Transfer divisibility to a... |
| dvdsabseq 15035 | If two integers divide eac... |
| dvdseq 15036 | If two nonnegative integer... |
| divconjdvds 15037 | If a nonzero integer ` M `... |
| dvdsdivcl 15038 | The complement of a diviso... |
| dvdsflip 15039 | An involution of the divis... |
| dvdsssfz1 15040 | The set of divisors of a n... |
| dvds1 15041 | The only nonnegative integ... |
| alzdvds 15042 | Only 0 is divisible by all... |
| dvdsext 15043 | Poset extensionality for d... |
| fzm1ndvds 15044 | No number between ` 1 ` an... |
| fzo0dvdseq 15045 | Zero is the only one of th... |
| fzocongeq 15046 | Two different elements of ... |
| addmodlteqALT 15047 | Two nonnegative integers l... |
| dvdsfac 15048 | A positive integer divides... |
| dvdsexp 15049 | A power divides a power wi... |
| dvdsmod 15050 | Any number ` K ` whose mod... |
| mulmoddvds 15051 | If an integer is divisible... |
| 3dvds 15052 | A rule for divisibility by... |
| 3dvdsOLD 15053 | Obsolete version of ~ 3dvd... |
| 3dvdsdec 15054 | A decimal number is divisi... |
| 3dvdsdecOLD 15055 | Obsolete proof of ~ 3dvdsd... |
| 3dvds2dec 15056 | A decimal number is divisi... |
| 3dvds2decOLD 15057 | Old version of ~ 3dvds2dec... |
| fprodfvdvdsd 15058 | A finite product of intege... |
| fproddvdsd 15059 | A finite product of intege... |
| evenelz 15060 | An even number is an integ... |
| zeo3 15061 | An integer is even or odd.... |
| zeo4 15062 | An integer is even or odd ... |
| zeneo 15063 | No even integer equals an ... |
| odd2np1lem 15064 | Lemma for ~ odd2np1 . (Co... |
| odd2np1 15065 | An integer is odd iff it i... |
| even2n 15066 | An integer is even iff it ... |
| oddm1even 15067 | An integer is odd iff its ... |
| oddp1even 15068 | An integer is odd iff its ... |
| oexpneg 15069 | The exponential of the neg... |
| mod2eq0even 15070 | An integer is 0 modulo 2 i... |
| mod2eq1n2dvds 15071 | An integer is 1 modulo 2 i... |
| oddnn02np1 15072 | A nonnegative integer is o... |
| oddge22np1 15073 | An integer greater than on... |
| evennn02n 15074 | A nonnegative integer is e... |
| evennn2n 15075 | A positive integer is even... |
| 2tp1odd 15076 | A number which is twice an... |
| mulsucdiv2z 15077 | An integer multiplied with... |
| sqoddm1div8z 15078 | A squared odd number minus... |
| 2teven 15079 | A number which is twice an... |
| zeo5 15080 | An integer is either even ... |
| evend2 15081 | An integer is even iff its... |
| oddp1d2 15082 | An integer is odd iff its ... |
| zob 15083 | Alternate characterization... |
| oddm1d2 15084 | An integer is odd iff its ... |
| ltoddhalfle 15085 | An integer is less than ha... |
| halfleoddlt 15086 | An integer is greater than... |
| opoe 15087 | The sum of two odds is eve... |
| omoe 15088 | The difference of two odds... |
| opeo 15089 | The sum of an odd and an e... |
| omeo 15090 | The difference of an odd a... |
| m1expe 15091 | Exponentiation of -1 by an... |
| m1expo 15092 | Exponentiation of -1 by an... |
| m1exp1 15093 | Exponentiation of negative... |
| nn0enne 15094 | A positive integer is an e... |
| nn0ehalf 15095 | The half of an even nonneg... |
| nnehalf 15096 | The half of an even positi... |
| nn0o1gt2 15097 | An odd nonnegative integer... |
| nno 15098 | An alternate characterizat... |
| nn0o 15099 | An alternate characterizat... |
| nn0ob 15100 | Alternate characterization... |
| nn0oddm1d2 15101 | A positive integer is odd ... |
| nnoddm1d2 15102 | A positive integer is odd ... |
| z0even 15103 | 0 is even. (Contributed b... |
| n2dvds1 15104 | 2 does not divide 1 (commo... |
| n2dvdsm1 15105 | 2 does not divide -1. Tha... |
| z2even 15106 | 2 is even. (Contributed b... |
| n2dvds3 15107 | 2 does not divide 3, i.e. ... |
| z4even 15108 | 4 is an even number. (Con... |
| 4dvdseven 15109 | An integer which is divisi... |
| sumeven 15110 | If every term in a sum is ... |
| sumodd 15111 | If every term in a sum is ... |
| evensumodd 15112 | If every term in a sum wit... |
| oddsumodd 15113 | If every term in a sum wit... |
| pwp1fsum 15114 | The n-th power of a number... |
| oddpwp1fsum 15115 | An odd power of a number i... |
| divalglem0 15116 | Lemma for ~ divalg . (Con... |
| divalglem1 15117 | Lemma for ~ divalg . (Con... |
| divalglem2 15118 | Lemma for ~ divalg . (Con... |
| divalglem4 15119 | Lemma for ~ divalg . (Con... |
| divalglem5 15120 | Lemma for ~ divalg . (Con... |
| divalglem6 15121 | Lemma for ~ divalg . (Con... |
| divalglem7 15122 | Lemma for ~ divalg . (Con... |
| divalglem8 15123 | Lemma for ~ divalg . (Con... |
| divalglem9 15124 | Lemma for ~ divalg . (Con... |
| divalglem10 15125 | Lemma for ~ divalg . (Con... |
| divalg 15126 | The division algorithm (th... |
| divalgb 15127 | Express the division algor... |
| divalg2 15128 | The division algorithm (th... |
| divalgmod 15129 | The result of the ` mod ` ... |
| divalgmodOLD 15130 | Obsolete proof of ~ divalg... |
| divalgmodcl 15131 | The result of the ` mod ` ... |
| modremain 15132 | The result of the modulo o... |
| ndvdssub 15133 | Corollary of the division ... |
| ndvdsadd 15134 | Corollary of the division ... |
| ndvdsp1 15135 | Special case of ~ ndvdsadd... |
| ndvdsi 15136 | A quick test for non-divis... |
| flodddiv4 15137 | The floor of an odd intege... |
| fldivndvdslt 15138 | The floor of an integer di... |
| flodddiv4lt 15139 | The floor of an odd number... |
| flodddiv4t2lthalf 15140 | The floor of an odd number... |
| bitsfval 15145 | Expand the definition of t... |
| bitsval 15146 | Expand the definition of t... |
| bitsval2 15147 | Expand the definition of t... |
| bitsss 15148 | The set of bits of an inte... |
| bitsf 15149 | The ` bits ` function is a... |
| bits0 15150 | Value of the zeroth bit. ... |
| bits0e 15151 | The zeroth bit of an even ... |
| bits0o 15152 | The zeroth bit of an odd n... |
| bitsp1 15153 | The ` M + 1 ` -th bit of `... |
| bitsp1e 15154 | The ` M + 1 ` -th bit of `... |
| bitsp1o 15155 | The ` M + 1 ` -th bit of `... |
| bitsfzolem 15156 | Lemma for ~ bitsfzo . (Co... |
| bitsfzo 15157 | The bits of a number are a... |
| bitsmod 15158 | Truncating the bit sequenc... |
| bitsfi 15159 | Every number is associated... |
| bitscmp 15160 | The bit complement of ` N ... |
| 0bits 15161 | The bits of zero. (Contri... |
| m1bits 15162 | The bits of negative one. ... |
| bitsinv1lem 15163 | Lemma for ~ bitsinv1 . (C... |
| bitsinv1 15164 | There is an explicit inver... |
| bitsinv2 15165 | There is an explicit inver... |
| bitsf1ocnv 15166 | The ` bits ` function rest... |
| bitsf1o 15167 | The ` bits ` function rest... |
| bitsf1 15168 | The ` bits ` function is a... |
| 2ebits 15169 | The bits of a power of two... |
| bitsinv 15170 | The inverse of the ` bits ... |
| bitsinvp1 15171 | Recursive definition of th... |
| sadadd2lem2 15172 | The core of the proof of ~... |
| sadfval 15174 | Define the addition of two... |
| sadcf 15175 | The carry sequence is a se... |
| sadc0 15176 | The initial element of the... |
| sadcp1 15177 | The carry sequence (which ... |
| sadval 15178 | The full adder sequence is... |
| sadcaddlem 15179 | Lemma for ~ sadcadd . (Co... |
| sadcadd 15180 | Non-recursive definition o... |
| sadadd2lem 15181 | Lemma for ~ sadadd2 . (Co... |
| sadadd2 15182 | Sum of initial segments of... |
| sadadd3 15183 | Sum of initial segments of... |
| sadcl 15184 | The sum of two sequences i... |
| sadcom 15185 | The adder sequence functio... |
| saddisjlem 15186 | Lemma for ~ sadadd . (Con... |
| saddisj 15187 | The sum of disjoint sequen... |
| sadaddlem 15188 | Lemma for ~ sadadd . (Con... |
| sadadd 15189 | For sequences that corresp... |
| sadid1 15190 | The adder sequence functio... |
| sadid2 15191 | The adder sequence functio... |
| sadasslem 15192 | Lemma for ~ sadass . (Con... |
| sadass 15193 | Sequence addition is assoc... |
| sadeq 15194 | Any element of a sequence ... |
| bitsres 15195 | Restrict the bits of a num... |
| bitsuz 15196 | The bits of a number are a... |
| bitsshft 15197 | Shifting a bit sequence to... |
| smufval 15199 | The multiplication of two ... |
| smupf 15200 | The sequence of partial su... |
| smup0 15201 | The initial element of the... |
| smupp1 15202 | The initial element of the... |
| smuval 15203 | Define the addition of two... |
| smuval2 15204 | The partial sum sequence s... |
| smupvallem 15205 | If ` A ` only has elements... |
| smucl 15206 | The product of two sequenc... |
| smu01lem 15207 | Lemma for ~ smu01 and ~ sm... |
| smu01 15208 | Multiplication of a sequen... |
| smu02 15209 | Multiplication of a sequen... |
| smupval 15210 | Rewrite the elements of th... |
| smup1 15211 | Rewrite ~ smupp1 using onl... |
| smueqlem 15212 | Any element of a sequence ... |
| smueq 15213 | Any element of a sequence ... |
| smumullem 15214 | Lemma for ~ smumul . (Con... |
| smumul 15215 | For sequences that corresp... |
| gcdval 15218 | The value of the ` gcd ` o... |
| gcd0val 15219 | The value, by convention, ... |
| gcdn0val 15220 | The value of the ` gcd ` o... |
| gcdcllem1 15221 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem2 15222 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem3 15223 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdn0cl 15224 | Closure of the ` gcd ` ope... |
| gcddvds 15225 | The gcd of two integers di... |
| dvdslegcd 15226 | An integer which divides b... |
| nndvdslegcd 15227 | A positive integer which d... |
| gcdcl 15228 | Closure of the ` gcd ` ope... |
| gcdnncl 15229 | Closure of the ` gcd ` ope... |
| gcdcld 15230 | Closure of the ` gcd ` ope... |
| gcd2n0cl 15231 | Closure of the ` gcd ` ope... |
| zeqzmulgcd 15232 | An integer is the product ... |
| divgcdz 15233 | An integer divided by the ... |
| gcdf 15234 | Domain and codomain of the... |
| gcdcom 15235 | The ` gcd ` operator is co... |
| divgcdnn 15236 | A positive integer divided... |
| divgcdnnr 15237 | A positive integer divided... |
| gcdeq0 15238 | The gcd of two integers is... |
| gcdn0gt0 15239 | The gcd of two integers is... |
| gcd0id 15240 | The gcd of 0 and an intege... |
| gcdid0 15241 | The gcd of an integer and ... |
| nn0gcdid0 15242 | The gcd of a nonnegative i... |
| gcdneg 15243 | Negating one operand of th... |
| neggcd 15244 | Negating one operand of th... |
| gcdaddmlem 15245 | Lemma for ~ gcdaddm . (Co... |
| gcdaddm 15246 | Adding a multiple of one o... |
| gcdadd 15247 | The GCD of two numbers is ... |
| gcdid 15248 | The gcd of a number and it... |
| gcd1 15249 | The gcd of a number with 1... |
| gcdabs 15250 | The gcd of two integers is... |
| gcdabs1 15251 | ` gcd ` of the absolute va... |
| gcdabs2 15252 | ` gcd ` of the absolute va... |
| modgcd 15253 | The gcd remains unchanged ... |
| 1gcd 15254 | The GCD of one and an inte... |
| 6gcd4e2 15255 | The greatest common diviso... |
| bezoutlem1 15256 | Lemma for ~ bezout . (Con... |
| bezoutlem2 15257 | Lemma for ~ bezout . (Con... |
| bezoutlem3 15258 | Lemma for ~ bezout . (Con... |
| bezoutlem4 15259 | Lemma for ~ bezout . (Con... |
| bezout 15260 | Bézout's identity: ... |
| dvdsgcd 15261 | An integer which divides e... |
| dvdsgcdb 15262 | Biconditional form of ~ dv... |
| dfgcd2 15263 | Alternate definition of th... |
| gcdass 15264 | Associative law for ` gcd ... |
| mulgcd 15265 | Distribute multiplication ... |
| absmulgcd 15266 | Distribute absolute value ... |
| mulgcdr 15267 | Reverse distribution law f... |
| gcddiv 15268 | Division law for GCD. (Con... |
| gcdmultiple 15269 | The GCD of a multiple of a... |
| gcdmultiplez 15270 | Extend ~ gcdmultiple so ` ... |
| gcdzeq 15271 | A positive integer ` A ` i... |
| gcdeq 15272 | ` A ` is equal to its gcd ... |
| dvdssqim 15273 | Unidirectional form of ~ d... |
| dvdsmulgcd 15274 | A divisibility equivalent ... |
| rpmulgcd 15275 | If ` K ` and ` M ` are rel... |
| rplpwr 15276 | If ` A ` and ` B ` are rel... |
| rppwr 15277 | If ` A ` and ` B ` are rel... |
| sqgcd 15278 | Square distributes over GC... |
| dvdssqlem 15279 | Lemma for ~ dvdssq . (Con... |
| dvdssq 15280 | Two numbers are divisible ... |
| bezoutr 15281 | Partial converse to ~ bezo... |
| bezoutr1 15282 | Converse of ~ bezout for w... |
| nn0seqcvgd 15283 | A strictly-decreasing nonn... |
| seq1st 15284 | A sequence whose iteration... |
| algr0 15285 | The value of the algorithm... |
| algrf 15286 | An algorithm is a step fun... |
| algrp1 15287 | The value of the algorithm... |
| alginv 15288 | If ` I ` is an invariant o... |
| algcvg 15289 | One way to prove that an a... |
| algcvgblem 15290 | Lemma for ~ algcvgb . (Co... |
| algcvgb 15291 | Two ways of expressing tha... |
| algcvga 15292 | The countdown function ` C... |
| algfx 15293 | If ` F ` reaches a fixed p... |
| eucalgval2 15294 | The value of the step func... |
| eucalgval 15295 | Euclid's Algorithm ~ eucal... |
| eucalgf 15296 | Domain and codomain of the... |
| eucalginv 15297 | The invariant of the step ... |
| eucalglt 15298 | The second member of the s... |
| eucalgcvga 15299 | Once Euclid's Algorithm ha... |
| eucalg 15300 | Euclid's Algorithm compute... |
| lcmval 15305 | Value of the ` lcm ` opera... |
| lcmcom 15306 | The ` lcm ` operator is co... |
| lcm0val 15307 | The value, by convention, ... |
| lcmn0val 15308 | The value of the ` lcm ` o... |
| lcmcllem 15309 | Lemma for ~ lcmn0cl and ~ ... |
| lcmn0cl 15310 | Closure of the ` lcm ` ope... |
| dvdslcm 15311 | The lcm of two integers is... |
| lcmledvds 15312 | A positive integer which b... |
| lcmeq0 15313 | The lcm of two integers is... |
| lcmcl 15314 | Closure of the ` lcm ` ope... |
| gcddvdslcm 15315 | The greatest common diviso... |
| lcmneg 15316 | Negating one operand of th... |
| neglcm 15317 | Negating one operand of th... |
| lcmabs 15318 | The lcm of two integers is... |
| lcmgcdlem 15319 | Lemma for ~ lcmgcd and ~ l... |
| lcmgcd 15320 | The product of two numbers... |
| lcmdvds 15321 | The lcm of two integers di... |
| lcmid 15322 | The lcm of an integer and ... |
| lcm1 15323 | The lcm of an integer and ... |
| lcmgcdnn 15324 | The product of two positiv... |
| lcmgcdeq 15325 | Two integers' absolute val... |
| lcmdvdsb 15326 | Biconditional form of ~ lc... |
| lcmass 15327 | Associative law for ` lcm ... |
| 3lcm2e6woprm 15328 | The least common multiple ... |
| 6lcm4e12 15329 | The least common multiple ... |
| absproddvds 15330 | The absolute value of the ... |
| absprodnn 15331 | The absolute value of the ... |
| fissn0dvds 15332 | For each finite subset of ... |
| fissn0dvdsn0 15333 | For each finite subset of ... |
| lcmfval 15334 | Value of the ` _lcm ` func... |
| lcmf0val 15335 | The value, by convention, ... |
| lcmfn0val 15336 | The value of the ` _lcm ` ... |
| lcmfnnval 15337 | The value of the ` _lcm ` ... |
| lcmfcllem 15338 | Lemma for ~ lcmfn0cl and ~... |
| lcmfn0cl 15339 | Closure of the ` _lcm ` fu... |
| lcmfpr 15340 | The value of the ` _lcm ` ... |
| lcmfcl 15341 | Closure of the ` _lcm ` fu... |
| lcmfnncl 15342 | Closure of the ` _lcm ` fu... |
| lcmfeq0b 15343 | The least common multiple ... |
| dvdslcmf 15344 | The least common multiple ... |
| lcmfledvds 15345 | A positive integer which i... |
| lcmf 15346 | Characterization of the le... |
| lcmf0 15347 | The least common multiple ... |
| lcmfsn 15348 | The least common multiple ... |
| lcmftp 15349 | The least common multiple ... |
| lcmfunsnlem1 15350 | Lemma for ~ lcmfdvds and ~... |
| lcmfunsnlem2lem1 15351 | Lemma 1 for ~ lcmfunsnlem2... |
| lcmfunsnlem2lem2 15352 | Lemma 2 for ~ lcmfunsnlem2... |
| lcmfunsnlem2 15353 | Lemma for ~ lcmfunsn and ~... |
| lcmfunsnlem 15354 | Lemma for ~ lcmfdvds and ~... |
| lcmfdvds 15355 | The least common multiple ... |
| lcmfdvdsb 15356 | Biconditional form of ~ lc... |
| lcmfunsn 15357 | The ` _lcm ` function for ... |
| lcmfun 15358 | The ` _lcm ` function for ... |
| lcmfass 15359 | Associative law for the ` ... |
| lcmf2a3a4e12 15360 | The least common multiple ... |
| lcmflefac 15361 | The least common multiple ... |
| coprmgcdb 15362 | Two positive integers are ... |
| ncoprmgcdne1b 15363 | Two positive integers are ... |
| ncoprmgcdgt1b 15364 | Two positive integers are ... |
| coprmdvds1 15365 | If two positive integers a... |
| coprmdvds 15366 | Euclid's Lemma (see ProofW... |
| coprmdvdsOLD 15367 | If an integer divides the ... |
| coprmdvds2 15368 | If an integer is divisible... |
| mulgcddvds 15369 | One half of ~ rpmulgcd2 , ... |
| rpmulgcd2 15370 | If ` M ` is relatively pri... |
| qredeq 15371 | Two equal reduced fraction... |
| qredeu 15372 | Every rational number has ... |
| rpmul 15373 | If ` K ` is relatively pri... |
| rpdvds 15374 | If ` K ` is relatively pri... |
| coprmprod 15375 | The product of the element... |
| coprmproddvdslem 15376 | Lemma for ~ coprmproddvds ... |
| coprmproddvds 15377 | If a positive integer is d... |
| congr 15378 | Definition of congruence b... |
| divgcdcoprm0 15379 | Integers divided by gcd ar... |
| divgcdcoprmex 15380 | Integers divided by gcd ar... |
| cncongr1 15381 | One direction of the bicon... |
| cncongr2 15382 | The other direction of the... |
| cncongr 15383 | Cancellability of Congruen... |
| cncongrcoprm 15384 | Corollary 1 of Cancellabil... |
| isprm 15387 | The predicate "is a prime ... |
| prmnn 15388 | A prime number is a positi... |
| prmz 15389 | A prime number is an integ... |
| prmssnn 15390 | The prime numbers are a su... |
| prmex 15391 | The set of prime numbers e... |
| 1nprm 15392 | 1 is not a prime number. ... |
| 1idssfct 15393 | The positive divisors of a... |
| isprm2lem 15394 | Lemma for ~ isprm2 . (Con... |
| isprm2 15395 | The predicate "is a prime ... |
| isprm3 15396 | The predicate "is a prime ... |
| isprm4 15397 | The predicate "is a prime ... |
| prmind2 15398 | A variation on ~ prmind as... |
| prmind 15399 | Perform induction over the... |
| dvdsprime 15400 | If ` M ` divides a prime, ... |
| nprm 15401 | A product of two integers ... |
| nprmi 15402 | An inference for composite... |
| dvdsnprmd 15403 | If a number is divisible b... |
| prm2orodd 15404 | A prime number is either 2... |
| 2prm 15405 | 2 is a prime number. (Con... |
| 3prm 15406 | 3 is a prime number. (Con... |
| 4nprm 15407 | 4 is not a prime number. ... |
| prmuz2 15408 | A prime number is an integ... |
| prmgt1 15409 | A prime number is an integ... |
| prmm2nn0 15410 | Subtracting 2 from a prime... |
| oddprmgt2 15411 | An odd prime is greater th... |
| oddprmge3 15412 | An odd prime is greater th... |
| prmn2uzge3OLD 15413 | Obsolete version of ~ oddp... |
| sqnprm 15414 | A square is never prime. ... |
| dvdsprm 15415 | An integer greater than or... |
| exprmfct 15416 | Every integer greater than... |
| prmdvdsfz 15417 | Each integer greater than ... |
| nprmdvds1 15418 | No prime number divides 1.... |
| isprm5 15419 | One need only check prime ... |
| isprm7 15420 | One need only check prime ... |
| maxprmfct 15421 | The set of prime factors o... |
| divgcdodd 15422 | Either ` A / ( A gcd B ) `... |
| coprm 15423 | A prime number either divi... |
| prmrp 15424 | Unequal prime numbers are ... |
| euclemma 15425 | Euclid's lemma. A prime n... |
| isprm6 15426 | A number is prime iff it s... |
| prmdvdsexp 15427 | A prime divides a positive... |
| prmdvdsexpb 15428 | A prime divides a positive... |
| prmdvdsexpr 15429 | If a prime divides a nonne... |
| prmexpb 15430 | Two positive prime powers ... |
| prmfac1 15431 | The factorial of a number ... |
| rpexp 15432 | If two numbers ` A ` and `... |
| rpexp1i 15433 | Relative primality passes ... |
| rpexp12i 15434 | Relative primality passes ... |
| prmndvdsfaclt 15435 | A prime number does not di... |
| ncoprmlnprm 15436 | If two positive integers a... |
| cncongrprm 15437 | Corollary 2 of Cancellabil... |
| isevengcd2 15438 | The predicate "is an even ... |
| isoddgcd1 15439 | The predicate "is an odd n... |
| 3lcm2e6 15440 | The least common multiple ... |
| qnumval 15445 | Value of the canonical num... |
| qdenval 15446 | Value of the canonical den... |
| qnumdencl 15447 | Lemma for ~ qnumcl and ~ q... |
| qnumcl 15448 | The canonical numerator of... |
| qdencl 15449 | The canonical denominator ... |
| fnum 15450 | Canonical numerator define... |
| fden 15451 | Canonical denominator defi... |
| qnumdenbi 15452 | Two numbers are the canoni... |
| qnumdencoprm 15453 | The canonical representati... |
| qeqnumdivden 15454 | Recover a rational number ... |
| qmuldeneqnum 15455 | Multiplying a rational by ... |
| divnumden 15456 | Calculate the reduced form... |
| divdenle 15457 | Reducing a quotient never ... |
| qnumgt0 15458 | A rational is positive iff... |
| qgt0numnn 15459 | A rational is positive iff... |
| nn0gcdsq 15460 | Squaring commutes with GCD... |
| zgcdsq 15461 | ~ nn0gcdsq extended to int... |
| numdensq 15462 | Squaring a rational square... |
| numsq 15463 | Square commutes with canon... |
| densq 15464 | Square commutes with canon... |
| qden1elz 15465 | A rational is an integer i... |
| zsqrtelqelz 15466 | If an integer has a ration... |
| nonsq 15467 | Any integer strictly betwe... |
| phival 15472 | Value of the Euler ` phi `... |
| phicl2 15473 | Bounds and closure for the... |
| phicl 15474 | Closure for the value of t... |
| phibndlem 15475 | Lemma for ~ phibnd . (Con... |
| phibnd 15476 | A slightly tighter bound o... |
| phicld 15477 | Closure for the value of t... |
| phi1 15478 | Value of the Euler ` phi `... |
| dfphi2 15479 | Alternate definition of th... |
| hashdvds 15480 | The number of numbers in a... |
| phiprmpw 15481 | Value of the Euler ` phi `... |
| phiprm 15482 | Value of the Euler ` phi `... |
| crth 15483 | The Chinese Remainder Theo... |
| phimullem 15484 | Lemma for ~ phimul . (Con... |
| phimul 15485 | The Euler ` phi ` function... |
| eulerthlem1 15486 | Lemma for ~ eulerth . (Co... |
| eulerthlem2 15487 | Lemma for ~ eulerth . (Co... |
| eulerth 15488 | Euler's theorem, a general... |
| fermltl 15489 | Fermat's little theorem. ... |
| prmdiv 15490 | Show an explicit expressio... |
| prmdiveq 15491 | The modular inverse of ` A... |
| prmdivdiv 15492 | The (modular) inverse of t... |
| hashgcdlem 15493 | A correspondence between e... |
| hashgcdeq 15494 | Number of initial positive... |
| phisum 15495 | The divisor sum identity o... |
| odzval 15496 | Value of the order functio... |
| odzcllem 15497 | - Lemma for ~ odzcl , show... |
| odzcl 15498 | The order of a group eleme... |
| odzid 15499 | Any element raised to the ... |
| odzdvds 15500 | The only powers of ` A ` t... |
| odzphi 15501 | The order of any group ele... |
| modprm1div 15502 | A prime number divides an ... |
| m1dvdsndvds 15503 | If an integer minus 1 is d... |
| modprminv 15504 | Show an explicit expressio... |
| modprminveq 15505 | The modular inverse of ` A... |
| vfermltl 15506 | Variant of Fermat's little... |
| vfermltlALT 15507 | Alternate proof of ~ vferm... |
| powm2modprm 15508 | If an integer minus 1 is d... |
| reumodprminv 15509 | For any prime number and f... |
| modprm0 15510 | For two positive integers ... |
| nnnn0modprm0 15511 | For a positive integer and... |
| modprmn0modprm0 15512 | For an integer not being 0... |
| coprimeprodsq 15513 | If three numbers are copri... |
| coprimeprodsq2 15514 | If three numbers are copri... |
| oddprm 15515 | A prime not equal to ` 2 `... |
| nnoddn2prm 15516 | A prime not equal to ` 2 `... |
| oddn2prm 15517 | A prime not equal to ` 2 `... |
| nnoddn2prmb 15518 | A number is a prime number... |
| prm23lt5 15519 | A prime less than 5 is eit... |
| prm23ge5 15520 | A prime is either 2 or 3 o... |
| pythagtriplem1 15521 | Lemma for ~ pythagtrip . ... |
| pythagtriplem2 15522 | Lemma for ~ pythagtrip . ... |
| pythagtriplem3 15523 | Lemma for ~ pythagtrip . ... |
| pythagtriplem4 15524 | Lemma for ~ pythagtrip . ... |
| pythagtriplem10 15525 | Lemma for ~ pythagtrip . ... |
| pythagtriplem6 15526 | Lemma for ~ pythagtrip . ... |
| pythagtriplem7 15527 | Lemma for ~ pythagtrip . ... |
| pythagtriplem8 15528 | Lemma for ~ pythagtrip . ... |
| pythagtriplem9 15529 | Lemma for ~ pythagtrip . ... |
| pythagtriplem11 15530 | Lemma for ~ pythagtrip . ... |
| pythagtriplem12 15531 | Lemma for ~ pythagtrip . ... |
| pythagtriplem13 15532 | Lemma for ~ pythagtrip . ... |
| pythagtriplem14 15533 | Lemma for ~ pythagtrip . ... |
| pythagtriplem15 15534 | Lemma for ~ pythagtrip . ... |
| pythagtriplem16 15535 | Lemma for ~ pythagtrip . ... |
| pythagtriplem17 15536 | Lemma for ~ pythagtrip . ... |
| pythagtriplem18 15537 | Lemma for ~ pythagtrip . ... |
| pythagtriplem19 15538 | Lemma for ~ pythagtrip . ... |
| pythagtrip 15539 | Parameterize the Pythagore... |
| iserodd 15540 | Collect the odd terms in a... |
| pclem 15543 | - Lemma for the prime powe... |
| pcprecl 15544 | Closure of the prime power... |
| pcprendvds 15545 | Non-divisibility property ... |
| pcprendvds2 15546 | Non-divisibility property ... |
| pcpre1 15547 | Value of the prime power p... |
| pcpremul 15548 | Multiplicative property of... |
| pcval 15549 | The value of the prime pow... |
| pceulem 15550 | Lemma for ~ pceu . (Contr... |
| pceu 15551 | Uniqueness for the prime p... |
| pczpre 15552 | Connect the prime count pr... |
| pczcl 15553 | Closure of the prime power... |
| pccl 15554 | Closure of the prime power... |
| pccld 15555 | Closure of the prime power... |
| pcmul 15556 | Multiplication property of... |
| pcdiv 15557 | Division property of the p... |
| pcqmul 15558 | Multiplication property of... |
| pc0 15559 | The value of the prime pow... |
| pc1 15560 | Value of the prime count f... |
| pcqcl 15561 | Closure of the general pri... |
| pcqdiv 15562 | Division property of the p... |
| pcrec 15563 | Prime power of a reciproca... |
| pcexp 15564 | Prime power of an exponent... |
| pcxcl 15565 | Extended real closure of t... |
| pcge0 15566 | The prime count of an inte... |
| pczdvds 15567 | Defining property of the p... |
| pcdvds 15568 | Defining property of the p... |
| pczndvds 15569 | Defining property of the p... |
| pcndvds 15570 | Defining property of the p... |
| pczndvds2 15571 | The remainder after dividi... |
| pcndvds2 15572 | The remainder after dividi... |
| pcdvdsb 15573 | ` P ^ A ` divides ` N ` if... |
| pcelnn 15574 | There are a positive numbe... |
| pceq0 15575 | There are zero powers of a... |
| pcidlem 15576 | The prime count of a prime... |
| pcid 15577 | The prime count of a prime... |
| pcneg 15578 | The prime count of a negat... |
| pcabs 15579 | The prime count of an abso... |
| pcdvdstr 15580 | The prime count increases ... |
| pcgcd1 15581 | The prime count of a GCD i... |
| pcgcd 15582 | The prime count of a GCD i... |
| pc2dvds 15583 | A characterization of divi... |
| pc11 15584 | The prime count function, ... |
| pcz 15585 | The prime count function c... |
| pcprmpw2 15586 | Self-referential expressio... |
| pcprmpw 15587 | Self-referential expressio... |
| dvdsprmpweq 15588 | If a positive integer divi... |
| dvdsprmpweqnn 15589 | If an integer greater than... |
| dvdsprmpweqle 15590 | If a positive integer divi... |
| difsqpwdvds 15591 | If the difference of two s... |
| pcaddlem 15592 | Lemma for ~ pcadd . The o... |
| pcadd 15593 | An inequality for the prim... |
| pcadd2 15594 | The inequality of ~ pcadd ... |
| pcmptcl 15595 | Closure for the prime powe... |
| pcmpt 15596 | Construct a function with ... |
| pcmpt2 15597 | Dividing two prime count m... |
| pcmptdvds 15598 | The partial products of th... |
| pcprod 15599 | The product of the primes ... |
| sumhash 15600 | The sum of 1 over a set is... |
| fldivp1 15601 | The difference between the... |
| pcfaclem 15602 | Lemma for ~ pcfac . (Cont... |
| pcfac 15603 | Calculate the prime count ... |
| pcbc 15604 | Calculate the prime count ... |
| qexpz 15605 | If a power of a rational n... |
| expnprm 15606 | A second or higher power o... |
| oddprmdvds 15607 | Every positive integer whi... |
| prmpwdvds 15608 | A relation involving divis... |
| pockthlem 15609 | Lemma for ~ pockthg . (Co... |
| pockthg 15610 | The generalized Pocklingto... |
| pockthi 15611 | Pocklington's theorem, whi... |
| unbenlem 15612 | Lemma for ~ unben . (Cont... |
| unben 15613 | An unbounded set of positi... |
| infpnlem1 15614 | Lemma for ~ infpn . The s... |
| infpnlem2 15615 | Lemma for ~ infpn . For a... |
| infpn 15616 | There exist infinitely man... |
| infpn2 15617 | There exist infinitely man... |
| prmunb 15618 | The primes are unbounded. ... |
| prminf 15619 | There are an infinite numb... |
| prmreclem1 15620 | Lemma for ~ prmrec . Prop... |
| prmreclem2 15621 | Lemma for ~ prmrec . Ther... |
| prmreclem3 15622 | Lemma for ~ prmrec . The ... |
| prmreclem4 15623 | Lemma for ~ prmrec . Show... |
| prmreclem5 15624 | Lemma for ~ prmrec . Here... |
| prmreclem6 15625 | Lemma for ~ prmrec . If t... |
| prmrec 15626 | The sum of the reciprocals... |
| 1arithlem1 15627 | Lemma for ~ 1arith . (Con... |
| 1arithlem2 15628 | Lemma for ~ 1arith . (Con... |
| 1arithlem3 15629 | Lemma for ~ 1arith . (Con... |
| 1arithlem4 15630 | Lemma for ~ 1arith . (Con... |
| 1arith 15631 | Fundamental theorem of ari... |
| 1arith2 15632 | Fundamental theorem of ari... |
| elgz 15635 | Elementhood in the gaussia... |
| gzcn 15636 | A gaussian integer is a co... |
| zgz 15637 | An integer is a gaussian i... |
| igz 15638 | ` _i ` is a gaussian integ... |
| gznegcl 15639 | The gaussian integers are ... |
| gzcjcl 15640 | The gaussian integers are ... |
| gzaddcl 15641 | The gaussian integers are ... |
| gzmulcl 15642 | The gaussian integers are ... |
| gzreim 15643 | Construct a gaussian integ... |
| gzsubcl 15644 | The gaussian integers are ... |
| gzabssqcl 15645 | The squared norm of a gaus... |
| 4sqlem5 15646 | Lemma for ~ 4sq . (Contri... |
| 4sqlem6 15647 | Lemma for ~ 4sq . (Contri... |
| 4sqlem7 15648 | Lemma for ~ 4sq . (Contri... |
| 4sqlem8 15649 | Lemma for ~ 4sq . (Contri... |
| 4sqlem9 15650 | Lemma for ~ 4sq . (Contri... |
| 4sqlem10 15651 | Lemma for ~ 4sq . (Contri... |
| 4sqlem1 15652 | Lemma for ~ 4sq . The set... |
| 4sqlem2 15653 | Lemma for ~ 4sq . Change ... |
| 4sqlem3 15654 | Lemma for ~ 4sq . Suffici... |
| 4sqlem4a 15655 | Lemma for ~ 4sqlem4 . (Co... |
| 4sqlem4 15656 | Lemma for ~ 4sq . We can ... |
| mul4sqlem 15657 | Lemma for ~ mul4sq : algeb... |
| mul4sq 15658 | Euler's four-square identi... |
| 4sqlem11 15659 | Lemma for ~ 4sq . Use the... |
| 4sqlem12 15660 | Lemma for ~ 4sq . For any... |
| 4sqlem13 15661 | Lemma for ~ 4sq . (Contri... |
| 4sqlem14 15662 | Lemma for ~ 4sq . (Contri... |
| 4sqlem15 15663 | Lemma for ~ 4sq . (Contri... |
| 4sqlem16 15664 | Lemma for ~ 4sq . (Contri... |
| 4sqlem17 15665 | Lemma for ~ 4sq . (Contri... |
| 4sqlem18 15666 | Lemma for ~ 4sq . Inducti... |
| 4sqlem19 15667 | Lemma for ~ 4sq . The pro... |
| 4sq 15668 | Lagrange's four-square the... |
| vdwapfval 15675 | Define the arithmetic prog... |
| vdwapf 15676 | The arithmetic progression... |
| vdwapval 15677 | Value of the arithmetic pr... |
| vdwapun 15678 | Remove the first element o... |
| vdwapid1 15679 | The first element of an ar... |
| vdwap0 15680 | Value of a length-1 arithm... |
| vdwap1 15681 | Value of a length-1 arithm... |
| vdwmc 15682 | The predicate " The ` <. R... |
| vdwmc2 15683 | Expand out the definition ... |
| vdwpc 15684 | The predicate " The colori... |
| vdwlem1 15685 | Lemma for ~ vdw . (Contri... |
| vdwlem2 15686 | Lemma for ~ vdw . (Contri... |
| vdwlem3 15687 | Lemma for ~ vdw . (Contri... |
| vdwlem4 15688 | Lemma for ~ vdw . (Contri... |
| vdwlem5 15689 | Lemma for ~ vdw . (Contri... |
| vdwlem6 15690 | Lemma for ~ vdw . (Contri... |
| vdwlem7 15691 | Lemma for ~ vdw . (Contri... |
| vdwlem8 15692 | Lemma for ~ vdw . (Contri... |
| vdwlem9 15693 | Lemma for ~ vdw . (Contri... |
| vdwlem10 15694 | Lemma for ~ vdw . Set up ... |
| vdwlem11 15695 | Lemma for ~ vdw . (Contri... |
| vdwlem12 15696 | Lemma for ~ vdw . ` K = 2 ... |
| vdwlem13 15697 | Lemma for ~ vdw . Main in... |
| vdw 15698 | Van der Waerden's theorem.... |
| vdwnnlem1 15699 | Corollary of ~ vdw , and l... |
| vdwnnlem2 15700 | Lemma for ~ vdwnn . The s... |
| vdwnnlem3 15701 | Lemma for ~ vdwnn . (Cont... |
| vdwnn 15702 | Van der Waerden's theorem,... |
| ramtlecl 15704 | The set ` T ` of numbers w... |
| hashbcval 15706 | Value of the "binomial set... |
| hashbccl 15707 | The binomial set is a fini... |
| hashbcss 15708 | Subset relation for the bi... |
| hashbc0 15709 | The set of subsets of size... |
| hashbc2 15710 | The size of the binomial s... |
| 0hashbc 15711 | There are no subsets of th... |
| ramval 15712 | The value of the Ramsey nu... |
| ramcl2lem 15713 | Lemma for extended real cl... |
| ramtcl 15714 | The Ramsey number has the ... |
| ramtcl2 15715 | The Ramsey number is an in... |
| ramtub 15716 | The Ramsey number is a low... |
| ramub 15717 | The Ramsey number is a low... |
| ramub2 15718 | It is sufficient to check ... |
| rami 15719 | The defining property of a... |
| ramcl2 15720 | The Ramsey number is eithe... |
| ramxrcl 15721 | The Ramsey number is an ex... |
| ramubcl 15722 | If the Ramsey number is up... |
| ramlb 15723 | Establish a lower bound on... |
| 0ram 15724 | The Ramsey number when ` M... |
| 0ram2 15725 | The Ramsey number when ` M... |
| ram0 15726 | The Ramsey number when ` R... |
| 0ramcl 15727 | Lemma for ~ ramcl : Exist... |
| ramz2 15728 | The Ramsey number when ` F... |
| ramz 15729 | The Ramsey number when ` F... |
| ramub1lem1 15730 | Lemma for ~ ramub1 . (Con... |
| ramub1lem2 15731 | Lemma for ~ ramub1 . (Con... |
| ramub1 15732 | Inductive step for Ramsey'... |
| ramcl 15733 | Ramsey's theorem: the Rams... |
| ramsey 15734 | Ramsey's theorem with the ... |
| prmoval 15737 | Value of the primorial fun... |
| prmocl 15738 | Closure of the primorial f... |
| prmone0 15739 | The primorial function is ... |
| prmo0 15740 | The primorial of 0. (Cont... |
| prmo1 15741 | The primorial of 1. (Cont... |
| prmop1 15742 | The primorial of a success... |
| prmonn2 15743 | Value of the primorial fun... |
| prmo2 15744 | The primorial of 2. (Cont... |
| prmo3 15745 | The primorial of 3. (Cont... |
| prmdvdsprmo 15746 | The primorial of a number ... |
| prmdvdsprmop 15747 | The primorial of a number ... |
| fvprmselelfz 15748 | The value of the prime sel... |
| fvprmselgcd1 15749 | The greatest common diviso... |
| prmolefac 15750 | The primorial of a positiv... |
| prmodvdslcmf 15751 | The primorial of a nonnega... |
| prmolelcmf 15752 | The primorial of a positiv... |
| prmgaplem1 15753 | Lemma for ~ prmgap : The ... |
| prmgaplem2 15754 | Lemma for ~ prmgap : The ... |
| prmgaplcmlem1 15755 | Lemma for ~ prmgaplcm : T... |
| prmgaplcmlem2 15756 | Lemma for ~ prmgaplcm : T... |
| prmgaplem3 15757 | Lemma for ~ prmgap . (Con... |
| prmgaplem4 15758 | Lemma for ~ prmgap . (Con... |
| prmgaplem5 15759 | Lemma for ~ prmgap : for e... |
| prmgaplem6 15760 | Lemma for ~ prmgap : for e... |
| prmgaplem7 15761 | Lemma for ~ prmgap . (Con... |
| prmgaplem8 15762 | Lemma for ~ prmgap . (Con... |
| prmgap 15763 | The prime gap theorem: for... |
| prmgaplcm 15764 | Alternate proof of ~ prmga... |
| prmgapprmolem 15765 | Lemma for ~ prmgapprmo : ... |
| prmgapprmo 15766 | Alternate proof of ~ prmga... |
| dec2dvds 15767 | Divisibility by two is obv... |
| dec5dvds 15768 | Divisibility by five is ob... |
| dec5dvds2 15769 | Divisibility by five is ob... |
| dec5nprm 15770 | Divisibility by five is ob... |
| dec2nprm 15771 | Divisibility by two is obv... |
| modxai 15772 | Add exponents in a power m... |
| mod2xi 15773 | Double exponents in a powe... |
| modxp1i 15774 | Add one to an exponent in ... |
| mod2xnegi 15775 | Version of ~ mod2xi with a... |
| modsubi 15776 | Subtract from within a mod... |
| gcdi 15777 | Calculate a GCD via Euclid... |
| gcdmodi 15778 | Calculate a GCD via Euclid... |
| decexp2 15779 | Calculate a power of two. ... |
| numexp0 15780 | Calculate an integer power... |
| numexp1 15781 | Calculate an integer power... |
| numexpp1 15782 | Calculate an integer power... |
| numexp2x 15783 | Double an integer power. ... |
| decsplit0b 15784 | Split a decimal number int... |
| decsplit0 15785 | Split a decimal number int... |
| decsplit1 15786 | Split a decimal number int... |
| decsplit 15787 | Split a decimal number int... |
| decsplit0bOLD 15788 | Obsolete version of ~ decs... |
| decsplit0OLD 15789 | Obsolete version of ~ decs... |
| decsplit1OLD 15790 | Obsolete version of ~ decs... |
| decsplitOLD 15791 | Obsolete version of ~ decs... |
| karatsuba 15792 | The Karatsuba multiplicati... |
| karatsubaOLD 15793 | Obsolete version of ~ kara... |
| 2exp4 15794 | Two to the fourth power is... |
| 2exp6 15795 | Two to the sixth power is ... |
| 2exp8 15796 | Two to the eighth power is... |
| 2exp16 15797 | Two to the sixteenth power... |
| 3exp3 15798 | Three to the third power i... |
| 2expltfac 15799 | The factorial grows faster... |
| cshwsidrepsw 15800 | If cyclically shifting a w... |
| cshwsidrepswmod0 15801 | If cyclically shifting a w... |
| cshwshashlem1 15802 | If cyclically shifting a w... |
| cshwshashlem2 15803 | If cyclically shifting a w... |
| cshwshashlem3 15804 | If cyclically shifting a w... |
| cshwsdisj 15805 | The singletons resulting b... |
| cshwsiun 15806 | The set of (different!) wo... |
| cshwsex 15807 | The class of (different!) ... |
| cshws0 15808 | The size of the set of (di... |
| cshwrepswhash1 15809 | The size of the set of (di... |
| cshwshashnsame 15810 | If a word (not consisting ... |
| cshwshash 15811 | If a word has a length bei... |
| prmlem0 15812 | Lemma for ~ prmlem1 and ~ ... |
| prmlem1a 15813 | A quick proof skeleton to ... |
| prmlem1 15814 | A quick proof skeleton to ... |
| 5prm 15815 | 5 is a prime number. (Con... |
| 6nprm 15816 | 6 is not a prime number. ... |
| 7prm 15817 | 7 is a prime number. (Con... |
| 8nprm 15818 | 8 is not a prime number. ... |
| 9nprm 15819 | 9 is not a prime number. ... |
| 10nprm 15820 | 10 is not a prime number. ... |
| 10nprmOLD 15821 | Obsolete version of ~ 10np... |
| 11prm 15822 | 11 is a prime number. (Co... |
| 13prm 15823 | 13 is a prime number. (Co... |
| 17prm 15824 | 17 is a prime number. (Co... |
| 19prm 15825 | 19 is a prime number. (Co... |
| 23prm 15826 | 23 is a prime number. (Co... |
| prmlem2 15827 | Our last proving session g... |
| 37prm 15828 | 37 is a prime number. (Co... |
| 43prm 15829 | 43 is a prime number. (Co... |
| 83prm 15830 | 83 is a prime number. (Co... |
| 139prm 15831 | 139 is a prime number. (C... |
| 163prm 15832 | 163 is a prime number. (C... |
| 317prm 15833 | 317 is a prime number. (C... |
| 631prm 15834 | 631 is a prime number. (C... |
| prmo4 15835 | The primorial of 4. (Cont... |
| prmo5 15836 | The primorial of 5. (Cont... |
| prmo6 15837 | The primorial of 6. (Cont... |
| 1259lem1 15838 | Lemma for ~ 1259prm . Cal... |
| 1259lem2 15839 | Lemma for ~ 1259prm . Cal... |
| 1259lem3 15840 | Lemma for ~ 1259prm . Cal... |
| 1259lem4 15841 | Lemma for ~ 1259prm . Cal... |
| 1259lem5 15842 | Lemma for ~ 1259prm . Cal... |
| 1259prm 15843 | 1259 is a prime number. (... |
| 2503lem1 15844 | Lemma for ~ 2503prm . Cal... |
| 2503lem2 15845 | Lemma for ~ 2503prm . Cal... |
| 2503lem3 15846 | Lemma for ~ 2503prm . Cal... |
| 2503prm 15847 | 2503 is a prime number. (... |
| 4001lem1 15848 | Lemma for ~ 4001prm . Cal... |
| 4001lem2 15849 | Lemma for ~ 4001prm . Cal... |
| 4001lem3 15850 | Lemma for ~ 4001prm . Cal... |
| 4001lem4 15851 | Lemma for ~ 4001prm . Cal... |
| 4001prm 15852 | 4001 is a prime number. (... |
| sloteq 15862 | Equality theorem for the `... |
| brstruct 15866 | The structure relation is ... |
| isstruct2 15867 | The property of being a st... |
| structex 15868 | A structure is a set. (Co... |
| structn0fun 15869 | A structure witout the emp... |
| isstruct 15870 | The property of being a st... |
| structcnvcnv 15871 | Two ways to express the re... |
| structfung 15872 | The converse of the conver... |
| structfun 15873 | Convert between two kinds ... |
| structfn 15874 | Convert between two kinds ... |
| slotfn 15875 | A slot is a function on se... |
| strfvnd 15876 | Deduction version of ~ str... |
| basfn 15877 | The base set extractor is ... |
| wunndx 15878 | Closure of the index extra... |
| strfvn 15879 | Value of a structure compo... |
| strfvss 15880 | A structure component extr... |
| wunstr 15881 | Closure of a structure ind... |
| ndxarg 15882 | Get the numeric argument f... |
| ndxid 15883 | A structure component extr... |
| ndxidOLD 15884 | Obsolete proof of ~ ndxid ... |
| strndxid 15885 | The value of a structure c... |
| reldmsets 15886 | The structure override ope... |
| setsvalg 15887 | Value of the structure rep... |
| setsval 15888 | Value of the structure rep... |
| setsidvald 15889 | Value of the structure rep... |
| fvsetsid 15890 | The value of the structure... |
| fsets 15891 | The structure replacement ... |
| setsdm 15892 | The domain of a structure ... |
| setsfun 15893 | A structure with replaceme... |
| setsfun0 15894 | A structure with replaceme... |
| setsn0fun 15895 | The value of the structure... |
| setsstruct2 15896 | An extensible structure wi... |
| setsexstruct2 15897 | An extensible structure wi... |
| setsstruct 15898 | An extensible structure wi... |
| setsstructOLD 15899 | Obsolete version of ~ sets... |
| wunsets 15900 | Closure of structure repla... |
| setsres 15901 | The structure replacement ... |
| setsabs 15902 | Replacing the same compone... |
| setscom 15903 | Component-setting is commu... |
| strfvd 15904 | Deduction version of ~ str... |
| strfv2d 15905 | Deduction version of ~ str... |
| strfv2 15906 | A variation on ~ strfv to ... |
| strfv 15907 | Extract a structure compon... |
| strfv3 15908 | Variant on ~ strfv for lar... |
| strssd 15909 | Deduction version of ~ str... |
| strss 15910 | Propagate component extrac... |
| str0 15911 | All components of the empt... |
| base0 15912 | The base set of the empty ... |
| strfvi 15913 | Structure slot extractors ... |
| setsid 15914 | Value of the structure rep... |
| setsnid 15915 | Value of the structure rep... |
| sbcie2s 15916 | A special version of class... |
| sbcie3s 15917 | A special version of class... |
| baseval 15918 | Value of the base set extr... |
| baseid 15919 | Utility theorem: index-ind... |
| elbasfv 15920 | Utility theorem: reverse c... |
| elbasov 15921 | Utility theorem: reverse c... |
| strov2rcl 15922 | Partial reverse closure fo... |
| basendx 15923 | Index value of the base se... |
| basendxnn 15924 | The index value of the bas... |
| basprssdmsets 15925 | The pair of the base index... |
| reldmress 15926 | The structure restriction ... |
| ressval 15927 | Value of structure restric... |
| ressid2 15928 | General behavior of trivia... |
| ressval2 15929 | Value of nontrivial struct... |
| ressbas 15930 | Base set of a structure re... |
| ressbas2 15931 | Base set of a structure re... |
| ressbasss 15932 | The base set of a restrict... |
| resslem 15933 | Other elements of a struct... |
| ress0 15934 | All restrictions of the nu... |
| ressid 15935 | Behavior of trivial restri... |
| ressinbas 15936 | Restriction only cares abo... |
| ressval3d 15937 | Value of structure restric... |
| ressress 15938 | Restriction composition la... |
| ressabs 15939 | Restriction absorption law... |
| wunress 15940 | Closure of structure restr... |
| dfpleOLD 15962 | Obsolete version of ~ df-p... |
| strlemor0OLD 15968 | Structure definition utili... |
| strlemor1OLD 15969 | Add one element to the end... |
| strlemor2OLD 15970 | Add two elements to the en... |
| strlemor3OLD 15971 | Add three elements to the ... |
| strleun 15972 | Combine two structures int... |
| strle1 15973 | Make a structure from a si... |
| strle2 15974 | Make a structure from a pa... |
| strle3 15975 | Make a structure from a tr... |
| plusgndx 15976 | Index value of the ~ df-pl... |
| plusgid 15977 | Utility theorem: index-ind... |
| opelstrbas 15978 | The base set of a structur... |
| 1strstr 15979 | A constructed one-slot str... |
| 1strbas 15980 | The base set of a construc... |
| 1strwunbndx 15981 | A constructed one-slot str... |
| 1strwun 15982 | A constructed one-slot str... |
| 2strstr 15983 | A constructed two-slot str... |
| 2strbas 15984 | The base set of a construc... |
| 2strop 15985 | The other slot of a constr... |
| 2strstr1 15986 | A constructed two-slot str... |
| 2strbas1 15987 | The base set of a construc... |
| 2strop1 15988 | The other slot of a constr... |
| basendxnplusgndx 15989 | The slot for the base set ... |
| grpstr 15990 | A constructed group is a s... |
| grpbase 15991 | The base set of a construc... |
| grpplusg 15992 | The operation of a constru... |
| ressplusg 15993 | ` +g ` is unaffected by re... |
| grpbasex 15994 | The base of an explicitly ... |
| grpplusgx 15995 | The operation of an explic... |
| mulrndx 15996 | Index value of the ~ df-mu... |
| mulrid 15997 | Utility theorem: index-ind... |
| plusgndxnmulrndx 15998 | The slot for the group (ad... |
| basendxnmulrndx 15999 | The slot for the base set ... |
| rngstr 16000 | A constructed ring is a st... |
| rngbase 16001 | The base set of a construc... |
| rngplusg 16002 | The additive operation of ... |
| rngmulr 16003 | The multiplicative operati... |
| starvndx 16004 | Index value of the ~ df-st... |
| starvid 16005 | Utility theorem: index-ind... |
| ressmulr 16006 | ` .r ` is unaffected by re... |
| ressstarv 16007 | ` *r ` is unaffected by re... |
| srngfn 16008 | A constructed star ring is... |
| srngbase 16009 | The base set of a construc... |
| srngplusg 16010 | The addition operation of ... |
| srngmulr 16011 | The multiplication operati... |
| srnginvl 16012 | The involution function of... |
| scandx 16013 | Index value of the ~ df-sc... |
| scaid 16014 | Utility theorem: index-ind... |
| vscandx 16015 | Index value of the ~ df-vs... |
| vscaid 16016 | Utility theorem: index-ind... |
| lmodstr 16017 | A constructed left module ... |
| lmodbase 16018 | The base set of a construc... |
| lmodplusg 16019 | The additive operation of ... |
| lmodsca 16020 | The set of scalars of a co... |
| lmodvsca 16021 | The scalar product operati... |
| ipndx 16022 | Index value of the ~ df-ip... |
| ipid 16023 | Utility theorem: index-ind... |
| ipsstr 16024 | Lemma to shorten proofs of... |
| ipsbase 16025 | The base set of a construc... |
| ipsaddg 16026 | The additive operation of ... |
| ipsmulr 16027 | The multiplicative operati... |
| ipssca 16028 | The set of scalars of a co... |
| ipsvsca 16029 | The scalar product operati... |
| ipsip 16030 | The multiplicative operati... |
| resssca 16031 | ` Scalar ` is unaffected b... |
| ressvsca 16032 | ` .s ` is unaffected by re... |
| ressip 16033 | The inner product is unaff... |
| phlstr 16034 | A constructed pre-Hilbert ... |
| phlbase 16035 | The base set of a construc... |
| phlplusg 16036 | The additive operation of ... |
| phlsca 16037 | The ring of scalars of a c... |
| phlvsca 16038 | The scalar product operati... |
| phlip 16039 | The inner product (Hermiti... |
| tsetndx 16040 | Index value of the ~ df-ts... |
| tsetid 16041 | Utility theorem: index-ind... |
| topgrpstr 16042 | A constructed topological ... |
| topgrpbas 16043 | The base set of a construc... |
| topgrpplusg 16044 | The additive operation of ... |
| topgrptset 16045 | The topology of a construc... |
| resstset 16046 | ` TopSet ` is unaffected b... |
| plendx 16047 | Index value of the ~ df-pl... |
| plendxOLD 16048 | Obsolete version of ~ df-p... |
| pleid 16049 | Utility theorem: self-refe... |
| pleidOLD 16050 | Obsolete version of ~ otps... |
| otpsstr 16051 | Functionality of a topolog... |
| otpsbas 16052 | The base set of a topologi... |
| otpstset 16053 | The open sets of a topolog... |
| otpsle 16054 | The order of a topological... |
| otpsstrOLD 16055 | Obsolete version of ~ otps... |
| otpsbasOLD 16056 | Obsolete version of ~ otps... |
| otpstsetOLD 16057 | Obsolete version of ~ otps... |
| otpsleOLD 16058 | Obsolete version of ~ otps... |
| ressle 16059 | ` le ` is unaffected by re... |
| ocndx 16060 | Index value of the ~ df-oc... |
| ocid 16061 | Utility theorem: index-ind... |
| dsndx 16062 | Index value of the ~ df-ds... |
| dsid 16063 | Utility theorem: index-ind... |
| unifndx 16064 | Index value of the ~ df-un... |
| unifid 16065 | Utility theorem: index-ind... |
| odrngstr 16066 | Functionality of an ordere... |
| odrngbas 16067 | The base set of an ordered... |
| odrngplusg 16068 | The addition operation of ... |
| odrngmulr 16069 | The multiplication operati... |
| odrngtset 16070 | The open sets of an ordere... |
| odrngle 16071 | The order of an ordered me... |
| odrngds 16072 | The metric of an ordered m... |
| ressds 16073 | ` dist ` is unaffected by ... |
| homndx 16074 | Index value of the ~ df-ho... |
| homid 16075 | Utility theorem: index-ind... |
| ccondx 16076 | Index value of the ~ df-cc... |
| ccoid 16077 | Utility theorem: index-ind... |
| resshom 16078 | ` Hom ` is unaffected by r... |
| ressco 16079 | ` comp ` is unaffected by ... |
| slotsbhcdif 16080 | The slots ` Base ` , ` Hom... |
| restfn 16085 | The subspace topology oper... |
| topnfn 16086 | The topology extractor fun... |
| restval 16087 | The subspace topology indu... |
| elrest 16088 | The predicate "is an open ... |
| elrestr 16089 | Sufficient condition for b... |
| 0rest 16090 | Value of the structure res... |
| restid2 16091 | The subspace topology over... |
| restsspw 16092 | The subspace topology is a... |
| firest 16093 | The finite intersections o... |
| restid 16094 | The subspace topology of t... |
| topnval 16095 | Value of the topology extr... |
| topnid 16096 | Value of the topology extr... |
| topnpropd 16097 | The topology extractor fun... |
| reldmprds 16109 | The structure product is a... |
| prdsbasex 16111 | Lemma for structure produc... |
| imasvalstr 16112 | Structure product value is... |
| prdsvalstr 16113 | Structure product value is... |
| prdsvallem 16114 | Lemma for ~ prdsbas and si... |
| prdsval 16115 | Value of the structure pro... |
| prdssca 16116 | Scalar ring of a structure... |
| prdsbas 16117 | Base set of a structure pr... |
| prdsplusg 16118 | Addition in a structure pr... |
| prdsmulr 16119 | Multiplication in a struct... |
| prdsvsca 16120 | Scalar multiplication in a... |
| prdsip 16121 | Inner product in a structu... |
| prdsle 16122 | Structure product weak ord... |
| prdsless 16123 | Closure of the order relat... |
| prdsds 16124 | Structure product distance... |
| prdsdsfn 16125 | Structure product distance... |
| prdstset 16126 | Structure product topology... |
| prdshom 16127 | Structure product hom-sets... |
| prdsco 16128 | Structure product composit... |
| prdsbas2 16129 | The base set of a structur... |
| prdsbasmpt 16130 | A constructed tuple is a p... |
| prdsbasfn 16131 | Points in the structure pr... |
| prdsbasprj 16132 | Each point in a structure ... |
| prdsplusgval 16133 | Value of a componentwise s... |
| prdsplusgfval 16134 | Value of a structure produ... |
| prdsmulrval 16135 | Value of a componentwise r... |
| prdsmulrfval 16136 | Value of a structure produ... |
| prdsleval 16137 | Value of the product order... |
| prdsdsval 16138 | Value of the metric in a s... |
| prdsvscaval 16139 | Scalar multiplication in a... |
| prdsvscafval 16140 | Scalar multiplication of a... |
| prdsbas3 16141 | The base set of an indexed... |
| prdsbasmpt2 16142 | A constructed tuple is a p... |
| prdsbascl 16143 | An element of the base has... |
| prdsdsval2 16144 | Value of the metric in a s... |
| prdsdsval3 16145 | Value of the metric in a s... |
| pwsval 16146 | Value of a structure power... |
| pwsbas 16147 | Base set of a structure po... |
| pwselbasb 16148 | Membership in the base set... |
| pwselbas 16149 | An element of a structure ... |
| pwsplusgval 16150 | Value of addition in a str... |
| pwsmulrval 16151 | Value of multiplication in... |
| pwsle 16152 | Ordering in a structure po... |
| pwsleval 16153 | Ordering in a structure po... |
| pwsvscafval 16154 | Scalar multiplication in a... |
| pwsvscaval 16155 | Scalar multiplication of a... |
| pwssca 16156 | The ring of scalars of a s... |
| pwsdiagel 16157 | Membership of diagonal ele... |
| pwssnf1o 16158 | Triviality of singleton po... |
| imasval 16171 | Value of an image structur... |
| imasbas 16172 | The base set of an image s... |
| imasds 16173 | The distance function of a... |
| imasdsfn 16174 | The distance function is a... |
| imasdsval 16175 | The distance function of a... |
| imasdsval2 16176 | The distance function of a... |
| imasplusg 16177 | The group operation in an ... |
| imasmulr 16178 | The ring multiplication in... |
| imassca 16179 | The scalar field of an ima... |
| imasvsca 16180 | The scalar multiplication ... |
| imasip 16181 | The inner product of an im... |
| imastset 16182 | The topology of an image s... |
| imasle 16183 | The ordering of an image s... |
| f1ocpbllem 16184 | Lemma for ~ f1ocpbl . (Co... |
| f1ocpbl 16185 | An injection is compatible... |
| f1ovscpbl 16186 | An injection is compatible... |
| f1olecpbl 16187 | An injection is compatible... |
| imasaddfnlem 16188 | The image structure operat... |
| imasaddvallem 16189 | The operation of an image ... |
| imasaddflem 16190 | The image set operations a... |
| imasaddfn 16191 | The image structure's grou... |
| imasaddval 16192 | The value of an image stru... |
| imasaddf 16193 | The image structure's grou... |
| imasmulfn 16194 | The image structure's ring... |
| imasmulval 16195 | The value of an image stru... |
| imasmulf 16196 | The image structure's ring... |
| imasvscafn 16197 | The image structure's scal... |
| imasvscaval 16198 | The value of an image stru... |
| imasvscaf 16199 | The image structure's scal... |
| imasless 16200 | The order relation defined... |
| imasleval 16201 | The value of the image str... |
| qusval 16202 | Value of a quotient struct... |
| quslem 16203 | The function in ~ qusval i... |
| qusin 16204 | Restrict the equivalence r... |
| qusbas 16205 | Base set of a quotient str... |
| quss 16206 | The scalar field of a quot... |
| divsfval 16207 | Value of the function in ~... |
| ercpbllem 16208 | Lemma for ~ ercpbl . (Con... |
| ercpbl 16209 | Translate the function com... |
| erlecpbl 16210 | Translate the relation com... |
| qusaddvallem 16211 | Value of an operation defi... |
| qusaddflem 16212 | The operation of a quotien... |
| qusaddval 16213 | The base set of an image s... |
| qusaddf 16214 | The base set of an image s... |
| qusmulval 16215 | The base set of an image s... |
| qusmulf 16216 | The base set of an image s... |
| xpsc 16217 | A short expression for the... |
| xpscg 16218 | A short expression for the... |
| xpscfn 16219 | The pair function is a fun... |
| xpsc0 16220 | The pair function maps ` 0... |
| xpsc1 16221 | The pair function maps ` 1... |
| xpscfv 16222 | The value of the pair func... |
| xpsfrnel 16223 | Elementhood in the target ... |
| xpsfeq 16224 | A function on ` 2o ` is de... |
| xpsfrnel2 16225 | Elementhood in the target ... |
| xpscf 16226 | Equivalent condition for t... |
| xpsfval 16227 | The value of the function ... |
| xpsff1o 16228 | The function appearing in ... |
| xpsfrn 16229 | A short expression for the... |
| xpsfrn2 16230 | A short expression for the... |
| xpsff1o2 16231 | The function appearing in ... |
| xpsval 16232 | Value of the binary struct... |
| xpslem 16233 | The indexed structure prod... |
| xpsbas 16234 | The base set of the binary... |
| xpsaddlem 16235 | Lemma for ~ xpsadd and ~ x... |
| xpsadd 16236 | Value of the addition oper... |
| xpsmul 16237 | Value of the multiplicatio... |
| xpssca 16238 | Value of the scalar field ... |
| xpsvsca 16239 | Value of the scalar multip... |
| xpsless 16240 | Closure of the ordering in... |
| xpsle 16241 | Value of the ordering in a... |
| ismre 16250 | Property of being a Moore ... |
| fnmre 16251 | The Moore collection gener... |
| mresspw 16252 | A Moore collection is a su... |
| mress 16253 | A Moore-closed subset is a... |
| mre1cl 16254 | In any Moore collection th... |
| mreintcl 16255 | A nonempty collection of c... |
| mreiincl 16256 | A nonempty indexed interse... |
| mrerintcl 16257 | The relative intersection ... |
| mreriincl 16258 | The relative intersection ... |
| mreincl 16259 | Two closed sets have a clo... |
| mreuni 16260 | Since the entire base set ... |
| mreunirn 16261 | Two ways to express the no... |
| ismred 16262 | Properties that determine ... |
| ismred2 16263 | Properties that determine ... |
| mremre 16264 | The Moore collections of s... |
| submre 16265 | The subcollection of a clo... |
| mrcflem 16266 | The domain and range of th... |
| fnmrc 16267 | Moore-closure is a well-be... |
| mrcfval 16268 | Value of the function expr... |
| mrcf 16269 | The Moore closure is a fun... |
| mrcval 16270 | Evaluation of the Moore cl... |
| mrccl 16271 | The Moore closure of a set... |
| mrcsncl 16272 | The Moore closure of a sin... |
| mrcid 16273 | The closure of a closed se... |
| mrcssv 16274 | The closure of a set is a ... |
| mrcidb 16275 | A set is closed iff it is ... |
| mrcss 16276 | Closure preserves subset o... |
| mrcssid 16277 | The closure of a set is a ... |
| mrcidb2 16278 | A set is closed iff it con... |
| mrcidm 16279 | The closure operation is i... |
| mrcsscl 16280 | The closure is the minimal... |
| mrcuni 16281 | Idempotence of closure und... |
| mrcun 16282 | Idempotence of closure und... |
| mrcssvd 16283 | The Moore closure of a set... |
| mrcssd 16284 | Moore closure preserves su... |
| mrcssidd 16285 | A set is contained in its ... |
| mrcidmd 16286 | Moore closure is idempoten... |
| mressmrcd 16287 | In a Moore system, if a se... |
| submrc 16288 | In a closure system which ... |
| mrieqvlemd 16289 | In a Moore system, if ` Y ... |
| mrisval 16290 | Value of the set of indepe... |
| ismri 16291 | Criterion for a set to be ... |
| ismri2 16292 | Criterion for a subset of ... |
| ismri2d 16293 | Criterion for a subset of ... |
| ismri2dd 16294 | Definition of independence... |
| mriss 16295 | An independent set of a Mo... |
| mrissd 16296 | An independent set of a Mo... |
| ismri2dad 16297 | Consequence of a set in a ... |
| mrieqvd 16298 | In a Moore system, a set i... |
| mrieqv2d 16299 | In a Moore system, a set i... |
| mrissmrcd 16300 | In a Moore system, if an i... |
| mrissmrid 16301 | In a Moore system, subsets... |
| mreexd 16302 | In a Moore system, the clo... |
| mreexmrid 16303 | In a Moore system whose cl... |
| mreexexlemd 16304 | This lemma is used to gene... |
| mreexexlem2d 16305 | Used in ~ mreexexlem4d to ... |
| mreexexlem3d 16306 | Base case of the induction... |
| mreexexlem4d 16307 | Induction step of the indu... |
| mreexexd 16308 | Exchange-type theorem. In... |
| mreexexdOLD 16309 | Obsolete proof of ~ mreexe... |
| mreexdomd 16310 | In a Moore system whose cl... |
| mreexfidimd 16311 | In a Moore system whose cl... |
| isacs 16312 | A set is an algebraic clos... |
| acsmre 16313 | Algebraic closure systems ... |
| isacs2 16314 | In the definition of an al... |
| acsfiel 16315 | A set is closed in an alge... |
| acsfiel2 16316 | A set is closed in an alge... |
| acsmred 16317 | An algebraic closure syste... |
| isacs1i 16318 | A closure system determine... |
| mreacs 16319 | Algebraicity is a composab... |
| acsfn 16320 | Algebraicity of a conditio... |
| acsfn0 16321 | Algebraicity of a point cl... |
| acsfn1 16322 | Algebraicity of a one-argu... |
| acsfn1c 16323 | Algebraicity of a one-argu... |
| acsfn2 16324 | Algebraicity of a two-argu... |
| iscat 16333 | The predicate "is a catego... |
| iscatd 16334 | Properties that determine ... |
| catidex 16335 | Each object in a category ... |
| catideu 16336 | Each object in a category ... |
| cidfval 16337 | Each object in a category ... |
| cidval 16338 | Each object in a category ... |
| cidffn 16339 | The identity arrow constru... |
| cidfn 16340 | The identity arrow operato... |
| catidd 16341 | Deduce the identity arrow ... |
| iscatd2 16342 | Version of ~ iscatd with a... |
| catidcl 16343 | Each object in a category ... |
| catlid 16344 | Left identity property of ... |
| catrid 16345 | Right identity property of... |
| catcocl 16346 | Closure of a composition a... |
| catass 16347 | Associativity of compositi... |
| 0catg 16348 | Any structure with an empt... |
| 0cat 16349 | The empty set is a categor... |
| homffval 16350 | Value of the functionalize... |
| fnhomeqhomf 16351 | If the Hom-set operation i... |
| homfval 16352 | Value of the functionalize... |
| homffn 16353 | The functionalized Hom-set... |
| homfeq 16354 | Condition for two categori... |
| homfeqd 16355 | If two structures have the... |
| homfeqbas 16356 | Deduce equality of base se... |
| homfeqval 16357 | Value of the functionalize... |
| comfffval 16358 | Value of the functionalize... |
| comffval 16359 | Value of the functionalize... |
| comfval 16360 | Value of the functionalize... |
| comfffval2 16361 | Value of the functionalize... |
| comffval2 16362 | Value of the functionalize... |
| comfval2 16363 | Value of the functionalize... |
| comfffn 16364 | The functionalized composi... |
| comffn 16365 | The functionalized composi... |
| comfeq 16366 | Condition for two categori... |
| comfeqd 16367 | Condition for two categori... |
| comfeqval 16368 | Equality of two compositio... |
| catpropd 16369 | Two structures with the sa... |
| cidpropd 16370 | Two structures with the sa... |
| oppcval 16373 | Value of the opposite cate... |
| oppchomfval 16374 | Hom-sets of the opposite c... |
| oppchom 16375 | Hom-sets of the opposite c... |
| oppccofval 16376 | Composition in the opposit... |
| oppcco 16377 | Composition in the opposit... |
| oppcbas 16378 | Base set of an opposite ca... |
| oppccatid 16379 | Lemma for ~ oppccat . (Co... |
| oppchomf 16380 | Hom-sets of the opposite c... |
| oppcid 16381 | Identity function of an op... |
| oppccat 16382 | An opposite category is a ... |
| 2oppcbas 16383 | The double opposite catego... |
| 2oppchomf 16384 | The double opposite catego... |
| 2oppccomf 16385 | The double opposite catego... |
| oppchomfpropd 16386 | If two categories have the... |
| oppccomfpropd 16387 | If two categories have the... |
| monfval 16392 | Definition of a monomorphi... |
| ismon 16393 | Definition of a monomorphi... |
| ismon2 16394 | Write out the monomorphism... |
| monhom 16395 | A monomorphism is a morphi... |
| moni 16396 | Property of a monomorphism... |
| monpropd 16397 | If two categories have the... |
| oppcmon 16398 | A monomorphism in the oppo... |
| oppcepi 16399 | An epimorphism in the oppo... |
| isepi 16400 | Definition of an epimorphi... |
| isepi2 16401 | Write out the epimorphism ... |
| epihom 16402 | An epimorphism is a morphi... |
| epii 16403 | Property of an epimorphism... |
| sectffval 16410 | Value of the section opera... |
| sectfval 16411 | Value of the section relat... |
| sectss 16412 | The section relation is a ... |
| issect 16413 | The property " ` F ` is a ... |
| issect2 16414 | Property of being a sectio... |
| sectcan 16415 | If ` G ` is a section of `... |
| sectco 16416 | Composition of two section... |
| isofval 16417 | Function value of the func... |
| invffval 16418 | Value of the inverse relat... |
| invfval 16419 | Value of the inverse relat... |
| isinv 16420 | Value of the inverse relat... |
| invss 16421 | The inverse relation is a ... |
| invsym 16422 | The inverse relation is sy... |
| invsym2 16423 | The inverse relation is sy... |
| invfun 16424 | The inverse relation is a ... |
| isoval 16425 | The isomorphisms are the d... |
| inviso1 16426 | If ` G ` is an inverse to ... |
| inviso2 16427 | If ` G ` is an inverse to ... |
| invf 16428 | The inverse relation is a ... |
| invf1o 16429 | The inverse relation is a ... |
| invinv 16430 | The inverse of the inverse... |
| invco 16431 | The composition of two iso... |
| dfiso2 16432 | Alternate definition of an... |
| dfiso3 16433 | Alternate definition of an... |
| inveq 16434 | If there are two inverses ... |
| isofn 16435 | The function value of the ... |
| isohom 16436 | An isomorphism is a homomo... |
| isoco 16437 | The composition of two iso... |
| oppcsect 16438 | A section in the opposite ... |
| oppcsect2 16439 | A section in the opposite ... |
| oppcinv 16440 | An inverse in the opposite... |
| oppciso 16441 | An isomorphism in the oppo... |
| sectmon 16442 | If ` F ` is a section of `... |
| monsect 16443 | If ` F ` is a monomorphism... |
| sectepi 16444 | If ` F ` is a section of `... |
| episect 16445 | If ` F ` is an epimorphism... |
| sectid 16446 | The identity is a section ... |
| invid 16447 | The inverse of the identit... |
| idiso 16448 | The identity is an isomorp... |
| idinv 16449 | The inverse of the identit... |
| invisoinvl 16450 | The inverse of an isomorph... |
| invisoinvr 16451 | The inverse of an isomorph... |
| invcoisoid 16452 | The inverse of an isomorph... |
| isocoinvid 16453 | The inverse of an isomorph... |
| rcaninv 16454 | Right cancellation of an i... |
| cicfval 16457 | The set of isomorphic obje... |
| brcic 16458 | The relation "is isomorphi... |
| cic 16459 | Objects ` X ` and ` Y ` in... |
| brcici 16460 | Prove that two objects are... |
| cicref 16461 | Isomorphism is reflexive. ... |
| ciclcl 16462 | Isomorphism implies the le... |
| cicrcl 16463 | Isomorphism implies the ri... |
| cicsym 16464 | Isomorphism is symmetric. ... |
| cictr 16465 | Isomorphism is transitive.... |
| cicer 16466 | Isomorphism is an equivale... |
| sscrel 16473 | The subcategory subset rel... |
| brssc 16474 | The subcategory subset rel... |
| sscpwex 16475 | An analogue of ~ pwex for ... |
| subcrcl 16476 | Reverse closure for the su... |
| sscfn1 16477 | The subcategory subset rel... |
| sscfn2 16478 | The subcategory subset rel... |
| ssclem 16479 | Lemma for ~ ssc1 and simil... |
| isssc 16480 | Value of the subcategory s... |
| ssc1 16481 | Infer subset relation on o... |
| ssc2 16482 | Infer subset relation on m... |
| sscres 16483 | Any function restricted to... |
| sscid 16484 | The subcategory subset rel... |
| ssctr 16485 | The subcategory subset rel... |
| ssceq 16486 | The subcategory subset rel... |
| rescval 16487 | Value of the category rest... |
| rescval2 16488 | Value of the category rest... |
| rescbas 16489 | Base set of the category r... |
| reschom 16490 | Hom-sets of the category r... |
| reschomf 16491 | Hom-sets of the category r... |
| rescco 16492 | Composition in the categor... |
| rescabs 16493 | Restriction absorption law... |
| rescabs2 16494 | Restriction absorption law... |
| issubc 16495 | Elementhood in the set of ... |
| issubc2 16496 | Elementhood in the set of ... |
| 0ssc 16497 | For any category ` C ` , t... |
| 0subcat 16498 | For any category ` C ` , t... |
| catsubcat 16499 | For any category ` C ` , `... |
| subcssc 16500 | An element in the set of s... |
| subcfn 16501 | An element in the set of s... |
| subcss1 16502 | The objects of a subcatego... |
| subcss2 16503 | The morphisms of a subcate... |
| subcidcl 16504 | The identity of the origin... |
| subccocl 16505 | A subcategory is closed un... |
| subccatid 16506 | A subcategory is a categor... |
| subcid 16507 | The identity in a subcateg... |
| subccat 16508 | A subcategory is a categor... |
| issubc3 16509 | Alternate definition of a ... |
| fullsubc 16510 | The full subcategory gener... |
| fullresc 16511 | The category formed by str... |
| resscat 16512 | A category restricted to a... |
| subsubc 16513 | A subcategory of a subcate... |
| relfunc 16522 | The set of functors is a r... |
| funcrcl 16523 | Reverse closure for a func... |
| isfunc 16524 | Value of the set of functo... |
| isfuncd 16525 | Deduce that an operation i... |
| funcf1 16526 | The object part of a funct... |
| funcixp 16527 | The morphism part of a fun... |
| funcf2 16528 | The morphism part of a fun... |
| funcfn2 16529 | The morphism part of a fun... |
| funcid 16530 | A functor maps each identi... |
| funcco 16531 | A functor maps composition... |
| funcsect 16532 | The image of a section und... |
| funcinv 16533 | The image of an inverse un... |
| funciso 16534 | The image of an isomorphis... |
| funcoppc 16535 | A functor on categories yi... |
| idfuval 16536 | Value of the identity func... |
| idfu2nd 16537 | Value of the morphism part... |
| idfu2 16538 | Value of the morphism part... |
| idfu1st 16539 | Value of the object part o... |
| idfu1 16540 | Value of the object part o... |
| idfucl 16541 | The identity functor is a ... |
| cofuval 16542 | Value of the composition o... |
| cofu1st 16543 | Value of the object part o... |
| cofu1 16544 | Value of the object part o... |
| cofu2nd 16545 | Value of the morphism part... |
| cofu2 16546 | Value of the morphism part... |
| cofuval2 16547 | Value of the composition o... |
| cofucl 16548 | The composition of two fun... |
| cofuass 16549 | Functor composition is ass... |
| cofulid 16550 | The identity functor is a ... |
| cofurid 16551 | The identity functor is a ... |
| resfval 16552 | Value of the functor restr... |
| resfval2 16553 | Value of the functor restr... |
| resf1st 16554 | Value of the functor restr... |
| resf2nd 16555 | Value of the functor restr... |
| funcres 16556 | A functor restricted to a ... |
| funcres2b 16557 | Condition for a functor to... |
| funcres2 16558 | A functor into a restricte... |
| wunfunc 16559 | A weak universe is closed ... |
| funcpropd 16560 | If two categories have the... |
| funcres2c 16561 | Condition for a functor to... |
| fullfunc 16566 | A full functor is a functo... |
| fthfunc 16567 | A faithful functor is a fu... |
| relfull 16568 | The set of full functors i... |
| relfth 16569 | The set of faithful functo... |
| isfull 16570 | Value of the set of full f... |
| isfull2 16571 | Equivalent condition for a... |
| fullfo 16572 | The morphism map of a full... |
| fulli 16573 | The morphism map of a full... |
| isfth 16574 | Value of the set of faithf... |
| isfth2 16575 | Equivalent condition for a... |
| isffth2 16576 | A fully faithful functor i... |
| fthf1 16577 | The morphism map of a fait... |
| fthi 16578 | The morphism map of a fait... |
| ffthf1o 16579 | The morphism map of a full... |
| fullpropd 16580 | If two categories have the... |
| fthpropd 16581 | If two categories have the... |
| fulloppc 16582 | The opposite functor of a ... |
| fthoppc 16583 | The opposite functor of a ... |
| ffthoppc 16584 | The opposite functor of a ... |
| fthsect 16585 | A faithful functor reflect... |
| fthinv 16586 | A faithful functor reflect... |
| fthmon 16587 | A faithful functor reflect... |
| fthepi 16588 | A faithful functor reflect... |
| ffthiso 16589 | A fully faithful functor r... |
| fthres2b 16590 | Condition for a faithful f... |
| fthres2c 16591 | Condition for a faithful f... |
| fthres2 16592 | A faithful functor into a ... |
| idffth 16593 | The identity functor is a ... |
| cofull 16594 | The composition of two ful... |
| cofth 16595 | The composition of two fai... |
| coffth 16596 | The composition of two ful... |
| rescfth 16597 | The inclusion functor from... |
| ressffth 16598 | The inclusion functor from... |
| fullres2c 16599 | Condition for a full funct... |
| ffthres2c 16600 | Condition for a fully fait... |
| fnfuc 16605 | The ` FuncCat ` operation ... |
| natfval 16606 | Value of the function givi... |
| isnat 16607 | Property of being a natura... |
| isnat2 16608 | Property of being a natura... |
| natffn 16609 | The natural transformation... |
| natrcl 16610 | Reverse closure for a natu... |
| nat1st2nd 16611 | Rewrite the natural transf... |
| natixp 16612 | A natural transformation i... |
| natcl 16613 | A component of a natural t... |
| natfn 16614 | A natural transformation i... |
| nati 16615 | Naturality property of a n... |
| wunnat 16616 | A weak universe is closed ... |
| catstr 16617 | A category structure is a ... |
| fucval 16618 | Value of the functor categ... |
| fuccofval 16619 | Value of the functor categ... |
| fucbas 16620 | The objects of the functor... |
| fuchom 16621 | The morphisms in the funct... |
| fucco 16622 | Value of the composition o... |
| fuccoval 16623 | Value of the functor categ... |
| fuccocl 16624 | The composition of two nat... |
| fucidcl 16625 | The identity natural trans... |
| fuclid 16626 | Left identity of natural t... |
| fucrid 16627 | Right identity of natural ... |
| fucass 16628 | Associativity of natural t... |
| fuccatid 16629 | The functor category is a ... |
| fuccat 16630 | The functor category is a ... |
| fucid 16631 | The identity morphism in t... |
| fucsect 16632 | Two natural transformation... |
| fucinv 16633 | Two natural transformation... |
| invfuc 16634 | If ` V ( x ) ` is an inver... |
| fuciso 16635 | A natural transformation i... |
| natpropd 16636 | If two categories have the... |
| fucpropd 16637 | If two categories have the... |
| initorcl 16644 | Reverse closure for an ini... |
| termorcl 16645 | Reverse closure for a term... |
| zeroorcl 16646 | Reverse closure for a zero... |
| initoval 16647 | The value of the initial o... |
| termoval 16648 | The value of the terminal ... |
| zerooval 16649 | The value of the zero obje... |
| isinito 16650 | The predicate "is an initi... |
| istermo 16651 | The predicate "is a termin... |
| iszeroo 16652 | The predicate "is a zero o... |
| isinitoi 16653 | Implication of a class bei... |
| istermoi 16654 | Implication of a class bei... |
| initoid 16655 | For an initial object, the... |
| termoid 16656 | For a terminal object, the... |
| initoo 16657 | An initial object is an ob... |
| termoo 16658 | A terminal object is an ob... |
| iszeroi 16659 | Implication of a class bei... |
| 2initoinv 16660 | Morphisms between two init... |
| initoeu1 16661 | Initial objects are essent... |
| initoeu1w 16662 | Initial objects are essent... |
| initoeu2lem0 16663 | Lemma 0 for ~ initoeu2 . ... |
| initoeu2lem1 16664 | Lemma 1 for ~ initoeu2 . ... |
| initoeu2lem2 16665 | Lemma 2 for ~ initoeu2 . ... |
| initoeu2 16666 | Initial objects are essent... |
| 2termoinv 16667 | Morphisms between two term... |
| termoeu1 16668 | Terminal objects are essen... |
| termoeu1w 16669 | Terminal objects are essen... |
| homarcl 16678 | Reverse closure for an arr... |
| homafval 16679 | Value of the disjointified... |
| homaf 16680 | Functionality of the disjo... |
| homaval 16681 | Value of the disjointified... |
| elhoma 16682 | Value of the disjointified... |
| elhomai 16683 | Produce an arrow from a mo... |
| elhomai2 16684 | Produce an arrow from a mo... |
| homarcl2 16685 | Reverse closure for the do... |
| homarel 16686 | An arrow is an ordered pai... |
| homa1 16687 | The first component of an ... |
| homahom2 16688 | The second component of an... |
| homahom 16689 | The second component of an... |
| homadm 16690 | The domain of an arrow wit... |
| homacd 16691 | The codomain of an arrow w... |
| homadmcd 16692 | Decompose an arrow into do... |
| arwval 16693 | The set of arrows is the u... |
| arwrcl 16694 | The first component of an ... |
| arwhoma 16695 | An arrow is contained in t... |
| homarw 16696 | A hom-set is a subset of t... |
| arwdm 16697 | The domain of an arrow is ... |
| arwcd 16698 | The codomain of an arrow i... |
| dmaf 16699 | The domain function is a f... |
| cdaf 16700 | The codomain function is a... |
| arwhom 16701 | The second component of an... |
| arwdmcd 16702 | Decompose an arrow into do... |
| idafval 16707 | Value of the identity arro... |
| idaval 16708 | Value of the identity arro... |
| ida2 16709 | Morphism part of the ident... |
| idahom 16710 | Domain and codomain of the... |
| idadm 16711 | Domain of the identity arr... |
| idacd 16712 | Codomain of the identity a... |
| idaf 16713 | The identity arrow functio... |
| coafval 16714 | The value of the compositi... |
| eldmcoa 16715 | A pair ` <. G , F >. ` is ... |
| dmcoass 16716 | The domain of composition ... |
| homdmcoa 16717 | If ` F : X --> Y ` and ` G... |
| coaval 16718 | Value of composition for c... |
| coa2 16719 | The morphism part of arrow... |
| coahom 16720 | The composition of two com... |
| coapm 16721 | Composition of arrows is a... |
| arwlid 16722 | Left identity of a categor... |
| arwrid 16723 | Right identity of a catego... |
| arwass 16724 | Associativity of compositi... |
| setcval 16727 | Value of the category of s... |
| setcbas 16728 | Set of objects of the cate... |
| setchomfval 16729 | Set of arrows of the categ... |
| setchom 16730 | Set of arrows of the categ... |
| elsetchom 16731 | A morphism of sets is a fu... |
| setccofval 16732 | Composition in the categor... |
| setcco 16733 | Composition in the categor... |
| setccatid 16734 | Lemma for ~ setccat . (Co... |
| setccat 16735 | The category of sets is a ... |
| setcid 16736 | The identity arrow in the ... |
| setcmon 16737 | A monomorphism of sets is ... |
| setcepi 16738 | An epimorphism of sets is ... |
| setcsect 16739 | A section in the category ... |
| setcinv 16740 | An inverse in the category... |
| setciso 16741 | An isomorphism in the cate... |
| resssetc 16742 | The restriction of the cat... |
| funcsetcres2 16743 | A functor into a smaller c... |
| catcval 16746 | Value of the category of c... |
| catcbas 16747 | Set of objects of the cate... |
| catchomfval 16748 | Set of arrows of the categ... |
| catchom 16749 | Set of arrows of the categ... |
| catccofval 16750 | Composition in the categor... |
| catcco 16751 | Composition in the categor... |
| catccatid 16752 | Lemma for ~ catccat . (Co... |
| catcid 16753 | The identity arrow in the ... |
| catccat 16754 | The category of categories... |
| resscatc 16755 | The restriction of the cat... |
| catcisolem 16756 | Lemma for ~ catciso . (Co... |
| catciso 16757 | A functor is an isomorphis... |
| catcoppccl 16758 | The category of categories... |
| catcfuccl 16759 | The category of categories... |
| fncnvimaeqv 16760 | The inverse images of the ... |
| bascnvimaeqv 16761 | The inverse image of the u... |
| estrcval 16764 | Value of the category of e... |
| estrcbas 16765 | Set of objects of the cate... |
| estrchomfval 16766 | Set of morphisms ("arrows"... |
| estrchom 16767 | The morphisms between exte... |
| elestrchom 16768 | A morphism between extensi... |
| estrccofval 16769 | Composition in the categor... |
| estrcco 16770 | Composition in the categor... |
| estrcbasbas 16771 | An element of the base set... |
| estrccatid 16772 | Lemma for ~ estrccat . (C... |
| estrccat 16773 | The category of extensible... |
| estrcid 16774 | The identity arrow in the ... |
| estrchomfn 16775 | The Hom-set operation in t... |
| estrchomfeqhom 16776 | The functionalized Hom-set... |
| estrreslem1 16777 | Lemma 1 for ~ estrres . (... |
| estrreslem2 16778 | Lemma 2 for ~ estrres . (... |
| estrres 16779 | Any restriction of a categ... |
| funcestrcsetclem1 16780 | Lemma 1 for ~ funcestrcset... |
| funcestrcsetclem2 16781 | Lemma 2 for ~ funcestrcset... |
| funcestrcsetclem3 16782 | Lemma 3 for ~ funcestrcset... |
| funcestrcsetclem4 16783 | Lemma 4 for ~ funcestrcset... |
| funcestrcsetclem5 16784 | Lemma 5 for ~ funcestrcset... |
| funcestrcsetclem6 16785 | Lemma 6 for ~ funcestrcset... |
| funcestrcsetclem7 16786 | Lemma 7 for ~ funcestrcset... |
| funcestrcsetclem8 16787 | Lemma 8 for ~ funcestrcset... |
| funcestrcsetclem9 16788 | Lemma 9 for ~ funcestrcset... |
| funcestrcsetc 16789 | The "natural forgetful fun... |
| fthestrcsetc 16790 | The "natural forgetful fun... |
| fullestrcsetc 16791 | The "natural forgetful fun... |
| equivestrcsetc 16792 | The "natural forgetful fun... |
| setc1strwun 16793 | A constructed one-slot str... |
| funcsetcestrclem1 16794 | Lemma 1 for ~ funcsetcestr... |
| funcsetcestrclem2 16795 | Lemma 2 for ~ funcsetcestr... |
| funcsetcestrclem3 16796 | Lemma 3 for ~ funcsetcestr... |
| embedsetcestrclem 16797 | Lemma for ~ embedsetcestrc... |
| funcsetcestrclem4 16798 | Lemma 4 for ~ funcsetcestr... |
| funcsetcestrclem5 16799 | Lemma 5 for ~ funcsetcestr... |
| funcsetcestrclem6 16800 | Lemma 6 for ~ funcsetcestr... |
| funcsetcestrclem7 16801 | Lemma 7 for ~ funcsetcestr... |
| funcsetcestrclem8 16802 | Lemma 8 for ~ funcsetcestr... |
| funcsetcestrclem9 16803 | Lemma 9 for ~ funcsetcestr... |
| funcsetcestrc 16804 | The "embedding functor" fr... |
| fthsetcestrc 16805 | The "embedding functor" fr... |
| fullsetcestrc 16806 | The "embedding functor" fr... |
| embedsetcestrc 16807 | The "embedding functor" fr... |
| fnxpc 16816 | The binary product of cate... |
| xpcval 16817 | Value of the binary produc... |
| xpcbas 16818 | Set of objects of the bina... |
| xpchomfval 16819 | Set of morphisms of the bi... |
| xpchom 16820 | Set of morphisms of the bi... |
| relxpchom 16821 | A hom-set in the binary pr... |
| xpccofval 16822 | Value of composition in th... |
| xpcco 16823 | Value of composition in th... |
| xpcco1st 16824 | Value of composition in th... |
| xpcco2nd 16825 | Value of composition in th... |
| xpchom2 16826 | Value of the set of morphi... |
| xpcco2 16827 | Value of composition in th... |
| xpccatid 16828 | The product of two categor... |
| xpcid 16829 | The identity morphism in t... |
| xpccat 16830 | The product of two categor... |
| 1stfval 16831 | Value of the first project... |
| 1stf1 16832 | Value of the first project... |
| 1stf2 16833 | Value of the first project... |
| 2ndfval 16834 | Value of the first project... |
| 2ndf1 16835 | Value of the first project... |
| 2ndf2 16836 | Value of the first project... |
| 1stfcl 16837 | The first projection funct... |
| 2ndfcl 16838 | The second projection func... |
| prfval 16839 | Value of the pairing funct... |
| prf1 16840 | Value of the pairing funct... |
| prf2fval 16841 | Value of the pairing funct... |
| prf2 16842 | Value of the pairing funct... |
| prfcl 16843 | The pairing of functors ` ... |
| prf1st 16844 | Cancellation of pairing wi... |
| prf2nd 16845 | Cancellation of pairing wi... |
| 1st2ndprf 16846 | Break a functor into a pro... |
| catcxpccl 16847 | The category of categories... |
| xpcpropd 16848 | If two categories have the... |
| evlfval 16857 | Value of the evaluation fu... |
| evlf2 16858 | Value of the evaluation fu... |
| evlf2val 16859 | Value of the evaluation na... |
| evlf1 16860 | Value of the evaluation fu... |
| evlfcllem 16861 | Lemma for ~ evlfcl . (Con... |
| evlfcl 16862 | The evaluation functor is ... |
| curfval 16863 | Value of the curry functor... |
| curf1fval 16864 | Value of the object part o... |
| curf1 16865 | Value of the object part o... |
| curf11 16866 | Value of the double evalua... |
| curf12 16867 | The partially evaluated cu... |
| curf1cl 16868 | The partially evaluated cu... |
| curf2 16869 | Value of the curry functor... |
| curf2val 16870 | Value of a component of th... |
| curf2cl 16871 | The curry functor at a mor... |
| curfcl 16872 | The curry functor of a fun... |
| curfpropd 16873 | If two categories have the... |
| uncfval 16874 | Value of the uncurry funct... |
| uncfcl 16875 | The uncurry operation take... |
| uncf1 16876 | Value of the uncurry funct... |
| uncf2 16877 | Value of the uncurry funct... |
| curfuncf 16878 | Cancellation of curry with... |
| uncfcurf 16879 | Cancellation of uncurry wi... |
| diagval 16880 | Define the diagonal functo... |
| diagcl 16881 | The diagonal functor is a ... |
| diag1cl 16882 | The constant functor of ` ... |
| diag11 16883 | Value of the constant func... |
| diag12 16884 | Value of the constant func... |
| diag2 16885 | Value of the diagonal func... |
| diag2cl 16886 | The diagonal functor at a ... |
| curf2ndf 16887 | As shown in ~ diagval , th... |
| hofval 16892 | Value of the Hom functor, ... |
| hof1fval 16893 | The object part of the Hom... |
| hof1 16894 | The object part of the Hom... |
| hof2fval 16895 | The morphism part of the H... |
| hof2val 16896 | The morphism part of the H... |
| hof2 16897 | The morphism part of the H... |
| hofcllem 16898 | Lemma for ~ hofcl . (Cont... |
| hofcl 16899 | Closure of the Hom functor... |
| oppchofcl 16900 | Closure of the opposite Ho... |
| yonval 16901 | Value of the Yoneda embedd... |
| yoncl 16902 | The Yoneda embedding is a ... |
| yon1cl 16903 | The Yoneda embedding at an... |
| yon11 16904 | Value of the Yoneda embedd... |
| yon12 16905 | Value of the Yoneda embedd... |
| yon2 16906 | Value of the Yoneda embedd... |
| hofpropd 16907 | If two categories have the... |
| yonpropd 16908 | If two categories have the... |
| oppcyon 16909 | Value of the opposite Yone... |
| oyoncl 16910 | The opposite Yoneda embedd... |
| oyon1cl 16911 | The opposite Yoneda embedd... |
| yonedalem1 16912 | Lemma for ~ yoneda . (Con... |
| yonedalem21 16913 | Lemma for ~ yoneda . (Con... |
| yonedalem3a 16914 | Lemma for ~ yoneda . (Con... |
| yonedalem4a 16915 | Lemma for ~ yoneda . (Con... |
| yonedalem4b 16916 | Lemma for ~ yoneda . (Con... |
| yonedalem4c 16917 | Lemma for ~ yoneda . (Con... |
| yonedalem22 16918 | Lemma for ~ yoneda . (Con... |
| yonedalem3b 16919 | Lemma for ~ yoneda . (Con... |
| yonedalem3 16920 | Lemma for ~ yoneda . (Con... |
| yonedainv 16921 | The Yoneda Lemma with expl... |
| yonffthlem 16922 | Lemma for ~ yonffth . (Co... |
| yoneda 16923 | The Yoneda Lemma. There i... |
| yonffth 16924 | The Yoneda Lemma. The Yon... |
| yoniso 16925 | If the codomain is recover... |
| isprs 16930 | Property of being a preord... |
| prslem 16931 | Lemma for ~ prsref and ~ p... |
| prsref 16932 | Less-or-equal is reflexive... |
| prstr 16933 | Less-or-equal is transitiv... |
| isdrs 16934 | Property of being a direct... |
| drsdir 16935 | Direction of a directed se... |
| drsprs 16936 | A directed set is a preset... |
| drsbn0 16937 | The base of a directed set... |
| drsdirfi 16938 | Any _finite_ number of ele... |
| isdrs2 16939 | Directed sets may be defin... |
| ispos 16947 | The predicate "is a poset.... |
| ispos2 16948 | A poset is an antisymmetri... |
| posprs 16949 | A poset is a preset. (Con... |
| posi 16950 | Lemma for poset properties... |
| posref 16951 | A poset ordering is reflex... |
| posasymb 16952 | A poset ordering is asymme... |
| postr 16953 | A poset ordering is transi... |
| 0pos 16954 | Technical lemma to simplif... |
| isposd 16955 | Properties that determine ... |
| isposi 16956 | Properties that determine ... |
| isposix 16957 | Properties that determine ... |
| pltfval 16959 | Value of the less-than rel... |
| pltval 16960 | Less-than relation. ( ~ d... |
| pltle 16961 | Less-than implies less-tha... |
| pltne 16962 | Less-than relation. ( ~ d... |
| pltirr 16963 | The less-than relation is ... |
| pleval2i 16964 | One direction of ~ pleval2... |
| pleval2 16965 | Less-than-or-equal in term... |
| pltnle 16966 | Less-than implies not inve... |
| pltval3 16967 | Alternate expression for l... |
| pltnlt 16968 | The less-than relation imp... |
| pltn2lp 16969 | The less-than relation has... |
| plttr 16970 | The less-than relation is ... |
| pltletr 16971 | Transitive law for chained... |
| plelttr 16972 | Transitive law for chained... |
| pospo 16973 | Write a poset structure in... |
| lubfval 16978 | Value of the least upper b... |
| lubdm 16979 | Domain of the least upper ... |
| lubfun 16980 | The LUB is a function. (C... |
| lubeldm 16981 | Member of the domain of th... |
| lubelss 16982 | A member of the domain of ... |
| lubeu 16983 | Unique existence proper of... |
| lubval 16984 | Value of the least upper b... |
| lubcl 16985 | The least upper bound func... |
| lubprop 16986 | Properties of greatest low... |
| luble 16987 | The greatest lower bound i... |
| lublecllem 16988 | Lemma for ~ lublecl and ~ ... |
| lublecl 16989 | The set of all elements le... |
| lubid 16990 | The LUB of elements less t... |
| glbfval 16991 | Value of the greatest lowe... |
| glbdm 16992 | Domain of the greatest low... |
| glbfun 16993 | The GLB is a function. (C... |
| glbeldm 16994 | Member of the domain of th... |
| glbelss 16995 | A member of the domain of ... |
| glbeu 16996 | Unique existence proper of... |
| glbval 16997 | Value of the greatest lowe... |
| glbcl 16998 | The least upper bound func... |
| glbprop 16999 | Properties of greatest low... |
| glble 17000 | The greatest lower bound i... |
| joinfval 17001 | Value of join function for... |
| joinfval2 17002 | Value of join function for... |
| joindm 17003 | Domain of join function fo... |
| joindef 17004 | Two ways to say that a joi... |
| joinval 17005 | Join value. Since both si... |
| joincl 17006 | Closure of join of element... |
| joindmss 17007 | Subset property of domain ... |
| joinval2lem 17008 | Lemma for ~ joinval2 and ~... |
| joinval2 17009 | Value of join for a poset ... |
| joineu 17010 | Uniqueness of join of elem... |
| joinlem 17011 | Lemma for join properties.... |
| lejoin1 17012 | A join's first argument is... |
| lejoin2 17013 | A join's second argument i... |
| joinle 17014 | A join is less than or equ... |
| meetfval 17015 | Value of meet function for... |
| meetfval2 17016 | Value of meet function for... |
| meetdm 17017 | Domain of meet function fo... |
| meetdef 17018 | Two ways to say that a mee... |
| meetval 17019 | Meet value. Since both si... |
| meetcl 17020 | Closure of meet of element... |
| meetdmss 17021 | Subset property of domain ... |
| meetval2lem 17022 | Lemma for ~ meetval2 and ~... |
| meetval2 17023 | Value of meet for a poset ... |
| meeteu 17024 | Uniqueness of meet of elem... |
| meetlem 17025 | Lemma for meet properties.... |
| lemeet1 17026 | A meet's first argument is... |
| lemeet2 17027 | A meet's second argument i... |
| meetle 17028 | A meet is less than or equ... |
| joincomALT 17029 | The join of a poset commut... |
| joincom 17030 | The join of a poset commut... |
| meetcomALT 17031 | The meet of a poset commut... |
| meetcom 17032 | The meet of a poset commut... |
| istos 17035 | The predicate "is a toset.... |
| tosso 17036 | Write the totally ordered ... |
| p0val 17041 | Value of poset zero. (Con... |
| p1val 17042 | Value of poset zero. (Con... |
| p0le 17043 | Any element is less than o... |
| ple1 17044 | Any element is less than o... |
| islat 17047 | The predicate "is a lattic... |
| latcl2 17048 | The join and meet of any t... |
| latlem 17049 | Lemma for lattice properti... |
| latpos 17050 | A lattice is a poset. (Co... |
| latjcl 17051 | Closure of join operation ... |
| latmcl 17052 | Closure of meet operation ... |
| latref 17053 | A lattice ordering is refl... |
| latasymb 17054 | A lattice ordering is asym... |
| latasym 17055 | A lattice ordering is asym... |
| lattr 17056 | A lattice ordering is tran... |
| latasymd 17057 | Deduce equality from latti... |
| lattrd 17058 | A lattice ordering is tran... |
| latjcom 17059 | The join of a lattice comm... |
| latlej1 17060 | A join's first argument is... |
| latlej2 17061 | A join's second argument i... |
| latjle12 17062 | A join is less than or equ... |
| latleeqj1 17063 | Less-than-or-equal-to in t... |
| latleeqj2 17064 | Less-than-or-equal-to in t... |
| latjlej1 17065 | Add join to both sides of ... |
| latjlej2 17066 | Add join to both sides of ... |
| latjlej12 17067 | Add join to both sides of ... |
| latnlej 17068 | An idiom to express that a... |
| latnlej1l 17069 | An idiom to express that a... |
| latnlej1r 17070 | An idiom to express that a... |
| latnlej2 17071 | An idiom to express that a... |
| latnlej2l 17072 | An idiom to express that a... |
| latnlej2r 17073 | An idiom to express that a... |
| latjidm 17074 | Lattice join is idempotent... |
| latmcom 17075 | The join of a lattice comm... |
| latmle1 17076 | A meet is less than or equ... |
| latmle2 17077 | A meet is less than or equ... |
| latlem12 17078 | An element is less than or... |
| latleeqm1 17079 | Less-than-or-equal-to in t... |
| latleeqm2 17080 | Less-than-or-equal-to in t... |
| latmlem1 17081 | Add meet to both sides of ... |
| latmlem2 17082 | Add meet to both sides of ... |
| latmlem12 17083 | Add join to both sides of ... |
| latnlemlt 17084 | Negation of less-than-or-e... |
| latnle 17085 | Equivalent expressions for... |
| latmidm 17086 | Lattice join is idempotent... |
| latabs1 17087 | Lattice absorption law. F... |
| latabs2 17088 | Lattice absorption law. F... |
| latledi 17089 | An ortholattice is distrib... |
| latmlej11 17090 | Ordering of a meet and joi... |
| latmlej12 17091 | Ordering of a meet and joi... |
| latmlej21 17092 | Ordering of a meet and joi... |
| latmlej22 17093 | Ordering of a meet and joi... |
| lubsn 17094 | The least upper bound of a... |
| latjass 17095 | Lattice join is associativ... |
| latj12 17096 | Swap 1st and 2nd members o... |
| latj32 17097 | Swap 2nd and 3rd members o... |
| latj13 17098 | Swap 1st and 3rd members o... |
| latj31 17099 | Swap 2nd and 3rd members o... |
| latjrot 17100 | Rotate lattice join of 3 c... |
| latj4 17101 | Rearrangement of lattice j... |
| latj4rot 17102 | Rotate lattice join of 4 c... |
| latjjdi 17103 | Lattice join distributes o... |
| latjjdir 17104 | Lattice join distributes o... |
| mod1ile 17105 | The weak direction of the ... |
| mod2ile 17106 | The weak direction of the ... |
| isclat 17109 | The predicate "is a comple... |
| clatpos 17110 | A complete lattice is a po... |
| clatlem 17111 | Lemma for properties of a ... |
| clatlubcl 17112 | Any subset of the base set... |
| clatlubcl2 17113 | Any subset of the base set... |
| clatglbcl 17114 | Any subset of the base set... |
| clatglbcl2 17115 | Any subset of the base set... |
| clatl 17116 | A complete lattice is a la... |
| isglbd 17117 | Properties that determine ... |
| lublem 17118 | Lemma for the least upper ... |
| lubub 17119 | The LUB of a complete latt... |
| lubl 17120 | The LUB of a complete latt... |
| lubss 17121 | Subset law for least upper... |
| lubel 17122 | An element of a set is les... |
| lubun 17123 | The LUB of a union. (Cont... |
| clatglb 17124 | Properties of greatest low... |
| clatglble 17125 | The greatest lower bound i... |
| clatleglb 17126 | Two ways of expressing "le... |
| clatglbss 17127 | Subset law for greatest lo... |
| oduval 17130 | Value of an order dual str... |
| oduleval 17131 | Value of the less-equal re... |
| oduleg 17132 | Truth of the less-equal re... |
| odubas 17133 | Base set of an order dual ... |
| pospropd 17134 | Posethood is determined on... |
| odupos 17135 | Being a poset is a self-du... |
| oduposb 17136 | Being a poset is a self-du... |
| meet0 17137 | Lemma for ~ odujoin . (Co... |
| join0 17138 | Lemma for ~ odumeet . (Co... |
| oduglb 17139 | Greatest lower bounds in a... |
| odumeet 17140 | Meets in a dual order are ... |
| odulub 17141 | Least upper bounds in a du... |
| odujoin 17142 | Joins in a dual order are ... |
| odulatb 17143 | Being a lattice is self-du... |
| oduclatb 17144 | Being a complete lattice i... |
| odulat 17145 | Being a lattice is self-du... |
| poslubmo 17146 | Least upper bounds in a po... |
| posglbmo 17147 | Greatest lower bounds in a... |
| poslubd 17148 | Properties which determine... |
| poslubdg 17149 | Properties which determine... |
| posglbd 17150 | Properties which determine... |
| ipostr 17153 | The structure of ~ df-ipo ... |
| ipoval 17154 | Value of the inclusion pos... |
| ipobas 17155 | Base set of the inclusion ... |
| ipolerval 17156 | Relation of the inclusion ... |
| ipotset 17157 | Topology of the inclusion ... |
| ipole 17158 | Weak order condition of th... |
| ipolt 17159 | Strict order condition of ... |
| ipopos 17160 | The inclusion poset on a f... |
| isipodrs 17161 | Condition for a family of ... |
| ipodrscl 17162 | Direction by inclusion as ... |
| ipodrsfi 17163 | Finite upper bound propert... |
| fpwipodrs 17164 | The finite subsets of any ... |
| ipodrsima 17165 | The monotone image of a di... |
| isacs3lem 17166 | An algebraic closure syste... |
| acsdrsel 17167 | An algebraic closure syste... |
| isacs4lem 17168 | In a closure system in whi... |
| isacs5lem 17169 | If closure commutes with d... |
| acsdrscl 17170 | In an algebraic closure sy... |
| acsficl 17171 | A closure in an algebraic ... |
| isacs5 17172 | A closure system is algebr... |
| isacs4 17173 | A closure system is algebr... |
| isacs3 17174 | A closure system is algebr... |
| acsficld 17175 | In an algebraic closure sy... |
| acsficl2d 17176 | In an algebraic closure sy... |
| acsfiindd 17177 | In an algebraic closure sy... |
| acsmapd 17178 | In an algebraic closure sy... |
| acsmap2d 17179 | In an algebraic closure sy... |
| acsinfd 17180 | In an algebraic closure sy... |
| acsdomd 17181 | In an algebraic closure sy... |
| acsinfdimd 17182 | In an algebraic closure sy... |
| acsexdimd 17183 | In an algebraic closure sy... |
| mrelatglb 17184 | Greatest lower bounds in a... |
| mrelatglb0 17185 | The empty intersection in ... |
| mrelatlub 17186 | Least upper bounds in a Mo... |
| mreclatBAD 17187 | A Moore space is a complet... |
| latmass 17188 | Lattice meet is associativ... |
| latdisdlem 17189 | Lemma for ~ latdisd . (Co... |
| latdisd 17190 | In a lattice, joins distri... |
| isdlat 17193 | Property of being a distri... |
| dlatmjdi 17194 | In a distributive lattice,... |
| dlatl 17195 | A distributive lattice is ... |
| odudlatb 17196 | The dual of a distributive... |
| dlatjmdi 17197 | In a distributive lattice,... |
| isps 17202 | The predicate "is a poset"... |
| psrel 17203 | A poset is a relation. (C... |
| psref2 17204 | A poset is antisymmetric a... |
| pstr2 17205 | A poset is transitive. (C... |
| pslem 17206 | Lemma for ~ psref and othe... |
| psdmrn 17207 | The domain and range of a ... |
| psref 17208 | A poset is reflexive. (Co... |
| psrn 17209 | The range of a poset equal... |
| psasym 17210 | A poset is antisymmetric. ... |
| pstr 17211 | A poset is transitive. (C... |
| cnvps 17212 | The converse of a poset is... |
| cnvpsb 17213 | The converse of a poset is... |
| psss 17214 | Any subset of a partially ... |
| psssdm2 17215 | Field of a subposet. (Con... |
| psssdm 17216 | Field of a subposet. (Con... |
| istsr 17217 | The predicate is a toset. ... |
| istsr2 17218 | The predicate is a toset. ... |
| tsrlin 17219 | A toset is a linear order.... |
| tsrlemax 17220 | Two ways of saying a numbe... |
| tsrps 17221 | A toset is a poset. (Cont... |
| cnvtsr 17222 | The converse of a toset is... |
| tsrss 17223 | Any subset of a totally or... |
| ledm 17224 | domain of ` <_ ` is ` RR* ... |
| lern 17225 | The range of ` <_ ` is ` R... |
| lefld 17226 | The field of the 'less or ... |
| letsr 17227 | The "less than or equal to... |
| isdir 17232 | A condition for a relation... |
| reldir 17233 | A direction is a relation.... |
| dirdm 17234 | A direction's domain is eq... |
| dirref 17235 | A direction is reflexive. ... |
| dirtr 17236 | A direction is transitive.... |
| dirge 17237 | For any two elements of a ... |
| tsrdir 17238 | A totally ordered set is a... |
| ismgm 17243 | The predicate "is a magma"... |
| ismgmn0 17244 | The predicate "is a magma"... |
| mgmcl 17245 | Closure of the operation o... |
| isnmgm 17246 | A condition for a structur... |
| plusffval 17247 | The group addition operati... |
| plusfval 17248 | The group addition operati... |
| plusfeq 17249 | If the addition operation ... |
| plusffn 17250 | The group addition operati... |
| mgmplusf 17251 | The group addition functio... |
| issstrmgm 17252 | Characterize a substructur... |
| intopsn 17253 | The internal operation for... |
| mgmb1mgm1 17254 | The only magma with a base... |
| mgm0 17255 | Any set with an empty base... |
| mgm0b 17256 | The structure with an empt... |
| mgm1 17257 | The structure with one ele... |
| opifismgm 17258 | A structure with a group a... |
| mgmidmo 17259 | A two-sided identity eleme... |
| grpidval 17260 | The value of the identity ... |
| grpidpropd 17261 | If two structures have the... |
| fn0g 17262 | The group zero extractor i... |
| 0g0 17263 | The identity element funct... |
| ismgmid 17264 | The identity element of a ... |
| mgmidcl 17265 | The identity element of a ... |
| mgmlrid 17266 | The identity element of a ... |
| ismgmid2 17267 | Show that a given element ... |
| grpidd 17268 | Deduce the identity elemen... |
| mgmidsssn0 17269 | Property of the set of ide... |
| gsumvalx 17270 | Expand out the substitutio... |
| gsumval 17271 | Expand out the substitutio... |
| gsumpropd 17272 | The group sum depends only... |
| gsumpropd2lem 17273 | Lemma for ~ gsumpropd2 . ... |
| gsumpropd2 17274 | A stronger version of ~ gs... |
| gsummgmpropd 17275 | A stronger version of ~ gs... |
| gsumress 17276 | The group sum in a substru... |
| gsumval1 17277 | Value of the group sum ope... |
| gsum0 17278 | Value of the empty group s... |
| gsumval2a 17279 | Value of the group sum ope... |
| gsumval2 17280 | Value of the group sum ope... |
| gsumprval 17281 | Value of the group sum ope... |
| gsumpr12val 17282 | Value of the group sum ope... |
| issgrp 17285 | The predicate "is a semigr... |
| issgrpv 17286 | The predicate "is a semigr... |
| issgrpn0 17287 | The predicate "is a semigr... |
| isnsgrp 17288 | A condition for a structur... |
| sgrpmgm 17289 | A semigroup is a magma. (... |
| sgrpass 17290 | A semigroup operation is a... |
| sgrp0 17291 | Any set with an empty base... |
| sgrp0b 17292 | The structure with an empt... |
| sgrp1 17293 | The structure with one ele... |
| ismnddef 17296 | The predicate "is a monoid... |
| ismnd 17297 | The predicate "is a monoid... |
| isnmnd 17298 | A condition for a structur... |
| mndsgrp 17299 | A monoid is a semigroup. ... |
| mndmgm 17300 | A monoid is a magma. (Con... |
| mndcl 17301 | Closure of the operation o... |
| mndass 17302 | A monoid operation is asso... |
| mndid 17303 | A monoid has a two-sided i... |
| mndideu 17304 | The two-sided identity ele... |
| mnd32g 17305 | Commutative/associative la... |
| mnd12g 17306 | Commutative/associative la... |
| mnd4g 17307 | Commutative/associative la... |
| mndidcl 17308 | The identity element of a ... |
| mndplusf 17309 | The group addition operati... |
| mndlrid 17310 | A monoid's identity elemen... |
| mndlid 17311 | The identity element of a ... |
| mndrid 17312 | The identity element of a ... |
| ismndd 17313 | Deduce a monoid from its p... |
| mndpfo 17314 | The addition operation of ... |
| mndfo 17315 | The addition operation of ... |
| mndpropd 17316 | If two structures have the... |
| mndprop 17317 | If two structures have the... |
| issubmnd 17318 | Characterize a submonoid b... |
| ress0g 17319 | ` 0g ` is unaffected by re... |
| submnd0 17320 | The zero of a submonoid is... |
| prdsplusgcl 17321 | Structure product pointwis... |
| prdsidlem 17322 | Characterization of identi... |
| prdsmndd 17323 | The product of a family of... |
| prds0g 17324 | Zero in a product of monoi... |
| pwsmnd 17325 | The structure power of a m... |
| pws0g 17326 | Zero in a product of monoi... |
| imasmnd2 17327 | The image structure of a m... |
| imasmnd 17328 | The image structure of a m... |
| imasmndf1 17329 | The image of a monoid unde... |
| xpsmnd 17330 | The binary product of mono... |
| mnd1 17331 | The (smallest) structure r... |
| mnd1id 17332 | The singleton element of a... |
| ismhm 17337 | Property of a monoid homom... |
| mhmrcl1 17338 | Reverse closure of a monoi... |
| mhmrcl2 17339 | Reverse closure of a monoi... |
| mhmf 17340 | A monoid homomorphism is a... |
| mhmpropd 17341 | Monoid homomorphism depend... |
| mhmlin 17342 | A monoid homomorphism comm... |
| mhm0 17343 | A monoid homomorphism pres... |
| idmhm 17344 | The identity homomorphism ... |
| mhmf1o 17345 | A monoid homomorphism is b... |
| submrcl 17346 | Reverse closure for submon... |
| issubm 17347 | Expand definition of a sub... |
| issubm2 17348 | Submonoids are subsets tha... |
| issubmd 17349 | Deduction for proving a su... |
| submss 17350 | Submonoids are subsets of ... |
| submid 17351 | Every monoid is trivially ... |
| subm0cl 17352 | Submonoids contain zero. ... |
| submcl 17353 | Submonoids are closed unde... |
| submmnd 17354 | Submonoids are themselves ... |
| submbas 17355 | The base set of a submonoi... |
| subm0 17356 | Submonoids have the same i... |
| subsubm 17357 | A submonoid of a submonoid... |
| 0mhm 17358 | The constant zero linear f... |
| resmhm 17359 | Restriction of a monoid ho... |
| resmhm2 17360 | One direction of ~ resmhm2... |
| resmhm2b 17361 | Restriction of the codomai... |
| mhmco 17362 | The composition of monoid ... |
| mhmima 17363 | The homomorphic image of a... |
| mhmeql 17364 | The equalizer of two monoi... |
| submacs 17365 | Submonoids are an algebrai... |
| mrcmndind 17366 | (( From SO's determinants ... |
| prdspjmhm 17367 | A projection from a produc... |
| pwspjmhm 17368 | A projection from a produc... |
| pwsdiagmhm 17369 | Diagonal monoid homomorphi... |
| pwsco1mhm 17370 | Right composition with a f... |
| pwsco2mhm 17371 | Left composition with a mo... |
| gsumvallem2 17372 | Lemma for properties of th... |
| gsumsubm 17373 | Evaluate a group sum in a ... |
| gsumz 17374 | Value of a group sum over ... |
| gsumwsubmcl 17375 | Closure of the composite i... |
| gsumws1 17376 | A singleton composite reco... |
| gsumwcl 17377 | Closure of the composite o... |
| gsumccat 17378 | Homomorphic property of co... |
| gsumws2 17379 | Valuation of a pair in a m... |
| gsumccatsn 17380 | Homomorphic property of co... |
| gsumspl 17381 | The primary purpose of the... |
| gsumwmhm 17382 | Behavior of homomorphisms ... |
| gsumwspan 17383 | The submonoid generated by... |
| frmdval 17388 | Value of the free monoid c... |
| frmdbas 17389 | The base set of a free mon... |
| frmdelbas 17390 | An element of the base set... |
| frmdplusg 17391 | The monoid operation of a ... |
| frmdadd 17392 | Value of the monoid operat... |
| vrmdfval 17393 | The canonical injection fr... |
| vrmdval 17394 | The value of the generatin... |
| vrmdf 17395 | The mapping from the index... |
| frmdmnd 17396 | A free monoid is a monoid.... |
| frmd0 17397 | The identity of the free m... |
| frmdsssubm 17398 | The set of words taking va... |
| frmdgsum 17399 | Any word in a free monoid ... |
| frmdss2 17400 | A subset of generators is ... |
| frmdup1 17401 | Any assignment of the gene... |
| frmdup2 17402 | The evaluation map has the... |
| frmdup3lem 17403 | Lemma for ~ frmdup3 . (Co... |
| frmdup3 17404 | Universal property of the ... |
| mgm2nsgrplem1 17405 | Lemma 1 for ~ mgm2nsgrp : ... |
| mgm2nsgrplem2 17406 | Lemma 2 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem3 17407 | Lemma 3 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem4 17408 | Lemma 4 for ~ mgm2nsgrp : ... |
| mgm2nsgrp 17409 | A small magma (with two el... |
| sgrp2nmndlem1 17410 | Lemma 1 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem2 17411 | Lemma 2 for ~ sgrp2nmnd . ... |
| sgrp2nmndlem3 17412 | Lemma 3 for ~ sgrp2nmnd . ... |
| sgrp2rid2 17413 | A small semigroup (with tw... |
| sgrp2rid2ex 17414 | A small semigroup (with tw... |
| sgrp2nmndlem4 17415 | Lemma 4 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem5 17416 | Lemma 5 for ~ sgrp2nmnd : ... |
| sgrp2nmnd 17417 | A small semigroup (with tw... |
| mgmnsgrpex 17418 | There is a magma which is ... |
| sgrpnmndex 17419 | There is a semigroup which... |
| sgrpssmgm 17420 | The class of all semigroup... |
| mndsssgrp 17421 | The class of all monoids i... |
| isgrp 17428 | The predicate "is a group.... |
| grpmnd 17429 | A group is a monoid. (Con... |
| grpcl 17430 | Closure of the operation o... |
| grpass 17431 | A group operation is assoc... |
| grpinvex 17432 | Every member of a group ha... |
| grpideu 17433 | The two-sided identity ele... |
| grpplusf 17434 | The group addition operati... |
| grpplusfo 17435 | The group addition operati... |
| resgrpplusfrn 17436 | The underlying set of a gr... |
| grppropd 17437 | If two structures have the... |
| grpprop 17438 | If two structures have the... |
| grppropstr 17439 | Generalize a specific 2-el... |
| grpss 17440 | Show that a structure exte... |
| isgrpd2e 17441 | Deduce a group from its pr... |
| isgrpd2 17442 | Deduce a group from its pr... |
| isgrpde 17443 | Deduce a group from its pr... |
| isgrpd 17444 | Deduce a group from its pr... |
| isgrpi 17445 | Properties that determine ... |
| grpsgrp 17446 | A group is a semigroup. (... |
| dfgrp2 17447 | Alternate definition of a ... |
| dfgrp2e 17448 | Alternate definition of a ... |
| isgrpix 17449 | Properties that determine ... |
| grpidcl 17450 | The identity element of a ... |
| grpbn0 17451 | The base set of a group is... |
| grplid 17452 | The identity element of a ... |
| grprid 17453 | The identity element of a ... |
| grpn0 17454 | A group is not empty. (Co... |
| grprcan 17455 | Right cancellation law for... |
| grpinveu 17456 | The left inverse element o... |
| grpid 17457 | Two ways of saying that an... |
| isgrpid2 17458 | Properties showing that an... |
| grpidd2 17459 | Deduce the identity elemen... |
| grpinvfval 17460 | The inverse function of a ... |
| grpinvval 17461 | The inverse of a group ele... |
| grpinvfn 17462 | Functionality of the group... |
| grpinvfvi 17463 | The group inverse function... |
| grpsubfval 17464 | Group subtraction (divisio... |
| grpsubval 17465 | Group subtraction (divisio... |
| grpinvf 17466 | The group inversion operat... |
| grpinvcl 17467 | A group element's inverse ... |
| grplinv 17468 | The left inverse of a grou... |
| grprinv 17469 | The right inverse of a gro... |
| grpinvid1 17470 | The inverse of a group ele... |
| grpinvid2 17471 | The inverse of a group ele... |
| isgrpinv 17472 | Properties showing that a ... |
| grplrinv 17473 | In a group, every member h... |
| grpidinv2 17474 | A group's properties using... |
| grpidinv 17475 | A group has a left and rig... |
| grpinvid 17476 | The inverse of the identit... |
| grplcan 17477 | Left cancellation law for ... |
| grpasscan1 17478 | An associative cancellatio... |
| grpasscan2 17479 | An associative cancellatio... |
| grpidrcan 17480 | If right adding an element... |
| grpidlcan 17481 | If left adding an element ... |
| grpinvinv 17482 | Double inverse law for gro... |
| grpinvcnv 17483 | The group inverse is its o... |
| grpinv11 17484 | The group inverse is one-t... |
| grpinvf1o 17485 | The group inverse is a one... |
| grpinvnz 17486 | The inverse of a nonzero g... |
| grpinvnzcl 17487 | The inverse of a nonzero g... |
| grpsubinv 17488 | Subtraction of an inverse.... |
| grplmulf1o 17489 | Left multiplication by a g... |
| grpinvpropd 17490 | If two structures have the... |
| grpidssd 17491 | If the base set of a group... |
| grpinvssd 17492 | If the base set of a group... |
| grpinvadd 17493 | The inverse of the group o... |
| grpsubf 17494 | Functionality of group sub... |
| grpsubcl 17495 | Closure of group subtracti... |
| grpsubrcan 17496 | Right cancellation law for... |
| grpinvsub 17497 | Inverse of a group subtrac... |
| grpinvval2 17498 | A ~ df-neg -like equation ... |
| grpsubid 17499 | Subtraction of a group ele... |
| grpsubid1 17500 | Subtraction of the identit... |
| grpsubeq0 17501 | If the difference between ... |
| grpsubadd0sub 17502 | Subtraction expressed as a... |
| grpsubadd 17503 | Relationship between group... |
| grpsubsub 17504 | Double group subtraction. ... |
| grpaddsubass 17505 | Associative-type law for g... |
| grppncan 17506 | Cancellation law for subtr... |
| grpnpcan 17507 | Cancellation law for subtr... |
| grpsubsub4 17508 | Double group subtraction (... |
| grppnpcan2 17509 | Cancellation law for mixed... |
| grpnpncan 17510 | Cancellation law for group... |
| grpnpncan0 17511 | Cancellation law for group... |
| grpnnncan2 17512 | Cancellation law for group... |
| dfgrp3lem 17513 | Lemma for ~ dfgrp3 . (Con... |
| dfgrp3 17514 | Alternate definition of a ... |
| dfgrp3e 17515 | Alternate definition of a ... |
| grplactfval 17516 | The left group action of e... |
| grplactval 17517 | The value of the left grou... |
| grplactcnv 17518 | The left group action of e... |
| grplactf1o 17519 | The left group action of e... |
| grpsubpropd 17520 | Weak property deduction fo... |
| grpsubpropd2 17521 | Strong property deduction ... |
| grp1 17522 | The (smallest) structure r... |
| grp1inv 17523 | The inverse function of th... |
| prdsinvlem 17524 | Characterization of invers... |
| prdsgrpd 17525 | The product of a family of... |
| prdsinvgd 17526 | Negation in a product of g... |
| pwsgrp 17527 | The product of a family of... |
| pwsinvg 17528 | Negation in a group power.... |
| pwssub 17529 | Subtraction in a group pow... |
| imasgrp2 17530 | The image structure of a g... |
| imasgrp 17531 | The image structure of a g... |
| imasgrpf1 17532 | The image of a group under... |
| qusgrp2 17533 | Prove that a quotient stru... |
| xpsgrp 17534 | The binary product of grou... |
| mhmlem 17535 | Lemma for ~ mhmmnd and ~ g... |
| mhmid 17536 | A surjective monoid morphi... |
| mhmmnd 17537 | The image of a monoid ` G ... |
| mhmfmhm 17538 | The function fulfilling th... |
| ghmgrp 17539 | The image of a group ` G `... |
| mulgfval 17542 | Group multiple (exponentia... |
| mulgval 17543 | Value of the group multipl... |
| mulgfn 17544 | Functionality of the group... |
| mulgfvi 17545 | The group multiple operati... |
| mulg0 17546 | Group multiple (exponentia... |
| mulgnn 17547 | Group multiple (exponentia... |
| mulg1 17548 | Group multiple (exponentia... |
| mulgnnp1 17549 | Group multiple (exponentia... |
| mulg2 17550 | Group multiple (exponentia... |
| mulgnegnn 17551 | Group multiple (exponentia... |
| mulgnn0p1 17552 | Group multiple (exponentia... |
| mulgnnsubcl 17553 | Closure of the group multi... |
| mulgnn0subcl 17554 | Closure of the group multi... |
| mulgsubcl 17555 | Closure of the group multi... |
| mulgnncl 17556 | Closure of the group multi... |
| mulgnnclOLD 17557 | Obsolete proof of ~ mulgnn... |
| mulgnn0cl 17558 | Closure of the group multi... |
| mulgcl 17559 | Closure of the group multi... |
| mulgneg 17560 | Group multiple (exponentia... |
| mulgnegneg 17561 | The inverse of a negative ... |
| mulgm1 17562 | Group multiple (exponentia... |
| mulgaddcomlem 17563 | Lemma for ~ mulgaddcom . ... |
| mulgaddcom 17564 | The group multiple operato... |
| mulginvcom 17565 | The group multiple operato... |
| mulginvinv 17566 | The group multiple operato... |
| mulgnn0z 17567 | A group multiple of the id... |
| mulgz 17568 | A group multiple of the id... |
| mulgnndir 17569 | Sum of group multiples, fo... |
| mulgnndirOLD 17570 | Obsolete proof of ~ mulgnn... |
| mulgnn0dir 17571 | Sum of group multiples, ge... |
| mulgdirlem 17572 | Lemma for ~ mulgdir . (Co... |
| mulgdir 17573 | Sum of group multiples, ge... |
| mulgp1 17574 | Group multiple (exponentia... |
| mulgneg2 17575 | Group multiple (exponentia... |
| mulgnnass 17576 | Product of group multiples... |
| mulgnnassOLD 17577 | Obsolete proof of ~ mulgnn... |
| mulgnn0ass 17578 | Product of group multiples... |
| mulgass 17579 | Product of group multiples... |
| mulgassr 17580 | Reversed product of group ... |
| mulgmodid 17581 | Casting out multiples of t... |
| mulgsubdir 17582 | Subtraction of a group ele... |
| mhmmulg 17583 | A homomorphism of monoids ... |
| mulgpropd 17584 | Two structures with the sa... |
| submmulgcl 17585 | Closure of the group multi... |
| submmulg 17586 | A group multiple is the sa... |
| pwsmulg 17587 | Value of a group multiple ... |
| issubg 17594 | The subgroup predicate. (... |
| subgss 17595 | A subgroup is a subset. (... |
| subgid 17596 | A group is a subgroup of i... |
| subggrp 17597 | A subgroup is a group. (C... |
| subgbas 17598 | The base of the restricted... |
| subgrcl 17599 | Reverse closure for the su... |
| subg0 17600 | A subgroup of a group must... |
| subginv 17601 | The inverse of an element ... |
| subg0cl 17602 | The group identity is an e... |
| subginvcl 17603 | The inverse of an element ... |
| subgcl 17604 | A subgroup is closed under... |
| subgsubcl 17605 | A subgroup is closed under... |
| subgsub 17606 | The subtraction of element... |
| subgmulgcl 17607 | Closure of the group multi... |
| subgmulg 17608 | A group multiple is the sa... |
| issubg2 17609 | Characterize the subgroups... |
| issubgrpd2 17610 | Prove a subgroup by closur... |
| issubgrpd 17611 | Prove a subgroup by closur... |
| issubg3 17612 | A subgroup is a symmetric ... |
| issubg4 17613 | A subgroup is a nonempty s... |
| grpissubg 17614 | If the base set of a group... |
| resgrpisgrp 17615 | If the base set of a group... |
| subgsubm 17616 | A subgroup is a submonoid.... |
| subsubg 17617 | A subgroup of a subgroup i... |
| subgint 17618 | The intersection of a none... |
| 0subg 17619 | The zero subgroup of an ar... |
| cycsubgcl 17620 | The set of integer powers ... |
| cycsubgss 17621 | The cyclic subgroup genera... |
| cycsubg 17622 | The cyclic group generated... |
| isnsg 17623 | Property of being a normal... |
| isnsg2 17624 | Weaken the condition of ~ ... |
| nsgbi 17625 | Defining property of a nor... |
| nsgsubg 17626 | A normal subgroup is a sub... |
| nsgconj 17627 | The conjugation of an elem... |
| isnsg3 17628 | A subgroup is normal iff t... |
| subgacs 17629 | Subgroups are an algebraic... |
| nsgacs 17630 | Normal subgroups form an a... |
| cycsubg2 17631 | The subgroup generated by ... |
| cycsubg2cl 17632 | Any multiple of an element... |
| elnmz 17633 | Elementhood in the normali... |
| nmzbi 17634 | Defining property of the n... |
| nmzsubg 17635 | The normalizer N_G(S) of a... |
| ssnmz 17636 | A subgroup is a subset of ... |
| isnsg4 17637 | A subgroup is normal iff i... |
| nmznsg 17638 | Any subgroup is a normal s... |
| 0nsg 17639 | The zero subgroup is norma... |
| nsgid 17640 | The whole group is a norma... |
| releqg 17641 | The left coset equivalence... |
| eqgfval 17642 | Value of the subgroup left... |
| eqgval 17643 | Value of the subgroup left... |
| eqger 17644 | The subgroup coset equival... |
| eqglact 17645 | A left coset can be expres... |
| eqgid 17646 | The left coset containing ... |
| eqgen 17647 | Each coset is equipotent t... |
| eqgcpbl 17648 | The subgroup coset equival... |
| qusgrp 17649 | If ` Y ` is a normal subgr... |
| quseccl 17650 | Closure of the quotient ma... |
| qusadd 17651 | Value of the group operati... |
| qus0 17652 | Value of the group identit... |
| qusinv 17653 | Value of the group inverse... |
| qussub 17654 | Value of the group subtrac... |
| lagsubg2 17655 | Lagrange's theorem for fin... |
| lagsubg 17656 | Lagrange theorem for Group... |
| reldmghm 17659 | Lemma for group homomorphi... |
| isghm 17660 | Property of being a homomo... |
| isghm3 17661 | Property of a group homomo... |
| ghmgrp1 17662 | A group homomorphism is on... |
| ghmgrp2 17663 | A group homomorphism is on... |
| ghmf 17664 | A group homomorphism is a ... |
| ghmlin 17665 | A homomorphism of groups i... |
| ghmid 17666 | A homomorphism of groups p... |
| ghminv 17667 | A homomorphism of groups p... |
| ghmsub 17668 | Linearity of subtraction t... |
| isghmd 17669 | Deduction for a group homo... |
| ghmmhm 17670 | A group homomorphism is a ... |
| ghmmhmb 17671 | Group homomorphisms and mo... |
| ghmmulg 17672 | A homomorphism of monoids ... |
| ghmrn 17673 | The range of a homomorphis... |
| 0ghm 17674 | The constant zero linear f... |
| idghm 17675 | The identity homomorphism ... |
| resghm 17676 | Restriction of a homomorph... |
| resghm2 17677 | One direction of ~ resghm2... |
| resghm2b 17678 | Restriction of the codomai... |
| ghmghmrn 17679 | A group homomorphism from ... |
| ghmco 17680 | The composition of group h... |
| ghmima 17681 | The image of a subgroup un... |
| ghmpreima 17682 | The inverse image of a sub... |
| ghmeql 17683 | The equalizer of two group... |
| ghmnsgima 17684 | The image of a normal subg... |
| ghmnsgpreima 17685 | The inverse image of a nor... |
| ghmker 17686 | The kernel of a homomorphi... |
| ghmeqker 17687 | Two source points map to t... |
| pwsdiagghm 17688 | Diagonal homomorphism into... |
| ghmf1 17689 | Two ways of saying a group... |
| ghmf1o 17690 | A bijective group homomorp... |
| conjghm 17691 | Conjugation is an automorp... |
| conjsubg 17692 | A conjugated subgroup is a... |
| conjsubgen 17693 | A conjugated subgroup is e... |
| conjnmz 17694 | A subgroup is unchanged un... |
| conjnmzb 17695 | Alternative condition for ... |
| conjnsg 17696 | A normal subgroup is uncha... |
| qusghm 17697 | If ` Y ` is a normal subgr... |
| ghmpropd 17698 | Group homomorphism depends... |
| gimfn 17703 | The group isomorphism func... |
| isgim 17704 | An isomorphism of groups i... |
| gimf1o 17705 | An isomorphism of groups i... |
| gimghm 17706 | An isomorphism of groups i... |
| isgim2 17707 | A group isomorphism is a h... |
| subggim 17708 | Behavior of subgroups unde... |
| gimcnv 17709 | The converse of a bijectiv... |
| gimco 17710 | The composition of group i... |
| brgic 17711 | The relation "is isomorphi... |
| brgici 17712 | Prove isomorphic by an exp... |
| gicref 17713 | Isomorphism is reflexive. ... |
| giclcl 17714 | Isomorphism implies the le... |
| gicrcl 17715 | Isomorphism implies the ri... |
| gicsym 17716 | Isomorphism is symmetric. ... |
| gictr 17717 | Isomorphism is transitive.... |
| gicer 17718 | Isomorphism is an equivale... |
| gicerOLD 17719 | Obsolete proof of ~ gicer ... |
| gicen 17720 | Isomorphic groups have equ... |
| gicsubgen 17721 | A less trivial example of ... |
| isga 17724 | The predicate "is a (left)... |
| gagrp 17725 | The left argument of a gro... |
| gaset 17726 | The right argument of a gr... |
| gagrpid 17727 | The identity of the group ... |
| gaf 17728 | The mapping of the group a... |
| gafo 17729 | A group action is onto its... |
| gaass 17730 | An "associative" property ... |
| ga0 17731 | The action of a group on t... |
| gaid 17732 | The trivial action of a gr... |
| subgga 17733 | A subgroup acts on its par... |
| gass 17734 | A subset of a group action... |
| gasubg 17735 | The restriction of a group... |
| gaid2 17736 | A group operation is a lef... |
| galcan 17737 | The action of a particular... |
| gacan 17738 | Group inverses cancel in a... |
| gapm 17739 | The action of a particular... |
| gaorb 17740 | The orbit equivalence rela... |
| gaorber 17741 | The orbit equivalence rela... |
| gastacl 17742 | The stabilizer subgroup in... |
| gastacos 17743 | Write the coset relation f... |
| orbstafun 17744 | Existence and uniqueness f... |
| orbstaval 17745 | Value of the function at a... |
| orbsta 17746 | The Orbit-Stabilizer theor... |
| orbsta2 17747 | Relation between the size ... |
| cntrval 17752 | Substitute definition of t... |
| cntzfval 17753 | First level substitution f... |
| cntzval 17754 | Definition substitution fo... |
| elcntz 17755 | Elementhood in the central... |
| cntzel 17756 | Membership in a centralize... |
| cntzsnval 17757 | Special substitution for t... |
| elcntzsn 17758 | Value of the centralizer o... |
| sscntz 17759 | A centralizer expression f... |
| cntzrcl 17760 | Reverse closure for elemen... |
| cntzssv 17761 | The centralizer is uncondi... |
| cntzi 17762 | Membership in a centralize... |
| cntri 17763 | Defining property of the c... |
| resscntz 17764 | Centralizer in a substruct... |
| cntz2ss 17765 | Centralizers reverse the s... |
| cntzrec 17766 | Reciprocity relationship f... |
| cntziinsn 17767 | Express any centralizer as... |
| cntzsubm 17768 | Centralizers in a monoid a... |
| cntzsubg 17769 | Centralizers in a group ar... |
| cntzidss 17770 | If the elements of ` S ` c... |
| cntzmhm 17771 | Centralizers in a monoid a... |
| cntzmhm2 17772 | Centralizers in a monoid a... |
| cntrsubgnsg 17773 | A central subgroup is norm... |
| cntrnsg 17774 | The center of a group is a... |
| oppgval 17777 | Value of the opposite grou... |
| oppgplusfval 17778 | Value of the addition oper... |
| oppgplus 17779 | Value of the addition oper... |
| oppglem 17780 | Lemma for ~ oppgbas . (Co... |
| oppgbas 17781 | Base set of an opposite gr... |
| oppgtset 17782 | Topology of an opposite gr... |
| oppgtopn 17783 | Topology of an opposite gr... |
| oppgmnd 17784 | The opposite of a monoid i... |
| oppgmndb 17785 | Bidirectional form of ~ op... |
| oppgid 17786 | Zero in a monoid is a symm... |
| oppggrp 17787 | The opposite of a group is... |
| oppggrpb 17788 | Bidirectional form of ~ op... |
| oppginv 17789 | Inverses in a group are a ... |
| invoppggim 17790 | The inverse is an antiauto... |
| oppggic 17791 | Every group is (naturally)... |
| oppgsubm 17792 | Being a submonoid is a sym... |
| oppgsubg 17793 | Being a subgroup is a symm... |
| oppgcntz 17794 | A centralizer in a group i... |
| oppgcntr 17795 | The center of a group is t... |
| gsumwrev 17796 | A sum in an opposite monoi... |
| symgval 17799 | The value of the symmetric... |
| symgbas 17800 | The base set of the symmet... |
| elsymgbas2 17801 | Two ways of saying a funct... |
| elsymgbas 17802 | Two ways of saying a funct... |
| symgbasf1o 17803 | Elements in the symmetric ... |
| symgbasf 17804 | A permutation (element of ... |
| symghash 17805 | The symmetric group on ` n... |
| symgbasfi 17806 | The symmetric group on a f... |
| symgfv 17807 | The function value of a pe... |
| symgfvne 17808 | The function values of a p... |
| symgplusg 17809 | The group operation of a s... |
| symgov 17810 | The value of the group ope... |
| symgcl 17811 | The group operation of the... |
| symgmov1 17812 | For a permutation of a set... |
| symgmov2 17813 | For a permutation of a set... |
| symgbas0 17814 | The base set of the symmet... |
| symg1hash 17815 | The symmetric group on a s... |
| symg1bas 17816 | The symmetric group on a s... |
| symg2hash 17817 | The symmetric group on a (... |
| symg2bas 17818 | The symmetric group on a p... |
| symgtset 17819 | The topology of the symmet... |
| symggrp 17820 | The symmetric group on a s... |
| symgid 17821 | The group identity element... |
| symginv 17822 | The group inverse in the s... |
| galactghm 17823 | The currying of a group ac... |
| lactghmga 17824 | The converse of ~ galactgh... |
| symgtopn 17825 | The topology of the symmet... |
| symgga 17826 | The symmetric group induce... |
| pgrpsubgsymgbi 17827 | Every permutation group is... |
| pgrpsubgsymg 17828 | Every permutation group is... |
| idresperm 17829 | The identity function rest... |
| idressubgsymg 17830 | The singleton containing o... |
| idrespermg 17831 | The structure with the sin... |
| cayleylem1 17832 | Lemma for ~ cayley . (Con... |
| cayleylem2 17833 | Lemma for ~ cayley . (Con... |
| cayley 17834 | Cayley's Theorem (construc... |
| cayleyth 17835 | Cayley's Theorem (existenc... |
| symgfix2 17836 | If a permutation does not ... |
| symgextf 17837 | The extension of a permuta... |
| symgextfv 17838 | The function value of the ... |
| symgextfve 17839 | The function value of the ... |
| symgextf1lem 17840 | Lemma for ~ symgextf1 . (... |
| symgextf1 17841 | The extension of a permuta... |
| symgextfo 17842 | The extension of a permuta... |
| symgextf1o 17843 | The extension of a permuta... |
| symgextsymg 17844 | The extension of a permuta... |
| symgextres 17845 | The restriction of the ext... |
| gsumccatsymgsn 17846 | Homomorphic property of co... |
| gsmsymgrfixlem1 17847 | Lemma 1 for ~ gsmsymgrfix ... |
| gsmsymgrfix 17848 | The composition of permuta... |
| fvcosymgeq 17849 | The values of two composit... |
| gsmsymgreqlem1 17850 | Lemma 1 for ~ gsmsymgreq .... |
| gsmsymgreqlem2 17851 | Lemma 2 for ~ gsmsymgreq .... |
| gsmsymgreq 17852 | Two combination of permuta... |
| symgfixelq 17853 | A permutation of a set fix... |
| symgfixels 17854 | The restriction of a permu... |
| symgfixelsi 17855 | The restriction of a permu... |
| symgfixf 17856 | The mapping of a permutati... |
| symgfixf1 17857 | The mapping of a permutati... |
| symgfixfolem1 17858 | Lemma 1 for ~ symgfixfo . ... |
| symgfixfo 17859 | The mapping of a permutati... |
| symgfixf1o 17860 | The mapping of a permutati... |
| f1omvdmvd 17863 | A permutation of any class... |
| f1omvdcnv 17864 | A permutation and its inve... |
| mvdco 17865 | Composing two permutations... |
| f1omvdconj 17866 | Conjugation of a permutati... |
| f1otrspeq 17867 | A transposition is charact... |
| f1omvdco2 17868 | If exactly one of two perm... |
| f1omvdco3 17869 | If a point is moved by exa... |
| pmtrfval 17870 | The function generating tr... |
| pmtrval 17871 | A generated transposition,... |
| pmtrfv 17872 | General value of mapping a... |
| pmtrprfv 17873 | In a transposition of two ... |
| pmtrprfv3 17874 | In a transposition of two ... |
| pmtrf 17875 | Functionality of a transpo... |
| pmtrmvd 17876 | A transposition moves prec... |
| pmtrrn 17877 | Transposing two points giv... |
| pmtrfrn 17878 | A transposition (as a kind... |
| pmtrffv 17879 | Mapping of a point under a... |
| pmtrrn2 17880 | For any transposition ther... |
| pmtrfinv 17881 | A transposition function i... |
| pmtrfmvdn0 17882 | A transposition moves at l... |
| pmtrff1o 17883 | A transposition function i... |
| pmtrfcnv 17884 | A transposition function i... |
| pmtrfb 17885 | An intrinsic characterizat... |
| pmtrfconj 17886 | Any conjugate of a transpo... |
| symgsssg 17887 | The symmetric group has su... |
| symgfisg 17888 | The symmetric group has a ... |
| symgtrf 17889 | Transpositions are element... |
| symggen 17890 | The span of the transposit... |
| symggen2 17891 | A finite permutation group... |
| symgtrinv 17892 | To invert a permutation re... |
| pmtr3ncomlem1 17893 | Lemma 1 for ~ pmtr3ncom . ... |
| pmtr3ncomlem2 17894 | Lemma 2 for ~ pmtr3ncom . ... |
| pmtr3ncom 17895 | Transpositions over sets w... |
| pmtrdifellem1 17896 | Lemma 1 for ~ pmtrdifel . ... |
| pmtrdifellem2 17897 | Lemma 2 for ~ pmtrdifel . ... |
| pmtrdifellem3 17898 | Lemma 3 for ~ pmtrdifel . ... |
| pmtrdifellem4 17899 | Lemma 4 for ~ pmtrdifel . ... |
| pmtrdifel 17900 | A transposition of element... |
| pmtrdifwrdellem1 17901 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem2 17902 | Lemma 2 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem3 17903 | Lemma 3 for ~ pmtrdifwrdel... |
| pmtrdifwrdel2lem1 17904 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdel 17905 | A sequence of transpositio... |
| pmtrdifwrdel2 17906 | A sequence of transpositio... |
| pmtrprfval 17907 | The transpositions on a pa... |
| pmtrprfvalrn 17908 | The range of the transposi... |
| psgnunilem1 17913 | Lemma for ~ psgnuni . Giv... |
| psgnunilem5 17914 | Lemma for ~ psgnuni . It ... |
| psgnunilem2 17915 | Lemma for ~ psgnuni . Ind... |
| psgnunilem3 17916 | Lemma for ~ psgnuni . Any... |
| psgnunilem4 17917 | Lemma for ~ psgnuni . An ... |
| m1expaddsub 17918 | Addition and subtraction o... |
| psgnuni 17919 | If the same permutation ca... |
| psgnfval 17920 | Function definition of the... |
| psgnfn 17921 | Functionality and domain o... |
| psgndmsubg 17922 | The finitary permutations ... |
| psgneldm 17923 | Property of being a finita... |
| psgneldm2 17924 | The finitary permutations ... |
| psgneldm2i 17925 | A sequence of transpositio... |
| psgneu 17926 | A finitary permutation has... |
| psgnval 17927 | Value of the permutation s... |
| psgnvali 17928 | A finitary permutation has... |
| psgnvalii 17929 | Any representation of a pe... |
| psgnpmtr 17930 | All transpositions are odd... |
| psgn0fv0 17931 | The permutation sign funct... |
| sygbasnfpfi 17932 | The class of non-fixed poi... |
| psgnfvalfi 17933 | Function definition of the... |
| psgnvalfi 17934 | Value of the permutation s... |
| psgnran 17935 | The range of the permutati... |
| gsmtrcl 17936 | The group sum of transposi... |
| psgnfitr 17937 | A permutation of a finite ... |
| psgnfieu 17938 | A permutation of a finite ... |
| pmtrsn 17939 | The value of the transposi... |
| psgnsn 17940 | The permutation sign funct... |
| psgnprfval 17941 | The permutation sign funct... |
| psgnprfval1 17942 | The permutation sign of th... |
| psgnprfval2 17943 | The permutation sign of th... |
| odfval 17952 | Value of the order functio... |
| odval 17953 | Second substitution for th... |
| odlem1 17954 | The group element order is... |
| odcl 17955 | The order of a group eleme... |
| odf 17956 | Functionality of the group... |
| odid 17957 | Any element to the power o... |
| odlem2 17958 | Any positive annihilator o... |
| odmodnn0 17959 | Reduce the argument of a g... |
| mndodconglem 17960 | Lemma for ~ mndodcong . (... |
| mndodcong 17961 | If two multipliers are con... |
| mndodcongi 17962 | If two multipliers are con... |
| oddvdsnn0 17963 | The only multiples of ` A ... |
| odnncl 17964 | If a nonzero multiple of a... |
| odmod 17965 | Reduce the argument of a g... |
| oddvds 17966 | The only multiples of ` A ... |
| oddvdsi 17967 | Any group element is annih... |
| odcong 17968 | If two multipliers are con... |
| odeq 17969 | The ~ oddvds property uniq... |
| odval2 17970 | A non-conditional definiti... |
| odmulgid 17971 | A relationship between the... |
| odmulg2 17972 | The order of a multiple di... |
| odmulg 17973 | Relationship between the o... |
| odmulgeq 17974 | A multiple of a point of f... |
| odbezout 17975 | If ` N ` is coprime to the... |
| od1 17976 | The order of the group ide... |
| odeq1 17977 | The group identity is the ... |
| odinv 17978 | The order of the inverse o... |
| odf1 17979 | The multiples of an elemen... |
| odinf 17980 | The multiples of an elemen... |
| dfod2 17981 | An alternative definition ... |
| odcl2 17982 | The order of an element of... |
| oddvds2 17983 | The order of an element of... |
| submod 17984 | The order of an element is... |
| subgod 17985 | The order of an element is... |
| odsubdvds 17986 | The order of an element of... |
| odf1o1 17987 | An element with zero order... |
| odf1o2 17988 | An element with nonzero or... |
| odhash 17989 | An element of zero order g... |
| odhash2 17990 | If an element has nonzero ... |
| odhash3 17991 | An element which generates... |
| odngen 17992 | A cyclic subgroup of size ... |
| gexval 17993 | Value of the exponent of a... |
| gexlem1 17994 | The group element order is... |
| gexcl 17995 | The exponent of a group is... |
| gexid 17996 | Any element to the power o... |
| gexlem2 17997 | Any positive annihilator o... |
| gexdvdsi 17998 | Any group element is annih... |
| gexdvds 17999 | The only ` N ` that annihi... |
| gexdvds2 18000 | An integer divides the gro... |
| gexod 18001 | Any group element is annih... |
| gexcl3 18002 | If the order of every grou... |
| gexnnod 18003 | Every group element has fi... |
| gexcl2 18004 | The exponent of a finite g... |
| gexdvds3 18005 | The exponent of a finite g... |
| gex1 18006 | A group or monoid has expo... |
| ispgp 18007 | A group is a ` P ` -group ... |
| pgpprm 18008 | Reverse closure for the fi... |
| pgpgrp 18009 | Reverse closure for the se... |
| pgpfi1 18010 | A finite group with order ... |
| pgp0 18011 | The identity subgroup is a... |
| subgpgp 18012 | A subgroup of a p-group is... |
| sylow1lem1 18013 | Lemma for ~ sylow1 . The ... |
| sylow1lem2 18014 | Lemma for ~ sylow1 . The ... |
| sylow1lem3 18015 | Lemma for ~ sylow1 . One ... |
| sylow1lem4 18016 | Lemma for ~ sylow1 . The ... |
| sylow1lem5 18017 | Lemma for ~ sylow1 . Usin... |
| sylow1 18018 | Sylow's first theorem. If... |
| odcau 18019 | Cauchy's theorem for the o... |
| pgpfi 18020 | The converse to ~ pgpfi1 .... |
| pgpfi2 18021 | Alternate version of ~ pgp... |
| pgphash 18022 | The order of a p-group. (... |
| isslw 18023 | The property of being a Sy... |
| slwprm 18024 | Reverse closure for the fi... |
| slwsubg 18025 | A Sylow ` P ` -subgroup is... |
| slwispgp 18026 | Defining property of a Syl... |
| slwpss 18027 | A proper superset of a Syl... |
| slwpgp 18028 | A Sylow ` P ` -subgroup is... |
| pgpssslw 18029 | Every ` P ` -subgroup is c... |
| slwn0 18030 | Every finite group contain... |
| subgslw 18031 | A Sylow subgroup that is c... |
| sylow2alem1 18032 | Lemma for ~ sylow2a . An ... |
| sylow2alem2 18033 | Lemma for ~ sylow2a . All... |
| sylow2a 18034 | A named lemma of Sylow's s... |
| sylow2blem1 18035 | Lemma for ~ sylow2b . Eva... |
| sylow2blem2 18036 | Lemma for ~ sylow2b . Lef... |
| sylow2blem3 18037 | Sylow's second theorem. P... |
| sylow2b 18038 | Sylow's second theorem. A... |
| slwhash 18039 | A sylow subgroup has cardi... |
| fislw 18040 | The sylow subgroups of a f... |
| sylow2 18041 | Sylow's second theorem. S... |
| sylow3lem1 18042 | Lemma for ~ sylow3 , first... |
| sylow3lem2 18043 | Lemma for ~ sylow3 , first... |
| sylow3lem3 18044 | Lemma for ~ sylow3 , first... |
| sylow3lem4 18045 | Lemma for ~ sylow3 , first... |
| sylow3lem5 18046 | Lemma for ~ sylow3 , secon... |
| sylow3lem6 18047 | Lemma for ~ sylow3 , secon... |
| sylow3 18048 | Sylow's third theorem. Th... |
| lsmfval 18053 | The subgroup sum function ... |
| lsmvalx 18054 | Subspace sum value (for a ... |
| lsmelvalx 18055 | Subspace sum membership (f... |
| lsmelvalix 18056 | Subspace sum membership (f... |
| oppglsm 18057 | The subspace sum operation... |
| lsmssv 18058 | Subgroup sum is a subset o... |
| lsmless1x 18059 | Subset implies subgroup su... |
| lsmless2x 18060 | Subset implies subgroup su... |
| lsmub1x 18061 | Subgroup sum is an upper b... |
| lsmub2x 18062 | Subgroup sum is an upper b... |
| lsmval 18063 | Subgroup sum value (for a ... |
| lsmelval 18064 | Subgroup sum membership (f... |
| lsmelvali 18065 | Subgroup sum membership (f... |
| lsmelvalm 18066 | Subgroup sum membership an... |
| lsmelvalmi 18067 | Membership of vector subtr... |
| lsmsubm 18068 | The sum of two commuting s... |
| lsmsubg 18069 | The sum of two commuting s... |
| lsmcom2 18070 | Subgroup sum commutes. (C... |
| lsmub1 18071 | Subgroup sum is an upper b... |
| lsmub2 18072 | Subgroup sum is an upper b... |
| lsmunss 18073 | Union of subgroups is a su... |
| lsmless1 18074 | Subset implies subgroup su... |
| lsmless2 18075 | Subset implies subgroup su... |
| lsmless12 18076 | Subset implies subgroup su... |
| lsmidm 18077 | Subgroup sum is idempotent... |
| lsmlub 18078 | The least upper bound prop... |
| lsmss1 18079 | Subgroup sum with a subset... |
| lsmss1b 18080 | Subgroup sum with a subset... |
| lsmss2 18081 | Subgroup sum with a subset... |
| lsmss2b 18082 | Subgroup sum with a subset... |
| lsmass 18083 | Subgroup sum is associativ... |
| lsm01 18084 | Subgroup sum with the zero... |
| lsm02 18085 | Subgroup sum with the zero... |
| subglsm 18086 | The subgroup sum evaluated... |
| lssnle 18087 | Equivalent expressions for... |
| lsmmod 18088 | The modular law holds for ... |
| lsmmod2 18089 | Modular law dual for subgr... |
| lsmpropd 18090 | If two structures have the... |
| cntzrecd 18091 | Commute the "subgroups com... |
| lsmcntz 18092 | The "subgroups commute" pr... |
| lsmcntzr 18093 | The "subgroups commute" pr... |
| lsmdisj 18094 | Disjointness from a subgro... |
| lsmdisj2 18095 | Association of the disjoin... |
| lsmdisj3 18096 | Association of the disjoin... |
| lsmdisjr 18097 | Disjointness from a subgro... |
| lsmdisj2r 18098 | Association of the disjoin... |
| lsmdisj3r 18099 | Association of the disjoin... |
| lsmdisj2a 18100 | Association of the disjoin... |
| lsmdisj2b 18101 | Association of the disjoin... |
| lsmdisj3a 18102 | Association of the disjoin... |
| lsmdisj3b 18103 | Association of the disjoin... |
| subgdisj1 18104 | Vectors belonging to disjo... |
| subgdisj2 18105 | Vectors belonging to disjo... |
| subgdisjb 18106 | Vectors belonging to disjo... |
| pj1fval 18107 | The left projection functi... |
| pj1val 18108 | The left projection functi... |
| pj1eu 18109 | Uniqueness of a left proje... |
| pj1f 18110 | The left projection functi... |
| pj2f 18111 | The right projection funct... |
| pj1id 18112 | Any element of a direct su... |
| pj1eq 18113 | Any element of a direct su... |
| pj1lid 18114 | The left projection functi... |
| pj1rid 18115 | The left projection functi... |
| pj1ghm 18116 | The left projection functi... |
| pj1ghm2 18117 | The left projection functi... |
| lsmhash 18118 | The order of the direct pr... |
| efgmval 18125 | Value of the formal invers... |
| efgmf 18126 | The formal inverse operati... |
| efgmnvl 18127 | The inversion function on ... |
| efgrcl 18128 | Lemma for ~ efgval . (Con... |
| efglem 18129 | Lemma for ~ efgval . (Con... |
| efgval 18130 | Value of the free group co... |
| efger 18131 | Value of the free group co... |
| efgi 18132 | Value of the free group co... |
| efgi0 18133 | Value of the free group co... |
| efgi1 18134 | Value of the free group co... |
| efgtf 18135 | Value of the free group co... |
| efgtval 18136 | Value of the extension fun... |
| efgval2 18137 | Value of the free group co... |
| efgi2 18138 | Value of the free group co... |
| efgtlen 18139 | Value of the free group co... |
| efginvrel2 18140 | The inverse of the reverse... |
| efginvrel1 18141 | The inverse of the reverse... |
| efgsf 18142 | Value of the auxiliary fun... |
| efgsdm 18143 | Elementhood in the domain ... |
| efgsval 18144 | Value of the auxiliary fun... |
| efgsdmi 18145 | Property of the last link ... |
| efgsval2 18146 | Value of the auxiliary fun... |
| efgsrel 18147 | The start and end of any e... |
| efgs1 18148 | A singleton of an irreduci... |
| efgs1b 18149 | Every extension sequence e... |
| efgsp1 18150 | If ` F ` is an extension s... |
| efgsres 18151 | An initial segment of an e... |
| efgsfo 18152 | For any word, there is a s... |
| efgredlema 18153 | The reduced word that form... |
| efgredlemf 18154 | Lemma for ~ efgredleme . ... |
| efgredlemg 18155 | Lemma for ~ efgred . (Con... |
| efgredleme 18156 | Lemma for ~ efgred . (Con... |
| efgredlemd 18157 | The reduced word that form... |
| efgredlemc 18158 | The reduced word that form... |
| efgredlemb 18159 | The reduced word that form... |
| efgredlem 18160 | The reduced word that form... |
| efgred 18161 | The reduced word that form... |
| efgrelexlema 18162 | If two words ` A , B ` are... |
| efgrelexlemb 18163 | If two words ` A , B ` are... |
| efgrelex 18164 | If two words ` A , B ` are... |
| efgredeu 18165 | There is a unique reduced ... |
| efgred2 18166 | Two extension sequences ha... |
| efgcpbllema 18167 | Lemma for ~ efgrelex . De... |
| efgcpbllemb 18168 | Lemma for ~ efgrelex . Sh... |
| efgcpbl 18169 | Two extension sequences ha... |
| efgcpbl2 18170 | Two extension sequences ha... |
| frgpval 18171 | Value of the free group co... |
| frgpcpbl 18172 | Compatibility of the group... |
| frgp0 18173 | The free group is a group.... |
| frgpeccl 18174 | Closure of the quotient ma... |
| frgpgrp 18175 | The free group is a group.... |
| frgpadd 18176 | Addition in the free group... |
| frgpinv 18177 | The inverse of an element ... |
| frgpmhm 18178 | The "natural map" from wor... |
| vrgpfval 18179 | The canonical injection fr... |
| vrgpval 18180 | The value of the generatin... |
| vrgpf 18181 | The mapping from the index... |
| vrgpinv 18182 | The inverse of a generatin... |
| frgpuptf 18183 | Any assignment of the gene... |
| frgpuptinv 18184 | Any assignment of the gene... |
| frgpuplem 18185 | Any assignment of the gene... |
| frgpupf 18186 | Any assignment of the gene... |
| frgpupval 18187 | Any assignment of the gene... |
| frgpup1 18188 | Any assignment of the gene... |
| frgpup2 18189 | The evaluation map has the... |
| frgpup3lem 18190 | The evaluation map has the... |
| frgpup3 18191 | Universal property of the ... |
| 0frgp 18192 | The free group on zero gen... |
| isabl 18197 | The predicate "is an Abeli... |
| ablgrp 18198 | An Abelian group is a grou... |
| ablcmn 18199 | An Abelian group is a comm... |
| iscmn 18200 | The predicate "is a commut... |
| isabl2 18201 | The predicate "is an Abeli... |
| cmnpropd 18202 | If two structures have the... |
| ablpropd 18203 | If two structures have the... |
| ablprop 18204 | If two structures have the... |
| iscmnd 18205 | Properties that determine ... |
| isabld 18206 | Properties that determine ... |
| isabli 18207 | Properties that determine ... |
| cmnmnd 18208 | A commutative monoid is a ... |
| cmncom 18209 | A commutative monoid is co... |
| ablcom 18210 | An Abelian group operation... |
| cmn32 18211 | Commutative/associative la... |
| cmn4 18212 | Commutative/associative la... |
| cmn12 18213 | Commutative/associative la... |
| abl32 18214 | Commutative/associative la... |
| ablinvadd 18215 | The inverse of an Abelian ... |
| ablsub2inv 18216 | Abelian group subtraction ... |
| ablsubadd 18217 | Relationship between Abeli... |
| ablsub4 18218 | Commutative/associative su... |
| abladdsub4 18219 | Abelian group addition/sub... |
| abladdsub 18220 | Associative-type law for g... |
| ablpncan2 18221 | Cancellation law for subtr... |
| ablpncan3 18222 | A cancellation law for com... |
| ablsubsub 18223 | Law for double subtraction... |
| ablsubsub4 18224 | Law for double subtraction... |
| ablpnpcan 18225 | Cancellation law for mixed... |
| ablnncan 18226 | Cancellation law for group... |
| ablsub32 18227 | Swap the second and third ... |
| ablnnncan 18228 | Cancellation law for group... |
| ablnnncan1 18229 | Cancellation law for group... |
| ablsubsub23 18230 | Swap subtrahend and result... |
| mulgnn0di 18231 | Group multiple of a sum, f... |
| mulgdi 18232 | Group multiple of a sum. ... |
| mulgmhm 18233 | The map from ` x ` to ` n ... |
| mulgghm 18234 | The map from ` x ` to ` n ... |
| mulgsubdi 18235 | Group multiple of a differ... |
| ghmfghm 18236 | The function fulfilling th... |
| ghmcmn 18237 | The image of a commutative... |
| ghmabl 18238 | The image of an abelian gr... |
| invghm 18239 | The inversion map is a gro... |
| eqgabl 18240 | Value of the subgroup cose... |
| subgabl 18241 | A subgroup of an abelian g... |
| subcmn 18242 | A submonoid of a commutati... |
| submcmn 18243 | A submonoid of a commutati... |
| submcmn2 18244 | A submonoid is commutative... |
| cntzcmn 18245 | The centralizer of any sub... |
| cntzcmnss 18246 | Any subset in a commutativ... |
| cntzspan 18247 | If the generators commute,... |
| cntzcmnf 18248 | Discharge the centralizer ... |
| ghmplusg 18249 | The pointwise sum of two l... |
| ablnsg 18250 | Every subgroup of an abeli... |
| odadd1 18251 | The order of a product in ... |
| odadd2 18252 | The order of a product in ... |
| odadd 18253 | The order of a product is ... |
| gex2abl 18254 | A group with exponent 2 (o... |
| gexexlem 18255 | Lemma for ~ gexex . (Cont... |
| gexex 18256 | In an abelian group with f... |
| torsubg 18257 | The set of all elements of... |
| oddvdssubg 18258 | The set of all elements wh... |
| lsmcomx 18259 | Subgroup sum commutes (ext... |
| ablcntzd 18260 | All subgroups in an abelia... |
| lsmcom 18261 | Subgroup sum commutes. (C... |
| lsmsubg2 18262 | The sum of two subgroups i... |
| lsm4 18263 | Commutative/associative la... |
| prdscmnd 18264 | The product of a family of... |
| prdsabld 18265 | The product of a family of... |
| pwscmn 18266 | The structure power on a c... |
| pwsabl 18267 | The structure power on an ... |
| qusabl 18268 | If ` Y ` is a subgroup of ... |
| abl1 18269 | The (smallest) structure r... |
| abln0 18270 | Abelian groups (and theref... |
| cnaddablx 18271 | The complex numbers are an... |
| cnaddabl 18272 | The complex numbers are an... |
| cnaddid 18273 | The group identity element... |
| cnaddinv 18274 | Value of the group inverse... |
| zaddablx 18275 | The integers are an Abelia... |
| frgpnabllem1 18276 | Lemma for ~ frgpnabl . (C... |
| frgpnabllem2 18277 | Lemma for ~ frgpnabl . (C... |
| frgpnabl 18278 | The free group on two or m... |
| iscyg 18281 | Definition of a cyclic gro... |
| iscyggen 18282 | The property of being a cy... |
| iscyggen2 18283 | The property of being a cy... |
| iscyg2 18284 | A cyclic group is a group ... |
| cyggeninv 18285 | The inverse of a cyclic ge... |
| cyggenod 18286 | An element is the generato... |
| cyggenod2 18287 | In an infinite cyclic grou... |
| iscyg3 18288 | Definition of a cyclic gro... |
| iscygd 18289 | Definition of a cyclic gro... |
| iscygodd 18290 | Show that a group with an ... |
| cyggrp 18291 | A cyclic group is a group.... |
| cygabl 18292 | A cyclic group is abelian.... |
| cygctb 18293 | A cyclic group is countabl... |
| 0cyg 18294 | The trivial group is cycli... |
| prmcyg 18295 | A group with prime order i... |
| lt6abl 18296 | A group with fewer than ` ... |
| ghmcyg 18297 | The image of a cyclic grou... |
| cyggex2 18298 | The exponent of a cyclic g... |
| cyggex 18299 | The exponent of a finite c... |
| cyggexb 18300 | A finite abelian group is ... |
| giccyg 18301 | Cyclicity is a group prope... |
| cycsubgcyg 18302 | The cyclic subgroup genera... |
| cycsubgcyg2 18303 | The cyclic subgroup genera... |
| gsumval3a 18304 | Value of the group sum ope... |
| gsumval3eu 18305 | The group sum as defined i... |
| gsumval3lem1 18306 | Lemma 1 for ~ gsumval3 . ... |
| gsumval3lem2 18307 | Lemma 2 for ~ gsumval3 . ... |
| gsumval3 18308 | Value of the group sum ope... |
| gsumcllem 18309 | Lemma for ~ gsumcl and rel... |
| gsumzres 18310 | Extend a finite group sum ... |
| gsumzcl2 18311 | Closure of a finite group ... |
| gsumzcl 18312 | Closure of a finite group ... |
| gsumzf1o 18313 | Re-index a finite group su... |
| gsumres 18314 | Extend a finite group sum ... |
| gsumcl2 18315 | Closure of a finite group ... |
| gsumcl 18316 | Closure of a finite group ... |
| gsumf1o 18317 | Re-index a finite group su... |
| gsumzsubmcl 18318 | Closure of a group sum in ... |
| gsumsubmcl 18319 | Closure of a group sum in ... |
| gsumsubgcl 18320 | Closure of a group sum in ... |
| gsumzaddlem 18321 | The sum of two group sums.... |
| gsumzadd 18322 | The sum of two group sums.... |
| gsumadd 18323 | The sum of two group sums.... |
| gsummptfsadd 18324 | The sum of two group sums ... |
| gsummptfidmadd 18325 | The sum of two group sums ... |
| gsummptfidmadd2 18326 | The sum of two group sums ... |
| gsumzsplit 18327 | Split a group sum into two... |
| gsumsplit 18328 | Split a group sum into two... |
| gsumsplit2 18329 | Split a group sum into two... |
| gsummptfidmsplit 18330 | Split a group sum expresse... |
| gsummptfidmsplitres 18331 | Split a group sum expresse... |
| gsummptfzsplit 18332 | Split a group sum expresse... |
| gsummptfzsplitl 18333 | Split a group sum expresse... |
| gsumconst 18334 | Sum of a constant series. ... |
| gsumconstf 18335 | Sum of a constant series. ... |
| gsummptshft 18336 | Index shift of a finite gr... |
| gsumzmhm 18337 | Apply a group homomorphism... |
| gsummhm 18338 | Apply a group homomorphism... |
| gsummhm2 18339 | Apply a group homomorphism... |
| gsummptmhm 18340 | Apply a group homomorphism... |
| gsummulglem 18341 | Lemma for ~ gsummulg and ~... |
| gsummulg 18342 | Nonnegative multiple of a ... |
| gsummulgz 18343 | Integer multiple of a grou... |
| gsumzoppg 18344 | The opposite of a group su... |
| gsumzinv 18345 | Inverse of a group sum. (... |
| gsuminv 18346 | Inverse of a group sum. (... |
| gsummptfidminv 18347 | Inverse of a group sum exp... |
| gsumsub 18348 | The difference of two grou... |
| gsummptfssub 18349 | The difference of two grou... |
| gsummptfidmsub 18350 | The difference of two grou... |
| gsumsnfd 18351 | Group sum of a singleton, ... |
| gsumsnd 18352 | Group sum of a singleton, ... |
| gsumsnf 18353 | Group sum of a singleton, ... |
| gsumsn 18354 | Group sum of a singleton. ... |
| gsumzunsnd 18355 | Append an element to a fin... |
| gsumunsnfd 18356 | Append an element to a fin... |
| gsumunsnd 18357 | Append an element to a fin... |
| gsumunsnf 18358 | Append an element to a fin... |
| gsumunsn 18359 | Append an element to a fin... |
| gsumdifsnd 18360 | Extract a summand from a f... |
| gsumpt 18361 | Sum of a family that is no... |
| gsummptf1o 18362 | Re-index a finite group su... |
| gsummptun 18363 | Group sum of a disjoint un... |
| gsummpt1n0 18364 | If only one summand in a f... |
| gsummptif1n0 18365 | If only one summand in a f... |
| gsummptcl 18366 | Closure of a finite group ... |
| gsummptfif1o 18367 | Re-index a finite group su... |
| gsummptfzcl 18368 | Closure of a finite group ... |
| gsum2dlem1 18369 | Lemma 1 for ~ gsum2d . (C... |
| gsum2dlem2 18370 | Lemma for ~ gsum2d . (Con... |
| gsum2d 18371 | Write a sum over a two-dim... |
| gsum2d2lem 18372 | Lemma for ~ gsum2d2 : show... |
| gsum2d2 18373 | Write a group sum over a t... |
| gsumcom2 18374 | Two-dimensional commutatio... |
| gsumxp 18375 | Write a group sum over a c... |
| gsumcom 18376 | Commute the arguments of a... |
| prdsgsum 18377 | Finite commutative sums in... |
| pwsgsum 18378 | Finite commutative sums in... |
| fsfnn0gsumfsffz 18379 | Replacing a finitely suppo... |
| nn0gsumfz 18380 | Replacing a finitely suppo... |
| nn0gsumfz0 18381 | Replacing a finitely suppo... |
| gsummptnn0fz 18382 | A final group sum over a f... |
| gsummptnn0fzv 18383 | A final group sum over a f... |
| gsummptnn0fzfv 18384 | A final group sum over a f... |
| telgsumfzslem 18385 | Lemma for ~ telgsumfzs (in... |
| telgsumfzs 18386 | Telescoping group sum rang... |
| telgsumfz 18387 | Telescoping group sum rang... |
| telgsumfz0s 18388 | Telescoping finite group s... |
| telgsumfz0 18389 | Telescoping finite group s... |
| telgsums 18390 | Telescoping finitely suppo... |
| telgsum 18391 | Telescoping finitely suppo... |
| reldmdprd 18396 | The domain of the internal... |
| dmdprd 18397 | The domain of definition o... |
| dmdprdd 18398 | Show that a given family i... |
| dprddomprc 18399 | A family of subgroups inde... |
| dprddomcld 18400 | If a family of subgroups i... |
| dprdval0prc 18401 | The internal direct produc... |
| dprdval 18402 | The value of the internal ... |
| eldprd 18403 | A class ` A ` is an intern... |
| dprdgrp 18404 | Reverse closure for the in... |
| dprdf 18405 | The function ` S ` is a fa... |
| dprdf2 18406 | The function ` S ` is a fa... |
| dprdcntz 18407 | The function ` S ` is a fa... |
| dprddisj 18408 | The function ` S ` is a fa... |
| dprdw 18409 | The property of being a fi... |
| dprdwd 18410 | A mapping being a finitely... |
| dprdff 18411 | A finitely supported funct... |
| dprdfcl 18412 | A finitely supported funct... |
| dprdffsupp 18413 | A finitely supported funct... |
| dprdfcntz 18414 | A function on the elements... |
| dprdssv 18415 | The internal direct produc... |
| dprdfid 18416 | A function mapping all but... |
| eldprdi 18417 | The domain of definition o... |
| dprdfinv 18418 | Take the inverse of a grou... |
| dprdfadd 18419 | Take the sum of group sums... |
| dprdfsub 18420 | Take the difference of gro... |
| dprdfeq0 18421 | The zero function is the o... |
| dprdf11 18422 | Two group sums over a dire... |
| dprdsubg 18423 | The internal direct produc... |
| dprdub 18424 | Each factor is a subset of... |
| dprdlub 18425 | The direct product is smal... |
| dprdspan 18426 | The direct product is the ... |
| dprdres 18427 | Restriction of a direct pr... |
| dprdss 18428 | Create a direct product by... |
| dprdz 18429 | A family consisting entire... |
| dprd0 18430 | The empty family is an int... |
| dprdf1o 18431 | Rearrange the index set of... |
| dprdf1 18432 | Rearrange the index set of... |
| subgdmdprd 18433 | A direct product in a subg... |
| subgdprd 18434 | A direct product in a subg... |
| dprdsn 18435 | A singleton family is an i... |
| dmdprdsplitlem 18436 | Lemma for ~ dmdprdsplit . ... |
| dprdcntz2 18437 | The function ` S ` is a fa... |
| dprddisj2 18438 | The function ` S ` is a fa... |
| dprd2dlem2 18439 | The direct product of a co... |
| dprd2dlem1 18440 | The direct product of a co... |
| dprd2da 18441 | The direct product of a co... |
| dprd2db 18442 | The direct product of a co... |
| dprd2d2 18443 | The direct product of a co... |
| dmdprdsplit2lem 18444 | Lemma for ~ dmdprdsplit . ... |
| dmdprdsplit2 18445 | The direct product splits ... |
| dmdprdsplit 18446 | The direct product splits ... |
| dprdsplit 18447 | The direct product is the ... |
| dmdprdpr 18448 | A singleton family is an i... |
| dprdpr 18449 | A singleton family is an i... |
| dpjlem 18450 | Lemma for theorems about d... |
| dpjcntz 18451 | The two subgroups that app... |
| dpjdisj 18452 | The two subgroups that app... |
| dpjlsm 18453 | The two subgroups that app... |
| dpjfval 18454 | Value of the direct produc... |
| dpjval 18455 | Value of the direct produc... |
| dpjf 18456 | The ` X ` -th index projec... |
| dpjidcl 18457 | The key property of projec... |
| dpjeq 18458 | Decompose a group sum into... |
| dpjid 18459 | The key property of projec... |
| dpjlid 18460 | The ` X ` -th index projec... |
| dpjrid 18461 | The ` Y ` -th index projec... |
| dpjghm 18462 | The direct product is the ... |
| dpjghm2 18463 | The direct product is the ... |
| ablfacrplem 18464 | Lemma for ~ ablfacrp2 . (... |
| ablfacrp 18465 | A finite abelian group who... |
| ablfacrp2 18466 | The factors ` K , L ` of ~... |
| ablfac1lem 18467 | Lemma for ~ ablfac1b . Sa... |
| ablfac1a 18468 | The factors of ~ ablfac1b ... |
| ablfac1b 18469 | Any abelian group is the d... |
| ablfac1c 18470 | The factors of ~ ablfac1b ... |
| ablfac1eulem 18471 | Lemma for ~ ablfac1eu . (... |
| ablfac1eu 18472 | The factorization of ~ abl... |
| pgpfac1lem1 18473 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem2 18474 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3a 18475 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3 18476 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem4 18477 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem5 18478 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1 18479 | Factorization of a finite ... |
| pgpfaclem1 18480 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem2 18481 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem3 18482 | Lemma for ~ pgpfac . (Con... |
| pgpfac 18483 | Full factorization of a fi... |
| ablfaclem1 18484 | Lemma for ~ ablfac . (Con... |
| ablfaclem2 18485 | Lemma for ~ ablfac . (Con... |
| ablfaclem3 18486 | Lemma for ~ ablfac . (Con... |
| ablfac 18487 | The Fundamental Theorem of... |
| ablfac2 18488 | Choose generators for each... |
| fnmgp 18491 | The multiplicative group o... |
| mgpval 18492 | Value of the multiplicatio... |
| mgpplusg 18493 | Value of the group operati... |
| mgplem 18494 | Lemma for ~ mgpbas . (Con... |
| mgpbas 18495 | Base set of the multiplica... |
| mgpsca 18496 | The multiplication monoid ... |
| mgptset 18497 | Topology component of the ... |
| mgptopn 18498 | Topology of the multiplica... |
| mgpds 18499 | Distance function of the m... |
| mgpress 18500 | Subgroup commutes with the... |
| ringidval 18503 | The value of the unity ele... |
| dfur2 18504 | The multiplicative identit... |
| issrg 18507 | The predicate "is a semiri... |
| srgcmn 18508 | A semiring is a commutativ... |
| srgmnd 18509 | A semiring is a monoid. (... |
| srgmgp 18510 | A semiring is a monoid und... |
| srgi 18511 | Properties of a semiring. ... |
| srgcl 18512 | Closure of the multiplicat... |
| srgass 18513 | Associative law for the mu... |
| srgideu 18514 | The unit element of a semi... |
| srgfcl 18515 | Functionality of the multi... |
| srgdi 18516 | Distributive law for the m... |
| srgdir 18517 | Distributive law for the m... |
| srgidcl 18518 | The unit element of a semi... |
| srg0cl 18519 | The zero element of a semi... |
| srgidmlem 18520 | Lemma for ~ srglidm and ~ ... |
| srglidm 18521 | The unit element of a semi... |
| srgridm 18522 | The unit element of a semi... |
| issrgid 18523 | Properties showing that an... |
| srgacl 18524 | Closure of the addition op... |
| srgcom 18525 | Commutativity of the addit... |
| srgrz 18526 | The zero of a semiring is ... |
| srglz 18527 | The zero of a semiring is ... |
| srgisid 18528 | In a semiring, the only le... |
| srg1zr 18529 | The only semiring with a b... |
| srgen1zr 18530 | The only semiring with one... |
| srgmulgass 18531 | An associative property be... |
| srgpcomp 18532 | If two elements of a semir... |
| srgpcompp 18533 | If two elements of a semir... |
| srgpcomppsc 18534 | If two elements of a semir... |
| srglmhm 18535 | Left-multiplication in a s... |
| srgrmhm 18536 | Right-multiplication in a ... |
| srgsummulcr 18537 | A finite semiring sum mult... |
| sgsummulcl 18538 | A finite semiring sum mult... |
| srg1expzeq1 18539 | The exponentiation (by a n... |
| srgbinomlem1 18540 | Lemma 1 for ~ srgbinomlem ... |
| srgbinomlem2 18541 | Lemma 2 for ~ srgbinomlem ... |
| srgbinomlem3 18542 | Lemma 3 for ~ srgbinomlem ... |
| srgbinomlem4 18543 | Lemma 4 for ~ srgbinomlem ... |
| srgbinomlem 18544 | Lemma for ~ srgbinom . In... |
| srgbinom 18545 | The binomial theorem for c... |
| csrgbinom 18546 | The binomial theorem for c... |
| isring 18551 | The predicate "is a (unita... |
| ringgrp 18552 | A ring is a group. (Contr... |
| ringmgp 18553 | A ring is a monoid under m... |
| iscrng 18554 | A commutative ring is a ri... |
| crngmgp 18555 | A commutative ring's multi... |
| ringmnd 18556 | A ring is a monoid under a... |
| ringmgm 18557 | A ring is a magma. (Contr... |
| crngring 18558 | A commutative ring is a ri... |
| mgpf 18559 | Restricted functionality o... |
| ringi 18560 | Properties of a unital rin... |
| ringcl 18561 | Closure of the multiplicat... |
| crngcom 18562 | A commutative ring's multi... |
| iscrng2 18563 | A commutative ring is a ri... |
| ringass 18564 | Associative law for the mu... |
| ringideu 18565 | The unit element of a ring... |
| ringdi 18566 | Distributive law for the m... |
| ringdir 18567 | Distributive law for the m... |
| ringidcl 18568 | The unit element of a ring... |
| ring0cl 18569 | The zero element of a ring... |
| ringidmlem 18570 | Lemma for ~ ringlidm and ~... |
| ringlidm 18571 | The unit element of a ring... |
| ringridm 18572 | The unit element of a ring... |
| isringid 18573 | Properties showing that an... |
| ringid 18574 | The multiplication operati... |
| ringadd2 18575 | A ring element plus itself... |
| rngo2times 18576 | A ring element plus itself... |
| ringidss 18577 | A subset of the multiplica... |
| ringacl 18578 | Closure of the addition op... |
| ringcom 18579 | Commutativity of the addit... |
| ringabl 18580 | A ring is an Abelian group... |
| ringcmn 18581 | A ring is a commutative mo... |
| ringpropd 18582 | If two structures have the... |
| crngpropd 18583 | If two structures have the... |
| ringprop 18584 | If two structures have the... |
| isringd 18585 | Properties that determine ... |
| iscrngd 18586 | Properties that determine ... |
| ringlz 18587 | The zero of a unital ring ... |
| ringrz 18588 | The zero of a unital ring ... |
| ringsrg 18589 | Any ring is also a semirin... |
| ring1eq0 18590 | If one and zero are equal,... |
| ring1ne0 18591 | If a ring has at least two... |
| ringinvnz1ne0 18592 | In a unitary ring, a left ... |
| ringinvnzdiv 18593 | In a unitary ring, a left ... |
| ringnegl 18594 | Negation in a ring is the ... |
| rngnegr 18595 | Negation in a ring is the ... |
| ringmneg1 18596 | Negation of a product in a... |
| ringmneg2 18597 | Negation of a product in a... |
| ringm2neg 18598 | Double negation of a produ... |
| ringsubdi 18599 | Ring multiplication distri... |
| rngsubdir 18600 | Ring multiplication distri... |
| mulgass2 18601 | An associative property be... |
| ring1 18602 | The (smallest) structure r... |
| ringn0 18603 | Rings exist. (Contributed... |
| ringlghm 18604 | Left-multiplication in a r... |
| ringrghm 18605 | Right-multiplication in a ... |
| gsummulc1 18606 | A finite ring sum multipli... |
| gsummulc2 18607 | A finite ring sum multipli... |
| gsummgp0 18608 | If one factor in a finite ... |
| gsumdixp 18609 | Distribute a binary produc... |
| prdsmgp 18610 | The multiplicative monoid ... |
| prdsmulrcl 18611 | A structure product of rin... |
| prdsringd 18612 | A product of rings is a ri... |
| prdscrngd 18613 | A product of commutative r... |
| prds1 18614 | Value of the ring unit in ... |
| pwsring 18615 | A structure power of a rin... |
| pws1 18616 | Value of the ring unit in ... |
| pwscrng 18617 | A structure power of a com... |
| pwsmgp 18618 | The multiplicative group o... |
| imasring 18619 | The image structure of a r... |
| qusring2 18620 | The quotient structure of ... |
| crngbinom 18621 | The binomial theorem for c... |
| opprval 18624 | Value of the opposite ring... |
| opprmulfval 18625 | Value of the multiplicatio... |
| opprmul 18626 | Value of the multiplicatio... |
| crngoppr 18627 | In a commutative ring, the... |
| opprlem 18628 | Lemma for ~ opprbas and ~ ... |
| opprbas 18629 | Base set of an opposite ri... |
| oppradd 18630 | Addition operation of an o... |
| opprring 18631 | An opposite ring is a ring... |
| opprringb 18632 | Bidirectional form of ~ op... |
| oppr0 18633 | Additive identity of an op... |
| oppr1 18634 | Multiplicative identity of... |
| opprneg 18635 | The negative function in a... |
| opprsubg 18636 | Being a subgroup is a symm... |
| mulgass3 18637 | An associative property be... |
| reldvdsr 18644 | The divides relation is a ... |
| dvdsrval 18645 | Value of the divides relat... |
| dvdsr 18646 | Value of the divides relat... |
| dvdsr2 18647 | Value of the divides relat... |
| dvdsrmul 18648 | A left-multiple of ` X ` i... |
| dvdsrcl 18649 | Closure of a dividing elem... |
| dvdsrcl2 18650 | Closure of a dividing elem... |
| dvdsrid 18651 | An element in a (unital) r... |
| dvdsrtr 18652 | Divisibility is transitive... |
| dvdsrmul1 18653 | The divisibility relation ... |
| dvdsrneg 18654 | An element divides its neg... |
| dvdsr01 18655 | In a ring, zero is divisib... |
| dvdsr02 18656 | Only zero is divisible by ... |
| isunit 18657 | Property of being a unit o... |
| 1unit 18658 | The multiplicative identit... |
| unitcl 18659 | A unit is an element of th... |
| unitss 18660 | The set of units is contai... |
| opprunit 18661 | Being a unit is a symmetri... |
| crngunit 18662 | Property of being a unit i... |
| dvdsunit 18663 | A divisor of a unit is a u... |
| unitmulcl 18664 | The product of units is a ... |
| unitmulclb 18665 | Reversal of ~ unitmulcl in... |
| unitgrpbas 18666 | The base set of the group ... |
| unitgrp 18667 | The group of units is a gr... |
| unitabl 18668 | The group of units of a co... |
| unitgrpid 18669 | The identity of the multip... |
| unitsubm 18670 | The group of units is a su... |
| invrfval 18673 | Multiplicative inverse fun... |
| unitinvcl 18674 | The inverse of a unit exis... |
| unitinvinv 18675 | The inverse of the inverse... |
| ringinvcl 18676 | The inverse of a unit is a... |
| unitlinv 18677 | A unit times its inverse i... |
| unitrinv 18678 | A unit times its inverse i... |
| 1rinv 18679 | The inverse of the identit... |
| 0unit 18680 | The additive identity is a... |
| unitnegcl 18681 | The negative of a unit is ... |
| dvrfval 18684 | Division operation in a ri... |
| dvrval 18685 | Division operation in a ri... |
| dvrcl 18686 | Closure of division operat... |
| unitdvcl 18687 | The units are closed under... |
| dvrid 18688 | A cancellation law for div... |
| dvr1 18689 | A cancellation law for div... |
| dvrass 18690 | An associative law for div... |
| dvrcan1 18691 | A cancellation law for div... |
| dvrcan3 18692 | A cancellation law for div... |
| dvreq1 18693 | A cancellation law for div... |
| ringinvdv 18694 | Write the inverse function... |
| rngidpropd 18695 | The ring identity depends ... |
| dvdsrpropd 18696 | The divisibility relation ... |
| unitpropd 18697 | The set of units depends o... |
| invrpropd 18698 | The ring inverse function ... |
| isirred 18699 | An irreducible element of ... |
| isnirred 18700 | The property of being a no... |
| isirred2 18701 | Expand out the class diffe... |
| opprirred 18702 | Irreducibility is symmetri... |
| irredn0 18703 | The additive identity is n... |
| irredcl 18704 | An irreducible element is ... |
| irrednu 18705 | An irreducible element is ... |
| irredn1 18706 | The multiplicative identit... |
| irredrmul 18707 | The product of an irreduci... |
| irredlmul 18708 | The product of a unit and ... |
| irredmul 18709 | If product of two elements... |
| irredneg 18710 | The negative of an irreduc... |
| irrednegb 18711 | An element is irreducible ... |
| dfrhm2 18717 | The property of a ring hom... |
| rhmrcl1 18719 | Reverse closure of a ring ... |
| rhmrcl2 18720 | Reverse closure of a ring ... |
| isrhm 18721 | A function is a ring homom... |
| rhmmhm 18722 | A ring homomorphism is a h... |
| isrim0 18723 | An isomorphism of rings is... |
| rimrcl 18724 | Reverse closure for an iso... |
| rhmghm 18725 | A ring homomorphism is an ... |
| rhmf 18726 | A ring homomorphism is a f... |
| rhmmul 18727 | A homomorphism of rings pr... |
| isrhm2d 18728 | Demonstration of ring homo... |
| isrhmd 18729 | Demonstration of ring homo... |
| rhm1 18730 | Ring homomorphisms are req... |
| idrhm 18731 | The identity homomorphism ... |
| rhmf1o 18732 | A ring homomorphism is bij... |
| isrim 18733 | An isomorphism of rings is... |
| rimf1o 18734 | An isomorphism of rings is... |
| rimrhm 18735 | An isomorphism of rings is... |
| rimgim 18736 | An isomorphism of rings is... |
| rhmco 18737 | The composition of ring ho... |
| pwsco1rhm 18738 | Right composition with a f... |
| pwsco2rhm 18739 | Left composition with a ri... |
| f1rhm0to0 18740 | If a ring homomorphism ` F... |
| f1rhm0to0ALT 18741 | Alternate proof for ~ f1rh... |
| rim0to0 18742 | A ring isomorphism maps th... |
| kerf1hrm 18743 | A ring homomorphism ` F ` ... |
| brric 18744 | The relation "is isomorphi... |
| brric2 18745 | The relation "is isomorphi... |
| ricgic 18746 | If two rings are (ring) is... |
| isdrng 18751 | The predicate "is a divisi... |
| drngunit 18752 | Elementhood in the set of ... |
| drngui 18753 | The set of units of a divi... |
| drngring 18754 | A division ring is a ring.... |
| drnggrp 18755 | A division ring is a group... |
| isfld 18756 | A field is a commutative d... |
| isdrng2 18757 | A division ring can equiva... |
| drngprop 18758 | If two structures have the... |
| drngmgp 18759 | A division ring contains a... |
| drngmcl 18760 | The product of two nonzero... |
| drngid 18761 | A division ring's unit is ... |
| drngunz 18762 | A division ring's unit is ... |
| drngid2 18763 | Properties showing that an... |
| drnginvrcl 18764 | Closure of the multiplicat... |
| drnginvrn0 18765 | The multiplicative inverse... |
| drnginvrl 18766 | Property of the multiplica... |
| drnginvrr 18767 | Property of the multiplica... |
| drngmul0or 18768 | A product is zero iff one ... |
| drngmulne0 18769 | A product is nonzero iff b... |
| drngmuleq0 18770 | An element is zero iff its... |
| opprdrng 18771 | The opposite of a division... |
| isdrngd 18772 | Properties that determine ... |
| isdrngrd 18773 | Properties that determine ... |
| drngpropd 18774 | If two structures have the... |
| fldpropd 18775 | If two structures have the... |
| issubrg 18780 | The subring predicate. (C... |
| subrgss 18781 | A subring is a subset. (C... |
| subrgid 18782 | Every ring is a subring of... |
| subrgring 18783 | A subring is a ring. (Con... |
| subrgcrng 18784 | A subring of a commutative... |
| subrgrcl 18785 | Reverse closure for a subr... |
| subrgsubg 18786 | A subring is a subgroup. ... |
| subrg0 18787 | A subring always has the s... |
| subrg1cl 18788 | A subring contains the mul... |
| subrgbas 18789 | Base set of a subring stru... |
| subrg1 18790 | A subring always has the s... |
| subrgacl 18791 | A subring is closed under ... |
| subrgmcl 18792 | A subgroup is closed under... |
| subrgsubm 18793 | A subring is a submonoid o... |
| subrgdvds 18794 | If an element divides anot... |
| subrguss 18795 | A unit of a subring is a u... |
| subrginv 18796 | A subring always has the s... |
| subrgdv 18797 | A subring always has the s... |
| subrgunit 18798 | An element of a ring is a ... |
| subrgugrp 18799 | The units of a subring for... |
| issubrg2 18800 | Characterize the subrings ... |
| opprsubrg 18801 | Being a subring is a symme... |
| subrgint 18802 | The intersection of a none... |
| subrgin 18803 | The intersection of two su... |
| subrgmre 18804 | The subrings of a ring are... |
| issubdrg 18805 | Characterize the subfields... |
| subsubrg 18806 | A subring of a subring is ... |
| subsubrg2 18807 | The set of subrings of a s... |
| issubrg3 18808 | A subring is an additive s... |
| resrhm 18809 | Restriction of a ring homo... |
| rhmeql 18810 | The equalizer of two ring ... |
| rhmima 18811 | The homomorphic image of a... |
| cntzsubr 18812 | Centralizers in a ring are... |
| pwsdiagrhm 18813 | Diagonal homomorphism into... |
| subrgpropd 18814 | If two structures have the... |
| rhmpropd 18815 | Ring homomorphism depends ... |
| abvfval 18818 | Value of the set of absolu... |
| isabv 18819 | Elementhood in the set of ... |
| isabvd 18820 | Properties that determine ... |
| abvrcl 18821 | Reverse closure for the ab... |
| abvfge0 18822 | An absolute value is a fun... |
| abvf 18823 | An absolute value is a fun... |
| abvcl 18824 | An absolute value is a fun... |
| abvge0 18825 | The absolute value of a nu... |
| abveq0 18826 | The value of an absolute v... |
| abvne0 18827 | The absolute value of a no... |
| abvgt0 18828 | The absolute value of a no... |
| abvmul 18829 | An absolute value distribu... |
| abvtri 18830 | An absolute value satisfie... |
| abv0 18831 | The absolute value of zero... |
| abv1z 18832 | The absolute value of one ... |
| abv1 18833 | The absolute value of one ... |
| abvneg 18834 | The absolute value of a ne... |
| abvsubtri 18835 | An absolute value satisfie... |
| abvrec 18836 | The absolute value distrib... |
| abvdiv 18837 | The absolute value distrib... |
| abvdom 18838 | Any ring with an absolute ... |
| abvres 18839 | The restriction of an abso... |
| abvtrivd 18840 | The trivial absolute value... |
| abvtriv 18841 | The trivial absolute value... |
| abvpropd 18842 | If two structures have the... |
| staffval 18847 | The functionalization of t... |
| stafval 18848 | The functionalization of t... |
| staffn 18849 | The functionalization is e... |
| issrng 18850 | The predicate "is a star r... |
| srngrhm 18851 | The involution function in... |
| srngring 18852 | A star ring is a ring. (C... |
| srngcnv 18853 | The involution function in... |
| srngf1o 18854 | The involution function in... |
| srngcl 18855 | The involution function in... |
| srngnvl 18856 | The involution function in... |
| srngadd 18857 | The involution function in... |
| srngmul 18858 | The involution function in... |
| srng1 18859 | The conjugate of the ring ... |
| srng0 18860 | The conjugate of the ring ... |
| issrngd 18861 | Properties that determine ... |
| idsrngd 18862 | A commutative ring is a st... |
| islmod 18867 | The predicate "is a left m... |
| lmodlema 18868 | Lemma for properties of a ... |
| islmodd 18869 | Properties that determine ... |
| lmodgrp 18870 | A left module is a group. ... |
| lmodring 18871 | The scalar component of a ... |
| lmodfgrp 18872 | The scalar component of a ... |
| lmodbn0 18873 | The base set of a left mod... |
| lmodacl 18874 | Closure of ring addition f... |
| lmodmcl 18875 | Closure of ring multiplica... |
| lmodsn0 18876 | The set of scalars in a le... |
| lmodvacl 18877 | Closure of vector addition... |
| lmodass 18878 | Left module vector sum is ... |
| lmodlcan 18879 | Left cancellation law for ... |
| lmodvscl 18880 | Closure of scalar product ... |
| scaffval 18881 | The scalar multiplication ... |
| scafval 18882 | The scalar multiplication ... |
| scafeq 18883 | If the scalar multiplicati... |
| scaffn 18884 | The scalar multiplication ... |
| lmodscaf 18885 | The scalar multiplication ... |
| lmodvsdi 18886 | Distributive law for scala... |
| lmodvsdir 18887 | Distributive law for scala... |
| lmodvsass 18888 | Associative law for scalar... |
| lmod0cl 18889 | The ring zero in a left mo... |
| lmod1cl 18890 | The ring unit in a left mo... |
| lmodvs1 18891 | Scalar product with ring u... |
| lmod0vcl 18892 | The zero vector is a vecto... |
| lmod0vlid 18893 | Left identity law for the ... |
| lmod0vrid 18894 | Right identity law for the... |
| lmod0vid 18895 | Identity equivalent to the... |
| lmod0vs 18896 | Zero times a vector is the... |
| lmodvs0 18897 | Anything times the zero ve... |
| lmodvsmmulgdi 18898 | Distributive law for a gro... |
| lmodfopnelem1 18899 | Lemma 1 for ~ lmodfopne . ... |
| lmodfopnelem2 18900 | Lemma 2 for ~ lmodfopne . ... |
| lmodfopne 18901 | The (functionalized) opera... |
| lcomf 18902 | A linear-combination sum i... |
| lcomfsupp 18903 | A linear-combination sum i... |
| lmodvnegcl 18904 | Closure of vector negative... |
| lmodvnegid 18905 | Addition of a vector with ... |
| lmodvneg1 18906 | Minus 1 times a vector is ... |
| lmodvsneg 18907 | Multiplication of a vector... |
| lmodvsubcl 18908 | Closure of vector subtract... |
| lmodcom 18909 | Left module vector sum is ... |
| lmodabl 18910 | A left module is an abelia... |
| lmodcmn 18911 | A left module is a commuta... |
| lmodnegadd 18912 | Distribute negation throug... |
| lmod4 18913 | Commutative/associative la... |
| lmodvsubadd 18914 | Relationship between vecto... |
| lmodvaddsub4 18915 | Vector addition/subtractio... |
| lmodvpncan 18916 | Addition/subtraction cance... |
| lmodvnpcan 18917 | Cancellation law for vecto... |
| lmodvsubval2 18918 | Value of vector subtractio... |
| lmodsubvs 18919 | Subtraction of a scalar pr... |
| lmodsubdi 18920 | Scalar multiplication dist... |
| lmodsubdir 18921 | Scalar multiplication dist... |
| lmodsubeq0 18922 | If the difference between ... |
| lmodsubid 18923 | Subtraction of a vector fr... |
| lmodvsghm 18924 | Scalar multiplication of t... |
| lmodprop2d 18925 | If two structures have the... |
| lmodpropd 18926 | If two structures have the... |
| gsumvsmul 18927 | Pull a scalar multiplicati... |
| mptscmfsupp0 18928 | A mapping to a scalar prod... |
| mptscmfsuppd 18929 | A function mapping to a sc... |
| rmodislmodlem 18930 | Lemma for ~ rmodislmod . ... |
| rmodislmod 18931 | The right module ` R ` ind... |
| lssset 18934 | The set of all (not necess... |
| islss 18935 | The predicate "is a subspa... |
| islssd 18936 | Properties that determine ... |
| lssss 18937 | A subspace is a set of vec... |
| lssel 18938 | A subspace member is a vec... |
| lss1 18939 | The set of vectors in a le... |
| lssuni 18940 | The union of all subspaces... |
| lssn0 18941 | A subspace is not empty. ... |
| 00lss 18942 | The empty structure has no... |
| lsscl 18943 | Closure property of a subs... |
| lssvsubcl 18944 | Closure of vector subtract... |
| lssvancl1 18945 | Non-closure: if one vector... |
| lssvancl2 18946 | Non-closure: if one vector... |
| lss0cl 18947 | The zero vector belongs to... |
| lsssn0 18948 | The singleton of the zero ... |
| lss0ss 18949 | The zero subspace is inclu... |
| lssle0 18950 | No subspace is smaller tha... |
| lssne0 18951 | A nonzero subspace has a n... |
| lssneln0 18952 | A vector which doesn't bel... |
| lssssr 18953 | Conclude subspace ordering... |
| lssvacl 18954 | Closure of vector addition... |
| lssvscl 18955 | Closure of scalar product ... |
| lssvnegcl 18956 | Closure of negative vector... |
| lsssubg 18957 | All subspaces are subgroup... |
| lsssssubg 18958 | All subspaces are subgroup... |
| islss3 18959 | A linear subspace of a mod... |
| lsslmod 18960 | A submodule is a module. ... |
| lsslss 18961 | The subspaces of a subspac... |
| islss4 18962 | A linear subspace is a sub... |
| lss1d 18963 | One-dimensional subspace (... |
| lssintcl 18964 | The intersection of a none... |
| lssincl 18965 | The intersection of two su... |
| lssmre 18966 | The subspaces of a module ... |
| lssacs 18967 | Submodules are an algebrai... |
| prdsvscacl 18968 | Pointwise scalar multiplic... |
| prdslmodd 18969 | The product of a family of... |
| pwslmod 18970 | The product of a family of... |
| lspfval 18973 | The span function for a le... |
| lspf 18974 | The span operator on a lef... |
| lspval 18975 | The span of a set of vecto... |
| lspcl 18976 | The span of a set of vecto... |
| lspsncl 18977 | The span of a singleton is... |
| lspprcl 18978 | The span of a pair is a su... |
| lsptpcl 18979 | The span of an unordered t... |
| lspsnsubg 18980 | The span of a singleton is... |
| 00lsp 18981 | ~ fvco4i lemma for linear ... |
| lspid 18982 | The span of a subspace is ... |
| lspssv 18983 | A span is a set of vectors... |
| lspss 18984 | Span preserves subset orde... |
| lspssid 18985 | A set of vectors is a subs... |
| lspidm 18986 | The span of a set of vecto... |
| lspun 18987 | The span of union is the s... |
| lspssp 18988 | If a set of vectors is a s... |
| mrclsp 18989 | Moore closure generalizes ... |
| lspsnss 18990 | The span of the singleton ... |
| lspsnel3 18991 | A member of the span of th... |
| lspprss 18992 | The span of a pair of vect... |
| lspsnid 18993 | A vector belongs to the sp... |
| lspsnel6 18994 | Relationship between a vec... |
| lspsnel5 18995 | Relationship between a vec... |
| lspsnel5a 18996 | Relationship between a vec... |
| lspprid1 18997 | A member of a pair of vect... |
| lspprid2 18998 | A member of a pair of vect... |
| lspprvacl 18999 | The sum of two vectors bel... |
| lssats2 19000 | A way to express atomistic... |
| lspsneli 19001 | A scalar product with a ve... |
| lspsn 19002 | Span of the singleton of a... |
| lspsnel 19003 | Member of span of the sing... |
| lspsnvsi 19004 | Span of a scalar product o... |
| lspsnss2 19005 | Comparable spans of single... |
| lspsnneg 19006 | Negation does not change t... |
| lspsnsub 19007 | Swapping subtraction order... |
| lspsn0 19008 | Span of the singleton of t... |
| lsp0 19009 | Span of the empty set. (C... |
| lspuni0 19010 | Union of the span of the e... |
| lspun0 19011 | The span of a union with t... |
| lspsneq0 19012 | Span of the singleton is t... |
| lspsneq0b 19013 | Equal singleton spans impl... |
| lmodindp1 19014 | Two independent (non-colin... |
| lsslsp 19015 | Spans in submodules corres... |
| lss0v 19016 | The zero vector in a submo... |
| lsspropd 19017 | If two structures have the... |
| lsppropd 19018 | If two structures have the... |
| reldmlmhm 19025 | Lemma for module homomorph... |
| lmimfn 19026 | Lemma for module isomorphi... |
| islmhm 19027 | Property of being a homomo... |
| islmhm3 19028 | Property of a module homom... |
| lmhmlem 19029 | Non-quantified consequence... |
| lmhmsca 19030 | A homomorphism of left mod... |
| lmghm 19031 | A homomorphism of left mod... |
| lmhmlmod2 19032 | A homomorphism of left mod... |
| lmhmlmod1 19033 | A homomorphism of left mod... |
| lmhmf 19034 | A homomorphism of left mod... |
| lmhmlin 19035 | A homomorphism of left mod... |
| lmodvsinv 19036 | Multiplication of a vector... |
| lmodvsinv2 19037 | Multiplying a negated vect... |
| islmhm2 19038 | A one-equation proof of li... |
| islmhmd 19039 | Deduction for a module hom... |
| 0lmhm 19040 | The constant zero linear f... |
| idlmhm 19041 | The identity function on a... |
| invlmhm 19042 | The negative function on a... |
| lmhmco 19043 | The composition of two mod... |
| lmhmplusg 19044 | The pointwise sum of two l... |
| lmhmvsca 19045 | The pointwise scalar produ... |
| lmhmf1o 19046 | A bijective module homomor... |
| lmhmima 19047 | The image of a subspace un... |
| lmhmpreima 19048 | The inverse image of a sub... |
| lmhmlsp 19049 | Homomorphisms preserve spa... |
| lmhmrnlss 19050 | The range of a homomorphis... |
| lmhmkerlss 19051 | The kernel of a homomorphi... |
| reslmhm 19052 | Restriction of a homomorph... |
| reslmhm2 19053 | Expansion of the codomain ... |
| reslmhm2b 19054 | Expansion of the codomain ... |
| lmhmeql 19055 | The equalizer of two modul... |
| lspextmo 19056 | A linear function is compl... |
| pwsdiaglmhm 19057 | Diagonal homomorphism into... |
| pwssplit0 19058 | Splitting for structure po... |
| pwssplit1 19059 | Splitting for structure po... |
| pwssplit2 19060 | Splitting for structure po... |
| pwssplit3 19061 | Splitting for structure po... |
| islmim 19062 | An isomorphism of left mod... |
| lmimf1o 19063 | An isomorphism of left mod... |
| lmimlmhm 19064 | An isomorphism of modules ... |
| lmimgim 19065 | An isomorphism of modules ... |
| islmim2 19066 | An isomorphism of left mod... |
| lmimcnv 19067 | The converse of a bijectiv... |
| brlmic 19068 | The relation "is isomorphi... |
| brlmici 19069 | Prove isomorphic by an exp... |
| lmiclcl 19070 | Isomorphism implies the le... |
| lmicrcl 19071 | Isomorphism implies the ri... |
| lmicsym 19072 | Module isomorphism is symm... |
| lmhmpropd 19073 | Module homomorphism depend... |
| islbs 19076 | The predicate " ` B ` is a... |
| lbsss 19077 | A basis is a set of vector... |
| lbsel 19078 | An element of a basis is a... |
| lbssp 19079 | The span of a basis is the... |
| lbsind 19080 | A basis is linearly indepe... |
| lbsind2 19081 | A basis is linearly indepe... |
| lbspss 19082 | No proper subset of a basi... |
| lsmcl 19083 | The sum of two subspaces i... |
| lsmspsn 19084 | Member of subspace sum of ... |
| lsmelval2 19085 | Subspace sum membership in... |
| lsmsp 19086 | Subspace sum in terms of s... |
| lsmsp2 19087 | Subspace sum of spans of s... |
| lsmssspx 19088 | Subspace sum (in its exten... |
| lsmpr 19089 | The span of a pair of vect... |
| lsppreli 19090 | A vector expressed as a su... |
| lsmelpr 19091 | Two ways to say that a vec... |
| lsppr0 19092 | The span of a vector paire... |
| lsppr 19093 | Span of a pair of vectors.... |
| lspprel 19094 | Member of the span of a pa... |
| lspprabs 19095 | Absorption of vector sum i... |
| lspvadd 19096 | The span of a vector sum i... |
| lspsntri 19097 | Triangle-type inequality f... |
| lspsntrim 19098 | Triangle-type inequality f... |
| lbspropd 19099 | If two structures have the... |
| pj1lmhm 19100 | The left projection functi... |
| pj1lmhm2 19101 | The left projection functi... |
| islvec 19104 | The predicate "is a left v... |
| lvecdrng 19105 | The set of scalars of a le... |
| lveclmod 19106 | A left vector space is a l... |
| lsslvec 19107 | A vector subspace is a vec... |
| lvecvs0or 19108 | If a scalar product is zer... |
| lvecvsn0 19109 | A scalar product is nonzer... |
| lssvs0or 19110 | If a scalar product belong... |
| lvecvscan 19111 | Cancellation law for scala... |
| lvecvscan2 19112 | Cancellation law for scala... |
| lvecinv 19113 | Invert coefficient of scal... |
| lspsnvs 19114 | A nonzero scalar product d... |
| lspsneleq 19115 | Membership relation that i... |
| lspsncmp 19116 | Comparable spans of nonzer... |
| lspsnne1 19117 | Two ways to express that v... |
| lspsnne2 19118 | Two ways to express that v... |
| lspsnnecom 19119 | Swap two vectors with diff... |
| lspabs2 19120 | Absorption law for span of... |
| lspabs3 19121 | Absorption law for span of... |
| lspsneq 19122 | Equal spans of singletons ... |
| lspsneu 19123 | Nonzero vectors with equal... |
| lspsnel4 19124 | A member of the span of th... |
| lspdisj 19125 | The span of a vector not i... |
| lspdisjb 19126 | A nonzero vector is not in... |
| lspdisj2 19127 | Unequal spans are disjoint... |
| lspfixed 19128 | Show membership in the spa... |
| lspexch 19129 | Exchange property for span... |
| lspexchn1 19130 | Exchange property for span... |
| lspexchn2 19131 | Exchange property for span... |
| lspindpi 19132 | Partial independence prope... |
| lspindp1 19133 | Alternate way to say 3 vec... |
| lspindp2l 19134 | Alternate way to say 3 vec... |
| lspindp2 19135 | Alternate way to say 3 vec... |
| lspindp3 19136 | Independence of 2 vectors ... |
| lspindp4 19137 | (Partial) independence of ... |
| lvecindp 19138 | Compute the ` X ` coeffici... |
| lvecindp2 19139 | Sums of independent vector... |
| lspsnsubn0 19140 | Unequal singleton spans im... |
| lsmcv 19141 | Subspace sum has the cover... |
| lspsolvlem 19142 | Lemma for ~ lspsolv . (Co... |
| lspsolv 19143 | If ` X ` is in the span of... |
| lssacsex 19144 | In a vector space, subspac... |
| lspsnat 19145 | There is no subspace stric... |
| lspsncv0 19146 | The span of a singleton co... |
| lsppratlem1 19147 | Lemma for ~ lspprat . Let... |
| lsppratlem2 19148 | Lemma for ~ lspprat . Sho... |
| lsppratlem3 19149 | Lemma for ~ lspprat . In ... |
| lsppratlem4 19150 | Lemma for ~ lspprat . In ... |
| lsppratlem5 19151 | Lemma for ~ lspprat . Com... |
| lsppratlem6 19152 | Lemma for ~ lspprat . Neg... |
| lspprat 19153 | A proper subspace of the s... |
| islbs2 19154 | An equivalent formulation ... |
| islbs3 19155 | An equivalent formulation ... |
| lbsacsbs 19156 | Being a basis in a vector ... |
| lvecdim 19157 | The dimension theorem for ... |
| lbsextlem1 19158 | Lemma for ~ lbsext . The ... |
| lbsextlem2 19159 | Lemma for ~ lbsext . Sinc... |
| lbsextlem3 19160 | Lemma for ~ lbsext . A ch... |
| lbsextlem4 19161 | Lemma for ~ lbsext . ~ lbs... |
| lbsextg 19162 | For any linearly independe... |
| lbsext 19163 | For any linearly independe... |
| lbsexg 19164 | Every vector space has a b... |
| lbsex 19165 | Every vector space has a b... |
| lvecprop2d 19166 | If two structures have the... |
| lvecpropd 19167 | If two structures have the... |
| sraval 19176 | Lemma for ~ srabase throug... |
| sralem 19177 | Lemma for ~ srabase and si... |
| srabase 19178 | Base set of a subring alge... |
| sraaddg 19179 | Additive operation of a su... |
| sramulr 19180 | Multiplicative operation o... |
| srasca 19181 | The set of scalars of a su... |
| sravsca 19182 | The scalar product operati... |
| sraip 19183 | The inner product operatio... |
| sratset 19184 | Topology component of a su... |
| sratopn 19185 | Topology component of a su... |
| srads 19186 | Distance function of a sub... |
| sralmod 19187 | The subring algebra is a l... |
| sralmod0 19188 | The subring module inherit... |
| issubrngd2 19189 | Prove a subring by closure... |
| rlmfn 19190 | ` ringLMod ` is a function... |
| rlmval 19191 | Value of the ring module. ... |
| lidlval 19192 | Value of the set of ring i... |
| rspval 19193 | Value of the ring span fun... |
| rlmval2 19194 | Value of the ring module e... |
| rlmbas 19195 | Base set of the ring modul... |
| rlmplusg 19196 | Vector addition in the rin... |
| rlm0 19197 | Zero vector in the ring mo... |
| rlmsub 19198 | Subtraction in the ring mo... |
| rlmmulr 19199 | Ring multiplication in the... |
| rlmsca 19200 | Scalars in the ring module... |
| rlmsca2 19201 | Scalars in the ring module... |
| rlmvsca 19202 | Scalar multiplication in t... |
| rlmtopn 19203 | Topology component of the ... |
| rlmds 19204 | Metric component of the ri... |
| rlmlmod 19205 | The ring module is a modul... |
| rlmlvec 19206 | The ring module over a div... |
| rlmvneg 19207 | Vector negation in the rin... |
| rlmscaf 19208 | Functionalized scalar mult... |
| ixpsnbasval 19209 | The value of an infinite C... |
| lidlss 19210 | An ideal is a subset of th... |
| islidl 19211 | Predicate of being a (left... |
| lidl0cl 19212 | An ideal contains 0. (Con... |
| lidlacl 19213 | An ideal is closed under a... |
| lidlnegcl 19214 | An ideal contains negative... |
| lidlsubg 19215 | An ideal is a subgroup of ... |
| lidlsubcl 19216 | An ideal is closed under s... |
| lidlmcl 19217 | An ideal is closed under l... |
| lidl1el 19218 | An ideal contains 1 iff it... |
| lidl0 19219 | Every ring contains a zero... |
| lidl1 19220 | Every ring contains a unit... |
| lidlacs 19221 | The ideal system is an alg... |
| rspcl 19222 | The span of a set of ring ... |
| rspssid 19223 | The span of a set of ring ... |
| rsp1 19224 | The span of the identity e... |
| rsp0 19225 | The span of the zero eleme... |
| rspssp 19226 | The ideal span of a set of... |
| mrcrsp 19227 | Moore closure generalizes ... |
| lidlnz 19228 | A nonzero ideal contains a... |
| drngnidl 19229 | A division ring has only t... |
| lidlrsppropd 19230 | The left ideals and ring s... |
| 2idlval 19233 | Definition of a two-sided ... |
| 2idlcpbl 19234 | The coset equivalence rela... |
| qus1 19235 | The multiplicative identit... |
| qusring 19236 | If ` S ` is a two-sided id... |
| qusrhm 19237 | If ` S ` is a two-sided id... |
| crngridl 19238 | In a commutative ring, the... |
| crng2idl 19239 | In a commutative ring, a t... |
| quscrng 19240 | The quotient of a commutat... |
| lpival 19245 | Value of the set of princi... |
| islpidl 19246 | Property of being a princi... |
| lpi0 19247 | The zero ideal is always p... |
| lpi1 19248 | The unit ideal is always p... |
| islpir 19249 | Principal ideal rings are ... |
| lpiss 19250 | Principal ideals are a sub... |
| islpir2 19251 | Principal ideal rings are ... |
| lpirring 19252 | Principal ideal rings are ... |
| drnglpir 19253 | Division rings are princip... |
| rspsn 19254 | Membership in principal id... |
| lidldvgen 19255 | An element generates an id... |
| lpigen 19256 | An ideal is principal iff ... |
| isnzr 19259 | Property of a nonzero ring... |
| nzrnz 19260 | One and zero are different... |
| nzrring 19261 | A nonzero ring is a ring. ... |
| drngnzr 19262 | All division rings are non... |
| isnzr2 19263 | Equivalent characterizatio... |
| isnzr2hash 19264 | Equivalent characterizatio... |
| opprnzr 19265 | The opposite of a nonzero ... |
| ringelnzr 19266 | A ring is nonzero if it ha... |
| nzrunit 19267 | A unit is nonzero in any n... |
| subrgnzr 19268 | A subring of a nonzero rin... |
| 0ringnnzr 19269 | A ring is a zero ring iff ... |
| 0ring 19270 | If a ring has only one ele... |
| 0ring01eq 19271 | In a ring with only one el... |
| 01eq0ring 19272 | If the zero and the identi... |
| 0ring01eqbi 19273 | In a unital ring the zero ... |
| rng1nnzr 19274 | The (smallest) structure r... |
| ring1zr 19275 | The only (unital) ring wit... |
| rngen1zr 19276 | The only (unital) ring wit... |
| ringen1zr 19277 | The only unital ring with ... |
| rng1nfld 19278 | The zero ring is not a fie... |
| rrgval 19287 | Value of the set or left-r... |
| isrrg 19288 | Membership in the set of l... |
| rrgeq0i 19289 | Property of a left-regular... |
| rrgeq0 19290 | Left-multiplication by a l... |
| rrgsupp 19291 | Left multiplication by a l... |
| rrgss 19292 | Left-regular elements are ... |
| unitrrg 19293 | Units are regular elements... |
| isdomn 19294 | Expand definition of a dom... |
| domnnzr 19295 | A domain is a nonzero ring... |
| domnring 19296 | A domain is a ring. (Cont... |
| domneq0 19297 | In a domain, a product is ... |
| domnmuln0 19298 | In a domain, a product of ... |
| isdomn2 19299 | A ring is a domain iff all... |
| domnrrg 19300 | In a domain, any nonzero e... |
| opprdomn 19301 | The opposite of a domain i... |
| abvn0b 19302 | Another characterization o... |
| drngdomn 19303 | A division ring is a domai... |
| isidom 19304 | An integral domain is a co... |
| fldidom 19305 | A field is an integral dom... |
| fidomndrnglem 19306 | Lemma for ~ fidomndrng . ... |
| fidomndrng 19307 | A finite domain is a divis... |
| fiidomfld 19308 | A finite integral domain i... |
| isassa 19315 | The properties of an assoc... |
| assalem 19316 | The properties of an assoc... |
| assaass 19317 | Left-associative property ... |
| assaassr 19318 | Right-associative property... |
| assalmod 19319 | An associative algebra is ... |
| assaring 19320 | An associative algebra is ... |
| assasca 19321 | An associative algebra's s... |
| assa2ass 19322 | Left- and right-associativ... |
| isassad 19323 | Sufficient condition for b... |
| issubassa 19324 | The subalgebras of an asso... |
| sraassa 19325 | The subring algebra over a... |
| rlmassa 19326 | The ring module over a com... |
| assapropd 19327 | If two structures have the... |
| aspval 19328 | Value of the algebraic clo... |
| asplss 19329 | The algebraic span of a se... |
| aspid 19330 | The algebraic span of a su... |
| aspsubrg 19331 | The algebraic span of a se... |
| aspss 19332 | Span preserves subset orde... |
| aspssid 19333 | A set of vectors is a subs... |
| asclfval 19334 | Function value of the alge... |
| asclval 19335 | Value of a mapped algebra ... |
| asclfn 19336 | Unconditional functionalit... |
| asclf 19337 | The algebra scalars functi... |
| asclghm 19338 | The algebra scalars functi... |
| asclmul1 19339 | Left multiplication by a l... |
| asclmul2 19340 | Right multiplication by a ... |
| asclinvg 19341 | The group inverse (negatio... |
| asclrhm 19342 | The scalar injection is a ... |
| rnascl 19343 | The set of injected scalar... |
| ressascl 19344 | The injection of scalars i... |
| issubassa2 19345 | A subring of a unital alge... |
| asclpropd 19346 | If two structures have the... |
| aspval2 19347 | The algebraic closure is t... |
| assamulgscmlem1 19348 | Lemma 1 for ~ assamulgscm ... |
| assamulgscmlem2 19349 | Lemma for ~ assamulgscm (i... |
| assamulgscm 19350 | Exponentiation of a scalar... |
| reldmpsr 19361 | The multivariate power ser... |
| psrval 19362 | Value of the multivariate ... |
| psrvalstr 19363 | The multivariate power ser... |
| psrbag 19364 | Elementhood in the set of ... |
| psrbagf 19365 | A finite bag is a function... |
| snifpsrbag 19366 | A bag containing one eleme... |
| fczpsrbag 19367 | The constant function equa... |
| psrbaglesupp 19368 | The support of a dominated... |
| psrbaglecl 19369 | The set of finite bags is ... |
| psrbagaddcl 19370 | The sum of two finite bags... |
| psrbagcon 19371 | The analogue of the statem... |
| psrbaglefi 19372 | There are finitely many ba... |
| psrbagconcl 19373 | The complement of a bag is... |
| psrbagconf1o 19374 | Bag complementation is a b... |
| gsumbagdiaglem 19375 | Lemma for ~ gsumbagdiag . ... |
| gsumbagdiag 19376 | Two-dimensional commutatio... |
| psrass1lem 19377 | A group sum commutation us... |
| psrbas 19378 | The base set of the multiv... |
| psrelbas 19379 | An element of the set of p... |
| psrelbasfun 19380 | An element of the set of p... |
| psrplusg 19381 | The addition operation of ... |
| psradd 19382 | The addition operation of ... |
| psraddcl 19383 | Closure of the power serie... |
| psrmulr 19384 | The multiplication operati... |
| psrmulfval 19385 | The multiplication operati... |
| psrmulval 19386 | The multiplication operati... |
| psrmulcllem 19387 | Closure of the power serie... |
| psrmulcl 19388 | Closure of the power serie... |
| psrsca 19389 | The scalar field of the mu... |
| psrvscafval 19390 | The scalar multiplication ... |
| psrvsca 19391 | The scalar multiplication ... |
| psrvscaval 19392 | The scalar multiplication ... |
| psrvscacl 19393 | Closure of the power serie... |
| psr0cl 19394 | The zero element of the ri... |
| psr0lid 19395 | The zero element of the ri... |
| psrnegcl 19396 | The negative function in t... |
| psrlinv 19397 | The negative function in t... |
| psrgrp 19398 | The ring of power series i... |
| psr0 19399 | The zero element of the ri... |
| psrneg 19400 | The negative function of t... |
| psrlmod 19401 | The ring of power series i... |
| psr1cl 19402 | The identity element of th... |
| psrlidm 19403 | The identity element of th... |
| psrridm 19404 | The identity element of th... |
| psrass1 19405 | Associative identity for t... |
| psrdi 19406 | Distributive law for the r... |
| psrdir 19407 | Distributive law for the r... |
| psrass23l 19408 | Associative identity for t... |
| psrcom 19409 | Commutative law for the ri... |
| psrass23 19410 | Associative identities for... |
| psrring 19411 | The ring of power series i... |
| psr1 19412 | The identity element of th... |
| psrcrng 19413 | The ring of power series i... |
| psrassa 19414 | The ring of power series i... |
| resspsrbas 19415 | A restricted power series ... |
| resspsradd 19416 | A restricted power series ... |
| resspsrmul 19417 | A restricted power series ... |
| resspsrvsca 19418 | A restricted power series ... |
| subrgpsr 19419 | A subring of the base ring... |
| mvrfval 19420 | Value of the generating el... |
| mvrval 19421 | Value of the generating el... |
| mvrval2 19422 | Value of the generating el... |
| mvrid 19423 | The ` X i ` -th coefficien... |
| mvrf 19424 | The power series variable ... |
| mvrf1 19425 | The power series variable ... |
| mvrcl2 19426 | A power series variable is... |
| reldmmpl 19427 | The multivariate polynomia... |
| mplval 19428 | Value of the set of multiv... |
| mplbas 19429 | Base set of the set of mul... |
| mplelbas 19430 | Property of being a polyno... |
| mplval2 19431 | Self-referential expressio... |
| mplbasss 19432 | The set of polynomials is ... |
| mplelf 19433 | A polynomial is defined as... |
| mplsubglem 19434 | If ` A ` is an ideal of se... |
| mpllsslem 19435 | If ` A ` is an ideal of su... |
| mplsubglem2 19436 | Lemma for ~ mplsubg and ~ ... |
| mplsubg 19437 | The set of polynomials is ... |
| mpllss 19438 | The set of polynomials is ... |
| mplsubrglem 19439 | Lemma for ~ mplsubrg . (C... |
| mplsubrg 19440 | The set of polynomials is ... |
| mpl0 19441 | The zero polynomial. (Con... |
| mpladd 19442 | The addition operation on ... |
| mplmul 19443 | The multiplication operati... |
| mpl1 19444 | The identity element of th... |
| mplsca 19445 | The scalar field of a mult... |
| mplvsca2 19446 | The scalar multiplication ... |
| mplvsca 19447 | The scalar multiplication ... |
| mplvscaval 19448 | The scalar multiplication ... |
| mvrcl 19449 | A power series variable is... |
| mplgrp 19450 | The polynomial ring is a g... |
| mpllmod 19451 | The polynomial ring is a l... |
| mplring 19452 | The polynomial ring is a r... |
| mplcrng 19453 | The polynomial ring is a c... |
| mplassa 19454 | The polynomial ring is an ... |
| ressmplbas2 19455 | The base set of a restrict... |
| ressmplbas 19456 | A restricted polynomial al... |
| ressmpladd 19457 | A restricted polynomial al... |
| ressmplmul 19458 | A restricted polynomial al... |
| ressmplvsca 19459 | A restricted power series ... |
| subrgmpl 19460 | A subring of the base ring... |
| subrgmvr 19461 | The variables in a subring... |
| subrgmvrf 19462 | The variables in a polynom... |
| mplmon 19463 | A monomial is a polynomial... |
| mplmonmul 19464 | The product of two monomia... |
| mplcoe1 19465 | Decompose a polynomial int... |
| mplcoe3 19466 | Decompose a monomial in on... |
| mplcoe5lem 19467 | Lemma for ~ mplcoe4 . (Co... |
| mplcoe5 19468 | Decompose a monomial into ... |
| mplcoe2 19469 | Decompose a monomial into ... |
| mplbas2 19470 | An alternative expression ... |
| ltbval 19471 | Value of the well-order on... |
| ltbwe 19472 | The finite bag order is a ... |
| reldmopsr 19473 | Lemma for ordered power se... |
| opsrval 19474 | The value of the "ordered ... |
| opsrle 19475 | An alternative expression ... |
| opsrval2 19476 | Self-referential expressio... |
| opsrbaslem 19477 | Get a component of the ord... |
| opsrbaslemOLD 19478 | Obsolete version of ~ opsr... |
| opsrbas 19479 | The base set of the ordere... |
| opsrplusg 19480 | The addition operation of ... |
| opsrmulr 19481 | The multiplication operati... |
| opsrvsca 19482 | The scalar product operati... |
| opsrsca 19483 | The scalar ring of the ord... |
| opsrtoslem1 19484 | Lemma for ~ opsrtos . (Co... |
| opsrtoslem2 19485 | Lemma for ~ opsrtos . (Co... |
| opsrtos 19486 | The ordered power series s... |
| opsrso 19487 | The ordered power series s... |
| opsrcrng 19488 | The ring of ordered power ... |
| opsrassa 19489 | The ring of ordered power ... |
| mplrcl 19490 | Reverse closure for the po... |
| mplelsfi 19491 | A polynomial treated as a ... |
| mvrf2 19492 | The power series/polynomia... |
| mplmon2 19493 | Express a scaled monomial.... |
| psrbag0 19494 | The empty bag is a bag. (... |
| psrbagsn 19495 | A singleton bag is a bag. ... |
| mplascl 19496 | Value of the scalar inject... |
| mplasclf 19497 | The scalar injection is a ... |
| subrgascl 19498 | The scalar injection funct... |
| subrgasclcl 19499 | The scalars in a polynomia... |
| mplmon2cl 19500 | A scaled monomial is a pol... |
| mplmon2mul 19501 | Product of scaled monomial... |
| mplind 19502 | Prove a property of polyno... |
| mplcoe4 19503 | Decompose a polynomial int... |
| evlslem4 19508 | The support of a tensor pr... |
| psrbagfsupp 19509 | Finite bags have finite no... |
| psrbagev1 19510 | A bag of multipliers provi... |
| psrbagev2 19511 | Closure of a sum using a b... |
| evlslem2 19512 | A linear function on the p... |
| evlslem6 19513 | Lemma for ~ evlseu . Fini... |
| evlslem3 19514 | Lemma for ~ evlseu . Poly... |
| evlslem1 19515 | Lemma for ~ evlseu , give ... |
| evlseu 19516 | For a given interpretation... |
| reldmevls 19517 | Well-behaved binary operat... |
| mpfrcl 19518 | Reverse closure for the se... |
| evlsval 19519 | Value of the polynomial ev... |
| evlsval2 19520 | Characterizing properties ... |
| evlsrhm 19521 | Polynomial evaluation is a... |
| evlssca 19522 | Polynomial evaluation maps... |
| evlsvar 19523 | Polynomial evaluation maps... |
| evlval 19524 | Value of the simple/same r... |
| evlrhm 19525 | The simple evaluation map ... |
| evlsscasrng 19526 | The evaluation of a scalar... |
| evlsca 19527 | Simple polynomial evaluati... |
| evlsvarsrng 19528 | The evaluation of the vari... |
| evlvar 19529 | Simple polynomial evaluati... |
| mpfconst 19530 | Constants are multivariate... |
| mpfproj 19531 | Projections are multivaria... |
| mpfsubrg 19532 | Polynomial functions are a... |
| mpff 19533 | Polynomial functions are f... |
| mpfaddcl 19534 | The sum of multivariate po... |
| mpfmulcl 19535 | The product of multivariat... |
| mpfind 19536 | Prove a property of polyno... |
| psr1baslem 19555 | The set of finite bags on ... |
| psr1val 19556 | Value of the ring of univa... |
| psr1crng 19557 | The ring of univariate pow... |
| psr1assa 19558 | The ring of univariate pow... |
| psr1tos 19559 | The ordered power series s... |
| psr1bas2 19560 | The base set of the ring o... |
| psr1bas 19561 | The base set of the ring o... |
| vr1val 19562 | The value of the generator... |
| vr1cl2 19563 | The variable ` X ` is a me... |
| ply1val 19564 | The value of the set of un... |
| ply1bas 19565 | The value of the base set ... |
| ply1lss 19566 | Univariate polynomials for... |
| ply1subrg 19567 | Univariate polynomials for... |
| ply1crng 19568 | The ring of univariate pol... |
| ply1assa 19569 | The ring of univariate pol... |
| psr1bascl 19570 | A univariate power series ... |
| psr1basf 19571 | Univariate power series ba... |
| ply1basf 19572 | Univariate polynomial base... |
| ply1bascl 19573 | A univariate polynomial is... |
| ply1bascl2 19574 | A univariate polynomial is... |
| coe1fval 19575 | Value of the univariate po... |
| coe1fv 19576 | Value of an evaluated coef... |
| fvcoe1 19577 | Value of a multivariate co... |
| coe1fval3 19578 | Univariate power series co... |
| coe1f2 19579 | Functionality of univariat... |
| coe1fval2 19580 | Univariate polynomial coef... |
| coe1f 19581 | Functionality of univariat... |
| coe1fvalcl 19582 | A coefficient of a univari... |
| coe1sfi 19583 | Finite support of univaria... |
| coe1fsupp 19584 | The coefficient vector of ... |
| mptcoe1fsupp 19585 | A mapping involving coeffi... |
| coe1ae0 19586 | The coefficient vector of ... |
| vr1cl 19587 | The generator of a univari... |
| opsr0 19588 | Zero in the ordered power ... |
| opsr1 19589 | One in the ordered power s... |
| mplplusg 19590 | Value of addition in a pol... |
| mplmulr 19591 | Value of multiplication in... |
| psr1plusg 19592 | Value of addition in a uni... |
| psr1vsca 19593 | Value of scalar multiplica... |
| psr1mulr 19594 | Value of multiplication in... |
| ply1plusg 19595 | Value of addition in a uni... |
| ply1vsca 19596 | Value of scalar multiplica... |
| ply1mulr 19597 | Value of multiplication in... |
| ressply1bas2 19598 | The base set of a restrict... |
| ressply1bas 19599 | A restricted polynomial al... |
| ressply1add 19600 | A restricted polynomial al... |
| ressply1mul 19601 | A restricted polynomial al... |
| ressply1vsca 19602 | A restricted power series ... |
| subrgply1 19603 | A subring of the base ring... |
| gsumply1subr 19604 | Evaluate a group sum in a ... |
| psrbaspropd 19605 | Property deduction for pow... |
| psrplusgpropd 19606 | Property deduction for pow... |
| mplbaspropd 19607 | Property deduction for pol... |
| psropprmul 19608 | Reversing multiplication i... |
| ply1opprmul 19609 | Reversing multiplication i... |
| 00ply1bas 19610 | Lemma for ~ ply1basfvi and... |
| ply1basfvi 19611 | Protection compatibility o... |
| ply1plusgfvi 19612 | Protection compatibility o... |
| ply1baspropd 19613 | Property deduction for uni... |
| ply1plusgpropd 19614 | Property deduction for uni... |
| opsrring 19615 | Ordered power series form ... |
| opsrlmod 19616 | Ordered power series form ... |
| psr1ring 19617 | Univariate power series fo... |
| ply1ring 19618 | Univariate polynomials for... |
| psr1lmod 19619 | Univariate power series fo... |
| psr1sca 19620 | Scalars of a univariate po... |
| psr1sca2 19621 | Scalars of a univariate po... |
| ply1lmod 19622 | Univariate polynomials for... |
| ply1sca 19623 | Scalars of a univariate po... |
| ply1sca2 19624 | Scalars of a univariate po... |
| ply1mpl0 19625 | The univariate polynomial ... |
| ply10s0 19626 | Zero times a univariate po... |
| ply1mpl1 19627 | The univariate polynomial ... |
| ply1ascl 19628 | The univariate polynomial ... |
| subrg1ascl 19629 | The scalar injection funct... |
| subrg1asclcl 19630 | The scalars in a polynomia... |
| subrgvr1 19631 | The variables in a subring... |
| subrgvr1cl 19632 | The variables in a polynom... |
| coe1z 19633 | The coefficient vector of ... |
| coe1add 19634 | The coefficient vector of ... |
| coe1addfv 19635 | A particular coefficient o... |
| coe1subfv 19636 | A particular coefficient o... |
| coe1mul2lem1 19637 | An equivalence for ~ coe1m... |
| coe1mul2lem2 19638 | An equivalence for ~ coe1m... |
| coe1mul2 19639 | The coefficient vector of ... |
| coe1mul 19640 | The coefficient vector of ... |
| ply1moncl 19641 | Closure of the expression ... |
| ply1tmcl 19642 | Closure of the expression ... |
| coe1tm 19643 | Coefficient vector of a po... |
| coe1tmfv1 19644 | Nonzero coefficient of a p... |
| coe1tmfv2 19645 | Zero coefficient of a poly... |
| coe1tmmul2 19646 | Coefficient vector of a po... |
| coe1tmmul 19647 | Coefficient vector of a po... |
| coe1tmmul2fv 19648 | Function value of a right-... |
| coe1pwmul 19649 | Coefficient vector of a po... |
| coe1pwmulfv 19650 | Function value of a right-... |
| ply1scltm 19651 | A scalar is a term with ze... |
| coe1sclmul 19652 | Coefficient vector of a po... |
| coe1sclmulfv 19653 | A single coefficient of a ... |
| coe1sclmul2 19654 | Coefficient vector of a po... |
| ply1sclf 19655 | A scalar polynomial is a p... |
| ply1sclcl 19656 | The value of the algebra s... |
| coe1scl 19657 | Coefficient vector of a sc... |
| ply1sclid 19658 | Recover the base scalar fr... |
| ply1sclf1 19659 | The polynomial scalar func... |
| ply1scl0 19660 | The zero scalar is zero. ... |
| ply1scln0 19661 | Nonzero scalars create non... |
| ply1scl1 19662 | The one scalar is the unit... |
| ply1idvr1 19663 | The identity of a polynomi... |
| cply1mul 19664 | The product of two constan... |
| ply1coefsupp 19665 | The decomposition of a uni... |
| ply1coe 19666 | Decompose a univariate pol... |
| eqcoe1ply1eq 19667 | Two polynomials over the s... |
| ply1coe1eq 19668 | Two polynomials over the s... |
| cply1coe0 19669 | All but the first coeffici... |
| cply1coe0bi 19670 | A polynomial is constant (... |
| coe1fzgsumdlem 19671 | Lemma for ~ coe1fzgsumd (i... |
| coe1fzgsumd 19672 | Value of an evaluated coef... |
| gsumsmonply1 19673 | A finite group sum of scal... |
| gsummoncoe1 19674 | A coefficient of the polyn... |
| gsumply1eq 19675 | Two univariate polynomials... |
| lply1binom 19676 | The binomial theorem for l... |
| lply1binomsc 19677 | The binomial theorem for l... |
| reldmevls1 19682 | Well-behaved binary operat... |
| ply1frcl 19683 | Reverse closure for the se... |
| evls1fval 19684 | Value of the univariate po... |
| evls1val 19685 | Value of the univariate po... |
| evls1rhmlem 19686 | Lemma for ~ evl1rhm and ~ ... |
| evls1rhm 19687 | Polynomial evaluation is a... |
| evls1sca 19688 | Univariate polynomial eval... |
| evls1gsumadd 19689 | Univariate polynomial eval... |
| evls1gsummul 19690 | Univariate polynomial eval... |
| evls1varpw 19691 | Univariate polynomial eval... |
| evl1fval 19692 | Value of the simple/same r... |
| evl1val 19693 | Value of the simple/same r... |
| evl1fval1lem 19694 | Lemma for ~ evl1fval1 . (... |
| evl1fval1 19695 | Value of the simple/same r... |
| evl1rhm 19696 | Polynomial evaluation is a... |
| fveval1fvcl 19697 | The function value of the ... |
| evl1sca 19698 | Polynomial evaluation maps... |
| evl1scad 19699 | Polynomial evaluation buil... |
| evl1var 19700 | Polynomial evaluation maps... |
| evl1vard 19701 | Polynomial evaluation buil... |
| evls1var 19702 | Univariate polynomial eval... |
| evls1scasrng 19703 | The evaluation of a scalar... |
| evls1varsrng 19704 | The evaluation of the vari... |
| evl1addd 19705 | Polynomial evaluation buil... |
| evl1subd 19706 | Polynomial evaluation buil... |
| evl1muld 19707 | Polynomial evaluation buil... |
| evl1vsd 19708 | Polynomial evaluation buil... |
| evl1expd 19709 | Polynomial evaluation buil... |
| pf1const 19710 | Constants are polynomial f... |
| pf1id 19711 | The identity is a polynomi... |
| pf1subrg 19712 | Polynomial functions are a... |
| pf1rcl 19713 | Reverse closure for the se... |
| pf1f 19714 | Polynomial functions are f... |
| mpfpf1 19715 | Convert a multivariate pol... |
| pf1mpf 19716 | Convert a univariate polyn... |
| pf1addcl 19717 | The sum of multivariate po... |
| pf1mulcl 19718 | The product of multivariat... |
| pf1ind 19719 | Prove a property of polyno... |
| evl1gsumdlem 19720 | Lemma for ~ evl1gsumd (ind... |
| evl1gsumd 19721 | Polynomial evaluation buil... |
| evl1gsumadd 19722 | Univariate polynomial eval... |
| evl1gsumaddval 19723 | Value of a univariate poly... |
| evl1gsummul 19724 | Univariate polynomial eval... |
| evl1varpw 19725 | Univariate polynomial eval... |
| evl1varpwval 19726 | Value of a univariate poly... |
| evl1scvarpw 19727 | Univariate polynomial eval... |
| evl1scvarpwval 19728 | Value of a univariate poly... |
| evl1gsummon 19729 | Value of a univariate poly... |
| cnfldstr 19748 | The field of complex numbe... |
| cnfldex 19749 | The field of complex numbe... |
| cnfldbas 19750 | The base set of the field ... |
| cnfldadd 19751 | The addition operation of ... |
| cnfldmul 19752 | The multiplication operati... |
| cnfldcj 19753 | The conjugation operation ... |
| cnfldtset 19754 | The topology component of ... |
| cnfldle 19755 | The ordering of the field ... |
| cnfldds 19756 | The metric of the field of... |
| cnfldunif 19757 | The uniform structure comp... |
| cnfldfun 19758 | The field of complex numbe... |
| cnfldfunALT 19759 | Alternate proof of ~ cnfld... |
| xrsstr 19760 | The extended real structur... |
| xrsex 19761 | The extended real structur... |
| xrsbas 19762 | The base set of the extend... |
| xrsadd 19763 | The addition operation of ... |
| xrsmul 19764 | The multiplication operati... |
| xrstset 19765 | The topology component of ... |
| xrsle 19766 | The ordering of the extend... |
| cncrng 19767 | The complex numbers form a... |
| cnring 19768 | The complex numbers form a... |
| xrsmcmn 19769 | The multiplicative group o... |
| cnfld0 19770 | The zero element of the fi... |
| cnfld1 19771 | The unit element of the fi... |
| cnfldneg 19772 | The additive inverse in th... |
| cnfldplusf 19773 | The functionalized additio... |
| cnfldsub 19774 | The subtraction operator i... |
| cndrng 19775 | The complex numbers form a... |
| cnflddiv 19776 | The division operation in ... |
| cnfldinv 19777 | The multiplicative inverse... |
| cnfldmulg 19778 | The group multiple functio... |
| cnfldexp 19779 | The exponentiation operato... |
| cnsrng 19780 | The complex numbers form a... |
| xrsmgm 19781 | The (additive group of the... |
| xrsnsgrp 19782 | The (additive group of the... |
| xrsmgmdifsgrp 19783 | The (additive group of the... |
| xrs1mnd 19784 | The extended real numbers,... |
| xrs10 19785 | The zero of the extended r... |
| xrs1cmn 19786 | The extended real numbers ... |
| xrge0subm 19787 | The nonnegative extended r... |
| xrge0cmn 19788 | The nonnegative extended r... |
| xrsds 19789 | The metric of the extended... |
| xrsdsval 19790 | The metric of the extended... |
| xrsdsreval 19791 | The metric of the extended... |
| xrsdsreclblem 19792 | Lemma for ~ xrsdsreclb . ... |
| xrsdsreclb 19793 | The metric of the extended... |
| cnsubmlem 19794 | Lemma for ~ nn0subm and fr... |
| cnsubglem 19795 | Lemma for ~ resubdrg and f... |
| cnsubrglem 19796 | Lemma for ~ resubdrg and f... |
| cnsubdrglem 19797 | Lemma for ~ resubdrg and f... |
| qsubdrg 19798 | The rational numbers form ... |
| zsubrg 19799 | The integers form a subrin... |
| gzsubrg 19800 | The gaussian integers form... |
| nn0subm 19801 | The nonnegative integers f... |
| rege0subm 19802 | The nonnegative reals form... |
| absabv 19803 | The regular absolute value... |
| zsssubrg 19804 | The integers are a subset ... |
| qsssubdrg 19805 | The rational numbers are a... |
| cnsubrg 19806 | There are no subrings of t... |
| cnmgpabl 19807 | The unit group of the comp... |
| cnmgpid 19808 | The group identity element... |
| cnmsubglem 19809 | Lemma for ~ rpmsubg and fr... |
| rpmsubg 19810 | The positive reals form a ... |
| gzrngunitlem 19811 | Lemma for ~ gzrngunit . (... |
| gzrngunit 19812 | The units on ` ZZ [ _i ] `... |
| gsumfsum 19813 | Relate a group sum on ` CC... |
| regsumfsum 19814 | Relate a group sum on ` ( ... |
| expmhm 19815 | Exponentiation is a monoid... |
| nn0srg 19816 | The nonnegative integers f... |
| rge0srg 19817 | The nonnegative real numbe... |
| zringcrng 19820 | The ring of integers is a ... |
| zringring 19821 | The ring of integers is a ... |
| zringabl 19822 | The ring of integers is an... |
| zringgrp 19823 | The ring of integers is an... |
| zringbas 19824 | The integers are the base ... |
| zringplusg 19825 | The addition operation of ... |
| zringmulg 19826 | The multiplication (group ... |
| zringmulr 19827 | The multiplication operati... |
| zring0 19828 | The neutral element of the... |
| zring1 19829 | The multiplicative neutral... |
| zringnzr 19830 | The ring of integers is a ... |
| dvdsrzring 19831 | Ring divisibility in the r... |
| zringlpirlem1 19832 | Lemma for ~ zringlpir . A... |
| zringlpirlem2 19833 | Lemma for ~ zringlpir . A... |
| zringlpirlem3 19834 | Lemma for ~ zringlpir . A... |
| zringinvg 19835 | The additive inverse of an... |
| zringunit 19836 | The units of ` ZZ ` are th... |
| zringlpir 19837 | The integers are a princip... |
| zringndrg 19838 | The integers are not a div... |
| zringcyg 19839 | The integers are a cyclic ... |
| zringmpg 19840 | The multiplication group o... |
| prmirredlem 19841 | A positive integer is irre... |
| dfprm2 19842 | The positive irreducible e... |
| prmirred 19843 | The irreducible elements o... |
| expghm 19844 | Exponentiation is a group ... |
| mulgghm2 19845 | The powers of a group elem... |
| mulgrhm 19846 | The powers of the element ... |
| mulgrhm2 19847 | The powers of the element ... |
| zrhval 19856 | Define the unique homomorp... |
| zrhval2 19857 | Alternate value of the ` Z... |
| zrhmulg 19858 | Value of the ` ZRHom ` hom... |
| zrhrhmb 19859 | The ` ZRHom ` homomorphism... |
| zrhrhm 19860 | The ` ZRHom ` homomorphism... |
| zrh1 19861 | Interpretation of 1 in a r... |
| zrh0 19862 | Interpretation of 0 in a r... |
| zrhpropd 19863 | The ` ZZ ` ring homomorphi... |
| zlmval 19864 | Augment an abelian group w... |
| zlmlem 19865 | Lemma for ~ zlmbas and ~ z... |
| zlmbas 19866 | Base set of a ` ZZ ` -modu... |
| zlmplusg 19867 | Group operation of a ` ZZ ... |
| zlmmulr 19868 | Ring operation of a ` ZZ `... |
| zlmsca 19869 | Scalar ring of a ` ZZ ` -m... |
| zlmvsca 19870 | Scalar multiplication oper... |
| zlmlmod 19871 | The ` ZZ ` -module operati... |
| zlmassa 19872 | The ` ZZ ` -module operati... |
| chrval 19873 | Definition substitution of... |
| chrcl 19874 | Closure of the characteris... |
| chrid 19875 | The canonical ` ZZ ` ring ... |
| chrdvds 19876 | The ` ZZ ` ring homomorphi... |
| chrcong 19877 | If two integers are congru... |
| chrnzr 19878 | Nonzero rings are precisel... |
| chrrhm 19879 | The characteristic restric... |
| domnchr 19880 | The characteristic of a do... |
| znlidl 19881 | The set ` n ZZ ` is an ide... |
| zncrng2 19882 | The value of the ` Z/nZ ` ... |
| znval 19883 | The value of the ` Z/nZ ` ... |
| znle 19884 | The value of the ` Z/nZ ` ... |
| znval2 19885 | Self-referential expressio... |
| znbaslem 19886 | Lemma for ~ znbas . (Cont... |
| znbaslemOLD 19887 | Obsolete version of ~ znba... |
| znbas2 19888 | The base set of ` Z/nZ ` i... |
| znadd 19889 | The additive structure of ... |
| znmul 19890 | The multiplicative structu... |
| znzrh 19891 | The ` ZZ ` ring homomorphi... |
| znbas 19892 | The base set of ` Z/nZ ` s... |
| zncrng 19893 | ` Z/nZ ` is a commutative ... |
| znzrh2 19894 | The ` ZZ ` ring homomorphi... |
| znzrhval 19895 | The ` ZZ ` ring homomorphi... |
| znzrhfo 19896 | The ` ZZ ` ring homomorphi... |
| zncyg 19897 | The group ` ZZ / n ZZ ` is... |
| zndvds 19898 | Express equality of equiva... |
| zndvds0 19899 | Special case of ~ zndvds w... |
| znf1o 19900 | The function ` F ` enumera... |
| zzngim 19901 | The ` ZZ ` ring homomorphi... |
| znle2 19902 | The ordering of the ` Z/nZ... |
| znleval 19903 | The ordering of the ` Z/nZ... |
| znleval2 19904 | The ordering of the ` Z/nZ... |
| zntoslem 19905 | Lemma for ~ zntos . (Cont... |
| zntos 19906 | The ` Z/nZ ` structure is ... |
| znhash 19907 | The ` Z/nZ ` structure has... |
| znfi 19908 | The ` Z/nZ ` structure is ... |
| znfld 19909 | The ` Z/nZ ` structure is ... |
| znidomb 19910 | The ` Z/nZ ` structure is ... |
| znchr 19911 | Cyclic rings are defined b... |
| znunit 19912 | The units of ` Z/nZ ` are ... |
| znunithash 19913 | The size of the unit group... |
| znrrg 19914 | The regular elements of ` ... |
| cygznlem1 19915 | Lemma for ~ cygzn . (Cont... |
| cygznlem2a 19916 | Lemma for ~ cygzn . (Cont... |
| cygznlem2 19917 | Lemma for ~ cygzn . (Cont... |
| cygznlem3 19918 | A cyclic group with ` n ` ... |
| cygzn 19919 | A cyclic group with ` n ` ... |
| cygth 19920 | The "fundamental theorem o... |
| cyggic 19921 | Cyclic groups are isomorph... |
| frgpcyg 19922 | A free group is cyclic iff... |
| cnmsgnsubg 19923 | The signs form a multiplic... |
| cnmsgnbas 19924 | The base set of the sign s... |
| cnmsgngrp 19925 | The group of signs under m... |
| psgnghm 19926 | The sign is a homomorphism... |
| psgnghm2 19927 | The sign is a homomorphism... |
| psgninv 19928 | The sign of a permutation ... |
| psgnco 19929 | Multiplicativity of the pe... |
| zrhpsgnmhm 19930 | Embedding of permutation s... |
| zrhpsgninv 19931 | The embedded sign of a per... |
| evpmss 19932 | Even permutations are perm... |
| psgnevpmb 19933 | A class is an even permuta... |
| psgnodpm 19934 | A permutation which is odd... |
| psgnevpm 19935 | A permutation which is eve... |
| psgnodpmr 19936 | If a permutation has sign ... |
| zrhpsgnevpm 19937 | The sign of an even permut... |
| zrhpsgnodpm 19938 | The sign of an odd permuta... |
| zrhcofipsgn 19939 | Composition of a ` ZRHom `... |
| zrhpsgnelbas 19940 | Embedding of permutation s... |
| zrhcopsgnelbas 19941 | Embedding of permutation s... |
| evpmodpmf1o 19942 | The function for performin... |
| pmtrodpm 19943 | A transposition is an odd ... |
| psgnfix1 19944 | A permutation of a finite ... |
| psgnfix2 19945 | A permutation of a finite ... |
| psgndiflemB 19946 | Lemma 1 for ~ psgndif . (... |
| psgndiflemA 19947 | Lemma 2 for ~ psgndif . (... |
| psgndif 19948 | Embedding of permutation s... |
| zrhcopsgndif 19949 | Embedding of permutation s... |
| rebase 19952 | The base of the field of r... |
| remulg 19953 | The multiplication (group ... |
| resubdrg 19954 | The real numbers form a di... |
| resubgval 19955 | Subtraction in the field o... |
| replusg 19956 | The addition operation of ... |
| remulr 19957 | The multiplication operati... |
| re0g 19958 | The neutral element of the... |
| re1r 19959 | The multiplicative neutral... |
| rele2 19960 | The ordering relation of t... |
| relt 19961 | The ordering relation of t... |
| reds 19962 | The distance of the field ... |
| redvr 19963 | The division operation of ... |
| retos 19964 | The real numbers are a tot... |
| refld 19965 | The real numbers form a fi... |
| refldcj 19966 | The conjugation operation ... |
| recrng 19967 | The real numbers form a st... |
| regsumsupp 19968 | The group sum over the rea... |
| isphl 19973 | The predicate "is a genera... |
| phllvec 19974 | A pre-Hilbert space is a l... |
| phllmod 19975 | A pre-Hilbert space is a l... |
| phlsrng 19976 | The scalar ring of a pre-H... |
| phllmhm 19977 | The inner product of a pre... |
| ipcl 19978 | Closure of the inner produ... |
| ipcj 19979 | Conjugate of an inner prod... |
| iporthcom 19980 | Orthogonality (meaning inn... |
| ip0l 19981 | Inner product with a zero ... |
| ip0r 19982 | Inner product with a zero ... |
| ipeq0 19983 | The inner product of a vec... |
| ipdir 19984 | Distributive law for inner... |
| ipdi 19985 | Distributive law for inner... |
| ip2di 19986 | Distributive law for inner... |
| ipsubdir 19987 | Distributive law for inner... |
| ipsubdi 19988 | Distributive law for inner... |
| ip2subdi 19989 | Distributive law for inner... |
| ipass 19990 | Associative law for inner ... |
| ipassr 19991 | "Associative" law for seco... |
| ipassr2 19992 | "Associative" law for inne... |
| ipffval 19993 | The inner product operatio... |
| ipfval 19994 | The inner product operatio... |
| ipfeq 19995 | If the inner product opera... |
| ipffn 19996 | The inner product operatio... |
| phlipf 19997 | The inner product operatio... |
| ip2eq 19998 | Two vectors are equal iff ... |
| isphld 19999 | Properties that determine ... |
| phlpropd 20000 | If two structures have the... |
| ssipeq 20001 | The inner product on a sub... |
| phssipval 20002 | The inner product on a sub... |
| phssip 20003 | The inner product (as a fu... |
| ocvfval 20010 | The orthocomplement operat... |
| ocvval 20011 | Value of the orthocompleme... |
| elocv 20012 | Elementhood in the orthoco... |
| ocvi 20013 | Property of a member of th... |
| ocvss 20014 | The orthocomplement of a s... |
| ocvocv 20015 | A set is contained in its ... |
| ocvlss 20016 | The orthocomplement of a s... |
| ocv2ss 20017 | Orthocomplements reverse s... |
| ocvin 20018 | An orthocomplement has tri... |
| ocvsscon 20019 | Two ways to say that ` S `... |
| ocvlsp 20020 | The orthocomplement of a l... |
| ocv0 20021 | The orthocomplement of the... |
| ocvz 20022 | The orthocomplement of the... |
| ocv1 20023 | The orthocomplement of the... |
| unocv 20024 | The orthocomplement of a u... |
| iunocv 20025 | The orthocomplement of an ... |
| cssval 20026 | The set of closed subspace... |
| iscss 20027 | The predicate "is a closed... |
| cssi 20028 | Property of a closed subsp... |
| cssss 20029 | A closed subspace is a sub... |
| iscss2 20030 | It is sufficient to prove ... |
| ocvcss 20031 | The orthocomplement of any... |
| cssincl 20032 | The zero subspace is a clo... |
| css0 20033 | The zero subspace is a clo... |
| css1 20034 | The whole space is a close... |
| csslss 20035 | A closed subspace of a pre... |
| lsmcss 20036 | A subset of a pre-Hilbert ... |
| cssmre 20037 | The closed subspaces of a ... |
| mrccss 20038 | The Moore closure correspo... |
| thlval 20039 | Value of the Hilbert latti... |
| thlbas 20040 | Base set of the Hilbert la... |
| thlle 20041 | Ordering on the Hilbert la... |
| thlleval 20042 | Ordering on the Hilbert la... |
| thloc 20043 | Orthocomplement on the Hil... |
| pjfval 20050 | The value of the projectio... |
| pjdm 20051 | A subspace is in the domai... |
| pjpm 20052 | The projection map is a pa... |
| pjfval2 20053 | Value of the projection ma... |
| pjval 20054 | Value of the projection ma... |
| pjdm2 20055 | A subspace is in the domai... |
| pjff 20056 | A projection is a linear o... |
| pjf 20057 | A projection is a function... |
| pjf2 20058 | A projection is a function... |
| pjfo 20059 | A projection is a surjecti... |
| pjcss 20060 | A projection subspace is a... |
| ocvpj 20061 | The orthocomplement of a p... |
| ishil 20062 | The predicate "is a Hilber... |
| ishil2 20063 | The predicate "is a Hilber... |
| isobs 20064 | The predicate "is an ortho... |
| obsip 20065 | The inner product of two e... |
| obsipid 20066 | A basis element has unit l... |
| obsrcl 20067 | Reverse closure for an ort... |
| obsss 20068 | An orthonormal basis is a ... |
| obsne0 20069 | A basis element is nonzero... |
| obsocv 20070 | An orthonormal basis has t... |
| obs2ocv 20071 | The double orthocomplement... |
| obselocv 20072 | A basis element is in the ... |
| obs2ss 20073 | A basis has no proper subs... |
| obslbs 20074 | An orthogonal basis is a l... |
| reldmdsmm 20077 | The direct sum is a well-b... |
| dsmmval 20078 | Value of the module direct... |
| dsmmbase 20079 | Base set of the module dir... |
| dsmmval2 20080 | Self-referential definitio... |
| dsmmbas2 20081 | Base set of the direct sum... |
| dsmmfi 20082 | For finite products, the d... |
| dsmmelbas 20083 | Membership in the finitely... |
| dsmm0cl 20084 | The all-zero vector is con... |
| dsmmacl 20085 | The finite hull is closed ... |
| prdsinvgd2 20086 | Negation of a single coord... |
| dsmmsubg 20087 | The finite hull of a produ... |
| dsmmlss 20088 | The finite hull of a produ... |
| dsmmlmod 20089 | The direct sum of a family... |
| frlmval 20092 | Value of the free module. ... |
| frlmlmod 20093 | The free module is a modul... |
| frlmpws 20094 | The free module as a restr... |
| frlmlss 20095 | The base set of the free m... |
| frlmpwsfi 20096 | The finite free module is ... |
| frlmsca 20097 | The ring of scalars of a f... |
| frlm0 20098 | Zero in a free module (rin... |
| frlmbas 20099 | Base set of the free modul... |
| frlmelbas 20100 | Membership in the base set... |
| frlmrcl 20101 | If a free module is inhabi... |
| frlmbasfsupp 20102 | Elements of the free modul... |
| frlmbasmap 20103 | Elements of the free modul... |
| frlmbasf 20104 | Elements of the free modul... |
| frlmfibas 20105 | The base set of the finite... |
| elfrlmbasn0 20106 | If the dimension of a free... |
| frlmplusgval 20107 | Addition in a free module.... |
| frlmsubgval 20108 | Subtraction in a free modu... |
| frlmvscafval 20109 | Scalar multiplication in a... |
| frlmvscaval 20110 | Scalar multiplication in a... |
| frlmgsum 20111 | Finite commutative sums in... |
| frlmsplit2 20112 | Restriction is homomorphic... |
| frlmsslss 20113 | A subset of a free module ... |
| frlmsslss2 20114 | A subset of a free module ... |
| frlmbas3 20115 | An element of the base set... |
| mpt2frlmd 20116 | Elements of the free modul... |
| frlmip 20117 | The inner product of a fre... |
| frlmipval 20118 | The inner product of a fre... |
| frlmphllem 20119 | Lemma for ~ frlmphl . (Co... |
| frlmphl 20120 | Conditions for a free modu... |
| uvcfval 20123 | Value of the unit-vector g... |
| uvcval 20124 | Value of a single unit vec... |
| uvcvval 20125 | Value of a unit vector coo... |
| uvcvvcl 20126 | A coodinate of a unit vect... |
| uvcvvcl2 20127 | A unit vector coordinate i... |
| uvcvv1 20128 | The unit vector is one at ... |
| uvcvv0 20129 | The unit vector is zero at... |
| uvcff 20130 | Domain and range of the un... |
| uvcf1 20131 | In a nonzero ring, each un... |
| uvcresum 20132 | Any element of a free modu... |
| frlmssuvc1 20133 | A scalar multiple of a uni... |
| frlmssuvc2 20134 | A nonzero scalar multiple ... |
| frlmsslsp 20135 | A subset of a free module ... |
| frlmlbs 20136 | The unit vectors comprise ... |
| frlmup1 20137 | Any assignment of unit vec... |
| frlmup2 20138 | The evaluation map has the... |
| frlmup3 20139 | The range of such an evalu... |
| frlmup4 20140 | Universal property of the ... |
| ellspd 20141 | The elements of the span o... |
| elfilspd 20142 | Simplified version of ~ el... |
| rellindf 20147 | The independent-family pre... |
| islinds 20148 | Property of an independent... |
| linds1 20149 | An independent set of vect... |
| linds2 20150 | An independent set of vect... |
| islindf 20151 | Property of an independent... |
| islinds2 20152 | Expanded property of an in... |
| islindf2 20153 | Property of an independent... |
| lindff 20154 | Functional property of a l... |
| lindfind 20155 | A linearly independent fam... |
| lindsind 20156 | A linearly independent set... |
| lindfind2 20157 | In a linearly independent ... |
| lindsind2 20158 | In a linearly independent ... |
| lindff1 20159 | A linearly independent fam... |
| lindfrn 20160 | The range of an independen... |
| f1lindf 20161 | Rearranging and deleting e... |
| lindfres 20162 | Any restriction of an inde... |
| lindsss 20163 | Any subset of an independe... |
| f1linds 20164 | A family constructed from ... |
| islindf3 20165 | In a nonzero ring, indepen... |
| lindfmm 20166 | Linear independence of a f... |
| lindsmm 20167 | Linear independence of a s... |
| lindsmm2 20168 | The monomorphic image of a... |
| lsslindf 20169 | Linear independence is unc... |
| lsslinds 20170 | Linear independence is unc... |
| islbs4 20171 | A basis is an independent ... |
| lbslinds 20172 | A basis is independent. (... |
| islinds3 20173 | A subset is linearly indep... |
| islinds4 20174 | A set is independent in a ... |
| lmimlbs 20175 | The isomorphic image of a ... |
| lmiclbs 20176 | Having a basis is an isomo... |
| islindf4 20177 | A family is independent if... |
| islindf5 20178 | A family is independent if... |
| indlcim 20179 | An independent, spanning f... |
| lbslcic 20180 | A module with a basis is i... |
| lmisfree 20181 | A module has a basis iff i... |
| lvecisfrlm 20182 | Every vector space is isom... |
| lmimco 20183 | The composition of two iso... |
| lmictra 20184 | Module isomorphism is tran... |
| uvcf1o 20185 | In a nonzero ring, the map... |
| uvcendim 20186 | In a nonzero ring, the num... |
| frlmisfrlm 20187 | A free module is isomorphi... |
| frlmiscvec 20188 | Every free module is isomo... |
| mamufval 20191 | Functional value of the ma... |
| mamuval 20192 | Multiplication of two matr... |
| mamufv 20193 | A cell in the multiplicati... |
| mamudm 20194 | The domain of the matrix m... |
| mamufacex 20195 | Every solution of the equa... |
| mamures 20196 | Rows in a matrix product a... |
| mndvcl 20197 | Tuple-wise additive closur... |
| mndvass 20198 | Tuple-wise associativity i... |
| mndvlid 20199 | Tuple-wise left identity i... |
| mndvrid 20200 | Tuple-wise right identity ... |
| grpvlinv 20201 | Tuple-wise left inverse in... |
| grpvrinv 20202 | Tuple-wise right inverse i... |
| mhmvlin 20203 | Tuple extension of monoid ... |
| ringvcl 20204 | Tuple-wise multiplication ... |
| gsumcom3 20205 | A commutative law for fini... |
| gsumcom3fi 20206 | A commutative law for fini... |
| mamucl 20207 | Operation closure of matri... |
| mamuass 20208 | Matrix multiplication is a... |
| mamudi 20209 | Matrix multiplication dist... |
| mamudir 20210 | Matrix multiplication dist... |
| mamuvs1 20211 | Matrix multiplication dist... |
| mamuvs2 20212 | Matrix multiplication dist... |
| matbas0pc 20215 | There is no matrix with a ... |
| matbas0 20216 | There is no matrix for a n... |
| matval 20217 | Value of the matrix algebr... |
| matrcl 20218 | Reverse closure for the ma... |
| matbas 20219 | The matrix ring has the sa... |
| matplusg 20220 | The matrix ring has the sa... |
| matsca 20221 | The matrix ring has the sa... |
| matvsca 20222 | The matrix ring has the sa... |
| mat0 20223 | The matrix ring has the sa... |
| matinvg 20224 | The matrix ring has the sa... |
| mat0op 20225 | Value of a zero matrix as ... |
| matsca2 20226 | The scalars of the matrix ... |
| matbas2 20227 | The base set of the matrix... |
| matbas2i 20228 | A matrix is a function. (... |
| matbas2d 20229 | The base set of the matrix... |
| eqmat 20230 | Two square matrices of the... |
| matecl 20231 | Each entry (according to W... |
| matecld 20232 | Each entry (according to W... |
| matplusg2 20233 | Addition in the matrix rin... |
| matvsca2 20234 | Scalar multiplication in t... |
| matlmod 20235 | The matrix ring is a linea... |
| matgrp 20236 | The matrix ring is a group... |
| matvscl 20237 | Closure of the scalar mult... |
| matsubg 20238 | The matrix ring has the sa... |
| matplusgcell 20239 | Addition in the matrix rin... |
| matsubgcell 20240 | Subtraction in the matrix ... |
| matinvgcell 20241 | Additive inversion in the ... |
| matvscacell 20242 | Scalar multiplication in t... |
| matgsum 20243 | Finite commutative sums in... |
| matmulr 20244 | Multiplication in the matr... |
| mamumat1cl 20245 | The identity matrix (as op... |
| mat1comp 20246 | The components of the iden... |
| mamulid 20247 | The identity matrix (as op... |
| mamurid 20248 | The identity matrix (as op... |
| matring 20249 | Existence of the matrix ri... |
| matassa 20250 | Existence of the matrix al... |
| matmulcell 20251 | Multiplication in the matr... |
| mpt2matmul 20252 | Multiplication of two N x ... |
| mat1 20253 | Value of an identity matri... |
| mat1ov 20254 | Entries of an identity mat... |
| mat1bas 20255 | The identity matrix is a m... |
| matsc 20256 | The identity matrix multip... |
| ofco2 20257 | Distribution law for the f... |
| oftpos 20258 | The transposition of the v... |
| mattposcl 20259 | The transpose of a square ... |
| mattpostpos 20260 | The transpose of the trans... |
| mattposvs 20261 | The transposition of a mat... |
| mattpos1 20262 | The transposition of the i... |
| tposmap 20263 | The transposition of an I ... |
| mamutpos 20264 | Behavior of transposes in ... |
| mattposm 20265 | Multiplying two transposed... |
| matgsumcl 20266 | Closure of a group sum ove... |
| madetsumid 20267 | The identity summand in th... |
| matepmcl 20268 | Each entry of a matrix wit... |
| matepm2cl 20269 | Each entry of a matrix wit... |
| madetsmelbas 20270 | A summand of the determina... |
| madetsmelbas2 20271 | A summand of the determina... |
| mat0dimbas0 20272 | The empty set is the one a... |
| mat0dim0 20273 | The zero of the algebra of... |
| mat0dimid 20274 | The identity of the algebr... |
| mat0dimscm 20275 | The scalar multiplication ... |
| mat0dimcrng 20276 | The algebra of matrices wi... |
| mat1dimelbas 20277 | A matrix with dimension 1 ... |
| mat1dimbas 20278 | A matrix with dimension 1 ... |
| mat1dim0 20279 | The zero of the algebra of... |
| mat1dimid 20280 | The identity of the algebr... |
| mat1dimscm 20281 | The scalar multiplication ... |
| mat1dimmul 20282 | The ring multiplication in... |
| mat1dimcrng 20283 | The algebra of matrices wi... |
| mat1f1o 20284 | There is a 1-1 function fr... |
| mat1rhmval 20285 | The value of the ring homo... |
| mat1rhmelval 20286 | The value of the ring homo... |
| mat1rhmcl 20287 | The value of the ring homo... |
| mat1f 20288 | There is a function from a... |
| mat1ghm 20289 | There is a group homomorph... |
| mat1mhm 20290 | There is a monoid homomorp... |
| mat1rhm 20291 | There is a ring homomorphi... |
| mat1rngiso 20292 | There is a ring isomorphis... |
| mat1ric 20293 | A ring is isomorphic to th... |
| dmatval 20298 | The set of ` N ` x ` N ` d... |
| dmatel 20299 | A ` N ` x ` N ` diagonal m... |
| dmatmat 20300 | An ` N ` x ` N ` diagonal ... |
| dmatid 20301 | The identity matrix is a d... |
| dmatelnd 20302 | An extradiagonal entry of ... |
| dmatmul 20303 | The product of two diagona... |
| dmatsubcl 20304 | The difference of two diag... |
| dmatsgrp 20305 | The set of diagonal matric... |
| dmatmulcl 20306 | The product of two diagona... |
| dmatsrng 20307 | The set of diagonal matric... |
| dmatcrng 20308 | The subring of diagonal ma... |
| dmatscmcl 20309 | The multiplication of a di... |
| scmatval 20310 | The set of ` N ` x ` N ` s... |
| scmatel 20311 | An ` N ` x ` N ` scalar ma... |
| scmatscmid 20312 | A scalar matrix can be exp... |
| scmatscmide 20313 | An entry of a scalar matri... |
| scmatscmiddistr 20314 | Distributive law for scala... |
| scmatmat 20315 | An ` N ` x ` N ` scalar ma... |
| scmate 20316 | An entry of an ` N ` x ` N... |
| scmatmats 20317 | The set of an ` N ` x ` N ... |
| scmateALT 20318 | Alternate proof of ~ scmat... |
| scmatscm 20319 | The multiplication of a ma... |
| scmatid 20320 | The identity matrix is a s... |
| scmatdmat 20321 | A scalar matrix is a diago... |
| scmataddcl 20322 | The sum of two scalar matr... |
| scmatsubcl 20323 | The difference of two scal... |
| scmatmulcl 20324 | The product of two scalar ... |
| scmatsgrp 20325 | The set of scalar matrices... |
| scmatsrng 20326 | The set of scalar matrices... |
| scmatcrng 20327 | The subring of scalar matr... |
| scmatsgrp1 20328 | The set of scalar matrices... |
| scmatsrng1 20329 | The set of scalar matrices... |
| smatvscl 20330 | Closure of the scalar mult... |
| scmatlss 20331 | The set of scalar matrices... |
| scmatstrbas 20332 | The set of scalar matrices... |
| scmatrhmval 20333 | The value of the ring homo... |
| scmatrhmcl 20334 | The value of the ring homo... |
| scmatf 20335 | There is a function from a... |
| scmatfo 20336 | There is a function from a... |
| scmatf1 20337 | There is a 1-1 function fr... |
| scmatf1o 20338 | There is a bijection betwe... |
| scmatghm 20339 | There is a group homomorph... |
| scmatmhm 20340 | There is a monoid homomorp... |
| scmatrhm 20341 | There is a ring homomorphi... |
| scmatrngiso 20342 | There is a ring isomorphis... |
| scmatric 20343 | A ring is isomorphic to ev... |
| mat0scmat 20344 | The empty matrix over a ri... |
| mat1scmat 20345 | A 1-dimensional matrix ove... |
| mvmulfval 20348 | Functional value of the ma... |
| mvmulval 20349 | Multiplication of a vector... |
| mvmulfv 20350 | A cell/element in the vect... |
| mavmulval 20351 | Multiplication of a vector... |
| mavmulfv 20352 | A cell/element in the vect... |
| mavmulcl 20353 | Multiplication of an NxN m... |
| 1mavmul 20354 | Multiplication of the iden... |
| mavmulass 20355 | Associativity of the multi... |
| mavmuldm 20356 | The domain of the matrix v... |
| mavmulsolcl 20357 | Every solution of the equa... |
| mavmul0 20358 | Multiplication of a 0-dime... |
| mavmul0g 20359 | The result of the 0-dimens... |
| mvmumamul1 20360 | The multiplication of an M... |
| mavmumamul1 20361 | The multiplication of an N... |
| marrepfval 20366 | First substitution for the... |
| marrepval0 20367 | Second substitution for th... |
| marrepval 20368 | Third substitution for the... |
| marrepeval 20369 | An entry of a matrix with ... |
| marrepcl 20370 | Closure of the row replace... |
| marepvfval 20371 | First substitution for the... |
| marepvval0 20372 | Second substitution for th... |
| marepvval 20373 | Third substitution for the... |
| marepveval 20374 | An entry of a matrix with ... |
| marepvcl 20375 | Closure of the column repl... |
| ma1repvcl 20376 | Closure of the column repl... |
| ma1repveval 20377 | An entry of an identity ma... |
| mulmarep1el 20378 | Element by element multipl... |
| mulmarep1gsum1 20379 | The sum of element by elem... |
| mulmarep1gsum2 20380 | The sum of element by elem... |
| 1marepvmarrepid 20381 | Replacing the ith row by 0... |
| submabas 20384 | Any subset of the index se... |
| submafval 20385 | First substitution for a s... |
| submaval0 20386 | Second substitution for a ... |
| submaval 20387 | Third substitution for a s... |
| submaeval 20388 | An entry of a submatrix of... |
| 1marepvsma1 20389 | The submatrix of the ident... |
| mdetfval 20392 | First substitution for the... |
| mdetleib 20393 | Full substitution of our d... |
| mdetleib2 20394 | Leibniz' formula can also ... |
| nfimdetndef 20395 | The determinant is not def... |
| mdetfval1 20396 | First substitution of an a... |
| mdetleib1 20397 | Full substitution of an al... |
| mdet0pr 20398 | The determinant for 0-dime... |
| mdet0f1o 20399 | The determinant for 0-dime... |
| mdet0fv0 20400 | The determinant of a 0-dim... |
| mdetf 20401 | Functionality of the deter... |
| mdetcl 20402 | The determinant evaluates ... |
| m1detdiag 20403 | The determinant of a 1-dim... |
| mdetdiaglem 20404 | Lemma for ~ mdetdiag . Pr... |
| mdetdiag 20405 | The determinant of a diago... |
| mdetdiagid 20406 | The determinant of a diago... |
| mdet1 20407 | The determinant of the ide... |
| mdetrlin 20408 | The determinant function i... |
| mdetrsca 20409 | The determinant function i... |
| mdetrsca2 20410 | The determinant function i... |
| mdetr0 20411 | The determinant of a matri... |
| mdet0 20412 | The determinant of the zer... |
| mdetrlin2 20413 | The determinant function i... |
| mdetralt 20414 | The determinant function i... |
| mdetralt2 20415 | The determinant function i... |
| mdetero 20416 | The determinant function i... |
| mdettpos 20417 | Determinant is invariant u... |
| mdetunilem1 20418 | Lemma for ~ mdetuni . (Co... |
| mdetunilem2 20419 | Lemma for ~ mdetuni . (Co... |
| mdetunilem3 20420 | Lemma for ~ mdetuni . (Co... |
| mdetunilem4 20421 | Lemma for ~ mdetuni . (Co... |
| mdetunilem5 20422 | Lemma for ~ mdetuni . (Co... |
| mdetunilem6 20423 | Lemma for ~ mdetuni . (Co... |
| mdetunilem7 20424 | Lemma for ~ mdetuni . (Co... |
| mdetunilem8 20425 | Lemma for ~ mdetuni . (Co... |
| mdetunilem9 20426 | Lemma for ~ mdetuni . (Co... |
| mdetuni0 20427 | Lemma for ~ mdetuni . (Co... |
| mdetuni 20428 | According to the definitio... |
| mdetmul 20429 | Multiplicativity of the de... |
| m2detleiblem1 20430 | Lemma 1 for ~ m2detleib . ... |
| m2detleiblem5 20431 | Lemma 5 for ~ m2detleib . ... |
| m2detleiblem6 20432 | Lemma 6 for ~ m2detleib . ... |
| m2detleiblem7 20433 | Lemma 7 for ~ m2detleib . ... |
| m2detleiblem2 20434 | Lemma 2 for ~ m2detleib . ... |
| m2detleiblem3 20435 | Lemma 3 for ~ m2detleib . ... |
| m2detleiblem4 20436 | Lemma 4 for ~ m2detleib . ... |
| m2detleib 20437 | Leibniz' Formula for 2x2-m... |
| mndifsplit 20442 | Lemma for ~ maducoeval2 . ... |
| madufval 20443 | First substitution for the... |
| maduval 20444 | Second substitution for th... |
| maducoeval 20445 | An entry of the adjunct (c... |
| maducoeval2 20446 | An entry of the adjunct (c... |
| maduf 20447 | Creating the adjunct of ma... |
| madutpos 20448 | The adjuct of a transposed... |
| madugsum 20449 | The determinant of a matri... |
| madurid 20450 | Multiplying a matrix with ... |
| madulid 20451 | Multiplying the adjunct of... |
| minmar1fval 20452 | First substitution for the... |
| minmar1val0 20453 | Second substitution for th... |
| minmar1val 20454 | Third substitution for the... |
| minmar1eval 20455 | An entry of a matrix for a... |
| minmar1marrep 20456 | The minor matrix is a spec... |
| minmar1cl 20457 | Closure of the row replace... |
| maducoevalmin1 20458 | The coefficients of an adj... |
| symgmatr01lem 20459 | Lemma for ~ symgmatr01 . ... |
| symgmatr01 20460 | Applying a permutation tha... |
| gsummatr01lem1 20461 | Lemma A for ~ gsummatr01 .... |
| gsummatr01lem2 20462 | Lemma B for ~ gsummatr01 .... |
| gsummatr01lem3 20463 | Lemma 1 for ~ gsummatr01 .... |
| gsummatr01lem4 20464 | Lemma 2 for ~ gsummatr01 .... |
| gsummatr01 20465 | Lemma 1 for ~ smadiadetlem... |
| marep01ma 20466 | Replacing a row of a squar... |
| smadiadetlem0 20467 | Lemma 0 for ~ smadiadet : ... |
| smadiadetlem1 20468 | Lemma 1 for ~ smadiadet : ... |
| smadiadetlem1a 20469 | Lemma 1a for ~ smadiadet :... |
| smadiadetlem2 20470 | Lemma 2 for ~ smadiadet : ... |
| smadiadetlem3lem0 20471 | Lemma 0 for ~ smadiadetlem... |
| smadiadetlem3lem1 20472 | Lemma 1 for ~ smadiadetlem... |
| smadiadetlem3lem2 20473 | Lemma 2 for ~ smadiadetlem... |
| smadiadetlem3 20474 | Lemma 3 for ~ smadiadet . ... |
| smadiadetlem4 20475 | Lemma 4 for ~ smadiadet . ... |
| smadiadet 20476 | The determinant of a subma... |
| smadiadetglem1 20477 | Lemma 1 for ~ smadiadetg .... |
| smadiadetglem2 20478 | Lemma 2 for ~ smadiadetg .... |
| smadiadetg 20479 | The determinant of a squar... |
| smadiadetg0 20480 | Lemma for ~ smadiadetr : v... |
| smadiadetr 20481 | The determinant of a squar... |
| invrvald 20482 | If a matrix multiplied wit... |
| matinv 20483 | The inverse of a matrix is... |
| matunit 20484 | A matrix is a unit in the ... |
| slesolvec 20485 | Every solution of a system... |
| slesolinv 20486 | The solution of a system o... |
| slesolinvbi 20487 | The solution of a system o... |
| slesolex 20488 | Every system of linear equ... |
| cramerimplem1 20489 | Lemma 1 for ~ cramerimp : ... |
| cramerimplem2 20490 | Lemma 2 for ~ cramerimp : ... |
| cramerimplem3 20491 | Lemma 3 for ~ cramerimp : ... |
| cramerimp 20492 | One direction of Cramer's ... |
| cramerlem1 20493 | Lemma 1 for ~ cramer . (C... |
| cramerlem2 20494 | Lemma 2 for ~ cramer . (C... |
| cramerlem3 20495 | Lemma 3 for ~ cramer . (C... |
| cramer0 20496 | Special case of Cramer's r... |
| cramer 20497 | Cramer's rule. According ... |
| pmatring 20498 | The set of polynomial matr... |
| pmatlmod 20499 | The set of polynomial matr... |
| pmat0op 20500 | The zero polynomial matrix... |
| pmat1op 20501 | The identity polynomial ma... |
| pmat1ovd 20502 | Entries of the identity po... |
| pmat0opsc 20503 | The zero polynomial matrix... |
| pmat1opsc 20504 | The identity polynomial ma... |
| pmat1ovscd 20505 | Entries of the identity po... |
| pmatcoe1fsupp 20506 | For a polynomial matrix th... |
| 1pmatscmul 20507 | The scalar product of the ... |
| cpmat 20514 | Value of the constructor o... |
| cpmatpmat 20515 | A constant polynomial matr... |
| cpmatel 20516 | Property of a constant pol... |
| cpmatelimp 20517 | Implication of a set being... |
| cpmatel2 20518 | Another property of a cons... |
| cpmatelimp2 20519 | Another implication of a s... |
| 1elcpmat 20520 | The identity of the ring o... |
| cpmatacl 20521 | The set of all constant po... |
| cpmatinvcl 20522 | The set of all constant po... |
| cpmatmcllem 20523 | Lemma for ~ cpmatmcl . (C... |
| cpmatmcl 20524 | The set of all constant po... |
| cpmatsubgpmat 20525 | The set of all constant po... |
| cpmatsrgpmat 20526 | The set of all constant po... |
| 0elcpmat 20527 | The zero of the ring of al... |
| mat2pmatfval 20528 | Value of the matrix transf... |
| mat2pmatval 20529 | The result of a matrix tra... |
| mat2pmatvalel 20530 | A (matrix) element of the ... |
| mat2pmatbas 20531 | The result of a matrix tra... |
| mat2pmatbas0 20532 | The result of a matrix tra... |
| mat2pmatf 20533 | The matrix transformation ... |
| mat2pmatf1 20534 | The matrix transformation ... |
| mat2pmatghm 20535 | The transformation of matr... |
| mat2pmatmul 20536 | The transformation of matr... |
| mat2pmat1 20537 | The transformation of the ... |
| mat2pmatmhm 20538 | The transformation of matr... |
| mat2pmatrhm 20539 | The transformation of matr... |
| mat2pmatlin 20540 | The transformation of matr... |
| 0mat2pmat 20541 | The transformed zero matri... |
| idmatidpmat 20542 | The transformed identity m... |
| d0mat2pmat 20543 | The transformed empty set ... |
| d1mat2pmat 20544 | The transformation of a ma... |
| mat2pmatscmxcl 20545 | A transformed matrix multi... |
| m2cpm 20546 | The result of a matrix tra... |
| m2cpmf 20547 | The matrix transformation ... |
| m2cpmf1 20548 | The matrix transformation ... |
| m2cpmghm 20549 | The transformation of matr... |
| m2cpmmhm 20550 | The transformation of matr... |
| m2cpmrhm 20551 | The transformation of matr... |
| m2pmfzmap 20552 | The transformed values of ... |
| m2pmfzgsumcl 20553 | Closure of the sum of scal... |
| cpm2mfval 20554 | Value of the inverse matri... |
| cpm2mval 20555 | The result of an inverse m... |
| cpm2mvalel 20556 | A (matrix) element of the ... |
| cpm2mf 20557 | The inverse matrix transfo... |
| m2cpminvid 20558 | The inverse transformation... |
| m2cpminvid2lem 20559 | Lemma for ~ m2cpminvid2 . ... |
| m2cpminvid2 20560 | The transformation applied... |
| m2cpmfo 20561 | The matrix transformation ... |
| m2cpmf1o 20562 | The matrix transformation ... |
| m2cpmrngiso 20563 | The transformation of matr... |
| matcpmric 20564 | The ring of matrices over ... |
| m2cpminv 20565 | The inverse matrix transfo... |
| m2cpminv0 20566 | The inverse matrix transfo... |
| decpmatval0 20569 | The matrix consisting of t... |
| decpmatval 20570 | The matrix consisting of t... |
| decpmate 20571 | An entry of the matrix con... |
| decpmatcl 20572 | Closure of the decompositi... |
| decpmataa0 20573 | The matrix consisting of t... |
| decpmatfsupp 20574 | The mapping to the matrice... |
| decpmatid 20575 | The matrix consisting of t... |
| decpmatmullem 20576 | Lemma for ~ decpmatmul . ... |
| decpmatmul 20577 | The matrix consisting of t... |
| decpmatmulsumfsupp 20578 | Lemma 0 for ~ pm2mpmhm . ... |
| pmatcollpw1lem1 20579 | Lemma 1 for ~ pmatcollpw1 ... |
| pmatcollpw1lem2 20580 | Lemma 2 for ~ pmatcollpw1 ... |
| pmatcollpw1 20581 | Write a polynomial matrix ... |
| pmatcollpw2lem 20582 | Lemma for ~ pmatcollpw2 . ... |
| pmatcollpw2 20583 | Write a polynomial matrix ... |
| monmatcollpw 20584 | The matrix consisting of t... |
| pmatcollpwlem 20585 | Lemma for ~ pmatcollpw . ... |
| pmatcollpw 20586 | Write a polynomial matrix ... |
| pmatcollpwfi 20587 | Write a polynomial matrix ... |
| pmatcollpw3lem 20588 | Lemma for ~ pmatcollpw3 an... |
| pmatcollpw3 20589 | Write a polynomial matrix ... |
| pmatcollpw3fi 20590 | Write a polynomial matrix ... |
| pmatcollpw3fi1lem1 20591 | Lemma 1 for ~ pmatcollpw3f... |
| pmatcollpw3fi1lem2 20592 | Lemma 2 for ~ pmatcollpw3f... |
| pmatcollpw3fi1 20593 | Write a polynomial matrix ... |
| pmatcollpwscmatlem1 20594 | Lemma 1 for ~ pmatcollpwsc... |
| pmatcollpwscmatlem2 20595 | Lemma 2 for ~ pmatcollpwsc... |
| pmatcollpwscmat 20596 | Write a scalar matrix over... |
| pm2mpf1lem 20599 | Lemma for ~ pm2mpf1 . (Co... |
| pm2mpval 20600 | Value of the transformatio... |
| pm2mpfval 20601 | A polynomial matrix transf... |
| pm2mpcl 20602 | The transformation of poly... |
| pm2mpf 20603 | The transformation of poly... |
| pm2mpf1 20604 | The transformation of poly... |
| pm2mpcoe1 20605 | A coefficient of the polyn... |
| idpm2idmp 20606 | The transformation of the ... |
| mptcoe1matfsupp 20607 | The mapping extracting the... |
| mply1topmatcllem 20608 | Lemma for ~ mply1topmatcl ... |
| mply1topmatval 20609 | A polynomial over matrices... |
| mply1topmatcl 20610 | A polynomial over matrices... |
| mp2pm2mplem1 20611 | Lemma 1 for ~ mp2pm2mp . ... |
| mp2pm2mplem2 20612 | Lemma 2 for ~ mp2pm2mp . ... |
| mp2pm2mplem3 20613 | Lemma 3 for ~ mp2pm2mp . ... |
| mp2pm2mplem4 20614 | Lemma 4 for ~ mp2pm2mp . ... |
| mp2pm2mplem5 20615 | Lemma 5 for ~ mp2pm2mp . ... |
| mp2pm2mp 20616 | A polynomial over matrices... |
| pm2mpghmlem2 20617 | Lemma 2 for ~ pm2mpghm . ... |
| pm2mpghmlem1 20618 | Lemma 1 for pm2mpghm . (C... |
| pm2mpfo 20619 | The transformation of poly... |
| pm2mpf1o 20620 | The transformation of poly... |
| pm2mpghm 20621 | The transformation of poly... |
| pm2mpgrpiso 20622 | The transformation of poly... |
| pm2mpmhmlem1 20623 | Lemma 1 for ~ pm2mpmhm . ... |
| pm2mpmhmlem2 20624 | Lemma 2 for ~ pm2mpmhm . ... |
| pm2mpmhm 20625 | The transformation of poly... |
| pm2mprhm 20626 | The transformation of poly... |
| pm2mprngiso 20627 | The transformation of poly... |
| pmmpric 20628 | The ring of polynomial mat... |
| monmat2matmon 20629 | The transformation of a po... |
| pm2mp 20630 | The transformation of a su... |
| chmatcl 20633 | Closure of the characteris... |
| chmatval 20634 | The entries of the charact... |
| chpmatfval 20635 | Value of the characteristi... |
| chpmatval 20636 | The characteristic polynom... |
| chpmatply1 20637 | The characteristic polynom... |
| chpmatval2 20638 | The characteristic polynom... |
| chpmat0d 20639 | The characteristic polynom... |
| chpmat1dlem 20640 | Lemma for ~ chpmat1d . (C... |
| chpmat1d 20641 | The characteristic polynom... |
| chpdmatlem0 20642 | Lemma 0 for ~ chpdmat . (... |
| chpdmatlem1 20643 | Lemma 1 for ~ chpdmat . (... |
| chpdmatlem2 20644 | Lemma 2 for ~ chpdmat . (... |
| chpdmatlem3 20645 | Lemma 3 for ~ chpdmat . (... |
| chpdmat 20646 | The characteristic polynom... |
| chpscmat 20647 | The characteristic polynom... |
| chpscmat0 20648 | The characteristic polynom... |
| chpscmatgsumbin 20649 | The characteristic polynom... |
| chpscmatgsummon 20650 | The characteristic polynom... |
| chp0mat 20651 | The characteristic polynom... |
| chpidmat 20652 | The characteristic polynom... |
| chmaidscmat 20653 | The characteristic polynom... |
| fvmptnn04if 20654 | The function values of a m... |
| fvmptnn04ifa 20655 | The function value of a ma... |
| fvmptnn04ifb 20656 | The function value of a ma... |
| fvmptnn04ifc 20657 | The function value of a ma... |
| fvmptnn04ifd 20658 | The function value of a ma... |
| chfacfisf 20659 | The "characteristic factor... |
| chfacfisfcpmat 20660 | The "characteristic factor... |
| chfacffsupp 20661 | The "characteristic factor... |
| chfacfscmulcl 20662 | Closure of a scaled value ... |
| chfacfscmul0 20663 | A scaled value of the "cha... |
| chfacfscmulfsupp 20664 | A mapping of scaled values... |
| chfacfscmulgsum 20665 | Breaking up a sum of value... |
| chfacfpmmulcl 20666 | Closure of the value of th... |
| chfacfpmmul0 20667 | The value of the "characte... |
| chfacfpmmulfsupp 20668 | A mapping of values of the... |
| chfacfpmmulgsum 20669 | Breaking up a sum of value... |
| chfacfpmmulgsum2 20670 | Breaking up a sum of value... |
| cayhamlem1 20671 | Lemma 1 for ~ cayleyhamilt... |
| cpmadurid 20672 | The right-hand fundamental... |
| cpmidgsum 20673 | Representation of the iden... |
| cpmidgsumm2pm 20674 | Representation of the iden... |
| cpmidpmatlem1 20675 | Lemma 1 for ~ cpmidpmat . ... |
| cpmidpmatlem2 20676 | Lemma 2 for ~ cpmidpmat . ... |
| cpmidpmatlem3 20677 | Lemma 3 for ~ cpmidpmat . ... |
| cpmidpmat 20678 | Representation of the iden... |
| cpmadugsumlemB 20679 | Lemma B for ~ cpmadugsum .... |
| cpmadugsumlemC 20680 | Lemma C for ~ cpmadugsum .... |
| cpmadugsumlemF 20681 | Lemma F for ~ cpmadugsum .... |
| cpmadugsumfi 20682 | The product of the charact... |
| cpmadugsum 20683 | The product of the charact... |
| cpmidgsum2 20684 | Representation of the iden... |
| cpmidg2sum 20685 | Equality of two sums repre... |
| cpmadumatpolylem1 20686 | Lemma 1 for ~ cpmadumatpol... |
| cpmadumatpolylem2 20687 | Lemma 2 for ~ cpmadumatpol... |
| cpmadumatpoly 20688 | The product of the charact... |
| cayhamlem2 20689 | Lemma for ~ cayhamlem3 . ... |
| chcoeffeqlem 20690 | Lemma for ~ chcoeffeq . (... |
| chcoeffeq 20691 | The coefficients of the ch... |
| cayhamlem3 20692 | Lemma for ~ cayhamlem4 . ... |
| cayhamlem4 20693 | Lemma for ~ cayleyhamilton... |
| cayleyhamilton0 20694 | The Cayley-Hamilton theore... |
| cayleyhamilton 20695 | The Cayley-Hamilton theore... |
| cayleyhamiltonALT 20696 | Alternate proof of ~ cayle... |
| cayleyhamilton1 20697 | The Cayley-Hamilton theore... |
| istopg 20700 | Express the predicate " ` ... |
| istop2g 20701 | Express the predicate " ` ... |
| uniopn 20702 | The union of a subset of a... |
| iunopn 20703 | The indexed union of a sub... |
| inopn 20704 | The intersection of two op... |
| fitop 20705 | A topology is closed under... |
| fiinopn 20706 | The intersection of a none... |
| iinopn 20707 | The intersection of a none... |
| unopn 20708 | The union of two open sets... |
| 0opn 20709 | The empty set is an open s... |
| 0ntop 20710 | The empty set is not a top... |
| topopn 20711 | The underlying set of a to... |
| eltopss 20712 | A member of a topology is ... |
| riinopn 20713 | A finite indexed relative ... |
| rintopn 20714 | A finite relative intersec... |
| istopon 20717 | Property of being a topolo... |
| topontop 20718 | A topology on a given base... |
| toponuni 20719 | The base set of a topology... |
| topontopi 20720 | A topology on a given base... |
| toponunii 20721 | The base set of a topology... |
| toptopon 20722 | Alternative definition of ... |
| toptopon2 20723 | A topology is the same thi... |
| topontopon 20724 | A topology on a set is a t... |
| funtopon 20725 | The class ` TopOn ` is a f... |
| toponsspwpw 20726 | The set of topologies on a... |
| dmtopon 20727 | The domain of ` TopOn ` is... |
| fntopon 20728 | The class ` TopOn ` is a f... |
| toprntopon 20729 | A topology is the same thi... |
| toponmax 20730 | The base set of a topology... |
| toponss 20731 | A member of a topology is ... |
| toponcom 20732 | If ` K ` is a topology on ... |
| toponcomb 20733 | Biconditional form of ~ to... |
| topgele 20734 | The topologies over the sa... |
| topsn 20735 | The only topology on a sin... |
| istps 20738 | Express the predicate "is ... |
| istps2 20739 | Express the predicate "is ... |
| tpsuni 20740 | The base set of a topologi... |
| tpstop 20741 | The topology extractor on ... |
| tpspropd 20742 | A topological space depend... |
| tpsprop2d 20743 | A topological space depend... |
| topontopn 20744 | Express the predicate "is ... |
| tsettps 20745 | If the topology component ... |
| istpsi 20746 | Properties that determine ... |
| eltpsg 20747 | Properties that determine ... |
| eltpsi 20748 | Properties that determine ... |
| isbasisg 20751 | Express the predicate " ` ... |
| isbasis2g 20752 | Express the predicate " ` ... |
| isbasis3g 20753 | Express the predicate " ` ... |
| basis1 20754 | Property of a basis. (Con... |
| basis2 20755 | Property of a basis. (Con... |
| fiinbas 20756 | If a set is closed under f... |
| basdif0 20757 | A basis is not affected by... |
| baspartn 20758 | A disjoint system of sets ... |
| tgval 20759 | The topology generated by ... |
| tgval2 20760 | Definition of a topology g... |
| eltg 20761 | Membership in a topology g... |
| eltg2 20762 | Membership in a topology g... |
| eltg2b 20763 | Membership in a topology g... |
| eltg4i 20764 | An open set in a topology ... |
| eltg3i 20765 | The union of a set of basi... |
| eltg3 20766 | Membership in a topology g... |
| tgval3 20767 | Alternate expression for t... |
| tg1 20768 | Property of a member of a ... |
| tg2 20769 | Property of a member of a ... |
| bastg 20770 | A member of a basis is a s... |
| unitg 20771 | The topology generated by ... |
| tgss 20772 | Subset relation for genera... |
| tgcl 20773 | Show that a basis generate... |
| tgclb 20774 | The property ~ tgcl can be... |
| tgtopon 20775 | A basis generates a topolo... |
| topbas 20776 | A topology is its own basi... |
| tgtop 20777 | A topology is its own basi... |
| eltop 20778 | Membership in a topology, ... |
| eltop2 20779 | Membership in a topology. ... |
| eltop3 20780 | Membership in a topology. ... |
| fibas 20781 | A collection of finite int... |
| tgdom 20782 | A space has no more open s... |
| tgiun 20783 | The indexed union of a set... |
| tgidm 20784 | The topology generator fun... |
| bastop 20785 | Two ways to express that a... |
| tgtop11 20786 | The topology generation fu... |
| 0top 20787 | The singleton of the empty... |
| en1top 20788 | ` { (/) } ` is the only to... |
| en2top 20789 | If a topology has two elem... |
| tgss3 20790 | A criterion for determinin... |
| tgss2 20791 | A criterion for determinin... |
| basgen 20792 | Given a topology ` J ` , s... |
| basgen2 20793 | Given a topology ` J ` , s... |
| 2basgen 20794 | Conditions that determine ... |
| tgfiss 20795 | If a subbase is included i... |
| tgdif0 20796 | A generated topology is no... |
| bastop1 20797 | A subset of a topology is ... |
| bastop2 20798 | A version of ~ bastop1 tha... |
| distop 20799 | The discrete topology on a... |
| topnex 20800 | The class of all topologie... |
| distopon 20801 | The discrete topology on a... |
| sn0topon 20802 | The singleton of the empty... |
| sn0top 20803 | The singleton of the empty... |
| indislem 20804 | A lemma to eliminate some ... |
| indistopon 20805 | The indiscrete topology on... |
| indistop 20806 | The indiscrete topology on... |
| indisuni 20807 | The base set of the indisc... |
| fctop 20808 | The finite complement topo... |
| fctop2 20809 | The finite complement topo... |
| cctop 20810 | The countable complement t... |
| ppttop 20811 | The particular point topol... |
| pptbas 20812 | The particular point topol... |
| epttop 20813 | The excluded point topolog... |
| indistpsx 20814 | The indiscrete topology on... |
| indistps 20815 | The indiscrete topology on... |
| indistps2 20816 | The indiscrete topology on... |
| indistpsALT 20817 | The indiscrete topology on... |
| indistps2ALT 20818 | The indiscrete topology on... |
| distps 20819 | The discrete topology on a... |
| fncld 20826 | The closed-set generator i... |
| cldval 20827 | The set of closed sets of ... |
| ntrfval 20828 | The interior function on t... |
| clsfval 20829 | The closure function on th... |
| cldrcl 20830 | Reverse closure of the clo... |
| iscld 20831 | The predicate " ` S ` is a... |
| iscld2 20832 | A subset of the underlying... |
| cldss 20833 | A closed set is a subset o... |
| cldss2 20834 | The set of closed sets is ... |
| cldopn 20835 | The complement of a closed... |
| isopn2 20836 | A subset of the underlying... |
| opncld 20837 | The complement of an open ... |
| difopn 20838 | The difference of a closed... |
| topcld 20839 | The underlying set of a to... |
| ntrval 20840 | The interior of a subset o... |
| clsval 20841 | The closure of a subset of... |
| 0cld 20842 | The empty set is closed. ... |
| iincld 20843 | The indexed intersection o... |
| intcld 20844 | The intersection of a set ... |
| uncld 20845 | The union of two closed se... |
| cldcls 20846 | A closed subset equals its... |
| incld 20847 | The intersection of two cl... |
| riincld 20848 | An indexed relative inters... |
| iuncld 20849 | A finite indexed union of ... |
| unicld 20850 | A finite union of closed s... |
| clscld 20851 | The closure of a subset of... |
| clsf 20852 | The closure function is a ... |
| ntropn 20853 | The interior of a subset o... |
| clsval2 20854 | Express closure in terms o... |
| ntrval2 20855 | Interior expressed in term... |
| ntrdif 20856 | An interior of a complemen... |
| clsdif 20857 | A closure of a complement ... |
| clsss 20858 | Subset relationship for cl... |
| ntrss 20859 | Subset relationship for in... |
| sscls 20860 | A subset of a topology's u... |
| ntrss2 20861 | A subset includes its inte... |
| ssntr 20862 | An open subset of a set is... |
| clsss3 20863 | The closure of a subset of... |
| ntrss3 20864 | The interior of a subset o... |
| ntrin 20865 | A pairwise intersection of... |
| cmclsopn 20866 | The complement of a closur... |
| cmntrcld 20867 | The complement of an inter... |
| iscld3 20868 | A subset is closed iff it ... |
| iscld4 20869 | A subset is closed iff it ... |
| isopn3 20870 | A subset is open iff it eq... |
| clsidm 20871 | The closure operation is i... |
| ntridm 20872 | The interior operation is ... |
| clstop 20873 | The closure of a topology'... |
| ntrtop 20874 | The interior of a topology... |
| 0ntr 20875 | A subset with an empty int... |
| clsss2 20876 | If a subset is included in... |
| elcls 20877 | Membership in a closure. ... |
| elcls2 20878 | Membership in a closure. ... |
| clsndisj 20879 | Any open set containing a ... |
| ntrcls0 20880 | A subset whose closure has... |
| ntreq0 20881 | Two ways to say that a sub... |
| cldmre 20882 | The closed sets of a topol... |
| mrccls 20883 | Moore closure generalizes ... |
| cls0 20884 | The closure of the empty s... |
| ntr0 20885 | The interior of the empty ... |
| isopn3i 20886 | An open subset equals its ... |
| elcls3 20887 | Membership in a closure in... |
| opncldf1 20888 | A bijection useful for con... |
| opncldf2 20889 | The values of the open-clo... |
| opncldf3 20890 | The values of the converse... |
| isclo 20891 | A set ` A ` is clopen iff ... |
| isclo2 20892 | A set ` A ` is clopen iff ... |
| discld 20893 | The open sets of a discret... |
| sn0cld 20894 | The closed sets of the top... |
| indiscld 20895 | The closed sets of an indi... |
| mretopd 20896 | A Moore collection which i... |
| toponmre 20897 | The topologies over a give... |
| cldmreon 20898 | The closed sets of a topol... |
| iscldtop 20899 | A family is the closed set... |
| mreclatdemoBAD 20900 | The closed subspaces of a ... |
| neifval 20903 | The neighborhood function ... |
| neif 20904 | The neighborhood function ... |
| neiss2 20905 | A set with a neighborhood ... |
| neival 20906 | The set of neighborhoods o... |
| isnei 20907 | The predicate " ` N ` is a... |
| neiint 20908 | An intuitive definition of... |
| isneip 20909 | The predicate " ` N ` is a... |
| neii1 20910 | A neighborhood is included... |
| neisspw 20911 | The neighborhoods of any s... |
| neii2 20912 | Property of a neighborhood... |
| neiss 20913 | Any neighborhood of a set ... |
| ssnei 20914 | A set is included in its n... |
| elnei 20915 | A point belongs to any of ... |
| 0nnei 20916 | The empty set is not a nei... |
| neips 20917 | A neighborhood of a set is... |
| opnneissb 20918 | An open set is a neighborh... |
| opnssneib 20919 | Any superset of an open se... |
| ssnei2 20920 | Any subset of ` X ` contai... |
| neindisj 20921 | Any neighborhood of an ele... |
| opnneiss 20922 | An open set is a neighborh... |
| opnneip 20923 | An open set is a neighborh... |
| opnnei 20924 | A set is open iff it is a ... |
| tpnei 20925 | The underlying set of a to... |
| neiuni 20926 | The union of the neighborh... |
| neindisj2 20927 | A point ` P ` belongs to t... |
| topssnei 20928 | A finer topology has more ... |
| innei 20929 | The intersection of two ne... |
| opnneiid 20930 | Only an open set is a neig... |
| neissex 20931 | For any neighborhood ` N `... |
| 0nei 20932 | The empty set is a neighbo... |
| neipeltop 20933 | Lemma for ~ neiptopreu . ... |
| neiptopuni 20934 | Lemma for ~ neiptopreu . ... |
| neiptoptop 20935 | Lemma for ~ neiptopreu . ... |
| neiptopnei 20936 | Lemma for ~ neiptopreu . ... |
| neiptopreu 20937 | If, to each element ` P ` ... |
| lpfval 20942 | The limit point function o... |
| lpval 20943 | The set of limit points of... |
| islp 20944 | The predicate " ` P ` is a... |
| lpsscls 20945 | The limit points of a subs... |
| lpss 20946 | The limit points of a subs... |
| lpdifsn 20947 | ` P ` is a limit point of ... |
| lpss3 20948 | Subset relationship for li... |
| islp2 20949 | The predicate " ` P ` is a... |
| islp3 20950 | The predicate " ` P ` is a... |
| maxlp 20951 | A point is a limit point o... |
| clslp 20952 | The closure of a subset of... |
| islpi 20953 | A point belonging to a set... |
| cldlp 20954 | A subset of a topological ... |
| isperf 20955 | Definition of a perfect sp... |
| isperf2 20956 | Definition of a perfect sp... |
| isperf3 20957 | A perfect space is a topol... |
| perflp 20958 | The limit points of a perf... |
| perfi 20959 | Property of a perfect spac... |
| perftop 20960 | A perfect space is a topol... |
| restrcl 20961 | Reverse closure for the su... |
| restbas 20962 | A subspace topology basis ... |
| tgrest 20963 | A subspace can be generate... |
| resttop 20964 | A subspace topology is a t... |
| resttopon 20965 | A subspace topology is a t... |
| restuni 20966 | The underlying set of a su... |
| stoig 20967 | The topological space buil... |
| restco 20968 | Composition of subspaces. ... |
| restabs 20969 | Equivalence of being a sub... |
| restin 20970 | When the subspace region i... |
| restuni2 20971 | The underlying set of a su... |
| resttopon2 20972 | The underlying set of a su... |
| rest0 20973 | The subspace topology indu... |
| restsn 20974 | The only subspace topology... |
| restsn2 20975 | The subspace topology indu... |
| restcld 20976 | A closed set of a subspace... |
| restcldi 20977 | A closed set is closed in ... |
| restcldr 20978 | A set which is closed in t... |
| restopnb 20979 | If ` B ` is an open subset... |
| ssrest 20980 | If ` K ` is a finer topolo... |
| restopn2 20981 | The if ` A ` is open, then... |
| restdis 20982 | A subspace of a discrete t... |
| restfpw 20983 | The restriction of the set... |
| neitr 20984 | The neighborhood of a trac... |
| restcls 20985 | A closure in a subspace to... |
| restntr 20986 | An interior in a subspace ... |
| restlp 20987 | The limit points of a subs... |
| restperf 20988 | Perfection of a subspace. ... |
| perfopn 20989 | An open subset of a perfec... |
| resstopn 20990 | The topology of a restrict... |
| resstps 20991 | A restricted topological s... |
| ordtbaslem 20992 | Lemma for ~ ordtbas . In ... |
| ordtval 20993 | Value of the order topolog... |
| ordtuni 20994 | Value of the order topolog... |
| ordtbas2 20995 | Lemma for ~ ordtbas . (Co... |
| ordtbas 20996 | In a total order, the fini... |
| ordttopon 20997 | Value of the order topolog... |
| ordtopn1 20998 | An upward ray ` ( P , +oo ... |
| ordtopn2 20999 | A downward ray ` ( -oo , P... |
| ordtopn3 21000 | An open interval ` ( A , B... |
| ordtcld1 21001 | A downward ray ` ( -oo , P... |
| ordtcld2 21002 | An upward ray ` [ P , +oo ... |
| ordtcld3 21003 | A closed interval ` [ A , ... |
| ordttop 21004 | The order topology is a to... |
| ordtcnv 21005 | The order dual generates t... |
| ordtrest 21006 | The subspace topology of a... |
| ordtrest2lem 21007 | Lemma for ~ ordtrest2 . (... |
| ordtrest2 21008 | An interval-closed set ` A... |
| letopon 21009 | The topology of the extend... |
| letop 21010 | The topology of the extend... |
| letopuni 21011 | The topology of the extend... |
| xrstopn 21012 | The topology component of ... |
| xrstps 21013 | The extended real number s... |
| leordtvallem1 21014 | Lemma for ~ leordtval . (... |
| leordtvallem2 21015 | Lemma for ~ leordtval . (... |
| leordtval2 21016 | The topology of the extend... |
| leordtval 21017 | The topology of the extend... |
| iccordt 21018 | A closed interval is close... |
| iocpnfordt 21019 | An unbounded above open in... |
| icomnfordt 21020 | An unbounded above open in... |
| iooordt 21021 | An open interval is open i... |
| reordt 21022 | The real numbers are an op... |
| lecldbas 21023 | The set of closed interval... |
| pnfnei 21024 | A neighborhood of ` +oo ` ... |
| mnfnei 21025 | A neighborhood of ` -oo ` ... |
| ordtrestixx 21026 | The restriction of the les... |
| ordtresticc 21027 | The restriction of the les... |
| lmrel 21034 | The topological space conv... |
| lmrcl 21035 | Reverse closure for the co... |
| lmfval 21036 | The relation "sequence ` f... |
| cnfval 21037 | The set of all continuous ... |
| cnpfval 21038 | The function mapping the p... |
| iscn 21039 | The predicate " ` F ` is a... |
| cnpval 21040 | The set of all functions f... |
| iscnp 21041 | The predicate " ` F ` is a... |
| iscn2 21042 | The predicate " ` F ` is a... |
| iscnp2 21043 | The predicate " ` F ` is a... |
| cntop1 21044 | Reverse closure for a cont... |
| cntop2 21045 | Reverse closure for a cont... |
| cnptop1 21046 | Reverse closure for a func... |
| cnptop2 21047 | Reverse closure for a func... |
| iscnp3 21048 | The predicate " ` F ` is a... |
| cnprcl 21049 | Reverse closure for a func... |
| cnf 21050 | A continuous function is a... |
| cnpf 21051 | A continuous function at p... |
| cnpcl 21052 | The value of a continuous ... |
| cnf2 21053 | A continuous function is a... |
| cnpf2 21054 | A continuous function at p... |
| cnprcl2 21055 | Reverse closure for a func... |
| tgcn 21056 | The continuity predicate w... |
| tgcnp 21057 | The "continuous at a point... |
| subbascn 21058 | The continuity predicate w... |
| ssidcn 21059 | The identity function is a... |
| cnpimaex 21060 | Property of a function con... |
| idcn 21061 | A restricted identity func... |
| lmbr 21062 | Express the binary relatio... |
| lmbr2 21063 | Express the binary relatio... |
| lmbrf 21064 | Express the binary relatio... |
| lmconst 21065 | A constant sequence conver... |
| lmcvg 21066 | Convergence property of a ... |
| iscnp4 21067 | The predicate " ` F ` is a... |
| cnpnei 21068 | A condition for continuity... |
| cnima 21069 | An open subset of the codo... |
| cnco 21070 | The composition of two con... |
| cnpco 21071 | The composition of two con... |
| cnclima 21072 | A closed subset of the cod... |
| iscncl 21073 | A definition of a continuo... |
| cncls2i 21074 | Property of the preimage o... |
| cnntri 21075 | Property of the preimage o... |
| cnclsi 21076 | Property of the image of a... |
| cncls2 21077 | Continuity in terms of clo... |
| cncls 21078 | Continuity in terms of clo... |
| cnntr 21079 | Continuity in terms of int... |
| cnss1 21080 | If the topology ` K ` is f... |
| cnss2 21081 | If the topology ` K ` is f... |
| cncnpi 21082 | A continuous function is c... |
| cnsscnp 21083 | The set of continuous func... |
| cncnp 21084 | A continuous function is c... |
| cncnp2 21085 | A continuous function is c... |
| cnnei 21086 | Continuity in terms of nei... |
| cnconst2 21087 | A constant function is con... |
| cnconst 21088 | A constant function is con... |
| cnrest 21089 | Continuity of a restrictio... |
| cnrest2 21090 | Equivalence of continuity ... |
| cnrest2r 21091 | Equivalence of continuity ... |
| cnpresti 21092 | One direction of ~ cnprest... |
| cnprest 21093 | Equivalence of continuity ... |
| cnprest2 21094 | Equivalence of point-conti... |
| cndis 21095 | Every function is continuo... |
| cnindis 21096 | Every function is continuo... |
| cnpdis 21097 | If ` A ` is an isolated po... |
| paste 21098 | Pasting lemma. If ` A ` a... |
| lmfpm 21099 | If ` F ` converges, then `... |
| lmfss 21100 | Inclusion of a function ha... |
| lmcl 21101 | Closure of a limit. (Cont... |
| lmss 21102 | Limit on a subspace. (Con... |
| sslm 21103 | A finer topology has fewer... |
| lmres 21104 | A function converges iff i... |
| lmff 21105 | If ` F ` converges, there ... |
| lmcls 21106 | Any convergent sequence of... |
| lmcld 21107 | Any convergent sequence of... |
| lmcnp 21108 | The image of a convergent ... |
| lmcn 21109 | The image of a convergent ... |
| ist0 21124 | The predicate "is a T_0 sp... |
| ist1 21125 | The predicate ` J ` is T_1... |
| ishaus 21126 | Express the predicate " ` ... |
| iscnrm 21127 | The property of being comp... |
| t0sep 21128 | Any two topologically indi... |
| t0dist 21129 | Any two distinct points in... |
| t1sncld 21130 | In a T_1 space, one-point ... |
| t1ficld 21131 | In a T_1 space, finite set... |
| hausnei 21132 | Neighborhood property of a... |
| t0top 21133 | A T_0 space is a topologic... |
| t1top 21134 | A T_1 space is a topologic... |
| haustop 21135 | A Hausdorff space is a top... |
| isreg 21136 | The predicate "is a regula... |
| regtop 21137 | A regular space is a topol... |
| regsep 21138 | In a regular space, every ... |
| isnrm 21139 | The predicate "is a normal... |
| nrmtop 21140 | A normal space is a topolo... |
| cnrmtop 21141 | A completely normal space ... |
| iscnrm2 21142 | The property of being comp... |
| ispnrm 21143 | The property of being perf... |
| pnrmnrm 21144 | A perfectly normal space i... |
| pnrmtop 21145 | A perfectly normal space i... |
| pnrmcld 21146 | A closed set in a perfectl... |
| pnrmopn 21147 | An open set in a perfectly... |
| ist0-2 21148 | The predicate "is a T_0 sp... |
| ist0-3 21149 | The predicate "is a T_0 sp... |
| cnt0 21150 | The preimage of a T_0 topo... |
| ist1-2 21151 | An alternate characterizat... |
| t1t0 21152 | A T_1 space is a T_0 space... |
| ist1-3 21153 | A space is T_1 iff every p... |
| cnt1 21154 | The preimage of a T_1 topo... |
| ishaus2 21155 | Express the predicate " ` ... |
| haust1 21156 | A Hausdorff space is a T_1... |
| hausnei2 21157 | The Hausdorff condition st... |
| cnhaus 21158 | The preimage of a Hausdorf... |
| nrmsep3 21159 | In a normal space, given a... |
| nrmsep2 21160 | In a normal space, any two... |
| nrmsep 21161 | In a normal space, disjoin... |
| isnrm2 21162 | An alternate characterizat... |
| isnrm3 21163 | A topological space is nor... |
| cnrmi 21164 | A subspace of a completely... |
| cnrmnrm 21165 | A completely normal space ... |
| restcnrm 21166 | A subspace of a completely... |
| resthauslem 21167 | Lemma for ~ resthaus and s... |
| lpcls 21168 | The limit points of the cl... |
| perfcls 21169 | A subset of a perfect spac... |
| restt0 21170 | A subspace of a T_0 topolo... |
| restt1 21171 | A subspace of a T_1 topolo... |
| resthaus 21172 | A subspace of a Hausdorff ... |
| t1sep2 21173 | Any two points in a T_1 sp... |
| t1sep 21174 | Any two distinct points in... |
| sncld 21175 | A singleton is closed in a... |
| sshauslem 21176 | Lemma for ~ sshaus and sim... |
| sst0 21177 | A topology finer than a T_... |
| sst1 21178 | A topology finer than a T_... |
| sshaus 21179 | A topology finer than a Ha... |
| regsep2 21180 | In a regular space, a clos... |
| isreg2 21181 | A topological space is reg... |
| dnsconst 21182 | If a continuous mapping to... |
| ordtt1 21183 | The order topology is T_1 ... |
| lmmo 21184 | A sequence in a Hausdorff ... |
| lmfun 21185 | The convergence relation i... |
| dishaus 21186 | A discrete topology is Hau... |
| ordthauslem 21187 | Lemma for ~ ordthaus . (C... |
| ordthaus 21188 | The order topology of a to... |
| iscmp 21191 | The predicate "is a compac... |
| cmpcov 21192 | An open cover of a compact... |
| cmpcov2 21193 | Rewrite ~ cmpcov for the c... |
| cmpcovf 21194 | Combine ~ cmpcov with ~ ac... |
| cncmp 21195 | Compactness is respected b... |
| fincmp 21196 | A finite topology is compa... |
| 0cmp 21197 | The singleton of the empty... |
| cmptop 21198 | A compact topology is a to... |
| rncmp 21199 | The image of a compact set... |
| imacmp 21200 | The image of a compact set... |
| discmp 21201 | A discrete topology is com... |
| cmpsublem 21202 | Lemma for ~ cmpsub . (Con... |
| cmpsub 21203 | Two equivalent ways of des... |
| tgcmp 21204 | A topology generated by a ... |
| cmpcld 21205 | A closed subset of a compa... |
| uncmp 21206 | The union of two compact s... |
| fiuncmp 21207 | A finite union of compact ... |
| sscmp 21208 | A subset of a compact topo... |
| hauscmplem 21209 | Lemma for ~ hauscmp . (Co... |
| hauscmp 21210 | A compact subspace of a T2... |
| cmpfi 21211 | If a topology is compact a... |
| cmpfii 21212 | In a compact topology, a s... |
| bwth 21213 | The glorious Bolzano-Weier... |
| isconn 21216 | The predicate ` J ` is a c... |
| isconn2 21217 | The predicate ` J ` is a c... |
| connclo 21218 | The only nonempty clopen s... |
| conndisj 21219 | If a topology is connected... |
| conntop 21220 | A connected topology is a ... |
| indisconn 21221 | The indiscrete topology (o... |
| dfconn2 21222 | An alternate definition of... |
| connsuba 21223 | Connectedness for a subspa... |
| connsub 21224 | Two equivalent ways of say... |
| cnconn 21225 | Connectedness is respected... |
| nconnsubb 21226 | Disconnectedness for a sub... |
| connsubclo 21227 | If a clopen set meets a co... |
| connima 21228 | The image of a connected s... |
| conncn 21229 | A continuous function from... |
| iunconnlem 21230 | Lemma for ~ iunconn . (Co... |
| iunconn 21231 | The indexed union of conne... |
| unconn 21232 | The union of two connected... |
| clsconn 21233 | The closure of a connected... |
| conncompid 21234 | The connected component co... |
| conncompconn 21235 | The connected component co... |
| conncompss 21236 | The connected component co... |
| conncompcld 21237 | The connected component co... |
| conncompclo 21238 | The connected component co... |
| t1connperf 21239 | A connected T_1 space is p... |
| is1stc 21244 | The predicate "is a first-... |
| is1stc2 21245 | An equivalent way of sayin... |
| 1stctop 21246 | A first-countable topology... |
| 1stcclb 21247 | A property of points in a ... |
| 1stcfb 21248 | For any point ` A ` in a f... |
| is2ndc 21249 | The property of being seco... |
| 2ndctop 21250 | A second-countable topolog... |
| 2ndci 21251 | A countable basis generate... |
| 2ndcsb 21252 | Having a countable subbase... |
| 2ndcredom 21253 | A second-countable space h... |
| 2ndc1stc 21254 | A second-countable space i... |
| 1stcrestlem 21255 | Lemma for ~ 1stcrest . (C... |
| 1stcrest 21256 | A subspace of a first-coun... |
| 2ndcrest 21257 | A subspace of a second-cou... |
| 2ndcctbss 21258 | If a topology is second-co... |
| 2ndcdisj 21259 | Any disjoint family of ope... |
| 2ndcdisj2 21260 | Any disjoint collection of... |
| 2ndcomap 21261 | A surjective continuous op... |
| 2ndcsep 21262 | A second-countable topolog... |
| dis2ndc 21263 | A discrete space is second... |
| 1stcelcls 21264 | A point belongs to the clo... |
| 1stccnp 21265 | A mapping is continuous at... |
| 1stccn 21266 | A mapping ` X --> Y ` , wh... |
| islly 21271 | The property of being a lo... |
| isnlly 21272 | The property of being an n... |
| llyeq 21273 | Equality theorem for the `... |
| nllyeq 21274 | Equality theorem for the `... |
| llytop 21275 | A locally ` A ` space is a... |
| nllytop 21276 | A locally ` A ` space is a... |
| llyi 21277 | The property of a locally ... |
| nllyi 21278 | The property of an n-local... |
| nlly2i 21279 | Eliminate the neighborhood... |
| llynlly 21280 | A locally ` A ` space is n... |
| llyssnlly 21281 | A locally ` A ` space is n... |
| llyss 21282 | The "locally" predicate re... |
| nllyss 21283 | The "n-locally" predicate ... |
| subislly 21284 | The property of a subspace... |
| restnlly 21285 | If the property ` A ` pass... |
| restlly 21286 | If the property ` A ` pass... |
| islly2 21287 | An alternative expression ... |
| llyrest 21288 | An open subspace of a loca... |
| nllyrest 21289 | An open subspace of an n-l... |
| loclly 21290 | If ` A ` is a local proper... |
| llyidm 21291 | Idempotence of the "locall... |
| nllyidm 21292 | Idempotence of the "n-loca... |
| toplly 21293 | A topology is locally a to... |
| topnlly 21294 | A topology is n-locally a ... |
| hauslly 21295 | A Hausdorff space is local... |
| hausnlly 21296 | A Hausdorff space is n-loc... |
| hausllycmp 21297 | A compact Hausdorff space ... |
| cldllycmp 21298 | A closed subspace of a loc... |
| lly1stc 21299 | First-countability is a lo... |
| dislly 21300 | The discrete space ` ~P X ... |
| disllycmp 21301 | A discrete space is locall... |
| dis1stc 21302 | A discrete space is first-... |
| hausmapdom 21303 | If ` X ` is a first-counta... |
| hauspwdom 21304 | Simplify the cardinal ` A ... |
| refrel 21311 | Refinement is a relation. ... |
| isref 21312 | The property of being a re... |
| refbas 21313 | A refinement covers the sa... |
| refssex 21314 | Every set in a refinement ... |
| ssref 21315 | A subcover is a refinement... |
| refref 21316 | Reflexivity of refinement.... |
| reftr 21317 | Refinement is transitive. ... |
| refun0 21318 | Adding the empty set prese... |
| isptfin 21319 | The statement "is a point-... |
| islocfin 21320 | The statement "is a locall... |
| finptfin 21321 | A finite cover is a point-... |
| ptfinfin 21322 | A point covered by a point... |
| finlocfin 21323 | A finite cover of a topolo... |
| locfintop 21324 | A locally finite cover cov... |
| locfinbas 21325 | A locally finite cover mus... |
| locfinnei 21326 | A point covered by a local... |
| lfinpfin 21327 | A locally finite cover is ... |
| lfinun 21328 | Adding a finite set preser... |
| locfincmp 21329 | For a compact space, the l... |
| unisngl 21330 | Taking the union of the se... |
| dissnref 21331 | The set of singletons is a... |
| dissnlocfin 21332 | The set of singletons is l... |
| locfindis 21333 | The locally finite covers ... |
| locfincf 21334 | A locally finite cover in ... |
| comppfsc 21335 | A space where every open c... |
| kgenval 21338 | Value of the compact gener... |
| elkgen 21339 | Value of the compact gener... |
| kgeni 21340 | Property of the open sets ... |
| kgentopon 21341 | The compact generator gene... |
| kgenuni 21342 | The base set of the compac... |
| kgenftop 21343 | The compact generator gene... |
| kgenf 21344 | The compact generator is a... |
| kgentop 21345 | A compactly generated spac... |
| kgenss 21346 | The compact generator gene... |
| kgenhaus 21347 | The compact generator gene... |
| kgencmp 21348 | The compact generator topo... |
| kgencmp2 21349 | The compact generator topo... |
| kgenidm 21350 | The compact generator is i... |
| iskgen2 21351 | A space is compactly gener... |
| iskgen3 21352 | Derive the usual definitio... |
| llycmpkgen2 21353 | A locally compact space is... |
| cmpkgen 21354 | A compact space is compact... |
| llycmpkgen 21355 | A locally compact space is... |
| 1stckgenlem 21356 | The one-point compactifica... |
| 1stckgen 21357 | A first-countable space is... |
| kgen2ss 21358 | The compact generator pres... |
| kgencn 21359 | A function from a compactl... |
| kgencn2 21360 | A function ` F : J --> K `... |
| kgencn3 21361 | The set of continuous func... |
| kgen2cn 21362 | A continuous function is a... |
| txval 21367 | Value of the binary topolo... |
| txuni2 21368 | The underlying set of the ... |
| txbasex 21369 | The basis for the product ... |
| txbas 21370 | The set of Cartesian produ... |
| eltx 21371 | A set in a product is open... |
| txtop 21372 | The product of two topolog... |
| ptval 21373 | The value of the product t... |
| ptpjpre1 21374 | The preimage of a projecti... |
| elpt 21375 | Elementhood in the bases o... |
| elptr 21376 | A basic open set in the pr... |
| elptr2 21377 | A basic open set in the pr... |
| ptbasid 21378 | The base set of the produc... |
| ptuni2 21379 | The base set for the produ... |
| ptbasin 21380 | The basis for a product to... |
| ptbasin2 21381 | The basis for a product to... |
| ptbas 21382 | The basis for a product to... |
| ptpjpre2 21383 | The basis for a product to... |
| ptbasfi 21384 | The basis for the product ... |
| pttop 21385 | The product topology is a ... |
| ptopn 21386 | A basic open set in the pr... |
| ptopn2 21387 | A sub-basic open set in th... |
| xkotf 21388 | Functionality of function ... |
| xkobval 21389 | Alternative expression for... |
| xkoval 21390 | Value of the compact-open ... |
| xkotop 21391 | The compact-open topology ... |
| xkoopn 21392 | A basic open set of the co... |
| txtopi 21393 | The product of two topolog... |
| txtopon 21394 | The underlying set of the ... |
| txuni 21395 | The underlying set of the ... |
| txunii 21396 | The underlying set of the ... |
| ptuni 21397 | The base set for the produ... |
| ptunimpt 21398 | Base set of a product topo... |
| pttopon 21399 | The base set for the produ... |
| pttoponconst 21400 | The base set for a product... |
| ptuniconst 21401 | The base set for a product... |
| xkouni 21402 | The base set of the compac... |
| xkotopon 21403 | The base set of the compac... |
| ptval2 21404 | The value of the product t... |
| txopn 21405 | The product of two open se... |
| txcld 21406 | The product of two closed ... |
| txcls 21407 | Closure of a rectangle in ... |
| txss12 21408 | Subset property of the top... |
| txbasval 21409 | It is sufficient to consid... |
| neitx 21410 | The Cartesian product of t... |
| txcnpi 21411 | Continuity of a two-argume... |
| tx1cn 21412 | Continuity of the first pr... |
| tx2cn 21413 | Continuity of the second p... |
| ptpjcn 21414 | Continuity of a projection... |
| ptpjopn 21415 | The projection map is an o... |
| ptcld 21416 | A closed box in the produc... |
| ptcldmpt 21417 | A closed box in the produc... |
| ptclsg 21418 | The closure of a box in th... |
| ptcls 21419 | The closure of a box in th... |
| dfac14lem 21420 | Lemma for ~ dfac14 . By e... |
| dfac14 21421 | Theorem ~ ptcls is an equi... |
| xkoccn 21422 | The "constant function" fu... |
| txcnp 21423 | If two functions are conti... |
| ptcnplem 21424 | Lemma for ~ ptcnp . (Cont... |
| ptcnp 21425 | If every projection of a f... |
| upxp 21426 | Universal property of the ... |
| txcnmpt 21427 | A map into the product of ... |
| uptx 21428 | Universal property of the ... |
| txcn 21429 | A map into the product of ... |
| ptcn 21430 | If every projection of a f... |
| prdstopn 21431 | Topology of a structure pr... |
| prdstps 21432 | A structure product of top... |
| pwstps 21433 | A structure product of top... |
| txrest 21434 | The subspace of a topologi... |
| txdis 21435 | The topological product of... |
| txindislem 21436 | Lemma for ~ txindis . (Co... |
| txindis 21437 | The topological product of... |
| txdis1cn 21438 | A function is jointly cont... |
| txlly 21439 | If the property ` A ` is p... |
| txnlly 21440 | If the property ` A ` is p... |
| pthaus 21441 | The product of a collectio... |
| ptrescn 21442 | Restriction is a continuou... |
| txtube 21443 | The "tube lemma". If ` X ... |
| txcmplem1 21444 | Lemma for ~ txcmp . (Cont... |
| txcmplem2 21445 | Lemma for ~ txcmp . (Cont... |
| txcmp 21446 | The topological product of... |
| txcmpb 21447 | The topological product of... |
| hausdiag 21448 | A topology is Hausdorff if... |
| hauseqlcld 21449 | In a Hausdorff topology, t... |
| txhaus 21450 | The topological product of... |
| txlm 21451 | Two sequences converge iff... |
| lmcn2 21452 | The image of a convergent ... |
| tx1stc 21453 | The topological product of... |
| tx2ndc 21454 | The topological product of... |
| txkgen 21455 | The topological product of... |
| xkohaus 21456 | If the codomain space is H... |
| xkoptsub 21457 | The compact-open topology ... |
| xkopt 21458 | The compact-open topology ... |
| xkopjcn 21459 | Continuity of a projection... |
| xkoco1cn 21460 | If ` F ` is a continuous f... |
| xkoco2cn 21461 | If ` F ` is a continuous f... |
| xkococnlem 21462 | Continuity of the composit... |
| xkococn 21463 | Continuity of the composit... |
| cnmptid 21464 | The identity function is c... |
| cnmptc 21465 | A constant function is con... |
| cnmpt11 21466 | The composition of continu... |
| cnmpt11f 21467 | The composition of continu... |
| cnmpt1t 21468 | The composition of continu... |
| cnmpt12f 21469 | The composition of continu... |
| cnmpt12 21470 | The composition of continu... |
| cnmpt1st 21471 | The projection onto the fi... |
| cnmpt2nd 21472 | The projection onto the se... |
| cnmpt2c 21473 | A constant function is con... |
| cnmpt21 21474 | The composition of continu... |
| cnmpt21f 21475 | The composition of continu... |
| cnmpt2t 21476 | The composition of continu... |
| cnmpt22 21477 | The composition of continu... |
| cnmpt22f 21478 | The composition of continu... |
| cnmpt1res 21479 | The restriction of a conti... |
| cnmpt2res 21480 | The restriction of a conti... |
| cnmptcom 21481 | The argument converse of a... |
| cnmptkc 21482 | The curried first projecti... |
| cnmptkp 21483 | The evaluation of the inne... |
| cnmptk1 21484 | The composition of a curri... |
| cnmpt1k 21485 | The composition of a one-a... |
| cnmptkk 21486 | The composition of two cur... |
| xkofvcn 21487 | Joint continuity of the fu... |
| cnmptk1p 21488 | The evaluation of a currie... |
| cnmptk2 21489 | The uncurrying of a currie... |
| xkoinjcn 21490 | Continuity of "injection",... |
| cnmpt2k 21491 | The currying of a two-argu... |
| txconn 21492 | The topological product of... |
| imasnopn 21493 | If a relation graph is ope... |
| imasncld 21494 | If a relation graph is clo... |
| imasncls 21495 | If a relation graph is clo... |
| qtopval 21498 | Value of the quotient topo... |
| qtopval2 21499 | Value of the quotient topo... |
| elqtop 21500 | Value of the quotient topo... |
| qtopres 21501 | The quotient topology is u... |
| qtoptop2 21502 | The quotient topology is a... |
| qtoptop 21503 | The quotient topology is a... |
| elqtop2 21504 | Value of the quotient topo... |
| qtopuni 21505 | The base set of the quotie... |
| elqtop3 21506 | Value of the quotient topo... |
| qtoptopon 21507 | The base set of the quotie... |
| qtopid 21508 | A quotient map is a contin... |
| idqtop 21509 | The quotient topology indu... |
| qtopcmplem 21510 | Lemma for ~ qtopcmp and ~ ... |
| qtopcmp 21511 | A quotient of a compact sp... |
| qtopconn 21512 | A quotient of a connected ... |
| qtopkgen 21513 | A quotient of a compactly ... |
| basqtop 21514 | An injection maps bases to... |
| tgqtop 21515 | An injection maps generate... |
| qtopcld 21516 | The property of being a cl... |
| qtopcn 21517 | Universal property of a qu... |
| qtopss 21518 | A surjective continuous fu... |
| qtopeu 21519 | Universal property of the ... |
| qtoprest 21520 | If ` A ` is a saturated op... |
| qtopomap 21521 | If ` F ` is a surjective c... |
| qtopcmap 21522 | If ` F ` is a surjective c... |
| imastopn 21523 | The topology of an image s... |
| imastps 21524 | The image of a topological... |
| qustps 21525 | A quotient structure is a ... |
| kqfval 21526 | Value of the function appe... |
| kqfeq 21527 | Two points in the Kolmogor... |
| kqffn 21528 | The topological indistingu... |
| kqval 21529 | Value of the quotient topo... |
| kqtopon 21530 | The Kolmogorov quotient is... |
| kqid 21531 | The topological indistingu... |
| ist0-4 21532 | The topological indistingu... |
| kqfvima 21533 | When the image set is open... |
| kqsat 21534 | Any open set is saturated ... |
| kqdisj 21535 | A version of ~ imain for t... |
| kqcldsat 21536 | Any closed set is saturate... |
| kqopn 21537 | The topological indistingu... |
| kqcld 21538 | The topological indistingu... |
| kqt0lem 21539 | Lemma for ~ kqt0 . (Contr... |
| isr0 21540 | The property " ` J ` is an... |
| r0cld 21541 | The analogue of the T_1 ax... |
| regr1lem 21542 | Lemma for ~ regr1 . (Cont... |
| regr1lem2 21543 | A Kolmogorov quotient of a... |
| kqreglem1 21544 | A Kolmogorov quotient of a... |
| kqreglem2 21545 | If the Kolmogorov quotient... |
| kqnrmlem1 21546 | A Kolmogorov quotient of a... |
| kqnrmlem2 21547 | If the Kolmogorov quotient... |
| kqtop 21548 | The Kolmogorov quotient is... |
| kqt0 21549 | The Kolmogorov quotient is... |
| kqf 21550 | The Kolmogorov quotient is... |
| r0sep 21551 | The separation property of... |
| nrmr0reg 21552 | A normal R_0 space is also... |
| regr1 21553 | A regular space is R_1, wh... |
| kqreg 21554 | The Kolmogorov quotient of... |
| kqnrm 21555 | The Kolmogorov quotient of... |
| hmeofn 21560 | The set of homeomorphisms ... |
| hmeofval 21561 | The set of all the homeomo... |
| ishmeo 21562 | The predicate F is a homeo... |
| hmeocn 21563 | A homeomorphism is continu... |
| hmeocnvcn 21564 | The converse of a homeomor... |
| hmeocnv 21565 | The converse of a homeomor... |
| hmeof1o2 21566 | A homeomorphism is a 1-1-o... |
| hmeof1o 21567 | A homeomorphism is a 1-1-o... |
| hmeoima 21568 | The image of an open set b... |
| hmeoopn 21569 | Homeomorphisms preserve op... |
| hmeocld 21570 | Homeomorphisms preserve cl... |
| hmeocls 21571 | Homeomorphisms preserve cl... |
| hmeontr 21572 | Homeomorphisms preserve in... |
| hmeoimaf1o 21573 | The function mapping open ... |
| hmeores 21574 | The restriction of a homeo... |
| hmeoco 21575 | The composite of two homeo... |
| idhmeo 21576 | The identity function is a... |
| hmeocnvb 21577 | The converse of a homeomor... |
| hmeoqtop 21578 | A homeomorphism is a quoti... |
| hmph 21579 | Express the predicate ` J ... |
| hmphi 21580 | If there is a homeomorphis... |
| hmphtop 21581 | Reverse closure for the ho... |
| hmphtop1 21582 | The relation "being homeom... |
| hmphtop2 21583 | The relation "being homeom... |
| hmphref 21584 | "Is homeomorphic to" is re... |
| hmphsym 21585 | "Is homeomorphic to" is sy... |
| hmphtr 21586 | "Is homeomorphic to" is tr... |
| hmpher 21587 | "Is homeomorphic to" is an... |
| hmphen 21588 | Homeomorphisms preserve th... |
| hmphsymb 21589 | "Is homeomorphic to" is sy... |
| haushmphlem 21590 | Lemma for ~ haushmph and s... |
| cmphmph 21591 | Compactness is a topologic... |
| connhmph 21592 | Connectedness is a topolog... |
| t0hmph 21593 | T_0 is a topological prope... |
| t1hmph 21594 | T_1 is a topological prope... |
| haushmph 21595 | Hausdorff-ness is a topolo... |
| reghmph 21596 | Regularity is a topologica... |
| nrmhmph 21597 | Normality is a topological... |
| hmph0 21598 | A topology homeomorphic to... |
| hmphdis 21599 | Homeomorphisms preserve to... |
| hmphindis 21600 | Homeomorphisms preserve to... |
| indishmph 21601 | Equinumerous sets equipped... |
| hmphen2 21602 | Homeomorphisms preserve th... |
| cmphaushmeo 21603 | A continuous bijection fro... |
| ordthmeolem 21604 | Lemma for ~ ordthmeo . (C... |
| ordthmeo 21605 | An order isomorphism is a ... |
| txhmeo 21606 | Lift a pair of homeomorphi... |
| txswaphmeolem 21607 | Show inverse for the "swap... |
| txswaphmeo 21608 | There is a homeomorphism f... |
| pt1hmeo 21609 | The canonical homeomorphis... |
| ptuncnv 21610 | Exhibit the converse funct... |
| ptunhmeo 21611 | Define a homeomorphism fro... |
| xpstopnlem1 21612 | The function ` F ` used in... |
| xpstps 21613 | A binary product of topolo... |
| xpstopnlem2 21614 | Lemma for ~ xpstopn . (Co... |
| xpstopn 21615 | The topology on a binary p... |
| ptcmpfi 21616 | A topological product of f... |
| xkocnv 21617 | The inverse of the "curryi... |
| xkohmeo 21618 | The Exponential Law for to... |
| qtopf1 21619 | If a quotient map is injec... |
| qtophmeo 21620 | If two functions on a base... |
| t0kq 21621 | A topological space is T_0... |
| kqhmph 21622 | A topological space is T_0... |
| ist1-5lem 21623 | Lemma for ~ ist1-5 and sim... |
| t1r0 21624 | A T_1 space is R_0. That ... |
| ist1-5 21625 | A topological space is T_1... |
| ishaus3 21626 | A topological space is Hau... |
| nrmreg 21627 | A normal T_1 space is regu... |
| reghaus 21628 | A regular T_0 space is Hau... |
| nrmhaus 21629 | A T_1 normal space is Haus... |
| elmptrab 21630 | Membership in a one-parame... |
| elmptrab2OLD 21631 | Obsolete version of ~ elmp... |
| elmptrab2 21632 | Membership in a one-parame... |
| isfbas 21633 | The predicate " ` F ` is a... |
| fbasne0 21634 | There are no empty filter ... |
| 0nelfb 21635 | No filter base contains th... |
| fbsspw 21636 | A filter base on a set is ... |
| fbelss 21637 | An element of the filter b... |
| fbdmn0 21638 | The domain of a filter bas... |
| isfbas2 21639 | The predicate " ` F ` is a... |
| fbasssin 21640 | A filter base contains sub... |
| fbssfi 21641 | A filter base contains sub... |
| fbssint 21642 | A filter base contains sub... |
| fbncp 21643 | A filter base does not con... |
| fbun 21644 | A necessary and sufficient... |
| fbfinnfr 21645 | No filter base containing ... |
| opnfbas 21646 | The collection of open sup... |
| trfbas2 21647 | Conditions for the trace o... |
| trfbas 21648 | Conditions for the trace o... |
| isfil 21651 | The predicate "is a filter... |
| filfbas 21652 | A filter is a filter base.... |
| 0nelfil 21653 | The empty set doesn't belo... |
| fileln0 21654 | An element of a filter is ... |
| filsspw 21655 | A filter is a subset of th... |
| filelss 21656 | An element of a filter is ... |
| filss 21657 | A filter is closed under t... |
| filin 21658 | A filter is closed under t... |
| filtop 21659 | The underlying set belongs... |
| isfil2 21660 | Derive the standard axioms... |
| isfildlem 21661 | Lemma for ~ isfild . (Con... |
| isfild 21662 | Sufficient condition for a... |
| filfi 21663 | A filter is closed under t... |
| filinn0 21664 | The intersection of two el... |
| filintn0 21665 | A filter has the finite in... |
| filn0 21666 | The empty set is not a fil... |
| infil 21667 | The intersection of two fi... |
| snfil 21668 | A singleton is a filter. ... |
| fbasweak 21669 | A filter base on any set i... |
| snfbas 21670 | Condition for a singleton ... |
| fsubbas 21671 | A condition for a set to g... |
| fbasfip 21672 | A filter base has the fini... |
| fbunfip 21673 | A helpful lemma for showin... |
| fgval 21674 | The filter generating clas... |
| elfg 21675 | A condition for elements o... |
| ssfg 21676 | A filter base is a subset ... |
| fgss 21677 | A bigger base generates a ... |
| fgss2 21678 | A condition for a filter t... |
| fgfil 21679 | A filter generates itself.... |
| elfilss 21680 | An element belongs to a fi... |
| filfinnfr 21681 | No filter containing a fin... |
| fgcl 21682 | A generated filter is a fi... |
| fgabs 21683 | Absorption law for filter ... |
| neifil 21684 | The neighborhoods of a non... |
| filunibas 21685 | Recover the base set from ... |
| filunirn 21686 | Two ways to express a filt... |
| filconn 21687 | A filter gives rise to a c... |
| fbasrn 21688 | Given a filter on a domain... |
| filuni 21689 | The union of a nonempty se... |
| trfil1 21690 | Conditions for the trace o... |
| trfil2 21691 | Conditions for the trace o... |
| trfil3 21692 | Conditions for the trace o... |
| trfilss 21693 | If ` A ` is a member of th... |
| fgtr 21694 | If ` A ` is a member of th... |
| trfg 21695 | The trace operation and th... |
| trnei 21696 | The trace, over a set ` A ... |
| cfinfil 21697 | Relative complements of th... |
| csdfil 21698 | The set of all elements wh... |
| supfil 21699 | The supersets of a nonempt... |
| zfbas 21700 | The set of upper sets of i... |
| uzrest 21701 | The restriction of the set... |
| uzfbas 21702 | The set of upper sets of i... |
| isufil 21707 | The property of being an u... |
| ufilfil 21708 | An ultrafilter is a filter... |
| ufilss 21709 | For any subset of the base... |
| ufilb 21710 | The complement is in an ul... |
| ufilmax 21711 | Any filter finer than an u... |
| isufil2 21712 | The maximal property of an... |
| ufprim 21713 | An ultrafilter is a prime ... |
| trufil 21714 | Conditions for the trace o... |
| filssufilg 21715 | A filter is contained in s... |
| filssufil 21716 | A filter is contained in s... |
| isufl 21717 | Define the (strong) ultraf... |
| ufli 21718 | Property of a set that sat... |
| numufl 21719 | Consequence of ~ filssufil... |
| fiufl 21720 | A finite set satisfies the... |
| acufl 21721 | The axiom of choice implie... |
| ssufl 21722 | If ` Y ` is a subset of ` ... |
| ufileu 21723 | If the ultrafilter contain... |
| filufint 21724 | A filter is equal to the i... |
| uffix 21725 | Lemma for ~ fixufil and ~ ... |
| fixufil 21726 | The condition describing a... |
| uffixfr 21727 | An ultrafilter is either f... |
| uffix2 21728 | A classification of fixed ... |
| uffixsn 21729 | The singleton of the gener... |
| ufildom1 21730 | An ultrafilter is generate... |
| uffinfix 21731 | An ultrafilter containing ... |
| cfinufil 21732 | An ultrafilter is free iff... |
| ufinffr 21733 | An infinite subset is cont... |
| ufilen 21734 | Any infinite set has an ul... |
| ufildr 21735 | An ultrafilter gives rise ... |
| fin1aufil 21736 | There are no definable fre... |
| fmval 21747 | Introduce a function that ... |
| fmfil 21748 | A mapping filter is a filt... |
| fmf 21749 | Pushing-forward via a func... |
| fmss 21750 | A finer filter produces a ... |
| elfm 21751 | An element of a mapping fi... |
| elfm2 21752 | An element of a mapping fi... |
| fmfg 21753 | The image filter of a filt... |
| elfm3 21754 | An alternate formulation o... |
| imaelfm 21755 | An image of a filter eleme... |
| rnelfmlem 21756 | Lemma for ~ rnelfm . (Con... |
| rnelfm 21757 | A condition for a filter t... |
| fmfnfmlem1 21758 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem2 21759 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem3 21760 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem4 21761 | Lemma for ~ fmfnfm . (Con... |
| fmfnfm 21762 | A filter finer than an ima... |
| fmufil 21763 | An image filter of an ultr... |
| fmid 21764 | The filter map applied to ... |
| fmco 21765 | Composition of image filte... |
| ufldom 21766 | The ultrafilter lemma prop... |
| flimval 21767 | The set of limit points of... |
| elflim2 21768 | The predicate "is a limit ... |
| flimtop 21769 | Reverse closure for the li... |
| flimneiss 21770 | A filter contains the neig... |
| flimnei 21771 | A filter contains all of t... |
| flimelbas 21772 | A limit point of a filter ... |
| flimfil 21773 | Reverse closure for the li... |
| flimtopon 21774 | Reverse closure for the li... |
| elflim 21775 | The predicate "is a limit ... |
| flimss2 21776 | A limit point of a filter ... |
| flimss1 21777 | A limit point of a filter ... |
| neiflim 21778 | A point is a limit point o... |
| flimopn 21779 | The condition for being a ... |
| fbflim 21780 | A condition for a filter t... |
| fbflim2 21781 | A condition for a filter b... |
| flimclsi 21782 | The convergent points of a... |
| hausflimlem 21783 | If ` A ` and ` B ` are bot... |
| hausflimi 21784 | One direction of ~ hausfli... |
| hausflim 21785 | A condition for a topology... |
| flimcf 21786 | Fineness is properly chara... |
| flimrest 21787 | The set of limit points in... |
| flimclslem 21788 | Lemma for ~ flimcls . (Co... |
| flimcls 21789 | Closure in terms of filter... |
| flimsncls 21790 | If ` A ` is a limit point ... |
| hauspwpwf1 21791 | Lemma for ~ hauspwpwdom . ... |
| hauspwpwdom 21792 | If ` X ` is a Hausdorff sp... |
| flffval 21793 | Given a topology and a fil... |
| flfval 21794 | Given a function from a fi... |
| flfnei 21795 | The property of being a li... |
| flfneii 21796 | A neighborhood of a limit ... |
| isflf 21797 | The property of being a li... |
| flfelbas 21798 | A limit point of a functio... |
| flffbas 21799 | Limit points of a function... |
| flftg 21800 | Limit points of a function... |
| hausflf 21801 | If a function has its valu... |
| hausflf2 21802 | If a convergent function h... |
| cnpflfi 21803 | Forward direction of ~ cnp... |
| cnpflf2 21804 | ` F ` is continuous at poi... |
| cnpflf 21805 | Continuity of a function a... |
| cnflf 21806 | A function is continuous i... |
| cnflf2 21807 | A function is continuous i... |
| flfcnp 21808 | A continuous function pres... |
| lmflf 21809 | The topological limit rela... |
| txflf 21810 | Two sequences converge in ... |
| flfcnp2 21811 | The image of a convergent ... |
| fclsval 21812 | The set of all cluster poi... |
| isfcls 21813 | A cluster point of a filte... |
| fclsfil 21814 | Reverse closure for the cl... |
| fclstop 21815 | Reverse closure for the cl... |
| fclstopon 21816 | Reverse closure for the cl... |
| isfcls2 21817 | A cluster point of a filte... |
| fclsopn 21818 | Write the cluster point co... |
| fclsopni 21819 | An open neighborhood of a ... |
| fclselbas 21820 | A cluster point is in the ... |
| fclsneii 21821 | A neighborhood of a cluste... |
| fclssscls 21822 | The set of cluster points ... |
| fclsnei 21823 | Cluster points in terms of... |
| supnfcls 21824 | The filter of supersets of... |
| fclsbas 21825 | Cluster points in terms of... |
| fclsss1 21826 | A finer topology has fewer... |
| fclsss2 21827 | A finer filter has fewer c... |
| fclsrest 21828 | The set of cluster points ... |
| fclscf 21829 | Characterization of finene... |
| flimfcls 21830 | A limit point is a cluster... |
| fclsfnflim 21831 | A filter clusters at a poi... |
| flimfnfcls 21832 | A filter converges to a po... |
| fclscmpi 21833 | Forward direction of ~ fcl... |
| fclscmp 21834 | A space is compact iff eve... |
| uffclsflim 21835 | The cluster points of an u... |
| ufilcmp 21836 | A space is compact iff eve... |
| fcfval 21837 | The set of cluster points ... |
| isfcf 21838 | The property of being a cl... |
| fcfnei 21839 | The property of being a cl... |
| fcfelbas 21840 | A cluster point of a funct... |
| fcfneii 21841 | A neighborhood of a cluste... |
| flfssfcf 21842 | A limit point of a functio... |
| uffcfflf 21843 | If the domain filter is an... |
| cnpfcfi 21844 | Lemma for ~ cnpfcf . If a... |
| cnpfcf 21845 | A function ` F ` is contin... |
| cnfcf 21846 | Continuity of a function i... |
| flfcntr 21847 | A continuous function's va... |
| alexsublem 21848 | Lemma for ~ alexsub . (Co... |
| alexsub 21849 | The Alexander Subbase Theo... |
| alexsubb 21850 | Biconditional form of the ... |
| alexsubALTlem1 21851 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem2 21852 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem3 21853 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem4 21854 | Lemma for ~ alexsubALT . ... |
| alexsubALT 21855 | The Alexander Subbase Theo... |
| ptcmplem1 21856 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem2 21857 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem3 21858 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem4 21859 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem5 21860 | Lemma for ~ ptcmp . (Cont... |
| ptcmpg 21861 | Tychonoff's theorem: The ... |
| ptcmp 21862 | Tychonoff's theorem: The ... |
| cnextval 21865 | The function applying cont... |
| cnextfval 21866 | The continuous extension o... |
| cnextrel 21867 | In the general case, a con... |
| cnextfun 21868 | If the target space is Hau... |
| cnextfvval 21869 | The value of the continuou... |
| cnextf 21870 | Extension by continuity. ... |
| cnextcn 21871 | Extension by continuity. ... |
| cnextfres1 21872 | ` F ` and its extension by... |
| cnextfres 21873 | ` F ` and its extension by... |
| istmd 21878 | The predicate "is a topolo... |
| tmdmnd 21879 | A topological monoid is a ... |
| tmdtps 21880 | A topological monoid is a ... |
| istgp 21881 | The predicate "is a topolo... |
| tgpgrp 21882 | A topological group is a g... |
| tgptmd 21883 | A topological group is a t... |
| tgptps 21884 | A topological group is a t... |
| tmdtopon 21885 | The topology of a topologi... |
| tgptopon 21886 | The topology of a topologi... |
| tmdcn 21887 | In a topological monoid, t... |
| tgpcn 21888 | In a topological group, th... |
| tgpinv 21889 | In a topological group, th... |
| grpinvhmeo 21890 | The inverse function in a ... |
| cnmpt1plusg 21891 | Continuity of the group su... |
| cnmpt2plusg 21892 | Continuity of the group su... |
| tmdcn2 21893 | Write out the definition o... |
| tgpsubcn 21894 | In a topological group, th... |
| istgp2 21895 | A group with a topology is... |
| tmdmulg 21896 | In a topological monoid, t... |
| tgpmulg 21897 | In a topological group, th... |
| tgpmulg2 21898 | In a topological monoid, t... |
| tmdgsum 21899 | In a topological monoid, t... |
| tmdgsum2 21900 | For any neighborhood ` U `... |
| oppgtmd 21901 | The opposite of a topologi... |
| oppgtgp 21902 | The opposite of a topologi... |
| distgp 21903 | Any group equipped with th... |
| indistgp 21904 | Any group equipped with th... |
| symgtgp 21905 | The symmetric group is a t... |
| tmdlactcn 21906 | The left group action of e... |
| tgplacthmeo 21907 | The left group action of e... |
| submtmd 21908 | A submonoid of a topologic... |
| subgtgp 21909 | A subgroup of a topologica... |
| subgntr 21910 | A subgroup of a topologica... |
| opnsubg 21911 | An open subgroup of a topo... |
| clssubg 21912 | The closure of a subgroup ... |
| clsnsg 21913 | The closure of a normal su... |
| cldsubg 21914 | A subgroup of finite index... |
| tgpconncompeqg 21915 | The connected component co... |
| tgpconncomp 21916 | The identity component, th... |
| tgpconncompss 21917 | The identity component is ... |
| ghmcnp 21918 | A group homomorphism on to... |
| snclseqg 21919 | The coset of the closure o... |
| tgphaus 21920 | A topological group is Hau... |
| tgpt1 21921 | Hausdorff and T1 are equiv... |
| tgpt0 21922 | Hausdorff and T0 are equiv... |
| qustgpopn 21923 | A quotient map in a topolo... |
| qustgplem 21924 | Lemma for ~ qustgp . (Con... |
| qustgp 21925 | The quotient of a topologi... |
| qustgphaus 21926 | The quotient of a topologi... |
| prdstmdd 21927 | The product of a family of... |
| prdstgpd 21928 | The product of a family of... |
| tsmsfbas 21931 | The collection of all sets... |
| tsmslem1 21932 | The finite partial sums of... |
| tsmsval2 21933 | Definition of the topologi... |
| tsmsval 21934 | Definition of the topologi... |
| tsmspropd 21935 | The group sum depends only... |
| eltsms 21936 | The property of being a su... |
| tsmsi 21937 | The property of being a su... |
| tsmscl 21938 | A sum in a topological gro... |
| haustsms 21939 | In a Hausdorff topological... |
| haustsms2 21940 | In a Hausdorff topological... |
| tsmscls 21941 | One half of ~ tgptsmscls ,... |
| tsmsgsum 21942 | The convergent points of a... |
| tsmsid 21943 | If a sum is finite, the us... |
| haustsmsid 21944 | In a Hausdorff topological... |
| tsms0 21945 | The sum of zero is zero. ... |
| tsmssubm 21946 | Evaluate an infinite group... |
| tsmsres 21947 | Extend an infinite group s... |
| tsmsf1o 21948 | Re-index an infinite group... |
| tsmsmhm 21949 | Apply a continuous group h... |
| tsmsadd 21950 | The sum of two infinite gr... |
| tsmsinv 21951 | Inverse of an infinite gro... |
| tsmssub 21952 | The difference of two infi... |
| tgptsmscls 21953 | A sum in a topological gro... |
| tgptsmscld 21954 | The set of limit points to... |
| tsmssplit 21955 | Split a topological group ... |
| tsmsxplem1 21956 | Lemma for ~ tsmsxp . (Con... |
| tsmsxplem2 21957 | Lemma for ~ tsmsxp . (Con... |
| tsmsxp 21958 | Write a sum over a two-dim... |
| istrg 21967 | Express the predicate " ` ... |
| trgtmd 21968 | The multiplicative monoid ... |
| istdrg 21969 | Express the predicate " ` ... |
| tdrgunit 21970 | The unit group of a topolo... |
| trgtgp 21971 | A topological ring is a to... |
| trgtmd2 21972 | A topological ring is a to... |
| trgtps 21973 | A topological ring is a to... |
| trgring 21974 | A topological ring is a ri... |
| trggrp 21975 | A topological ring is a gr... |
| tdrgtrg 21976 | A topological division rin... |
| tdrgdrng 21977 | A topological division rin... |
| tdrgring 21978 | A topological division rin... |
| tdrgtmd 21979 | A topological division rin... |
| tdrgtps 21980 | A topological division rin... |
| istdrg2 21981 | A topological-ring divisio... |
| mulrcn 21982 | The functionalization of t... |
| invrcn2 21983 | The multiplicative inverse... |
| invrcn 21984 | The multiplicative inverse... |
| cnmpt1mulr 21985 | Continuity of ring multipl... |
| cnmpt2mulr 21986 | Continuity of ring multipl... |
| dvrcn 21987 | The division function is c... |
| istlm 21988 | The predicate " ` W ` is a... |
| vscacn 21989 | The scalar multiplication ... |
| tlmtmd 21990 | A topological module is a ... |
| tlmtps 21991 | A topological module is a ... |
| tlmlmod 21992 | A topological module is a ... |
| tlmtrg 21993 | The scalar ring of a topol... |
| tlmscatps 21994 | The scalar ring of a topol... |
| istvc 21995 | A topological vector space... |
| tvctdrg 21996 | The scalar field of a topo... |
| cnmpt1vsca 21997 | Continuity of scalar multi... |
| cnmpt2vsca 21998 | Continuity of scalar multi... |
| tlmtgp 21999 | A topological vector space... |
| tvctlm 22000 | A topological vector space... |
| tvclmod 22001 | A topological vector space... |
| tvclvec 22002 | A topological vector space... |
| ustfn 22005 | The defined uniform struct... |
| ustval 22006 | The class of all uniform s... |
| isust 22007 | The predicate " ` U ` is a... |
| ustssxp 22008 | Entourages are subsets of ... |
| ustssel 22009 | A uniform structure is upw... |
| ustbasel 22010 | The full set is always an ... |
| ustincl 22011 | A uniform structure is clo... |
| ustdiag 22012 | The diagonal set is includ... |
| ustinvel 22013 | If ` V ` is an entourage, ... |
| ustexhalf 22014 | For each entourage ` V ` t... |
| ustrel 22015 | The elements of uniform st... |
| ustfilxp 22016 | A uniform structure on a n... |
| ustne0 22017 | A uniform structure cannot... |
| ustssco 22018 | In an uniform structure, a... |
| ustexsym 22019 | In an uniform structure, f... |
| ustex2sym 22020 | In an uniform structure, f... |
| ustex3sym 22021 | In an uniform structure, f... |
| ustref 22022 | Any element of the base se... |
| ust0 22023 | The unique uniform structu... |
| ustn0 22024 | The empty set is not an un... |
| ustund 22025 | If two intersecting sets `... |
| ustelimasn 22026 | Any point ` A ` is near en... |
| ustneism 22027 | For a point ` A ` in ` X `... |
| elrnust 22028 | First direction for ~ ustb... |
| ustbas2 22029 | Second direction for ~ ust... |
| ustuni 22030 | The set union of a uniform... |
| ustbas 22031 | Recover the base of an uni... |
| ustimasn 22032 | Lemma for ~ ustuqtop . (C... |
| trust 22033 | The trace of a uniform str... |
| utopval 22036 | The topology induced by a ... |
| elutop 22037 | Open sets in the topology ... |
| utoptop 22038 | The topology induced by a ... |
| utopbas 22039 | The base of the topology i... |
| utoptopon 22040 | Topology induced by a unif... |
| restutop 22041 | Restriction of a topology ... |
| restutopopn 22042 | The restriction of the top... |
| ustuqtoplem 22043 | Lemma for ~ ustuqtop . (C... |
| ustuqtop0 22044 | Lemma for ~ ustuqtop . (C... |
| ustuqtop1 22045 | Lemma for ~ ustuqtop , sim... |
| ustuqtop2 22046 | Lemma for ~ ustuqtop . (C... |
| ustuqtop3 22047 | Lemma for ~ ustuqtop , sim... |
| ustuqtop4 22048 | Lemma for ~ ustuqtop . (C... |
| ustuqtop5 22049 | Lemma for ~ ustuqtop . (C... |
| ustuqtop 22050 | For a given uniform struct... |
| utopsnneiplem 22051 | The neighborhoods of a poi... |
| utopsnneip 22052 | The neighborhoods of a poi... |
| utopsnnei 22053 | Images of singletons by en... |
| utop2nei 22054 | For any symmetrical entour... |
| utop3cls 22055 | Relation between a topolog... |
| utopreg 22056 | All Hausdorff uniform spac... |
| ussval 22063 | The uniform structure on u... |
| ussid 22064 | In case the base of the ` ... |
| isusp 22065 | The predicate ` W ` is a u... |
| ressunif 22066 | ` UnifSet ` is unaffected ... |
| ressuss 22067 | Value of the uniform struc... |
| ressust 22068 | The uniform structure of a... |
| ressusp 22069 | The restriction of a unifo... |
| tusval 22070 | The value of the uniform s... |
| tuslem 22071 | Lemma for ~ tusbas , ~ tus... |
| tusbas 22072 | The base set of a construc... |
| tusunif 22073 | The uniform structure of a... |
| tususs 22074 | The uniform structure of a... |
| tustopn 22075 | The topology induced by a ... |
| tususp 22076 | A constructed uniform spac... |
| tustps 22077 | A constructed uniform spac... |
| uspreg 22078 | If a uniform space is Haus... |
| ucnval 22081 | The set of all uniformly c... |
| isucn 22082 | The predicate " ` F ` is a... |
| isucn2 22083 | The predicate " ` F ` is a... |
| ucnimalem 22084 | Reformulate the ` G ` func... |
| ucnima 22085 | An equivalent statement of... |
| ucnprima 22086 | The preimage by a uniforml... |
| iducn 22087 | The identity is uniformly ... |
| cstucnd 22088 | A constant function is uni... |
| ucncn 22089 | Uniform continuity implies... |
| iscfilu 22092 | The predicate " ` F ` is a... |
| cfilufbas 22093 | A Cauchy filter base is a ... |
| cfiluexsm 22094 | For a Cauchy filter base a... |
| fmucndlem 22095 | Lemma for ~ fmucnd . (Con... |
| fmucnd 22096 | The image of a Cauchy filt... |
| cfilufg 22097 | The filter generated by a ... |
| trcfilu 22098 | Condition for the trace of... |
| cfiluweak 22099 | A Cauchy filter base is al... |
| neipcfilu 22100 | In an uniform space, a nei... |
| iscusp 22103 | The predicate " ` W ` is a... |
| cuspusp 22104 | A complete uniform space i... |
| cuspcvg 22105 | In a complete uniform spac... |
| iscusp2 22106 | The predicate " ` W ` is a... |
| cnextucn 22107 | Extension by continuity. ... |
| ucnextcn 22108 | Extension by continuity. ... |
| ispsmet 22109 | Express the predicate " ` ... |
| psmetdmdm 22110 | Recover the base set from ... |
| psmetf 22111 | The distance function of a... |
| psmetcl 22112 | Closure of the distance fu... |
| psmet0 22113 | The distance function of a... |
| psmettri2 22114 | Triangle inequality for th... |
| psmetsym 22115 | The distance function of a... |
| psmettri 22116 | Triangle inequality for th... |
| psmetge0 22117 | The distance function of a... |
| psmetxrge0 22118 | The distance function of a... |
| psmetres2 22119 | Restriction of a pseudomet... |
| psmetlecl 22120 | Real closure of an extende... |
| distspace 22121 | A structure ` G ` with a d... |
| ismet 22128 | Express the predicate " ` ... |
| isxmet 22129 | Express the predicate " ` ... |
| ismeti 22130 | Properties that determine ... |
| isxmetd 22131 | Properties that determine ... |
| isxmet2d 22132 | It is safe to only require... |
| metflem 22133 | Lemma for ~ metf and other... |
| xmetf 22134 | Mapping of the distance fu... |
| metf 22135 | Mapping of the distance fu... |
| xmetcl 22136 | Closure of the distance fu... |
| metcl 22137 | Closure of the distance fu... |
| ismet2 22138 | An extended metric is a me... |
| metxmet 22139 | A metric is an extended me... |
| xmetdmdm 22140 | Recover the base set from ... |
| metdmdm 22141 | Recover the base set from ... |
| xmetunirn 22142 | Two ways to express an ext... |
| xmeteq0 22143 | The value of an extended m... |
| meteq0 22144 | The value of a metric is z... |
| xmettri2 22145 | Triangle inequality for th... |
| mettri2 22146 | Triangle inequality for th... |
| xmet0 22147 | The distance function of a... |
| met0 22148 | The distance function of a... |
| xmetge0 22149 | The distance function of a... |
| metge0 22150 | The distance function of a... |
| xmetlecl 22151 | Real closure of an extende... |
| xmetsym 22152 | The distance function of a... |
| xmetpsmet 22153 | An extended metric is a ps... |
| xmettpos 22154 | The distance function of a... |
| metsym 22155 | The distance function of a... |
| xmettri 22156 | Triangle inequality for th... |
| mettri 22157 | Triangle inequality for th... |
| xmettri3 22158 | Triangle inequality for th... |
| mettri3 22159 | Triangle inequality for th... |
| xmetrtri 22160 | One half of the reverse tr... |
| xmetrtri2 22161 | The reverse triangle inequ... |
| metrtri 22162 | Reverse triangle inequalit... |
| xmetgt0 22163 | The distance function of a... |
| metgt0 22164 | The distance function of a... |
| metn0 22165 | A metric space is nonempty... |
| xmetres2 22166 | Restriction of an extended... |
| metreslem 22167 | Lemma for ~ metres . (Con... |
| metres2 22168 | Lemma for ~ metres . (Con... |
| xmetres 22169 | A restriction of an extend... |
| metres 22170 | A restriction of a metric ... |
| 0met 22171 | The empty metric. (Contri... |
| prdsdsf 22172 | The product metric is a fu... |
| prdsxmetlem 22173 | The product metric is an e... |
| prdsxmet 22174 | The product metric is an e... |
| prdsmet 22175 | The product metric is a me... |
| ressprdsds 22176 | Restriction of a product m... |
| resspwsds 22177 | Restriction of a product m... |
| imasdsf1olem 22178 | Lemma for ~ imasdsf1o . (... |
| imasdsf1o 22179 | The distance function is t... |
| imasf1oxmet 22180 | The image of an extended m... |
| imasf1omet 22181 | The image of a metric is a... |
| xpsdsfn 22182 | Closure of the metric in a... |
| xpsdsfn2 22183 | Closure of the metric in a... |
| xpsxmetlem 22184 | Lemma for ~ xpsxmet . (Co... |
| xpsxmet 22185 | A product metric of extend... |
| xpsdsval 22186 | Value of the metric in a b... |
| xpsmet 22187 | The direct product of two ... |
| blfvalps 22188 | The value of the ball func... |
| blfval 22189 | The value of the ball func... |
| blvalps 22190 | The ball around a point ` ... |
| blval 22191 | The ball around a point ` ... |
| elblps 22192 | Membership in a ball. (Co... |
| elbl 22193 | Membership in a ball. (Co... |
| elbl2ps 22194 | Membership in a ball. (Co... |
| elbl2 22195 | Membership in a ball. (Co... |
| elbl3ps 22196 | Membership in a ball, with... |
| elbl3 22197 | Membership in a ball, with... |
| blcomps 22198 | Commute the arguments to t... |
| blcom 22199 | Commute the arguments to t... |
| xblpnfps 22200 | The infinity ball in an ex... |
| xblpnf 22201 | The infinity ball in an ex... |
| blpnf 22202 | The infinity ball in a sta... |
| bldisj 22203 | Two balls are disjoint if ... |
| blgt0 22204 | A nonempty ball implies th... |
| bl2in 22205 | Two balls are disjoint if ... |
| xblss2ps 22206 | One ball is contained in a... |
| xblss2 22207 | One ball is contained in a... |
| blss2ps 22208 | One ball is contained in a... |
| blss2 22209 | One ball is contained in a... |
| blhalf 22210 | A ball of radius ` R / 2 `... |
| blfps 22211 | Mapping of a ball. (Contr... |
| blf 22212 | Mapping of a ball. (Contr... |
| blrnps 22213 | Membership in the range of... |
| blrn 22214 | Membership in the range of... |
| xblcntrps 22215 | A ball contains its center... |
| xblcntr 22216 | A ball contains its center... |
| blcntrps 22217 | A ball contains its center... |
| blcntr 22218 | A ball contains its center... |
| xbln0 22219 | A ball is nonempty iff the... |
| bln0 22220 | A ball is not empty. (Con... |
| blelrnps 22221 | A ball belongs to the set ... |
| blelrn 22222 | A ball belongs to the set ... |
| blssm 22223 | A ball is a subset of the ... |
| unirnblps 22224 | The union of the set of ba... |
| unirnbl 22225 | The union of the set of ba... |
| blin 22226 | The intersection of two ba... |
| ssblps 22227 | The size of a ball increas... |
| ssbl 22228 | The size of a ball increas... |
| blssps 22229 | Any point ` P ` in a ball ... |
| blss 22230 | Any point ` P ` in a ball ... |
| blssexps 22231 | Two ways to express the ex... |
| blssex 22232 | Two ways to express the ex... |
| ssblex 22233 | A nested ball exists whose... |
| blin2 22234 | Given any two balls and a ... |
| blbas 22235 | The balls of a metric spac... |
| blres 22236 | A ball in a restricted met... |
| xmeterval 22237 | Value of the "finitely sep... |
| xmeter 22238 | The "finitely separated" r... |
| xmetec 22239 | The equivalence classes un... |
| blssec 22240 | A ball centered at ` P ` i... |
| blpnfctr 22241 | The infinity ball in an ex... |
| xmetresbl 22242 | An extended metric restric... |
| mopnval 22243 | An open set is a subset of... |
| mopntopon 22244 | The set of open sets of a ... |
| mopntop 22245 | The set of open sets of a ... |
| mopnuni 22246 | The union of all open sets... |
| elmopn 22247 | The defining property of a... |
| mopnfss 22248 | The family of open sets of... |
| mopnm 22249 | The base set of a metric s... |
| elmopn2 22250 | A defining property of an ... |
| mopnss 22251 | An open set of a metric sp... |
| isxms 22252 | Express the predicate " ` ... |
| isxms2 22253 | Express the predicate " ` ... |
| isms 22254 | Express the predicate " ` ... |
| isms2 22255 | Express the predicate " ` ... |
| xmstopn 22256 | The topology component of ... |
| mstopn 22257 | The topology component of ... |
| xmstps 22258 | A metric space is a topolo... |
| msxms 22259 | A metric space is a topolo... |
| mstps 22260 | A metric space is a topolo... |
| xmsxmet 22261 | The distance function, sui... |
| msmet 22262 | The distance function, sui... |
| msf 22263 | Mapping of the distance fu... |
| xmsxmet2 22264 | The distance function, sui... |
| msmet2 22265 | The distance function, sui... |
| mscl 22266 | Closure of the distance fu... |
| xmscl 22267 | Closure of the distance fu... |
| xmsge0 22268 | The distance function in a... |
| xmseq0 22269 | The distance function in a... |
| xmssym 22270 | The distance function in a... |
| xmstri2 22271 | Triangle inequality for th... |
| mstri2 22272 | Triangle inequality for th... |
| xmstri 22273 | Triangle inequality for th... |
| mstri 22274 | Triangle inequality for th... |
| xmstri3 22275 | Triangle inequality for th... |
| mstri3 22276 | Triangle inequality for th... |
| msrtri 22277 | Reverse triangle inequalit... |
| xmspropd 22278 | Property deduction for an ... |
| mspropd 22279 | Property deduction for a m... |
| setsmsbas 22280 | The base set of a construc... |
| setsmsds 22281 | The distance function of a... |
| setsmstset 22282 | The topology of a construc... |
| setsmstopn 22283 | The topology of a construc... |
| setsxms 22284 | The constructed metric spa... |
| setsms 22285 | The constructed metric spa... |
| tmsval 22286 | For any metric there is an... |
| tmslem 22287 | Lemma for ~ tmsbas , ~ tms... |
| tmsbas 22288 | The base set of a construc... |
| tmsds 22289 | The metric of a constructe... |
| tmstopn 22290 | The topology of a construc... |
| tmsxms 22291 | The constructed metric spa... |
| tmsms 22292 | The constructed metric spa... |
| imasf1obl 22293 | The image of a metric spac... |
| imasf1oxms 22294 | The image of a metric spac... |
| imasf1oms 22295 | The image of a metric spac... |
| prdsbl 22296 | A ball in the product metr... |
| mopni 22297 | An open set of a metric sp... |
| mopni2 22298 | An open set of a metric sp... |
| mopni3 22299 | An open set of a metric sp... |
| blssopn 22300 | The balls of a metric spac... |
| unimopn 22301 | The union of a collection ... |
| mopnin 22302 | The intersection of two op... |
| mopn0 22303 | The empty set is an open s... |
| rnblopn 22304 | A ball of a metric space i... |
| blopn 22305 | A ball of a metric space i... |
| neibl 22306 | The neighborhoods around a... |
| blnei 22307 | A ball around a point is a... |
| lpbl 22308 | Every ball around a limit ... |
| blsscls2 22309 | A smaller closed ball is c... |
| blcld 22310 | A "closed ball" in a metri... |
| blcls 22311 | The closure of an open bal... |
| blsscls 22312 | If two concentric balls ha... |
| metss 22313 | Two ways of saying that me... |
| metequiv 22314 | Two ways of saying that tw... |
| metequiv2 22315 | If there is a sequence of ... |
| metss2lem 22316 | Lemma for ~ metss2 . (Con... |
| metss2 22317 | If the metric ` D ` is "st... |
| comet 22318 | The composition of an exte... |
| stdbdmetval 22319 | Value of the standard boun... |
| stdbdxmet 22320 | The standard bounded metri... |
| stdbdmet 22321 | The standard bounded metri... |
| stdbdbl 22322 | The standard bounded metri... |
| stdbdmopn 22323 | The standard bounded metri... |
| mopnex 22324 | The topology generated by ... |
| methaus 22325 | The topology generated by ... |
| met1stc 22326 | The topology generated by ... |
| met2ndci 22327 | A separable metric space (... |
| met2ndc 22328 | A metric space is second-c... |
| metrest 22329 | Two alternate formulations... |
| ressxms 22330 | The restriction of a metri... |
| ressms 22331 | The restriction of a metri... |
| prdsmslem1 22332 | Lemma for ~ prdsms . The ... |
| prdsxmslem1 22333 | Lemma for ~ prdsms . The ... |
| prdsxmslem2 22334 | Lemma for ~ prdsxms . The... |
| prdsxms 22335 | The indexed product struct... |
| prdsms 22336 | The indexed product struct... |
| pwsxms 22337 | The product of a finite fa... |
| pwsms 22338 | The product of a finite fa... |
| xpsxms 22339 | A binary product of metric... |
| xpsms 22340 | A binary product of metric... |
| tmsxps 22341 | Express the product of two... |
| tmsxpsmopn 22342 | Express the product of two... |
| tmsxpsval 22343 | Value of the product of tw... |
| tmsxpsval2 22344 | Value of the product of tw... |
| metcnp3 22345 | Two ways to express that `... |
| metcnp 22346 | Two ways to say a mapping ... |
| metcnp2 22347 | Two ways to say a mapping ... |
| metcn 22348 | Two ways to say a mapping ... |
| metcnpi 22349 | Epsilon-delta property of ... |
| metcnpi2 22350 | Epsilon-delta property of ... |
| metcnpi3 22351 | Epsilon-delta property of ... |
| txmetcnp 22352 | Continuity of a binary ope... |
| txmetcn 22353 | Continuity of a binary ope... |
| metuval 22354 | Value of the uniform struc... |
| metustel 22355 | Define a filter base ` F `... |
| metustss 22356 | Range of the elements of t... |
| metustrel 22357 | Elements of the filter bas... |
| metustto 22358 | Any two elements of the fi... |
| metustid 22359 | The identity diagonal is i... |
| metustsym 22360 | Elements of the filter bas... |
| metustexhalf 22361 | For any element ` A ` of t... |
| metustfbas 22362 | The filter base generated ... |
| metust 22363 | The uniform structure gene... |
| cfilucfil 22364 | Given a metric ` D ` and a... |
| metuust 22365 | The uniform structure gene... |
| cfilucfil2 22366 | Given a metric ` D ` and a... |
| blval2 22367 | The ball around a point ` ... |
| elbl4 22368 | Membership in a ball, alte... |
| metuel 22369 | Elementhood in the uniform... |
| metuel2 22370 | Elementhood in the uniform... |
| metustbl 22371 | The "section" image of an ... |
| psmetutop 22372 | The topology induced by a ... |
| xmetutop 22373 | The topology induced by a ... |
| xmsusp 22374 | If the uniform set of a me... |
| restmetu 22375 | The uniform structure gene... |
| metucn 22376 | Uniform continuity in metr... |
| dscmet 22377 | The discrete metric on any... |
| dscopn 22378 | The discrete metric genera... |
| nrmmetd 22379 | Show that a group norm gen... |
| abvmet 22380 | An absolute value ` F ` ge... |
| nmfval 22393 | The value of the norm func... |
| nmval 22394 | The value of the norm func... |
| nmfval2 22395 | The value of the norm func... |
| nmval2 22396 | The value of the norm func... |
| nmf2 22397 | The norm is a function fro... |
| nmpropd 22398 | Weak property deduction fo... |
| nmpropd2 22399 | Strong property deduction ... |
| isngp 22400 | The property of being a no... |
| isngp2 22401 | The property of being a no... |
| isngp3 22402 | The property of being a no... |
| ngpgrp 22403 | A normed group is a group.... |
| ngpms 22404 | A normed group is a metric... |
| ngpxms 22405 | A normed group is a metric... |
| ngptps 22406 | A normed group is a topolo... |
| ngpmet 22407 | The (induced) metric of a ... |
| ngpds 22408 | Value of the distance func... |
| ngpdsr 22409 | Value of the distance func... |
| ngpds2 22410 | Write the distance between... |
| ngpds2r 22411 | Write the distance between... |
| ngpds3 22412 | Write the distance between... |
| ngpds3r 22413 | Write the distance between... |
| ngprcan 22414 | Cancel right addition insi... |
| ngplcan 22415 | Cancel left addition insid... |
| isngp4 22416 | Express the property of be... |
| ngpinvds 22417 | Two elements are the same ... |
| ngpsubcan 22418 | Cancel right subtraction i... |
| nmf 22419 | The norm on a normed group... |
| nmcl 22420 | The norm of a normed group... |
| nmge0 22421 | The norm of a normed group... |
| nmeq0 22422 | The identity is the only e... |
| nmne0 22423 | The norm of a nonzero elem... |
| nmrpcl 22424 | The norm of a nonzero elem... |
| nminv 22425 | The norm of a negated elem... |
| nmmtri 22426 | The triangle inequality fo... |
| nmsub 22427 | The norm of the difference... |
| nmrtri 22428 | Reverse triangle inequalit... |
| nm2dif 22429 | Inequality for the differe... |
| nmtri 22430 | The triangle inequality fo... |
| nmtri2 22431 | Triangle inequality for th... |
| ngpi 22432 | The properties of a normed... |
| nm0 22433 | Norm of the identity eleme... |
| nmgt0 22434 | The norm of a nonzero elem... |
| sgrim 22435 | The induced metric on a su... |
| sgrimval 22436 | The induced metric on a su... |
| subgnm 22437 | The norm in a subgroup. (... |
| subgnm2 22438 | A substructure assigns the... |
| subgngp 22439 | A normed group restricted ... |
| ngptgp 22440 | A normed abelian group is ... |
| ngppropd 22441 | Property deduction for a n... |
| reldmtng 22442 | The function ` toNrmGrp ` ... |
| tngval 22443 | Value of the function whic... |
| tnglem 22444 | Lemma for ~ tngbas and sim... |
| tngbas 22445 | The base set of a structur... |
| tngplusg 22446 | The group addition of a st... |
| tng0 22447 | The group identity of a st... |
| tngmulr 22448 | The ring multiplication of... |
| tngsca 22449 | The scalar ring of a struc... |
| tngvsca 22450 | The scalar multiplication ... |
| tngip 22451 | The inner product operatio... |
| tngds 22452 | The metric function of a s... |
| tngtset 22453 | The topology generated by ... |
| tngtopn 22454 | The topology generated by ... |
| tngnm 22455 | The topology generated by ... |
| tngngp2 22456 | A norm turns a group into ... |
| tngngpd 22457 | Derive the axioms for a no... |
| tngngp 22458 | Derive the axioms for a no... |
| tnggrpr 22459 | If a structure equipped wi... |
| tngngp3 22460 | Alternate definition of a ... |
| nrmtngdist 22461 | The augmentation of a norm... |
| nrmtngnrm 22462 | The augmentation of a norm... |
| tngngpim 22463 | The induced metric of a no... |
| isnrg 22464 | A normed ring is a ring wi... |
| nrgabv 22465 | The norm of a normed ring ... |
| nrgngp 22466 | A normed ring is a normed ... |
| nrgring 22467 | A normed ring is a ring. ... |
| nmmul 22468 | The norm of a product in a... |
| nrgdsdi 22469 | Distribute a distance calc... |
| nrgdsdir 22470 | Distribute a distance calc... |
| nm1 22471 | The norm of one in a nonze... |
| unitnmn0 22472 | The norm of a unit is nonz... |
| nminvr 22473 | The norm of an inverse in ... |
| nmdvr 22474 | The norm of a division in ... |
| nrgdomn 22475 | A nonzero normed ring is a... |
| nrgtgp 22476 | A normed ring is a topolog... |
| subrgnrg 22477 | A normed ring restricted t... |
| tngnrg 22478 | Given any absolute value o... |
| isnlm 22479 | A normed (left) module is ... |
| nmvs 22480 | Defining property of a nor... |
| nlmngp 22481 | A normed module is a norme... |
| nlmlmod 22482 | A normed module is a left ... |
| nlmnrg 22483 | The scalar component of a ... |
| nlmngp2 22484 | The scalar component of a ... |
| nlmdsdi 22485 | Distribute a distance calc... |
| nlmdsdir 22486 | Distribute a distance calc... |
| nlmmul0or 22487 | If a scalar product is zer... |
| sranlm 22488 | The subring algebra over a... |
| nlmvscnlem2 22489 | Lemma for ~ nlmvscn . Com... |
| nlmvscnlem1 22490 | Lemma for ~ nlmvscn . (Co... |
| nlmvscn 22491 | The scalar multiplication ... |
| rlmnlm 22492 | The ring module over a nor... |
| rlmnm 22493 | The norm function in the r... |
| nrgtrg 22494 | A normed ring is a topolog... |
| nrginvrcnlem 22495 | Lemma for ~ nrginvrcn . C... |
| nrginvrcn 22496 | The ring inverse function ... |
| nrgtdrg 22497 | A normed division ring is ... |
| nlmtlm 22498 | A normed module is a topol... |
| isnvc 22499 | A normed vector space is j... |
| nvcnlm 22500 | A normed vector space is a... |
| nvclvec 22501 | A normed vector space is a... |
| nvclmod 22502 | A normed vector space is a... |
| isnvc2 22503 | A normed vector space is j... |
| nvctvc 22504 | A normed vector space is a... |
| lssnlm 22505 | A subspace of a normed mod... |
| lssnvc 22506 | A subspace of a normed vec... |
| rlmnvc 22507 | The ring module over a nor... |
| ngpocelbl 22508 | Membership of an off-cente... |
| nmoffn 22515 | The function producing ope... |
| reldmnghm 22516 | Lemma for normed group hom... |
| reldmnmhm 22517 | Lemma for module homomorph... |
| nmofval 22518 | Value of the operator norm... |
| nmoval 22519 | Value of the operator norm... |
| nmogelb 22520 | Property of the operator n... |
| nmolb 22521 | Any upper bound on the val... |
| nmolb2d 22522 | Any upper bound on the val... |
| nmof 22523 | The operator norm is a fun... |
| nmocl 22524 | The operator norm of an op... |
| nmoge0 22525 | The operator norm of an op... |
| nghmfval 22526 | A normed group homomorphis... |
| isnghm 22527 | A normed group homomorphis... |
| isnghm2 22528 | A normed group homomorphis... |
| isnghm3 22529 | A normed group homomorphis... |
| bddnghm 22530 | A bounded group homomorphi... |
| nghmcl 22531 | A normed group homomorphis... |
| nmoi 22532 | The operator norm achieves... |
| nmoix 22533 | The operator norm is a bou... |
| nmoi2 22534 | The operator norm is a bou... |
| nmoleub 22535 | The operator norm, defined... |
| nghmrcl1 22536 | Reverse closure for a norm... |
| nghmrcl2 22537 | Reverse closure for a norm... |
| nghmghm 22538 | A normed group homomorphis... |
| nmo0 22539 | The operator norm of the z... |
| nmoeq0 22540 | The operator norm is zero ... |
| nmoco 22541 | An upper bound on the oper... |
| nghmco 22542 | The composition of normed ... |
| nmotri 22543 | Triangle inequality for th... |
| nghmplusg 22544 | The sum of two bounded lin... |
| 0nghm 22545 | The zero operator is a nor... |
| nmoid 22546 | The operator norm of the i... |
| idnghm 22547 | The identity operator is a... |
| nmods 22548 | Upper bound for the distan... |
| nghmcn 22549 | A normed group homomorphis... |
| isnmhm 22550 | A normed module homomorphi... |
| nmhmrcl1 22551 | Reverse closure for a norm... |
| nmhmrcl2 22552 | Reverse closure for a norm... |
| nmhmlmhm 22553 | A normed module homomorphi... |
| nmhmnghm 22554 | A normed module homomorphi... |
| nmhmghm 22555 | A normed module homomorphi... |
| isnmhm2 22556 | A normed module homomorphi... |
| nmhmcl 22557 | A normed module homomorphi... |
| idnmhm 22558 | The identity operator is a... |
| 0nmhm 22559 | The zero operator is a bou... |
| nmhmco 22560 | The composition of bounded... |
| nmhmplusg 22561 | The sum of two bounded lin... |
| qtopbaslem 22562 | The set of open intervals ... |
| qtopbas 22563 | The set of open intervals ... |
| retopbas 22564 | A basis for the standard t... |
| retop 22565 | The standard topology on t... |
| uniretop 22566 | The underlying set of the ... |
| retopon 22567 | The standard topology on t... |
| retps 22568 | The standard topological s... |
| iooretop 22569 | Open intervals are open se... |
| icccld 22570 | Closed intervals are close... |
| icopnfcld 22571 | Right-unbounded closed int... |
| iocmnfcld 22572 | Left-unbounded closed inte... |
| qdensere 22573 | ` QQ ` is dense in the sta... |
| cnmetdval 22574 | Value of the distance func... |
| cnmet 22575 | The absolute value metric ... |
| cnxmet 22576 | The absolute value metric ... |
| cnbl0 22577 | Two ways to write the open... |
| cnblcld 22578 | Two ways to write the clos... |
| cnfldms 22579 | The complex number field i... |
| cnfldxms 22580 | The complex number field i... |
| cnfldtps 22581 | The complex number field i... |
| cnfldnm 22582 | The norm of the field of c... |
| cnngp 22583 | The complex numbers form a... |
| cnnrg 22584 | The complex numbers form a... |
| cnfldtopn 22585 | The topology of the comple... |
| cnfldtopon 22586 | The topology of the comple... |
| cnfldtop 22587 | The topology of the comple... |
| cnfldhaus 22588 | The topology of the comple... |
| unicntop 22589 | The underlying set of the ... |
| cnopn 22590 | The set of complex numbers... |
| zringnrg 22591 | The ring of integers is a ... |
| remetdval 22592 | Value of the distance func... |
| remet 22593 | The absolute value metric ... |
| rexmet 22594 | The absolute value metric ... |
| bl2ioo 22595 | A ball in terms of an open... |
| ioo2bl 22596 | An open interval of reals ... |
| ioo2blex 22597 | An open interval of reals ... |
| blssioo 22598 | The balls of the standard ... |
| tgioo 22599 | The topology generated by ... |
| qdensere2 22600 | ` QQ ` is dense in ` RR ` ... |
| blcvx 22601 | An open ball in the comple... |
| rehaus 22602 | The standard topology on t... |
| tgqioo 22603 | The topology generated by ... |
| re2ndc 22604 | The standard topology on t... |
| resubmet 22605 | The subspace topology indu... |
| tgioo2 22606 | The standard topology on t... |
| rerest 22607 | The subspace topology indu... |
| tgioo3 22608 | The standard topology on t... |
| xrtgioo 22609 | The topology on the extend... |
| xrrest 22610 | The subspace topology indu... |
| xrrest2 22611 | The subspace topology indu... |
| xrsxmet 22612 | The metric on the extended... |
| xrsdsre 22613 | The metric on the extended... |
| xrsblre 22614 | Any ball of the metric of ... |
| xrsmopn 22615 | The metric on the extended... |
| zcld 22616 | The integers are a closed ... |
| recld2 22617 | The real numbers are a clo... |
| zcld2 22618 | The integers are a closed ... |
| zdis 22619 | The integers are a discret... |
| sszcld 22620 | Every subset of the intege... |
| reperflem 22621 | A subset of the real numbe... |
| reperf 22622 | The real numbers are a per... |
| cnperf 22623 | The complex numbers are a ... |
| iccntr 22624 | The interior of a closed i... |
| icccmplem1 22625 | Lemma for ~ icccmp . (Con... |
| icccmplem2 22626 | Lemma for ~ icccmp . (Con... |
| icccmplem3 22627 | Lemma for ~ icccmp . (Con... |
| icccmp 22628 | A closed interval in ` RR ... |
| reconnlem1 22629 | Lemma for ~ reconn . Conn... |
| reconnlem2 22630 | Lemma for ~ reconn . (Con... |
| reconn 22631 | A subset of the reals is c... |
| retopconn 22632 | Corollary of ~ reconn . T... |
| iccconn 22633 | A closed interval is conne... |
| opnreen 22634 | Every nonempty open set is... |
| rectbntr0 22635 | A countable subset of the ... |
| xrge0gsumle 22636 | A finite sum in the nonneg... |
| xrge0tsms 22637 | Any finite or infinite sum... |
| xrge0tsms2 22638 | Any finite or infinite sum... |
| metdcnlem 22639 | The metric function of a m... |
| xmetdcn2 22640 | The metric function of an ... |
| xmetdcn 22641 | The metric function of an ... |
| metdcn2 22642 | The metric function of a m... |
| metdcn 22643 | The metric function of a m... |
| msdcn 22644 | The metric function of a m... |
| cnmpt1ds 22645 | Continuity of the metric f... |
| cnmpt2ds 22646 | Continuity of the metric f... |
| nmcn 22647 | The norm of a normed group... |
| ngnmcncn 22648 | The norm of a normed group... |
| abscn 22649 | The absolute value functio... |
| metdsval 22650 | Value of the "distance to ... |
| metdsf 22651 | The distance from a point ... |
| metdsge 22652 | The distance from the poin... |
| metds0 22653 | If a point is in a set, it... |
| metdstri 22654 | A generalization of the tr... |
| metdsle 22655 | The distance from a point ... |
| metdsre 22656 | The distance from a point ... |
| metdseq0 22657 | The distance from a point ... |
| metdscnlem 22658 | Lemma for ~ metdscn . (Co... |
| metdscn 22659 | The function ` F ` which g... |
| metdscn2 22660 | The function ` F ` which g... |
| metnrmlem1a 22661 | Lemma for ~ metnrm . (Con... |
| metnrmlem1 22662 | Lemma for ~ metnrm . (Con... |
| metnrmlem2 22663 | Lemma for ~ metnrm . (Con... |
| metnrmlem3 22664 | Lemma for ~ metnrm . (Con... |
| metnrm 22665 | A metric space is normal. ... |
| metreg 22666 | A metric space is regular.... |
| addcnlem 22667 | Lemma for ~ addcn , ~ subc... |
| addcn 22668 | Complex number addition is... |
| subcn 22669 | Complex number subtraction... |
| mulcn 22670 | Complex number multiplicat... |
| divcn 22671 | Complex number division is... |
| cnfldtgp 22672 | The complex numbers form a... |
| fsumcn 22673 | A finite sum of functions ... |
| fsum2cn 22674 | Version of ~ fsumcn for tw... |
| expcn 22675 | The power function on comp... |
| divccn 22676 | Division by a nonzero cons... |
| sqcn 22677 | The square function on com... |
| iitopon 22682 | The unit interval is a top... |
| iitop 22683 | The unit interval is a top... |
| iiuni 22684 | The base set of the unit i... |
| dfii2 22685 | Alternate definition of th... |
| dfii3 22686 | Alternate definition of th... |
| dfii4 22687 | Alternate definition of th... |
| dfii5 22688 | The unit interval expresse... |
| iicmp 22689 | The unit interval is compa... |
| iiconn 22690 | The unit interval is conne... |
| cncfval 22691 | The value of the continuou... |
| elcncf 22692 | Membership in the set of c... |
| elcncf2 22693 | Version of ~ elcncf with a... |
| cncfrss 22694 | Reverse closure of the con... |
| cncfrss2 22695 | Reverse closure of the con... |
| cncff 22696 | A continuous complex funct... |
| cncfi 22697 | Defining property of a con... |
| elcncf1di 22698 | Membership in the set of c... |
| elcncf1ii 22699 | Membership in the set of c... |
| rescncf 22700 | A continuous complex funct... |
| cncffvrn 22701 | Change the codomain of a c... |
| cncfss 22702 | The set of continuous func... |
| climcncf 22703 | Image of a limit under a c... |
| abscncf 22704 | Absolute value is continuo... |
| recncf 22705 | Real part is continuous. ... |
| imcncf 22706 | Imaginary part is continuo... |
| cjcncf 22707 | Complex conjugate is conti... |
| mulc1cncf 22708 | Multiplication by a consta... |
| divccncf 22709 | Division by a constant is ... |
| cncfco 22710 | The composition of two con... |
| cncfmet 22711 | Relate complex function co... |
| cncfcn 22712 | Relate complex function co... |
| cncfcn1 22713 | Relate complex function co... |
| cncfmptc 22714 | A constant function is a c... |
| cncfmptid 22715 | The identity function is a... |
| cncfmpt1f 22716 | Composition of continuous ... |
| cncfmpt2f 22717 | Composition of continuous ... |
| cncfmpt2ss 22718 | Composition of continuous ... |
| addccncf 22719 | Adding a constant is a con... |
| cdivcncf 22720 | Division with a constant n... |
| negcncf 22721 | The negative function is c... |
| negfcncf 22722 | The negative of a continuo... |
| abscncfALT 22723 | Absolute value is continuo... |
| cncfcnvcn 22724 | Rewrite ~ cmphaushmeo for ... |
| expcncf 22725 | The power function on comp... |
| cnmptre 22726 | Lemma for ~ iirevcn and re... |
| cnmpt2pc 22727 | Piecewise definition of a ... |
| iirev 22728 | Reverse the unit interval.... |
| iirevcn 22729 | The reversion function is ... |
| iihalf1 22730 | Map the first half of ` II... |
| iihalf1cn 22731 | The first half function is... |
| iihalf2 22732 | Map the second half of ` I... |
| iihalf2cn 22733 | The second half function i... |
| elii1 22734 | Divide the unit interval i... |
| elii2 22735 | Divide the unit interval i... |
| iimulcl 22736 | The unit interval is close... |
| iimulcn 22737 | Multiplication is a contin... |
| icoopnst 22738 | A half-open interval start... |
| iocopnst 22739 | A half-open interval endin... |
| icchmeo 22740 | The natural bijection from... |
| icopnfcnv 22741 | Define a bijection from ` ... |
| icopnfhmeo 22742 | The defined bijection from... |
| iccpnfcnv 22743 | Define a bijection from ` ... |
| iccpnfhmeo 22744 | The defined bijection from... |
| xrhmeo 22745 | The bijection from ` [ -u ... |
| xrhmph 22746 | The extended reals are hom... |
| xrcmp 22747 | The topology of the extend... |
| xrconn 22748 | The topology of the extend... |
| icccvx 22749 | A linear combination of tw... |
| oprpiece1res1 22750 | Restriction to the first p... |
| oprpiece1res2 22751 | Restriction to the second ... |
| cnrehmeo 22752 | The canonical bijection fr... |
| cnheiborlem 22753 | Lemma for ~ cnheibor . (C... |
| cnheibor 22754 | Heine-Borel theorem for co... |
| cnllycmp 22755 | The topology on the comple... |
| rellycmp 22756 | The topology on the reals ... |
| bndth 22757 | The Boundedness Theorem. ... |
| evth 22758 | The Extreme Value Theorem.... |
| evth2 22759 | The Extreme Value Theorem,... |
| lebnumlem1 22760 | Lemma for ~ lebnum . The ... |
| lebnumlem2 22761 | Lemma for ~ lebnum . As a... |
| lebnumlem3 22762 | Lemma for ~ lebnum . By t... |
| lebnum 22763 | The Lebesgue number lemma,... |
| xlebnum 22764 | Generalize ~ lebnum to ext... |
| lebnumii 22765 | Specialize the Lebesgue nu... |
| ishtpy 22771 | Membership in the class of... |
| htpycn 22772 | A homotopy is a continuous... |
| htpyi 22773 | A homotopy evaluated at it... |
| ishtpyd 22774 | Deduction for membership i... |
| htpycom 22775 | Given a homotopy from ` F ... |
| htpyid 22776 | A homotopy from a function... |
| htpyco1 22777 | Compose a homotopy with a ... |
| htpyco2 22778 | Compose a homotopy with a ... |
| htpycc 22779 | Concatenate two homotopies... |
| isphtpy 22780 | Membership in the class of... |
| phtpyhtpy 22781 | A path homotopy is a homot... |
| phtpycn 22782 | A path homotopy is a conti... |
| phtpyi 22783 | Membership in the class of... |
| phtpy01 22784 | Two path-homotopic paths h... |
| isphtpyd 22785 | Deduction for membership i... |
| isphtpy2d 22786 | Deduction for membership i... |
| phtpycom 22787 | Given a homotopy from ` F ... |
| phtpyid 22788 | A homotopy from a path to ... |
| phtpyco2 22789 | Compose a path homotopy wi... |
| phtpycc 22790 | Concatenate two path homot... |
| phtpcrel 22792 | The path homotopy relation... |
| isphtpc 22793 | The relation "is path homo... |
| phtpcer 22794 | Path homotopy is an equiva... |
| phtpcerOLD 22795 | Obsolete proof of ~ phtpce... |
| phtpc01 22796 | Path homotopic paths have ... |
| reparphti 22797 | Lemma for ~ reparpht . (C... |
| reparpht 22798 | Reparametrization lemma. ... |
| phtpcco2 22799 | Compose a path homotopy wi... |
| pcofval 22810 | The value of the path conc... |
| pcoval 22811 | The concatenation of two p... |
| pcovalg 22812 | Evaluate the concatenation... |
| pcoval1 22813 | Evaluate the concatenation... |
| pco0 22814 | The starting point of a pa... |
| pco1 22815 | The ending point of a path... |
| pcoval2 22816 | Evaluate the concatenation... |
| pcocn 22817 | The concatenation of two p... |
| copco 22818 | The composition of a conca... |
| pcohtpylem 22819 | Lemma for ~ pcohtpy . (Co... |
| pcohtpy 22820 | Homotopy invariance of pat... |
| pcoptcl 22821 | A constant function is a p... |
| pcopt 22822 | Concatenation with a point... |
| pcopt2 22823 | Concatenation with a point... |
| pcoass 22824 | Order of concatenation doe... |
| pcorevcl 22825 | Closure for a reversed pat... |
| pcorevlem 22826 | Lemma for ~ pcorev . Prov... |
| pcorev 22827 | Concatenation with the rev... |
| pcorev2 22828 | Concatenation with the rev... |
| pcophtb 22829 | The path homotopy equivale... |
| om1val 22830 | The definition of the loop... |
| om1bas 22831 | The base set of the loop s... |
| om1elbas 22832 | Elementhood in the base se... |
| om1addcl 22833 | Closure of the group opera... |
| om1plusg 22834 | The group operation (which... |
| om1tset 22835 | The topology of the loop s... |
| om1opn 22836 | The topology of the loop s... |
| pi1val 22837 | The definition of the fund... |
| pi1bas 22838 | The base set of the fundam... |
| pi1blem 22839 | Lemma for ~ pi1buni . (Co... |
| pi1buni 22840 | Another way to write the l... |
| pi1bas2 22841 | The base set of the fundam... |
| pi1eluni 22842 | Elementhood in the base se... |
| pi1bas3 22843 | The base set of the fundam... |
| pi1cpbl 22844 | The group operation, loop ... |
| elpi1 22845 | The elements of the fundam... |
| elpi1i 22846 | The elements of the fundam... |
| pi1addf 22847 | The group operation of ` p... |
| pi1addval 22848 | The concatenation of two p... |
| pi1grplem 22849 | Lemma for ~ pi1grp . (Con... |
| pi1grp 22850 | The fundamental group is a... |
| pi1id 22851 | The identity element of th... |
| pi1inv 22852 | An inverse in the fundamen... |
| pi1xfrf 22853 | Functionality of the loop ... |
| pi1xfrval 22854 | The value of the loop tran... |
| pi1xfr 22855 | Given a path ` F ` and its... |
| pi1xfrcnvlem 22856 | Given a path ` F ` between... |
| pi1xfrcnv 22857 | Given a path ` F ` between... |
| pi1xfrgim 22858 | The mapping ` G ` between ... |
| pi1cof 22859 | Functionality of the loop ... |
| pi1coval 22860 | The value of the loop tran... |
| pi1coghm 22861 | The mapping ` G ` between ... |
| isclm 22864 | A subcomplex module is a l... |
| clmsca 22865 | The ring of scalars ` F ` ... |
| clmsubrg 22866 | The base set of the ring o... |
| clmlmod 22867 | A subcomplex module is a l... |
| clmgrp 22868 | A subcomplex module is an ... |
| clmabl 22869 | A subcomplex module is an ... |
| clmring 22870 | The scalar ring of a subco... |
| clmfgrp 22871 | The scalar ring of a subco... |
| clm0 22872 | The zero of the scalar rin... |
| clm1 22873 | The identity of the scalar... |
| clmadd 22874 | The addition of the scalar... |
| clmmul 22875 | The multiplication of the ... |
| clmcj 22876 | The conjugation of the sca... |
| isclmi 22877 | Reverse direction of ~ isc... |
| clmzss 22878 | The scalar ring of a subco... |
| clmsscn 22879 | The scalar ring of a subco... |
| clmsub 22880 | Subtraction in the scalar ... |
| clmneg 22881 | Negation in the scalar rin... |
| clmneg1 22882 | Minus one is in the scalar... |
| clmabs 22883 | Norm in the scalar ring of... |
| clmacl 22884 | Closure of ring addition f... |
| clmmcl 22885 | Closure of ring multiplica... |
| clmsubcl 22886 | Closure of ring subtractio... |
| lmhmclm 22887 | The domain of a linear ope... |
| clmvscl 22888 | Closure of scalar product ... |
| clmvsass 22889 | Associative law for scalar... |
| clmvscom 22890 | Commutative law for the sc... |
| clmvsdir 22891 | Distributive law for scala... |
| clmvsdi 22892 | Distributive law for scala... |
| clmvs1 22893 | Scalar product with ring u... |
| clmvs2 22894 | A vector plus itself is tw... |
| clm0vs 22895 | Zero times a vector is the... |
| clmopfne 22896 | The (functionalized) opera... |
| isclmp 22897 | The predicate "is a subcom... |
| isclmi0 22898 | Properties that determine ... |
| clmvneg1 22899 | Minus 1 times a vector is ... |
| clmvsneg 22900 | Multiplication of a vector... |
| clmmulg 22901 | The group multiple functio... |
| clmsubdir 22902 | Scalar multiplication dist... |
| clmpm1dir 22903 | Subtractive distributive l... |
| clmnegneg 22904 | Double negative of a vecto... |
| clmnegsubdi2 22905 | Distribution of negative o... |
| clmsub4 22906 | Rearrangement of 4 terms i... |
| clmvsrinv 22907 | A vector minus itself. (C... |
| clmvslinv 22908 | Minus a vector plus itself... |
| clmvsubval 22909 | Value of vector subtractio... |
| clmvsubval2 22910 | Value of vector subtractio... |
| clmvz 22911 | Two ways to express the ne... |
| zlmclm 22912 | The ` ZZ ` -module operati... |
| clmzlmvsca 22913 | The scalar product of a su... |
| nmoleub2lem 22914 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem3 22915 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem2 22916 | Lemma for ~ nmoleub2a and ... |
| nmoleub2a 22917 | The operator norm is the s... |
| nmoleub2b 22918 | The operator norm is the s... |
| nmoleub3 22919 | The operator norm is the s... |
| nmhmcn 22920 | A linear operator over a n... |
| cmodscexp 22921 | The powers of ` _i ` belon... |
| cmodscmulexp 22922 | The scalar product of a ve... |
| cvslvec 22925 | A subcomplex vector space ... |
| cvsclm 22926 | A subcomplex vector space ... |
| iscvs 22927 | A subcomplex vector space ... |
| iscvsp 22928 | The predicate "is a subcom... |
| iscvsi 22929 | Properties that determine ... |
| cvsi 22930 | The properties of a subcom... |
| cvsunit 22931 | Unit group of the scalar r... |
| cvsdiv 22932 | Division of the scalar rin... |
| cvsdivcl 22933 | The scalar field of a subc... |
| cvsmuleqdivd 22934 | An equality involving rati... |
| cvsdiveqd 22935 | An equality involving rati... |
| cnlmodlem1 22936 | Lemma 1 for ~ cnlmod . (C... |
| cnlmodlem2 22937 | Lemma 2 for ~ cnlmod . (C... |
| cnlmodlem3 22938 | Lemma 3 for ~ cnlmod . (C... |
| cnlmod4 22939 | Lemma 4 for ~ cnlmod . (C... |
| cnlmod 22940 | The set of complex numbers... |
| cnstrcvs 22941 | The set of complex numbers... |
| cnrbas 22942 | The set of complex numbers... |
| cnrlmod 22943 | The complex left module of... |
| cnrlvec 22944 | The complex left module of... |
| cncvs 22945 | The complex left module of... |
| recvs 22946 | The field of the real numb... |
| qcvs 22947 | The field of rational numb... |
| zclmncvs 22948 | The ring of integers as le... |
| isncvsngp 22949 | A normed subcomplex vector... |
| isncvsngpd 22950 | Properties that determine ... |
| ncvsi 22951 | The properties of a normed... |
| ncvsprp 22952 | Proportionality property o... |
| ncvsge0 22953 | The norm of a scalar produ... |
| ncvsm1 22954 | The norm of the negative o... |
| ncvsdif 22955 | The norm of the difference... |
| ncvspi 22956 | The norm of a vector plus ... |
| ncvs1 22957 | From any nonzero vector, c... |
| cnrnvc 22958 | The set of complex numbers... |
| cnncvs 22959 | The set of complex numbers... |
| cnnm 22960 | The norm operation of the ... |
| ncvspds 22961 | Value of the distance func... |
| cnindmet 22962 | The metric induced on the ... |
| cnncvsaddassdemo 22963 | Derive the associative law... |
| cnncvsmulassdemo 22964 | Derive the associative law... |
| cnncvsabsnegdemo 22965 | Derive the absolute value ... |
| iscph 22970 | A subcomplex pre-Hilbert s... |
| cphphl 22971 | A subcomplex pre-Hilbert s... |
| cphnlm 22972 | A subcomplex pre-Hilbert s... |
| cphngp 22973 | A subcomplex pre-Hilbert s... |
| cphlmod 22974 | A subcomplex pre-Hilbert s... |
| cphlvec 22975 | A subcomplex pre-Hilbert s... |
| cphnvc 22976 | A subcomplex pre-Hilbert s... |
| cphsubrglem 22977 | Lemma for ~ cphsubrg . (C... |
| cphreccllem 22978 | Lemma for ~ cphreccl . (C... |
| cphsca 22979 | A subcomplex pre-Hilbert s... |
| cphsubrg 22980 | The scalar field of a subc... |
| cphreccl 22981 | The scalar field of a subc... |
| cphdivcl 22982 | The scalar field of a subc... |
| cphcjcl 22983 | The scalar field of a subc... |
| cphsqrtcl 22984 | The scalar field of a subc... |
| cphabscl 22985 | The scalar field of a subc... |
| cphsqrtcl2 22986 | The scalar field of a subc... |
| cphsqrtcl3 22987 | If the scalar field contai... |
| cphqss 22988 | The scalar field of a subc... |
| cphclm 22989 | A subcomplex pre-Hilbert s... |
| cphnmvs 22990 | Norm of a scalar product. ... |
| cphipcl 22991 | An inner product is a memb... |
| cphnmfval 22992 | The value of the norm in a... |
| cphnm 22993 | The square of the norm is ... |
| nmsq 22994 | The square of the norm is ... |
| cphnmf 22995 | The norm of a vector is a ... |
| cphnmcl 22996 | The norm of a vector is a ... |
| reipcl 22997 | An inner product of an ele... |
| ipge0 22998 | The inner product in a sub... |
| cphipcj 22999 | Conjugate of an inner prod... |
| cphipipcj 23000 | An inner product times its... |
| cphorthcom 23001 | Orthogonality (meaning inn... |
| cphip0l 23002 | Inner product with a zero ... |
| cphip0r 23003 | Inner product with a zero ... |
| cphipeq0 23004 | The inner product of a vec... |
| cphdir 23005 | Distributive law for inner... |
| cphdi 23006 | Distributive law for inner... |
| cph2di 23007 | Distributive law for inner... |
| cphsubdir 23008 | Distributive law for inner... |
| cphsubdi 23009 | Distributive law for inner... |
| cph2subdi 23010 | Distributive law for inner... |
| cphass 23011 | Associative law for inner ... |
| cphassr 23012 | "Associative" law for seco... |
| cph2ass 23013 | Move scalar multiplication... |
| cphassi 23014 | Associative law for the fi... |
| cphassir 23015 | "Associative" law for the ... |
| tchex 23016 | Lemma for ~ tchbas and sim... |
| tchval 23017 | Define a function to augme... |
| tchbas 23018 | The base set of a subcompl... |
| tchplusg 23019 | The addition operation of ... |
| tchsub 23020 | The subtraction operation ... |
| tchmulr 23021 | The ring operation of a su... |
| tchsca 23022 | The scalar field of a subc... |
| tchvsca 23023 | The scalar multiplication ... |
| tchip 23024 | The inner product of a sub... |
| tchtopn 23025 | The topology of a subcompl... |
| tchphl 23026 | Augmentation of a subcompl... |
| tchnmfval 23027 | The norm of a subcomplex p... |
| tchnmval 23028 | The norm of a subcomplex p... |
| cphtchnm 23029 | The norm of a norm-augment... |
| tchds 23030 | The distance of a pre-Hilb... |
| tchclm 23031 | Lemma for ~ tchcph . (Con... |
| tchcphlem3 23032 | Lemma for ~ tchcph : real ... |
| ipcau2 23033 | The Cauchy-Schwarz inequal... |
| tchcphlem1 23034 | Lemma for ~ tchcph : the t... |
| tchcphlem2 23035 | Lemma for ~ tchcph : homog... |
| tchcph 23036 | The standard definition of... |
| ipcau 23037 | The Cauchy-Schwarz inequal... |
| nmparlem 23038 | Lemma for ~ nmpar . (Cont... |
| nmpar 23039 | A subcomplex pre-Hilbert s... |
| cphipval2 23040 | Value of the inner product... |
| 4cphipval2 23041 | Four times the inner produ... |
| cphipval 23042 | Value of the inner product... |
| ipcnlem2 23043 | The inner product operatio... |
| ipcnlem1 23044 | The inner product operatio... |
| ipcn 23045 | The inner product operatio... |
| cnmpt1ip 23046 | Continuity of inner produc... |
| cnmpt2ip 23047 | Continuity of inner produc... |
| csscld 23048 | A "closed subspace" in a s... |
| clsocv 23049 | The orthogonal complement ... |
| lmmbr 23056 | Express the binary relatio... |
| lmmbr2 23057 | Express the binary relatio... |
| lmmbr3 23058 | Express the binary relatio... |
| lmmcvg 23059 | Convergence property of a ... |
| lmmbrf 23060 | Express the binary relatio... |
| lmnn 23061 | A condition that implies c... |
| cfilfval 23062 | The set of Cauchy filters ... |
| iscfil 23063 | The property of being a Ca... |
| iscfil2 23064 | The property of being a Ca... |
| cfilfil 23065 | A Cauchy filter is a filte... |
| cfili 23066 | Property of a Cauchy filte... |
| cfil3i 23067 | A Cauchy filter contains b... |
| cfilss 23068 | A filter finer than a Cauc... |
| fgcfil 23069 | The Cauchy filter conditio... |
| fmcfil 23070 | The Cauchy filter conditio... |
| iscfil3 23071 | A filter is Cauchy iff it ... |
| cfilfcls 23072 | Similar to ultrafilters ( ... |
| caufval 23073 | The set of Cauchy sequence... |
| iscau 23074 | Express the property " ` F... |
| iscau2 23075 | Express the property " ` F... |
| iscau3 23076 | Express the Cauchy sequenc... |
| iscau4 23077 | Express the property " ` F... |
| iscauf 23078 | Express the property " ` F... |
| caun0 23079 | A metric with a Cauchy seq... |
| caufpm 23080 | Inclusion of a Cauchy sequ... |
| caucfil 23081 | A Cauchy sequence predicat... |
| iscmet 23082 | The property " ` D ` is a ... |
| cmetcvg 23083 | The convergence of a Cauch... |
| cmetmet 23084 | A complete metric space is... |
| cmetmeti 23085 | A complete metric space is... |
| cmetcaulem 23086 | Lemma for ~ cmetcau . (Co... |
| cmetcau 23087 | The convergence of a Cauch... |
| iscmet3lem3 23088 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem1 23089 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem2 23090 | Lemma for ~ iscmet3 . (Co... |
| iscmet3 23091 | The property " ` D ` is a ... |
| iscmet2 23092 | A metric ` D ` is complete... |
| cfilresi 23093 | A Cauchy filter on a metri... |
| cfilres 23094 | Cauchy filter on a metric ... |
| caussi 23095 | Cauchy sequence on a metri... |
| causs 23096 | Cauchy sequence on a metri... |
| equivcfil 23097 | If the metric ` D ` is "st... |
| equivcau 23098 | If the metric ` D ` is "st... |
| lmle 23099 | If the distance from each ... |
| nglmle 23100 | If the norm of each member... |
| lmclim 23101 | Relate a limit on the metr... |
| lmclimf 23102 | Relate a limit on the metr... |
| metelcls 23103 | A point belongs to the clo... |
| metcld 23104 | A subset of a metric space... |
| metcld2 23105 | A subset of a metric space... |
| caubl 23106 | Sufficient condition to en... |
| caublcls 23107 | The convergent point of a ... |
| metcnp4 23108 | Two ways to say a mapping ... |
| metcn4 23109 | Two ways to say a mapping ... |
| iscmet3i 23110 | Properties that determine ... |
| lmcau 23111 | Every convergent sequence ... |
| flimcfil 23112 | Every convergent filter in... |
| cmetss 23113 | A subspace of a complete m... |
| equivcmet 23114 | If two metrics are strongl... |
| relcmpcmet 23115 | If ` D ` is a metric space... |
| cmpcmet 23116 | A compact metric space is ... |
| cfilucfil3 23117 | Given a metric ` D ` and a... |
| cfilucfil4 23118 | Given a metric ` D ` and a... |
| cncmet 23119 | The set of complex numbers... |
| recmet 23120 | The real numbers are a com... |
| bcthlem1 23121 | Lemma for ~ bcth . Substi... |
| bcthlem2 23122 | Lemma for ~ bcth . The ba... |
| bcthlem3 23123 | Lemma for ~ bcth . The li... |
| bcthlem4 23124 | Lemma for ~ bcth . Given ... |
| bcthlem5 23125 | Lemma for ~ bcth . The pr... |
| bcth 23126 | Baire's Category Theorem. ... |
| bcth2 23127 | Baire's Category Theorem, ... |
| bcth3 23128 | Baire's Category Theorem, ... |
| isbn 23135 | A Banach space is a normed... |
| bnsca 23136 | The scalar field of a Bana... |
| bnnvc 23137 | A Banach space is a normed... |
| bnnlm 23138 | A Banach space is a normed... |
| bnngp 23139 | A Banach space is a normed... |
| bnlmod 23140 | A Banach space is a left m... |
| bncms 23141 | A Banach space is a comple... |
| iscms 23142 | A complete metric space is... |
| cmscmet 23143 | The induced metric on a co... |
| bncmet 23144 | The induced metric on Bana... |
| cmsms 23145 | A complete metric space is... |
| cmspropd 23146 | Property deduction for a c... |
| cmsss 23147 | The restriction of a compl... |
| lssbn 23148 | A subspace of a Banach spa... |
| cmetcusp1 23149 | If the uniform set of a co... |
| cmetcusp 23150 | The uniform space generate... |
| cncms 23151 | The field of complex numbe... |
| cnflduss 23152 | The uniform structure of t... |
| cnfldcusp 23153 | The field of complex numbe... |
| resscdrg 23154 | The real numbers are a sub... |
| cncdrg 23155 | The only complete subfield... |
| srabn 23156 | The subring algebra over a... |
| rlmbn 23157 | The ring module over a com... |
| ishl 23158 | The predicate "is a subcom... |
| hlbn 23159 | Every subcomplex Hilbert s... |
| hlcph 23160 | Every subcomplex Hilbert s... |
| hlphl 23161 | Every subcomplex Hilbert s... |
| hlcms 23162 | Every subcomplex Hilbert s... |
| hlprlem 23163 | Lemma for ~ hlpr . (Contr... |
| hlress 23164 | The scalar field of a subc... |
| hlpr 23165 | The scalar field of a subc... |
| ishl2 23166 | A Hilbert space is a compl... |
| retopn 23167 | The topology of the real n... |
| recms 23168 | The real numbers form a co... |
| reust 23169 | The Uniform structure of t... |
| recusp 23170 | The real numbers form a co... |
| rrxval 23175 | Value of the generalized E... |
| rrxbase 23176 | The base of the generalize... |
| rrxprds 23177 | Expand the definition of t... |
| rrxip 23178 | The inner product of the g... |
| rrxnm 23179 | The norm of the generalize... |
| rrxcph 23180 | Generalized Euclidean real... |
| rrxds 23181 | The distance over generali... |
| csbren 23182 | Cauchy-Schwarz-Bunjakovsky... |
| trirn 23183 | Triangle inequality in R^n... |
| rrxf 23184 | Euclidean vectors as funct... |
| rrxfsupp 23185 | Euclidean vectors are of f... |
| rrxsuppss 23186 | Support of Euclidean vecto... |
| rrxmvallem 23187 | Support of the function us... |
| rrxmval 23188 | The value of the Euclidean... |
| rrxmfval 23189 | The value of the Euclidean... |
| rrxmetlem 23190 | Lemma for ~ rrxmet . (Con... |
| rrxmet 23191 | Euclidean space is a metri... |
| rrxdstprj1 23192 | The distance between two p... |
| ehlval 23193 | Value of the Euclidean spa... |
| ehlbase 23194 | The base of the Euclidean ... |
| minveclem1 23195 | Lemma for ~ minvec . The ... |
| minveclem4c 23196 | Lemma for ~ minvec . The ... |
| minveclem2 23197 | Lemma for ~ minvec . Any ... |
| minveclem3a 23198 | Lemma for ~ minvec . ` D `... |
| minveclem3b 23199 | Lemma for ~ minvec . The ... |
| minveclem3 23200 | Lemma for ~ minvec . The ... |
| minveclem4a 23201 | Lemma for ~ minvec . ` F `... |
| minveclem4b 23202 | Lemma for ~ minvec . The ... |
| minveclem4 23203 | Lemma for ~ minvec . The ... |
| minveclem5 23204 | Lemma for ~ minvec . Disc... |
| minveclem6 23205 | Lemma for ~ minvec . Any ... |
| minveclem7 23206 | Lemma for ~ minvec . Sinc... |
| minvec 23207 | Minimizing vector theorem,... |
| pjthlem1 23208 | Lemma for ~ pjth . (Contr... |
| pjthlem2 23209 | Lemma for ~ pjth . (Contr... |
| pjth 23210 | Projection Theorem: Any H... |
| pjth2 23211 | Projection Theorem with ab... |
| cldcss 23212 | Corollary of the Projectio... |
| cldcss2 23213 | Corollary of the Projectio... |
| hlhil 23214 | Corollary of the Projectio... |
| mulcncf 23215 | The multiplication of two ... |
| divcncf 23216 | The quotient of two contin... |
| pmltpclem1 23217 | Lemma for ~ pmltpc . (Con... |
| pmltpclem2 23218 | Lemma for ~ pmltpc . (Con... |
| pmltpc 23219 | Any function on the reals ... |
| ivthlem1 23220 | Lemma for ~ ivth . The se... |
| ivthlem2 23221 | Lemma for ~ ivth . Show t... |
| ivthlem3 23222 | Lemma for ~ ivth , the int... |
| ivth 23223 | The intermediate value the... |
| ivth2 23224 | The intermediate value the... |
| ivthle 23225 | The intermediate value the... |
| ivthle2 23226 | The intermediate value the... |
| ivthicc 23227 | The interval between any t... |
| evthicc 23228 | Specialization of the Extr... |
| evthicc2 23229 | Combine ~ ivthicc with ~ e... |
| cniccbdd 23230 | A continuous function on a... |
| ovolfcl 23235 | Closure for the interval e... |
| ovolfioo 23236 | Unpack the interval coveri... |
| ovolficc 23237 | Unpack the interval coveri... |
| ovolficcss 23238 | Any (closed) interval cove... |
| ovolfsval 23239 | The value of the interval ... |
| ovolfsf 23240 | Closure for the interval l... |
| ovolsf 23241 | Closure for the partial su... |
| ovolval 23242 | The value of the outer mea... |
| elovolm 23243 | Elementhood in the set ` M... |
| elovolmr 23244 | Sufficient condition for e... |
| ovolmge0 23245 | The set ` M ` is composed ... |
| ovolcl 23246 | The volume of a set is an ... |
| ovollb 23247 | The outer volume is a lowe... |
| ovolgelb 23248 | The outer volume is the gr... |
| ovolge0 23249 | The volume of a set is alw... |
| ovolf 23250 | The domain and range of th... |
| ovollecl 23251 | If an outer volume is boun... |
| ovolsslem 23252 | Lemma for ~ ovolss . (Con... |
| ovolss 23253 | The volume of a set is mon... |
| ovolsscl 23254 | If a set is contained in a... |
| ovolssnul 23255 | A subset of a nullset is n... |
| ovollb2lem 23256 | Lemma for ~ ovollb2 . (Co... |
| ovollb2 23257 | It is often more convenien... |
| ovolctb 23258 | The volume of a denumerabl... |
| ovolq 23259 | The rational numbers have ... |
| ovolctb2 23260 | The volume of a countable ... |
| ovol0 23261 | The empty set has 0 outer ... |
| ovolfi 23262 | A finite set has 0 outer L... |
| ovolsn 23263 | A singleton has 0 outer Le... |
| ovolunlem1a 23264 | Lemma for ~ ovolun . (Con... |
| ovolunlem1 23265 | Lemma for ~ ovolun . (Con... |
| ovolunlem2 23266 | Lemma for ~ ovolun . (Con... |
| ovolun 23267 | The Lebesgue outer measure... |
| ovolunnul 23268 | Adding a nullset does not ... |
| ovolfiniun 23269 | The Lebesgue outer measure... |
| ovoliunlem1 23270 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem2 23271 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem3 23272 | Lemma for ~ ovoliun . (Co... |
| ovoliun 23273 | The Lebesgue outer measure... |
| ovoliun2 23274 | The Lebesgue outer measure... |
| ovoliunnul 23275 | A countable union of nulls... |
| shft2rab 23276 | If ` B ` is a shift of ` A... |
| ovolshftlem1 23277 | Lemma for ~ ovolshft . (C... |
| ovolshftlem2 23278 | Lemma for ~ ovolshft . (C... |
| ovolshft 23279 | The Lebesgue outer measure... |
| sca2rab 23280 | If ` B ` is a scale of ` A... |
| ovolscalem1 23281 | Lemma for ~ ovolsca . (Co... |
| ovolscalem2 23282 | Lemma for ~ ovolshft . (C... |
| ovolsca 23283 | The Lebesgue outer measure... |
| ovolicc1 23284 | The measure of a closed in... |
| ovolicc2lem1 23285 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem2 23286 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem3 23287 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem4 23288 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem5 23289 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2 23290 | The measure of a closed in... |
| ovolicc 23291 | The measure of a closed in... |
| ovolicopnf 23292 | The measure of a right-unb... |
| ovolre 23293 | The measure of the real nu... |
| ismbl 23294 | The predicate " ` A ` is L... |
| ismbl2 23295 | From ~ ovolun , it suffice... |
| volres 23296 | A self-referencing abbrevi... |
| volf 23297 | The domain and range of th... |
| mblvol 23298 | The volume of a measurable... |
| mblss 23299 | A measurable set is a subs... |
| mblsplit 23300 | The defining property of m... |
| volss 23301 | The Lebesgue measure is mo... |
| cmmbl 23302 | The complement of a measur... |
| nulmbl 23303 | A nullset is measurable. ... |
| nulmbl2 23304 | A set of outer measure zer... |
| unmbl 23305 | A union of measurable sets... |
| shftmbl 23306 | A shift of a measurable se... |
| 0mbl 23307 | The empty set is measurabl... |
| rembl 23308 | The set of all real number... |
| unidmvol 23309 | The union of the Lebesgue ... |
| inmbl 23310 | An intersection of measura... |
| difmbl 23311 | A difference of measurable... |
| finiunmbl 23312 | A finite union of measurab... |
| volun 23313 | The Lebesgue measure funct... |
| volinun 23314 | Addition of non-disjoint s... |
| volfiniun 23315 | The volume of a disjoint f... |
| iundisj 23316 | Rewrite a countable union ... |
| iundisj2 23317 | A disjoint union is disjoi... |
| voliunlem1 23318 | Lemma for ~ voliun . (Con... |
| voliunlem2 23319 | Lemma for ~ voliun . (Con... |
| voliunlem3 23320 | Lemma for ~ voliun . (Con... |
| iunmbl 23321 | The measurable sets are cl... |
| voliun 23322 | The Lebesgue measure funct... |
| volsuplem 23323 | Lemma for ~ volsup . (Con... |
| volsup 23324 | The volume of the limit of... |
| iunmbl2 23325 | The measurable sets are cl... |
| ioombl1lem1 23326 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem2 23327 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem3 23328 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem4 23329 | Lemma for ~ ioombl1 . (Co... |
| ioombl1 23330 | An open right-unbounded in... |
| icombl1 23331 | A closed unbounded-above i... |
| icombl 23332 | A closed-below, open-above... |
| ioombl 23333 | An open real interval is m... |
| iccmbl 23334 | A closed real interval is ... |
| iccvolcl 23335 | A closed real interval has... |
| ovolioo 23336 | The measure of an open int... |
| volioo 23337 | The measure of an open int... |
| ioovolcl 23338 | An open real interval has ... |
| ovolfs2 23339 | Alternative expression for... |
| ioorcl2 23340 | An open interval with fini... |
| ioorf 23341 | Define a function from ope... |
| ioorval 23342 | Define a function from ope... |
| ioorinv2 23343 | The function ` F ` is an "... |
| ioorinv 23344 | The function ` F ` is an "... |
| ioorcl 23345 | The function ` F ` does no... |
| uniiccdif 23346 | A union of closed interval... |
| uniioovol 23347 | A disjoint union of open i... |
| uniiccvol 23348 | An almost-disjoint union o... |
| uniioombllem1 23349 | Lemma for ~ uniioombl . (... |
| uniioombllem2a 23350 | Lemma for ~ uniioombl . (... |
| uniioombllem2 23351 | Lemma for ~ uniioombl . (... |
| uniioombllem3a 23352 | Lemma for ~ uniioombl . (... |
| uniioombllem3 23353 | Lemma for ~ uniioombl . (... |
| uniioombllem4 23354 | Lemma for ~ uniioombl . (... |
| uniioombllem5 23355 | Lemma for ~ uniioombl . (... |
| uniioombllem6 23356 | Lemma for ~ uniioombl . (... |
| uniioombl 23357 | A disjoint union of open i... |
| uniiccmbl 23358 | An almost-disjoint union o... |
| dyadf 23359 | The function ` F ` returns... |
| dyadval 23360 | Value of the dyadic ration... |
| dyadovol 23361 | Volume of a dyadic rationa... |
| dyadss 23362 | Two closed dyadic rational... |
| dyaddisjlem 23363 | Lemma for ~ dyaddisj . (C... |
| dyaddisj 23364 | Two closed dyadic rational... |
| dyadmaxlem 23365 | Lemma for ~ dyadmax . (Co... |
| dyadmax 23366 | Any nonempty set of dyadic... |
| dyadmbllem 23367 | Lemma for ~ dyadmbl . (Co... |
| dyadmbl 23368 | Any union of dyadic ration... |
| opnmbllem 23369 | Lemma for ~ opnmbl . (Con... |
| opnmbl 23370 | All open sets are measurab... |
| opnmblALT 23371 | All open sets are measurab... |
| subopnmbl 23372 | Sets which are open in a m... |
| volsup2 23373 | The volume of ` A ` is the... |
| volcn 23374 | The function formed by res... |
| volivth 23375 | The Intermediate Value The... |
| vitalilem1 23376 | Lemma for ~ vitali . (Con... |
| vitalilem1OLD 23377 | Obsolete proof of ~ vitali... |
| vitalilem2 23378 | Lemma for ~ vitali . (Con... |
| vitalilem3 23379 | Lemma for ~ vitali . (Con... |
| vitalilem4 23380 | Lemma for ~ vitali . (Con... |
| vitalilem5 23381 | Lemma for ~ vitali . (Con... |
| vitali 23382 | If the reals can be well-o... |
| ismbf1 23393 | The predicate " ` F ` is a... |
| mbff 23394 | A measurable function is a... |
| mbfdm 23395 | The domain of a measurable... |
| mbfconstlem 23396 | Lemma for ~ mbfconst . (C... |
| ismbf 23397 | The predicate " ` F ` is a... |
| ismbfcn 23398 | A complex function is meas... |
| mbfima 23399 | Definitional property of a... |
| mbfimaicc 23400 | The preimage of any closed... |
| mbfimasn 23401 | The preimage of a point un... |
| mbfconst 23402 | A constant function is mea... |
| mbfid 23403 | The identity function is m... |
| mbfmptcl 23404 | Lemma for the ` MblFn ` pr... |
| mbfdm2 23405 | The domain of a measurable... |
| ismbfcn2 23406 | A complex function is meas... |
| ismbfd 23407 | Deduction to prove measura... |
| ismbf2d 23408 | Deduction to prove measura... |
| mbfeqalem 23409 | Lemma for ~ mbfeqa . (Con... |
| mbfeqa 23410 | If two functions are equal... |
| mbfres 23411 | The restriction of a measu... |
| mbfres2 23412 | Measurability of a piecewi... |
| mbfss 23413 | Change the domain of a mea... |
| mbfmulc2lem 23414 | Multiplication by a consta... |
| mbfmulc2re 23415 | Multiplication by a consta... |
| mbfmax 23416 | The maximum of two functio... |
| mbfneg 23417 | The negative of a measurab... |
| mbfpos 23418 | The positive part of a mea... |
| mbfposr 23419 | Converse to ~ mbfpos . (C... |
| mbfposb 23420 | A function is measurable i... |
| ismbf3d 23421 | Simplified form of ~ ismbf... |
| mbfimaopnlem 23422 | Lemma for ~ mbfimaopn . (... |
| mbfimaopn 23423 | The preimage of any open s... |
| mbfimaopn2 23424 | The preimage of any set op... |
| cncombf 23425 | The composition of a conti... |
| cnmbf 23426 | A continuous function is m... |
| mbfaddlem 23427 | The sum of two measurable ... |
| mbfadd 23428 | The sum of two measurable ... |
| mbfsub 23429 | The difference of two meas... |
| mbfmulc2 23430 | A complex constant times a... |
| mbfsup 23431 | The supremum of a sequence... |
| mbfinf 23432 | The infimum of a sequence ... |
| mbflimsup 23433 | The limit supremum of a se... |
| mbflimlem 23434 | The pointwise limit of a s... |
| mbflim 23435 | The pointwise limit of a s... |
| 0pval 23438 | The zero function evaluate... |
| 0plef 23439 | Two ways to say that the f... |
| 0pledm 23440 | Adjust the domain of the l... |
| isi1f 23441 | The predicate " ` F ` is a... |
| i1fmbf 23442 | Simple functions are measu... |
| i1ff 23443 | A simple function is a fun... |
| i1frn 23444 | A simple function has fini... |
| i1fima 23445 | Any preimage of a simple f... |
| i1fima2 23446 | Any preimage of a simple f... |
| i1fima2sn 23447 | Preimage of a singleton. ... |
| i1fd 23448 | A simplified set of assump... |
| i1f0rn 23449 | Any simple function takes ... |
| itg1val 23450 | The value of the integral ... |
| itg1val2 23451 | The value of the integral ... |
| itg1cl 23452 | Closure of the integral on... |
| itg1ge0 23453 | Closure of the integral on... |
| i1f0 23454 | The zero function is simpl... |
| itg10 23455 | The zero function has zero... |
| i1f1lem 23456 | Lemma for ~ i1f1 and ~ itg... |
| i1f1 23457 | Base case simple functions... |
| itg11 23458 | The integral of an indicat... |
| itg1addlem1 23459 | Decompose a preimage, whic... |
| i1faddlem 23460 | Decompose the preimage of ... |
| i1fmullem 23461 | Decompose the preimage of ... |
| i1fadd 23462 | The sum of two simple func... |
| i1fmul 23463 | The pointwise product of t... |
| itg1addlem2 23464 | Lemma for ~ itg1add . The... |
| itg1addlem3 23465 | Lemma for ~ itg1add . (Co... |
| itg1addlem4 23466 | Lemma for itg1add . (Cont... |
| itg1addlem5 23467 | Lemma for itg1add . (Cont... |
| itg1add 23468 | The integral of a sum of s... |
| i1fmulclem 23469 | Decompose the preimage of ... |
| i1fmulc 23470 | A nonnegative constant tim... |
| itg1mulc 23471 | The integral of a constant... |
| i1fres 23472 | The "restriction" of a sim... |
| i1fpos 23473 | The positive part of a sim... |
| i1fposd 23474 | Deduction form of ~ i1fpos... |
| i1fsub 23475 | The difference of two simp... |
| itg1sub 23476 | The integral of a differen... |
| itg10a 23477 | The integral of a simple f... |
| itg1ge0a 23478 | The integral of an almost ... |
| itg1lea 23479 | Approximate version of ~ i... |
| itg1le 23480 | If one simple function dom... |
| itg1climres 23481 | Restricting the simple fun... |
| mbfi1fseqlem1 23482 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem2 23483 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem3 23484 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem4 23485 | Lemma for ~ mbfi1fseq . T... |
| mbfi1fseqlem5 23486 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseqlem6 23487 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseq 23488 | A characterization of meas... |
| mbfi1flimlem 23489 | Lemma for ~ mbfi1flim . (... |
| mbfi1flim 23490 | Any real measurable functi... |
| mbfmullem2 23491 | Lemma for ~ mbfmul . (Con... |
| mbfmullem 23492 | Lemma for ~ mbfmul . (Con... |
| mbfmul 23493 | The product of two measura... |
| itg2lcl 23494 | The set of lower sums is a... |
| itg2val 23495 | Value of the integral on n... |
| itg2l 23496 | Elementhood in the set ` L... |
| itg2lr 23497 | Sufficient condition for e... |
| xrge0f 23498 | A real function is a nonne... |
| itg2cl 23499 | The integral of a nonnegat... |
| itg2ub 23500 | The integral of a nonnegat... |
| itg2leub 23501 | Any upper bound on the int... |
| itg2ge0 23502 | The integral of a nonnegat... |
| itg2itg1 23503 | The integral of a nonnegat... |
| itg20 23504 | The integral of the zero f... |
| itg2lecl 23505 | If an ` S.2 ` integral is ... |
| itg2le 23506 | If one function dominates ... |
| itg2const 23507 | Integral of a constant fun... |
| itg2const2 23508 | When the base set of a con... |
| itg2seq 23509 | Definitional property of t... |
| itg2uba 23510 | Approximate version of ~ i... |
| itg2lea 23511 | Approximate version of ~ i... |
| itg2eqa 23512 | Approximate equality of in... |
| itg2mulclem 23513 | Lemma for ~ itg2mulc . (C... |
| itg2mulc 23514 | The integral of a nonnegat... |
| itg2splitlem 23515 | Lemma for ~ itg2split . (... |
| itg2split 23516 | The ` S.2 ` integral split... |
| itg2monolem1 23517 | Lemma for ~ itg2mono . We... |
| itg2monolem2 23518 | Lemma for ~ itg2mono . (C... |
| itg2monolem3 23519 | Lemma for ~ itg2mono . (C... |
| itg2mono 23520 | The Monotone Convergence T... |
| itg2i1fseqle 23521 | Subject to the conditions ... |
| itg2i1fseq 23522 | Subject to the conditions ... |
| itg2i1fseq2 23523 | In an extension to the res... |
| itg2i1fseq3 23524 | Special case of ~ itg2i1fs... |
| itg2addlem 23525 | Lemma for ~ itg2add . (Co... |
| itg2add 23526 | The ` S.2 ` integral is li... |
| itg2gt0 23527 | If the function ` F ` is s... |
| itg2cnlem1 23528 | Lemma for ~ itgcn . (Cont... |
| itg2cnlem2 23529 | Lemma for ~ itgcn . (Cont... |
| itg2cn 23530 | A sort of absolute continu... |
| ibllem 23531 | Conditioned equality theor... |
| isibl 23532 | The predicate " ` F ` is i... |
| isibl2 23533 | The predicate " ` F ` is i... |
| iblmbf 23534 | An integrable function is ... |
| iblitg 23535 | If a function is integrabl... |
| dfitg 23536 | Evaluate the class substit... |
| itgex 23537 | An integral is a set. (Co... |
| itgeq1f 23538 | Equality theorem for an in... |
| itgeq1 23539 | Equality theorem for an in... |
| nfitg1 23540 | Bound-variable hypothesis ... |
| nfitg 23541 | Bound-variable hypothesis ... |
| cbvitg 23542 | Change bound variable in a... |
| cbvitgv 23543 | Change bound variable in a... |
| itgeq2 23544 | Equality theorem for an in... |
| itgresr 23545 | The domain of an integral ... |
| itg0 23546 | The integral of anything o... |
| itgz 23547 | The integral of zero on an... |
| itgeq2dv 23548 | Equality theorem for an in... |
| itgmpt 23549 | Change bound variable in a... |
| itgcl 23550 | The integral of an integra... |
| itgvallem 23551 | Substitution lemma. (Cont... |
| itgvallem3 23552 | Lemma for ~ itgposval and ... |
| ibl0 23553 | The zero function is integ... |
| iblcnlem1 23554 | Lemma for ~ iblcnlem . (C... |
| iblcnlem 23555 | Expand out the forall in ~... |
| itgcnlem 23556 | Expand out the sum in ~ df... |
| iblrelem 23557 | Integrability of a real fu... |
| iblposlem 23558 | Lemma for ~ iblpos . (Con... |
| iblpos 23559 | Integrability of a nonnega... |
| iblre 23560 | Integrability of a real fu... |
| itgrevallem1 23561 | Lemma for ~ itgposval and ... |
| itgposval 23562 | The integral of a nonnegat... |
| itgreval 23563 | Decompose the integral of ... |
| itgrecl 23564 | Real closure of an integra... |
| iblcn 23565 | Integrability of a complex... |
| itgcnval 23566 | Decompose the integral of ... |
| itgre 23567 | Real part of an integral. ... |
| itgim 23568 | Imaginary part of an integ... |
| iblneg 23569 | The negative of an integra... |
| itgneg 23570 | Negation of an integral. ... |
| iblss 23571 | A subset of an integrable ... |
| iblss2 23572 | Change the domain of an in... |
| itgitg2 23573 | Transfer an integral using... |
| i1fibl 23574 | A simple function is integ... |
| itgitg1 23575 | Transfer an integral using... |
| itgle 23576 | Monotonicity of an integra... |
| itgge0 23577 | The integral of a positive... |
| itgss 23578 | Expand the set of an integ... |
| itgss2 23579 | Expand the set of an integ... |
| itgeqa 23580 | Approximate equality of in... |
| itgss3 23581 | Expand the set of an integ... |
| itgioo 23582 | Equality of integrals on o... |
| itgless 23583 | Expand the integral of a n... |
| iblconst 23584 | A constant function is int... |
| itgconst 23585 | Integral of a constant fun... |
| ibladdlem 23586 | Lemma for ~ ibladd . (Con... |
| ibladd 23587 | Add two integrals over the... |
| iblsub 23588 | Subtract two integrals ove... |
| itgaddlem1 23589 | Lemma for ~ itgadd . (Con... |
| itgaddlem2 23590 | Lemma for ~ itgadd . (Con... |
| itgadd 23591 | Add two integrals over the... |
| itgsub 23592 | Subtract two integrals ove... |
| itgfsum 23593 | Take a finite sum of integ... |
| iblabslem 23594 | Lemma for ~ iblabs . (Con... |
| iblabs 23595 | The absolute value of an i... |
| iblabsr 23596 | A measurable function is i... |
| iblmulc2 23597 | Multiply an integral by a ... |
| itgmulc2lem1 23598 | Lemma for ~ itgmulc2 : pos... |
| itgmulc2lem2 23599 | Lemma for ~ itgmulc2 : rea... |
| itgmulc2 23600 | Multiply an integral by a ... |
| itgabs 23601 | The triangle inequality fo... |
| itgsplit 23602 | The ` S. ` integral splits... |
| itgspliticc 23603 | The ` S. ` integral splits... |
| itgsplitioo 23604 | The ` S. ` integral splits... |
| bddmulibl 23605 | A bounded function times a... |
| bddibl 23606 | A bounded function is inte... |
| cniccibl 23607 | A continuous function on a... |
| itggt0 23608 | The integral of a strictly... |
| itgcn 23609 | Transfer ~ itg2cn to the f... |
| ditgeq1 23612 | Equality theorem for the d... |
| ditgeq2 23613 | Equality theorem for the d... |
| ditgeq3 23614 | Equality theorem for the d... |
| ditgeq3dv 23615 | Equality theorem for the d... |
| ditgex 23616 | A directed integral is a s... |
| ditg0 23617 | Value of the directed inte... |
| cbvditg 23618 | Change bound variable in a... |
| cbvditgv 23619 | Change bound variable in a... |
| ditgpos 23620 | Value of the directed inte... |
| ditgneg 23621 | Value of the directed inte... |
| ditgcl 23622 | Closure of a directed inte... |
| ditgswap 23623 | Reverse a directed integra... |
| ditgsplitlem 23624 | Lemma for ~ ditgsplit . (... |
| ditgsplit 23625 | This theorem is the raison... |
| reldv 23634 | The derivative function is... |
| limcvallem 23635 | Lemma for ~ ellimc . (Con... |
| limcfval 23636 | Value and set bounds on th... |
| ellimc 23637 | Value of the limit predica... |
| limcrcl 23638 | Reverse closure for the li... |
| limccl 23639 | Closure of the limit opera... |
| limcdif 23640 | It suffices to consider fu... |
| ellimc2 23641 | Write the definition of a ... |
| limcnlp 23642 | If ` B ` is not a limit po... |
| ellimc3 23643 | Write the epsilon-delta de... |
| limcflflem 23644 | Lemma for ~ limcflf . (Co... |
| limcflf 23645 | The limit operator can be ... |
| limcmo 23646 | If ` B ` is a limit point ... |
| limcmpt 23647 | Express the limit operator... |
| limcmpt2 23648 | Express the limit operator... |
| limcresi 23649 | Any limit of ` F ` is also... |
| limcres 23650 | If ` B ` is an interior po... |
| cnplimc 23651 | A function is continuous a... |
| cnlimc 23652 | ` F ` is a continuous func... |
| cnlimci 23653 | If ` F ` is a continuous f... |
| cnmptlimc 23654 | If ` F ` is a continuous f... |
| limccnp 23655 | If the limit of ` F ` at `... |
| limccnp2 23656 | The image of a convergent ... |
| limcco 23657 | Composition of two limits.... |
| limciun 23658 | A point is a limit of ` F ... |
| limcun 23659 | A point is a limit of ` F ... |
| dvlem 23660 | Closure for a difference q... |
| dvfval 23661 | Value and set bounds on th... |
| eldv 23662 | The differentiable predica... |
| dvcl 23663 | The derivative function ta... |
| dvbssntr 23664 | The set of differentiable ... |
| dvbss 23665 | The set of differentiable ... |
| dvbsss 23666 | The set of differentiable ... |
| perfdvf 23667 | The derivative is a functi... |
| recnprss 23668 | Both ` RR ` and ` CC ` are... |
| recnperf 23669 | Both ` RR ` and ` CC ` are... |
| dvfg 23670 | Explicitly write out the f... |
| dvf 23671 | The derivative is a functi... |
| dvfcn 23672 | The derivative is a functi... |
| dvreslem 23673 | Lemma for ~ dvres . (Cont... |
| dvres2lem 23674 | Lemma for ~ dvres2 . (Con... |
| dvres 23675 | Restriction of a derivativ... |
| dvres2 23676 | Restriction of the base se... |
| dvres3 23677 | Restriction of a complex d... |
| dvres3a 23678 | Restriction of a complex d... |
| dvidlem 23679 | Lemma for ~ dvid and ~ dvc... |
| dvconst 23680 | Derivative of a constant f... |
| dvid 23681 | Derivative of the identity... |
| dvcnp 23682 | The difference quotient is... |
| dvcnp2 23683 | A function is continuous a... |
| dvcn 23684 | A differentiable function ... |
| dvnfval 23685 | Value of the iterated deri... |
| dvnff 23686 | The iterated derivative is... |
| dvn0 23687 | Zero times iterated deriva... |
| dvnp1 23688 | Successor iterated derivat... |
| dvn1 23689 | One times iterated derivat... |
| dvnf 23690 | The N-times derivative is ... |
| dvnbss 23691 | The set of N-times differe... |
| dvnadd 23692 | The ` N ` -th derivative o... |
| dvn2bss 23693 | An N-times differentiable ... |
| dvnres 23694 | Multiple derivative versio... |
| cpnfval 23695 | Condition for n-times cont... |
| fncpn 23696 | The ` C^n ` object is a fu... |
| elcpn 23697 | Condition for n-times cont... |
| cpnord 23698 | ` C^n ` conditions are ord... |
| cpncn 23699 | A ` C^n ` function is cont... |
| cpnres 23700 | The restriction of a ` C^n... |
| dvaddbr 23701 | The sum rule for derivativ... |
| dvmulbr 23702 | The product rule for deriv... |
| dvadd 23703 | The sum rule for derivativ... |
| dvmul 23704 | The product rule for deriv... |
| dvaddf 23705 | The sum rule for everywher... |
| dvmulf 23706 | The product rule for every... |
| dvcmul 23707 | The product rule when one ... |
| dvcmulf 23708 | The product rule when one ... |
| dvcobr 23709 | The chain rule for derivat... |
| dvco 23710 | The chain rule for derivat... |
| dvcof 23711 | The chain rule for everywh... |
| dvcjbr 23712 | The derivative of the conj... |
| dvcj 23713 | The derivative of the conj... |
| dvfre 23714 | The derivative of a real f... |
| dvnfre 23715 | The ` N ` -th derivative o... |
| dvexp 23716 | Derivative of a power func... |
| dvexp2 23717 | Derivative of an exponenti... |
| dvrec 23718 | Derivative of the reciproc... |
| dvmptres3 23719 | Function-builder for deriv... |
| dvmptid 23720 | Function-builder for deriv... |
| dvmptc 23721 | Function-builder for deriv... |
| dvmptcl 23722 | Closure lemma for ~ dvmptc... |
| dvmptadd 23723 | Function-builder for deriv... |
| dvmptmul 23724 | Function-builder for deriv... |
| dvmptres2 23725 | Function-builder for deriv... |
| dvmptres 23726 | Function-builder for deriv... |
| dvmptcmul 23727 | Function-builder for deriv... |
| dvmptdivc 23728 | Function-builder for deriv... |
| dvmptneg 23729 | Function-builder for deriv... |
| dvmptsub 23730 | Function-builder for deriv... |
| dvmptcj 23731 | Function-builder for deriv... |
| dvmptre 23732 | Function-builder for deriv... |
| dvmptim 23733 | Function-builder for deriv... |
| dvmptntr 23734 | Function-builder for deriv... |
| dvmptco 23735 | Function-builder for deriv... |
| dvrecg 23736 | Derivative of the reciproc... |
| dvmptdiv 23737 | Function-builder for deriv... |
| dvmptfsum 23738 | Function-builder for deriv... |
| dvcnvlem 23739 | Lemma for ~ dvcnvre . (Co... |
| dvcnv 23740 | A weak version of ~ dvcnvr... |
| dvexp3 23741 | Derivative of an exponenti... |
| dveflem 23742 | Derivative of the exponent... |
| dvef 23743 | Derivative of the exponent... |
| dvsincos 23744 | Derivative of the sine and... |
| dvsin 23745 | Derivative of the sine fun... |
| dvcos 23746 | Derivative of the cosine f... |
| dvferm1lem 23747 | Lemma for ~ dvferm . (Con... |
| dvferm1 23748 | One-sided version of ~ dvf... |
| dvferm2lem 23749 | Lemma for ~ dvferm . (Con... |
| dvferm2 23750 | One-sided version of ~ dvf... |
| dvferm 23751 | Fermat's theorem on statio... |
| rollelem 23752 | Lemma for ~ rolle . (Cont... |
| rolle 23753 | Rolle's theorem. If ` F `... |
| cmvth 23754 | Cauchy's Mean Value Theore... |
| mvth 23755 | The Mean Value Theorem. I... |
| dvlip 23756 | A function with derivative... |
| dvlipcn 23757 | A complex function with de... |
| dvlip2 23758 | Combine the results of ~ d... |
| c1liplem1 23759 | Lemma for ~ c1lip1 . (Con... |
| c1lip1 23760 | C1 functions are Lipschitz... |
| c1lip2 23761 | C1 functions are Lipschitz... |
| c1lip3 23762 | C1 functions are Lipschitz... |
| dveq0 23763 | If a continuous function h... |
| dv11cn 23764 | Two functions defined on a... |
| dvgt0lem1 23765 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0lem2 23766 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0 23767 | A function on a closed int... |
| dvlt0 23768 | A function on a closed int... |
| dvge0 23769 | A function on a closed int... |
| dvle 23770 | If ` A ( x ) , C ( x ) ` a... |
| dvivthlem1 23771 | Lemma for ~ dvivth . (Con... |
| dvivthlem2 23772 | Lemma for ~ dvivth . (Con... |
| dvivth 23773 | Darboux' theorem, or the i... |
| dvne0 23774 | A function on a closed int... |
| dvne0f1 23775 | A function on a closed int... |
| lhop1lem 23776 | Lemma for ~ lhop1 . (Cont... |
| lhop1 23777 | L'Hôpital's Rule for... |
| lhop2 23778 | L'Hôpital's Rule for... |
| lhop 23779 | L'Hôpital's Rule. I... |
| dvcnvrelem1 23780 | Lemma for ~ dvcnvre . (Co... |
| dvcnvrelem2 23781 | Lemma for ~ dvcnvre . (Co... |
| dvcnvre 23782 | The derivative rule for in... |
| dvcvx 23783 | A real function with stric... |
| dvfsumle 23784 | Compare a finite sum to an... |
| dvfsumge 23785 | Compare a finite sum to an... |
| dvfsumabs 23786 | Compare a finite sum to an... |
| dvmptrecl 23787 | Real closure of a derivati... |
| dvfsumrlimf 23788 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem1 23789 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2 23790 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem3 23791 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem4 23792 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlimge0 23793 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlim 23794 | Compare a finite sum to an... |
| dvfsumrlim2 23795 | Compare a finite sum to an... |
| dvfsumrlim3 23796 | Conjoin the statements of ... |
| dvfsum2 23797 | The reverse of ~ dvfsumrli... |
| ftc1lem1 23798 | Lemma for ~ ftc1a and ~ ft... |
| ftc1lem2 23799 | Lemma for ~ ftc1 . (Contr... |
| ftc1a 23800 | The Fundamental Theorem of... |
| ftc1lem3 23801 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem4 23802 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem5 23803 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem6 23804 | Lemma for ~ ftc1 . (Contr... |
| ftc1 23805 | The Fundamental Theorem of... |
| ftc1cn 23806 | Strengthen the assumptions... |
| ftc2 23807 | The Fundamental Theorem of... |
| ftc2ditglem 23808 | Lemma for ~ ftc2ditg . (C... |
| ftc2ditg 23809 | Directed integral analogue... |
| itgparts 23810 | Integration by parts. If ... |
| itgsubstlem 23811 | Lemma for ~ itgsubst . (C... |
| itgsubst 23812 | Integration by ` u ` -subs... |
| reldmmdeg 23817 | Multivariate degree is a b... |
| tdeglem1 23818 | Functionality of the total... |
| tdeglem3 23819 | Additivity of the total de... |
| tdeglem4 23820 | There is only one multi-in... |
| tdeglem2 23821 | Simplification of total de... |
| mdegfval 23822 | Value of the multivariate ... |
| mdegval 23823 | Value of the multivariate ... |
| mdegleb 23824 | Property of being of limit... |
| mdeglt 23825 | If there is an upper limit... |
| mdegldg 23826 | A nonzero polynomial has s... |
| mdegxrcl 23827 | Closure of polynomial degr... |
| mdegxrf 23828 | Functionality of polynomia... |
| mdegcl 23829 | Sharp closure for multivar... |
| mdeg0 23830 | Degree of the zero polynom... |
| mdegnn0cl 23831 | Degree of a nonzero polyno... |
| degltlem1 23832 | Theorem on arithmetic of e... |
| degltp1le 23833 | Theorem on arithmetic of e... |
| mdegaddle 23834 | The degree of a sum is at ... |
| mdegvscale 23835 | The degree of a scalar mul... |
| mdegvsca 23836 | The degree of a scalar mul... |
| mdegle0 23837 | A polynomial has nonpositi... |
| mdegmullem 23838 | Lemma for ~ mdegmulle2 . ... |
| mdegmulle2 23839 | The multivariate degree of... |
| deg1fval 23840 | Relate univariate polynomi... |
| deg1xrf 23841 | Functionality of univariat... |
| deg1xrcl 23842 | Closure of univariate poly... |
| deg1cl 23843 | Sharp closure of univariat... |
| mdegpropd 23844 | Property deduction for pol... |
| deg1fvi 23845 | Univariate polynomial degr... |
| deg1propd 23846 | Property deduction for pol... |
| deg1z 23847 | Degree of the zero univari... |
| deg1nn0cl 23848 | Degree of a nonzero univar... |
| deg1n0ima 23849 | Degree image of a set of p... |
| deg1nn0clb 23850 | A polynomial is nonzero if... |
| deg1lt0 23851 | A polynomial is zero iff i... |
| deg1ldg 23852 | A nonzero univariate polyn... |
| deg1ldgn 23853 | An index at which a polyno... |
| deg1ldgdomn 23854 | A nonzero univariate polyn... |
| deg1leb 23855 | Property of being of limit... |
| deg1val 23856 | Value of the univariate de... |
| deg1lt 23857 | If the degree of a univari... |
| deg1ge 23858 | Conversely, a nonzero coef... |
| coe1mul3 23859 | The coefficient vector of ... |
| coe1mul4 23860 | Value of the "leading" coe... |
| deg1addle 23861 | The degree of a sum is at ... |
| deg1addle2 23862 | If both factors have degre... |
| deg1add 23863 | Exact degree of a sum of t... |
| deg1vscale 23864 | The degree of a scalar tim... |
| deg1vsca 23865 | The degree of a scalar tim... |
| deg1invg 23866 | The degree of the negated ... |
| deg1suble 23867 | The degree of a difference... |
| deg1sub 23868 | Exact degree of a differen... |
| deg1mulle2 23869 | Produce a bound on the pro... |
| deg1sublt 23870 | Subtraction of two polynom... |
| deg1le0 23871 | A polynomial has nonpositi... |
| deg1sclle 23872 | A scalar polynomial has no... |
| deg1scl 23873 | A nonzero scalar polynomia... |
| deg1mul2 23874 | Degree of multiplication o... |
| deg1mul3 23875 | Degree of multiplication o... |
| deg1mul3le 23876 | Degree of multiplication o... |
| deg1tmle 23877 | Limiting degree of a polyn... |
| deg1tm 23878 | Exact degree of a polynomi... |
| deg1pwle 23879 | Limiting degree of a varia... |
| deg1pw 23880 | Exact degree of a variable... |
| ply1nz 23881 | Univariate polynomials ove... |
| ply1nzb 23882 | Univariate polynomials are... |
| ply1domn 23883 | Corollary of ~ deg1mul2 : ... |
| ply1idom 23884 | The ring of univariate pol... |
| ply1divmo 23895 | Uniqueness of a quotient i... |
| ply1divex 23896 | Lemma for ~ ply1divalg : e... |
| ply1divalg 23897 | The division algorithm for... |
| ply1divalg2 23898 | Reverse the order of multi... |
| uc1pval 23899 | Value of the set of unitic... |
| isuc1p 23900 | Being a unitic polynomial.... |
| mon1pval 23901 | Value of the set of monic ... |
| ismon1p 23902 | Being a monic polynomial. ... |
| uc1pcl 23903 | Unitic polynomials are pol... |
| mon1pcl 23904 | Monic polynomials are poly... |
| uc1pn0 23905 | Unitic polynomials are not... |
| mon1pn0 23906 | Monic polynomials are not ... |
| uc1pdeg 23907 | Unitic polynomials have no... |
| uc1pldg 23908 | Unitic polynomials have un... |
| mon1pldg 23909 | Unitic polynomials have on... |
| mon1puc1p 23910 | Monic polynomials are unit... |
| uc1pmon1p 23911 | Make a unitic polynomial m... |
| deg1submon1p 23912 | The difference of two moni... |
| q1pval 23913 | Value of the univariate po... |
| q1peqb 23914 | Characterizing property of... |
| q1pcl 23915 | Closure of the quotient by... |
| r1pval 23916 | Value of the polynomial re... |
| r1pcl 23917 | Closure of remainder follo... |
| r1pdeglt 23918 | The remainder has a degree... |
| r1pid 23919 | Express the original polyn... |
| dvdsq1p 23920 | Divisibility in a polynomi... |
| dvdsr1p 23921 | Divisibility in a polynomi... |
| ply1remlem 23922 | A term of the form ` x - N... |
| ply1rem 23923 | The polynomial remainder t... |
| facth1 23924 | The factor theorem and its... |
| fta1glem1 23925 | Lemma for ~ fta1g . (Cont... |
| fta1glem2 23926 | Lemma for ~ fta1g . (Cont... |
| fta1g 23927 | The one-sided fundamental ... |
| fta1blem 23928 | Lemma for ~ fta1b . (Cont... |
| fta1b 23929 | The assumption that ` R ` ... |
| drnguc1p 23930 | Over a division ring, all ... |
| ig1peu 23931 | There is a unique monic po... |
| ig1pval 23932 | Substitutions for the poly... |
| ig1pval2 23933 | Generator of the zero idea... |
| ig1pval3 23934 | Characterizing properties ... |
| ig1pcl 23935 | The monic generator of an ... |
| ig1pdvds 23936 | The monic generator of an ... |
| ig1prsp 23937 | Any ideal of polynomials o... |
| ply1lpir 23938 | The ring of polynomials ov... |
| ply1pid 23939 | The polynomials over a fie... |
| plyco0 23948 | Two ways to say that a fun... |
| plyval 23949 | Value of the polynomial se... |
| plybss 23950 | Reverse closure of the par... |
| elply 23951 | Definition of a polynomial... |
| elply2 23952 | The coefficient function c... |
| plyun0 23953 | The set of polynomials is ... |
| plyf 23954 | The polynomial is a functi... |
| plyss 23955 | The polynomial set functio... |
| plyssc 23956 | Every polynomial ring is c... |
| elplyr 23957 | Sufficient condition for e... |
| elplyd 23958 | Sufficient condition for e... |
| ply1termlem 23959 | Lemma for ~ ply1term . (C... |
| ply1term 23960 | A one-term polynomial. (C... |
| plypow 23961 | A power is a polynomial. ... |
| plyconst 23962 | A constant function is a p... |
| ne0p 23963 | A test to show that a poly... |
| ply0 23964 | The zero function is a pol... |
| plyid 23965 | The identity function is a... |
| plyeq0lem 23966 | Lemma for ~ plyeq0 . If `... |
| plyeq0 23967 | If a polynomial is zero at... |
| plypf1 23968 | Write the set of complex p... |
| plyaddlem1 23969 | Derive the coefficient fun... |
| plymullem1 23970 | Derive the coefficient fun... |
| plyaddlem 23971 | Lemma for ~ plyadd . (Con... |
| plymullem 23972 | Lemma for ~ plymul . (Con... |
| plyadd 23973 | The sum of two polynomials... |
| plymul 23974 | The product of two polynom... |
| plysub 23975 | The difference of two poly... |
| plyaddcl 23976 | The sum of two polynomials... |
| plymulcl 23977 | The product of two polynom... |
| plysubcl 23978 | The difference of two poly... |
| coeval 23979 | Value of the coefficient f... |
| coeeulem 23980 | Lemma for ~ coeeu . (Cont... |
| coeeu 23981 | Uniqueness of the coeffici... |
| coelem 23982 | Lemma for properties of th... |
| coeeq 23983 | If ` A ` satisfies the pro... |
| dgrval 23984 | Value of the degree functi... |
| dgrlem 23985 | Lemma for ~ dgrcl and simi... |
| coef 23986 | The domain and range of th... |
| coef2 23987 | The domain and range of th... |
| coef3 23988 | The domain and range of th... |
| dgrcl 23989 | The degree of any polynomi... |
| dgrub 23990 | If the ` M ` -th coefficie... |
| dgrub2 23991 | All the coefficients above... |
| dgrlb 23992 | If all the coefficients ab... |
| coeidlem 23993 | Lemma for ~ coeid . (Cont... |
| coeid 23994 | Reconstruct a polynomial a... |
| coeid2 23995 | Reconstruct a polynomial a... |
| coeid3 23996 | Reconstruct a polynomial a... |
| plyco 23997 | The composition of two pol... |
| coeeq2 23998 | Compute the coefficient fu... |
| dgrle 23999 | Given an explicit expressi... |
| dgreq 24000 | If the highest term in a p... |
| 0dgr 24001 | A constant function has de... |
| 0dgrb 24002 | A function has degree zero... |
| dgrnznn 24003 | A nonzero polynomial with ... |
| coefv0 24004 | The result of evaluating a... |
| coeaddlem 24005 | Lemma for ~ coeadd and ~ d... |
| coemullem 24006 | Lemma for ~ coemul and ~ d... |
| coeadd 24007 | The coefficient function o... |
| coemul 24008 | A coefficient of a product... |
| coe11 24009 | The coefficient function i... |
| coemulhi 24010 | The leading coefficient of... |
| coemulc 24011 | The coefficient function i... |
| coe0 24012 | The coefficients of the ze... |
| coesub 24013 | The coefficient function o... |
| coe1termlem 24014 | The coefficient function o... |
| coe1term 24015 | The coefficient function o... |
| dgr1term 24016 | The degree of a monomial. ... |
| plycn 24017 | A polynomial is a continuo... |
| dgr0 24018 | The degree of the zero pol... |
| coeidp 24019 | The coefficients of the id... |
| dgrid 24020 | The degree of the identity... |
| dgreq0 24021 | The leading coefficient of... |
| dgrlt 24022 | Two ways to say that the d... |
| dgradd 24023 | The degree of a sum of pol... |
| dgradd2 24024 | The degree of a sum of pol... |
| dgrmul2 24025 | The degree of a product of... |
| dgrmul 24026 | The degree of a product of... |
| dgrmulc 24027 | Scalar multiplication by a... |
| dgrsub 24028 | The degree of a difference... |
| dgrcolem1 24029 | The degree of a compositio... |
| dgrcolem2 24030 | Lemma for ~ dgrco . (Cont... |
| dgrco 24031 | The degree of a compositio... |
| plycjlem 24032 | Lemma for ~ plycj and ~ co... |
| plycj 24033 | The double conjugation of ... |
| coecj 24034 | Double conjugation of a po... |
| plyrecj 24035 | A polynomial with real coe... |
| plymul0or 24036 | Polynomial multiplication ... |
| ofmulrt 24037 | The set of roots of a prod... |
| plyreres 24038 | Real-coefficient polynomia... |
| dvply1 24039 | Derivative of a polynomial... |
| dvply2g 24040 | The derivative of a polyno... |
| dvply2 24041 | The derivative of a polyno... |
| dvnply2 24042 | Polynomials have polynomia... |
| dvnply 24043 | Polynomials have polynomia... |
| plycpn 24044 | Polynomials are smooth. (... |
| quotval 24047 | Value of the quotient func... |
| plydivlem1 24048 | Lemma for ~ plydivalg . (... |
| plydivlem2 24049 | Lemma for ~ plydivalg . (... |
| plydivlem3 24050 | Lemma for ~ plydivex . Ba... |
| plydivlem4 24051 | Lemma for ~ plydivex . In... |
| plydivex 24052 | Lemma for ~ plydivalg . (... |
| plydiveu 24053 | Lemma for ~ plydivalg . (... |
| plydivalg 24054 | The division algorithm on ... |
| quotlem 24055 | Lemma for properties of th... |
| quotcl 24056 | The quotient of two polyno... |
| quotcl2 24057 | Closure of the quotient fu... |
| quotdgr 24058 | Remainder property of the ... |
| plyremlem 24059 | Closure of a linear factor... |
| plyrem 24060 | The polynomial remainder t... |
| facth 24061 | The factor theorem. If a ... |
| fta1lem 24062 | Lemma for ~ fta1 . (Contr... |
| fta1 24063 | The easy direction of the ... |
| quotcan 24064 | Exact division with a mult... |
| vieta1lem1 24065 | Lemma for ~ vieta1 . (Con... |
| vieta1lem2 24066 | Lemma for ~ vieta1 : induc... |
| vieta1 24067 | The first-order Vieta's fo... |
| plyexmo 24068 | An infinite set of values ... |
| elaa 24071 | Elementhood in the set of ... |
| aacn 24072 | An algebraic number is a c... |
| aasscn 24073 | The algebraic numbers are ... |
| elqaalem1 24074 | Lemma for ~ elqaa . The f... |
| elqaalem2 24075 | Lemma for ~ elqaa . (Cont... |
| elqaalem3 24076 | Lemma for ~ elqaa . (Cont... |
| elqaa 24077 | The set of numbers generat... |
| qaa 24078 | Every rational number is a... |
| qssaa 24079 | The rational numbers are c... |
| iaa 24080 | The imaginary unit is alge... |
| aareccl 24081 | The reciprocal of an algeb... |
| aacjcl 24082 | The conjugate of an algebr... |
| aannenlem1 24083 | Lemma for ~ aannen . (Con... |
| aannenlem2 24084 | Lemma for ~ aannen . (Con... |
| aannenlem3 24085 | The algebraic numbers are ... |
| aannen 24086 | The algebraic numbers are ... |
| aalioulem1 24087 | Lemma for ~ aaliou . An i... |
| aalioulem2 24088 | Lemma for ~ aaliou . (Con... |
| aalioulem3 24089 | Lemma for ~ aaliou . (Con... |
| aalioulem4 24090 | Lemma for ~ aaliou . (Con... |
| aalioulem5 24091 | Lemma for ~ aaliou . (Con... |
| aalioulem6 24092 | Lemma for ~ aaliou . (Con... |
| aaliou 24093 | Liouville's theorem on dio... |
| geolim3 24094 | Geometric series convergen... |
| aaliou2 24095 | Liouville's approximation ... |
| aaliou2b 24096 | Liouville's approximation ... |
| aaliou3lem1 24097 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem2 24098 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem3 24099 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem8 24100 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem4 24101 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem5 24102 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem6 24103 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem7 24104 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem9 24105 | Example of a "Liouville nu... |
| aaliou3 24106 | Example of a "Liouville nu... |
| taylfvallem1 24111 | Lemma for ~ taylfval . (C... |
| taylfvallem 24112 | Lemma for ~ taylfval . (C... |
| taylfval 24113 | Define the Taylor polynomi... |
| eltayl 24114 | Value of the Taylor series... |
| taylf 24115 | The Taylor series defines ... |
| tayl0 24116 | The Taylor series is alway... |
| taylplem1 24117 | Lemma for ~ taylpfval and ... |
| taylplem2 24118 | Lemma for ~ taylpfval and ... |
| taylpfval 24119 | Define the Taylor polynomi... |
| taylpf 24120 | The Taylor polynomial is a... |
| taylpval 24121 | Value of the Taylor polyno... |
| taylply2 24122 | The Taylor polynomial is a... |
| taylply 24123 | The Taylor polynomial is a... |
| dvtaylp 24124 | The derivative of the Tayl... |
| dvntaylp 24125 | The ` M ` -th derivative o... |
| dvntaylp0 24126 | The first ` N ` derivative... |
| taylthlem1 24127 | Lemma for ~ taylth . This... |
| taylthlem2 24128 | Lemma for ~ taylth . (Con... |
| taylth 24129 | Taylor's theorem. The Tay... |
| ulmrel 24132 | The uniform limit relation... |
| ulmscl 24133 | Closure of the base set in... |
| ulmval 24134 | Express the predicate: Th... |
| ulmcl 24135 | Closure of a uniform limit... |
| ulmf 24136 | Closure of a uniform limit... |
| ulmpm 24137 | Closure of a uniform limit... |
| ulmf2 24138 | Closure of a uniform limit... |
| ulm2 24139 | Simplify ~ ulmval when ` F... |
| ulmi 24140 | The uniform limit property... |
| ulmclm 24141 | A uniform limit of functio... |
| ulmres 24142 | A sequence of functions co... |
| ulmshftlem 24143 | Lemma for ~ ulmshft . (Co... |
| ulmshft 24144 | A sequence of functions co... |
| ulm0 24145 | Every function converges u... |
| ulmuni 24146 | An sequence of functions u... |
| ulmdm 24147 | Two ways to express that a... |
| ulmcaulem 24148 | Lemma for ~ ulmcau and ~ u... |
| ulmcau 24149 | A sequence of functions co... |
| ulmcau2 24150 | A sequence of functions co... |
| ulmss 24151 | A uniform limit of functio... |
| ulmbdd 24152 | A uniform limit of bounded... |
| ulmcn 24153 | A uniform limit of continu... |
| ulmdvlem1 24154 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem2 24155 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem3 24156 | Lemma for ~ ulmdv . (Cont... |
| ulmdv 24157 | If ` F ` is a sequence of ... |
| mtest 24158 | The Weierstrass M-test. I... |
| mtestbdd 24159 | Given the hypotheses of th... |
| mbfulm 24160 | A uniform limit of measura... |
| iblulm 24161 | A uniform limit of integra... |
| itgulm 24162 | A uniform limit of integra... |
| itgulm2 24163 | A uniform limit of integra... |
| pserval 24164 | Value of the function ` G ... |
| pserval2 24165 | Value of the function ` G ... |
| psergf 24166 | The sequence of terms in t... |
| radcnvlem1 24167 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem2 24168 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem3 24169 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnv0 24170 | Zero is always a convergen... |
| radcnvcl 24171 | The radius of convergence ... |
| radcnvlt1 24172 | If ` X ` is within the ope... |
| radcnvlt2 24173 | If ` X ` is within the ope... |
| radcnvle 24174 | If ` X ` is a convergent p... |
| dvradcnv 24175 | The radius of convergence ... |
| pserulm 24176 | If ` S ` is a region conta... |
| psercn2 24177 | Since by ~ pserulm the ser... |
| psercnlem2 24178 | Lemma for ~ psercn . (Con... |
| psercnlem1 24179 | Lemma for ~ psercn . (Con... |
| psercn 24180 | An infinite series converg... |
| pserdvlem1 24181 | Lemma for ~ pserdv . (Con... |
| pserdvlem2 24182 | Lemma for ~ pserdv . (Con... |
| pserdv 24183 | The derivative of a power ... |
| pserdv2 24184 | The derivative of a power ... |
| abelthlem1 24185 | Lemma for ~ abelth . (Con... |
| abelthlem2 24186 | Lemma for ~ abelth . The ... |
| abelthlem3 24187 | Lemma for ~ abelth . (Con... |
| abelthlem4 24188 | Lemma for ~ abelth . (Con... |
| abelthlem5 24189 | Lemma for ~ abelth . (Con... |
| abelthlem6 24190 | Lemma for ~ abelth . (Con... |
| abelthlem7a 24191 | Lemma for ~ abelth . (Con... |
| abelthlem7 24192 | Lemma for ~ abelth . (Con... |
| abelthlem8 24193 | Lemma for ~ abelth . (Con... |
| abelthlem9 24194 | Lemma for ~ abelth . By a... |
| abelth 24195 | Abel's theorem. If the po... |
| abelth2 24196 | Abel's theorem, restricted... |
| efcn 24197 | The exponential function i... |
| sincn 24198 | Sine is continuous. (Cont... |
| coscn 24199 | Cosine is continuous. (Co... |
| reeff1olem 24200 | Lemma for ~ reeff1o . (Co... |
| reeff1o 24201 | The real exponential funct... |
| reefiso 24202 | The exponential function o... |
| efcvx 24203 | The exponential function o... |
| reefgim 24204 | The exponential function i... |
| pilem1 24205 | Lemma for ~ pire , ~ pigt2... |
| pilem2 24206 | Lemma for ~ pire , ~ pigt2... |
| pilem3 24207 | Lemma for ~ pire , ~ pigt2... |
| pigt2lt4 24208 | ` _pi ` is between 2 and 4... |
| sinpi 24209 | The sine of ` _pi ` is 0. ... |
| pire 24210 | ` _pi ` is a real number. ... |
| picn 24211 | ` _pi ` is a complex numbe... |
| pipos 24212 | ` _pi ` is positive. (Con... |
| pirp 24213 | ` _pi ` is a positive real... |
| negpicn 24214 | ` -u _pi ` is a real numbe... |
| sinhalfpilem 24215 | Lemma for ~ sinhalfpi and ... |
| halfpire 24216 | ` _pi / 2 ` is real. (Con... |
| neghalfpire 24217 | ` -u _pi / 2 ` is real. (... |
| neghalfpirx 24218 | ` -u _pi / 2 ` is an exten... |
| pidiv2halves 24219 | Adding ` _pi / 2 ` to itse... |
| sinhalfpi 24220 | The sine of ` _pi / 2 ` is... |
| coshalfpi 24221 | The cosine of ` _pi / 2 ` ... |
| cosneghalfpi 24222 | The cosine of ` -u _pi / 2... |
| efhalfpi 24223 | The exponential of ` _i _p... |
| cospi 24224 | The cosine of ` _pi ` is `... |
| efipi 24225 | The exponential of ` _i x.... |
| eulerid 24226 | Euler's identity. (Contri... |
| sin2pi 24227 | The sine of ` 2 _pi ` is 0... |
| cos2pi 24228 | The cosine of ` 2 _pi ` is... |
| ef2pi 24229 | The exponential of ` 2 _pi... |
| ef2kpi 24230 | The exponential of ` 2 K _... |
| efper 24231 | The exponential function i... |
| sinperlem 24232 | Lemma for ~ sinper and ~ c... |
| sinper 24233 | The sine function is perio... |
| cosper 24234 | The cosine function is per... |
| sin2kpi 24235 | If ` K ` is an integer, th... |
| cos2kpi 24236 | If ` K ` is an integer, th... |
| sin2pim 24237 | Sine of a number subtracte... |
| cos2pim 24238 | Cosine of a number subtrac... |
| sinmpi 24239 | Sine of a number less ` _p... |
| cosmpi 24240 | Cosine of a number less ` ... |
| sinppi 24241 | Sine of a number plus ` _p... |
| cosppi 24242 | Cosine of a complex number... |
| efimpi 24243 | The exponential function o... |
| sinhalfpip 24244 | The sine of ` _pi / 2 ` pl... |
| sinhalfpim 24245 | The sine of ` _pi / 2 ` mi... |
| coshalfpip 24246 | The cosine of ` _pi / 2 ` ... |
| coshalfpim 24247 | The cosine of ` _pi / 2 ` ... |
| ptolemy 24248 | Ptolemy's Theorem. This t... |
| sincosq1lem 24249 | Lemma for ~ sincosq1sgn . ... |
| sincosq1sgn 24250 | The signs of the sine and ... |
| sincosq2sgn 24251 | The signs of the sine and ... |
| sincosq3sgn 24252 | The signs of the sine and ... |
| sincosq4sgn 24253 | The signs of the sine and ... |
| coseq00topi 24254 | Location of the zeroes of ... |
| coseq0negpitopi 24255 | Location of the zeroes of ... |
| tanrpcl 24256 | Positive real closure of t... |
| tangtx 24257 | The tangent function is gr... |
| tanabsge 24258 | The tangent function is gr... |
| sinq12gt0 24259 | The sine of a number stric... |
| sinq12ge0 24260 | The sine of a number betwe... |
| sinq34lt0t 24261 | The sine of a number stric... |
| cosq14gt0 24262 | The cosine of a number str... |
| cosq14ge0 24263 | The cosine of a number bet... |
| sincosq1eq 24264 | Complementarity of the sin... |
| sincos4thpi 24265 | The sine and cosine of ` _... |
| tan4thpi 24266 | The tangent of ` _pi / 4 `... |
| sincos6thpi 24267 | The sine and cosine of ` _... |
| sincos3rdpi 24268 | The sine and cosine of ` _... |
| pige3 24269 | ` _pi ` is greater or equa... |
| abssinper 24270 | The absolute value of sine... |
| sinkpi 24271 | The sine of an integer mul... |
| coskpi 24272 | The absolute value of the ... |
| sineq0 24273 | A complex number whose sin... |
| coseq1 24274 | A complex number whose cos... |
| efeq1 24275 | A complex number whose exp... |
| cosne0 24276 | The cosine function has no... |
| cosordlem 24277 | Lemma for ~ cosord . (Con... |
| cosord 24278 | Cosine is decreasing over ... |
| cos11 24279 | Cosine is one-to-one over ... |
| sinord 24280 | Sine is increasing over th... |
| recosf1o 24281 | The cosine function is a b... |
| resinf1o 24282 | The sine function is a bij... |
| tanord1 24283 | The tangent function is st... |
| tanord 24284 | The tangent function is st... |
| tanregt0 24285 | The positivity of ` tan ( ... |
| negpitopissre 24286 | ` ( -u _pi (,] _pi ) ` is ... |
| efgh 24287 | The exponential function o... |
| efif1olem1 24288 | Lemma for ~ efif1o . (Con... |
| efif1olem2 24289 | Lemma for ~ efif1o . (Con... |
| efif1olem3 24290 | Lemma for ~ efif1o . (Con... |
| efif1olem4 24291 | The exponential function o... |
| efif1o 24292 | The exponential function o... |
| efifo 24293 | The exponential function o... |
| eff1olem 24294 | The exponential function m... |
| eff1o 24295 | The exponential function m... |
| efabl 24296 | The image of a subgroup of... |
| efsubm 24297 | The image of a subgroup of... |
| circgrp 24298 | The circle group ` T ` is ... |
| circsubm 24299 | The circle group ` T ` is ... |
| rzgrp 24300 | The quotient group R/Z is ... |
| logrn 24305 | The range of the natural l... |
| ellogrn 24306 | Write out the property ` A... |
| dflog2 24307 | The natural logarithm func... |
| relogrn 24308 | The range of the natural l... |
| logrncn 24309 | The range of the natural l... |
| eff1o2 24310 | The exponential function r... |
| logf1o 24311 | The natural logarithm func... |
| dfrelog 24312 | The natural logarithm func... |
| relogf1o 24313 | The natural logarithm func... |
| logrncl 24314 | Closure of the natural log... |
| logcl 24315 | Closure of the natural log... |
| logimcl 24316 | Closure of the imaginary p... |
| logcld 24317 | The logarithm of a nonzero... |
| logimcld 24318 | The imaginary part of the ... |
| logimclad 24319 | The imaginary part of the ... |
| abslogimle 24320 | The imaginary part of the ... |
| logrnaddcl 24321 | The range of the natural l... |
| relogcl 24322 | Closure of the natural log... |
| eflog 24323 | Relationship between the n... |
| logeq0im1 24324 | If the logarithm of a numb... |
| logccne0 24325 | The logarithm isn't 0 if i... |
| logne0 24326 | Logarithm of a non-1 posit... |
| reeflog 24327 | Relationship between the n... |
| logef 24328 | Relationship between the n... |
| relogef 24329 | Relationship between the n... |
| logeftb 24330 | Relationship between the n... |
| relogeftb 24331 | Relationship between the n... |
| log1 24332 | The natural logarithm of `... |
| loge 24333 | The natural logarithm of `... |
| logneg 24334 | The natural logarithm of a... |
| logm1 24335 | The natural logarithm of n... |
| lognegb 24336 | If a number has imaginary ... |
| relogoprlem 24337 | Lemma for ~ relogmul and ~... |
| relogmul 24338 | The natural logarithm of t... |
| relogdiv 24339 | The natural logarithm of t... |
| explog 24340 | Exponentiation of a nonzer... |
| reexplog 24341 | Exponentiation of a positi... |
| relogexp 24342 | The natural logarithm of p... |
| relog 24343 | Real part of a logarithm. ... |
| relogiso 24344 | The natural logarithm func... |
| reloggim 24345 | The natural logarithm is a... |
| logltb 24346 | The natural logarithm func... |
| logfac 24347 | The logarithm of a factori... |
| eflogeq 24348 | Solve an equation involvin... |
| logleb 24349 | Natural logarithm preserve... |
| rplogcl 24350 | Closure of the logarithm f... |
| logge0 24351 | The logarithm of a number ... |
| logcj 24352 | The natural logarithm dist... |
| efiarg 24353 | The exponential of the "ar... |
| cosargd 24354 | The cosine of the argument... |
| cosarg0d 24355 | The cosine of the argument... |
| argregt0 24356 | Closure of the argument of... |
| argrege0 24357 | Closure of the argument of... |
| argimgt0 24358 | Closure of the argument of... |
| argimlt0 24359 | Closure of the argument of... |
| logimul 24360 | Multiplying a number by ` ... |
| logneg2 24361 | The logarithm of the negat... |
| logmul2 24362 | Generalization of ~ relogm... |
| logdiv2 24363 | Generalization of ~ relogd... |
| abslogle 24364 | Bound on the magnitude of ... |
| tanarg 24365 | The basic relation between... |
| logdivlti 24366 | The ` log x / x ` function... |
| logdivlt 24367 | The ` log x / x ` function... |
| logdivle 24368 | The ` log x / x ` function... |
| relogcld 24369 | Closure of the natural log... |
| reeflogd 24370 | Relationship between the n... |
| relogmuld 24371 | The natural logarithm of t... |
| relogdivd 24372 | The natural logarithm of t... |
| logled 24373 | Natural logarithm preserve... |
| relogefd 24374 | Relationship between the n... |
| rplogcld 24375 | Closure of the logarithm f... |
| logge0d 24376 | The logarithm of a number ... |
| logge0b 24377 | The logarithm of a number ... |
| loggt0b 24378 | The logarithm of a number ... |
| logle1b 24379 | The logarithm of a number ... |
| loglt1b 24380 | The logarithm of a number ... |
| divlogrlim 24381 | The inverse logarithm func... |
| logno1 24382 | The logarithm function is ... |
| dvrelog 24383 | The derivative of the real... |
| relogcn 24384 | The real logarithm functio... |
| ellogdm 24385 | Elementhood in the "contin... |
| logdmn0 24386 | A number in the continuous... |
| logdmnrp 24387 | A number in the continuous... |
| logdmss 24388 | The continuity domain of `... |
| logcnlem2 24389 | Lemma for ~ logcn . (Cont... |
| logcnlem3 24390 | Lemma for ~ logcn . (Cont... |
| logcnlem4 24391 | Lemma for ~ logcn . (Cont... |
| logcnlem5 24392 | Lemma for ~ logcn . (Cont... |
| logcn 24393 | The logarithm function is ... |
| dvloglem 24394 | Lemma for ~ dvlog . (Cont... |
| logdmopn 24395 | The "continuous domain" of... |
| logf1o2 24396 | The logarithm maps its con... |
| dvlog 24397 | The derivative of the comp... |
| dvlog2lem 24398 | Lemma for ~ dvlog2 . (Con... |
| dvlog2 24399 | The derivative of the comp... |
| advlog 24400 | The antiderivative of the ... |
| advlogexp 24401 | The antiderivative of a po... |
| efopnlem1 24402 | Lemma for ~ efopn . (Cont... |
| efopnlem2 24403 | Lemma for ~ efopn . (Cont... |
| efopn 24404 | The exponential map is an ... |
| logtayllem 24405 | Lemma for ~ logtayl . (Co... |
| logtayl 24406 | The Taylor series for ` -u... |
| logtaylsum 24407 | The Taylor series for ` -u... |
| logtayl2 24408 | Power series expression fo... |
| logccv 24409 | The natural logarithm func... |
| cxpval 24410 | Value of the complex power... |
| cxpef 24411 | Value of the complex power... |
| 0cxp 24412 | Value of the complex power... |
| cxpexpz 24413 | Relate the complex power f... |
| cxpexp 24414 | Relate the complex power f... |
| logcxp 24415 | Logarithm of a complex pow... |
| cxp0 24416 | Value of the complex power... |
| cxp1 24417 | Value of the complex power... |
| 1cxp 24418 | Value of the complex power... |
| ecxp 24419 | Write the exponential func... |
| cxpcl 24420 | Closure of the complex pow... |
| recxpcl 24421 | Real closure of the comple... |
| rpcxpcl 24422 | Positive real closure of t... |
| cxpne0 24423 | Complex exponentiation is ... |
| cxpeq0 24424 | Complex exponentiation is ... |
| cxpadd 24425 | Sum of exponents law for c... |
| cxpp1 24426 | Value of a nonzero complex... |
| cxpneg 24427 | Value of a complex number ... |
| cxpsub 24428 | Exponent subtraction law f... |
| cxpge0 24429 | Nonnegative exponentiation... |
| mulcxplem 24430 | Lemma for ~ mulcxp . (Con... |
| mulcxp 24431 | Complex exponentiation of ... |
| cxprec 24432 | Complex exponentiation of ... |
| divcxp 24433 | Complex exponentiation of ... |
| cxpmul 24434 | Product of exponents law f... |
| cxpmul2 24435 | Product of exponents law f... |
| cxproot 24436 | The complex power function... |
| cxpmul2z 24437 | Generalize ~ cxpmul2 to ne... |
| abscxp 24438 | Absolute value of a power,... |
| abscxp2 24439 | Absolute value of a power,... |
| cxplt 24440 | Ordering property for comp... |
| cxple 24441 | Ordering property for comp... |
| cxplea 24442 | Ordering property for comp... |
| cxple2 24443 | Ordering property for comp... |
| cxplt2 24444 | Ordering property for comp... |
| cxple2a 24445 | Ordering property for comp... |
| cxplt3 24446 | Ordering property for comp... |
| cxple3 24447 | Ordering property for comp... |
| cxpsqrtlem 24448 | Lemma for ~ cxpsqrt . (Co... |
| cxpsqrt 24449 | The complex exponential fu... |
| logsqrt 24450 | Logarithm of a square root... |
| cxp0d 24451 | Value of the complex power... |
| cxp1d 24452 | Value of the complex power... |
| 1cxpd 24453 | Value of the complex power... |
| cxpcld 24454 | Closure of the complex pow... |
| cxpmul2d 24455 | Product of exponents law f... |
| 0cxpd 24456 | Value of the complex power... |
| cxpexpzd 24457 | Relate the complex power f... |
| cxpefd 24458 | Value of the complex power... |
| cxpne0d 24459 | Complex exponentiation is ... |
| cxpp1d 24460 | Value of a nonzero complex... |
| cxpnegd 24461 | Value of a complex number ... |
| cxpmul2zd 24462 | Generalize ~ cxpmul2 to ne... |
| cxpaddd 24463 | Sum of exponents law for c... |
| cxpsubd 24464 | Exponent subtraction law f... |
| cxpltd 24465 | Ordering property for comp... |
| cxpled 24466 | Ordering property for comp... |
| cxplead 24467 | Ordering property for comp... |
| divcxpd 24468 | Complex exponentiation of ... |
| recxpcld 24469 | Positive real closure of t... |
| cxpge0d 24470 | Nonnegative exponentiation... |
| cxple2ad 24471 | Ordering property for comp... |
| cxplt2d 24472 | Ordering property for comp... |
| cxple2d 24473 | Ordering property for comp... |
| mulcxpd 24474 | Complex exponentiation of ... |
| cxprecd 24475 | Complex exponentiation of ... |
| rpcxpcld 24476 | Positive real closure of t... |
| logcxpd 24477 | Logarithm of a complex pow... |
| cxplt3d 24478 | Ordering property for comp... |
| cxple3d 24479 | Ordering property for comp... |
| cxpmuld 24480 | Product of exponents law f... |
| dvcxp1 24481 | The derivative of a comple... |
| dvcxp2 24482 | The derivative of a comple... |
| dvsqrt 24483 | The derivative of the real... |
| dvcncxp1 24484 | Derivative of complex powe... |
| dvcnsqrt 24485 | Derivative of square root ... |
| cxpcn 24486 | Domain of continuity of th... |
| cxpcn2 24487 | Continuity of the complex ... |
| cxpcn3lem 24488 | Lemma for ~ cxpcn3 . (Con... |
| cxpcn3 24489 | Extend continuity of the c... |
| resqrtcn 24490 | Continuity of the real squ... |
| sqrtcn 24491 | Continuity of the square r... |
| cxpaddlelem 24492 | Lemma for ~ cxpaddle . (C... |
| cxpaddle 24493 | Ordering property for comp... |
| abscxpbnd 24494 | Bound on the absolute valu... |
| root1id 24495 | Property of an ` N ` -th r... |
| root1eq1 24496 | The only powers of an ` N ... |
| root1cj 24497 | Within the ` N ` -th roots... |
| cxpeq 24498 | Solve an equation involvin... |
| loglesqrt 24499 | An upper bound on the loga... |
| logreclem 24500 | Symmetry of the natural lo... |
| logrec 24501 | Logarithm of a reciprocal ... |
| logbval 24504 | Define the value of the ` ... |
| logbcl 24505 | General logarithm closure.... |
| logbid1 24506 | General logarithm is 1 whe... |
| logb1 24507 | The logarithm of ` 1 ` to ... |
| elogb 24508 | The general logarithm of a... |
| logbchbase 24509 | Change of base for logarit... |
| relogbval 24510 | Value of the general logar... |
| relogbcl 24511 | Closure of the general log... |
| relogbzcl 24512 | Closure of the general log... |
| relogbreexp 24513 | Power law for the general ... |
| relogbzexp 24514 | Power law for the general ... |
| relogbmul 24515 | The logarithm of the produ... |
| relogbmulexp 24516 | The logarithm of the produ... |
| relogbdiv 24517 | The logarithm of the quoti... |
| relogbexp 24518 | Identity law for general l... |
| nnlogbexp 24519 | Identity law for general l... |
| logbrec 24520 | Logarithm of a reciprocal ... |
| logbleb 24521 | The general logarithm func... |
| logblt 24522 | The general logarithm func... |
| relogbcxp 24523 | Identity law for the gener... |
| cxplogb 24524 | Identity law for the gener... |
| relogbcxpb 24525 | The logarithm is the inver... |
| logbmpt 24526 | The general logarithm to a... |
| logbf 24527 | The general logarithm to a... |
| logbfval 24528 | The general logarithm of a... |
| relogbf 24529 | The general logarithm to a... |
| logblog 24530 | The general logarithm to t... |
| angval 24531 | Define the angle function,... |
| angcan 24532 | Cancel a constant multipli... |
| angneg 24533 | Cancel a negative sign in ... |
| angvald 24534 | The (signed) angle between... |
| angcld 24535 | The (signed) angle between... |
| angrteqvd 24536 | Two vectors are at a right... |
| cosangneg2d 24537 | The cosine of the angle be... |
| angrtmuld 24538 | Perpendicularity of two ve... |
| ang180lem1 24539 | Lemma for ~ ang180 . Show... |
| ang180lem2 24540 | Lemma for ~ ang180 . Show... |
| ang180lem3 24541 | Lemma for ~ ang180 . Sinc... |
| ang180lem4 24542 | Lemma for ~ ang180 . Redu... |
| ang180lem5 24543 | Lemma for ~ ang180 : Redu... |
| ang180 24544 | The sum of angles ` m A B ... |
| lawcoslem1 24545 | Lemma for ~ lawcos . Here... |
| lawcos 24546 | Law of cosines (also known... |
| pythag 24547 | Pythagorean theorem. Give... |
| isosctrlem1 24548 | Lemma for ~ isosctr . (Co... |
| isosctrlem2 24549 | Lemma for ~ isosctr . Cor... |
| isosctrlem3 24550 | Lemma for ~ isosctr . Cor... |
| isosctr 24551 | Isosceles triangle theorem... |
| ssscongptld 24552 | If two triangles have equa... |
| affineequiv 24553 | Equivalence between two wa... |
| affineequiv2 24554 | Equivalence between two wa... |
| angpieqvdlem 24555 | Equivalence used in the pr... |
| angpieqvdlem2 24556 | Equivalence used in ~ angp... |
| angpined 24557 | If the angle at ABC is ` _... |
| angpieqvd 24558 | The angle ABC is ` _pi ` i... |
| chordthmlem 24559 | If M is the midpoint of AB... |
| chordthmlem2 24560 | If M is the midpoint of AB... |
| chordthmlem3 24561 | If M is the midpoint of AB... |
| chordthmlem4 24562 | If P is on the segment AB ... |
| chordthmlem5 24563 | If P is on the segment AB ... |
| chordthm 24564 | The intersecting chords th... |
| heron 24565 | Heron's formula gives the ... |
| quad2 24566 | The quadratic equation, wi... |
| quad 24567 | The quadratic equation. (... |
| 1cubrlem 24568 | The cube roots of unity. ... |
| 1cubr 24569 | The cube roots of unity. ... |
| dcubic1lem 24570 | Lemma for ~ dcubic1 and ~ ... |
| dcubic2 24571 | Reverse direction of ~ dcu... |
| dcubic1 24572 | Forward direction of ~ dcu... |
| dcubic 24573 | Solutions to the depressed... |
| mcubic 24574 | Solutions to a monic cubic... |
| cubic2 24575 | The solution to the genera... |
| cubic 24576 | The cubic equation, which ... |
| binom4 24577 | Work out a quartic binomia... |
| dquartlem1 24578 | Lemma for ~ dquart . (Con... |
| dquartlem2 24579 | Lemma for ~ dquart . (Con... |
| dquart 24580 | Solve a depressed quartic ... |
| quart1cl 24581 | Closure lemmas for ~ quart... |
| quart1lem 24582 | Lemma for ~ quart1 . (Con... |
| quart1 24583 | Depress a quartic equation... |
| quartlem1 24584 | Lemma for ~ quart . (Cont... |
| quartlem2 24585 | Closure lemmas for ~ quart... |
| quartlem3 24586 | Closure lemmas for ~ quart... |
| quartlem4 24587 | Closure lemmas for ~ quart... |
| quart 24588 | The quartic equation, writ... |
| asinlem 24595 | The argument to the logari... |
| asinlem2 24596 | The argument to the logari... |
| asinlem3a 24597 | Lemma for ~ asinlem3 . (C... |
| asinlem3 24598 | The argument to the logari... |
| asinf 24599 | Domain and range of the ar... |
| asincl 24600 | Closure for the arcsin fun... |
| acosf 24601 | Domain and range of the ar... |
| acoscl 24602 | Closure for the arccos fun... |
| atandm 24603 | Since the property is a li... |
| atandm2 24604 | This form of ~ atandm is a... |
| atandm3 24605 | A compact form of ~ atandm... |
| atandm4 24606 | A compact form of ~ atandm... |
| atanf 24607 | Domain and range of the ar... |
| atancl 24608 | Closure for the arctan fun... |
| asinval 24609 | Value of the arcsin functi... |
| acosval 24610 | Value of the arccos functi... |
| atanval 24611 | Value of the arctan functi... |
| atanre 24612 | A real number is in the do... |
| asinneg 24613 | The arcsine function is od... |
| acosneg 24614 | The negative symmetry rela... |
| efiasin 24615 | The exponential of the arc... |
| sinasin 24616 | The arcsine function is an... |
| cosacos 24617 | The arccosine function is ... |
| asinsinlem 24618 | Lemma for ~ asinsin . (Co... |
| asinsin 24619 | The arcsine function compo... |
| acoscos 24620 | The arccosine function is ... |
| asin1 24621 | The arcsine of ` 1 ` is ` ... |
| acos1 24622 | The arcsine of ` 1 ` is ` ... |
| reasinsin 24623 | The arcsine function compo... |
| asinsinb 24624 | Relationship between sine ... |
| acoscosb 24625 | Relationship between sine ... |
| asinbnd 24626 | The arcsine function has r... |
| acosbnd 24627 | The arccosine function has... |
| asinrebnd 24628 | Bounds on the arcsine func... |
| asinrecl 24629 | The arcsine function is re... |
| acosrecl 24630 | The arccosine function is ... |
| cosasin 24631 | The cosine of the arcsine ... |
| sinacos 24632 | The sine of the arccosine ... |
| atandmneg 24633 | The domain of the arctange... |
| atanneg 24634 | The arctangent function is... |
| atan0 24635 | The arctangent of zero is ... |
| atandmcj 24636 | The arctangent function di... |
| atancj 24637 | The arctangent function di... |
| atanrecl 24638 | The arctangent function is... |
| efiatan 24639 | Value of the exponential o... |
| atanlogaddlem 24640 | Lemma for ~ atanlogadd . ... |
| atanlogadd 24641 | The rule ` sqrt ( z w ) = ... |
| atanlogsublem 24642 | Lemma for ~ atanlogsub . ... |
| atanlogsub 24643 | A variation on ~ atanlogad... |
| efiatan2 24644 | Value of the exponential o... |
| 2efiatan 24645 | Value of the exponential o... |
| tanatan 24646 | The arctangent function is... |
| atandmtan 24647 | The tangent function has r... |
| cosatan 24648 | The cosine of an arctangen... |
| cosatanne0 24649 | The arctangent function ha... |
| atantan 24650 | The arctangent function is... |
| atantanb 24651 | Relationship between tange... |
| atanbndlem 24652 | Lemma for ~ atanbnd . (Co... |
| atanbnd 24653 | The arctangent function is... |
| atanord 24654 | The arctangent function is... |
| atan1 24655 | The arctangent of ` 1 ` is... |
| bndatandm 24656 | A point in the open unit d... |
| atans 24657 | The "domain of continuity"... |
| atans2 24658 | It suffices to show that `... |
| atansopn 24659 | The domain of continuity o... |
| atansssdm 24660 | The domain of continuity o... |
| ressatans 24661 | The real number line is a ... |
| dvatan 24662 | The derivative of the arct... |
| atancn 24663 | The arctangent is a contin... |
| atantayl 24664 | The Taylor series for ` ar... |
| atantayl2 24665 | The Taylor series for ` ar... |
| atantayl3 24666 | The Taylor series for ` ar... |
| leibpilem1 24667 | Lemma for ~ leibpi . (Con... |
| leibpilem2 24668 | The Leibniz formula for ` ... |
| leibpi 24669 | The Leibniz formula for ` ... |
| leibpisum 24670 | The Leibniz formula for ` ... |
| log2cnv 24671 | Using the Taylor series fo... |
| log2tlbnd 24672 | Bound the error term in th... |
| log2ublem1 24673 | Lemma for ~ log2ub . The ... |
| log2ublem2 24674 | Lemma for ~ log2ub . (Con... |
| log2ublem3 24675 | Lemma for ~ log2ub . In d... |
| log2ub 24676 | ` log 2 ` is less than ` 2... |
| log2le1 24677 | ` log 2 ` is less than ` 1... |
| birthdaylem1 24678 | Lemma for ~ birthday . (C... |
| birthdaylem2 24679 | For general ` N ` and ` K ... |
| birthdaylem3 24680 | For general ` N ` and ` K ... |
| birthday 24681 | The Birthday Problem. The... |
| dmarea 24684 | The domain of the area fun... |
| areambl 24685 | The fibers of a measurable... |
| areass 24686 | A measurable region is a s... |
| dfarea 24687 | Rewrite ~ df-area self-ref... |
| areaf 24688 | Area measurement is a func... |
| areacl 24689 | The area of a measurable r... |
| areage0 24690 | The area of a measurable r... |
| areaval 24691 | The area of a measurable r... |
| rlimcnp 24692 | Relate a limit of a real-v... |
| rlimcnp2 24693 | Relate a limit of a real-v... |
| rlimcnp3 24694 | Relate a limit of a real-v... |
| xrlimcnp 24695 | Relate a limit of a real-v... |
| efrlim 24696 | The limit of the sequence ... |
| dfef2 24697 | The limit of the sequence ... |
| cxplim 24698 | A power to a negative expo... |
| sqrtlim 24699 | The inverse square root fu... |
| rlimcxp 24700 | Any power to a positive ex... |
| o1cxp 24701 | An eventually bounded func... |
| cxp2limlem 24702 | A linear factor grows slow... |
| cxp2lim 24703 | Any power grows slower tha... |
| cxploglim 24704 | The logarithm grows slower... |
| cxploglim2 24705 | Every power of the logarit... |
| divsqrtsumlem 24706 | Lemma for ~ divsqrsum and ... |
| divsqrsumf 24707 | The function ` F ` used in... |
| divsqrsum 24708 | The sum ` sum_ n <_ x ( 1 ... |
| divsqrtsum2 24709 | A bound on the distance of... |
| divsqrtsumo1 24710 | The sum ` sum_ n <_ x ( 1 ... |
| cvxcl 24711 | Closure of a 0-1 linear co... |
| scvxcvx 24712 | A strictly convex function... |
| jensenlem1 24713 | Lemma for ~ jensen . (Con... |
| jensenlem2 24714 | Lemma for ~ jensen . (Con... |
| jensen 24715 | Jensen's inequality, a fin... |
| amgmlem 24716 | Lemma for ~ amgm . (Contr... |
| amgm 24717 | Inequality of arithmetic a... |
| logdifbnd 24720 | Bound on the difference of... |
| logdiflbnd 24721 | Lower bound on the differe... |
| emcllem1 24722 | Lemma for ~ emcl . The se... |
| emcllem2 24723 | Lemma for ~ emcl . ` F ` i... |
| emcllem3 24724 | Lemma for ~ emcl . The fu... |
| emcllem4 24725 | Lemma for ~ emcl . The di... |
| emcllem5 24726 | Lemma for ~ emcl . The pa... |
| emcllem6 24727 | Lemma for ~ emcl . By the... |
| emcllem7 24728 | Lemma for ~ emcl and ~ har... |
| emcl 24729 | Closure and bounds for the... |
| harmonicbnd 24730 | A bound on the harmonic se... |
| harmonicbnd2 24731 | A bound on the harmonic se... |
| emre 24732 | The Euler-Mascheroni const... |
| emgt0 24733 | The Euler-Mascheroni const... |
| harmonicbnd3 24734 | A bound on the harmonic se... |
| harmoniclbnd 24735 | A bound on the harmonic se... |
| harmonicubnd 24736 | A bound on the harmonic se... |
| harmonicbnd4 24737 | The asymptotic behavior of... |
| fsumharmonic 24738 | Bound a finite sum based o... |
| zetacvg 24741 | The zeta series is converg... |
| eldmgm 24748 | Elementhood in the set of ... |
| dmgmaddn0 24749 | If ` A ` is not a nonposit... |
| dmlogdmgm 24750 | If ` A ` is in the continu... |
| rpdmgm 24751 | A positive real number is ... |
| dmgmn0 24752 | If ` A ` is not a nonposit... |
| dmgmaddnn0 24753 | If ` A ` is not a nonposit... |
| dmgmdivn0 24754 | Lemma for ~ lgamf . (Cont... |
| lgamgulmlem1 24755 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem2 24756 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem3 24757 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem4 24758 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem5 24759 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem6 24760 | The series ` G ` is unifor... |
| lgamgulm 24761 | The series ` G ` is unifor... |
| lgamgulm2 24762 | Rewrite the limit of the s... |
| lgambdd 24763 | The log-Gamma function is ... |
| lgamucov 24764 | The ` U ` regions used in ... |
| lgamucov2 24765 | The ` U ` regions used in ... |
| lgamcvglem 24766 | Lemma for ~ lgamf and ~ lg... |
| lgamcl 24767 | The log-Gamma function is ... |
| lgamf 24768 | The log-Gamma function is ... |
| gamf 24769 | The Gamma function is a co... |
| gamcl 24770 | The exponential of the log... |
| eflgam 24771 | The exponential of the log... |
| gamne0 24772 | The Gamma function is neve... |
| igamval 24773 | Value of the inverse Gamma... |
| igamz 24774 | Value of the inverse Gamma... |
| igamgam 24775 | Value of the inverse Gamma... |
| igamlgam 24776 | Value of the inverse Gamma... |
| igamf 24777 | Closure of the inverse Gam... |
| igamcl 24778 | Closure of the inverse Gam... |
| gamigam 24779 | The Gamma function is the ... |
| lgamcvg 24780 | The series ` G ` converges... |
| lgamcvg2 24781 | The series ` G ` converges... |
| gamcvg 24782 | The pointwise exponential ... |
| lgamp1 24783 | The functional equation of... |
| gamp1 24784 | The functional equation of... |
| gamcvg2lem 24785 | Lemma for ~ gamcvg2 . (Co... |
| gamcvg2 24786 | An infinite product expres... |
| regamcl 24787 | The Gamma function is real... |
| relgamcl 24788 | The log-Gamma function is ... |
| rpgamcl 24789 | The log-Gamma function is ... |
| lgam1 24790 | The log-Gamma function at ... |
| gam1 24791 | The log-Gamma function at ... |
| facgam 24792 | The Gamma function general... |
| gamfac 24793 | The Gamma function general... |
| wilthlem1 24794 | The only elements that are... |
| wilthlem2 24795 | Lemma for ~ wilth : induct... |
| wilthlem3 24796 | Lemma for ~ wilth . Here ... |
| wilth 24797 | Wilson's theorem. A numbe... |
| wilthimp 24798 | The forward implication of... |
| ftalem1 24799 | Lemma for ~ fta : "growth... |
| ftalem2 24800 | Lemma for ~ fta . There e... |
| ftalem3 24801 | Lemma for ~ fta . There e... |
| ftalem4 24802 | Lemma for ~ fta : Closure... |
| ftalem5 24803 | Lemma for ~ fta : Main pr... |
| ftalem6 24804 | Lemma for ~ fta : Dischar... |
| ftalem7 24805 | Lemma for ~ fta . Shift t... |
| fta 24806 | The Fundamental Theorem of... |
| basellem1 24807 | Lemma for ~ basel . Closu... |
| basellem2 24808 | Lemma for ~ basel . Show ... |
| basellem3 24809 | Lemma for ~ basel . Using... |
| basellem4 24810 | Lemma for ~ basel . By ~ ... |
| basellem5 24811 | Lemma for ~ basel . Using... |
| basellem6 24812 | Lemma for ~ basel . The f... |
| basellem7 24813 | Lemma for ~ basel . The f... |
| basellem8 24814 | Lemma for ~ basel . The f... |
| basellem9 24815 | Lemma for ~ basel . Since... |
| basel 24816 | The sum of the inverse squ... |
| efnnfsumcl 24829 | Finite sum closure in the ... |
| ppisval 24830 | The set of primes less tha... |
| ppisval2 24831 | The set of primes less tha... |
| ppifi 24832 | The set of primes less tha... |
| prmdvdsfi 24833 | The set of prime divisors ... |
| chtf 24834 | Domain and range of the Ch... |
| chtcl 24835 | Real closure of the Chebys... |
| chtval 24836 | Value of the Chebyshev fun... |
| efchtcl 24837 | The Chebyshev function is ... |
| chtge0 24838 | The Chebyshev function is ... |
| vmaval 24839 | Value of the von Mangoldt ... |
| isppw 24840 | Two ways to say that ` A `... |
| isppw2 24841 | Two ways to say that ` A `... |
| vmappw 24842 | Value of the von Mangoldt ... |
| vmaprm 24843 | Value of the von Mangoldt ... |
| vmacl 24844 | Closure for the von Mangol... |
| vmaf 24845 | Functionality of the von M... |
| efvmacl 24846 | The von Mangoldt is closed... |
| vmage0 24847 | The von Mangoldt function ... |
| chpval 24848 | Value of the second Chebys... |
| chpf 24849 | Functionality of the secon... |
| chpcl 24850 | Closure for the second Che... |
| efchpcl 24851 | The second Chebyshev funct... |
| chpge0 24852 | The second Chebyshev funct... |
| ppival 24853 | Value of the prime-countin... |
| ppival2 24854 | Value of the prime-countin... |
| ppival2g 24855 | Value of the prime-countin... |
| ppif 24856 | Domain and range of the pr... |
| ppicl 24857 | Real closure of the prime-... |
| muval 24858 | The value of the Möbi... |
| muval1 24859 | The value of the Möbi... |
| muval2 24860 | The value of the Möbi... |
| isnsqf 24861 | Two ways to say that a num... |
| issqf 24862 | Two ways to say that a num... |
| sqfpc 24863 | The prime count of a squar... |
| dvdssqf 24864 | A divisor of a squarefree ... |
| sqf11 24865 | A squarefree number is com... |
| muf 24866 | The Möbius function i... |
| mucl 24867 | Closure of the Möbius... |
| sgmval 24868 | The value of the divisor f... |
| sgmval2 24869 | The value of the divisor f... |
| 0sgm 24870 | The value of the sum-of-di... |
| sgmf 24871 | The divisor function is a ... |
| sgmcl 24872 | Closure of the divisor fun... |
| sgmnncl 24873 | Closure of the divisor fun... |
| mule1 24874 | The Möbius function t... |
| chtfl 24875 | The Chebyshev function doe... |
| chpfl 24876 | The second Chebyshev funct... |
| ppiprm 24877 | The prime-counting functio... |
| ppinprm 24878 | The prime-counting functio... |
| chtprm 24879 | The Chebyshev function at ... |
| chtnprm 24880 | The Chebyshev function at ... |
| chpp1 24881 | The second Chebyshev funct... |
| chtwordi 24882 | The Chebyshev function is ... |
| chpwordi 24883 | The second Chebyshev funct... |
| chtdif 24884 | The difference of the Cheb... |
| efchtdvds 24885 | The exponentiated Chebyshe... |
| ppifl 24886 | The prime-counting functio... |
| ppip1le 24887 | The prime-counting functio... |
| ppiwordi 24888 | The prime-counting functio... |
| ppidif 24889 | The difference of the prim... |
| ppi1 24890 | The prime-counting functio... |
| cht1 24891 | The Chebyshev function at ... |
| vma1 24892 | The von Mangoldt function ... |
| chp1 24893 | The second Chebyshev funct... |
| ppi1i 24894 | Inference form of ~ ppiprm... |
| ppi2i 24895 | Inference form of ~ ppinpr... |
| ppi2 24896 | The prime-counting functio... |
| ppi3 24897 | The prime-counting functio... |
| cht2 24898 | The Chebyshev function at ... |
| cht3 24899 | The Chebyshev function at ... |
| ppinncl 24900 | Closure of the prime-count... |
| chtrpcl 24901 | Closure of the Chebyshev f... |
| ppieq0 24902 | The prime-counting functio... |
| ppiltx 24903 | The prime-counting functio... |
| prmorcht 24904 | Relate the primorial (prod... |
| mumullem1 24905 | Lemma for ~ mumul . A mul... |
| mumullem2 24906 | Lemma for ~ mumul . The p... |
| mumul 24907 | The Möbius function i... |
| sqff1o 24908 | There is a bijection from ... |
| fsumdvdsdiaglem 24909 | A "diagonal commutation" o... |
| fsumdvdsdiag 24910 | A "diagonal commutation" o... |
| fsumdvdscom 24911 | A double commutation of di... |
| dvdsppwf1o 24912 | A bijection from the divis... |
| dvdsflf1o 24913 | A bijection from the numbe... |
| dvdsflsumcom 24914 | A sum commutation from ` s... |
| fsumfldivdiaglem 24915 | Lemma for ~ fsumfldivdiag ... |
| fsumfldivdiag 24916 | The right-hand side of ~ d... |
| musum 24917 | The sum of the Möbius... |
| musumsum 24918 | Evaluate a collapsing sum ... |
| muinv 24919 | The Möbius inversion ... |
| dvdsmulf1o 24920 | If ` M ` and ` N ` are two... |
| fsumdvdsmul 24921 | Product of two divisor sum... |
| sgmppw 24922 | The value of the divisor f... |
| 0sgmppw 24923 | A prime power ` P ^ K ` ha... |
| 1sgmprm 24924 | The sum of divisors for a ... |
| 1sgm2ppw 24925 | The sum of the divisors of... |
| sgmmul 24926 | The divisor function for f... |
| ppiublem1 24927 | Lemma for ~ ppiub . (Cont... |
| ppiublem2 24928 | A prime greater than ` 3 `... |
| ppiub 24929 | An upper bound on the prim... |
| vmalelog 24930 | The von Mangoldt function ... |
| chtlepsi 24931 | The first Chebyshev functi... |
| chprpcl 24932 | Closure of the second Cheb... |
| chpeq0 24933 | The second Chebyshev funct... |
| chteq0 24934 | The first Chebyshev functi... |
| chtleppi 24935 | Upper bound on the ` theta... |
| chtublem 24936 | Lemma for ~ chtub . (Cont... |
| chtub 24937 | An upper bound on the Cheb... |
| fsumvma 24938 | Rewrite a sum over the von... |
| fsumvma2 24939 | Apply ~ fsumvma for the co... |
| pclogsum 24940 | The logarithmic analogue o... |
| vmasum 24941 | The sum of the von Mangold... |
| logfac2 24942 | Another expression for the... |
| chpval2 24943 | Express the second Chebysh... |
| chpchtsum 24944 | The second Chebyshev funct... |
| chpub 24945 | An upper bound on the seco... |
| logfacubnd 24946 | A simple upper bound on th... |
| logfaclbnd 24947 | A lower bound on the logar... |
| logfacbnd3 24948 | Show the stronger statemen... |
| logfacrlim 24949 | Combine the estimates ~ lo... |
| logexprlim 24950 | The sum ` sum_ n <_ x , lo... |
| logfacrlim2 24951 | Write out ~ logfacrlim as ... |
| mersenne 24952 | A Mersenne prime is a prim... |
| perfect1 24953 | Euclid's contribution to t... |
| perfectlem1 24954 | Lemma for ~ perfect . (Co... |
| perfectlem2 24955 | Lemma for ~ perfect . (Co... |
| perfect 24956 | The Euclid-Euler theorem, ... |
| dchrval 24959 | Value of the group of Diri... |
| dchrbas 24960 | Base set of the group of D... |
| dchrelbas 24961 | A Dirichlet character is a... |
| dchrelbas2 24962 | A Dirichlet character is a... |
| dchrelbas3 24963 | A Dirichlet character is a... |
| dchrelbasd 24964 | A Dirichlet character is a... |
| dchrrcl 24965 | Reverse closure for a Diri... |
| dchrmhm 24966 | A Dirichlet character is a... |
| dchrf 24967 | A Dirichlet character is a... |
| dchrelbas4 24968 | A Dirichlet character is a... |
| dchrzrh1 24969 | Value of a Dirichlet chara... |
| dchrzrhcl 24970 | A Dirichlet character take... |
| dchrzrhmul 24971 | A Dirichlet character is c... |
| dchrplusg 24972 | Group operation on the gro... |
| dchrmul 24973 | Group operation on the gro... |
| dchrmulcl 24974 | Closure of the group opera... |
| dchrn0 24975 | A Dirichlet character is n... |
| dchr1cl 24976 | Closure of the principal D... |
| dchrmulid2 24977 | Left identity for the prin... |
| dchrinvcl 24978 | Closure of the group inver... |
| dchrabl 24979 | The set of Dirichlet chara... |
| dchrfi 24980 | The group of Dirichlet cha... |
| dchrghm 24981 | A Dirichlet character rest... |
| dchr1 24982 | Value of the principal Dir... |
| dchreq 24983 | A Dirichlet character is d... |
| dchrresb 24984 | A Dirichlet character is d... |
| dchrabs 24985 | A Dirichlet character take... |
| dchrinv 24986 | The inverse of a Dirichlet... |
| dchrabs2 24987 | A Dirichlet character take... |
| dchr1re 24988 | The principal Dirichlet ch... |
| dchrptlem1 24989 | Lemma for ~ dchrpt . (Con... |
| dchrptlem2 24990 | Lemma for ~ dchrpt . (Con... |
| dchrptlem3 24991 | Lemma for ~ dchrpt . (Con... |
| dchrpt 24992 | For any element other than... |
| dchrsum2 24993 | An orthogonality relation ... |
| dchrsum 24994 | An orthogonality relation ... |
| sumdchr2 24995 | Lemma for ~ sumdchr . (Co... |
| dchrhash 24996 | There are exactly ` phi ( ... |
| sumdchr 24997 | An orthogonality relation ... |
| dchr2sum 24998 | An orthogonality relation ... |
| sum2dchr 24999 | An orthogonality relation ... |
| bcctr 25000 | Value of the central binom... |
| pcbcctr 25001 | Prime count of a central b... |
| bcmono 25002 | The binomial coefficient i... |
| bcmax 25003 | The binomial coefficient t... |
| bcp1ctr 25004 | Ratio of two central binom... |
| bclbnd 25005 | A bound on the binomial co... |
| efexple 25006 | Convert a bound on a power... |
| bpos1lem 25007 | Lemma for ~ bpos1 . (Cont... |
| bpos1 25008 | Bertrand's postulate, chec... |
| bposlem1 25009 | An upper bound on the prim... |
| bposlem2 25010 | There are no odd primes in... |
| bposlem3 25011 | Lemma for ~ bpos . Since ... |
| bposlem4 25012 | Lemma for ~ bpos . (Contr... |
| bposlem5 25013 | Lemma for ~ bpos . Bound ... |
| bposlem6 25014 | Lemma for ~ bpos . By usi... |
| bposlem7 25015 | Lemma for ~ bpos . The fu... |
| bposlem8 25016 | Lemma for ~ bpos . Evalua... |
| bposlem9 25017 | Lemma for ~ bpos . Derive... |
| bpos 25018 | Bertrand's postulate: ther... |
| zabsle1 25021 | ` { -u 1 , 0 , 1 } ` is th... |
| lgslem1 25022 | When ` a ` is coprime to t... |
| lgslem2 25023 | The set ` Z ` of all integ... |
| lgslem3 25024 | The set ` Z ` of all integ... |
| lgslem4 25025 | The function ` F ` is clos... |
| lgsval 25026 | Value of the Legendre symb... |
| lgsfval 25027 | Value of the function ` F ... |
| lgsfcl2 25028 | The function ` F ` is clos... |
| lgscllem 25029 | The Legendre symbol is an ... |
| lgsfcl 25030 | Closure of the function ` ... |
| lgsfle1 25031 | The function ` F ` has mag... |
| lgsval2lem 25032 | Lemma for ~ lgsval2 . (Co... |
| lgsval4lem 25033 | Lemma for ~ lgsval4 . (Co... |
| lgscl2 25034 | The Legendre symbol is an ... |
| lgs0 25035 | The Legendre symbol when t... |
| lgscl 25036 | The Legendre symbol is an ... |
| lgsle1 25037 | The Legendre symbol has ab... |
| lgsval2 25038 | The Legendre symbol at a p... |
| lgs2 25039 | The Legendre symbol at ` 2... |
| lgsval3 25040 | The Legendre symbol at an ... |
| lgsvalmod 25041 | The Legendre symbol is equ... |
| lgsval4 25042 | Restate ~ lgsval for nonze... |
| lgsfcl3 25043 | Closure of the function ` ... |
| lgsval4a 25044 | Same as ~ lgsval4 for posi... |
| lgscl1 25045 | The value of the Legendre ... |
| lgsneg 25046 | The Legendre symbol is eit... |
| lgsneg1 25047 | The Legendre symbol for no... |
| lgsmod 25048 | The Legendre (Jacobi) symb... |
| lgsdilem 25049 | Lemma for ~ lgsdi and ~ lg... |
| lgsdir2lem1 25050 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem2 25051 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem3 25052 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem4 25053 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem5 25054 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2 25055 | The Legendre symbol is com... |
| lgsdirprm 25056 | The Legendre symbol is com... |
| lgsdir 25057 | The Legendre symbol is com... |
| lgsdilem2 25058 | Lemma for ~ lgsdi . (Cont... |
| lgsdi 25059 | The Legendre symbol is com... |
| lgsne0 25060 | The Legendre symbol is non... |
| lgsabs1 25061 | The Legendre symbol is non... |
| lgssq 25062 | The Legendre symbol at a s... |
| lgssq2 25063 | The Legendre symbol at a s... |
| lgsprme0 25064 | The Legendre symbol at any... |
| 1lgs 25065 | The Legendre symbol at ` 1... |
| lgs1 25066 | The Legendre symbol at ` 1... |
| lgsmodeq 25067 | The Legendre (Jacobi) symb... |
| lgsmulsqcoprm 25068 | The Legendre (Jacobi) symb... |
| lgsdirnn0 25069 | Variation on ~ lgsdir vali... |
| lgsdinn0 25070 | Variation on ~ lgsdi valid... |
| lgsqrlem1 25071 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem2 25072 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem3 25073 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem4 25074 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem5 25075 | Lemma for ~ lgsqr . (Cont... |
| lgsqr 25076 | The Legendre symbol for od... |
| lgsqrmod 25077 | If the Legendre symbol of ... |
| lgsqrmodndvds 25078 | If the Legendre symbol of ... |
| lgsdchrval 25079 | The Legendre symbol functi... |
| lgsdchr 25080 | The Legendre symbol functi... |
| gausslemma2dlem0a 25081 | Auxiliary lemma 1 for ~ ga... |
| gausslemma2dlem0b 25082 | Auxiliary lemma 2 for ~ ga... |
| gausslemma2dlem0c 25083 | Auxiliary lemma 3 for ~ ga... |
| gausslemma2dlem0d 25084 | Auxiliary lemma 4 for ~ ga... |
| gausslemma2dlem0e 25085 | Auxiliary lemma 5 for ~ ga... |
| gausslemma2dlem0f 25086 | Auxiliary lemma 6 for ~ ga... |
| gausslemma2dlem0g 25087 | Auxiliary lemma 7 for ~ ga... |
| gausslemma2dlem0h 25088 | Auxiliary lemma 8 for ~ ga... |
| gausslemma2dlem0i 25089 | Auxiliary lemma 9 for ~ ga... |
| gausslemma2dlem1a 25090 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem1 25091 | Lemma 1 for ~ gausslemma2d... |
| gausslemma2dlem2 25092 | Lemma 2 for ~ gausslemma2d... |
| gausslemma2dlem3 25093 | Lemma 3 for ~ gausslemma2d... |
| gausslemma2dlem4 25094 | Lemma 4 for ~ gausslemma2d... |
| gausslemma2dlem5a 25095 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem5 25096 | Lemma 5 for ~ gausslemma2d... |
| gausslemma2dlem6 25097 | Lemma 6 for ~ gausslemma2d... |
| gausslemma2dlem7 25098 | Lemma 7 for ~ gausslemma2d... |
| gausslemma2d 25099 | Gauss' Lemma (see also the... |
| lgseisenlem1 25100 | Lemma for ~ lgseisen . If... |
| lgseisenlem2 25101 | Lemma for ~ lgseisen . Th... |
| lgseisenlem3 25102 | Lemma for ~ lgseisen . (C... |
| lgseisenlem4 25103 | Lemma for ~ lgseisen . Th... |
| lgseisen 25104 | Eisenstein's lemma, an exp... |
| lgsquadlem1 25105 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem2 25106 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem3 25107 | Lemma for ~ lgsquad . (Co... |
| lgsquad 25108 | The Law of Quadratic Recip... |
| lgsquad2lem1 25109 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2lem2 25110 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2 25111 | Extend ~ lgsquad to coprim... |
| lgsquad3 25112 | Extend ~ lgsquad2 to integ... |
| m1lgs 25113 | The first supplement to th... |
| 2lgslem1a1 25114 | Lemma 1 for ~ 2lgslem1a . ... |
| 2lgslem1a2 25115 | Lemma 2 for ~ 2lgslem1a . ... |
| 2lgslem1a 25116 | Lemma 1 for ~ 2lgslem1 . ... |
| 2lgslem1b 25117 | Lemma 2 for ~ 2lgslem1 . ... |
| 2lgslem1c 25118 | Lemma 3 for ~ 2lgslem1 . ... |
| 2lgslem1 25119 | Lemma 1 for ~ 2lgs . (Con... |
| 2lgslem2 25120 | Lemma 2 for ~ 2lgs . (Con... |
| 2lgslem3a 25121 | Lemma for ~ 2lgslem3a1 . ... |
| 2lgslem3b 25122 | Lemma for ~ 2lgslem3b1 . ... |
| 2lgslem3c 25123 | Lemma for ~ 2lgslem3c1 . ... |
| 2lgslem3d 25124 | Lemma for ~ 2lgslem3d1 . ... |
| 2lgslem3a1 25125 | Lemma 1 for ~ 2lgslem3 . ... |
| 2lgslem3b1 25126 | Lemma 2 for ~ 2lgslem3 . ... |
| 2lgslem3c1 25127 | Lemma 3 for ~ 2lgslem3 . ... |
| 2lgslem3d1 25128 | Lemma 4 for ~ 2lgslem3 . ... |
| 2lgslem3 25129 | Lemma 3 for ~ 2lgs . (Con... |
| 2lgs2 25130 | The Legendre symbol for ` ... |
| 2lgslem4 25131 | Lemma 4 for ~ 2lgs : speci... |
| 2lgs 25132 | The second supplement to t... |
| 2lgsoddprmlem1 25133 | Lemma 1 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem2 25134 | Lemma 2 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem3a 25135 | Lemma 1 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3b 25136 | Lemma 2 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3c 25137 | Lemma 3 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3d 25138 | Lemma 4 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3 25139 | Lemma 3 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem4 25140 | Lemma 4 for ~ 2lgsoddprm .... |
| 2lgsoddprm 25141 | The second supplement to t... |
| 2sqlem1 25142 | Lemma for ~ 2sq . (Contri... |
| 2sqlem2 25143 | Lemma for ~ 2sq . (Contri... |
| mul2sq 25144 | Fibonacci's identity (actu... |
| 2sqlem3 25145 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem4 25146 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem5 25147 | Lemma for ~ 2sq . If a nu... |
| 2sqlem6 25148 | Lemma for ~ 2sq . If a nu... |
| 2sqlem7 25149 | Lemma for ~ 2sq . (Contri... |
| 2sqlem8a 25150 | Lemma for ~ 2sqlem8 . (Co... |
| 2sqlem8 25151 | Lemma for ~ 2sq . (Contri... |
| 2sqlem9 25152 | Lemma for ~ 2sq . (Contri... |
| 2sqlem10 25153 | Lemma for ~ 2sq . Every f... |
| 2sqlem11 25154 | Lemma for ~ 2sq . (Contri... |
| 2sq 25155 | All primes of the form ` 4... |
| 2sqblem 25156 | The converse to ~ 2sq . (... |
| 2sqb 25157 | The converse to ~ 2sq . (... |
| chebbnd1lem1 25158 | Lemma for ~ chebbnd1 : sho... |
| chebbnd1lem2 25159 | Lemma for ~ chebbnd1 : Sh... |
| chebbnd1lem3 25160 | Lemma for ~ chebbnd1 : get... |
| chebbnd1 25161 | The Chebyshev bound: The ... |
| chtppilimlem1 25162 | Lemma for ~ chtppilim . (... |
| chtppilimlem2 25163 | Lemma for ~ chtppilim . (... |
| chtppilim 25164 | The ` theta ` function is ... |
| chto1ub 25165 | The ` theta ` function is ... |
| chebbnd2 25166 | The Chebyshev bound, part ... |
| chto1lb 25167 | The ` theta ` function is ... |
| chpchtlim 25168 | The ` psi ` and ` theta ` ... |
| chpo1ub 25169 | The ` psi ` function is up... |
| chpo1ubb 25170 | The ` psi ` function is up... |
| vmadivsum 25171 | The sum of the von Mangold... |
| vmadivsumb 25172 | Give a total bound on the ... |
| rplogsumlem1 25173 | Lemma for ~ rplogsum . (C... |
| rplogsumlem2 25174 | Lemma for ~ rplogsum . Eq... |
| dchrisum0lem1a 25175 | Lemma for ~ dchrisum0lem1 ... |
| rpvmasumlem 25176 | Lemma for ~ rpvmasum . Ca... |
| dchrisumlema 25177 | Lemma for ~ dchrisum . Le... |
| dchrisumlem1 25178 | Lemma for ~ dchrisum . Le... |
| dchrisumlem2 25179 | Lemma for ~ dchrisum . Le... |
| dchrisumlem3 25180 | Lemma for ~ dchrisum . Le... |
| dchrisum 25181 | If ` n e. [ M , +oo ) |-> ... |
| dchrmusumlema 25182 | Lemma for ~ dchrmusum and ... |
| dchrmusum2 25183 | The sum of the Möbius... |
| dchrvmasumlem1 25184 | An alternative expression ... |
| dchrvmasum2lem 25185 | Give an expression for ` l... |
| dchrvmasum2if 25186 | Combine the results of ~ d... |
| dchrvmasumlem2 25187 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlem3 25188 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlema 25189 | Lemma for ~ dchrvmasum and... |
| dchrvmasumiflem1 25190 | Lemma for ~ dchrvmasumif .... |
| dchrvmasumiflem2 25191 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumif 25192 | An asymptotic approximatio... |
| dchrvmaeq0 25193 | The set ` W ` is the colle... |
| dchrisum0fval 25194 | Value of the function ` F ... |
| dchrisum0fmul 25195 | The function ` F ` , the d... |
| dchrisum0ff 25196 | The function ` F ` is a re... |
| dchrisum0flblem1 25197 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flblem2 25198 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flb 25199 | The divisor sum of a real ... |
| dchrisum0fno1 25200 | The sum ` sum_ k <_ x , F ... |
| rpvmasum2 25201 | A partial result along the... |
| dchrisum0re 25202 | Suppose ` X ` is a non-pri... |
| dchrisum0lema 25203 | Lemma for ~ dchrisum0 . A... |
| dchrisum0lem1b 25204 | Lemma for ~ dchrisum0lem1 ... |
| dchrisum0lem1 25205 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2a 25206 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2 25207 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem3 25208 | Lemma for ~ dchrisum0 . (... |
| dchrisum0 25209 | The sum ` sum_ n e. NN , X... |
| dchrisumn0 25210 | The sum ` sum_ n e. NN , X... |
| dchrmusumlem 25211 | The sum of the Möbius... |
| dchrvmasumlem 25212 | The sum of the Möbius... |
| dchrmusum 25213 | The sum of the Möbius... |
| dchrvmasum 25214 | The sum of the von Mangold... |
| rpvmasum 25215 | The sum of the von Mangold... |
| rplogsum 25216 | The sum of ` log p / p ` o... |
| dirith2 25217 | Dirichlet's theorem: there... |
| dirith 25218 | Dirichlet's theorem: there... |
| mudivsum 25219 | Asymptotic formula for ` s... |
| mulogsumlem 25220 | Lemma for ~ mulogsum . (C... |
| mulogsum 25221 | Asymptotic formula for ... |
| logdivsum 25222 | Asymptotic analysis of ... |
| mulog2sumlem1 25223 | Asymptotic formula for ... |
| mulog2sumlem2 25224 | Lemma for ~ mulog2sum . (... |
| mulog2sumlem3 25225 | Lemma for ~ mulog2sum . (... |
| mulog2sum 25226 | Asymptotic formula for ... |
| vmalogdivsum2 25227 | The sum ` sum_ n <_ x , La... |
| vmalogdivsum 25228 | The sum ` sum_ n <_ x , La... |
| 2vmadivsumlem 25229 | Lemma for ~ 2vmadivsum . ... |
| 2vmadivsum 25230 | The sum ` sum_ m n <_ x , ... |
| logsqvma 25231 | A formula for ` log ^ 2 ( ... |
| logsqvma2 25232 | The Möbius inverse of... |
| log2sumbnd 25233 | Bound on the difference be... |
| selberglem1 25234 | Lemma for ~ selberg . Est... |
| selberglem2 25235 | Lemma for ~ selberg . (Co... |
| selberglem3 25236 | Lemma for ~ selberg . Est... |
| selberg 25237 | Selberg's symmetry formula... |
| selbergb 25238 | Convert eventual boundedne... |
| selberg2lem 25239 | Lemma for ~ selberg2 . Eq... |
| selberg2 25240 | Selberg's symmetry formula... |
| selberg2b 25241 | Convert eventual boundedne... |
| chpdifbndlem1 25242 | Lemma for ~ chpdifbnd . (... |
| chpdifbndlem2 25243 | Lemma for ~ chpdifbnd . (... |
| chpdifbnd 25244 | A bound on the difference ... |
| logdivbnd 25245 | A bound on a sum of logs, ... |
| selberg3lem1 25246 | Introduce a log weighting ... |
| selberg3lem2 25247 | Lemma for ~ selberg3 . Eq... |
| selberg3 25248 | Introduce a log weighting ... |
| selberg4lem1 25249 | Lemma for ~ selberg4 . Eq... |
| selberg4 25250 | The Selberg symmetry formu... |
| pntrval 25251 | Define the residual of the... |
| pntrf 25252 | Functionality of the resid... |
| pntrmax 25253 | There is a bound on the re... |
| pntrsumo1 25254 | A bound on a sum over ` R ... |
| pntrsumbnd 25255 | A bound on a sum over ` R ... |
| pntrsumbnd2 25256 | A bound on a sum over ` R ... |
| selbergr 25257 | Selberg's symmetry formula... |
| selberg3r 25258 | Selberg's symmetry formula... |
| selberg4r 25259 | Selberg's symmetry formula... |
| selberg34r 25260 | The sum of ~ selberg3r and... |
| pntsval 25261 | Define the "Selberg functi... |
| pntsf 25262 | Functionality of the Selbe... |
| selbergs 25263 | Selberg's symmetry formula... |
| selbergsb 25264 | Selberg's symmetry formula... |
| pntsval2 25265 | The Selberg function can b... |
| pntrlog2bndlem1 25266 | The sum of ~ selberg3r and... |
| pntrlog2bndlem2 25267 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem3 25268 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem4 25269 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem5 25270 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem6a 25271 | Lemma for ~ pntrlog2bndlem... |
| pntrlog2bndlem6 25272 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bnd 25273 | A bound on ` R ( x ) log ^... |
| pntpbnd1a 25274 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd1 25275 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd2 25276 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd 25277 | Lemma for ~ pnt . Establi... |
| pntibndlem1 25278 | Lemma for ~ pntibnd . (Co... |
| pntibndlem2a 25279 | Lemma for ~ pntibndlem2 . ... |
| pntibndlem2 25280 | Lemma for ~ pntibnd . The... |
| pntibndlem3 25281 | Lemma for ~ pntibnd . Pac... |
| pntibnd 25282 | Lemma for ~ pnt . Establi... |
| pntlemd 25283 | Lemma for ~ pnt . Closure... |
| pntlemc 25284 | Lemma for ~ pnt . Closure... |
| pntlema 25285 | Lemma for ~ pnt . Closure... |
| pntlemb 25286 | Lemma for ~ pnt . Unpack ... |
| pntlemg 25287 | Lemma for ~ pnt . Closure... |
| pntlemh 25288 | Lemma for ~ pnt . Bounds ... |
| pntlemn 25289 | Lemma for ~ pnt . The "na... |
| pntlemq 25290 | Lemma for ~ pntlemj . (Co... |
| pntlemr 25291 | Lemma for ~ pntlemj . (Co... |
| pntlemj 25292 | Lemma for ~ pnt . The ind... |
| pntlemi 25293 | Lemma for ~ pnt . Elimina... |
| pntlemf 25294 | Lemma for ~ pnt . Add up ... |
| pntlemk 25295 | Lemma for ~ pnt . Evaluat... |
| pntlemo 25296 | Lemma for ~ pnt . Combine... |
| pntleme 25297 | Lemma for ~ pnt . Package... |
| pntlem3 25298 | Lemma for ~ pnt . Equatio... |
| pntlemp 25299 | Lemma for ~ pnt . Wrappin... |
| pntleml 25300 | Lemma for ~ pnt . Equatio... |
| pnt3 25301 | The Prime Number Theorem, ... |
| pnt2 25302 | The Prime Number Theorem, ... |
| pnt 25303 | The Prime Number Theorem: ... |
| abvcxp 25304 | Raising an absolute value ... |
| padicfval 25305 | Value of the p-adic absolu... |
| padicval 25306 | Value of the p-adic absolu... |
| ostth2lem1 25307 | Lemma for ~ ostth2 , altho... |
| qrngbas 25308 | The base set of the field ... |
| qdrng 25309 | The rationals form a divis... |
| qrng0 25310 | The zero element of the fi... |
| qrng1 25311 | The unit element of the fi... |
| qrngneg 25312 | The additive inverse in th... |
| qrngdiv 25313 | The division operation in ... |
| qabvle 25314 | By using induction on ` N ... |
| qabvexp 25315 | Induct the product rule ~ ... |
| ostthlem1 25316 | Lemma for ~ ostth . If tw... |
| ostthlem2 25317 | Lemma for ~ ostth . Refin... |
| qabsabv 25318 | The regular absolute value... |
| padicabv 25319 | The p-adic absolute value ... |
| padicabvf 25320 | The p-adic absolute value ... |
| padicabvcxp 25321 | All positive powers of the... |
| ostth1 25322 | - Lemma for ~ ostth : triv... |
| ostth2lem2 25323 | Lemma for ~ ostth2 . (Con... |
| ostth2lem3 25324 | Lemma for ~ ostth2 . (Con... |
| ostth2lem4 25325 | Lemma for ~ ostth2 . (Con... |
| ostth2 25326 | - Lemma for ~ ostth : regu... |
| ostth3 25327 | - Lemma for ~ ostth : p-ad... |
| ostth 25328 | Ostrowski's theorem, which... |
| itvndx 25339 | Index value of the Interva... |
| lngndx 25340 | Index value of the "line" ... |
| itvid 25341 | Utility theorem: index-ind... |
| lngid 25342 | Utility theorem: index-ind... |
| trkgstr 25343 | Functionality of a Tarski ... |
| trkgbas 25344 | The base set of a Tarski g... |
| trkgdist 25345 | The measure of a distance ... |
| trkgitv 25346 | The congruence relation in... |
| istrkgc 25353 | Property of being a Tarski... |
| istrkgb 25354 | Property of being a Tarski... |
| istrkgcb 25355 | Property of being a Tarski... |
| istrkge 25356 | Property of fulfilling Euc... |
| istrkgl 25357 | Building lines from the se... |
| istrkgld 25358 | Property of fulfilling the... |
| istrkg2ld 25359 | Property of fulfilling the... |
| istrkg3ld 25360 | Property of fulfilling the... |
| axtgcgrrflx 25361 | Axiom of reflexivity of co... |
| axtgcgrid 25362 | Axiom of identity of congr... |
| axtgsegcon 25363 | Axiom of segment construct... |
| axtg5seg 25364 | Five segments axiom, Axiom... |
| axtgbtwnid 25365 | Identity of Betweenness. ... |
| axtgpasch 25366 | Axiom of (Inner) Pasch, Ax... |
| axtgcont1 25367 | Axiom of Continuity. Axio... |
| axtgcont 25368 | Axiom of Continuity. Axio... |
| axtglowdim2 25369 | Lower dimension axiom for ... |
| axtgupdim2 25370 | Upper dimension axiom for ... |
| axtgeucl 25371 | Euclid's Axiom. Axiom A10... |
| tgcgrcomimp 25372 | Congruence commutes on the... |
| tgcgrcomr 25373 | Congruence commutes on the... |
| tgcgrcoml 25374 | Congruence commutes on the... |
| tgcgrcomlr 25375 | Congruence commutes on bot... |
| tgcgreqb 25376 | Congruence and equality. ... |
| tgcgreq 25377 | Congruence and equality. ... |
| tgcgrneq 25378 | Congruence and equality. ... |
| tgcgrtriv 25379 | Degenerate segments are co... |
| tgcgrextend 25380 | Link congruence over a pai... |
| tgsegconeq 25381 | Two points that satisfy th... |
| tgbtwntriv2 25382 | Betweenness always holds f... |
| tgbtwncom 25383 | Betweenness commutes. The... |
| tgbtwncomb 25384 | Betweenness commutes, bico... |
| tgbtwnne 25385 | Betweenness and inequality... |
| tgbtwntriv1 25386 | Betweenness always holds f... |
| tgbtwnswapid 25387 | If you can swap the first ... |
| tgbtwnintr 25388 | Inner transitivity law for... |
| tgbtwnexch3 25389 | Exchange the first endpoin... |
| tgbtwnouttr2 25390 | Outer transitivity law for... |
| tgbtwnexch2 25391 | Exchange the outer point o... |
| tgbtwnouttr 25392 | Outer transitivity law for... |
| tgbtwnexch 25393 | Outer transitivity law for... |
| tgtrisegint 25394 | A line segment between two... |
| tglowdim1 25395 | Lower dimension axiom for ... |
| tglowdim1i 25396 | Lower dimension axiom for ... |
| tgldimor 25397 | Excluded-middle like state... |
| tgldim0eq 25398 | In dimension zero, any two... |
| tgldim0itv 25399 | In dimension zero, any two... |
| tgldim0cgr 25400 | In dimension zero, any two... |
| tgbtwndiff 25401 | There is always a ` c ` di... |
| tgdim01 25402 | In geometries of dimension... |
| tgifscgr 25403 | Inner five segment congrue... |
| tgcgrsub 25404 | Removing identical parts f... |
| iscgrg 25407 | The congruence property fo... |
| iscgrgd 25408 | The property for two seque... |
| iscgrglt 25409 | The property for two seque... |
| trgcgrg 25410 | The property for two trian... |
| trgcgr 25411 | Triangle congruence. (Con... |
| ercgrg 25412 | The shape congruence relat... |
| tgcgrxfr 25413 | A line segment can be divi... |
| cgr3id 25414 | Reflexivity law for three-... |
| cgr3simp1 25415 | Deduce segment congruence ... |
| cgr3simp2 25416 | Deduce segment congruence ... |
| cgr3simp3 25417 | Deduce segment congruence ... |
| cgr3swap12 25418 | Permutation law for three-... |
| cgr3swap23 25419 | Permutation law for three-... |
| cgr3swap13 25420 | Permutation law for three-... |
| cgr3rotr 25421 | Permutation law for three-... |
| cgr3rotl 25422 | Permutation law for three-... |
| trgcgrcom 25423 | Commutative law for three-... |
| cgr3tr 25424 | Transitivity law for three... |
| tgbtwnxfr 25425 | A condition for extending ... |
| tgcgr4 25426 | Two quadrilaterals to be c... |
| isismt 25429 | Property of being an isome... |
| ismot 25430 | Property of being an isome... |
| motcgr 25431 | Property of a motion: dist... |
| idmot 25432 | The identity is a motion. ... |
| motf1o 25433 | Motions are bijections. (... |
| motcl 25434 | Closure of motions. (Cont... |
| motco 25435 | The composition of two mot... |
| cnvmot 25436 | The converse of a motion i... |
| motplusg 25437 | The operation for motions ... |
| motgrp 25438 | The motions of a geometry ... |
| motcgrg 25439 | Property of a motion: dist... |
| motcgr3 25440 | Property of a motion: dist... |
| tglng 25441 | Lines of a Tarski Geometry... |
| tglnfn 25442 | Lines as functions. (Cont... |
| tglnunirn 25443 | Lines are sets of points. ... |
| tglnpt 25444 | Lines are sets of points. ... |
| tglngne 25445 | It takes two different poi... |
| tglngval 25446 | The line going through poi... |
| tglnssp 25447 | Lines are subset of the ge... |
| tgellng 25448 | Property of lying on the l... |
| tgcolg 25449 | We choose the notation ` (... |
| btwncolg1 25450 | Betweenness implies coline... |
| btwncolg2 25451 | Betweenness implies coline... |
| btwncolg3 25452 | Betweenness implies coline... |
| colcom 25453 | Swapping the points defini... |
| colrot1 25454 | Rotating the points defini... |
| colrot2 25455 | Rotating the points defini... |
| ncolcom 25456 | Swapping non-colinear poin... |
| ncolrot1 25457 | Rotating non-colinear poin... |
| ncolrot2 25458 | Rotating non-colinear poin... |
| tgdim01ln 25459 | In geometries of dimension... |
| ncoltgdim2 25460 | If there are 3 non-colinea... |
| lnxfr 25461 | Transfer law for colineari... |
| lnext 25462 | Extend a line with a missi... |
| tgfscgr 25463 | Congruence law for the gen... |
| lncgr 25464 | Congruence rule for lines.... |
| lnid 25465 | Identity law for points on... |
| tgidinside 25466 | Law for finding a point in... |
| tgbtwnconn1lem1 25467 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem2 25468 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem3 25469 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1 25470 | Connectivity law for betwe... |
| tgbtwnconn2 25471 | Another connectivity law f... |
| tgbtwnconn3 25472 | Inner connectivity law for... |
| tgbtwnconnln3 25473 | Derive colinearity from be... |
| tgbtwnconn22 25474 | Double connectivity law fo... |
| tgbtwnconnln1 25475 | Derive colinearity from be... |
| tgbtwnconnln2 25476 | Derive colinearity from be... |
| legval 25479 | Value of the less-than rel... |
| legov 25480 | Value of the less-than rel... |
| legov2 25481 | An equivalent definition o... |
| legid 25482 | Reflexivity of the less-th... |
| btwnleg 25483 | Betweenness implies less-t... |
| legtrd 25484 | Transitivity of the less-t... |
| legtri3 25485 | Equality from the less-tha... |
| legtrid 25486 | Trichotomy law for the les... |
| leg0 25487 | Degenerated (zero-length) ... |
| legeq 25488 | Deduce equality from "less... |
| legbtwn 25489 | Deduce betweenness from "l... |
| tgcgrsub2 25490 | Removing identical parts f... |
| ltgseg 25491 | The set ` E ` denotes the ... |
| ltgov 25492 | Strict "shorter than" geom... |
| legov3 25493 | An equivalent definition o... |
| legso 25494 | The shorter-than relations... |
| ishlg 25497 | Rays : Definition 6.1 of ... |
| hlcomb 25498 | The half-line relation com... |
| hlcomd 25499 | The half-line relation com... |
| hlne1 25500 | The half-line relation imp... |
| hlne2 25501 | The half-line relation imp... |
| hlln 25502 | The half-line relation imp... |
| hleqnid 25503 | The endpoint does not belo... |
| hlid 25504 | The half-line relation is ... |
| hltr 25505 | The half-line relation is ... |
| hlbtwn 25506 | Betweenness is a sufficien... |
| btwnhl1 25507 | Deduce half-line from betw... |
| btwnhl2 25508 | Deduce half-line from betw... |
| btwnhl 25509 | Swap betweenness for a hal... |
| lnhl 25510 | Either a point ` C ` on th... |
| hlcgrex 25511 | Construct a point on a hal... |
| hlcgreulem 25512 | Lemma for ~ hlcgreu . (Co... |
| hlcgreu 25513 | The point constructed in ~... |
| btwnlng1 25514 | Betweenness implies coline... |
| btwnlng2 25515 | Betweenness implies coline... |
| btwnlng3 25516 | Betweenness implies coline... |
| lncom 25517 | Swapping the points defini... |
| lnrot1 25518 | Rotating the points defini... |
| lnrot2 25519 | Rotating the points defini... |
| ncolne1 25520 | Non-colinear points are di... |
| ncolne2 25521 | Non-colinear points are di... |
| tgisline 25522 | The property of being a pr... |
| tglnne 25523 | It takes two different poi... |
| tglndim0 25524 | There are no lines in dime... |
| tgelrnln 25525 | The property of being a pr... |
| tglineeltr 25526 | Transitivity law for lines... |
| tglineelsb2 25527 | If ` S ` lies on PQ , then... |
| tglinerflx1 25528 | Reflexivity law for line m... |
| tglinerflx2 25529 | Reflexivity law for line m... |
| tglinecom 25530 | Commutativity law for line... |
| tglinethru 25531 | If ` A ` is a line contain... |
| tghilberti1 25532 | There is a line through an... |
| tghilberti2 25533 | There is at most one line ... |
| tglinethrueu 25534 | There is a unique line goi... |
| tglnne0 25535 | A line ` A ` has at least ... |
| tglnpt2 25536 | Find a second point on a l... |
| tglineintmo 25537 | Two distinct lines interse... |
| tglineineq 25538 | Two distinct lines interse... |
| tglineneq 25539 | Given three non-colinear p... |
| tglineinteq 25540 | Two distinct lines interse... |
| ncolncol 25541 | Deduce non-colinearity fro... |
| coltr 25542 | A transitivity law for col... |
| coltr3 25543 | A transitivity law for col... |
| colline 25544 | Three points are colinear ... |
| tglowdim2l 25545 | Reformulation of the lower... |
| tglowdim2ln 25546 | There is always one point ... |
| mirreu3 25549 | Existential uniqueness of ... |
| mirval 25550 | Value of the point inversi... |
| mirfv 25551 | Value of the point inversi... |
| mircgr 25552 | Property of the image by t... |
| mirbtwn 25553 | Property of the image by t... |
| ismir 25554 | Property of the image by t... |
| mirf 25555 | Point inversion as functio... |
| mircl 25556 | Closure of the point inver... |
| mirmir 25557 | The point inversion functi... |
| mircom 25558 | Variation on ~ mirmir . (... |
| mirreu 25559 | Any point has a unique ant... |
| mireq 25560 | Equality deduction for poi... |
| mirinv 25561 | The only invariant point o... |
| mirne 25562 | Mirror of non-center point... |
| mircinv 25563 | The center point is invari... |
| mirf1o 25564 | The point inversion functi... |
| miriso 25565 | The point inversion functi... |
| mirbtwni 25566 | Point inversion preserves ... |
| mirbtwnb 25567 | Point inversion preserves ... |
| mircgrs 25568 | Point inversion preserves ... |
| mirmir2 25569 | Point inversion of a point... |
| mirmot 25570 | Point investion is a motio... |
| mirln 25571 | If two points are on the s... |
| mirln2 25572 | If a point and its mirror ... |
| mirconn 25573 | Point inversion of connect... |
| mirhl 25574 | If two points ` X ` and ` ... |
| mirbtwnhl 25575 | If the center of the point... |
| mirhl2 25576 | Deduce half-line relation ... |
| mircgrextend 25577 | Link congruence over a pai... |
| mirtrcgr 25578 | Point inversion of one poi... |
| mirauto 25579 | Point inversion preserves ... |
| miduniq 25580 | Unicity of the middle poin... |
| miduniq1 25581 | Unicity of the middle poin... |
| miduniq2 25582 | If two point inversions co... |
| colmid 25583 | Colinearity and equidistan... |
| symquadlem 25584 | Lemma of the symetrial qua... |
| krippenlem 25585 | Lemma for ~ krippen . We ... |
| krippen 25586 | Krippenlemma (German for c... |
| midexlem 25587 | Lemma for the existence of... |
| israg 25592 | Property for 3 points A, B... |
| ragcom 25593 | Commutative rule for right... |
| ragcol 25594 | The right angle property i... |
| ragmir 25595 | Right angle property is pr... |
| mirrag 25596 | Right angle is conserved b... |
| ragtrivb 25597 | Trivial right angle. Theo... |
| ragflat2 25598 | Deduce equality from two r... |
| ragflat 25599 | Deduce equality from two r... |
| ragtriva 25600 | Trivial right angle. Theo... |
| ragflat3 25601 | Right angle and colinearit... |
| ragcgr 25602 | Right angle and colinearit... |
| motrag 25603 | Right angles are preserved... |
| ragncol 25604 | Right angle implies non-co... |
| perpln1 25605 | Derive a line from perpend... |
| perpln2 25606 | Derive a line from perpend... |
| isperp 25607 | Property for 2 lines A, B ... |
| perpcom 25608 | The "perpendicular" relati... |
| perpneq 25609 | Two perpendicular lines ar... |
| isperp2 25610 | Property for 2 lines A, B,... |
| isperp2d 25611 | One direction of ~ isperp2... |
| ragperp 25612 | Deduce that two lines are ... |
| footex 25613 | Lemma for ~ foot : existen... |
| foot 25614 | From a point ` C ` outside... |
| footne 25615 | Uniqueness of the foot poi... |
| footeq 25616 | Uniqueness of the foot poi... |
| hlperpnel 25617 | A point on a half-line whi... |
| perprag 25618 | Deduce a right angle from ... |
| perpdragALT 25619 | Deduce a right angle from ... |
| perpdrag 25620 | Deduce a right angle from ... |
| colperp 25621 | Deduce a perpendicularity ... |
| colperpexlem1 25622 | Lemma for ~ colperp . Fir... |
| colperpexlem2 25623 | Lemma for ~ colperpex . S... |
| colperpexlem3 25624 | Lemma for ~ colperpex . C... |
| colperpex 25625 | In dimension 2 and above, ... |
| mideulem2 25626 | Lemma for ~ opphllem , whi... |
| opphllem 25627 | Lemma 8.24 of [Schwabhause... |
| mideulem 25628 | Lemma for ~ mideu . We ca... |
| midex 25629 | Existence of the midpoint,... |
| mideu 25630 | Existence and uniqueness o... |
| islnopp 25631 | The property for two point... |
| islnoppd 25632 | Deduce that ` A ` and ` B ... |
| oppne1 25633 | Points lying on opposite s... |
| oppne2 25634 | Points lying on opposite s... |
| oppne3 25635 | Points lying on opposite s... |
| oppcom 25636 | Commutativity rule for "op... |
| opptgdim2 25637 | If two points opposite to ... |
| oppnid 25638 | The "opposite to a line" r... |
| opphllem1 25639 | Lemma for ~ opphl . (Cont... |
| opphllem2 25640 | Lemma for ~ opphl . Lemma... |
| opphllem3 25641 | Lemma for ~ opphl : We as... |
| opphllem4 25642 | Lemma for ~ opphl . (Cont... |
| opphllem5 25643 | Second part of Lemma 9.4 o... |
| opphllem6 25644 | First part of Lemma 9.4 of... |
| oppperpex 25645 | Restating ~ colperpex usin... |
| opphl 25646 | If two points ` A ` and ` ... |
| outpasch 25647 | Axiom of Pasch, outer form... |
| hlpasch 25648 | An application of the axio... |
| ishpg 25651 | Value of the half-plane re... |
| hpgbr 25652 | Half-planes : property for... |
| hpgne1 25653 | Points on the open half pl... |
| hpgne2 25654 | Points on the open half pl... |
| lnopp2hpgb 25655 | Theorem 9.8 of [Schwabhaus... |
| lnoppnhpg 25656 | If two points lie on the o... |
| hpgerlem 25657 | Lemma for the proof that t... |
| hpgid 25658 | The half-plane relation is... |
| hpgcom 25659 | The half-plane relation co... |
| hpgtr 25660 | The half-plane relation is... |
| colopp 25661 | Opposite sides of a line f... |
| colhp 25662 | Half-plane relation for co... |
| hphl 25663 | If two points are on the s... |
| midf 25668 | Midpoint as a function. (... |
| midcl 25669 | Closure of the midpoint. ... |
| ismidb 25670 | Property of the midpoint. ... |
| midbtwn 25671 | Betweenness of midpoint. ... |
| midcgr 25672 | Congruence of midpoint. (... |
| midid 25673 | Midpoint of a null segment... |
| midcom 25674 | Commutativity rule for the... |
| mirmid 25675 | Point inversion preserves ... |
| lmieu 25676 | Uniqueness of the line mir... |
| lmif 25677 | Line mirror as a function.... |
| lmicl 25678 | Closure of the line mirror... |
| islmib 25679 | Property of the line mirro... |
| lmicom 25680 | The line mirroring functio... |
| lmilmi 25681 | Line mirroring is an invol... |
| lmireu 25682 | Any point has a unique ant... |
| lmieq 25683 | Equality deduction for lin... |
| lmiinv 25684 | The invariants of the line... |
| lmicinv 25685 | The mirroring line is an i... |
| lmimid 25686 | If we have a right angle, ... |
| lmif1o 25687 | The line mirroring functio... |
| lmiisolem 25688 | Lemma for ~ lmiiso . (Con... |
| lmiiso 25689 | The line mirroring functio... |
| lmimot 25690 | Line mirroring is a motion... |
| hypcgrlem1 25691 | Lemma for ~ hypcgr , case ... |
| hypcgrlem2 25692 | Lemma for ~ hypcgr , case ... |
| hypcgr 25693 | If the catheti of two righ... |
| lmiopp 25694 | Line mirroring produces po... |
| lnperpex 25695 | Existence of a perpendicul... |
| trgcopy 25696 | Triangle construction: a c... |
| trgcopyeulem 25697 | Lemma for ~ trgcopyeu . (... |
| trgcopyeu 25698 | Triangle construction: a c... |
| iscgra 25701 | Property for two angles AB... |
| iscgra1 25702 | A special version of ~ isc... |
| iscgrad 25703 | Sufficient conditions for ... |
| cgrane1 25704 | Angles imply inequality. ... |
| cgrane2 25705 | Angles imply inequality. ... |
| cgrane3 25706 | Angles imply inequality. ... |
| cgrane4 25707 | Angles imply inequality. ... |
| cgrahl1 25708 | Angle congruence is indepe... |
| cgrahl2 25709 | Angle congruence is indepe... |
| cgracgr 25710 | First direction of proposi... |
| cgraid 25711 | Angle congruence is reflex... |
| cgraswap 25712 | Swap rays in a congruence ... |
| cgrcgra 25713 | Triangle congruence implie... |
| cgracom 25714 | Angle congruence commutes.... |
| cgratr 25715 | Angle congruence is transi... |
| cgraswaplr 25716 | Swap both side of angle co... |
| cgrabtwn 25717 | Angle congruence preserves... |
| cgrahl 25718 | Angle congruence preserves... |
| cgracol 25719 | Angle congruence preserves... |
| cgrancol 25720 | Angle congruence preserves... |
| dfcgra2 25721 | This is the full statement... |
| sacgr 25722 | Supplementary angles of co... |
| oacgr 25723 | Vertical angle theorem. V... |
| acopy 25724 | Angle construction. Theor... |
| acopyeu 25725 | Angle construction. Theor... |
| isinag 25729 | Property for point ` X ` t... |
| inagswap 25730 | Swap the order of the half... |
| inaghl 25731 | The "point lie in angle" r... |
| isleag 25733 | Geometrical "less than" pr... |
| cgrg3col4 25734 | Lemma 11.28 of [Schwabhaus... |
| tgsas1 25735 | First congruence theorem: ... |
| tgsas 25736 | First congruence theorem: ... |
| tgsas2 25737 | First congruence theorem: ... |
| tgsas3 25738 | First congruence theorem: ... |
| tgasa1 25739 | Second congruence theorem:... |
| tgasa 25740 | Second congruence theorem:... |
| tgsss1 25741 | Third congruence theorem: ... |
| tgsss2 25742 | Third congruence theorem: ... |
| tgsss3 25743 | Third congruence theorem: ... |
| isoas 25744 | Congruence theorem for iso... |
| iseqlg 25747 | Property of a triangle bei... |
| iseqlgd 25748 | Condition for a triangle t... |
| f1otrgds 25749 | Convenient lemma for ~ f1o... |
| f1otrgitv 25750 | Convenient lemma for ~ f1o... |
| f1otrg 25751 | A bijection between bases ... |
| f1otrge 25752 | A bijection between bases ... |
| ttgval 25755 | Define a function to augme... |
| ttglem 25756 | Lemma for ~ ttgbas and ~ t... |
| ttgbas 25757 | The base set of a subcompl... |
| ttgplusg 25758 | The addition operation of ... |
| ttgsub 25759 | The subtraction operation ... |
| ttgvsca 25760 | The scalar product of a su... |
| ttgds 25761 | The metric of a subcomplex... |
| ttgitvval 25762 | Betweenness for a subcompl... |
| ttgelitv 25763 | Betweenness for a subcompl... |
| ttgbtwnid 25764 | Any subcomplex module equi... |
| ttgcontlem1 25765 | Lemma for % ttgcont . (Co... |
| xmstrkgc 25766 | Any metric space fulfills ... |
| cchhllem 25767 | Lemma for chlbas and chlvs... |
| elee 25774 | Membership in a Euclidean ... |
| mptelee 25775 | A condition for a mapping ... |
| eleenn 25776 | If ` A ` is in ` ( EE `` N... |
| eleei 25777 | The forward direction of ~... |
| eedimeq 25778 | A point belongs to at most... |
| brbtwn 25779 | The binary relation form o... |
| brcgr 25780 | The binary relation form o... |
| fveere 25781 | The function value of a po... |
| fveecn 25782 | The function value of a po... |
| eqeefv 25783 | Two points are equal iff t... |
| eqeelen 25784 | Two points are equal iff t... |
| brbtwn2 25785 | Alternate characterization... |
| colinearalglem1 25786 | Lemma for ~ colinearalg . ... |
| colinearalglem2 25787 | Lemma for ~ colinearalg . ... |
| colinearalglem3 25788 | Lemma for ~ colinearalg . ... |
| colinearalglem4 25789 | Lemma for ~ colinearalg . ... |
| colinearalg 25790 | An algebraic characterizat... |
| eleesub 25791 | Membership of a subtractio... |
| eleesubd 25792 | Membership of a subtractio... |
| axdimuniq 25793 | The unique dimension axiom... |
| axcgrrflx 25794 | ` A ` is as far from ` B `... |
| axcgrtr 25795 | Congruence is transitive. ... |
| axcgrid 25796 | If there is no distance be... |
| axsegconlem1 25797 | Lemma for ~ axsegcon . Ha... |
| axsegconlem2 25798 | Lemma for ~ axsegcon . Sh... |
| axsegconlem3 25799 | Lemma for ~ axsegcon . Sh... |
| axsegconlem4 25800 | Lemma for ~ axsegcon . Sh... |
| axsegconlem5 25801 | Lemma for ~ axsegcon . Sh... |
| axsegconlem6 25802 | Lemma for ~ axsegcon . Sh... |
| axsegconlem7 25803 | Lemma for ~ axsegcon . Sh... |
| axsegconlem8 25804 | Lemma for ~ axsegcon . Sh... |
| axsegconlem9 25805 | Lemma for ~ axsegcon . Sh... |
| axsegconlem10 25806 | Lemma for ~ axsegcon . Sh... |
| axsegcon 25807 | Any segment ` A B ` can be... |
| ax5seglem1 25808 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem2 25809 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem3a 25810 | Lemma for ~ ax5seg . (Con... |
| ax5seglem3 25811 | Lemma for ~ ax5seg . Comb... |
| ax5seglem4 25812 | Lemma for ~ ax5seg . Give... |
| ax5seglem5 25813 | Lemma for ~ ax5seg . If `... |
| ax5seglem6 25814 | Lemma for ~ ax5seg . Give... |
| ax5seglem7 25815 | Lemma for ~ ax5seg . An a... |
| ax5seglem8 25816 | Lemma for ~ ax5seg . Use ... |
| ax5seglem9 25817 | Lemma for ~ ax5seg . Take... |
| ax5seg 25818 | The five segment axiom. T... |
| axbtwnid 25819 | Points are indivisible. T... |
| axpaschlem 25820 | Lemma for ~ axpasch . Set... |
| axpasch 25821 | The inner Pasch axiom. Ta... |
| axlowdimlem1 25822 | Lemma for ~ axlowdim . Es... |
| axlowdimlem2 25823 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem3 25824 | Lemma for ~ axlowdim . Se... |
| axlowdimlem4 25825 | Lemma for ~ axlowdim . Se... |
| axlowdimlem5 25826 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem6 25827 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem7 25828 | Lemma for ~ axlowdim . Se... |
| axlowdimlem8 25829 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem9 25830 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem10 25831 | Lemma for ~ axlowdim . Se... |
| axlowdimlem11 25832 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem12 25833 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem13 25834 | Lemma for ~ axlowdim . Es... |
| axlowdimlem14 25835 | Lemma for ~ axlowdim . Ta... |
| axlowdimlem15 25836 | Lemma for ~ axlowdim . Se... |
| axlowdimlem16 25837 | Lemma for ~ axlowdim . Se... |
| axlowdimlem17 25838 | Lemma for ~ axlowdim . Es... |
| axlowdim1 25839 | The lower dimension axiom ... |
| axlowdim2 25840 | The lower two-dimensional ... |
| axlowdim 25841 | The general lower dimensio... |
| axeuclidlem 25842 | Lemma for ~ axeuclid . Ha... |
| axeuclid 25843 | Euclid's axiom. Take an a... |
| axcontlem1 25844 | Lemma for ~ axcont . Chan... |
| axcontlem2 25845 | Lemma for ~ axcont . The ... |
| axcontlem3 25846 | Lemma for ~ axcont . Give... |
| axcontlem4 25847 | Lemma for ~ axcont . Give... |
| axcontlem5 25848 | Lemma for ~ axcont . Comp... |
| axcontlem6 25849 | Lemma for ~ axcont . Stat... |
| axcontlem7 25850 | Lemma for ~ axcont . Give... |
| axcontlem8 25851 | Lemma for ~ axcont . A po... |
| axcontlem9 25852 | Lemma for ~ axcont . Give... |
| axcontlem10 25853 | Lemma for ~ axcont . Give... |
| axcontlem11 25854 | Lemma for ~ axcont . Elim... |
| axcontlem12 25855 | Lemma for ~ axcont . Elim... |
| axcont 25856 | The axiom of continuity. ... |
| eengv 25859 | The value of the Euclidean... |
| eengstr 25860 | The Euclidean geometry as ... |
| eengbas 25861 | The Base of the Euclidean ... |
| ebtwntg 25862 | The betweenness relation u... |
| ecgrtg 25863 | The congruence relation us... |
| elntg 25864 | The line definition in the... |
| eengtrkg 25865 | The geometry structure for... |
| eengtrkge 25866 | The geometry structure for... |
| edgfid 25869 | Utility theorem: index-ind... |
| edgfndxnn 25870 | The index value of the edg... |
| edgfndxid 25871 | The value of the edge func... |
| baseltedgf 25872 | The index value of the ` B... |
| slotsbaseefdif 25873 | The slots ` Base ` and ` .... |
| vtxval 25878 | The set of vertices of a g... |
| iedgval 25879 | The set of indexed edges o... |
| vtxvalOLD 25880 | Obsolete version of ~ vtxv... |
| iedgvalOLD 25881 | Obsolete version of ~ iedg... |
| 1vgrex 25882 | A graph with at least one ... |
| opvtxval 25883 | The set of vertices of a g... |
| opvtxfv 25884 | The set of vertices of a g... |
| opvtxov 25885 | The set of vertices of a g... |
| opiedgval 25886 | The set of indexed edges o... |
| opiedgfv 25887 | The set of indexed edges o... |
| opiedgov 25888 | The set of indexed edges o... |
| opvtxfvi 25889 | The set of vertices of a g... |
| opiedgfvi 25890 | The set of indexed edges o... |
| funvtxdmge2val 25891 | The set of vertices of an ... |
| funiedgdmge2val 25892 | The set of indexed edges o... |
| funvtxdm2val 25893 | The set of vertices of an ... |
| funiedgdm2val 25894 | The set of indexed edges o... |
| funvtxdm2valOLD 25895 | Obsolete version of ~ funv... |
| funiedgdm2valOLD 25896 | Obsolete version of ~ funi... |
| funvtxval0 25897 | The set of vertices of an ... |
| funvtxval0OLD 25898 | Obsolete version of ~ funv... |
| funvtxdmge2valOLD 25899 | Obsolete version of ~ funv... |
| funiedgdmge2valOLD 25900 | Obsolete version of ~ funi... |
| basvtxval 25901 | The set of vertices of a g... |
| edgfiedgval 25902 | The set of indexed edges o... |
| basvtxvalOLD 25903 | Obsolete version of ~ basv... |
| edgfiedgvalOLD 25904 | Obsolete version of ~ edgf... |
| funvtxval 25905 | The set of vertices of a g... |
| funiedgval 25906 | The set of indexed edges o... |
| funvtxvalOLD 25907 | Obsolete version of ~ funv... |
| funiedgvalOLD 25908 | Obsolete version of ~ funi... |
| structvtxvallem 25909 | Lemma for ~ structvtxval a... |
| structvtxval 25910 | The set of vertices of an ... |
| structiedg0val 25911 | The set of indexed edges o... |
| structgrssvtxlem 25912 | Lemma for ~ structgrssvtx ... |
| structgrssvtx 25913 | The set of vertices of a g... |
| structgrssiedg 25914 | The set of indexed edges o... |
| structgrssvtxlemOLD 25915 | Obsolete version of ~ stru... |
| structgrssvtxOLD 25916 | Obsolete version of ~ stru... |
| structgrssiedgOLD 25917 | Obsolete version of ~ stru... |
| struct2grstr 25918 | A graph represented as an ... |
| struct2grvtx 25919 | The set of vertices of a g... |
| struct2griedg 25920 | The set of indexed edges o... |
| graop 25921 | Any representation of a gr... |
| grastruct 25922 | Any representation of a gr... |
| gropd 25923 | If any representation of a... |
| grstructd 25924 | If any representation of a... |
| gropeld 25925 | If any representation of a... |
| grstructeld 25926 | If any representation of a... |
| setsvtx 25927 | The vertices of a structur... |
| setsiedg 25928 | The (indexed) edges of a s... |
| snstrvtxval 25929 | The set of vertices of a g... |
| snstriedgval 25930 | The set of indexed edges o... |
| vtxval0 25931 | Degenerated case 1 for ver... |
| iedgval0 25932 | Degenerated case 1 for edg... |
| vtxvalsnop 25933 | Degenerated case 2 for ver... |
| iedgvalsnop 25934 | Degenerated case 2 for edg... |
| vtxval3sn 25935 | Degenerated case 3 for ver... |
| iedgval3sn 25936 | Degenerated case 3 for edg... |
| vtxvalprc 25937 | Degenerated case 4 for ver... |
| iedgvalprc 25938 | Degenerated case 4 for edg... |
| edgval 25941 | The edges of a graph. (Co... |
| edgvalOLD 25942 | Obsolete version of ~ edgv... |
| iedgedg 25943 | An indexed edge is an edge... |
| edgopval 25944 | The edges of a graph repre... |
| edgov 25945 | The edges of a graph repre... |
| edgstruct 25946 | The edges of a graph repre... |
| edgiedgb 25947 | A set is an edge iff it is... |
| edgiedgbOLD 25948 | Obsolete version of ~ edgi... |
| edg0iedg0 25949 | There is no edge in a grap... |
| edg0iedg0OLD 25950 | Obsolete version of ~ edg0... |
| isuhgr 25955 | The predicate "is an undir... |
| isushgr 25956 | The predicate "is an undir... |
| uhgrf 25957 | The edge function of an un... |
| ushgrf 25958 | The edge function of an un... |
| uhgrss 25959 | An edge is a subset of ver... |
| uhgreq12g 25960 | If two sets have the same ... |
| uhgrfun 25961 | The edge function of an un... |
| uhgrn0 25962 | An edge is a nonempty subs... |
| lpvtx 25963 | The endpoints of a loop (w... |
| ushgruhgr 25964 | An undirected simple hyper... |
| isuhgrop 25965 | The property of being an u... |
| uhgr0e 25966 | The empty graph, with vert... |
| uhgr0vb 25967 | The null graph, with no ve... |
| uhgr0 25968 | The null graph represented... |
| uhgrun 25969 | The union ` U ` of two (un... |
| uhgrunop 25970 | The union of two (undirect... |
| ushgrun 25971 | The union ` U ` of two (un... |
| ushgrunop 25972 | The union of two (undirect... |
| uhgrstrrepe 25973 | Replacing (or adding) the ... |
| incistruhgr 25974 | An _incidence structure_ `... |
| isupgr 25979 | The property of being an u... |
| wrdupgr 25980 | The property of being an u... |
| upgrf 25981 | The edge function of an un... |
| upgrfn 25982 | The edge function of an un... |
| upgrss 25983 | An edge is a subset of ver... |
| upgrn0 25984 | An edge is a nonempty subs... |
| upgrle 25985 | An edge of an undirected p... |
| upgrfi 25986 | An edge is a finite subset... |
| upgrex 25987 | An edge is an unordered pa... |
| upgrbi 25988 | Show that an unordered pai... |
| upgrop 25989 | A pseudograph represented ... |
| isumgr 25990 | The property of being an u... |
| isumgrs 25991 | The simplified property of... |
| wrdumgr 25992 | The property of being an u... |
| umgrf 25993 | The edge function of an un... |
| umgrfn 25994 | The edge function of an un... |
| umgredg2 25995 | An edge of a multigraph ha... |
| umgrbi 25996 | Show that an unordered pai... |
| upgruhgr 25997 | An undirected pseudograph ... |
| umgrupgr 25998 | An undirected multigraph i... |
| umgruhgr 25999 | An undirected multigraph i... |
| upgrle2 26000 | An edge of an undirected p... |
| umgrnloopv 26001 | In a multigraph, there is ... |
| umgredgprv 26002 | In a multigraph, an edge i... |
| umgrnloop 26003 | In a multigraph, there is ... |
| umgrnloop0 26004 | A multigraph has no loops.... |
| umgr0e 26005 | The empty graph, with vert... |
| upgr0e 26006 | The empty graph, with vert... |
| upgr1elem 26007 | Lemma for ~ upgr1e and ~ u... |
| upgr1e 26008 | A pseudograph with one edg... |
| upgr0eop 26009 | The empty graph, with vert... |
| upgr1eop 26010 | A pseudograph with one edg... |
| upgr0eopALT 26011 | Alternate proof of ~ upgr0... |
| upgr1eopALT 26012 | Alternate proof of ~ upgr1... |
| upgrun 26013 | The union ` U ` of two pse... |
| upgrunop 26014 | The union of two pseudogra... |
| umgrun 26015 | The union ` U ` of two mul... |
| umgrunop 26016 | The union of two multigrap... |
| umgrislfupgrlem 26017 | Lemma for ~ umgrislfupgr a... |
| umgrislfupgr 26018 | A multigraph is a loop-fre... |
| lfgredgge2 26019 | An edge of a loop-free gra... |
| lfgrnloop 26020 | A loop-free graph has no l... |
| uhgredgiedgb 26021 | In a hypergraph, a set is ... |
| uhgriedg0edg0 26022 | A hypergraph has no edges ... |
| uhgredgn0 26023 | An edge of a hypergraph is... |
| edguhgr 26024 | An edge of a hypergraph is... |
| uhgredgrnv 26025 | An edge of a hypergraph co... |
| uhgredgss 26026 | The set of edges of a hype... |
| upgredgss 26027 | The set of edges of a pseu... |
| umgredgss 26028 | The set of edges of a mult... |
| edgupgr 26029 | Properties of an edge of a... |
| edgumgr 26030 | Properties of an edge of a... |
| uhgrvtxedgiedgb 26031 | In a hypergraph, a vertex ... |
| upgredg 26032 | For each edge in a pseudog... |
| umgredg 26033 | For each edge in a multigr... |
| upgrpredgv 26034 | An edge of a pseudograph a... |
| umgrpredgv 26035 | An edge of a multigraph al... |
| upgredg2vtx 26036 | For a vertex incident to a... |
| upgredgpr 26037 | If a proper pair (of verti... |
| edglnl 26038 | The edges incident with a ... |
| numedglnl 26039 | The number of edges incide... |
| umgredgne 26040 | An edge of a multigraph al... |
| umgrnloop2 26041 | A multigraph has no loops.... |
| umgredgnlp 26042 | An edge of a multigraph is... |
| isuspgr 26047 | The property of being a si... |
| isusgr 26048 | The property of being a si... |
| uspgrf 26049 | The edge function of a sim... |
| usgrf 26050 | The edge function of a sim... |
| isusgrs 26051 | The property of being a si... |
| usgrfs 26052 | The edge function of a sim... |
| usgrfun 26053 | The edge function of a sim... |
| usgredgss 26054 | The set of edges of a simp... |
| edgusgr 26055 | An edge of a simple graph ... |
| isuspgrop 26056 | The property of being an u... |
| isusgrop 26057 | The property of being an u... |
| usgrop 26058 | A simple graph represented... |
| isausgr 26059 | The property of an unorder... |
| ausgrusgrb 26060 | The equivalence of the def... |
| usgrausgri 26061 | A simple graph represented... |
| ausgrumgri 26062 | If an alternatively define... |
| ausgrusgri 26063 | The equivalence of the def... |
| usgrausgrb 26064 | The equivalence of the def... |
| usgredgop 26065 | An edge of a simple graph ... |
| usgrf1o 26066 | The edge function of a sim... |
| usgrf1 26067 | The edge function of a sim... |
| uspgrf1oedg 26068 | The edge function of a sim... |
| usgrss 26069 | An edge is a subset of ver... |
| uspgrushgr 26070 | A simple pseudograph is an... |
| uspgrupgr 26071 | A simple pseudograph is an... |
| uspgrupgrushgr 26072 | A graph is a simple pseudo... |
| usgruspgr 26073 | A simple graph is a simple... |
| usgrumgr 26074 | A simple graph is an undir... |
| usgrumgruspgr 26075 | A graph is a simple graph ... |
| usgruspgrb 26076 | A class is a simple graph ... |
| usgrupgr 26077 | A simple graph is an undir... |
| usgruhgr 26078 | A simple graph is an undir... |
| usgrislfuspgr 26079 | A simple graph is a loop-f... |
| uspgrun 26080 | The union ` U ` of two sim... |
| uspgrunop 26081 | The union of two simple ps... |
| usgrun 26082 | The union ` U ` of two sim... |
| usgrunop 26083 | The union of two simple gr... |
| usgredg2 26084 | The value of the "edge fun... |
| usgredg2ALT 26085 | Alternate proof of ~ usgre... |
| usgredgprv 26086 | In a simple graph, an edge... |
| usgredgprvALT 26087 | Alternate proof of ~ usgre... |
| usgredgppr 26088 | An edge of a simple graph ... |
| usgrpredgv 26089 | An edge of a simple graph ... |
| edgssv2 26090 | An edge of a simple graph ... |
| usgredg 26091 | For each edge in a simple ... |
| usgrnloopv 26092 | In a simple graph, there i... |
| usgrnloopvALT 26093 | Alternate proof of ~ usgrn... |
| usgrnloop 26094 | In a simple graph, there i... |
| usgrnloopALT 26095 | Alternate proof of ~ usgrn... |
| usgrnloop0 26096 | A simple graph has no loop... |
| usgrnloop0ALT 26097 | Alternate proof of ~ usgrn... |
| usgredgne 26098 | An edge of a simple graph ... |
| usgrf1oedg 26099 | The edge function of a sim... |
| uhgr2edg 26100 | If a vertex is adjacent to... |
| umgr2edg 26101 | If a vertex is adjacent to... |
| usgr2edg 26102 | If a vertex is adjacent to... |
| umgr2edg1 26103 | If a vertex is adjacent to... |
| usgr2edg1 26104 | If a vertex is adjacent to... |
| umgrvad2edg 26105 | If a vertex is adjacent to... |
| umgr2edgneu 26106 | If a vertex is adjacent to... |
| usgrsizedg 26107 | In a simple graph, the siz... |
| usgredg3 26108 | The value of the "edge fun... |
| usgredg4 26109 | For a vertex incident to a... |
| usgredgreu 26110 | For a vertex incident to a... |
| usgredg2vtx 26111 | For a vertex incident to a... |
| uspgredg2vtxeu 26112 | For a vertex incident to a... |
| usgredg2vtxeu 26113 | For a vertex incident to a... |
| usgredg2vtxeuALT 26114 | Alternate proof of ~ usgre... |
| uspgredg2vlem 26115 | Lemma for ~ uspgredg2v . ... |
| uspgredg2v 26116 | In a simple pseudograph, t... |
| usgredg2vlem1 26117 | Lemma 1 for ~ usgredg2v . ... |
| usgredg2vlem2 26118 | Lemma 2 for ~ usgredg2v . ... |
| usgredg2v 26119 | In a simple graph, the map... |
| usgriedgleord 26120 | Alternate version of ~ usg... |
| ushgredgedg 26121 | In a simple hypergraph the... |
| usgredgedg 26122 | In a simple graph there is... |
| ushgredgedgloop 26123 | In a simple hypergraph the... |
| uspgredgleord 26124 | In a simple pseudograph th... |
| usgredgleord 26125 | In a simple graph the numb... |
| usgredgleordALT 26126 | Alternate proof for ~ usgr... |
| usgrstrrepe 26127 | Replacing (or adding) the ... |
| usgr0e 26128 | The empty graph, with vert... |
| usgr0vb 26129 | The null graph, with no ve... |
| uhgr0v0e 26130 | The null graph, with no ve... |
| uhgr0vsize0 26131 | The size of a hypergraph w... |
| uhgr0edgfi 26132 | A graph of order 0 (i.e. w... |
| usgr0v 26133 | The null graph, with no ve... |
| uhgr0vusgr 26134 | The null graph, with no ve... |
| usgr0 26135 | The null graph represented... |
| uspgr1e 26136 | A simple pseudograph with ... |
| usgr1e 26137 | A simple graph with one ed... |
| usgr0eop 26138 | The empty graph, with vert... |
| uspgr1eop 26139 | A simple pseudograph with ... |
| uspgr1ewop 26140 | A simple pseudograph with ... |
| uspgr1v1eop 26141 | A simple pseudograph with ... |
| usgr1eop 26142 | A simple graph with (at le... |
| uspgr2v1e2w 26143 | A simple pseudograph with ... |
| usgr2v1e2w 26144 | A simple graph with two ve... |
| edg0usgr 26145 | A class without edges is a... |
| lfuhgr1v0e 26146 | A loop-free hypergraph wit... |
| usgr1vr 26147 | A simple graph with one ve... |
| usgr1v 26148 | A class with one (or no) v... |
| usgr1v0edg 26149 | A class with one (or no) v... |
| usgrexmpldifpr 26150 | Lemma for ~ usgrexmpledg :... |
| usgrexmplef 26151 | Lemma for ~ usgrexmpl . (... |
| usgrexmpllem 26152 | Lemma for ~ usgrexmpl . (... |
| usgrexmplvtx 26153 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpledg 26154 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl 26155 | ` G ` is a simple graph of... |
| griedg0prc 26156 | The class of empty graphs ... |
| griedg0ssusgr 26157 | The class of all simple gr... |
| usgrprc 26158 | The class of simple graphs... |
| relsubgr 26161 | The class of the subgraph ... |
| subgrv 26162 | If a class is a subgraph o... |
| issubgr 26163 | The property of a set to b... |
| issubgr2 26164 | The property of a set to b... |
| subgrprop 26165 | The properties of a subgra... |
| subgrprop2 26166 | The properties of a subgra... |
| uhgrissubgr 26167 | The property of a hypergra... |
| subgrprop3 26168 | The properties of a subgra... |
| egrsubgr 26169 | An empty graph consisting ... |
| 0grsubgr 26170 | The null graph (represente... |
| 0uhgrsubgr 26171 | The null graph (as hypergr... |
| uhgrsubgrself 26172 | A hypergraph is a subgraph... |
| subgrfun 26173 | The edge function of a sub... |
| subgruhgrfun 26174 | The edge function of a sub... |
| subgreldmiedg 26175 | An element of the domain o... |
| subgruhgredgd 26176 | An edge of a subgraph of a... |
| subumgredg2 26177 | An edge of a subgraph of a... |
| subuhgr 26178 | A subgraph of a hypergraph... |
| subupgr 26179 | A subgraph of a pseudograp... |
| subumgr 26180 | A subgraph of a multigraph... |
| subusgr 26181 | A subgraph of a simple gra... |
| uhgrspansubgrlem 26182 | Lemma for ~ uhgrspansubgr ... |
| uhgrspansubgr 26183 | A spanning subgraph ` S ` ... |
| uhgrspan 26184 | A spanning subgraph ` S ` ... |
| upgrspan 26185 | A spanning subgraph ` S ` ... |
| umgrspan 26186 | A spanning subgraph ` S ` ... |
| usgrspan 26187 | A spanning subgraph ` S ` ... |
| uhgrspanop 26188 | A spanning subgraph of a h... |
| upgrspanop 26189 | A spanning subgraph of a p... |
| umgrspanop 26190 | A spanning subgraph of a m... |
| usgrspanop 26191 | A spanning subgraph of a s... |
| uhgrspan1lem1 26192 | Lemma 1 for ~ uhgrspan1 . ... |
| uhgrspan1lem2 26193 | Lemma 2 for ~ uhgrspan1 . ... |
| uhgrspan1lem3 26194 | Lemma 3 for ~ uhgrspan1 . ... |
| uhgrspan1 26195 | The induced subgraph ` S `... |
| upgrreslem 26196 | Lemma for ~ upgrres . (Co... |
| umgrreslem 26197 | Lemma for ~ umgrres and ~ ... |
| upgrres 26198 | A subgraph obtained by rem... |
| umgrres 26199 | A subgraph obtained by rem... |
| usgrres 26200 | A subgraph obtained by rem... |
| upgrres1lem1 26201 | Lemma 1 for ~ upgrres1 . ... |
| umgrres1lem 26202 | Lemma for ~ umgrres1 . (C... |
| upgrres1lem2 26203 | Lemma 2 for ~ upgrres1 . ... |
| upgrres1lem3 26204 | Lemma 3 for ~ upgrres1 . ... |
| upgrres1 26205 | A pseudograph obtained by ... |
| umgrres1 26206 | A multigraph obtained by r... |
| usgrres1 26207 | Restricting a simple graph... |
| isfusgr 26210 | The property of being a fi... |
| fusgrvtxfi 26211 | A finite simple graph has ... |
| isfusgrf1 26212 | The property of being a fi... |
| isfusgrcl 26213 | The property of being a fi... |
| fusgrusgr 26214 | A finite simple graph is a... |
| opfusgr 26215 | A finite simple graph repr... |
| usgredgffibi 26216 | The number of edges in a s... |
| fusgredgfi 26217 | In a finite simple graph t... |
| usgr1v0e 26218 | The size of a (finite) sim... |
| usgrfilem 26219 | In a finite simple graph, ... |
| fusgrfisbase 26220 | Induction base for ~ fusgr... |
| fusgrfisstep 26221 | Induction step in ~ fusgrf... |
| fusgrfis 26222 | A finite simple graph is o... |
| fusgrfupgrfs 26223 | A finite simple graph is a... |
| nbgrprc0 26229 | The set of neighbors is em... |
| nbgrcl 26233 | If a class has at least on... |
| nbgrval 26234 | The set of neighbors of a ... |
| dfnbgr2 26235 | Alternate definition of th... |
| dfnbgr3 26236 | Alternate definition of th... |
| nbgrnvtx0 26237 | There are no neighbors of ... |
| nbgrel 26238 | Characterization of a neig... |
| nbuhgr 26239 | The set of neighbors of a ... |
| nbupgr 26240 | The set of neighbors of a ... |
| nbupgrel 26241 | A neighbor of a vertex in ... |
| nbumgrvtx 26242 | The set of neighbors of a ... |
| nbumgr 26243 | The set of neighbors of an... |
| nbusgrvtx 26244 | The set of neighbors of a ... |
| nbusgr 26245 | The set of neighbors of an... |
| nbgr2vtx1edg 26246 | If a graph has two vertice... |
| nbuhgr2vtx1edgblem 26247 | Lemma for ~ nbuhgr2vtx1edg... |
| nbuhgr2vtx1edgb 26248 | If a hypergraph has two ve... |
| nbusgreledg 26249 | A class/vertex is a neighb... |
| uhgrnbgr0nb 26250 | A vertex which is not endp... |
| nbgr0vtxlem 26251 | Lemma for ~ nbgr0vtx and ~... |
| nbgr0vtx 26252 | In a null graph (with no v... |
| nbgr0edg 26253 | In an empty graph (with no... |
| nbgr1vtx 26254 | In a graph with one vertex... |
| nbgrisvtx 26255 | Every neighbor of a class/... |
| nbgrssvtx 26256 | The neighbors of a vertex ... |
| nbgrnself 26257 | A vertex in a graph is not... |
| usgrnbnself 26258 | A vertex in a simple graph... |
| nbgrnself2 26259 | A class is not a neighbor ... |
| nbgrssovtx 26260 | The neighbors of a vertex ... |
| nbgrssvwo2 26261 | The neighbors of a vertex ... |
| usgrnbnself2 26262 | In a simple graph, a class... |
| usgrnbssovtx 26263 | The neighbors of a vertex ... |
| usgrnbssvwo2 26264 | The neighbors of a vertex ... |
| nbgrsym 26265 | A vertex in a graph is a n... |
| nbupgrres 26266 | The neighborhood of a vert... |
| usgrnbcnvfv 26267 | Applying the edge function... |
| nbusgredgeu 26268 | For each neighbor of a ver... |
| edgnbusgreu 26269 | For each edge incident to ... |
| nbusgredgeu0 26270 | For each neighbor of a ver... |
| nbusgrf1o0 26271 | The mapping of neighbors o... |
| nbusgrf1o1 26272 | The set of neighbors of a ... |
| nbusgrf1o 26273 | The set of neighbors of a ... |
| nbedgusgr 26274 | The number of neighbors of... |
| edgusgrnbfin 26275 | The number of neighbors of... |
| nbusgrfi 26276 | The class of neighbors of ... |
| nbfiusgrfi 26277 | The class of neighbors of ... |
| hashnbusgrnn0 26278 | The number of neighbors of... |
| nbfusgrlevtxm1 26279 | The number of neighbors of... |
| nbfusgrlevtxm2 26280 | If there is a vertex which... |
| nbusgrvtxm1 26281 | If the number of neighbors... |
| nb3grprlem1 26282 | Lemma 1 for ~ nb3grpr . (... |
| nb3grprlem2 26283 | Lemma 2 for ~ nb3grpr . (... |
| nb3grpr 26284 | The neighbors of a vertex ... |
| nb3grpr2 26285 | The neighbors of a vertex ... |
| nb3gr2nb 26286 | If the neighbors of two ve... |
| uvtxaval 26287 | The set of all universal v... |
| uvtxael 26288 | A universal vertex, i.e. a... |
| uvtxaisvtx 26289 | A universal vertex is a ve... |
| uvtxassvtx 26290 | The set of the universal v... |
| vtxnbuvtx 26291 | A universal vertex has all... |
| uvtxanbgr 26292 | A universal vertex has all... |
| uvtxanbgrvtx 26293 | A universal vertex is neig... |
| uvtxa0 26294 | There is no universal vert... |
| isuvtxa 26295 | The set of all universal v... |
| uvtxael1 26296 | A universal vertex, i.e. a... |
| uvtxa01vtx0 26297 | If a graph/class has no ed... |
| uvtxa01vtx 26298 | If a graph/class has no ed... |
| uvtx2vtx1edg 26299 | If a graph has two vertice... |
| uvtx2vtx1edgb 26300 | If a hypergraph has two ve... |
| uvtxnbgr 26301 | A universal vertex has all... |
| uvtxnbgrb 26302 | A vertex is universal iff ... |
| uvtxusgr 26303 | The set of all universal v... |
| uvtxusgrel 26304 | A universal vertex, i.e. a... |
| uvtxanm1nbgr 26305 | A universal vertex has ` n... |
| nbusgrvtxm1uvtx 26306 | If the number of neighbors... |
| uvtxnbvtxm1 26307 | A universal vertex has ` n... |
| nbupgruvtxres 26308 | The neighborhood of a univ... |
| uvtxupgrres 26309 | A universal vertex is univ... |
| iscplgr 26310 | The property of being a co... |
| cplgruvtxb 26311 | An graph is complete iff e... |
| iscplgrnb 26312 | A graph is complete iff al... |
| iscplgredg 26313 | A graph is complete iff al... |
| iscusgr 26314 | The property of being a co... |
| cusgrusgr 26315 | A complete simple graph is... |
| cusgrcplgr 26316 | A complete simple graph is... |
| iscusgrvtx 26317 | A simple graph is complete... |
| cusgruvtxb 26318 | A simple graph is complete... |
| iscusgredg 26319 | A simple graph is complete... |
| cusgredg 26320 | In a complete simple graph... |
| cplgr0 26321 | The null graph (with no ve... |
| cusgr0 26322 | The null graph (with no ve... |
| cplgr0v 26323 | A null graph (with no vert... |
| cusgr0v 26324 | A graph with no vertices a... |
| cplgr1vlem 26325 | Lemma for ~ cplgr1v and ~ ... |
| cplgr1v 26326 | A graph with one vertex is... |
| cusgr1v 26327 | A graph with one vertex an... |
| cplgr2v 26328 | An undirected hypergraph w... |
| cplgr2vpr 26329 | An undirected hypergraph w... |
| nbcplgr 26330 | In a complete graph, each ... |
| cplgr3v 26331 | A pseudograph with three (... |
| cusgr3vnbpr 26332 | The neighbors of a vertex ... |
| cplgrop 26333 | A complete graph represent... |
| cusgrop 26334 | A complete simple graph re... |
| cusgrexilem1 26335 | Lemma 1 for ~ cusgrexi . ... |
| usgrexilem 26336 | Lemma for ~ usgrexi . (Co... |
| usgrexi 26337 | An arbitrary set regarded ... |
| cusgrexilem2 26338 | Lemma 2 for ~ cusgrexi . ... |
| cusgrexi 26339 | An arbitrary set regarded ... |
| cusgrexg 26340 | For each set there is a se... |
| structtousgr 26341 | Any (extensible) structure... |
| structtocusgr 26342 | Any (extensible) structure... |
| cffldtocusgr 26343 | The field of complex numbe... |
| cusgrres 26344 | Restricting a complete sim... |
| cusgrsizeindb0 26345 | Base case of the induction... |
| cusgrsizeindb1 26346 | Base case of the induction... |
| cusgrsizeindslem 26347 | Lemma for ~ cusgrsizeinds ... |
| cusgrsizeinds 26348 | Part 1 of induction step i... |
| cusgrsize2inds 26349 | Induction step in ~ cusgrs... |
| cusgrsize 26350 | The size of a finite compl... |
| cusgrfilem1 26351 | Lemma 1 for ~ cusgrfi . (... |
| cusgrfilem2 26352 | Lemma 2 for ~ cusgrfi . (... |
| cusgrfilem3 26353 | Lemma 3 for ~ cusgrfi . (... |
| cusgrfi 26354 | If the size of a complete ... |
| usgredgsscusgredg 26355 | A simple graph is a subgra... |
| usgrsscusgr 26356 | A simple graph is a subgra... |
| sizusglecusglem1 26357 | Lemma 1 for ~ sizusglecusg... |
| sizusglecusglem2 26358 | Lemma 2 for ~ sizusglecusg... |
| sizusglecusg 26359 | The size of a simple graph... |
| fusgrmaxsize 26360 | The maximum size of a fini... |
| vtxdgfval 26363 | The value of the vertex de... |
| vtxdgval 26364 | The degree of a vertex. (... |
| vtxdgfival 26365 | The degree of a vertex for... |
| vtxdgop 26366 | The vertex degree expresse... |
| vtxdgf 26367 | The vertex degree function... |
| vtxdgelxnn0 26368 | The degree of a vertex is ... |
| vtxdg0v 26369 | The degree of a vertex in ... |
| vtxdg0e 26370 | The degree of a vertex in ... |
| vtxdgfisnn0 26371 | The degree of a vertex in ... |
| vtxdgfisf 26372 | The vertex degree function... |
| vtxdeqd 26373 | Equality theorem for the v... |
| vtxduhgr0e 26374 | The degree of a vertex in ... |
| vtxdlfuhgr1v 26375 | The degree of the vertex i... |
| vdumgr0 26376 | A vertex in a multigraph h... |
| vtxdun 26377 | The degree of a vertex in ... |
| vtxdfiun 26378 | The degree of a vertex in ... |
| vtxduhgrun 26379 | The degree of a vertex in ... |
| vtxduhgrfiun 26380 | The degree of a vertex in ... |
| vtxdlfgrval 26381 | The value of the vertex de... |
| vtxdumgrval 26382 | The value of the vertex de... |
| vtxdusgrval 26383 | The value of the vertex de... |
| vtxd0nedgb 26384 | A vertex has degree 0 iff ... |
| vtxdushgrfvedglem 26385 | Lemma for ~ vtxdushgrfvedg... |
| vtxdushgrfvedg 26386 | The value of the vertex de... |
| vtxdusgrfvedg 26387 | The value of the vertex de... |
| vtxduhgr0nedg 26388 | If a vertex in a hypergrap... |
| vtxdumgr0nedg 26389 | If a vertex in a multigrap... |
| vtxduhgr0edgnel 26390 | A vertex in a hypergraph h... |
| vtxdusgr0edgnel 26391 | A vertex in a simple graph... |
| vtxdusgr0edgnelALT 26392 | Alternate proof of ~ vtxdu... |
| vtxdgfusgrf 26393 | The vertex degree function... |
| vtxdgfusgr 26394 | In a finite simple graph, ... |
| fusgrn0degnn0 26395 | In a nonempty, finite grap... |
| 1loopgruspgr 26396 | A graph with one edge whic... |
| 1loopgredg 26397 | The set of edges in a grap... |
| 1loopgrnb0 26398 | In a graph (simple pseudog... |
| 1loopgrvd2 26399 | The vertex degree of a one... |
| 1loopgrvd0 26400 | The vertex degree of a one... |
| 1hevtxdg0 26401 | The vertex degree of verte... |
| 1hevtxdg1 26402 | The vertex degree of verte... |
| 1hegrvtxdg1 26403 | The vertex degree of a gra... |
| 1hegrvtxdg1r 26404 | The vertex degree of a gra... |
| 1egrvtxdg1 26405 | The vertex degree of a one... |
| 1egrvtxdg1r 26406 | The vertex degree of a one... |
| 1egrvtxdg0 26407 | The vertex degree of a one... |
| p1evtxdeqlem 26408 | Lemma for ~ p1evtxdeq and ... |
| p1evtxdeq 26409 | If an edge ` E ` which doe... |
| p1evtxdp1 26410 | If an edge ` E ` (not bein... |
| uspgrloopvtx 26411 | The set of vertices in a g... |
| uspgrloopvtxel 26412 | A vertex in a graph (simpl... |
| uspgrloopiedg 26413 | The set of edges in a grap... |
| uspgrloopedg 26414 | The set of edges in a grap... |
| uspgrloopnb0 26415 | In a graph (simple pseudog... |
| uspgrloopvd2 26416 | The vertex degree of a one... |
| umgr2v2evtx 26417 | The set of vertices in a m... |
| umgr2v2evtxel 26418 | A vertex in a multigraph w... |
| umgr2v2eiedg 26419 | The edge function in a mul... |
| umgr2v2eedg 26420 | The set of edges in a mult... |
| umgr2v2e 26421 | A multigraph with two edge... |
| umgr2v2enb1 26422 | In a multigraph with two e... |
| umgr2v2evd2 26423 | In a multigraph with two e... |
| hashnbusgrvd 26424 | In a simple graph, the num... |
| usgruvtxvdb 26425 | In a finite simple graph w... |
| vdiscusgrb 26426 | A finite simple graph with... |
| vdiscusgr 26427 | In a finite complete simpl... |
| vtxdusgradjvtx 26428 | The degree of a vertex in ... |
| usgrvd0nedg 26429 | If a vertex in a simple gr... |
| uhgrvd00 26430 | If every vertex in a hyper... |
| usgrvd00 26431 | If every vertex in a simpl... |
| vdegp1ai 26432 | The induction step for a v... |
| vdegp1bi 26433 | The induction step for a v... |
| vdegp1ci 26434 | The induction step for a v... |
| vtxdginducedm1lem1 26435 | Lemma 1 for ~ vtxdginduced... |
| vtxdginducedm1lem2 26436 | Lemma 2 for ~ vtxdginduced... |
| vtxdginducedm1lem3 26437 | Lemma 3 for ~ vtxdginduced... |
| vtxdginducedm1lem4 26438 | Lemma 4 for ~ vtxdginduced... |
| vtxdginducedm1 26439 | The degree of a vertex ` v... |
| vtxdginducedm1fi 26440 | The degree of a vertex ` v... |
| finsumvtxdg2ssteplem1 26441 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem2 26442 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem3 26443 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem4 26444 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2sstep 26445 | Induction step of ~ finsum... |
| finsumvtxdg2size 26446 | The sum of the degrees of ... |
| fusgr1th 26447 | The sum of the degrees of ... |
| finsumvtxdgeven 26448 | The sum of the degrees of ... |
| vtxdgoddnumeven 26449 | The number of vertices of ... |
| fusgrvtxdgonume 26450 | The number of vertices of ... |
| isrgr 26455 | The property of a class be... |
| rgrprop 26456 | The properties of a k-regu... |
| isrusgr 26457 | The property of being a k-... |
| rusgrprop 26458 | The properties of a k-regu... |
| rusgrrgr 26459 | A k-regular simple graph i... |
| rusgrusgr 26460 | A k-regular simple graph i... |
| finrusgrfusgr 26461 | A finite regular simple gr... |
| isrusgr0 26462 | The property of being a k-... |
| rusgrprop0 26463 | The properties of a k-regu... |
| usgreqdrusgr 26464 | If all vertices in a simpl... |
| fusgrregdegfi 26465 | In a nonempty finite simpl... |
| fusgrn0eqdrusgr 26466 | If all vertices in a nonem... |
| frusgrnn0 26467 | In a nonempty finite k-reg... |
| 0edg0rgr 26468 | A graph is 0-regular if it... |
| uhgr0edg0rgr 26469 | A hypergraph is 0-regular ... |
| uhgr0edg0rgrb 26470 | A hypergraph is 0-regular ... |
| usgr0edg0rusgr 26471 | A simple graph is 0-regula... |
| 0vtxrgr 26472 | A null graph (with no vert... |
| 0vtxrusgr 26473 | A graph with no vertices a... |
| 0uhgrrusgr 26474 | The null graph as hypergra... |
| 0grrusgr 26475 | The null graph represented... |
| 0grrgr 26476 | The null graph represented... |
| cusgrrusgr 26477 | A complete simple graph wi... |
| cusgrm1rusgr 26478 | A finite simple graph with... |
| rusgrpropnb 26479 | The properties of a k-regu... |
| rusgrpropedg 26480 | The properties of a k-regu... |
| rusgrpropadjvtx 26481 | The properties of a k-regu... |
| rusgrnumwrdl2 26482 | In a k-regular simple grap... |
| rusgr1vtxlem 26483 | Lemma for ~ rusgr1vtx . (... |
| rusgr1vtx 26484 | If a k-regular simple grap... |
| rgrusgrprc 26485 | The class of 0-regular sim... |
| rusgrprc 26486 | The class of 0-regular sim... |
| rgrprc 26487 | The class of 0-regular gra... |
| rgrprcx 26488 | The class of 0-regular gra... |
| rgrx0ndm 26489 | 0 is not in the domain of ... |
| rgrx0nd 26490 | The potentially alternativ... |
| ewlksfval 26497 | The set of s-walks of edge... |
| isewlk 26498 | Conditions for a function ... |
| ewlkprop 26499 | Properties of an s-walk of... |
| ewlkinedg 26500 | The intersection (common v... |
| ewlkle 26501 | An s-walk of edges is also... |
| upgrewlkle2 26502 | In a pseudograph, there is... |
| wkslem1 26503 | Lemma 1 for walks to subst... |
| wkslem2 26504 | Lemma 2 for walks to subst... |
| wksfval 26505 | The set of walks (in an un... |
| iswlk 26506 | Properties of a pair of fu... |
| wlkprop 26507 | Properties of a walk. (Co... |
| wlkv 26508 | The classes involved in a ... |
| iswlkg 26509 | Generalisation of ~ iswlk ... |
| wlkf 26510 | The mapping enumerating th... |
| wlkcl 26511 | A walk has length ` # ( F ... |
| wlkp 26512 | The mapping enumerating th... |
| wlkpwrd 26513 | The sequence of vertices o... |
| wlklenvp1 26514 | The number of vertices of ... |
| wksv 26515 | The class of walks is a se... |
| wlkn0 26516 | The sequence of vertices o... |
| wlklenvm1 26517 | The number of edges of a w... |
| ifpsnprss 26518 | Lemma for ~ wlkvtxeledg : ... |
| wlkvtxeledg 26519 | Each pair of adjacent vert... |
| wlkvtxiedg 26520 | The vertices of a walk are... |
| relwlk 26521 | The set ` ( Walks `` G ) `... |
| wlkvv 26522 | If there is at least one w... |
| wlkop 26523 | A walk is an ordered pair.... |
| wlkcpr 26524 | A walk as class with two c... |
| wlk2f 26525 | If there is a walk ` W ` t... |
| wlkcomp 26526 | A walk expressed by proper... |
| wlkcompim 26527 | Implications for the prope... |
| wlkelwrd 26528 | The components of a walk a... |
| wlkeq 26529 | Conditions for two walks (... |
| edginwlk 26530 | The value of the edge func... |
| edginwlkOLD 26531 | Obsolete version of ~ edgi... |
| upgredginwlk 26532 | The value of the edge func... |
| iedginwlk 26533 | The value of the edge func... |
| wlkl1loop 26534 | A walk of length 1 from a ... |
| wlk1walk 26535 | A walk is a 1-walk "on the... |
| wlk1ewlk 26536 | A walk is an s-walk "on th... |
| upgriswlk 26537 | Properties of a pair of fu... |
| upgrwlkedg 26538 | The edges of a walk in a p... |
| upgrwlkcompim 26539 | Implications for the prope... |
| wlkvtxedg 26540 | The vertices of a walk are... |
| upgrwlkvtxedg 26541 | The pairs of connected ver... |
| uspgr2wlkeq 26542 | Conditions for two walks w... |
| uspgr2wlkeq2 26543 | Conditions for two walks w... |
| uspgr2wlkeqi 26544 | Conditions for two walks w... |
| umgrwlknloop 26545 | In a multigraph, each walk... |
| wlkRes 26546 | Restrictions of walks (i.e... |
| wlkv0 26547 | If there is a walk in the ... |
| g0wlk0 26548 | There is no walk in a null... |
| 0wlk0 26549 | There is no walk for the e... |
| wlk0prc 26550 | There is no walk in a null... |
| wlklenvclwlk 26551 | The number of vertices in ... |
| wlkson 26552 | The set of walks between t... |
| iswlkon 26553 | Properties of a pair of fu... |
| wlkonprop 26554 | Properties of a walk betwe... |
| wlkpvtx 26555 | A walk connects vertices. ... |
| wlkepvtx 26556 | The endpoints of a walk ar... |
| wlkoniswlk 26557 | A walk between two vertice... |
| wlkonwlk 26558 | A walk is a walk between i... |
| wlkonwlk1l 26559 | A walk is a walk from its ... |
| wlksoneq1eq2 26560 | Two walks with identical s... |
| wlkonl1iedg 26561 | If there is a walk between... |
| wlkon2n0 26562 | The length of a walk betwe... |
| 2wlklem 26563 | Lemma for theorems for wal... |
| upgr2wlk 26564 | Properties of a pair of fu... |
| wlkreslem0 26565 | Lemma for ~ wlkres . TODO... |
| wlkreslem 26566 | Lemma for ~ wlkres . (Con... |
| wlkres 26567 | The restriction ` <. H , Q... |
| redwlklem 26568 | Lemma for ~ redwlk . (Con... |
| redwlk 26569 | A walk ending at the last ... |
| wlkp1lem1 26570 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem2 26571 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem3 26572 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem4 26573 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem5 26574 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem6 26575 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem7 26576 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem8 26577 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1 26578 | Append one path segment (e... |
| wlkdlem1 26579 | Lemma 1 for ~ wlkd . (Con... |
| wlkdlem2 26580 | Lemma 2 for ~ wlkd . (Con... |
| wlkdlem3 26581 | Lemma 3 for ~ wlkd . (Con... |
| wlkdlem4 26582 | Lemma 4 for ~ wlkd . (Con... |
| wlkd 26583 | Two words representing a w... |
| lfgrwlkprop 26584 | Two adjacent vertices in a... |
| lfgriswlk 26585 | Conditions for a pair of f... |
| lfgrwlknloop 26586 | In a loop-free graph, each... |
| reltrls 26591 | The set ` ( Trails `` G ) ... |
| trlsfval 26592 | The set of trails (in an u... |
| istrl 26593 | Conditions for a pair of c... |
| trliswlk 26594 | A trail is a walk. (Contr... |
| trlf1 26595 | The enumeration ` F ` of a... |
| trlreslem 26596 | Lemma for ~ trlres . Form... |
| trlres 26597 | The restriction ` <. H , Q... |
| upgrtrls 26598 | The set of trails in a pse... |
| upgristrl 26599 | Properties of a pair of fu... |
| upgrf1istrl 26600 | Properties of a pair of a ... |
| wksonproplem 26601 | Lemma for theorems for pro... |
| trlsonfval 26602 | The set of trails between ... |
| istrlson 26603 | Properties of a pair of fu... |
| trlsonprop 26604 | Properties of a trail betw... |
| trlsonistrl 26605 | A trail between two vertic... |
| trlsonwlkon 26606 | A trail between two vertic... |
| trlontrl 26607 | A trail is a trail between... |
| relpths 26616 | The set ` ( Paths `` G ) `... |
| pthsfval 26617 | The set of paths (in an un... |
| spthsfval 26618 | The set of simple paths (i... |
| ispth 26619 | Conditions for a pair of c... |
| isspth 26620 | Conditions for a pair of c... |
| pthistrl 26621 | A path is a trail (in an u... |
| spthispth 26622 | A simple path is a path (i... |
| pthiswlk 26623 | A path is a walk (in an un... |
| spthiswlk 26624 | A simple path is a walk (i... |
| pthdivtx 26625 | The inner vertices of a pa... |
| pthdadjvtx 26626 | The adjacent vertices of a... |
| 2pthnloop 26627 | A path of length at least ... |
| upgr2pthnlp 26628 | A path of length at least ... |
| spthdifv 26629 | The vertices of a simple p... |
| spthdep 26630 | A simple path (at least of... |
| pthdepisspth 26631 | A path with different star... |
| upgrwlkdvdelem 26632 | Lemma for ~ upgrwlkdvde . ... |
| upgrwlkdvde 26633 | In a pseudograph, all edge... |
| upgrspthswlk 26634 | The set of simple paths in... |
| upgrwlkdvspth 26635 | A walk consisting of diffe... |
| pthsonfval 26636 | The set of paths between t... |
| spthson 26637 | The set of simple paths be... |
| ispthson 26638 | Properties of a pair of fu... |
| isspthson 26639 | Properties of a pair of fu... |
| pthsonprop 26640 | Properties of a path betwe... |
| spthonprop 26641 | Properties of a simple pat... |
| pthonispth 26642 | A path between two vertice... |
| pthontrlon 26643 | A path between two vertice... |
| pthonpth 26644 | A path is a path between i... |
| isspthonpth 26645 | A pair of functions is a s... |
| spthonisspth 26646 | A simple path between to v... |
| spthonpthon 26647 | A simple path between two ... |
| spthonepeq 26648 | The endpoints of a simple ... |
| uhgrwkspthlem1 26649 | Lemma 1 for ~ uhgrwkspth .... |
| uhgrwkspthlem2 26650 | Lemma 2 for ~ uhgrwkspth .... |
| uhgrwkspth 26651 | Any walk of length 1 betwe... |
| usgr2wlkneq 26652 | The vertices and edges are... |
| usgr2wlkspthlem1 26653 | Lemma 1 for ~ usgr2wlkspth... |
| usgr2wlkspthlem2 26654 | Lemma 2 for ~ usgr2wlkspth... |
| usgr2wlkspth 26655 | In a simple graph, any wal... |
| usgr2trlncl 26656 | In a simple graph, any tra... |
| usgr2trlspth 26657 | In a simple graph, any tra... |
| usgr2pthspth 26658 | In a simple graph, any pat... |
| usgr2pthlem 26659 | Lemma for ~ usgr2pth . (C... |
| usgr2pth 26660 | In a simple graph, there i... |
| usgr2pth0 26661 | In a simply graph, there i... |
| pthdlem1 26662 | Lemma 1 for ~ pthd . (Con... |
| pthdlem2lem 26663 | Lemma for ~ pthdlem2 . (C... |
| pthdlem2 26664 | Lemma 2 for ~ pthd . (Con... |
| pthd 26665 | Two words representing a t... |
| clwlks 26668 | The set of closed walks (i... |
| isclwlk 26669 | A pair of functions repres... |
| clwlkiswlk 26670 | A closed walk is a walk (i... |
| clwlkwlk 26671 | Closed walks are walks (in... |
| clwlkswks 26672 | Closed walks are walks (in... |
| isclwlke 26673 | Properties of a pair of fu... |
| isclwlkupgr 26674 | Properties of a pair of fu... |
| clwlkcomp 26675 | A closed walk expressed by... |
| clwlkcompim 26676 | Implications for the prope... |
| upgrclwlkcompim 26677 | Implications for the prope... |
| clwlkl1loop 26678 | A closed walk of length 1 ... |
| crcts 26683 | The set of circuits (in an... |
| cycls 26684 | The set of cycles (in an u... |
| iscrct 26685 | Sufficient and necessary c... |
| iscycl 26686 | Sufficient and necessary c... |
| crctprop 26687 | The properties of a circui... |
| cyclprop 26688 | The properties of a cycle:... |
| crctisclwlk 26689 | A circuit is a closed walk... |
| crctistrl 26690 | A circuit is a trail. (Co... |
| crctiswlk 26691 | A circuit is a walk. (Con... |
| cyclispth 26692 | A cycle is a path. (Contr... |
| cycliswlk 26693 | A cycle is a walk. (Contr... |
| cycliscrct 26694 | A cycle is a circuit. (Co... |
| cyclnspth 26695 | A (non trivial) cycle is n... |
| cyclispthon 26696 | A cycle is a path starting... |
| lfgrn1cycl 26697 | In a loop-free graph there... |
| usgr2trlncrct 26698 | In a simple graph, any tra... |
| umgrn1cycl 26699 | In a multigraph graph (wit... |
| uspgrn2crct 26700 | In a simple pseudograph th... |
| usgrn2cycl 26701 | In a simple graph there ar... |
| crctcshwlkn0lem1 26702 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem2 26703 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem3 26704 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem4 26705 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem5 26706 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem6 26707 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem7 26708 | Lemma for ~ crctcshwlkn0 .... |
| crctcshlem1 26709 | Lemma for ~ crctcsh . (Co... |
| crctcshlem2 26710 | Lemma for ~ crctcsh . (Co... |
| crctcshlem3 26711 | Lemma for ~ crctcsh . (Co... |
| crctcshlem4 26712 | Lemma for ~ crctcsh . (Co... |
| crctcshwlkn0 26713 | Cyclically shifting the in... |
| crctcshwlk 26714 | Cyclically shifting the in... |
| crctcshtrl 26715 | Cyclically shifting the in... |
| crctcsh 26716 | Cyclically shifting the in... |
| wwlks 26727 | The set of walks (in an un... |
| iswwlks 26728 | A word over the set of ver... |
| wwlksn 26729 | The set of walks (in an un... |
| iswwlksn 26730 | A word over the set of ver... |
| iswwlksnx 26731 | Properties of a word to re... |
| wwlkbp 26732 | Basic properties of a walk... |
| wwlknbp 26733 | Basic properties of a walk... |
| wwlknp 26734 | Properties of a set being ... |
| wspthsn 26735 | The set of simple paths of... |
| iswspthn 26736 | An element of the set of s... |
| wspthnp 26737 | Properties of a set being ... |
| wwlksnon 26738 | The set of walks of a fixe... |
| wspthsnon 26739 | The set of simple paths of... |
| iswwlksnon 26740 | The set of walks of a fixe... |
| iswspthsnon 26741 | The set of simple paths of... |
| wwlknon 26742 | An element of the set of w... |
| wspthnon 26743 | An element of the set of s... |
| wspthnonp 26744 | Properties of a set being ... |
| wspthneq1eq2 26745 | Two simple paths with iden... |
| wwlksn0s 26746 | The set of all walks as wo... |
| wwlkssswrd 26747 | Walks (represented by word... |
| wwlksn0 26748 | A walk of length 0 is repr... |
| 0enwwlksnge1 26749 | In graphs without edges, t... |
| wwlkswwlksn 26750 | A walk of a fixed length a... |
| wwlkssswwlksn 26751 | The walks of a fixed lengt... |
| wwlknbp2 26752 | Other basic properties of ... |
| wlkiswwlks1 26753 | The sequence of vertices i... |
| wlklnwwlkln1 26754 | The sequence of vertices i... |
| wlkiswwlks2lem1 26755 | Lemma 1 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem2 26756 | Lemma 2 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem3 26757 | Lemma 3 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem4 26758 | Lemma 4 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem5 26759 | Lemma 5 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem6 26760 | Lemma 6 for ~ wlkiswwlks2 ... |
| wlkiswwlks2 26761 | A walk as word corresponds... |
| wlkiswwlks 26762 | A walk as word corresponds... |
| wlkiswwlksupgr2 26763 | A walk as word corresponds... |
| wlkiswwlkupgr 26764 | A walk as word corresponds... |
| wlkpwwlkf1ouspgr 26765 | The mapping of (ordinary) ... |
| wlkisowwlkupgr 26766 | The set of walks as words ... |
| wwlksm1edg 26767 | Removing the trailing edge... |
| wlklnwwlkln2lem 26768 | Lemma for ~ wlklnwwlkln2 a... |
| wlklnwwlkln2 26769 | A walk of length ` N ` as ... |
| wlklnwwlkn 26770 | A walk of length ` N ` as ... |
| wlklnwwlklnupgr2 26771 | A walk of length ` N ` as ... |
| wlklnwwlknupgr 26772 | A walk of length ` N ` as ... |
| wlknewwlksn 26773 | If a walk in a pseudograph... |
| wlknwwlksnfun 26774 | Lemma 1 for ~ wlknwwlksnbi... |
| wlknwwlksninj 26775 | Lemma 2 for ~ wlknwwlksnbi... |
| wlknwwlksnsur 26776 | Lemma 3 for ~ wlknwwlksnbi... |
| wlknwwlksnbij 26777 | Lemma 4 for ~ wlknwwlksnbi... |
| wlknwwlksnbij2 26778 | There is a bijection betwe... |
| wlknwwlksnen 26779 | In a simple pseudograph, t... |
| wlknwwlksneqs 26780 | The set of walks of a fixe... |
| wlkwwlkfun 26781 | Lemma 1 for ~ wlkwwlkbij2 ... |
| wlkwwlkinj 26782 | Lemma 2 for ~ wlkwwlkbij2 ... |
| wlkwwlksur 26783 | Lemma 3 for ~ wlkwwlkbij2 ... |
| wlkwwlkbij 26784 | Lemma 4 for ~ wlkwwlkbij2 ... |
| wlkwwlkbij2 26785 | There is a bijection betwe... |
| wwlkseq 26786 | Equality of two walks (as ... |
| wwlksnred 26787 | Reduction of a walk (as wo... |
| wwlksnext 26788 | Extension of a walk (as wo... |
| wwlksnextbi 26789 | Extension of a walk (as wo... |
| wwlksnredwwlkn 26790 | For each walk (as word) of... |
| wwlksnredwwlkn0 26791 | For each walk (as word) of... |
| wwlksnextwrd 26792 | Lemma for ~ wwlksnextbij .... |
| wwlksnextfun 26793 | Lemma for ~ wwlksnextbij .... |
| wwlksnextinj 26794 | Lemma for ~ wwlksnextbij .... |
| wwlksnextsur 26795 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij0 26796 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij 26797 | There is a bijection betwe... |
| wwlksnexthasheq 26798 | The number of the extensio... |
| disjxwwlksn 26799 | Sets of walks (as words) e... |
| wwlksnndef 26800 | Conditions for ` WWalksN `... |
| wwlksnfi 26801 | The number of walks repres... |
| wlksnfi 26802 | The number of walks of fix... |
| wlksnwwlknvbij 26803 | There is a bijection betwe... |
| wwlksnextproplem1 26804 | Lemma 1 for ~ wwlksnextpro... |
| wwlksnextproplem2 26805 | Lemma 2 for ~ wwlksnextpro... |
| wwlksnextproplem3 26806 | Lemma 3 for ~ wwlksnextpro... |
| wwlksnextprop 26807 | Adding additional properti... |
| disjxwwlkn 26808 | Sets of walks (as words) e... |
| hashwwlksnext 26809 | Number of walks (as words)... |
| wwlksnwwlksnon 26810 | A walk of fixed length is ... |
| wspthsnwspthsnon 26811 | A simple path of fixed len... |
| wwlksnon0 26812 | Conditions for a set of wa... |
| wspthsnonn0vne 26813 | If the set of simple paths... |
| wspthsswwlkn 26814 | The set of simple paths of... |
| wspthnfi 26815 | In a finite graph, the set... |
| wwlksnonfi 26816 | In a finite graph, the set... |
| wspthsswwlknon 26817 | The set of simple paths of... |
| wspthnonfi 26818 | In a finite graph, the set... |
| wspniunwspnon 26819 | The set of nonempty simple... |
| wspn0 26820 | If there are no vertices, ... |
| 2wlkdlem1 26821 | Lemma 1 for ~ 2wlkd . (Co... |
| 2wlkdlem2 26822 | Lemma 2 for ~ 2wlkd . (Co... |
| 2wlkdlem3 26823 | Lemma 3 for ~ 2wlkd . (Co... |
| 2wlkdlem4 26824 | Lemma 4 for ~ 2wlkd . (Co... |
| 2wlkdlem5 26825 | Lemma 5 for ~ 2wlkd . (Co... |
| 2pthdlem1 26826 | Lemma 1 for ~ 2pthd . (Co... |
| 2wlkdlem6 26827 | Lemma 6 for ~ 2wlkd . (Co... |
| 2wlkdlem7 26828 | Lemma 7 for ~ 2wlkd . (Co... |
| 2wlkdlem8 26829 | Lemma 8 for ~ 2wlkd . (Co... |
| 2wlkdlem9 26830 | Lemma 9 for ~ 2wlkd . (Co... |
| 2wlkdlem10 26831 | Lemma 10 for ~ 3wlkd . (C... |
| 2wlkd 26832 | Construction of a walk fro... |
| 2wlkond 26833 | A walk of length 2 from on... |
| 2trld 26834 | Construction of a trail fr... |
| 2trlond 26835 | A trail of length 2 from o... |
| 2pthd 26836 | A path of length 2 from on... |
| 2spthd 26837 | A simple path of length 2 ... |
| 2pthond 26838 | A simple path of length 2 ... |
| 2pthon3v 26839 | For a vertex adjacent to t... |
| umgr2adedgwlklem 26840 | Lemma for ~ umgr2adedgwlk ... |
| umgr2adedgwlk 26841 | In a multigraph, two adjac... |
| umgr2adedgwlkon 26842 | In a multigraph, two adjac... |
| umgr2adedgwlkonALT 26843 | Alternate proof for ~ umgr... |
| umgr2adedgspth 26844 | In a multigraph, two adjac... |
| umgr2wlk 26845 | In a multigraph, there is ... |
| umgr2wlkon 26846 | For each pair of adjacent ... |
| wwlks2onv 26847 | If a length 3 string repre... |
| elwwlks2ons3 26848 | For each walk of length 2 ... |
| s3wwlks2on 26849 | A length 3 string which re... |
| umgrwwlks2on 26850 | A walk of length 2 between... |
| wwlks2onsym 26851 | There is a walk of length ... |
| elwwlks2on 26852 | A walk of length 2 between... |
| elwspths2on 26853 | A simple path of length 2 ... |
| wpthswwlks2on 26854 | For two different vertices... |
| 2wspdisj 26855 | All simple paths of length... |
| 2wspiundisj 26856 | All simple paths of length... |
| usgr2wspthons3 26857 | A simple path of length 2 ... |
| usgr2wspthon 26858 | A simple path of length 2 ... |
| elwwlks2s3 26859 | A walk of length 2 between... |
| midwwlks2s3 26860 | There is a vertex between ... |
| elwwlks2 26861 | A walk of length 2 between... |
| elwspths2spth 26862 | A simple path of length 2 ... |
| rusgrnumwwlkl1 26863 | In a k-regular graph, ther... |
| rusgrnumwwlkslem 26864 | Lemma for ~ rusgrnumwwlks ... |
| rusgrnumwwlklem 26865 | Lemma for ~ rusgrnumwwlk e... |
| rusgrnumwwlkb0 26866 | Induction base 0 for ~ rus... |
| rusgrnumwwlkb1 26867 | Induction base 1 for ~ rus... |
| rusgr0edg 26868 | Special case for graphs wi... |
| rusgrnumwwlks 26869 | Induction step for ~ rusgr... |
| rusgrnumwwlk 26870 | In a ` K `-regular graph, ... |
| rusgrnumwwlkg 26871 | In a ` K `-regular graph, ... |
| rusgrnumwlkg 26872 | In a k-regular graph, the ... |
| clwwlknclwwlkdifs 26873 | The set of walks of length... |
| clwwlknclwwlkdifnum 26874 | In a k-regular graph, the ... |
| clwwlks 26879 | The set of closed walks (i... |
| isclwwlks 26880 | Properties of a word to re... |
| clwwlksn 26881 | The set of closed walks (i... |
| isclwwlksn 26882 | A word over the set of ver... |
| clwwlkbp 26883 | Basic properties of a clos... |
| clwwlknbp0 26884 | Basic properties of a clos... |
| clwwlknbp 26885 | Basic properties of a clos... |
| clwwlksnwrd 26886 | A closed walk of a fixed l... |
| clwwlknp 26887 | Properties of a set being ... |
| isclwwlksng 26888 | Properties of a word to re... |
| isclwwlksnx 26889 | Properties of a word to re... |
| clwwlksnndef 26890 | Conditions for ` ClWWalksN... |
| clwwlkclwwlkn 26891 | A closed walk of a fixed l... |
| clwwlkssclwwlksn 26892 | The closed walks of a fixe... |
| clwlkclwwlklem2a1 26893 | Lemma 1 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a2 26894 | Lemma 2 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a3 26895 | Lemma 3 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2fv1 26896 | Lemma 4a for ~ clwlkclwwlk... |
| clwlkclwwlklem2fv2 26897 | Lemma 4b for ~ clwlkclwwlk... |
| clwlkclwwlklem2a4 26898 | Lemma 4 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a 26899 | Lemma for ~ clwlkclwwlklem... |
| clwlkclwwlklem1 26900 | Lemma 1 for ~ clwlkclwwlk ... |
| clwlkclwwlklem2 26901 | Lemma 2 for ~ clwlkclwwlk ... |
| clwlkclwwlklem3 26902 | Lemma 3 for ~ clwlkclwwlk ... |
| clwlkclwwlk 26903 | A closed walk as word of l... |
| clwlkclwwlk2 26904 | A closed walk corresponds ... |
| clwwlkinwwlk 26905 | If the initial vertex of a... |
| clwwlksgt0 26906 | There is no empty closed w... |
| clwwlksn0 26907 | There is no closed walk of... |
| clwwlks1loop 26908 | A closed walk of length 1 ... |
| clwwlksn1loop 26909 | A closed walk of length 1 ... |
| clwwlksn2 26910 | A closed walk of length 2 ... |
| clwwlkssswrd 26911 | Closed walks (represented ... |
| umgrclwwlksge2 26912 | A closed walk in a multigr... |
| clwwlksnfi 26913 | If there is only a finite ... |
| clwwlksel 26914 | Obtaining a closed walk (a... |
| clwwlksf 26915 | Lemma 1 for ~ clwwlksbij :... |
| clwwlksfv 26916 | Lemma 2 for ~ clwwlksbij :... |
| clwwlksf1 26917 | Lemma 3 for ~ clwwlksbij :... |
| clwwlksfo 26918 | Lemma 4 for ~ clwwlksbij :... |
| clwwlksf1o 26919 | Lemma 5 for ~ clwwlksbij :... |
| clwwlksbij 26920 | There is a bijection betwe... |
| clwwlksnwwlkncl 26921 | Obtaining a closed walk (a... |
| clwwlksvbij 26922 | There is a bijection betwe... |
| clwwlksext2edg 26923 | If a word concatenated wit... |
| wwlksext2clwwlk 26924 | If a word represents a wal... |
| wwlksubclwwlks 26925 | Any prefix of a word repre... |
| clwwisshclwwslemlem 26926 | Lemma for ~ clwwisshclwwsl... |
| clwwisshclwwslem 26927 | Lemma for ~ clwwisshclwws ... |
| clwwisshclwws 26928 | Cyclically shifting a clos... |
| clwwisshclwwsn 26929 | Cyclically shifting a clos... |
| clwwnisshclwwsn 26930 | Cyclically shifting a clos... |
| erclwwlksrel 26931 | ` .~ ` is a relation. (Co... |
| erclwwlkseq 26932 | Two classes are equivalent... |
| erclwwlkseqlen 26933 | If two classes are equival... |
| erclwwlksref 26934 | ` .~ ` is a reflexive rela... |
| erclwwlkssym 26935 | ` .~ ` is a symmetric rela... |
| erclwwlkstr 26936 | ` .~ ` is a transitive rel... |
| erclwwlks 26937 | ` .~ ` is an equivalence r... |
| eleclclwwlksnlem1 26938 | Lemma 1 for ~ eleclclwwlks... |
| eleclclwwlksnlem2 26939 | Lemma 2 for ~ eleclclwwlks... |
| clwwlksnscsh 26940 | The set of cyclical shifts... |
| umgr2cwwk2dif 26941 | If a word represents a clo... |
| umgr2cwwkdifex 26942 | If a word represents a clo... |
| erclwwlksnrel 26943 | ` .~ ` is a relation. (Co... |
| erclwwlksneq 26944 | Two classes are equivalent... |
| erclwwlksneqlen 26945 | If two classes are equival... |
| erclwwlksnref 26946 | ` .~ ` is a reflexive rela... |
| erclwwlksnsym 26947 | ` .~ ` is a symmetric rela... |
| erclwwlksntr 26948 | ` .~ ` is a transitive rel... |
| erclwwlksn 26949 | ` .~ ` is an equivalence r... |
| qerclwwlksnfi 26950 | The quotient set of the se... |
| hashclwwlksn0 26951 | The number of closed walks... |
| eclclwwlksn1 26952 | An equivalence class accor... |
| eleclclwwlksn 26953 | A member of an equivalence... |
| hashecclwwlksn1 26954 | The size of every equivale... |
| umgrhashecclwwlk 26955 | The size of every equivale... |
| fusgrhashclwwlkn 26956 | The size of the set of clo... |
| clwwlksndivn 26957 | The size of the set of clo... |
| clwlksfclwwlk2wrd 26958 | The second component of a ... |
| clwlksfclwwlk1hashn 26959 | The size of the first comp... |
| clwlksfclwwlk1hash 26960 | The size of the first comp... |
| clwlksfclwwlk2sswd 26961 | The size of a subword of t... |
| clwlksfclwwlk 26962 | There is a function betwee... |
| clwlksfoclwwlk 26963 | There is an onto function ... |
| clwlksf1clwwlklem0 26964 | Lemma 1 for ~ clwlksf1clww... |
| clwlksf1clwwlklem1 26965 | Lemma 1 for ~ clwlksf1clww... |
| clwlksf1clwwlklem2 26966 | Lemma 2 for ~ clwlksf1clww... |
| clwlksf1clwwlklem3 26967 | Lemma 3 for ~ clwlksf1clww... |
| clwlksf1clwwlklem 26968 | Lemma for ~ clwlksf1clwwlk... |
| clwlksf1clwwlk 26969 | There is a one-to-one func... |
| clwlksf1oclwwlk 26970 | There is a one-to-one onto... |
| clwlkssizeeq 26971 | The size of the set of clo... |
| clwlksndivn 26972 | The size of the set of clo... |
| clwwlksndisj 26973 | The sets of closed walks s... |
| clwwlksnun 26974 | The set of closed walks of... |
| 0ewlk 26975 | The empty set (empty seque... |
| 1ewlk 26976 | A sequence of 1 edge is an... |
| 0wlk 26977 | A pair of an empty set (of... |
| is0wlk 26978 | A pair of an empty set (of... |
| 0wlkonlem1 26979 | Lemma 1 for ~ 0wlkon and ~... |
| 0wlkonlem2 26980 | Lemma 2 for ~ 0wlkon and ~... |
| 0wlkon 26981 | A walk of length 0 from a ... |
| 0wlkons1 26982 | A walk of length 0 from a ... |
| 0trl 26983 | A pair of an empty set (of... |
| is0trl 26984 | A pair of an empty set (of... |
| 0trlon 26985 | A trail of length 0 from a... |
| 0pth 26986 | A pair of an empty set (of... |
| 0spth 26987 | A pair of an empty set (of... |
| 0pthon 26988 | A path of length 0 from a ... |
| 0pthon1 26989 | A path of length 0 from a ... |
| 0pthonv 26990 | For each vertex there is a... |
| 0clwlk 26991 | A pair of an empty set (of... |
| 0clwlk0 26992 | There is no closed walk in... |
| 0crct 26993 | A pair of an empty set (of... |
| 0cycl 26994 | A pair of an empty set (of... |
| 1pthdlem1 26995 | Lemma 1 for ~ 1pthd . (Co... |
| 1pthdlem2 26996 | Lemma 2 for ~ 1pthd . (Co... |
| 1wlkdlem1 26997 | Lemma 1 for ~ 1wlkd . (Co... |
| 1wlkdlem2 26998 | Lemma 2 for ~ 1wlkd . (Co... |
| 1wlkdlem3 26999 | Lemma 3 for ~ 1wlkd . (Co... |
| 1wlkdlem4 27000 | Lemma 4 for ~ 1wlkd . (Co... |
| 1wlkd 27001 | In a graph with two vertic... |
| 1trld 27002 | In a graph with two vertic... |
| 1pthd 27003 | In a graph with two vertic... |
| 1pthond 27004 | In a graph with two vertic... |
| upgr1wlkdlem1 27005 | Lemma 1 for ~ upgr1wlkd . ... |
| upgr1wlkdlem2 27006 | Lemma 2 for ~ upgr1wlkd . ... |
| upgr1wlkd 27007 | In a pseudograph with two ... |
| upgr1trld 27008 | In a pseudograph with two ... |
| upgr1pthd 27009 | In a pseudograph with two ... |
| upgr1pthond 27010 | In a pseudograph with two ... |
| lppthon 27011 | A loop (which is an edge a... |
| lp1cycl 27012 | A loop (which is an edge a... |
| 1pthon2v 27013 | For each pair of adjacent ... |
| 1pthon2ve 27014 | For each pair of adjacent ... |
| wlk2v2elem1 27015 | Lemma 1 for ~ wlk2v2e : ` ... |
| wlk2v2elem2 27016 | Lemma 2 for ~ wlk2v2e : T... |
| wlk2v2e 27017 | In a graph with two vertic... |
| ntrl2v2e 27018 | A walk which is not a trai... |
| 3wlkdlem1 27019 | Lemma 1 for ~ 3wlkd . (Co... |
| 3wlkdlem2 27020 | Lemma 2 for ~ 3wlkd . (Co... |
| 3wlkdlem3 27021 | Lemma 3 for ~ 3wlkd . (Co... |
| 3wlkdlem4 27022 | Lemma 4 for ~ 3wlkd . (Co... |
| 3wlkdlem5 27023 | Lemma 5 for ~ 3wlkd . (Co... |
| 3pthdlem1 27024 | Lemma 1 for ~ 3pthd . (Co... |
| 3wlkdlem6 27025 | Lemma 6 for ~ 3wlkd . (Co... |
| 3wlkdlem7 27026 | Lemma 7 for ~ 3wlkd . (Co... |
| 3wlkdlem8 27027 | Lemma 8 for ~ 3wlkd . (Co... |
| 3wlkdlem9 27028 | Lemma 9 for ~ 3wlkd . (Co... |
| 3wlkdlem10 27029 | Lemma 10 for ~ 3wlkd . (C... |
| 3wlkd 27030 | Construction of a walk fro... |
| 3wlkond 27031 | A walk of length 3 from on... |
| 3trld 27032 | Construction of a trail fr... |
| 3trlond 27033 | A trail of length 3 from o... |
| 3pthd 27034 | A path of length 3 from on... |
| 3pthond 27035 | A path of length 3 from on... |
| 3spthd 27036 | A simple path of length 3 ... |
| 3spthond 27037 | A simple path of length 3 ... |
| 3cycld 27038 | Construction of a 3-cycle ... |
| 3cyclpd 27039 | Construction of a 3-cycle ... |
| upgr3v3e3cycl 27040 | If there is a cycle of len... |
| uhgr3cyclexlem 27041 | Lemma for ~ uhgr3cyclex . ... |
| uhgr3cyclex 27042 | If there are three differe... |
| umgr3cyclex 27043 | If there are three (differ... |
| umgr3v3e3cycl 27044 | If and only if there is a ... |
| upgr4cycl4dv4e 27045 | If there is a cycle of len... |
| dfconngr1 27048 | Alternative definition of ... |
| isconngr 27049 | The property of being a co... |
| isconngr1 27050 | The property of being a co... |
| cusconngr 27051 | A complete hypergraph is c... |
| 0conngr 27052 | A graph without vertices i... |
| 0vconngr 27053 | A graph without vertices i... |
| 1conngr 27054 | A graph with (at most) one... |
| conngrv2edg 27055 | A vertex in a connected gr... |
| vdn0conngrumgrv2 27056 | A vertex in a connected mu... |
| releupth 27059 | The set ` ( EulerPaths `` ... |
| eupths 27060 | The Eulerian paths on the ... |
| iseupth 27061 | The property " ` <. F , P ... |
| iseupthf1o 27062 | The property " ` <. F , P ... |
| eupthi 27063 | Properties of an Eulerian ... |
| eupthf1o 27064 | The ` F ` function in an E... |
| eupthfi 27065 | Any graph with an Eulerian... |
| eupthseg 27066 | The ` N ` -th edge in an e... |
| upgriseupth 27067 | The property " ` <. F , P ... |
| upgreupthi 27068 | Properties of an Eulerian ... |
| upgreupthseg 27069 | The ` N ` -th edge in an e... |
| eupthcl 27070 | An Eulerian path has lengt... |
| eupthistrl 27071 | An Eulerian path is a trai... |
| eupthiswlk 27072 | An Eulerian path is a walk... |
| eupthpf 27073 | The ` P ` function in an E... |
| eupth0 27074 | There is an Eulerian path ... |
| eupthres 27075 | The restriction ` <. H , Q... |
| eupthp1 27076 | Append one path segment to... |
| eupth2eucrct 27077 | Append one path segment to... |
| eupth2lem1 27078 | Lemma for ~ eupth2 . (Con... |
| eupth2lem2 27079 | Lemma for ~ eupth2 . (Con... |
| trlsegvdeglem1 27080 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem2 27081 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem3 27082 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem4 27083 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem5 27084 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem6 27085 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem7 27086 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeg 27087 | Formerly part of proof of ... |
| eupth2lem3lem1 27088 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem2 27089 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem3 27090 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem4 27091 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem5 27092 | Lemma for ~ eupth2 . (Con... |
| eupth2lem3lem6 27093 | Formerly part of proof of ... |
| eupth2lem3lem7 27094 | Lemma for ~ eupth2lem3 : ... |
| eupthvdres 27095 | Formerly part of proof of ... |
| eupth2lem3 27096 | Lemma for ~ eupth2 . (Con... |
| eupth2lemb 27097 | Lemma for ~ eupth2 (induct... |
| eupth2lems 27098 | Lemma for ~ eupth2 (induct... |
| eupth2 27099 | The only vertices of odd d... |
| eulerpathpr 27100 | A graph with an Eulerian p... |
| eulerpath 27101 | A pseudograph with an Eule... |
| eulercrct 27102 | A pseudograph with an Eule... |
| eucrctshift 27103 | Cyclically shifting the in... |
| eucrct2eupth1 27104 | Removing one edge ` ( I ``... |
| eucrct2eupth 27105 | Removing one edge ` ( I ``... |
| konigsbergvtx 27106 | The set of vertices of the... |
| konigsbergiedg 27107 | The indexed edges of the K... |
| konigsbergiedgw 27108 | The indexed edges of the K... |
| konigsbergiedgwOLD 27109 | The indexed edges of the K... |
| konigsbergssiedgwpr 27110 | Each subset of the indexed... |
| konigsbergssiedgw 27111 | Each subset of the indexed... |
| konigsbergumgr 27112 | The Königsberg graph ... |
| konigsbergupgrOLD 27113 | The Königsberg graph ... |
| konigsberglem1 27114 | Lemma 1 for ~ konigsberg :... |
| konigsberglem2 27115 | Lemma 2 for ~ konigsberg :... |
| konigsberglem3 27116 | Lemma 3 for ~ konigsberg :... |
| konigsberglem4 27117 | Lemma 4 for ~ konigsberg :... |
| konigsberglem5 27118 | Lemma 5 for ~ konigsberg :... |
| konigsberg 27119 | The Königsberg Bridge... |
| isfrgr 27122 | The property of being a fr... |
| frgrusgrfrcond 27123 | A friendship graph is a si... |
| frgrusgr 27124 | A friendship graph is a si... |
| frgr0v 27125 | Any null graph (set with n... |
| frgr0vb 27126 | Any null graph (without ve... |
| frgruhgr0v 27127 | Any null graph (without ve... |
| frgr0 27128 | The null graph (graph with... |
| rspc2vd 27129 | Deduction version of 2-var... |
| frcond1 27130 | The friendship condition: ... |
| frcond2 27131 | The friendship condition: ... |
| frgreu 27132 | Variant of ~ frcond2 : An... |
| frcond3 27133 | The friendship condition, ... |
| frcond4 27134 | The friendship condition, ... |
| frgr1v 27135 | Any graph with (at most) o... |
| nfrgr2v 27136 | Any graph with two (differ... |
| frgr3vlem1 27137 | Lemma 1 for ~ frgr3v . (C... |
| frgr3vlem2 27138 | Lemma 2 for ~ frgr3v . (C... |
| frgr3v 27139 | Any graph with three verti... |
| 1vwmgr 27140 | Every graph with one verte... |
| 3vfriswmgrlem 27141 | Lemma for ~ 3vfriswmgr . ... |
| 3vfriswmgr 27142 | Every friendship graph wit... |
| 1to2vfriswmgr 27143 | Every friendship graph wit... |
| 1to3vfriswmgr 27144 | Every friendship graph wit... |
| 1to3vfriendship 27145 | The friendship theorem for... |
| 2pthfrgrrn 27146 | Between any two (different... |
| 2pthfrgrrn2 27147 | Between any two (different... |
| 2pthfrgr 27148 | Between any two (different... |
| 3cyclfrgrrn1 27149 | Every vertex in a friendsh... |
| 3cyclfrgrrn 27150 | Every vertex in a friendsh... |
| 3cyclfrgrrn2 27151 | Every vertex in a friendsh... |
| 3cyclfrgr 27152 | Every vertex in a friendsh... |
| 4cycl2v2nb 27153 | In a (maybe degenerated) 4... |
| 4cycl2vnunb 27154 | In a 4-cycle, two distinct... |
| n4cyclfrgr 27155 | There is no 4-cycle in a f... |
| 4cyclusnfrgr 27156 | A graph with a 4-cycle is ... |
| frgrnbnb 27157 | If two neighbors ` U ` and... |
| frgrconngr 27158 | A friendship graph is conn... |
| vdgn0frgrv2 27159 | A vertex in a friendship g... |
| vdgn1frgrv2 27160 | Any vertex in a friendship... |
| vdgn1frgrv3 27161 | Any vertex in a friendship... |
| vdgfrgrgt2 27162 | Any vertex in a friendship... |
| frgrncvvdeqlem1 27163 | Lemma 1 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem2 27164 | Lemma 2 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem3 27165 | Lemma 3 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem4 27166 | Lemma 4 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem5 27167 | Lemma 5 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem6 27168 | Lemma 6 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem7 27169 | Lemma 7 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem8 27170 | Lemma 8 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem9 27171 | Lemma 9 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem10 27172 | Lemma 10 for ~ frgrncvvdeq... |
| frgrncvvdeq 27173 | In a friendship graph, two... |
| frgrwopreglem4a 27174 | In a friendship graph any ... |
| frgrwopreglem5a 27175 | If a friendship graph has ... |
| frgrwopreglem1 27176 | Lemma 1 for ~ frgrwopreg :... |
| frgrwopreglem2 27177 | Lemma 2 for ~ frgrwopreg .... |
| frgrwopreglem3 27178 | Lemma 3 for ~ frgrwopreg .... |
| frgrwopreglem4 27179 | Lemma 4 for ~ frgrwopreg .... |
| frgrwopregasn 27180 | According to statement 5 i... |
| frgrwopregbsn 27181 | According to statement 5 i... |
| frgrwopreg1 27182 | According to statement 5 i... |
| frgrwopreg2 27183 | According to statement 5 i... |
| frgrwopreglem5lem 27184 | Lemma for ~ frgrwopreglem5... |
| frgrwopreglem5 27185 | Lemma 5 for ~ frgrwopreg .... |
| frgrwopreglem5ALT 27186 | Alternate direct proof of ... |
| frgrwopreg 27187 | In a friendship graph ther... |
| frgrregorufr0 27188 | In a friendship graph ther... |
| frgrregorufr 27189 | If there is a vertex havin... |
| frgrregorufrg 27190 | If there is a vertex havin... |
| frgr2wwlkeu 27191 | For two different vertices... |
| frgr2wwlkn0 27192 | In a friendship graph, the... |
| frgr2wwlk1 27193 | In a friendship graph, the... |
| frgr2wsp1 27194 | In a friendship graph, the... |
| frgr2wwlkeqm 27195 | If there is a (simple) pat... |
| frgrhash2wsp 27196 | The number of simple paths... |
| fusgreg2wsplem 27197 | Lemma for ~ fusgreg2wsp an... |
| fusgr2wsp2nb 27198 | The set of paths of length... |
| fusgreghash2wspv 27199 | According to statement 7 i... |
| fusgreg2wsp 27200 | In a finite simple graph, ... |
| 2wspmdisj 27201 | The sets of paths of lengt... |
| fusgreghash2wsp 27202 | In a finite k-regular grap... |
| frrusgrord0lem 27203 | Lemma for ~ frrusgrord0 . ... |
| frrusgrord0 27204 | If a nonempty finite frien... |
| frrusgrord 27205 | If a nonempty finite frien... |
| numclwlk3lem3 27206 | Lemma 3 for ~ numclwwlk3 .... |
| extwwlkfablem1 27207 | Lemma 1 for ~ extwwlkfab .... |
| clwwlkextfrlem1 27208 | Lemma for ~ numclwwlk2lem1... |
| clwwlksnwwlksn 27209 | A word representing a clos... |
| extwwlkfablem2 27210 | Lemma 2 for ~ extwwlkfab .... |
| numclwwlkovf2exlem1 27211 | Lemma 1 for ~ numclwwlkovf... |
| numclwwlkovf2exlem2 27212 | Lemma 2 for ~ numclwwlkovf... |
| numclwwlkovf 27213 | Value of operation ` F ` ,... |
| numclwwlkffin 27214 | In a finite graph, the val... |
| numclwwlkffin0 27215 | In a finite graph, the val... |
| numclwwlkovfel2 27216 | Properties of an element o... |
| numclwwlkovf2 27217 | Value of operation ` F ` f... |
| numclwwlkovf2num 27218 | In a ` K `-regular graph, ... |
| numclwwlkovf2ex 27219 | Extending a closed walk st... |
| numclwwlkovg 27220 | Value of operation ` C ` ,... |
| numclwwlkovgel 27221 | Properties of an element o... |
| numclwlk1lem2foalem 27222 | Lemma for ~ numclwlk1lem2f... |
| extwwlkfab 27223 | The set ` ( X C N ) ` of c... |
| numclwlk1lem2foa 27224 | Going forth and back form ... |
| numclwlk1lem2f 27225 | ` T ` is a function, mappi... |
| numclwlk1lem2fv 27226 | Value of the function ` T ... |
| numclwlk1lem2f1 27227 | ` T ` is a 1-1 function. ... |
| numclwlk1lem2fo 27228 | ` T ` is an onto function.... |
| numclwlk1lem2f1o 27229 | ` T ` is a 1-1 onto functi... |
| numclwlk1lem2 27230 | There is a bijection betwe... |
| numclwwlk1 27231 | Statement 9 in [Huneke] p.... |
| numclwwlkovq 27232 | Value of operation ` Q ` ,... |
| numclwwlkqhash 27233 | In a ` K `-regular graph, ... |
| numclwwlkovh 27234 | Value of operation ` H ` ,... |
| numclwwlk2lem1 27235 | In a friendship graph, for... |
| numclwlk2lem2f 27236 | ` R ` is a function mappin... |
| numclwlk2lem2fv 27237 | Value of the function R. (... |
| numclwlk2lem2f1o 27238 | R is a 1-1 onto function. ... |
| numclwwlk2lem3 27239 | In a friendship graph, the... |
| numclwwlk2 27240 | Statement 10 in [Huneke] p... |
| numclwwlk3lem 27241 | Lemma for ~ numclwwlk3 . ... |
| numclwwlk3OLD 27242 | Obsolete version of ~ numc... |
| numclwwlk3 27243 | Statement 12 in [Huneke] p... |
| numclwwlk4 27244 | The total number of closed... |
| numclwwlk5lem 27245 | Lemma for ~ numclwwlk5 . ... |
| numclwwlk5 27246 | Statement 13 in [Huneke] p... |
| numclwwlk7lem 27247 | Lemma for ~ numclwwlk7 , ~... |
| numclwwlk6 27248 | For a prime divisor ` P ` ... |
| numclwwlk7 27249 | Statement 14 in [Huneke] p... |
| numclwwlk8 27250 | The size of the set of clo... |
| frgrreggt1 27251 | If a finite nonempty frien... |
| frgrreg 27252 | If a finite nonempty frien... |
| frgrregord013 27253 | If a finite friendship gra... |
| frgrregord13 27254 | If a nonempty finite frien... |
| frgrogt3nreg 27255 | If a finite friendship gra... |
| friendshipgt3 27256 | The friendship theorem for... |
| friendship 27257 | The friendship theorem: I... |
| conventions 27258 |
... |
| conventions-label 27259 |
... |
| natded 27260 | Here are typical n... |
| ex-natded5.2 27261 | Theorem 5.2 of [Clemente] ... |
| ex-natded5.2-2 27262 | A more efficient proof of ... |
| ex-natded5.2i 27263 | The same as ~ ex-natded5.2... |
| ex-natded5.3 27264 | Theorem 5.3 of [Clemente] ... |
| ex-natded5.3-2 27265 | A more efficient proof of ... |
| ex-natded5.3i 27266 | The same as ~ ex-natded5.3... |
| ex-natded5.5 27267 | Theorem 5.5 of [Clemente] ... |
| ex-natded5.7 27268 | Theorem 5.7 of [Clemente] ... |
| ex-natded5.7-2 27269 | A more efficient proof of ... |
| ex-natded5.8 27270 | Theorem 5.8 of [Clemente] ... |
| ex-natded5.8-2 27271 | A more efficient proof of ... |
| ex-natded5.13 27272 | Theorem 5.13 of [Clemente]... |
| ex-natded5.13-2 27273 | A more efficient proof of ... |
| ex-natded9.20 27274 | Theorem 9.20 of [Clemente]... |
| ex-natded9.20-2 27275 | A more efficient proof of ... |
| ex-natded9.26 27276 | Theorem 9.26 of [Clemente]... |
| ex-natded9.26-2 27277 | A more efficient proof of ... |
| ex-or 27278 | Example for ~ df-or . Exa... |
| ex-an 27279 | Example for ~ df-an . Exa... |
| ex-dif 27280 | Example for ~ df-dif . Ex... |
| ex-un 27281 | Example for ~ df-un . Exa... |
| ex-in 27282 | Example for ~ df-in . Exa... |
| ex-uni 27283 | Example for ~ df-uni . Ex... |
| ex-ss 27284 | Example for ~ df-ss . Exa... |
| ex-pss 27285 | Example for ~ df-pss . Ex... |
| ex-pw 27286 | Example for ~ df-pw . Exa... |
| ex-pr 27287 | Example for ~ df-pr . (Co... |
| ex-br 27288 | Example for ~ df-br . Exa... |
| ex-opab 27289 | Example for ~ df-opab . E... |
| ex-eprel 27290 | Example for ~ df-eprel . ... |
| ex-id 27291 | Example for ~ df-id . Exa... |
| ex-po 27292 | Example for ~ df-po . Exa... |
| ex-xp 27293 | Example for ~ df-xp . Exa... |
| ex-cnv 27294 | Example for ~ df-cnv . Ex... |
| ex-co 27295 | Example for ~ df-co . Exa... |
| ex-dm 27296 | Example for ~ df-dm . Exa... |
| ex-rn 27297 | Example for ~ df-rn . Exa... |
| ex-res 27298 | Example for ~ df-res . Ex... |
| ex-ima 27299 | Example for ~ df-ima . Ex... |
| ex-fv 27300 | Example for ~ df-fv . Exa... |
| ex-1st 27301 | Example for ~ df-1st . Ex... |
| ex-2nd 27302 | Example for ~ df-2nd . Ex... |
| 1kp2ke3k 27303 | Example for ~ df-dec , 100... |
| ex-fl 27304 | Example for ~ df-fl . Exa... |
| ex-ceil 27305 | Example for ~ df-ceil . (... |
| ex-mod 27306 | Example for ~ df-mod . (C... |
| ex-exp 27307 | Example for ~ df-exp . (C... |
| ex-fac 27308 | Example for ~ df-fac . (C... |
| ex-bc 27309 | Example for ~ df-bc . (Co... |
| ex-hash 27310 | Example for ~ df-hash . (... |
| ex-sqrt 27311 | Example for ~ df-sqrt . (... |
| ex-abs 27312 | Example for ~ df-abs . (C... |
| ex-dvds 27313 | Example for ~ df-dvds : 3 ... |
| ex-gcd 27314 | Example for ~ df-gcd . (C... |
| ex-lcm 27315 | Example for ~ df-lcm . (C... |
| ex-prmo 27316 | Example for ~ df-prmo : ` ... |
| aevdemo 27317 | Proof illustrating the com... |
| ex-ind-dvds 27318 | Example of a proof by indu... |
| avril1 27319 | Poisson d'Avril's Theorem.... |
| 2bornot2b 27320 | The law of excluded middle... |
| helloworld 27321 | The classic "Hello world" ... |
| 1p1e2apr1 27322 | One plus one equals two. ... |
| eqid1 27323 | Law of identity (reflexivi... |
| 1div0apr 27324 | Division by zero is forbid... |
| topnfbey 27325 | Nothing seems to be imposs... |
| isplig 27328 | The predicate "is a planar... |
| ispligb 27329 | The predicate "is a planar... |
| tncp 27330 | In any planar incidence ge... |
| l2p 27331 | For any line in a planar i... |
| lpni 27332 | For any line in a planar i... |
| nsnlplig 27333 | There is no "one-point lin... |
| nsnlpligALT 27334 | Alternate version of ~ nsn... |
| n0lplig 27335 | There is no "empty line" i... |
| n0lpligALT 27336 | Alternate version of ~ n0l... |
| eulplig 27337 | Through two distinct point... |
| pliguhgr 27338 | Any planar incidence geome... |
| dummylink 27341 | Alias for ~ a1ii that may ... |
| id1 27342 | Alias for ~ idALT that may... |
| isgrpo 27351 | The predicate "is a group ... |
| isgrpoi 27352 | Properties that determine ... |
| grpofo 27353 | A group operation maps ont... |
| grpocl 27354 | Closure law for a group op... |
| grpolidinv 27355 | A group has a left identit... |
| grpon0 27356 | The base set of a group is... |
| grpoass 27357 | A group operation is assoc... |
| grpoidinvlem1 27358 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem2 27359 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem3 27360 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem4 27361 | Lemma for ~ grpoidinv . (... |
| grpoidinv 27362 | A group has a left and rig... |
| grpoideu 27363 | The left identity element ... |
| grporndm 27364 | A group's range in terms o... |
| 0ngrp 27365 | The empty set is not a gro... |
| gidval 27366 | The value of the identity ... |
| grpoidval 27367 | Lemma for ~ grpoidcl and o... |
| grpoidcl 27368 | The identity element of a ... |
| grpoidinv2 27369 | A group's properties using... |
| grpolid 27370 | The identity element of a ... |
| grporid 27371 | The identity element of a ... |
| grporcan 27372 | Right cancellation law for... |
| grpoinveu 27373 | The left inverse element o... |
| grpoid 27374 | Two ways of saying that an... |
| grporn 27375 | The range of a group opera... |
| grpoinvfval 27376 | The inverse function of a ... |
| grpoinvval 27377 | The inverse of a group ele... |
| grpoinvcl 27378 | A group element's inverse ... |
| grpoinv 27379 | The properties of a group ... |
| grpolinv 27380 | The left inverse of a grou... |
| grporinv 27381 | The right inverse of a gro... |
| grpoinvid1 27382 | The inverse of a group ele... |
| grpoinvid2 27383 | The inverse of a group ele... |
| grpolcan 27384 | Left cancellation law for ... |
| grpo2inv 27385 | Double inverse law for gro... |
| grpoinvf 27386 | Mapping of the inverse fun... |
| grpoinvop 27387 | The inverse of the group o... |
| grpodivfval 27388 | Group division (or subtrac... |
| grpodivval 27389 | Group division (or subtrac... |
| grpodivinv 27390 | Group division by an inver... |
| grpoinvdiv 27391 | Inverse of a group divisio... |
| grpodivf 27392 | Mapping for group division... |
| grpodivcl 27393 | Closure of group division ... |
| grpodivdiv 27394 | Double group division. (C... |
| grpomuldivass 27395 | Associative-type law for m... |
| grpodivid 27396 | Division of a group member... |
| grponpcan 27397 | Cancellation law for group... |
| isablo 27400 | The predicate "is an Abeli... |
| ablogrpo 27401 | An Abelian group operation... |
| ablocom 27402 | An Abelian group operation... |
| ablo32 27403 | Commutative/associative la... |
| ablo4 27404 | Commutative/associative la... |
| isabloi 27405 | Properties that determine ... |
| ablomuldiv 27406 | Law for group multiplicati... |
| ablodivdiv 27407 | Law for double group divis... |
| ablodivdiv4 27408 | Law for double group divis... |
| ablodiv32 27409 | Swap the second and third ... |
| ablonnncan 27410 | Cancellation law for group... |
| ablonncan 27411 | Cancellation law for group... |
| ablonnncan1 27412 | Cancellation law for group... |
| vcrel 27415 | The class of all complex v... |
| vciOLD 27416 | Obsolete version of ~ cvsi... |
| vcsm 27417 | Functionality of th scalar... |
| vccl 27418 | Closure of the scalar prod... |
| vcidOLD 27419 | Identity element for the s... |
| vcdi 27420 | Distributive law for the s... |
| vcdir 27421 | Distributive law for the s... |
| vcass 27422 | Associative law for the sc... |
| vc2OLD 27423 | A vector plus itself is tw... |
| vcablo 27424 | Vector addition is an Abel... |
| vcgrp 27425 | Vector addition is a group... |
| vclcan 27426 | Left cancellation law for ... |
| vczcl 27427 | The zero vector is a vecto... |
| vc0rid 27428 | The zero vector is a right... |
| vc0 27429 | Zero times a vector is the... |
| vcz 27430 | Anything times the zero ve... |
| vcm 27431 | Minus 1 times a vector is ... |
| isvclem 27432 | Lemma for ~ isvcOLD . (Co... |
| vcex 27433 | The components of a comple... |
| isvcOLD 27434 | The predicate "is a comple... |
| isvciOLD 27435 | Properties that determine ... |
| cnaddabloOLD 27436 | Obsolete as of 23-Jan-2020... |
| cnidOLD 27437 | Obsolete as of 23-Jan-2020... |
| cncvcOLD 27438 | Obsolete version of ~ cncv... |
| nvss 27448 | Structure of the class of ... |
| nvvcop 27449 | A normed complex vector sp... |
| nvrel 27457 | The class of all normed co... |
| vafval 27458 | Value of the function for ... |
| bafval 27459 | Value of the function for ... |
| smfval 27460 | Value of the function for ... |
| 0vfval 27461 | Value of the function for ... |
| nmcvfval 27462 | Value of the norm function... |
| nvop2 27463 | A normed complex vector sp... |
| nvvop 27464 | The vector space component... |
| isnvlem 27465 | Lemma for ~ isnv . (Contr... |
| nvex 27466 | The components of a normed... |
| isnv 27467 | The predicate "is a normed... |
| isnvi 27468 | Properties that determine ... |
| nvi 27469 | The properties of a normed... |
| nvvc 27470 | The vector space component... |
| nvablo 27471 | The vector addition operat... |
| nvgrp 27472 | The vector addition operat... |
| nvgf 27473 | Mapping for the vector add... |
| nvsf 27474 | Mapping for the scalar mul... |
| nvgcl 27475 | Closure law for the vector... |
| nvcom 27476 | The vector addition (group... |
| nvass 27477 | The vector addition (group... |
| nvadd32 27478 | Commutative/associative la... |
| nvrcan 27479 | Right cancellation law for... |
| nvadd4 27480 | Rearrangement of 4 terms i... |
| nvscl 27481 | Closure law for the scalar... |
| nvsid 27482 | Identity element for the s... |
| nvsass 27483 | Associative law for the sc... |
| nvscom 27484 | Commutative law for the sc... |
| nvdi 27485 | Distributive law for the s... |
| nvdir 27486 | Distributive law for the s... |
| nv2 27487 | A vector plus itself is tw... |
| vsfval 27488 | Value of the function for ... |
| nvzcl 27489 | Closure law for the zero v... |
| nv0rid 27490 | The zero vector is a right... |
| nv0lid 27491 | The zero vector is a left ... |
| nv0 27492 | Zero times a vector is the... |
| nvsz 27493 | Anything times the zero ve... |
| nvinv 27494 | Minus 1 times a vector is ... |
| nvinvfval 27495 | Function for the negative ... |
| nvm 27496 | Vector subtraction in term... |
| nvmval 27497 | Value of vector subtractio... |
| nvmval2 27498 | Value of vector subtractio... |
| nvmfval 27499 | Value of the function for ... |
| nvmf 27500 | Mapping for the vector sub... |
| nvmcl 27501 | Closure law for the vector... |
| nvnnncan1 27502 | Cancellation law for vecto... |
| nvmdi 27503 | Distributive law for scala... |
| nvnegneg 27504 | Double negative of a vecto... |
| nvmul0or 27505 | If a scalar product is zer... |
| nvrinv 27506 | A vector minus itself. (C... |
| nvlinv 27507 | Minus a vector plus itself... |
| nvpncan2 27508 | Cancellation law for vecto... |
| nvpncan 27509 | Cancellation law for vecto... |
| nvaddsub 27510 | Commutative/associative la... |
| nvnpcan 27511 | Cancellation law for a nor... |
| nvaddsub4 27512 | Rearrangement of 4 terms i... |
| nvmeq0 27513 | The difference between two... |
| nvmid 27514 | A vector minus itself is t... |
| nvf 27515 | Mapping for the norm funct... |
| nvcl 27516 | The norm of a normed compl... |
| nvcli 27517 | The norm of a normed compl... |
| nvs 27518 | Proportionality property o... |
| nvsge0 27519 | The norm of a scalar produ... |
| nvm1 27520 | The norm of the negative o... |
| nvdif 27521 | The norm of the difference... |
| nvpi 27522 | The norm of a vector plus ... |
| nvz0 27523 | The norm of a zero vector ... |
| nvz 27524 | The norm of a vector is ze... |
| nvtri 27525 | Triangle inequality for th... |
| nvmtri 27526 | Triangle inequality for th... |
| nvabs 27527 | Norm difference property o... |
| nvge0 27528 | The norm of a normed compl... |
| nvgt0 27529 | A nonzero norm is positive... |
| nv1 27530 | From any nonzero vector, c... |
| nvop 27531 | A complex inner product sp... |
| cnnv 27532 | The set of complex numbers... |
| cnnvg 27533 | The vector addition (group... |
| cnnvba 27534 | The base set of the normed... |
| cnnvs 27535 | The scalar product operati... |
| cnnvnm 27536 | The norm operation of the ... |
| cnnvm 27537 | The vector subtraction ope... |
| elimnv 27538 | Hypothesis elimination lem... |
| elimnvu 27539 | Hypothesis elimination lem... |
| imsval 27540 | Value of the induced metri... |
| imsdval 27541 | Value of the induced metri... |
| imsdval2 27542 | Value of the distance func... |
| nvnd 27543 | The norm of a normed compl... |
| imsdf 27544 | Mapping for the induced me... |
| imsmetlem 27545 | Lemma for ~ imsmet . (Con... |
| imsmet 27546 | The induced metric of a no... |
| imsxmet 27547 | The induced metric of a no... |
| cnims 27548 | The metric induced on the ... |
| vacn 27549 | Vector addition is jointly... |
| nmcvcn 27550 | The norm of a normed compl... |
| nmcnc 27551 | The norm of a normed compl... |
| smcnlem 27552 | Lemma for ~ smcn . (Contr... |
| smcn 27553 | Scalar multiplication is j... |
| vmcn 27554 | Vector subtraction is join... |
| dipfval 27557 | The inner product function... |
| ipval 27558 | Value of the inner product... |
| ipval2lem2 27559 | Lemma for ~ ipval3 . (Con... |
| ipval2lem3 27560 | Lemma for ~ ipval3 . (Con... |
| ipval2lem4 27561 | Lemma for ~ ipval3 . (Con... |
| ipval2 27562 | Expansion of the inner pro... |
| 4ipval2 27563 | Four times the inner produ... |
| ipval3 27564 | Expansion of the inner pro... |
| ipidsq 27565 | The inner product of a vec... |
| ipnm 27566 | Norm expressed in terms of... |
| dipcl 27567 | An inner product is a comp... |
| ipf 27568 | Mapping for the inner prod... |
| dipcj 27569 | The complex conjugate of a... |
| ipipcj 27570 | An inner product times its... |
| diporthcom 27571 | Orthogonality (meaning inn... |
| dip0r 27572 | Inner product with a zero ... |
| dip0l 27573 | Inner product with a zero ... |
| ipz 27574 | The inner product of a vec... |
| dipcn 27575 | Inner product is jointly c... |
| sspval 27578 | The set of all subspaces o... |
| isssp 27579 | The predicate "is a subspa... |
| sspid 27580 | A normed complex vector sp... |
| sspnv 27581 | A subspace is a normed com... |
| sspba 27582 | The base set of a subspace... |
| sspg 27583 | Vector addition on a subsp... |
| sspgval 27584 | Vector addition on a subsp... |
| ssps 27585 | Scalar multiplication on a... |
| sspsval 27586 | Scalar multiplication on a... |
| sspmlem 27587 | Lemma for ~ sspm and other... |
| sspmval 27588 | Vector addition on a subsp... |
| sspm 27589 | Vector subtraction on a su... |
| sspz 27590 | The zero vector of a subsp... |
| sspn 27591 | The norm on a subspace is ... |
| sspnval 27592 | The norm on a subspace in ... |
| sspimsval 27593 | The induced metric on a su... |
| sspims 27594 | The induced metric on a su... |
| lnoval 27607 | The set of linear operator... |
| islno 27608 | The predicate "is a linear... |
| lnolin 27609 | Basic linearity property o... |
| lnof 27610 | A linear operator is a map... |
| lno0 27611 | The value of a linear oper... |
| lnocoi 27612 | The composition of two lin... |
| lnoadd 27613 | Addition property of a lin... |
| lnosub 27614 | Subtraction property of a ... |
| lnomul 27615 | Scalar multiplication prop... |
| nvo00 27616 | Two ways to express a zero... |
| nmoofval 27617 | The operator norm function... |
| nmooval 27618 | The operator norm function... |
| nmosetre 27619 | The set in the supremum of... |
| nmosetn0 27620 | The set in the supremum of... |
| nmoxr 27621 | The norm of an operator is... |
| nmooge0 27622 | The norm of an operator is... |
| nmorepnf 27623 | The norm of an operator is... |
| nmoreltpnf 27624 | The norm of any operator i... |
| nmogtmnf 27625 | The norm of an operator is... |
| nmoolb 27626 | A lower bound for an opera... |
| nmoubi 27627 | An upper bound for an oper... |
| nmoub3i 27628 | An upper bound for an oper... |
| nmoub2i 27629 | An upper bound for an oper... |
| nmobndi 27630 | Two ways to express that a... |
| nmounbi 27631 | Two ways two express that ... |
| nmounbseqi 27632 | An unbounded operator dete... |
| nmounbseqiALT 27633 | Alternate shorter proof of... |
| nmobndseqi 27634 | A bounded sequence determi... |
| nmobndseqiALT 27635 | Alternate shorter proof of... |
| bloval 27636 | The class of bounded linea... |
| isblo 27637 | The predicate "is a bounde... |
| isblo2 27638 | The predicate "is a bounde... |
| bloln 27639 | A bounded operator is a li... |
| blof 27640 | A bounded operator is an o... |
| nmblore 27641 | The norm of a bounded oper... |
| 0ofval 27642 | The zero operator between ... |
| 0oval 27643 | Value of the zero operator... |
| 0oo 27644 | The zero operator is an op... |
| 0lno 27645 | The zero operator is linea... |
| nmoo0 27646 | The operator norm of the z... |
| 0blo 27647 | The zero operator is a bou... |
| nmlno0lem 27648 | Lemma for ~ nmlno0i . (Co... |
| nmlno0i 27649 | The norm of a linear opera... |
| nmlno0 27650 | The norm of a linear opera... |
| nmlnoubi 27651 | An upper bound for the ope... |
| nmlnogt0 27652 | The norm of a nonzero line... |
| lnon0 27653 | The domain of a nonzero li... |
| nmblolbii 27654 | A lower bound for the norm... |
| nmblolbi 27655 | A lower bound for the norm... |
| isblo3i 27656 | The predicate "is a bounde... |
| blo3i 27657 | Properties that determine ... |
| blometi 27658 | Upper bound for the distan... |
| blocnilem 27659 | Lemma for ~ blocni and ~ l... |
| blocni 27660 | A linear operator is conti... |
| lnocni 27661 | If a linear operator is co... |
| blocn 27662 | A linear operator is conti... |
| blocn2 27663 | A bounded linear operator ... |
| ajfval 27664 | The adjoint function. (Co... |
| hmoval 27665 | The set of Hermitian (self... |
| ishmo 27666 | The predicate "is a hermit... |
| phnv 27669 | Every complex inner produc... |
| phrel 27670 | The class of all complex i... |
| phnvi 27671 | Every complex inner produc... |
| isphg 27672 | The predicate "is a comple... |
| phop 27673 | A complex inner product sp... |
| cncph 27674 | The set of complex numbers... |
| elimph 27675 | Hypothesis elimination lem... |
| elimphu 27676 | Hypothesis elimination lem... |
| isph 27677 | The predicate "is an inner... |
| phpar2 27678 | The parallelogram law for ... |
| phpar 27679 | The parallelogram law for ... |
| ip0i 27680 | A slight variant of Equati... |
| ip1ilem 27681 | Lemma for ~ ip1i . (Contr... |
| ip1i 27682 | Equation 6.47 of [Ponnusam... |
| ip2i 27683 | Equation 6.48 of [Ponnusam... |
| ipdirilem 27684 | Lemma for ~ ipdiri . (Con... |
| ipdiri 27685 | Distributive law for inner... |
| ipasslem1 27686 | Lemma for ~ ipassi . Show... |
| ipasslem2 27687 | Lemma for ~ ipassi . Show... |
| ipasslem3 27688 | Lemma for ~ ipassi . Show... |
| ipasslem4 27689 | Lemma for ~ ipassi . Show... |
| ipasslem5 27690 | Lemma for ~ ipassi . Show... |
| ipasslem7 27691 | Lemma for ~ ipassi . Show... |
| ipasslem8 27692 | Lemma for ~ ipassi . By ~... |
| ipasslem9 27693 | Lemma for ~ ipassi . Conc... |
| ipasslem10 27694 | Lemma for ~ ipassi . Show... |
| ipasslem11 27695 | Lemma for ~ ipassi . Show... |
| ipassi 27696 | Associative law for inner ... |
| dipdir 27697 | Distributive law for inner... |
| dipdi 27698 | Distributive law for inner... |
| ip2dii 27699 | Inner product of two sums.... |
| dipass 27700 | Associative law for inner ... |
| dipassr 27701 | "Associative" law for seco... |
| dipassr2 27702 | "Associative" law for inne... |
| dipsubdir 27703 | Distributive law for inner... |
| dipsubdi 27704 | Distributive law for inner... |
| pythi 27705 | The Pythagorean theorem fo... |
| siilem1 27706 | Lemma for ~ sii . (Contri... |
| siilem2 27707 | Lemma for ~ sii . (Contri... |
| siii 27708 | Inference from ~ sii . (C... |
| sii 27709 | Schwarz inequality. Part ... |
| sspph 27710 | A subspace of an inner pro... |
| ipblnfi 27711 | A function ` F ` generated... |
| ip2eqi 27712 | Two vectors are equal iff ... |
| phoeqi 27713 | A condition implying that ... |
| ajmoi 27714 | Every operator has at most... |
| ajfuni 27715 | The adjoint function is a ... |
| ajfun 27716 | The adjoint function is a ... |
| ajval 27717 | Value of the adjoint funct... |
| iscbn 27720 | A complex Banach space is ... |
| cbncms 27721 | The induced metric on comp... |
| bnnv 27722 | Every complex Banach space... |
| bnrel 27723 | The class of all complex B... |
| bnsscmcl 27724 | A subspace of a Banach spa... |
| cnbn 27725 | The set of complex numbers... |
| ubthlem1 27726 | Lemma for ~ ubth . The fu... |
| ubthlem2 27727 | Lemma for ~ ubth . Given ... |
| ubthlem3 27728 | Lemma for ~ ubth . Prove ... |
| ubth 27729 | Uniform Boundedness Theore... |
| minvecolem1 27730 | Lemma for ~ minveco . The... |
| minvecolem2 27731 | Lemma for ~ minveco . Any... |
| minvecolem3 27732 | Lemma for ~ minveco . The... |
| minvecolem4a 27733 | Lemma for ~ minveco . ` F ... |
| minvecolem4b 27734 | Lemma for ~ minveco . The... |
| minvecolem4c 27735 | Lemma for ~ minveco . The... |
| minvecolem4 27736 | Lemma for ~ minveco . The... |
| minvecolem5 27737 | Lemma for ~ minveco . Dis... |
| minvecolem6 27738 | Lemma for ~ minveco . Any... |
| minvecolem7 27739 | Lemma for ~ minveco . Sin... |
| minveco 27740 | Minimizing vector theorem,... |
| ishlo 27743 | The predicate "is a comple... |
| hlobn 27744 | Every complex Hilbert spac... |
| hlph 27745 | Every complex Hilbert spac... |
| hlrel 27746 | The class of all complex H... |
| hlnv 27747 | Every complex Hilbert spac... |
| hlnvi 27748 | Every complex Hilbert spac... |
| hlvc 27749 | Every complex Hilbert spac... |
| hlcmet 27750 | The induced metric on a co... |
| hlmet 27751 | The induced metric on a co... |
| hlpar2 27752 | The parallelogram law sati... |
| hlpar 27753 | The parallelogram law sati... |
| hlex 27754 | The base set of a Hilbert ... |
| hladdf 27755 | Mapping for Hilbert space ... |
| hlcom 27756 | Hilbert space vector addit... |
| hlass 27757 | Hilbert space vector addit... |
| hl0cl 27758 | The Hilbert space zero vec... |
| hladdid 27759 | Hilbert space addition wit... |
| hlmulf 27760 | Mapping for Hilbert space ... |
| hlmulid 27761 | Hilbert space scalar multi... |
| hlmulass 27762 | Hilbert space scalar multi... |
| hldi 27763 | Hilbert space scalar multi... |
| hldir 27764 | Hilbert space scalar multi... |
| hlmul0 27765 | Hilbert space scalar multi... |
| hlipf 27766 | Mapping for Hilbert space ... |
| hlipcj 27767 | Conjugate law for Hilbert ... |
| hlipdir 27768 | Distributive law for Hilbe... |
| hlipass 27769 | Associative law for Hilber... |
| hlipgt0 27770 | The inner product of a Hil... |
| hlcompl 27771 | Completeness of a Hilbert ... |
| cnchl 27772 | The set of complex numbers... |
| ssphl 27773 | A Banach subspace of an in... |
| htthlem 27774 | Lemma for ~ htth . The co... |
| htth 27775 | Hellinger-Toeplitz Theorem... |
| The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
| h2hva 27831 | The group (addition) opera... |
| h2hsm 27832 | The scalar product operati... |
| h2hnm 27833 | The norm function of Hilbe... |
| h2hvs 27834 | The vector subtraction ope... |
| h2hmetdval 27835 | Value of the distance func... |
| h2hcau 27836 | The Cauchy sequences of Hi... |
| h2hlm 27837 | The limit sequences of Hil... |
| axhilex-zf 27838 | Derive axiom ~ ax-hilex fr... |
| axhfvadd-zf 27839 | Derive axiom ~ ax-hfvadd f... |
| axhvcom-zf 27840 | Derive axiom ~ ax-hvcom fr... |
| axhvass-zf 27841 | Derive axiom ~ ax-hvass fr... |
| axhv0cl-zf 27842 | Derive axiom ~ ax-hv0cl fr... |
| axhvaddid-zf 27843 | Derive axiom ~ ax-hvaddid ... |
| axhfvmul-zf 27844 | Derive axiom ~ ax-hfvmul f... |
| axhvmulid-zf 27845 | Derive axiom ~ ax-hvmulid ... |
| axhvmulass-zf 27846 | Derive axiom ~ ax-hvmulass... |
| axhvdistr1-zf 27847 | Derive axiom ~ ax-hvdistr1... |
| axhvdistr2-zf 27848 | Derive axiom ~ ax-hvdistr2... |
| axhvmul0-zf 27849 | Derive axiom ~ ax-hvmul0 f... |
| axhfi-zf 27850 | Derive axiom ~ ax-hfi from... |
| axhis1-zf 27851 | Derive axiom ~ ax-his1 fro... |
| axhis2-zf 27852 | Derive axiom ~ ax-his2 fro... |
| axhis3-zf 27853 | Derive axiom ~ ax-his3 fro... |
| axhis4-zf 27854 | Derive axiom ~ ax-his4 fro... |
| axhcompl-zf 27855 | Derive axiom ~ ax-hcompl f... |
| hvmulex 27868 | The Hilbert space scalar p... |
| hvaddcl 27869 | Closure of vector addition... |
| hvmulcl 27870 | Closure of scalar multipli... |
| hvmulcli 27871 | Closure inference for scal... |
| hvsubf 27872 | Mapping domain and codomai... |
| hvsubval 27873 | Value of vector subtractio... |
| hvsubcl 27874 | Closure of vector subtract... |
| hvaddcli 27875 | Closure of vector addition... |
| hvcomi 27876 | Commutation of vector addi... |
| hvsubvali 27877 | Value of vector subtractio... |
| hvsubcli 27878 | Closure of vector subtract... |
| ifhvhv0 27879 | Prove ` if ( A e. ~H , A ,... |
| hvaddid2 27880 | Addition with the zero vec... |
| hvmul0 27881 | Scalar multiplication with... |
| hvmul0or 27882 | If a scalar product is zer... |
| hvsubid 27883 | Subtraction of a vector fr... |
| hvnegid 27884 | Addition of negative of a ... |
| hv2neg 27885 | Two ways to express the ne... |
| hvaddid2i 27886 | Addition with the zero vec... |
| hvnegidi 27887 | Addition of negative of a ... |
| hv2negi 27888 | Two ways to express the ne... |
| hvm1neg 27889 | Convert minus one times a ... |
| hvaddsubval 27890 | Value of vector addition i... |
| hvadd32 27891 | Commutative/associative la... |
| hvadd12 27892 | Commutative/associative la... |
| hvadd4 27893 | Hilbert vector space addit... |
| hvsub4 27894 | Hilbert vector space addit... |
| hvaddsub12 27895 | Commutative/associative la... |
| hvpncan 27896 | Addition/subtraction cance... |
| hvpncan2 27897 | Addition/subtraction cance... |
| hvaddsubass 27898 | Associativity of sum and d... |
| hvpncan3 27899 | Subtraction and addition o... |
| hvmulcom 27900 | Scalar multiplication comm... |
| hvsubass 27901 | Hilbert vector space assoc... |
| hvsub32 27902 | Hilbert vector space commu... |
| hvmulassi 27903 | Scalar multiplication asso... |
| hvmulcomi 27904 | Scalar multiplication comm... |
| hvmul2negi 27905 | Double negative in scalar ... |
| hvsubdistr1 27906 | Scalar multiplication dist... |
| hvsubdistr2 27907 | Scalar multiplication dist... |
| hvdistr1i 27908 | Scalar multiplication dist... |
| hvsubdistr1i 27909 | Scalar multiplication dist... |
| hvassi 27910 | Hilbert vector space assoc... |
| hvadd32i 27911 | Hilbert vector space commu... |
| hvsubassi 27912 | Hilbert vector space assoc... |
| hvsub32i 27913 | Hilbert vector space commu... |
| hvadd12i 27914 | Hilbert vector space commu... |
| hvadd4i 27915 | Hilbert vector space addit... |
| hvsubsub4i 27916 | Hilbert vector space addit... |
| hvsubsub4 27917 | Hilbert vector space addit... |
| hv2times 27918 | Two times a vector. (Cont... |
| hvnegdii 27919 | Distribution of negative o... |
| hvsubeq0i 27920 | If the difference between ... |
| hvsubcan2i 27921 | Vector cancellation law. ... |
| hvaddcani 27922 | Cancellation law for vecto... |
| hvsubaddi 27923 | Relationship between vecto... |
| hvnegdi 27924 | Distribution of negative o... |
| hvsubeq0 27925 | If the difference between ... |
| hvaddeq0 27926 | If the sum of two vectors ... |
| hvaddcan 27927 | Cancellation law for vecto... |
| hvaddcan2 27928 | Cancellation law for vecto... |
| hvmulcan 27929 | Cancellation law for scala... |
| hvmulcan2 27930 | Cancellation law for scala... |
| hvsubcan 27931 | Cancellation law for vecto... |
| hvsubcan2 27932 | Cancellation law for vecto... |
| hvsub0 27933 | Subtraction of a zero vect... |
| hvsubadd 27934 | Relationship between vecto... |
| hvaddsub4 27935 | Hilbert vector space addit... |
| hicl 27937 | Closure of inner product. ... |
| hicli 27938 | Closure inference for inne... |
| his5 27943 | Associative law for inner ... |
| his52 27944 | Associative law for inner ... |
| his35 27945 | Move scalar multiplication... |
| his35i 27946 | Move scalar multiplication... |
| his7 27947 | Distributive law for inner... |
| hiassdi 27948 | Distributive/associative l... |
| his2sub 27949 | Distributive law for inner... |
| his2sub2 27950 | Distributive law for inner... |
| hire 27951 | A necessary and sufficient... |
| hiidrcl 27952 | Real closure of inner prod... |
| hi01 27953 | Inner product with the 0 v... |
| hi02 27954 | Inner product with the 0 v... |
| hiidge0 27955 | Inner product with self is... |
| his6 27956 | Zero inner product with se... |
| his1i 27957 | Conjugate law for inner pr... |
| abshicom 27958 | Commuted inner products ha... |
| hial0 27959 | A vector whose inner produ... |
| hial02 27960 | A vector whose inner produ... |
| hisubcomi 27961 | Two vector subtractions si... |
| hi2eq 27962 | Lemma used to prove equali... |
| hial2eq 27963 | Two vectors whose inner pr... |
| hial2eq2 27964 | Two vectors whose inner pr... |
| orthcom 27965 | Orthogonality commutes. (... |
| normlem0 27966 | Lemma used to derive prope... |
| normlem1 27967 | Lemma used to derive prope... |
| normlem2 27968 | Lemma used to derive prope... |
| normlem3 27969 | Lemma used to derive prope... |
| normlem4 27970 | Lemma used to derive prope... |
| normlem5 27971 | Lemma used to derive prope... |
| normlem6 27972 | Lemma used to derive prope... |
| normlem7 27973 | Lemma used to derive prope... |
| normlem8 27974 | Lemma used to derive prope... |
| normlem9 27975 | Lemma used to derive prope... |
| normlem7tALT 27976 | Lemma used to derive prope... |
| bcseqi 27977 | Equality case of Bunjakova... |
| normlem9at 27978 | Lemma used to derive prope... |
| dfhnorm2 27979 | Alternate definition of th... |
| normf 27980 | The norm function maps fro... |
| normval 27981 | The value of the norm of a... |
| normcl 27982 | Real closure of the norm o... |
| normge0 27983 | The norm of a vector is no... |
| normgt0 27984 | The norm of nonzero vector... |
| norm0 27985 | The norm of a zero vector.... |
| norm-i 27986 | Theorem 3.3(i) of [Beran] ... |
| normne0 27987 | A norm is nonzero iff its ... |
| normcli 27988 | Real closure of the norm o... |
| normsqi 27989 | The square of a norm. (Co... |
| norm-i-i 27990 | Theorem 3.3(i) of [Beran] ... |
| normsq 27991 | The square of a norm. (Co... |
| normsub0i 27992 | Two vectors are equal iff ... |
| normsub0 27993 | Two vectors are equal iff ... |
| norm-ii-i 27994 | Triangle inequality for no... |
| norm-ii 27995 | Triangle inequality for no... |
| norm-iii-i 27996 | Theorem 3.3(iii) of [Beran... |
| norm-iii 27997 | Theorem 3.3(iii) of [Beran... |
| normsubi 27998 | Negative doesn't change th... |
| normpythi 27999 | Analogy to Pythagorean the... |
| normsub 28000 | Swapping order of subtract... |
| normneg 28001 | The norm of a vector equal... |
| normpyth 28002 | Analogy to Pythagorean the... |
| normpyc 28003 | Corollary to Pythagorean t... |
| norm3difi 28004 | Norm of differences around... |
| norm3adifii 28005 | Norm of differences around... |
| norm3lem 28006 | Lemma involving norm of di... |
| norm3dif 28007 | Norm of differences around... |
| norm3dif2 28008 | Norm of differences around... |
| norm3lemt 28009 | Lemma involving norm of di... |
| norm3adifi 28010 | Norm of differences around... |
| normpari 28011 | Parallelogram law for norm... |
| normpar 28012 | Parallelogram law for norm... |
| normpar2i 28013 | Corollary of parallelogram... |
| polid2i 28014 | Generalized polarization i... |
| polidi 28015 | Polarization identity. Re... |
| polid 28016 | Polarization identity. Re... |
| hilablo 28017 | Hilbert space vector addit... |
| hilid 28018 | The group identity element... |
| hilvc 28019 | Hilbert space is a complex... |
| hilnormi 28020 | Hilbert space norm in term... |
| hilhhi 28021 | Deduce the structure of Hi... |
| hhnv 28022 | Hilbert space is a normed ... |
| hhva 28023 | The group (addition) opera... |
| hhba 28024 | The base set of Hilbert sp... |
| hh0v 28025 | The zero vector of Hilbert... |
| hhsm 28026 | The scalar product operati... |
| hhvs 28027 | The vector subtraction ope... |
| hhnm 28028 | The norm function of Hilbe... |
| hhims 28029 | The induced metric of Hilb... |
| hhims2 28030 | Hilbert space distance met... |
| hhmet 28031 | The induced metric of Hilb... |
| hhxmet 28032 | The induced metric of Hilb... |
| hhmetdval 28033 | Value of the distance func... |
| hhip 28034 | The inner product operatio... |
| hhph 28035 | The Hilbert space of the H... |
| bcsiALT 28036 | Bunjakovaskij-Cauchy-Schwa... |
| bcsiHIL 28037 | Bunjakovaskij-Cauchy-Schwa... |
| bcs 28038 | Bunjakovaskij-Cauchy-Schwa... |
| bcs2 28039 | Corollary of the Bunjakova... |
| bcs3 28040 | Corollary of the Bunjakova... |
| hcau 28041 | Member of the set of Cauch... |
| hcauseq 28042 | A Cauchy sequences on a Hi... |
| hcaucvg 28043 | A Cauchy sequence on a Hil... |
| seq1hcau 28044 | A sequence on a Hilbert sp... |
| hlimi 28045 | Express the predicate: Th... |
| hlimseqi 28046 | A sequence with a limit on... |
| hlimveci 28047 | Closure of the limit of a ... |
| hlimconvi 28048 | Convergence of a sequence ... |
| hlim2 28049 | The limit of a sequence on... |
| hlimadd 28050 | Limit of the sum of two se... |
| hilmet 28051 | The Hilbert space norm det... |
| hilxmet 28052 | The Hilbert space norm det... |
| hilmetdval 28053 | Value of the distance func... |
| hilims 28054 | Hilbert space distance met... |
| hhcau 28055 | The Cauchy sequences of Hi... |
| hhlm 28056 | The limit sequences of Hil... |
| hhcmpl 28057 | Lemma used for derivation ... |
| hilcompl 28058 | Lemma used for derivation ... |
| hhcms 28060 | The Hilbert space induced ... |
| hhhl 28061 | The Hilbert space structur... |
| hilcms 28062 | The Hilbert space norm det... |
| hilhl 28063 | The Hilbert space of the H... |
| issh 28065 | Subspace ` H ` of a Hilber... |
| issh2 28066 | Subspace ` H ` of a Hilber... |
| shss 28067 | A subspace is a subset of ... |
| shel 28068 | A member of a subspace of ... |
| shex 28069 | The set of subspaces of a ... |
| shssii 28070 | A closed subspace of a Hil... |
| sheli 28071 | A member of a subspace of ... |
| shelii 28072 | A member of a subspace of ... |
| sh0 28073 | The zero vector belongs to... |
| shaddcl 28074 | Closure of vector addition... |
| shmulcl 28075 | Closure of vector scalar m... |
| issh3 28076 | Subspace ` H ` of a Hilber... |
| shsubcl 28077 | Closure of vector subtract... |
| isch 28079 | Closed subspace ` H ` of a... |
| isch2 28080 | Closed subspace ` H ` of a... |
| chsh 28081 | A closed subspace is a sub... |
| chsssh 28082 | Closed subspaces are subsp... |
| chex 28083 | The set of closed subspace... |
| chshii 28084 | A closed subspace is a sub... |
| ch0 28085 | The zero vector belongs to... |
| chss 28086 | A closed subspace of a Hil... |
| chel 28087 | A member of a closed subsp... |
| chssii 28088 | A closed subspace of a Hil... |
| cheli 28089 | A member of a closed subsp... |
| chelii 28090 | A member of a closed subsp... |
| chlimi 28091 | The limit property of a cl... |
| hlim0 28092 | The zero sequence in Hilbe... |
| hlimcaui 28093 | If a sequence in Hilbert s... |
| hlimf 28094 | Function-like behavior of ... |
| hlimuni 28095 | A Hilbert space sequence c... |
| hlimreui 28096 | The limit of a Hilbert spa... |
| hlimeui 28097 | The limit of a Hilbert spa... |
| isch3 28098 | A Hilbert subspace is clos... |
| chcompl 28099 | Completeness of a closed s... |
| helch 28100 | The unit Hilbert lattice e... |
| ifchhv 28101 | Prove ` if ( A e. CH , A ,... |
| helsh 28102 | Hilbert space is a subspac... |
| shsspwh 28103 | Subspaces are subsets of H... |
| chsspwh 28104 | Closed subspaces are subse... |
| hsn0elch 28105 | The zero subspace belongs ... |
| norm1 28106 | From any nonzero Hilbert s... |
| norm1exi 28107 | A normalized vector exists... |
| norm1hex 28108 | A normalized vector can ex... |
| elch0 28111 | Membership in zero for clo... |
| h0elch 28112 | The zero subspace is a clo... |
| h0elsh 28113 | The zero subspace is a sub... |
| hhssva 28114 | The vector addition operat... |
| hhsssm 28115 | The scalar multiplication ... |
| hhssnm 28116 | The norm operation on a su... |
| issubgoilem 28117 | Lemma for ~ hhssabloilem .... |
| hhssabloilem 28118 | Lemma for ~ hhssabloi . F... |
| hhssabloi 28119 | Abelian group property of ... |
| hhssablo 28120 | Abelian group property of ... |
| hhssnv 28121 | Normed complex vector spac... |
| hhssnvt 28122 | Normed complex vector spac... |
| hhsst 28123 | A member of ` SH ` is a su... |
| hhshsslem1 28124 | Lemma for ~ hhsssh . (Con... |
| hhshsslem2 28125 | Lemma for ~ hhsssh . (Con... |
| hhsssh 28126 | The predicate " ` H ` is a... |
| hhsssh2 28127 | The predicate " ` H ` is a... |
| hhssba 28128 | The base set of a subspace... |
| hhssvs 28129 | The vector subtraction ope... |
| hhssvsf 28130 | Mapping of the vector subt... |
| hhssph 28131 | Inner product space proper... |
| hhssims 28132 | Induced metric of a subspa... |
| hhssims2 28133 | Induced metric of a subspa... |
| hhssmet 28134 | Induced metric of a subspa... |
| hhssmetdval 28135 | Value of the distance func... |
| hhsscms 28136 | The induced metric of a cl... |
| hhssbn 28137 | Banach space property of a... |
| hhsshl 28138 | Hilbert space property of ... |
| ocval 28139 | Value of orthogonal comple... |
| ocel 28140 | Membership in orthogonal c... |
| shocel 28141 | Membership in orthogonal c... |
| ocsh 28142 | The orthogonal complement ... |
| shocsh 28143 | The orthogonal complement ... |
| ocss 28144 | An orthogonal complement i... |
| shocss 28145 | An orthogonal complement i... |
| occon 28146 | Contraposition law for ort... |
| occon2 28147 | Double contraposition for ... |
| occon2i 28148 | Double contraposition for ... |
| oc0 28149 | The zero vector belongs to... |
| ocorth 28150 | Members of a subset and it... |
| shocorth 28151 | Members of a subspace and ... |
| ococss 28152 | Inclusion in complement of... |
| shococss 28153 | Inclusion in complement of... |
| shorth 28154 | Members of orthogonal subs... |
| ocin 28155 | Intersection of a Hilbert ... |
| occon3 28156 | Hilbert lattice contraposi... |
| ocnel 28157 | A nonzero vector in the co... |
| chocvali 28158 | Value of the orthogonal co... |
| shuni 28159 | Two subspaces with trivial... |
| chocunii 28160 | Lemma for uniqueness part ... |
| pjhthmo 28161 | Projection Theorem, unique... |
| occllem 28162 | Lemma for ~ occl . (Contr... |
| occl 28163 | Closure of complement of H... |
| shoccl 28164 | Closure of complement of H... |
| choccl 28165 | Closure of complement of H... |
| choccli 28166 | Closure of ` CH ` orthocom... |
| shsval 28171 | Value of subspace sum of t... |
| shsss 28172 | The subspace sum is a subs... |
| shsel 28173 | Membership in the subspace... |
| shsel3 28174 | Membership in the subspace... |
| shseli 28175 | Membership in subspace sum... |
| shscli 28176 | Closure of subspace sum. ... |
| shscl 28177 | Closure of subspace sum. ... |
| shscom 28178 | Commutative law for subspa... |
| shsva 28179 | Vector sum belongs to subs... |
| shsel1 28180 | A subspace sum contains a ... |
| shsel2 28181 | A subspace sum contains a ... |
| shsvs 28182 | Vector subtraction belongs... |
| shsub1 28183 | Subspace sum is an upper b... |
| shsub2 28184 | Subspace sum is an upper b... |
| choc0 28185 | The orthocomplement of the... |
| choc1 28186 | The orthocomplement of the... |
| chocnul 28187 | Orthogonal complement of t... |
| shintcli 28188 | Closure of intersection of... |
| shintcl 28189 | The intersection of a none... |
| chintcli 28190 | The intersection of a none... |
| chintcl 28191 | The intersection (infimum)... |
| spanval 28192 | Value of the linear span o... |
| hsupval 28193 | Value of supremum of set o... |
| chsupval 28194 | The value of the supremum ... |
| spancl 28195 | The span of a subset of Hi... |
| elspancl 28196 | A member of a span is a ve... |
| shsupcl 28197 | Closure of the subspace su... |
| hsupcl 28198 | Closure of supremum of set... |
| chsupcl 28199 | Closure of supremum of sub... |
| hsupss 28200 | Subset relation for suprem... |
| chsupss 28201 | Subset relation for suprem... |
| hsupunss 28202 | The union of a set of Hilb... |
| chsupunss 28203 | The union of a set of clos... |
| spanss2 28204 | A subset of Hilbert space ... |
| shsupunss 28205 | The union of a set of subs... |
| spanid 28206 | A subspace of Hilbert spac... |
| spanss 28207 | Ordering relationship for ... |
| spanssoc 28208 | The span of a subset of Hi... |
| sshjval 28209 | Value of join for subsets ... |
| shjval 28210 | Value of join in ` SH ` . ... |
| chjval 28211 | Value of join in ` CH ` . ... |
| chjvali 28212 | Value of join in ` CH ` . ... |
| sshjval3 28213 | Value of join for subsets ... |
| sshjcl 28214 | Closure of join for subset... |
| shjcl 28215 | Closure of join in ` SH ` ... |
| chjcl 28216 | Closure of join in ` CH ` ... |
| shjcom 28217 | Commutative law for Hilber... |
| shless 28218 | Subset implies subset of s... |
| shlej1 28219 | Add disjunct to both sides... |
| shlej2 28220 | Add disjunct to both sides... |
| shincli 28221 | Closure of intersection of... |
| shscomi 28222 | Commutative law for subspa... |
| shsvai 28223 | Vector sum belongs to subs... |
| shsel1i 28224 | A subspace sum contains a ... |
| shsel2i 28225 | A subspace sum contains a ... |
| shsvsi 28226 | Vector subtraction belongs... |
| shunssi 28227 | Union is smaller than subs... |
| shunssji 28228 | Union is smaller than Hilb... |
| shsleji 28229 | Subspace sum is smaller th... |
| shjcomi 28230 | Commutative law for join i... |
| shsub1i 28231 | Subspace sum is an upper b... |
| shsub2i 28232 | Subspace sum is an upper b... |
| shub1i 28233 | Hilbert lattice join is an... |
| shjcli 28234 | Closure of ` CH ` join. (... |
| shjshcli 28235 | ` SH ` closure of join. (... |
| shlessi 28236 | Subset implies subset of s... |
| shlej1i 28237 | Add disjunct to both sides... |
| shlej2i 28238 | Add disjunct to both sides... |
| shslej 28239 | Subspace sum is smaller th... |
| shincl 28240 | Closure of intersection of... |
| shub1 28241 | Hilbert lattice join is an... |
| shub2 28242 | A subspace is a subset of ... |
| shsidmi 28243 | Idempotent law for Hilbert... |
| shslubi 28244 | The least upper bound law ... |
| shlesb1i 28245 | Hilbert lattice ordering i... |
| shsval2i 28246 | An alternate way to expres... |
| shsval3i 28247 | An alternate way to expres... |
| shmodsi 28248 | The modular law holds for ... |
| shmodi 28249 | The modular law is implied... |
| pjhthlem1 28250 | Lemma for ~ pjhth . (Cont... |
| pjhthlem2 28251 | Lemma for ~ pjhth . (Cont... |
| pjhth 28252 | Projection Theorem: Any H... |
| pjhtheu 28253 | Projection Theorem: Any H... |
| pjhfval 28255 | The value of the projectio... |
| pjhval 28256 | Value of a projection. (C... |
| pjpreeq 28257 | Equality with a projection... |
| pjeq 28258 | Equality with a projection... |
| axpjcl 28259 | Closure of a projection in... |
| pjhcl 28260 | Closure of a projection in... |
| omlsilem 28261 | Lemma for orthomodular law... |
| omlsii 28262 | Subspace inference form of... |
| omlsi 28263 | Subspace form of orthomodu... |
| ococi 28264 | Complement of complement o... |
| ococ 28265 | Complement of complement o... |
| dfch2 28266 | Alternate definition of th... |
| ococin 28267 | The double complement is t... |
| hsupval2 28268 | Alternate definition of su... |
| chsupval2 28269 | The value of the supremum ... |
| sshjval2 28270 | Value of join in the set o... |
| chsupid 28271 | A subspace is the supremum... |
| chsupsn 28272 | Value of supremum of subse... |
| shlub 28273 | Hilbert lattice join is th... |
| shlubi 28274 | Hilbert lattice join is th... |
| pjhtheu2 28275 | Uniqueness of ` y ` for th... |
| pjcli 28276 | Closure of a projection in... |
| pjhcli 28277 | Closure of a projection in... |
| pjpjpre 28278 | Decomposition of a vector ... |
| axpjpj 28279 | Decomposition of a vector ... |
| pjclii 28280 | Closure of a projection in... |
| pjhclii 28281 | Closure of a projection in... |
| pjpj0i 28282 | Decomposition of a vector ... |
| pjpji 28283 | Decomposition of a vector ... |
| pjpjhth 28284 | Projection Theorem: Any H... |
| pjpjhthi 28285 | Projection Theorem: Any H... |
| pjop 28286 | Orthocomplement projection... |
| pjpo 28287 | Projection in terms of ort... |
| pjopi 28288 | Orthocomplement projection... |
| pjpoi 28289 | Projection in terms of ort... |
| pjoc1i 28290 | Projection of a vector in ... |
| pjchi 28291 | Projection of a vector in ... |
| pjoccl 28292 | The part of a vector that ... |
| pjoc1 28293 | Projection of a vector in ... |
| pjomli 28294 | Subspace form of orthomodu... |
| pjoml 28295 | Subspace form of orthomodu... |
| pjococi 28296 | Proof of orthocomplement t... |
| pjoc2i 28297 | Projection of a vector in ... |
| pjoc2 28298 | Projection of a vector in ... |
| sh0le 28299 | The zero subspace is the s... |
| ch0le 28300 | The zero subspace is the s... |
| shle0 28301 | No subspace is smaller tha... |
| chle0 28302 | No Hilbert lattice element... |
| chnlen0 28303 | A Hilbert lattice element ... |
| ch0pss 28304 | The zero subspace is a pro... |
| orthin 28305 | The intersection of orthog... |
| ssjo 28306 | The lattice join of a subs... |
| shne0i 28307 | A nonzero subspace has a n... |
| shs0i 28308 | Hilbert subspace sum with ... |
| shs00i 28309 | Two subspaces are zero iff... |
| ch0lei 28310 | The closed subspace zero i... |
| chle0i 28311 | No Hilbert closed subspace... |
| chne0i 28312 | A nonzero closed subspace ... |
| chocini 28313 | Intersection of a closed s... |
| chj0i 28314 | Join with lattice zero in ... |
| chm1i 28315 | Meet with lattice one in `... |
| chjcli 28316 | Closure of ` CH ` join. (... |
| chsleji 28317 | Subspace sum is smaller th... |
| chseli 28318 | Membership in subspace sum... |
| chincli 28319 | Closure of Hilbert lattice... |
| chsscon3i 28320 | Hilbert lattice contraposi... |
| chsscon1i 28321 | Hilbert lattice contraposi... |
| chsscon2i 28322 | Hilbert lattice contraposi... |
| chcon2i 28323 | Hilbert lattice contraposi... |
| chcon1i 28324 | Hilbert lattice contraposi... |
| chcon3i 28325 | Hilbert lattice contraposi... |
| chunssji 28326 | Union is smaller than ` CH... |
| chjcomi 28327 | Commutative law for join i... |
| chub1i 28328 | ` CH ` join is an upper bo... |
| chub2i 28329 | ` CH ` join is an upper bo... |
| chlubi 28330 | Hilbert lattice join is th... |
| chlubii 28331 | Hilbert lattice join is th... |
| chlej1i 28332 | Add join to both sides of ... |
| chlej2i 28333 | Add join to both sides of ... |
| chlej12i 28334 | Add join to both sides of ... |
| chlejb1i 28335 | Hilbert lattice ordering i... |
| chdmm1i 28336 | De Morgan's law for meet i... |
| chdmm2i 28337 | De Morgan's law for meet i... |
| chdmm3i 28338 | De Morgan's law for meet i... |
| chdmm4i 28339 | De Morgan's law for meet i... |
| chdmj1i 28340 | De Morgan's law for join i... |
| chdmj2i 28341 | De Morgan's law for join i... |
| chdmj3i 28342 | De Morgan's law for join i... |
| chdmj4i 28343 | De Morgan's law for join i... |
| chnlei 28344 | Equivalent expressions for... |
| chjassi 28345 | Associative law for Hilber... |
| chj00i 28346 | Two Hilbert lattice elemen... |
| chjoi 28347 | The join of a closed subsp... |
| chj1i 28348 | Join with Hilbert lattice ... |
| chm0i 28349 | Meet with Hilbert lattice ... |
| chm0 28350 | Meet with Hilbert lattice ... |
| shjshsi 28351 | Hilbert lattice join equal... |
| shjshseli 28352 | A closed subspace sum equa... |
| chne0 28353 | A nonzero closed subspace ... |
| chocin 28354 | Intersection of a closed s... |
| chssoc 28355 | A closed subspace less tha... |
| chj0 28356 | Join with Hilbert lattice ... |
| chslej 28357 | Subspace sum is smaller th... |
| chincl 28358 | Closure of Hilbert lattice... |
| chsscon3 28359 | Hilbert lattice contraposi... |
| chsscon1 28360 | Hilbert lattice contraposi... |
| chsscon2 28361 | Hilbert lattice contraposi... |
| chpsscon3 28362 | Hilbert lattice contraposi... |
| chpsscon1 28363 | Hilbert lattice contraposi... |
| chpsscon2 28364 | Hilbert lattice contraposi... |
| chjcom 28365 | Commutative law for Hilber... |
| chub1 28366 | Hilbert lattice join is gr... |
| chub2 28367 | Hilbert lattice join is gr... |
| chlub 28368 | Hilbert lattice join is th... |
| chlej1 28369 | Add join to both sides of ... |
| chlej2 28370 | Add join to both sides of ... |
| chlejb1 28371 | Hilbert lattice ordering i... |
| chlejb2 28372 | Hilbert lattice ordering i... |
| chnle 28373 | Equivalent expressions for... |
| chjo 28374 | The join of a closed subsp... |
| chabs1 28375 | Hilbert lattice absorption... |
| chabs2 28376 | Hilbert lattice absorption... |
| chabs1i 28377 | Hilbert lattice absorption... |
| chabs2i 28378 | Hilbert lattice absorption... |
| chjidm 28379 | Idempotent law for Hilbert... |
| chjidmi 28380 | Idempotent law for Hilbert... |
| chj12i 28381 | A rearrangement of Hilbert... |
| chj4i 28382 | Rearrangement of the join ... |
| chjjdiri 28383 | Hilbert lattice join distr... |
| chdmm1 28384 | De Morgan's law for meet i... |
| chdmm2 28385 | De Morgan's law for meet i... |
| chdmm3 28386 | De Morgan's law for meet i... |
| chdmm4 28387 | De Morgan's law for meet i... |
| chdmj1 28388 | De Morgan's law for join i... |
| chdmj2 28389 | De Morgan's law for join i... |
| chdmj3 28390 | De Morgan's law for join i... |
| chdmj4 28391 | De Morgan's law for join i... |
| chjass 28392 | Associative law for Hilber... |
| chj12 28393 | A rearrangement of Hilbert... |
| chj4 28394 | Rearrangement of the join ... |
| ledii 28395 | An ortholattice is distrib... |
| lediri 28396 | An ortholattice is distrib... |
| lejdii 28397 | An ortholattice is distrib... |
| lejdiri 28398 | An ortholattice is distrib... |
| ledi 28399 | An ortholattice is distrib... |
| spansn0 28400 | The span of the singleton ... |
| span0 28401 | The span of the empty set ... |
| elspani 28402 | Membership in the span of ... |
| spanuni 28403 | The span of a union is the... |
| spanun 28404 | The span of a union is the... |
| sshhococi 28405 | The join of two Hilbert sp... |
| hne0 28406 | Hilbert space has a nonzer... |
| chsup0 28407 | The supremum of the empty ... |
| h1deoi 28408 | Membership in orthocomplem... |
| h1dei 28409 | Membership in 1-dimensiona... |
| h1did 28410 | A generating vector belong... |
| h1dn0 28411 | A nonzero vector generates... |
| h1de2i 28412 | Membership in 1-dimensiona... |
| h1de2bi 28413 | Membership in 1-dimensiona... |
| h1de2ctlem 28414 | Lemma for ~ h1de2ci . (Co... |
| h1de2ci 28415 | Membership in 1-dimensiona... |
| spansni 28416 | The span of a singleton in... |
| elspansni 28417 | Membership in the span of ... |
| spansn 28418 | The span of a singleton in... |
| spansnch 28419 | The span of a Hilbert spac... |
| spansnsh 28420 | The span of a Hilbert spac... |
| spansnchi 28421 | The span of a singleton in... |
| spansnid 28422 | A vector belongs to the sp... |
| spansnmul 28423 | A scalar product with a ve... |
| elspansncl 28424 | A member of a span of a si... |
| elspansn 28425 | Membership in the span of ... |
| elspansn2 28426 | Membership in the span of ... |
| spansncol 28427 | The singletons of collinea... |
| spansneleqi 28428 | Membership relation implie... |
| spansneleq 28429 | Membership relation that i... |
| spansnss 28430 | The span of the singleton ... |
| elspansn3 28431 | A member of the span of th... |
| elspansn4 28432 | A span membership conditio... |
| elspansn5 28433 | A vector belonging to both... |
| spansnss2 28434 | The span of the singleton ... |
| normcan 28435 | Cancellation-type law that... |
| pjspansn 28436 | A projection on the span o... |
| spansnpji 28437 | A subset of Hilbert space ... |
| spanunsni 28438 | The span of the union of a... |
| spanpr 28439 | The span of a pair of vect... |
| h1datomi 28440 | A 1-dimensional subspace i... |
| h1datom 28441 | A 1-dimensional subspace i... |
| cmbr 28443 | Binary relation expressing... |
| pjoml2i 28444 | Variation of orthomodular ... |
| pjoml3i 28445 | Variation of orthomodular ... |
| pjoml4i 28446 | Variation of orthomodular ... |
| pjoml5i 28447 | The orthomodular law. Rem... |
| pjoml6i 28448 | An equivalent of the ortho... |
| cmbri 28449 | Binary relation expressing... |
| cmcmlem 28450 | Commutation is symmetric. ... |
| cmcmi 28451 | Commutation is symmetric. ... |
| cmcm2i 28452 | Commutation with orthocomp... |
| cmcm3i 28453 | Commutation with orthocomp... |
| cmcm4i 28454 | Commutation with orthocomp... |
| cmbr2i 28455 | Alternate definition of th... |
| cmcmii 28456 | Commutation is symmetric. ... |
| cmcm2ii 28457 | Commutation with orthocomp... |
| cmcm3ii 28458 | Commutation with orthocomp... |
| cmbr3i 28459 | Alternate definition for t... |
| cmbr4i 28460 | Alternate definition for t... |
| lecmi 28461 | Comparable Hilbert lattice... |
| lecmii 28462 | Comparable Hilbert lattice... |
| cmj1i 28463 | A Hilbert lattice element ... |
| cmj2i 28464 | A Hilbert lattice element ... |
| cmm1i 28465 | A Hilbert lattice element ... |
| cmm2i 28466 | A Hilbert lattice element ... |
| cmbr3 28467 | Alternate definition for t... |
| cm0 28468 | The zero Hilbert lattice e... |
| cmidi 28469 | The commutes relation is r... |
| pjoml2 28470 | Variation of orthomodular ... |
| pjoml3 28471 | Variation of orthomodular ... |
| pjoml5 28472 | The orthomodular law. Rem... |
| cmcm 28473 | Commutation is symmetric. ... |
| cmcm3 28474 | Commutation with orthocomp... |
| cmcm2 28475 | Commutation with orthocomp... |
| lecm 28476 | Comparable Hilbert lattice... |
| fh1 28477 | Foulis-Holland Theorem. I... |
| fh2 28478 | Foulis-Holland Theorem. I... |
| cm2j 28479 | A lattice element that com... |
| fh1i 28480 | Foulis-Holland Theorem. I... |
| fh2i 28481 | Foulis-Holland Theorem. I... |
| fh3i 28482 | Variation of the Foulis-Ho... |
| fh4i 28483 | Variation of the Foulis-Ho... |
| cm2ji 28484 | A lattice element that com... |
| cm2mi 28485 | A lattice element that com... |
| qlax1i 28486 | One of the equations showi... |
| qlax2i 28487 | One of the equations showi... |
| qlax3i 28488 | One of the equations showi... |
| qlax4i 28489 | One of the equations showi... |
| qlax5i 28490 | One of the equations showi... |
| qlaxr1i 28491 | One of the conditions show... |
| qlaxr2i 28492 | One of the conditions show... |
| qlaxr4i 28493 | One of the conditions show... |
| qlaxr5i 28494 | One of the conditions show... |
| qlaxr3i 28495 | A variation of the orthomo... |
| chscllem1 28496 | Lemma for ~ chscl . (Cont... |
| chscllem2 28497 | Lemma for ~ chscl . (Cont... |
| chscllem3 28498 | Lemma for ~ chscl . (Cont... |
| chscllem4 28499 | Lemma for ~ chscl . (Cont... |
| chscl 28500 | The subspace sum of two cl... |
| osumi 28501 | If two closed subspaces of... |
| osumcori 28502 | Corollary of ~ osumi . (C... |
| osumcor2i 28503 | Corollary of ~ osumi , sho... |
| osum 28504 | If two closed subspaces of... |
| spansnji 28505 | The subspace sum of a clos... |
| spansnj 28506 | The subspace sum of a clos... |
| spansnscl 28507 | The subspace sum of a clos... |
| sumspansn 28508 | The sum of two vectors bel... |
| spansnm0i 28509 | The meet of different one-... |
| nonbooli 28510 | A Hilbert lattice with two... |
| spansncvi 28511 | Hilbert space has the cove... |
| spansncv 28512 | Hilbert space has the cove... |
| 5oalem1 28513 | Lemma for orthoarguesian l... |
| 5oalem2 28514 | Lemma for orthoarguesian l... |
| 5oalem3 28515 | Lemma for orthoarguesian l... |
| 5oalem4 28516 | Lemma for orthoarguesian l... |
| 5oalem5 28517 | Lemma for orthoarguesian l... |
| 5oalem6 28518 | Lemma for orthoarguesian l... |
| 5oalem7 28519 | Lemma for orthoarguesian l... |
| 5oai 28520 | Orthoarguesian law 5OA. Th... |
| 3oalem1 28521 | Lemma for 3OA (weak) ortho... |
| 3oalem2 28522 | Lemma for 3OA (weak) ortho... |
| 3oalem3 28523 | Lemma for 3OA (weak) ortho... |
| 3oalem4 28524 | Lemma for 3OA (weak) ortho... |
| 3oalem5 28525 | Lemma for 3OA (weak) ortho... |
| 3oalem6 28526 | Lemma for 3OA (weak) ortho... |
| 3oai 28527 | 3OA (weak) orthoarguesian ... |
| pjorthi 28528 | Projection components on o... |
| pjch1 28529 | Property of identity proje... |
| pjo 28530 | The orthogonal projection.... |
| pjcompi 28531 | Component of a projection.... |
| pjidmi 28532 | A projection is idempotent... |
| pjadjii 28533 | A projection is self-adjoi... |
| pjaddii 28534 | Projection of vector sum i... |
| pjinormii 28535 | The inner product of a pro... |
| pjmulii 28536 | Projection of (scalar) pro... |
| pjsubii 28537 | Projection of vector diffe... |
| pjsslem 28538 | Lemma for subset relations... |
| pjss2i 28539 | Subset relationship for pr... |
| pjssmii 28540 | Projection meet property. ... |
| pjssge0ii 28541 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormii 28542 | Theorem 4.5(v)<->(vi) of [... |
| pjcji 28543 | The projection on a subspa... |
| pjadji 28544 | A projection is self-adjoi... |
| pjaddi 28545 | Projection of vector sum i... |
| pjinormi 28546 | The inner product of a pro... |
| pjsubi 28547 | Projection of vector diffe... |
| pjmuli 28548 | Projection of scalar produ... |
| pjige0i 28549 | The inner product of a pro... |
| pjige0 28550 | The inner product of a pro... |
| pjcjt2 28551 | The projection on a subspa... |
| pj0i 28552 | The projection of the zero... |
| pjch 28553 | Projection of a vector in ... |
| pjid 28554 | The projection of a vector... |
| pjvec 28555 | The set of vectors belongi... |
| pjocvec 28556 | The set of vectors belongi... |
| pjocini 28557 | Membership of projection i... |
| pjini 28558 | Membership of projection i... |
| pjjsi 28559 | A sufficient condition for... |
| pjfni 28560 | Functionality of a project... |
| pjrni 28561 | The range of a projection.... |
| pjfoi 28562 | A projection maps onto its... |
| pjfi 28563 | The mapping of a projectio... |
| pjvi 28564 | The value of a projection ... |
| pjhfo 28565 | A projection maps onto its... |
| pjrn 28566 | The range of a projection.... |
| pjhf 28567 | The mapping of a projectio... |
| pjfn 28568 | Functionality of a project... |
| pjsumi 28569 | The projection on a subspa... |
| pj11i 28570 | One-to-one correspondence ... |
| pjdsi 28571 | Vector decomposition into ... |
| pjds3i 28572 | Vector decomposition into ... |
| pj11 28573 | One-to-one correspondence ... |
| pjmfn 28574 | Functionality of the proje... |
| pjmf1 28575 | The projector function map... |
| pjoi0 28576 | The inner product of proje... |
| pjoi0i 28577 | The inner product of proje... |
| pjopythi 28578 | Pythagorean theorem for pr... |
| pjopyth 28579 | Pythagorean theorem for pr... |
| pjnormi 28580 | The norm of the projection... |
| pjpythi 28581 | Pythagorean theorem for pr... |
| pjneli 28582 | If a vector does not belon... |
| pjnorm 28583 | The norm of the projection... |
| pjpyth 28584 | Pythagorean theorem for pr... |
| pjnel 28585 | If a vector does not belon... |
| pjnorm2 28586 | A vector belongs to the su... |
| mayete3i 28587 | Mayet's equation E_3. Par... |
| mayetes3i 28588 | Mayet's equation E^*_3, de... |
| hosmval 28594 | Value of the sum of two Hi... |
| hommval 28595 | Value of the scalar produc... |
| hodmval 28596 | Value of the difference of... |
| hfsmval 28597 | Value of the sum of two Hi... |
| hfmmval 28598 | Value of the scalar produc... |
| hosval 28599 | Value of the sum of two Hi... |
| homval 28600 | Value of the scalar produc... |
| hodval 28601 | Value of the difference of... |
| hfsval 28602 | Value of the sum of two Hi... |
| hfmval 28603 | Value of the scalar produc... |
| hoscl 28604 | Closure of the sum of two ... |
| homcl 28605 | Closure of the scalar prod... |
| hodcl 28606 | Closure of the difference ... |
| ho0val 28609 | Value of the zero Hilbert ... |
| ho0f 28610 | Functionality of the zero ... |
| df0op2 28611 | Alternate definition of Hi... |
| dfiop2 28612 | Alternate definition of Hi... |
| hoif 28613 | Functionality of the Hilbe... |
| hoival 28614 | The value of the Hilbert s... |
| hoico1 28615 | Composition with the Hilbe... |
| hoico2 28616 | Composition with the Hilbe... |
| hoaddcl 28617 | The sum of Hilbert space o... |
| homulcl 28618 | The scalar product of a Hi... |
| hoeq 28619 | Equality of Hilbert space ... |
| hoeqi 28620 | Equality of Hilbert space ... |
| hoscli 28621 | Closure of Hilbert space o... |
| hodcli 28622 | Closure of Hilbert space o... |
| hocoi 28623 | Composition of Hilbert spa... |
| hococli 28624 | Closure of composition of ... |
| hocofi 28625 | Mapping of composition of ... |
| hocofni 28626 | Functionality of compositi... |
| hoaddcli 28627 | Mapping of sum of Hilbert ... |
| hosubcli 28628 | Mapping of difference of H... |
| hoaddfni 28629 | Functionality of sum of Hi... |
| hosubfni 28630 | Functionality of differenc... |
| hoaddcomi 28631 | Commutativity of sum of Hi... |
| hosubcl 28632 | Mapping of difference of H... |
| hoaddcom 28633 | Commutativity of sum of Hi... |
| hodsi 28634 | Relationship between Hilbe... |
| hoaddassi 28635 | Associativity of sum of Hi... |
| hoadd12i 28636 | Commutative/associative la... |
| hoadd32i 28637 | Commutative/associative la... |
| hocadddiri 28638 | Distributive law for Hilbe... |
| hocsubdiri 28639 | Distributive law for Hilbe... |
| ho2coi 28640 | Double composition of Hilb... |
| hoaddass 28641 | Associativity of sum of Hi... |
| hoadd32 28642 | Commutative/associative la... |
| hoadd4 28643 | Rearrangement of 4 terms i... |
| hocsubdir 28644 | Distributive law for Hilbe... |
| hoaddid1i 28645 | Sum of a Hilbert space ope... |
| hodidi 28646 | Difference of a Hilbert sp... |
| ho0coi 28647 | Composition of the zero op... |
| hoid1i 28648 | Composition of Hilbert spa... |
| hoid1ri 28649 | Composition of Hilbert spa... |
| hoaddid1 28650 | Sum of a Hilbert space ope... |
| hodid 28651 | Difference of a Hilbert sp... |
| hon0 28652 | A Hilbert space operator i... |
| hodseqi 28653 | Subtraction and addition o... |
| ho0subi 28654 | Subtraction of Hilbert spa... |
| honegsubi 28655 | Relationship between Hilbe... |
| ho0sub 28656 | Subtraction of Hilbert spa... |
| hosubid1 28657 | The zero operator subtract... |
| honegsub 28658 | Relationship between Hilbe... |
| homulid2 28659 | An operator equals its sca... |
| homco1 28660 | Associative law for scalar... |
| homulass 28661 | Scalar product associative... |
| hoadddi 28662 | Scalar product distributiv... |
| hoadddir 28663 | Scalar product reverse dis... |
| homul12 28664 | Swap first and second fact... |
| honegneg 28665 | Double negative of a Hilbe... |
| hosubneg 28666 | Relationship between opera... |
| hosubdi 28667 | Scalar product distributiv... |
| honegdi 28668 | Distribution of negative o... |
| honegsubdi 28669 | Distribution of negative o... |
| honegsubdi2 28670 | Distribution of negative o... |
| hosubsub2 28671 | Law for double subtraction... |
| hosub4 28672 | Rearrangement of 4 terms i... |
| hosubadd4 28673 | Rearrangement of 4 terms i... |
| hoaddsubass 28674 | Associative-type law for a... |
| hoaddsub 28675 | Law for operator addition ... |
| hosubsub 28676 | Law for double subtraction... |
| hosubsub4 28677 | Law for double subtraction... |
| ho2times 28678 | Two times a Hilbert space ... |
| hoaddsubassi 28679 | Associativity of sum and d... |
| hoaddsubi 28680 | Law for sum and difference... |
| hosd1i 28681 | Hilbert space operator sum... |
| hosd2i 28682 | Hilbert space operator sum... |
| hopncani 28683 | Hilbert space operator can... |
| honpcani 28684 | Hilbert space operator can... |
| hosubeq0i 28685 | If the difference between ... |
| honpncani 28686 | Hilbert space operator can... |
| ho01i 28687 | A condition implying that ... |
| ho02i 28688 | A condition implying that ... |
| hoeq1 28689 | A condition implying that ... |
| hoeq2 28690 | A condition implying that ... |
| adjmo 28691 | Every Hilbert space operat... |
| adjsym 28692 | Symmetry property of an ad... |
| eigrei 28693 | A necessary and sufficient... |
| eigre 28694 | A necessary and sufficient... |
| eigposi 28695 | A sufficient condition (fi... |
| eigorthi 28696 | A necessary and sufficient... |
| eigorth 28697 | A necessary and sufficient... |
| nmopval 28715 | Value of the norm of a Hil... |
| elcnop 28716 | Property defining a contin... |
| ellnop 28717 | Property defining a linear... |
| lnopf 28718 | A linear Hilbert space ope... |
| elbdop 28719 | Property defining a bounde... |
| bdopln 28720 | A bounded linear Hilbert s... |
| bdopf 28721 | A bounded linear Hilbert s... |
| nmopsetretALT 28722 | The set in the supremum of... |
| nmopsetretHIL 28723 | The set in the supremum of... |
| nmopsetn0 28724 | The set in the supremum of... |
| nmopxr 28725 | The norm of a Hilbert spac... |
| nmoprepnf 28726 | The norm of a Hilbert spac... |
| nmopgtmnf 28727 | The norm of a Hilbert spac... |
| nmopreltpnf 28728 | The norm of a Hilbert spac... |
| nmopre 28729 | The norm of a bounded oper... |
| elbdop2 28730 | Property defining a bounde... |
| elunop 28731 | Property defining a unitar... |
| elhmop 28732 | Property defining a Hermit... |
| hmopf 28733 | A Hermitian operator is a ... |
| hmopex 28734 | The class of Hermitian ope... |
| nmfnval 28735 | Value of the norm of a Hil... |
| nmfnsetre 28736 | The set in the supremum of... |
| nmfnsetn0 28737 | The set in the supremum of... |
| nmfnxr 28738 | The norm of any Hilbert sp... |
| nmfnrepnf 28739 | The norm of a Hilbert spac... |
| nlfnval 28740 | Value of the null space of... |
| elcnfn 28741 | Property defining a contin... |
| ellnfn 28742 | Property defining a linear... |
| lnfnf 28743 | A linear Hilbert space fun... |
| dfadj2 28744 | Alternate definition of th... |
| funadj 28745 | Functionality of the adjoi... |
| dmadjss 28746 | The domain of the adjoint ... |
| dmadjop 28747 | A member of the domain of ... |
| adjeu 28748 | Elementhood in the domain ... |
| adjval 28749 | Value of the adjoint funct... |
| adjval2 28750 | Value of the adjoint funct... |
| cnvadj 28751 | The adjoint function equal... |
| funcnvadj 28752 | The converse of the adjoin... |
| adj1o 28753 | The adjoint function maps ... |
| dmadjrn 28754 | The adjoint of an operator... |
| eigvecval 28755 | The set of eigenvectors of... |
| eigvalfval 28756 | The eigenvalues of eigenve... |
| specval 28757 | The value of the spectrum ... |
| speccl 28758 | The spectrum of an operato... |
| hhlnoi 28759 | The linear operators of Hi... |
| hhnmoi 28760 | The norm of an operator in... |
| hhbloi 28761 | A bounded linear operator ... |
| hh0oi 28762 | The zero operator in Hilbe... |
| hhcno 28763 | The continuous operators o... |
| hhcnf 28764 | The continuous functionals... |
| dmadjrnb 28765 | The adjoint of an operator... |
| nmoplb 28766 | A lower bound for an opera... |
| nmopub 28767 | An upper bound for an oper... |
| nmopub2tALT 28768 | An upper bound for an oper... |
| nmopub2tHIL 28769 | An upper bound for an oper... |
| nmopge0 28770 | The norm of any Hilbert sp... |
| nmopgt0 28771 | A linear Hilbert space ope... |
| cnopc 28772 | Basic continuity property ... |
| lnopl 28773 | Basic linearity property o... |
| unop 28774 | Basic inner product proper... |
| unopf1o 28775 | A unitary operator in Hilb... |
| unopnorm 28776 | A unitary operator is idem... |
| cnvunop 28777 | The inverse (converse) of ... |
| unopadj 28778 | The inverse (converse) of ... |
| unoplin 28779 | A unitary operator is line... |
| counop 28780 | The composition of two uni... |
| hmop 28781 | Basic inner product proper... |
| hmopre 28782 | The inner product of the v... |
| nmfnlb 28783 | A lower bound for a functi... |
| nmfnleub 28784 | An upper bound for the nor... |
| nmfnleub2 28785 | An upper bound for the nor... |
| nmfnge0 28786 | The norm of any Hilbert sp... |
| elnlfn 28787 | Membership in the null spa... |
| elnlfn2 28788 | Membership in the null spa... |
| cnfnc 28789 | Basic continuity property ... |
| lnfnl 28790 | Basic linearity property o... |
| adjcl 28791 | Closure of the adjoint of ... |
| adj1 28792 | Property of an adjoint Hil... |
| adj2 28793 | Property of an adjoint Hil... |
| adjeq 28794 | A property that determines... |
| adjadj 28795 | Double adjoint. Theorem 3... |
| adjvalval 28796 | Value of the value of the ... |
| unopadj2 28797 | The adjoint of a unitary o... |
| hmopadj 28798 | A Hermitian operator is se... |
| hmdmadj 28799 | Every Hermitian operator h... |
| hmopadj2 28800 | An operator is Hermitian i... |
| hmoplin 28801 | A Hermitian operator is li... |
| brafval 28802 | The bra of a vector, expre... |
| braval 28803 | A bra-ket juxtaposition, e... |
| braadd 28804 | Linearity property of bra ... |
| bramul 28805 | Linearity property of bra ... |
| brafn 28806 | The bra function is a func... |
| bralnfn 28807 | The Dirac bra function is ... |
| bracl 28808 | Closure of the bra functio... |
| bra0 28809 | The Dirac bra of the zero ... |
| brafnmul 28810 | Anti-linearity property of... |
| kbfval 28811 | The outer product of two v... |
| kbop 28812 | The outer product of two v... |
| kbval 28813 | The value of the operator ... |
| kbmul 28814 | Multiplication property of... |
| kbpj 28815 | If a vector ` A ` has norm... |
| eleigvec 28816 | Membership in the set of e... |
| eleigvec2 28817 | Membership in the set of e... |
| eleigveccl 28818 | Closure of an eigenvector ... |
| eigvalval 28819 | The eigenvalue of an eigen... |
| eigvalcl 28820 | An eigenvalue is a complex... |
| eigvec1 28821 | Property of an eigenvector... |
| eighmre 28822 | The eigenvalues of a Hermi... |
| eighmorth 28823 | Eigenvectors of a Hermitia... |
| nmopnegi 28824 | Value of the norm of the n... |
| lnop0 28825 | The value of a linear Hilb... |
| lnopmul 28826 | Multiplicative property of... |
| lnopli 28827 | Basic scalar product prope... |
| lnopfi 28828 | A linear Hilbert space ope... |
| lnop0i 28829 | The value of a linear Hilb... |
| lnopaddi 28830 | Additive property of a lin... |
| lnopmuli 28831 | Multiplicative property of... |
| lnopaddmuli 28832 | Sum/product property of a ... |
| lnopsubi 28833 | Subtraction property for a... |
| lnopsubmuli 28834 | Subtraction/product proper... |
| lnopmulsubi 28835 | Product/subtraction proper... |
| homco2 28836 | Move a scalar product out ... |
| idunop 28837 | The identity function (res... |
| 0cnop 28838 | The identically zero funct... |
| 0cnfn 28839 | The identically zero funct... |
| idcnop 28840 | The identity function (res... |
| idhmop 28841 | The Hilbert space identity... |
| 0hmop 28842 | The identically zero funct... |
| 0lnop 28843 | The identically zero funct... |
| 0lnfn 28844 | The identically zero funct... |
| nmop0 28845 | The norm of the zero opera... |
| nmfn0 28846 | The norm of the identicall... |
| hmopbdoptHIL 28847 | A Hermitian operator is a ... |
| hoddii 28848 | Distributive law for Hilbe... |
| hoddi 28849 | Distributive law for Hilbe... |
| nmop0h 28850 | The norm of any operator o... |
| idlnop 28851 | The identity function (res... |
| 0bdop 28852 | The identically zero opera... |
| adj0 28853 | Adjoint of the zero operat... |
| nmlnop0iALT 28854 | A linear operator with a z... |
| nmlnop0iHIL 28855 | A linear operator with a z... |
| nmlnopgt0i 28856 | A linear Hilbert space ope... |
| nmlnop0 28857 | A linear operator with a z... |
| nmlnopne0 28858 | A linear operator with a n... |
| lnopmi 28859 | The scalar product of a li... |
| lnophsi 28860 | The sum of two linear oper... |
| lnophdi 28861 | The difference of two line... |
| lnopcoi 28862 | The composition of two lin... |
| lnopco0i 28863 | The composition of a linea... |
| lnopeq0lem1 28864 | Lemma for ~ lnopeq0i . Ap... |
| lnopeq0lem2 28865 | Lemma for ~ lnopeq0i . (C... |
| lnopeq0i 28866 | A condition implying that ... |
| lnopeqi 28867 | Two linear Hilbert space o... |
| lnopeq 28868 | Two linear Hilbert space o... |
| lnopunilem1 28869 | Lemma for ~ lnopunii . (C... |
| lnopunilem2 28870 | Lemma for ~ lnopunii . (C... |
| lnopunii 28871 | If a linear operator (whos... |
| elunop2 28872 | An operator is unitary iff... |
| nmopun 28873 | Norm of a unitary Hilbert ... |
| unopbd 28874 | A unitary operator is a bo... |
| lnophmlem1 28875 | Lemma for ~ lnophmi . (Co... |
| lnophmlem2 28876 | Lemma for ~ lnophmi . (Co... |
| lnophmi 28877 | A linear operator is Hermi... |
| lnophm 28878 | A linear operator is Hermi... |
| hmops 28879 | The sum of two Hermitian o... |
| hmopm 28880 | The scalar product of a He... |
| hmopd 28881 | The difference of two Herm... |
| hmopco 28882 | The composition of two com... |
| nmbdoplbi 28883 | A lower bound for the norm... |
| nmbdoplb 28884 | A lower bound for the norm... |
| nmcexi 28885 | Lemma for ~ nmcopexi and ~... |
| nmcopexi 28886 | The norm of a continuous l... |
| nmcoplbi 28887 | A lower bound for the norm... |
| nmcopex 28888 | The norm of a continuous l... |
| nmcoplb 28889 | A lower bound for the norm... |
| nmophmi 28890 | The norm of the scalar pro... |
| bdophmi 28891 | The scalar product of a bo... |
| lnconi 28892 | Lemma for ~ lnopconi and ~... |
| lnopconi 28893 | A condition equivalent to ... |
| lnopcon 28894 | A condition equivalent to ... |
| lnopcnbd 28895 | A linear operator is conti... |
| lncnopbd 28896 | A continuous linear operat... |
| lncnbd 28897 | A continuous linear operat... |
| lnopcnre 28898 | A linear operator is conti... |
| lnfnli 28899 | Basic property of a linear... |
| lnfnfi 28900 | A linear Hilbert space fun... |
| lnfn0i 28901 | The value of a linear Hilb... |
| lnfnaddi 28902 | Additive property of a lin... |
| lnfnmuli 28903 | Multiplicative property of... |
| lnfnaddmuli 28904 | Sum/product property of a ... |
| lnfnsubi 28905 | Subtraction property for a... |
| lnfn0 28906 | The value of a linear Hilb... |
| lnfnmul 28907 | Multiplicative property of... |
| nmbdfnlbi 28908 | A lower bound for the norm... |
| nmbdfnlb 28909 | A lower bound for the norm... |
| nmcfnexi 28910 | The norm of a continuous l... |
| nmcfnlbi 28911 | A lower bound for the norm... |
| nmcfnex 28912 | The norm of a continuous l... |
| nmcfnlb 28913 | A lower bound of the norm ... |
| lnfnconi 28914 | A condition equivalent to ... |
| lnfncon 28915 | A condition equivalent to ... |
| lnfncnbd 28916 | A linear functional is con... |
| imaelshi 28917 | The image of a subspace un... |
| rnelshi 28918 | The range of a linear oper... |
| nlelshi 28919 | The null space of a linear... |
| nlelchi 28920 | The null space of a contin... |
| riesz3i 28921 | A continuous linear functi... |
| riesz4i 28922 | A continuous linear functi... |
| riesz4 28923 | A continuous linear functi... |
| riesz1 28924 | Part 1 of the Riesz repres... |
| riesz2 28925 | Part 2 of the Riesz repres... |
| cnlnadjlem1 28926 | Lemma for ~ cnlnadji (Theo... |
| cnlnadjlem2 28927 | Lemma for ~ cnlnadji . ` G... |
| cnlnadjlem3 28928 | Lemma for ~ cnlnadji . By... |
| cnlnadjlem4 28929 | Lemma for ~ cnlnadji . Th... |
| cnlnadjlem5 28930 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem6 28931 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem7 28932 | Lemma for ~ cnlnadji . He... |
| cnlnadjlem8 28933 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem9 28934 | Lemma for ~ cnlnadji . ` F... |
| cnlnadji 28935 | Every continuous linear op... |
| cnlnadjeui 28936 | Every continuous linear op... |
| cnlnadjeu 28937 | Every continuous linear op... |
| cnlnadj 28938 | Every continuous linear op... |
| cnlnssadj 28939 | Every continuous linear Hi... |
| bdopssadj 28940 | Every bounded linear Hilbe... |
| bdopadj 28941 | Every bounded linear Hilbe... |
| adjbdln 28942 | The adjoint of a bounded l... |
| adjbdlnb 28943 | An operator is bounded and... |
| adjbd1o 28944 | The mapping of adjoints of... |
| adjlnop 28945 | The adjoint of an operator... |
| adjsslnop 28946 | Every operator with an adj... |
| nmopadjlei 28947 | Property of the norm of an... |
| nmopadjlem 28948 | Lemma for ~ nmopadji . (C... |
| nmopadji 28949 | Property of the norm of an... |
| adjeq0 28950 | An operator is zero iff it... |
| adjmul 28951 | The adjoint of the scalar ... |
| adjadd 28952 | The adjoint of the sum of ... |
| nmoptrii 28953 | Triangle inequality for th... |
| nmopcoi 28954 | Upper bound for the norm o... |
| bdophsi 28955 | The sum of two bounded lin... |
| bdophdi 28956 | The difference between two... |
| bdopcoi 28957 | The composition of two bou... |
| nmoptri2i 28958 | Triangle-type inequality f... |
| adjcoi 28959 | The adjoint of a compositi... |
| nmopcoadji 28960 | The norm of an operator co... |
| nmopcoadj2i 28961 | The norm of an operator co... |
| nmopcoadj0i 28962 | An operator composed with ... |
| unierri 28963 | If we approximate a chain ... |
| branmfn 28964 | The norm of the bra functi... |
| brabn 28965 | The bra of a vector is a b... |
| rnbra 28966 | The set of bras equals the... |
| bra11 28967 | The bra function maps vect... |
| bracnln 28968 | A bra is a continuous line... |
| cnvbraval 28969 | Value of the converse of t... |
| cnvbracl 28970 | Closure of the converse of... |
| cnvbrabra 28971 | The converse bra of the br... |
| bracnvbra 28972 | The bra of the converse br... |
| bracnlnval 28973 | The vector that a continuo... |
| cnvbramul 28974 | Multiplication property of... |
| kbass1 28975 | Dirac bra-ket associative ... |
| kbass2 28976 | Dirac bra-ket associative ... |
| kbass3 28977 | Dirac bra-ket associative ... |
| kbass4 28978 | Dirac bra-ket associative ... |
| kbass5 28979 | Dirac bra-ket associative ... |
| kbass6 28980 | Dirac bra-ket associative ... |
| leopg 28981 | Ordering relation for posi... |
| leop 28982 | Ordering relation for oper... |
| leop2 28983 | Ordering relation for oper... |
| leop3 28984 | Operator ordering in terms... |
| leoppos 28985 | Binary relation defining a... |
| leoprf2 28986 | The ordering relation for ... |
| leoprf 28987 | The ordering relation for ... |
| leopsq 28988 | The square of a Hermitian ... |
| 0leop 28989 | The zero operator is a pos... |
| idleop 28990 | The identity operator is a... |
| leopadd 28991 | The sum of two positive op... |
| leopmuli 28992 | The scalar product of a no... |
| leopmul 28993 | The scalar product of a po... |
| leopmul2i 28994 | Scalar product applied to ... |
| leoptri 28995 | The positive operator orde... |
| leoptr 28996 | The positive operator orde... |
| leopnmid 28997 | A bounded Hermitian operat... |
| nmopleid 28998 | A nonzero, bounded Hermiti... |
| opsqrlem1 28999 | Lemma for opsqri . (Contr... |
| opsqrlem2 29000 | Lemma for opsqri . ` F `` ... |
| opsqrlem3 29001 | Lemma for opsqri . (Contr... |
| opsqrlem4 29002 | Lemma for opsqri . (Contr... |
| opsqrlem5 29003 | Lemma for opsqri . (Contr... |
| opsqrlem6 29004 | Lemma for opsqri . (Contr... |
| pjhmopi 29005 | A projector is a Hermitian... |
| pjlnopi 29006 | A projector is a linear op... |
| pjnmopi 29007 | The operator norm of a pro... |
| pjbdlni 29008 | A projector is a bounded l... |
| pjhmop 29009 | A projection is a Hermitia... |
| hmopidmchi 29010 | An idempotent Hermitian op... |
| hmopidmpji 29011 | An idempotent Hermitian op... |
| hmopidmch 29012 | An idempotent Hermitian op... |
| hmopidmpj 29013 | An idempotent Hermitian op... |
| pjsdii 29014 | Distributive law for Hilbe... |
| pjddii 29015 | Distributive law for Hilbe... |
| pjsdi2i 29016 | Chained distributive law f... |
| pjcoi 29017 | Composition of projections... |
| pjcocli 29018 | Closure of composition of ... |
| pjcohcli 29019 | Closure of composition of ... |
| pjadjcoi 29020 | Adjoint of composition of ... |
| pjcofni 29021 | Functionality of compositi... |
| pjss1coi 29022 | Subset relationship for pr... |
| pjss2coi 29023 | Subset relationship for pr... |
| pjssmi 29024 | Projection meet property. ... |
| pjssge0i 29025 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormi 29026 | Theorem 4.5(v)<->(vi) of [... |
| pjnormssi 29027 | Theorem 4.5(i)<->(vi) of [... |
| pjorthcoi 29028 | Composition of projections... |
| pjscji 29029 | The projection of orthogon... |
| pjssumi 29030 | The projection on a subspa... |
| pjssposi 29031 | Projector ordering can be ... |
| pjordi 29032 | The definition of projecto... |
| pjssdif2i 29033 | The projection subspace of... |
| pjssdif1i 29034 | A necessary and sufficient... |
| pjimai 29035 | The image of a projection.... |
| pjidmcoi 29036 | A projection is idempotent... |
| pjoccoi 29037 | Composition of projections... |
| pjtoi 29038 | Subspace sum of projection... |
| pjoci 29039 | Projection of orthocomplem... |
| pjidmco 29040 | A projection operator is i... |
| dfpjop 29041 | Definition of projection o... |
| pjhmopidm 29042 | Two ways to express the se... |
| elpjidm 29043 | A projection operator is i... |
| elpjhmop 29044 | A projection operator is H... |
| 0leopj 29045 | A projector is a positive ... |
| pjadj2 29046 | A projector is self-adjoin... |
| pjadj3 29047 | A projector is self-adjoin... |
| elpjch 29048 | Reconstruction of the subs... |
| elpjrn 29049 | Reconstruction of the subs... |
| pjinvari 29050 | A closed subspace ` H ` wi... |
| pjin1i 29051 | Lemma for Theorem 1.22 of ... |
| pjin2i 29052 | Lemma for Theorem 1.22 of ... |
| pjin3i 29053 | Lemma for Theorem 1.22 of ... |
| pjclem1 29054 | Lemma for projection commu... |
| pjclem2 29055 | Lemma for projection commu... |
| pjclem3 29056 | Lemma for projection commu... |
| pjclem4a 29057 | Lemma for projection commu... |
| pjclem4 29058 | Lemma for projection commu... |
| pjci 29059 | Two subspaces commute iff ... |
| pjcmul1i 29060 | A necessary and sufficient... |
| pjcmul2i 29061 | The projection subspace of... |
| pjcohocli 29062 | Closure of composition of ... |
| pjadj2coi 29063 | Adjoint of double composit... |
| pj2cocli 29064 | Closure of double composit... |
| pj3lem1 29065 | Lemma for projection tripl... |
| pj3si 29066 | Stronger projection triple... |
| pj3i 29067 | Projection triplet theorem... |
| pj3cor1i 29068 | Projection triplet corolla... |
| pjs14i 29069 | Theorem S-14 of Watanabe, ... |
| isst 29072 | Property of a state. (Con... |
| ishst 29073 | Property of a complex Hilb... |
| sticl 29074 | ` [ 0 , 1 ] ` closure of t... |
| stcl 29075 | Real closure of the value ... |
| hstcl 29076 | Closure of the value of a ... |
| hst1a 29077 | Unit value of a Hilbert-sp... |
| hstel2 29078 | Properties of a Hilbert-sp... |
| hstorth 29079 | Orthogonality property of ... |
| hstosum 29080 | Orthogonal sum property of... |
| hstoc 29081 | Sum of a Hilbert-space-val... |
| hstnmoc 29082 | Sum of norms of a Hilbert-... |
| stge0 29083 | The value of a state is no... |
| stle1 29084 | The value of a state is le... |
| hstle1 29085 | The norm of the value of a... |
| hst1h 29086 | The norm of a Hilbert-spac... |
| hst0h 29087 | The norm of a Hilbert-spac... |
| hstpyth 29088 | Pythagorean property of a ... |
| hstle 29089 | Ordering property of a Hil... |
| hstles 29090 | Ordering property of a Hil... |
| hstoh 29091 | A Hilbert-space-valued sta... |
| hst0 29092 | A Hilbert-space-valued sta... |
| sthil 29093 | The value of a state at th... |
| stj 29094 | The value of a state on a ... |
| sto1i 29095 | The state of a subspace pl... |
| sto2i 29096 | The state of the orthocomp... |
| stge1i 29097 | If a state is greater than... |
| stle0i 29098 | If a state is less than or... |
| stlei 29099 | Ordering law for states. ... |
| stlesi 29100 | Ordering law for states. ... |
| stji1i 29101 | Join of components of Sasa... |
| stm1i 29102 | State of component of unit... |
| stm1ri 29103 | State of component of unit... |
| stm1addi 29104 | Sum of states whose meet i... |
| staddi 29105 | If the sum of 2 states is ... |
| stm1add3i 29106 | Sum of states whose meet i... |
| stadd3i 29107 | If the sum of 3 states is ... |
| st0 29108 | The state of the zero subs... |
| strlem1 29109 | Lemma for strong state the... |
| strlem2 29110 | Lemma for strong state the... |
| strlem3a 29111 | Lemma for strong state the... |
| strlem3 29112 | Lemma for strong state the... |
| strlem4 29113 | Lemma for strong state the... |
| strlem5 29114 | Lemma for strong state the... |
| strlem6 29115 | Lemma for strong state the... |
| stri 29116 | Strong state theorem. The... |
| strb 29117 | Strong state theorem (bidi... |
| hstrlem2 29118 | Lemma for strong set of CH... |
| hstrlem3a 29119 | Lemma for strong set of CH... |
| hstrlem3 29120 | Lemma for strong set of CH... |
| hstrlem4 29121 | Lemma for strong set of CH... |
| hstrlem5 29122 | Lemma for strong set of CH... |
| hstrlem6 29123 | Lemma for strong set of CH... |
| hstri 29124 | Hilbert space admits a str... |
| hstrbi 29125 | Strong CH-state theorem (b... |
| largei 29126 | A Hilbert lattice admits a... |
| jplem1 29127 | Lemma for Jauch-Piron theo... |
| jplem2 29128 | Lemma for Jauch-Piron theo... |
| jpi 29129 | The function ` S ` , that ... |
| golem1 29130 | Lemma for Godowski's equat... |
| golem2 29131 | Lemma for Godowski's equat... |
| goeqi 29132 | Godowski's equation, shown... |
| stcltr1i 29133 | Property of a strong class... |
| stcltr2i 29134 | Property of a strong class... |
| stcltrlem1 29135 | Lemma for strong classical... |
| stcltrlem2 29136 | Lemma for strong classical... |
| stcltrthi 29137 | Theorem for classically st... |
| cvbr 29141 | Binary relation expressing... |
| cvbr2 29142 | Binary relation expressing... |
| cvcon3 29143 | Contraposition law for the... |
| cvpss 29144 | The covers relation implie... |
| cvnbtwn 29145 | The covers relation implie... |
| cvnbtwn2 29146 | The covers relation implie... |
| cvnbtwn3 29147 | The covers relation implie... |
| cvnbtwn4 29148 | The covers relation implie... |
| cvnsym 29149 | The covers relation is not... |
| cvnref 29150 | The covers relation is not... |
| cvntr 29151 | The covers relation is not... |
| spansncv2 29152 | Hilbert space has the cove... |
| mdbr 29153 | Binary relation expressing... |
| mdi 29154 | Consequence of the modular... |
| mdbr2 29155 | Binary relation expressing... |
| mdbr3 29156 | Binary relation expressing... |
| mdbr4 29157 | Binary relation expressing... |
| dmdbr 29158 | Binary relation expressing... |
| dmdmd 29159 | The dual modular pair prop... |
| mddmd 29160 | The modular pair property ... |
| dmdi 29161 | Consequence of the dual mo... |
| dmdbr2 29162 | Binary relation expressing... |
| dmdi2 29163 | Consequence of the dual mo... |
| dmdbr3 29164 | Binary relation expressing... |
| dmdbr4 29165 | Binary relation expressing... |
| dmdi4 29166 | Consequence of the dual mo... |
| dmdbr5 29167 | Binary relation expressing... |
| mddmd2 29168 | Relationship between modul... |
| mdsl0 29169 | A sublattice condition tha... |
| ssmd1 29170 | Ordering implies the modul... |
| ssmd2 29171 | Ordering implies the modul... |
| ssdmd1 29172 | Ordering implies the dual ... |
| ssdmd2 29173 | Ordering implies the dual ... |
| dmdsl3 29174 | Sublattice mapping for a d... |
| mdsl3 29175 | Sublattice mapping for a m... |
| mdslle1i 29176 | Order preservation of the ... |
| mdslle2i 29177 | Order preservation of the ... |
| mdslj1i 29178 | Join preservation of the o... |
| mdslj2i 29179 | Meet preservation of the r... |
| mdsl1i 29180 | If the modular pair proper... |
| mdsl2i 29181 | If the modular pair proper... |
| mdsl2bi 29182 | If the modular pair proper... |
| cvmdi 29183 | The covering property impl... |
| mdslmd1lem1 29184 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem2 29185 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem3 29186 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem4 29187 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1i 29188 | Preservation of the modula... |
| mdslmd2i 29189 | Preservation of the modula... |
| mdsldmd1i 29190 | Preservation of the dual m... |
| mdslmd3i 29191 | Modular pair conditions th... |
| mdslmd4i 29192 | Modular pair condition tha... |
| csmdsymi 29193 | Cross-symmetry implies M-s... |
| mdexchi 29194 | An exchange lemma for modu... |
| cvmd 29195 | The covering property impl... |
| cvdmd 29196 | The covering property impl... |
| ela 29198 | Atoms in a Hilbert lattice... |
| elat2 29199 | Expanded membership relati... |
| elatcv0 29200 | A Hilbert lattice element ... |
| atcv0 29201 | An atom covers the zero su... |
| atssch 29202 | Atoms are a subset of the ... |
| atelch 29203 | An atom is a Hilbert latti... |
| atne0 29204 | An atom is not the Hilbert... |
| atss 29205 | A lattice element smaller ... |
| atsseq 29206 | Two atoms in a subset rela... |
| atcveq0 29207 | A Hilbert lattice element ... |
| h1da 29208 | A 1-dimensional subspace i... |
| spansna 29209 | The span of the singleton ... |
| sh1dle 29210 | A 1-dimensional subspace i... |
| ch1dle 29211 | A 1-dimensional subspace i... |
| atom1d 29212 | The 1-dimensional subspace... |
| superpos 29213 | Superposition Principle. ... |
| chcv1 29214 | The Hilbert lattice has th... |
| chcv2 29215 | The Hilbert lattice has th... |
| chjatom 29216 | The join of a closed subsp... |
| shatomici 29217 | The lattice of Hilbert sub... |
| hatomici 29218 | The Hilbert lattice is ato... |
| hatomic 29219 | A Hilbert lattice is atomi... |
| shatomistici 29220 | The lattice of Hilbert sub... |
| hatomistici 29221 | ` CH ` is atomistic, i.e. ... |
| chpssati 29222 | Two Hilbert lattice elemen... |
| chrelati 29223 | The Hilbert lattice is rel... |
| chrelat2i 29224 | A consequence of relative ... |
| cvati 29225 | If a Hilbert lattice eleme... |
| cvbr4i 29226 | An alternate way to expres... |
| cvexchlem 29227 | Lemma for ~ cvexchi . (Co... |
| cvexchi 29228 | The Hilbert lattice satisf... |
| chrelat2 29229 | A consequence of relative ... |
| chrelat3 29230 | A consequence of relative ... |
| chrelat3i 29231 | A consequence of the relat... |
| chrelat4i 29232 | A consequence of relative ... |
| cvexch 29233 | The Hilbert lattice satisf... |
| cvp 29234 | The Hilbert lattice satisf... |
| atnssm0 29235 | The meet of a Hilbert latt... |
| atnemeq0 29236 | The meet of distinct atoms... |
| atssma 29237 | The meet with an atom's su... |
| atcv0eq 29238 | Two atoms covering the zer... |
| atcv1 29239 | Two atoms covering the zer... |
| atexch 29240 | The Hilbert lattice satisf... |
| atomli 29241 | An assertion holding in at... |
| atoml2i 29242 | An assertion holding in at... |
| atordi 29243 | An ordering law for a Hilb... |
| atcvatlem 29244 | Lemma for ~ atcvati . (Co... |
| atcvati 29245 | A nonzero Hilbert lattice ... |
| atcvat2i 29246 | A Hilbert lattice element ... |
| atord 29247 | An ordering law for a Hilb... |
| atcvat2 29248 | A Hilbert lattice element ... |
| chirredlem1 29249 | Lemma for ~ chirredi . (C... |
| chirredlem2 29250 | Lemma for ~ chirredi . (C... |
| chirredlem3 29251 | Lemma for ~ chirredi . (C... |
| chirredlem4 29252 | Lemma for ~ chirredi . (C... |
| chirredi 29253 | The Hilbert lattice is irr... |
| chirred 29254 | The Hilbert lattice is irr... |
| atcvat3i 29255 | A condition implying that ... |
| atcvat4i 29256 | A condition implying exist... |
| atdmd 29257 | Two Hilbert lattice elemen... |
| atmd 29258 | Two Hilbert lattice elemen... |
| atmd2 29259 | Two Hilbert lattice elemen... |
| atabsi 29260 | Absorption of an incompara... |
| atabs2i 29261 | Absorption of an incompara... |
| mdsymlem1 29262 | Lemma for ~ mdsymi . (Con... |
| mdsymlem2 29263 | Lemma for ~ mdsymi . (Con... |
| mdsymlem3 29264 | Lemma for ~ mdsymi . (Con... |
| mdsymlem4 29265 | Lemma for ~ mdsymi . This... |
| mdsymlem5 29266 | Lemma for ~ mdsymi . (Con... |
| mdsymlem6 29267 | Lemma for ~ mdsymi . This... |
| mdsymlem7 29268 | Lemma for ~ mdsymi . Lemm... |
| mdsymlem8 29269 | Lemma for ~ mdsymi . Lemm... |
| mdsymi 29270 | M-symmetry of the Hilbert ... |
| mdsym 29271 | M-symmetry of the Hilbert ... |
| dmdsym 29272 | Dual M-symmetry of the Hil... |
| atdmd2 29273 | Two Hilbert lattice elemen... |
| sumdmdii 29274 | If the subspace sum of two... |
| cmmdi 29275 | Commuting subspaces form a... |
| cmdmdi 29276 | Commuting subspaces form a... |
| sumdmdlem 29277 | Lemma for ~ sumdmdi . The... |
| sumdmdlem2 29278 | Lemma for ~ sumdmdi . (Co... |
| sumdmdi 29279 | The subspace sum of two Hi... |
| dmdbr4ati 29280 | Dual modular pair property... |
| dmdbr5ati 29281 | Dual modular pair property... |
| dmdbr6ati 29282 | Dual modular pair property... |
| dmdbr7ati 29283 | Dual modular pair property... |
| mdoc1i 29284 | Orthocomplements form a mo... |
| mdoc2i 29285 | Orthocomplements form a mo... |
| dmdoc1i 29286 | Orthocomplements form a du... |
| dmdoc2i 29287 | Orthocomplements form a du... |
| mdcompli 29288 | A condition equivalent to ... |
| dmdcompli 29289 | A condition equivalent to ... |
| mddmdin0i 29290 | If dual modular implies mo... |
| cdjreui 29291 | A member of the sum of dis... |
| cdj1i 29292 | Two ways to express " ` A ... |
| cdj3lem1 29293 | A property of " ` A ` and ... |
| cdj3lem2 29294 | Lemma for ~ cdj3i . Value... |
| cdj3lem2a 29295 | Lemma for ~ cdj3i . Closu... |
| cdj3lem2b 29296 | Lemma for ~ cdj3i . The f... |
| cdj3lem3 29297 | Lemma for ~ cdj3i . Value... |
| cdj3lem3a 29298 | Lemma for ~ cdj3i . Closu... |
| cdj3lem3b 29299 | Lemma for ~ cdj3i . The s... |
| cdj3i 29300 | Two ways to express " ` A ... |
| The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here | |
| mathbox 29301 | (_This theorem is a dummy ... |
| foo3 29302 | A theorem about the univer... |
| xfree 29303 | A partial converse to ~ 19... |
| xfree2 29304 | A partial converse to ~ 19... |
| addltmulALT 29305 | A proof readability experi... |
| bian1d 29306 | Adding a superfluous conju... |
| or3di 29307 | Distributive law for disju... |
| or3dir 29308 | Distributive law for disju... |
| 3o1cs 29309 | Deduction eliminating disj... |
| 3o2cs 29310 | Deduction eliminating disj... |
| 3o3cs 29311 | Deduction eliminating disj... |
| spc2ed 29312 | Existential specialization... |
| spc2d 29313 | Specialization with 2 quan... |
| vtocl2d 29314 | Implicit substitution of t... |
| eqri 29315 | Infer equality of classes ... |
| ralcom4f 29316 | Commutation of restricted ... |
| rexcom4f 29317 | Commutation of restricted ... |
| 19.9d2rf 29318 | A deduction version of one... |
| 19.9d2r 29319 | A deduction version of one... |
| r19.29ffa 29320 | A commonly used pattern ba... |
| sbceqbidf 29321 | Equality theorem for class... |
| sbcies 29322 | A special version of class... |
| moel 29323 | "At most one" element in a... |
| mo5f 29324 | Alternate definition of "a... |
| nmo 29325 | Negation of "at most one".... |
| moimd 29326 | "At most one" is preserved... |
| rmoeqALT 29327 | Equality's restricted exis... |
| 2reuswap2 29328 | A condition allowing swap ... |
| reuxfr3d 29329 | Transfer existential uniqu... |
| reuxfr4d 29330 | Transfer existential uniqu... |
| rexunirn 29331 | Restricted existential qua... |
| rmoxfrdOLD 29332 | Transfer "at most one" res... |
| rmoxfrd 29333 | Transfer "at most one" res... |
| ssrmo 29334 | "At most one" existential ... |
| rmo3f 29335 | Restricted "at most one" u... |
| rmo4fOLD 29336 | Restricted "at most one" u... |
| rmo4f 29337 | Restricted "at most one" u... |
| rabrab 29338 | Abstract builder restricte... |
| difrab2 29339 | Difference of two restrict... |
| rabexgfGS 29340 | Separation Scheme in terms... |
| rabsnel 29341 | Truth implied by equality ... |
| rabeqsnd 29342 | Conditions for a restricte... |
| foresf1o 29343 | From a surjective function... |
| rabfodom 29344 | Domination relation for re... |
| abrexdomjm 29345 | An indexed set is dominate... |
| abrexdom2jm 29346 | An indexed set is dominate... |
| abrexexd 29347 | Existence of a class abstr... |
| elabreximd 29348 | Class substitution in an i... |
| elabreximdv 29349 | Class substitution in an i... |
| abrexss 29350 | A necessary condition for ... |
| rabss3d 29351 | Subclass law for restricte... |
| inin 29352 | Intersection with an inter... |
| inindif 29353 | See ~ inundif . (Contribu... |
| difininv 29354 | Condition for the intersec... |
| difeq 29355 | Rewriting an equation with... |
| indifundif 29356 | A remarkable equation with... |
| elpwincl1 29357 | Closure of intersection wi... |
| elpwdifcl 29358 | Closure of class differenc... |
| elpwiuncl 29359 | Closure of indexed union w... |
| elpreq 29360 | Equality wihin a pair. (C... |
| ifeqeqx 29361 | An equality theorem tailor... |
| elimifd 29362 | Elimination of a condition... |
| elim2if 29363 | Elimination of two conditi... |
| elim2ifim 29364 | Elimination of two conditi... |
| ifeq3da 29365 | Given an expression ` C ` ... |
| uniinn0 29366 | Sufficient and necessary c... |
| uniin1 29367 | Union of intersection. Ge... |
| uniin2 29368 | Union of intersection. Ge... |
| difuncomp 29369 | Express a class difference... |
| pwuniss 29370 | Condition for a class unio... |
| elpwunicl 29371 | Closure of a set union wit... |
| cbviunf 29372 | Rule used to change the bo... |
| iuneq12daf 29373 | Equality deduction for ind... |
| iunin1f 29374 | Indexed union of intersect... |
| iunxsngf 29375 | A singleton index picks ou... |
| ssiun3 29376 | Subset equivalence for an ... |
| iinssiun 29377 | An indexed intersection is... |
| ssiun2sf 29378 | Subset relationship for an... |
| iuninc 29379 | The union of an increasing... |
| iundifdifd 29380 | The intersection of a set ... |
| iundifdif 29381 | The intersection of a set ... |
| iunrdx 29382 | Re-index an indexed union.... |
| iunpreima 29383 | Preimage of an indexed uni... |
| disjnf 29384 | In case ` x ` is not free ... |
| cbvdisjf 29385 | Change bound variables in ... |
| disjss1f 29386 | A subset of a disjoint col... |
| disjeq1f 29387 | Equality theorem for disjo... |
| disjdifprg 29388 | A trivial partition into a... |
| disjdifprg2 29389 | A trivial partition of a s... |
| disji2f 29390 | Property of a disjoint col... |
| disjif 29391 | Property of a disjoint col... |
| disjorf 29392 | Two ways to say that a col... |
| disjorsf 29393 | Two ways to say that a col... |
| disjif2 29394 | Property of a disjoint col... |
| disjabrex 29395 | Rewriting a disjoint colle... |
| disjabrexf 29396 | Rewriting a disjoint colle... |
| disjpreima 29397 | A preimage of a disjoint s... |
| disjrnmpt 29398 | Rewriting a disjoint colle... |
| disjin 29399 | If a collection is disjoin... |
| disjin2 29400 | If a collection is disjoin... |
| disjxpin 29401 | Derive a disjunction over ... |
| iundisjf 29402 | Rewrite a countable union ... |
| iundisj2f 29403 | A disjoint union is disjoi... |
| disjrdx 29404 | Re-index a disjunct collec... |
| disjex 29405 | Two ways to say that two c... |
| disjexc 29406 | A variant of ~ disjex , ap... |
| disjunsn 29407 | Append an element to a dis... |
| disjun0 29408 | Adding the empty element p... |
| disjiunel 29409 | A set of elements B of a d... |
| disjuniel 29410 | A set of elements B of a d... |
| xpdisjres 29411 | Restriction of a constant ... |
| opeldifid 29412 | Ordered pair elementhood o... |
| difres 29413 | Case when class difference... |
| imadifxp 29414 | Image of the difference wi... |
| relfi 29415 | A relation (set) is finite... |
| funresdm1 29416 | Restriction of a disjoint ... |
| fnunres1 29417 | Restriction of a disjoint ... |
| fcoinver 29418 | Build an equivalence relat... |
| fcoinvbr 29419 | Binary relation for the eq... |
| brabgaf 29420 | The law of concretion for ... |
| brelg 29421 | Two things in a binary rel... |
| br8d 29422 | Substitution for an eight-... |
| opabdm 29423 | Domain of an ordered-pair ... |
| opabrn 29424 | Range of an ordered-pair c... |
| ssrelf 29425 | A subclass relationship de... |
| eqrelrd2 29426 | A version of ~ eqrelrdv2 w... |
| erbr3b 29427 | Biconditional for equivale... |
| iunsnima 29428 | Image of a singleton by an... |
| ac6sf2 29429 | Alternate version of ~ ac6... |
| fnresin 29430 | Restriction of a function ... |
| f1o3d 29431 | Describe an implicit one-t... |
| rinvf1o 29432 | Sufficient conditions for ... |
| fresf1o 29433 | Conditions for a restricti... |
| fmptco1f1o 29434 | The action of composing (t... |
| f1mptrn 29435 | Express injection for a ma... |
| dfimafnf 29436 | Alternate definition of th... |
| funimass4f 29437 | Membership relation for th... |
| elimampt 29438 | Membership in the image of... |
| suppss2f 29439 | Show that the support of a... |
| fovcld 29440 | Closure law for an operati... |
| ofrn 29441 | The range of the function ... |
| ofrn2 29442 | The range of the function ... |
| off2 29443 | The function operation pro... |
| ofresid 29444 | Applying an operation rest... |
| fimarab 29445 | Expressing the image of a ... |
| unipreima 29446 | Preimage of a class union.... |
| sspreima 29447 | The preimage of a subset i... |
| opfv 29448 | Value of a function produc... |
| xppreima 29449 | The preimage of a Cartesia... |
| xppreima2 29450 | The preimage of a Cartesia... |
| elunirn2 29451 | Condition for the membersh... |
| abfmpunirn 29452 | Membership in a union of a... |
| rabfmpunirn 29453 | Membership in a union of a... |
| abfmpeld 29454 | Membership in an element o... |
| abfmpel 29455 | Membership in an element o... |
| fmptdF 29456 | Domain and co-domain of th... |
| fmptcof2 29457 | Composition of two functio... |
| fcomptf 29458 | Express composition of two... |
| acunirnmpt 29459 | Axiom of choice for the un... |
| acunirnmpt2 29460 | Axiom of choice for the un... |
| acunirnmpt2f 29461 | Axiom of choice for the un... |
| aciunf1lem 29462 | Choice in an index union. ... |
| aciunf1 29463 | Choice in an index union. ... |
| ofoprabco 29464 | Function operation as a co... |
| ofpreima 29465 | Express the preimage of a ... |
| ofpreima2 29466 | Express the preimage of a ... |
| funcnvmptOLD 29467 | Condition for a function i... |
| funcnvmpt 29468 | Condition for a function i... |
| funcnv5mpt 29469 | Two ways to say that a fun... |
| funcnv4mpt 29470 | Two ways to say that a fun... |
| fgreu 29471 | Exactly one point of a fun... |
| fcnvgreu 29472 | If the converse of a relat... |
| rnmpt2ss 29473 | The range of an operation ... |
| mptssALT 29474 | Deduce subset relation of ... |
| partfun 29475 | Rewrite a function defined... |
| dfcnv2 29476 | Alternative definition of ... |
| mpt2mptxf 29477 | Express a two-argument fun... |
| gtiso 29478 | Two ways to write a strict... |
| isoun 29479 | Infer an isomorphism from ... |
| disjdsct 29480 | A disjoint collection is d... |
| df1stres 29481 | Definition for a restricti... |
| df2ndres 29482 | Definition for a restricti... |
| 1stpreimas 29483 | The preimage of a singleto... |
| 1stpreima 29484 | The preimage by ` 1st ` is... |
| 2ndpreima 29485 | The preimage by ` 2nd ` is... |
| curry2ima 29486 | The image of a curried fun... |
| supssd 29487 | Inequality deduction for s... |
| infssd 29488 | Inequality deduction for i... |
| imafi2 29489 | The image by a finite set ... |
| unifi3 29490 | If a union is finite, then... |
| snct 29491 | A singleton is countable. ... |
| prct 29492 | An unordered pair is count... |
| mpt2cti 29493 | An operation is countable ... |
| abrexct 29494 | An image set of a countabl... |
| mptctf 29495 | A countable mapping set is... |
| abrexctf 29496 | An image set of a countabl... |
| padct 29497 | Index a countable set with... |
| cnvoprab 29498 | The converse of a class ab... |
| f1od2 29499 | Describe an implicit one-t... |
| fcobij 29500 | Composing functions with a... |
| fcobijfs 29501 | Composing finitely support... |
| suppss3 29502 | Deduce a function's suppor... |
| ffs2 29503 | Rewrite a function's suppo... |
| ffsrn 29504 | The range of a finitely su... |
| resf1o 29505 | Restriction of functions t... |
| maprnin 29506 | Restricting the range of t... |
| fpwrelmapffslem 29507 | Lemma for ~ fpwrelmapffs .... |
| fpwrelmap 29508 | Define a canonical mapping... |
| fpwrelmapffs 29509 | Define a canonical mapping... |
| addeq0 29510 | Two complex which add up t... |
| subeqxfrd 29511 | Transfer two terms of a su... |
| znsqcld 29512 | Squaring of nonzero relati... |
| nn0sqeq1 29513 | Integer square one. (Cont... |
| 1neg1t1neg1 29514 | An integer unit times itse... |
| nnmulge 29515 | Multiplying by an integer ... |
| lt2addrd 29516 | If the right-hand side of ... |
| xrlelttric 29517 | Trichotomy law for extende... |
| xaddeq0 29518 | Two extended reals which a... |
| xrinfm 29519 | The extended real numbers ... |
| le2halvesd 29520 | A sum is less than the who... |
| xraddge02 29521 | A number is less than or e... |
| xrge0addge 29522 | A number is less than or e... |
| xlt2addrd 29523 | If the right-hand side of ... |
| xrsupssd 29524 | Inequality deduction for s... |
| xrge0infss 29525 | Any subset of nonnegative ... |
| xrge0infssd 29526 | Inequality deduction for i... |
| xrge0addcld 29527 | Nonnegative extended reals... |
| xrge0subcld 29528 | Condition for closure of n... |
| infxrge0lb 29529 | A member of a set of nonne... |
| infxrge0glb 29530 | The infimum of a set of no... |
| infxrge0gelb 29531 | The infimum of a set of no... |
| dfrp2 29532 | Alternate definition of th... |
| xrofsup 29533 | The supremum is preserved ... |
| supxrnemnf 29534 | The supremum of a nonempty... |
| xrhaus 29535 | The topology of the extend... |
| joiniooico 29536 | Disjoint joining an open i... |
| ubico 29537 | A right-open interval does... |
| xeqlelt 29538 | Equality in terms of 'less... |
| eliccelico 29539 | Relate elementhood to a cl... |
| elicoelioo 29540 | Relate elementhood to a cl... |
| iocinioc2 29541 | Intersection between two o... |
| xrdifh 29542 | Class difference of a half... |
| iocinif 29543 | Relate intersection of two... |
| difioo 29544 | The difference between two... |
| difico 29545 | The difference between two... |
| uzssico 29546 | Upper integer sets are a s... |
| fz2ssnn0 29547 | A finite set of sequential... |
| nndiffz1 29548 | Upper set of the positive ... |
| ssnnssfz 29549 | For any finite subset of `... |
| fzspl 29550 | Split the last element of ... |
| fzdif2 29551 | Split the last element of ... |
| fzodif2 29552 | Split the last element of ... |
| fzsplit3 29553 | Split a finite interval of... |
| bcm1n 29554 | The proportion of one bino... |
| iundisjfi 29555 | Rewrite a countable union ... |
| iundisj2fi 29556 | A disjoint union is disjoi... |
| iundisjcnt 29557 | Rewrite a countable union ... |
| iundisj2cnt 29558 | A countable disjoint union... |
| f1ocnt 29559 | Given a countable set ` A ... |
| fz1nnct 29560 | NN and integer ranges star... |
| fz1nntr 29561 | NN and integer ranges star... |
| hashunif 29562 | The cardinality of a disjo... |
| numdenneg 29563 | Numerator and denominator ... |
| divnumden2 29564 | Calculate the reduced form... |
| nnindf 29565 | Principle of Mathematical ... |
| nnindd 29566 | Principle of Mathematical ... |
| nn0min 29567 | Extracting the minimum pos... |
| ltesubnnd 29568 | Subtracting an integer num... |
| fprodeq02 29569 | If one of the factors is z... |
| pr01ssre 29570 | The range of the indicator... |
| fprodex01 29571 | A product of factors equal... |
| prodpr 29572 | A product over a pair is t... |
| prodtp 29573 | A product over a triple is... |
| fsumub 29574 | An upper bound for a term ... |
| fsumiunle 29575 | Upper bound for a sum of n... |
| dfdec100 29576 | Split the hundreds from a ... |
| dfdp2OLD 29579 | Obsolete version of ~ df-d... |
| dp2eq1 29580 | Equality theorem for the d... |
| dp2eq2 29581 | Equality theorem for the d... |
| dp2eq1i 29582 | Equality theorem for the d... |
| dp2eq2i 29583 | Equality theorem for the d... |
| dp2eq12i 29584 | Equality theorem for the d... |
| dp20u 29585 | Add a zero in the tenths (... |
| dp20h 29586 | Add a zero in the unit pla... |
| dp2cl 29587 | Closure for the decimal fr... |
| dp2clq 29588 | Closure for a decimal frac... |
| rpdp2cl 29589 | Closure for a decimal frac... |
| rpdp2cl2 29590 | Closure for a decimal frac... |
| dp2lt10 29591 | Decimal fraction builds re... |
| dp2lt 29592 | Comparing two decimal frac... |
| dp2ltsuc 29593 | Comparing a decimal fracti... |
| dp2ltc 29594 | Comparing two decimal expa... |
| dpval 29597 | Define the value of the de... |
| dpcl 29598 | Prove that the closure of ... |
| dpfrac1 29599 | Prove a simple equivalence... |
| dpfrac1OLD 29600 | Obsolete version of ~ dpfr... |
| dpval2 29601 | Value of the decimal point... |
| dpval3 29602 | Value of the decimal point... |
| dpmul10 29603 | Multiply by 10 a decimal e... |
| decdiv10 29604 | Divide a decimal number by... |
| dpmul100 29605 | Multiply by 100 a decimal ... |
| dp3mul10 29606 | Multiply by 10 a decimal e... |
| dpmul1000 29607 | Multiply by 1000 a decimal... |
| dpval3rp 29608 | Value of the decimal point... |
| dp0u 29609 | Add a zero in the tenths p... |
| dp0h 29610 | Remove a zero in the units... |
| rpdpcl 29611 | Closure of the decimal poi... |
| dplt 29612 | Comparing two decimal expa... |
| dplti 29613 | Comparing a decimal expans... |
| dpgti 29614 | Comparing a decimal expans... |
| dpltc 29615 | Comparing two decimal inte... |
| dpexpp1 29616 | Add one zero to the mantis... |
| 0dp2dp 29617 | Multiply by 10 a decimal e... |
| dpadd2 29618 | Addition with one decimal,... |
| dpadd 29619 | Addition with one decimal.... |
| dpadd3 29620 | Addition with two decimals... |
| dpmul 29621 | Multiplication with one de... |
| dpmul4 29622 | An upper bound to multipli... |
| threehalves 29623 | Example theorem demonstrat... |
| 1mhdrd 29624 | Example theorem demonstrat... |
| xdivval 29627 | Value of division: the (un... |
| xrecex 29628 | Existence of reciprocal of... |
| xmulcand 29629 | Cancellation law for exten... |
| xreceu 29630 | Existential uniqueness of ... |
| xdivcld 29631 | Closure law for the extend... |
| xdivcl 29632 | Closure law for the extend... |
| xdivmul 29633 | Relationship between divis... |
| rexdiv 29634 | The extended real division... |
| xdivrec 29635 | Relationship between divis... |
| xdivid 29636 | A number divided by itself... |
| xdiv0 29637 | Division into zero is zero... |
| xdiv0rp 29638 | Division into zero is zero... |
| eliccioo 29639 | Membership in a closed int... |
| elxrge02 29640 | Elementhood in the set of ... |
| xdivpnfrp 29641 | Plus infinity divided by a... |
| rpxdivcld 29642 | Closure law for extended d... |
| xrpxdivcld 29643 | Closure law for extended d... |
| bhmafibid1 29644 | The Brahmagupta-Fibonacci ... |
| bhmafibid2 29645 | The Brahmagupta-Fibonacci ... |
| 2sqn0 29646 | If the sum of two squares ... |
| 2sqcoprm 29647 | If the sum of two squares ... |
| 2sqmod 29648 | Given two decompositions o... |
| 2sqmo 29649 | There exists at most one d... |
| ressplusf 29650 | The group operation functi... |
| ressnm 29651 | The norm in a restricted s... |
| abvpropd2 29652 | Weaker version of ~ abvpro... |
| oppgle 29653 | less-than relation of an o... |
| oppglt 29654 | less-than relation of an o... |
| ressprs 29655 | The restriction of a preor... |
| oduprs 29656 | Being a preset is a self-d... |
| posrasymb 29657 | A poset ordering is asymet... |
| tospos 29658 | A Toset is a Poset. (Cont... |
| resspos 29659 | The restriction of a Poset... |
| resstos 29660 | The restriction of a Toset... |
| tleile 29661 | In a Toset, two elements m... |
| tltnle 29662 | In a Toset, less-than is e... |
| odutos 29663 | Being a toset is a self-du... |
| tlt2 29664 | In a Toset, two elements m... |
| tlt3 29665 | In a Toset, two elements m... |
| trleile 29666 | In a Toset, two elements m... |
| toslublem 29667 | Lemma for ~ toslub and ~ x... |
| toslub 29668 | In a toset, the lowest upp... |
| tosglblem 29669 | Lemma for ~ tosglb and ~ x... |
| tosglb 29670 | Same theorem as ~ toslub ,... |
| clatp0cl 29671 | The poset zero of a comple... |
| clatp1cl 29672 | The poset one of a complet... |
| xrs0 29675 | The zero of the extended r... |
| xrslt 29676 | The "strictly less than" r... |
| xrsinvgval 29677 | The inversion operation in... |
| xrsmulgzz 29678 | The "multiple" function in... |
| xrstos 29679 | The extended real numbers ... |
| xrsclat 29680 | The extended real numbers ... |
| xrsp0 29681 | The poset 0 of the extende... |
| xrsp1 29682 | The poset 1 of the extende... |
| ressmulgnn 29683 | Values for the group multi... |
| ressmulgnn0 29684 | Values for the group multi... |
| xrge0base 29685 | The base of the extended n... |
| xrge00 29686 | The zero of the extended n... |
| xrge0plusg 29687 | The additive law of the ex... |
| xrge0le 29688 | The lower-or-equal relatio... |
| xrge0mulgnn0 29689 | The group multiple functio... |
| xrge0addass 29690 | Associativity of extended ... |
| xrge0addgt0 29691 | The sum of nonnegative and... |
| xrge0adddir 29692 | Right-distributivity of ex... |
| xrge0adddi 29693 | Left-distributivity of ext... |
| xrge0npcan 29694 | Extended nonnegative real ... |
| fsumrp0cl 29695 | Closure of a finite sum of... |
| abliso 29696 | The image of an Abelian gr... |
| isomnd 29701 | A (left) ordered monoid is... |
| isogrp 29702 | A (left) ordered group is ... |
| ogrpgrp 29703 | An left ordered group is a... |
| omndmnd 29704 | A left ordered monoid is a... |
| omndtos 29705 | A left ordered monoid is a... |
| omndadd 29706 | In an ordered monoid, the ... |
| omndaddr 29707 | In a right ordered monoid,... |
| omndadd2d 29708 | In a commutative left orde... |
| omndadd2rd 29709 | In a left- and right- orde... |
| submomnd 29710 | A submonoid of an ordered ... |
| xrge0omnd 29711 | The nonnegative extended r... |
| omndmul2 29712 | In an ordered monoid, the ... |
| omndmul3 29713 | In an ordered monoid, the ... |
| omndmul 29714 | In a commutative ordered m... |
| ogrpinvOLD 29715 | In an ordered group, the o... |
| ogrpinv0le 29716 | In an ordered group, the o... |
| ogrpsub 29717 | In an ordered group, the o... |
| ogrpaddlt 29718 | In an ordered group, stric... |
| ogrpaddltbi 29719 | In a right ordered group, ... |
| ogrpaddltrd 29720 | In a right ordered group, ... |
| ogrpaddltrbid 29721 | In a right ordered group, ... |
| ogrpsublt 29722 | In an ordered group, stric... |
| ogrpinv0lt 29723 | In an ordered group, the o... |
| ogrpinvlt 29724 | In an ordered group, the o... |
| sgnsv 29727 | The sign mapping. (Contri... |
| sgnsval 29728 | The sign value. (Contribu... |
| sgnsf 29729 | The sign function. (Contr... |
| inftmrel 29734 | The infinitesimal relation... |
| isinftm 29735 | Express ` x ` is infinites... |
| isarchi 29736 | Express the predicate " ` ... |
| pnfinf 29737 | Plus infinity is an infini... |
| xrnarchi 29738 | The completed real line is... |
| isarchi2 29739 | Alternative way to express... |
| submarchi 29740 | A submonoid is archimedean... |
| isarchi3 29741 | This is the usual definiti... |
| archirng 29742 | Property of Archimedean or... |
| archirngz 29743 | Property of Archimedean le... |
| archiexdiv 29744 | In an Archimedean group, g... |
| archiabllem1a 29745 | Lemma for ~ archiabl : In... |
| archiabllem1b 29746 | Lemma for ~ archiabl . (C... |
| archiabllem1 29747 | Archimedean ordered groups... |
| archiabllem2a 29748 | Lemma for ~ archiabl , whi... |
| archiabllem2c 29749 | Lemma for ~ archiabl . (C... |
| archiabllem2b 29750 | Lemma for ~ archiabl . (C... |
| archiabllem2 29751 | Archimedean ordered groups... |
| archiabl 29752 | Archimedean left- and righ... |
| isslmd 29755 | The predicate "is a semimo... |
| slmdlema 29756 | Lemma for properties of a ... |
| lmodslmd 29757 | Left semimodules generaliz... |
| slmdcmn 29758 | A semimodule is a commutat... |
| slmdmnd 29759 | A semimodule is a monoid. ... |
| slmdsrg 29760 | The scalar component of a ... |
| slmdbn0 29761 | The base set of a semimodu... |
| slmdacl 29762 | Closure of ring addition f... |
| slmdmcl 29763 | Closure of ring multiplica... |
| slmdsn0 29764 | The set of scalars in a se... |
| slmdvacl 29765 | Closure of vector addition... |
| slmdass 29766 | Semiring left module vecto... |
| slmdvscl 29767 | Closure of scalar product ... |
| slmdvsdi 29768 | Distributive law for scala... |
| slmdvsdir 29769 | Distributive law for scala... |
| slmdvsass 29770 | Associative law for scalar... |
| slmd0cl 29771 | The ring zero in a semimod... |
| slmd1cl 29772 | The ring unit in a semirin... |
| slmdvs1 29773 | Scalar product with ring u... |
| slmd0vcl 29774 | The zero vector is a vecto... |
| slmd0vlid 29775 | Left identity law for the ... |
| slmd0vrid 29776 | Right identity law for the... |
| slmd0vs 29777 | Zero times a vector is the... |
| slmdvs0 29778 | Anything times the zero ve... |
| gsumle 29779 | A finite sum in an ordered... |
| gsummpt2co 29780 | Split a finite sum into a ... |
| gsummpt2d 29781 | Express a finite sum over ... |
| gsumvsca1 29782 | Scalar product of a finite... |
| gsumvsca2 29783 | Scalar product of a finite... |
| gsummptres 29784 | Extend a finite group sum ... |
| xrge0tsmsd 29785 | Any finite or infinite sum... |
| xrge0tsmsbi 29786 | Any limit of a finite or i... |
| xrge0tsmseq 29787 | Any limit of a finite or i... |
| rngurd 29788 | Deduce the unit of a ring ... |
| ress1r 29789 | ` 1r ` is unaffected by re... |
| dvrdir 29790 | Distributive law for the d... |
| rdivmuldivd 29791 | Multiplication of two rati... |
| ringinvval 29792 | The ring inverse expressed... |
| dvrcan5 29793 | Cancellation law for commo... |
| subrgchr 29794 | If ` A ` is a subring of `... |
| isorng 29799 | An ordered ring is a ring ... |
| orngring 29800 | An ordered ring is a ring.... |
| orngogrp 29801 | An ordered ring is an orde... |
| isofld 29802 | An ordered field is a fiel... |
| orngmul 29803 | In an ordered ring, the or... |
| orngsqr 29804 | In an ordered ring, all sq... |
| ornglmulle 29805 | In an ordered ring, multip... |
| orngrmulle 29806 | In an ordered ring, multip... |
| ornglmullt 29807 | In an ordered ring, multip... |
| orngrmullt 29808 | In an ordered ring, multip... |
| orngmullt 29809 | In an ordered ring, the st... |
| ofldfld 29810 | An ordered field is a fiel... |
| ofldtos 29811 | An ordered field is a tota... |
| orng0le1 29812 | In an ordered ring, the ri... |
| ofldlt1 29813 | In an ordered field, the r... |
| ofldchr 29814 | The characteristic of an o... |
| suborng 29815 | Every subring of an ordere... |
| subofld 29816 | Every subfield of an order... |
| isarchiofld 29817 | Axiom of Archimedes : a ch... |
| rhmdvdsr 29818 | A ring homomorphism preser... |
| rhmopp 29819 | A ring homomorphism is als... |
| elrhmunit 29820 | Ring homomorphisms preserv... |
| rhmdvd 29821 | A ring homomorphism preser... |
| rhmunitinv 29822 | Ring homomorphisms preserv... |
| kerunit 29823 | If a unit element lies in ... |
| reldmresv 29826 | The scalar restriction is ... |
| resvval 29827 | Value of structure restric... |
| resvid2 29828 | General behavior of trivia... |
| resvval2 29829 | Value of nontrivial struct... |
| resvsca 29830 | Base set of a structure re... |
| resvlem 29831 | Other elements of a struct... |
| resvbas 29832 | ` Base ` is unaffected by ... |
| resvplusg 29833 | ` +g ` is unaffected by sc... |
| resvvsca 29834 | ` .s ` is unaffected by sc... |
| resvmulr 29835 | ` .s ` is unaffected by sc... |
| resv0g 29836 | ` 0g ` is unaffected by sc... |
| resv1r 29837 | ` 1r ` is unaffected by sc... |
| resvcmn 29838 | Scalar restriction preserv... |
| gzcrng 29839 | The gaussian integers form... |
| reofld 29840 | The real numbers form an o... |
| nn0omnd 29841 | The nonnegative integers f... |
| rearchi 29842 | The field of the real numb... |
| nn0archi 29843 | The monoid of the nonnegat... |
| xrge0slmod 29844 | The extended nonnegative r... |
| symgfcoeu 29845 | Uniqueness property of per... |
| psgndmfi 29846 | For a finite base set, the... |
| psgnid 29847 | Permutation sign of the id... |
| pmtrprfv2 29848 | In a transposition of two ... |
| pmtrto1cl 29849 | Useful lemma for the follo... |
| psgnfzto1stlem 29850 | Lemma for ~ psgnfzto1st . ... |
| fzto1stfv1 29851 | Value of our permutation `... |
| fzto1st1 29852 | Special case where the per... |
| fzto1st 29853 | The function moving one el... |
| fzto1stinvn 29854 | Value of the inverse of ou... |
| psgnfzto1st 29855 | The permutation sign for m... |
| pmtridf1o 29856 | Transpositions of ` X ` an... |
| pmtridfv1 29857 | Value at X of the transpos... |
| pmtridfv2 29858 | Value at Y of the transpos... |
| smatfval 29861 | Value of the submatrix. (... |
| smatrcl 29862 | Closure of the rectangular... |
| smatlem 29863 | Lemma for the next theorem... |
| smattl 29864 | Entries of a submatrix, to... |
| smattr 29865 | Entries of a submatrix, to... |
| smatbl 29866 | Entries of a submatrix, bo... |
| smatbr 29867 | Entries of a submatrix, bo... |
| smatcl 29868 | Closure of the square subm... |
| matmpt2 29869 | Write a square matrix as a... |
| 1smat1 29870 | The submatrix of the ident... |
| submat1n 29871 | One case where the submatr... |
| submatres 29872 | Special case where the sub... |
| submateqlem1 29873 | Lemma for ~ submateq . (C... |
| submateqlem2 29874 | Lemma for ~ submateq . (C... |
| submateq 29875 | Sufficient condition for t... |
| submatminr1 29876 | If we take a submatrix by ... |
| lmatval 29879 | Value of the literal matri... |
| lmatfval 29880 | Entries of a literal matri... |
| lmatfvlem 29881 | Useful lemma to extract li... |
| lmatcl 29882 | Closure of the literal mat... |
| lmat22lem 29883 | Lemma for ~ lmat22e11 and ... |
| lmat22e11 29884 | Entry of a 2x2 literal mat... |
| lmat22e12 29885 | Entry of a 2x2 literal mat... |
| lmat22e21 29886 | Entry of a 2x2 literal mat... |
| lmat22e22 29887 | Entry of a 2x2 literal mat... |
| lmat22det 29888 | The determinant of a liter... |
| mdetpmtr1 29889 | The determinant of a matri... |
| mdetpmtr2 29890 | The determinant of a matri... |
| mdetpmtr12 29891 | The determinant of a matri... |
| mdetlap1 29892 | A Laplace expansion of the... |
| madjusmdetlem1 29893 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem2 29894 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem3 29895 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem4 29896 | Lemma for ~ madjusmdet . ... |
| madjusmdet 29897 | Express the cofactor of th... |
| mdetlap 29898 | Laplace expansion of the d... |
| fvproj 29899 | Value of a function on pai... |
| fimaproj 29900 | Image of a cartesian produ... |
| txomap 29901 | Given two open maps ` F ` ... |
| qtopt1 29902 | If every equivalence class... |
| qtophaus 29903 | If an open map's graph in ... |
| circtopn 29904 | The topology of the unit c... |
| circcn 29905 | The function gluing the re... |
| reff 29906 | For any cover refinement, ... |
| locfinreflem 29907 | A locally finite refinemen... |
| locfinref 29908 | A locally finite refinemen... |
| iscref 29911 | The property that every op... |
| crefeq 29912 | Equality theorem for the "... |
| creftop 29913 | A space where every open c... |
| crefi 29914 | The property that every op... |
| crefdf 29915 | A formulation of ~ crefi e... |
| crefss 29916 | The "every open cover has ... |
| cmpcref 29917 | Equivalent definition of c... |
| cmpfiref 29918 | Every open cover of a Comp... |
| ldlfcntref 29921 | Every open cover of a Lind... |
| ispcmp 29924 | The predicate "is a paraco... |
| cmppcmp 29925 | Every compact space is par... |
| dispcmp 29926 | Every discrete space is pa... |
| pcmplfin 29927 | Given a paracompact topolo... |
| pcmplfinf 29928 | Given a paracompact topolo... |
| metidval 29933 | Value of the metric identi... |
| metidss 29934 | As a relation, the metric ... |
| metidv 29935 | ` A ` and ` B ` identify b... |
| metideq 29936 | Basic property of the metr... |
| metider 29937 | The metric identification ... |
| pstmval 29938 | Value of the metric induce... |
| pstmfval 29939 | Function value of the metr... |
| pstmxmet 29940 | The metric induced by a ps... |
| hauseqcn 29941 | In a Hausdorff topology, t... |
| unitsscn 29942 | The closed unit is a subse... |
| elunitrn 29943 | The closed unit is a subse... |
| elunitcn 29944 | The closed unit is a subse... |
| elunitge0 29945 | An element of the closed u... |
| unitssxrge0 29946 | The closed unit is a subse... |
| unitdivcld 29947 | Necessary conditions for a... |
| iistmd 29948 | The closed unit forms a to... |
| unicls 29949 | The union of the closed se... |
| tpr2tp 29950 | The usual topology on ` ( ... |
| tpr2uni 29951 | The usual topology on ` ( ... |
| xpinpreima 29952 | Rewrite the cartesian prod... |
| xpinpreima2 29953 | Rewrite the cartesian prod... |
| sqsscirc1 29954 | The complex square of side... |
| sqsscirc2 29955 | The complex square of side... |
| cnre2csqlem 29956 | Lemma for ~ cnre2csqima . ... |
| cnre2csqima 29957 | Image of a centered square... |
| tpr2rico 29958 | For any point of an open s... |
| cnvordtrestixx 29959 | The restriction of the 'gr... |
| prsdm 29960 | Domain of the relation of ... |
| prsrn 29961 | Range of the relation of a... |
| prsss 29962 | Relation of a subpreset. ... |
| prsssdm 29963 | Domain of a subpreset rela... |
| ordtprsval 29964 | Value of the order topolog... |
| ordtprsuni 29965 | Value of the order topolog... |
| ordtcnvNEW 29966 | The order dual generates t... |
| ordtrestNEW 29967 | The subspace topology of a... |
| ordtrest2NEWlem 29968 | Lemma for ~ ordtrest2NEW .... |
| ordtrest2NEW 29969 | An interval-closed set ` A... |
| ordtconnlem1 29970 | Connectedness in the order... |
| ordtconn 29971 | Connectedness in the order... |
| mndpluscn 29972 | A mapping that is both a h... |
| mhmhmeotmd 29973 | Deduce a Topological Monoi... |
| rmulccn 29974 | Multiplication by a real c... |
| raddcn 29975 | Addition in the real numbe... |
| xrmulc1cn 29976 | The operation multiplying ... |
| fmcncfil 29977 | The image of a Cauchy filt... |
| xrge0hmph 29978 | The extended nonnegative r... |
| xrge0iifcnv 29979 | Define a bijection from ` ... |
| xrge0iifcv 29980 | The defined function's val... |
| xrge0iifiso 29981 | The defined bijection from... |
| xrge0iifhmeo 29982 | Expose a homeomorphism fro... |
| xrge0iifhom 29983 | The defined function from ... |
| xrge0iif1 29984 | Condition for the defined ... |
| xrge0iifmhm 29985 | The defined function from ... |
| xrge0pluscn 29986 | The addition operation of ... |
| xrge0mulc1cn 29987 | The operation multiplying ... |
| xrge0tps 29988 | The extended nonnegative r... |
| xrge0topn 29989 | The topology of the extend... |
| xrge0haus 29990 | The topology of the extend... |
| xrge0tmdOLD 29991 | The extended nonnegative r... |
| xrge0tmd 29992 | The extended nonnegative r... |
| lmlim 29993 | Relate a limit in a given ... |
| lmlimxrge0 29994 | Relate a limit in the nonn... |
| rge0scvg 29995 | Implication of convergence... |
| fsumcvg4 29996 | A serie with finite suppor... |
| pnfneige0 29997 | A neighborhood of ` +oo ` ... |
| lmxrge0 29998 | Express "sequence ` F ` co... |
| lmdvg 29999 | If a monotonic sequence of... |
| lmdvglim 30000 | If a monotonic real number... |
| pl1cn 30001 | A univariate polynomial is... |
| zringnm 30004 | The norm (function) for a ... |
| zzsnm 30005 | The norm of the ring of th... |
| zlm0 30006 | Zero of a ` ZZ ` -module. ... |
| zlm1 30007 | Unit of a ` ZZ ` -module (... |
| zlmds 30008 | Distance in a ` ZZ ` -modu... |
| zlmtset 30009 | Topology in a ` ZZ ` -modu... |
| zlmnm 30010 | Norm of a ` ZZ ` -module (... |
| zhmnrg 30011 | The ` ZZ ` -module built f... |
| nmmulg 30012 | The norm of a group produc... |
| zrhnm 30013 | The norm of the image by `... |
| cnzh 30014 | The ` ZZ ` -module of ` CC... |
| rezh 30015 | The ` ZZ ` -module of ` RR... |
| qqhval 30018 | Value of the canonical hom... |
| zrhf1ker 30019 | The kernel of the homomorp... |
| zrhchr 30020 | The kernel of the homomorp... |
| zrhker 30021 | The kernel of the homomorp... |
| zrhunitpreima 30022 | The preimage by ` ZRHom ` ... |
| elzrhunit 30023 | Condition for the image by... |
| elzdif0 30024 | Lemma for ~ qqhval2 . (Co... |
| qqhval2lem 30025 | Lemma for ~ qqhval2 . (Co... |
| qqhval2 30026 | Value of the canonical hom... |
| qqhvval 30027 | Value of the canonical hom... |
| qqh0 30028 | The image of ` 0 ` by the ... |
| qqh1 30029 | The image of ` 1 ` by the ... |
| qqhf 30030 | ` QQHom ` as a function. ... |
| qqhvq 30031 | The image of a quotient by... |
| qqhghm 30032 | The ` QQHom ` homomorphism... |
| qqhrhm 30033 | The ` QQHom ` homomorphism... |
| qqhnm 30034 | The norm of the image by `... |
| qqhcn 30035 | The ` QQHom ` homomorphism... |
| qqhucn 30036 | The ` QQHom ` homomorphism... |
| rrhval 30040 | Value of the canonical hom... |
| rrhcn 30041 | If the topology of ` R ` i... |
| rrhf 30042 | If the topology of ` R ` i... |
| isrrext 30044 | Express the property " ` R... |
| rrextnrg 30045 | An extension of ` RR ` is ... |
| rrextdrg 30046 | An extension of ` RR ` is ... |
| rrextnlm 30047 | The norm of an extension o... |
| rrextchr 30048 | The ring characteristic of... |
| rrextcusp 30049 | An extension of ` RR ` is ... |
| rrexttps 30050 | An extension of ` RR ` is ... |
| rrexthaus 30051 | The topology of an extensi... |
| rrextust 30052 | The uniformity of an exten... |
| rerrext 30053 | The field of the real numb... |
| cnrrext 30054 | The field of the complex n... |
| qqtopn 30055 | The topology of the field ... |
| rrhfe 30056 | If ` R ` is an extension o... |
| rrhcne 30057 | If ` R ` is an extension o... |
| rrhqima 30058 | The ` RRHom ` homomorphism... |
| rrh0 30059 | The image of ` 0 ` by the ... |
| xrhval 30062 | The value of the embedding... |
| zrhre 30063 | The ` ZRHom ` homomorphism... |
| qqhre 30064 | The ` QQHom ` homomorphism... |
| rrhre 30065 | The ` RRHom ` homomorphism... |
| relmntop 30068 | Manifold is a relation. (... |
| ismntoplly 30069 | Property of being a manifo... |
| ismntop 30070 | Property of being a manifo... |
| nexple 30071 | A lower bound for an expon... |
| indv 30074 | Value of the indicator fun... |
| indval 30075 | Value of the indicator fun... |
| indval2 30076 | Alternate value of the ind... |
| indf 30077 | An indicator function as a... |
| indfval 30078 | Value of the indicator fun... |
| ind1 30079 | Value of the indicator fun... |
| ind0 30080 | Value of the indicator fun... |
| ind1a 30081 | Value of the indicator fun... |
| indpi1 30082 | Preimage of the singleton ... |
| indsum 30083 | Finite sum of a product wi... |
| indsumin 30084 | Finite sum of a product wi... |
| prodindf 30085 | The product of indicators ... |
| indf1o 30086 | The bijection between a po... |
| indpreima 30087 | A function with range ` { ... |
| indf1ofs 30088 | The bijection between fini... |
| esumex 30091 | An extended sum is a set b... |
| esumcl 30092 | Closure for extended sum i... |
| esumeq12dvaf 30093 | Equality deduction for ext... |
| esumeq12dva 30094 | Equality deduction for ext... |
| esumeq12d 30095 | Equality deduction for ext... |
| esumeq1 30096 | Equality theorem for an ex... |
| esumeq1d 30097 | Equality theorem for an ex... |
| esumeq2 30098 | Equality theorem for exten... |
| esumeq2d 30099 | Equality deduction for ext... |
| esumeq2dv 30100 | Equality deduction for ext... |
| esumeq2sdv 30101 | Equality deduction for ext... |
| nfesum1 30102 | Bound-variable hypothesis ... |
| nfesum2 30103 | Bound-variable hypothesis ... |
| cbvesum 30104 | Change bound variable in a... |
| cbvesumv 30105 | Change bound variable in a... |
| esumid 30106 | Identify the extended sum ... |
| esumgsum 30107 | A finite extended sum is t... |
| esumval 30108 | Develop the value of the e... |
| esumel 30109 | The extended sum is a limi... |
| esumnul 30110 | Extended sum over the empt... |
| esum0 30111 | Extended sum of zero. (Co... |
| esumf1o 30112 | Re-index an extended sum u... |
| esumc 30113 | Convert from the collectio... |
| esumrnmpt 30114 | Rewrite an extended sum in... |
| esumsplit 30115 | Split an extended sum into... |
| esummono 30116 | Extended sum is monotonic.... |
| esumpad 30117 | Extend an extended sum by ... |
| esumpad2 30118 | Remove zeroes from an exte... |
| esumadd 30119 | Addition of infinite sums.... |
| esumle 30120 | If all of the terms of an ... |
| gsumesum 30121 | Relate a group sum on ` ( ... |
| esumlub 30122 | The extended sum is the lo... |
| esumaddf 30123 | Addition of infinite sums.... |
| esumlef 30124 | If all of the terms of an ... |
| esumcst 30125 | The extended sum of a cons... |
| esumsnf 30126 | The extended sum of a sing... |
| esumsn 30127 | The extended sum of a sing... |
| esumpr 30128 | Extended sum over a pair. ... |
| esumpr2 30129 | Extended sum over a pair, ... |
| esumrnmpt2 30130 | Rewrite an extended sum in... |
| esumfzf 30131 | Formulating a partial exte... |
| esumfsup 30132 | Formulating an extended su... |
| esumfsupre 30133 | Formulating an extended su... |
| esumss 30134 | Change the index set to a ... |
| esumpinfval 30135 | The value of the extended ... |
| esumpfinvallem 30136 | Lemma for ~ esumpfinval . ... |
| esumpfinval 30137 | The value of the extended ... |
| esumpfinvalf 30138 | Same as ~ esumpfinval , mi... |
| esumpinfsum 30139 | The value of the extended ... |
| esumpcvgval 30140 | The value of the extended ... |
| esumpmono 30141 | The partial sums in an ext... |
| esumcocn 30142 | Lemma for ~ esummulc2 and ... |
| esummulc1 30143 | An extended sum multiplied... |
| esummulc2 30144 | An extended sum multiplied... |
| esumdivc 30145 | An extended sum divided by... |
| hashf2 30146 | Lemma for ~ hasheuni . (C... |
| hasheuni 30147 | The cardinality of a disjo... |
| esumcvg 30148 | The sequence of partial su... |
| esumcvg2 30149 | Simpler version of ~ esumc... |
| esumcvgsum 30150 | The value of the extended ... |
| esumsup 30151 | Express an extended sum as... |
| esumgect 30152 | "Send ` n ` to ` +oo ` " i... |
| esumcvgre 30153 | All terms of a converging ... |
| esum2dlem 30154 | Lemma for ~ esum2d (finite... |
| esum2d 30155 | Write a double extended su... |
| esumiun 30156 | Sum over a non necessarily... |
| ofceq 30159 | Equality theorem for funct... |
| ofcfval 30160 | Value of an operation appl... |
| ofcval 30161 | Evaluate a function/consta... |
| ofcfn 30162 | The function operation pro... |
| ofcfeqd2 30163 | Equality theorem for funct... |
| ofcfval3 30164 | General value of ` ( F oFC... |
| ofcf 30165 | The function/constant oper... |
| ofcfval2 30166 | The function operation exp... |
| ofcfval4 30167 | The function/constant oper... |
| ofcc 30168 | Left operation by a consta... |
| ofcof 30169 | Relate function operation ... |
| sigaex 30172 | Lemma for ~ issiga and ~ i... |
| sigaval 30173 | The set of sigma-algebra w... |
| issiga 30174 | An alternative definition ... |
| isrnsigaOLD 30175 | The property of being a si... |
| isrnsiga 30176 | The property of being a si... |
| 0elsiga 30177 | A sigma-algebra contains t... |
| baselsiga 30178 | A sigma-algebra contains i... |
| sigasspw 30179 | A sigma-algebra is a set o... |
| sigaclcu 30180 | A sigma-algebra is closed ... |
| sigaclcuni 30181 | A sigma-algebra is closed ... |
| sigaclfu 30182 | A sigma-algebra is closed ... |
| sigaclcu2 30183 | A sigma-algebra is closed ... |
| sigaclfu2 30184 | A sigma-algebra is closed ... |
| sigaclcu3 30185 | A sigma-algebra is closed ... |
| issgon 30186 | Property of being a sigma-... |
| sgon 30187 | A sigma-algebra is a sigma... |
| elsigass 30188 | An element of a sigma-alge... |
| elrnsiga 30189 | Dropping the base informat... |
| isrnsigau 30190 | The property of being a si... |
| unielsiga 30191 | A sigma-algebra contains i... |
| dmvlsiga 30192 | Lebesgue-measurable subset... |
| pwsiga 30193 | Any power set forms a sigm... |
| prsiga 30194 | The smallest possible sigm... |
| sigaclci 30195 | A sigma-algebra is closed ... |
| difelsiga 30196 | A sigma-algebra is closed ... |
| unelsiga 30197 | A sigma-algebra is closed ... |
| inelsiga 30198 | A sigma-algebra is closed ... |
| sigainb 30199 | Building a sigma-algebra f... |
| insiga 30200 | The intersection of a coll... |
| sigagenval 30203 | Value of the generated sig... |
| sigagensiga 30204 | A generated sigma-algebra ... |
| sgsiga 30205 | A generated sigma-algebra ... |
| unisg 30206 | The sigma-algebra generate... |
| dmsigagen 30207 | A sigma-algebra can be gen... |
| sssigagen 30208 | A set is a subset of the s... |
| sssigagen2 30209 | A subset of the generating... |
| elsigagen 30210 | Any element of a set is al... |
| elsigagen2 30211 | Any countable union of ele... |
| sigagenss 30212 | The generated sigma-algebr... |
| sigagenss2 30213 | Sufficient condition for i... |
| sigagenid 30214 | The sigma-algebra generate... |
| ispisys 30215 | The property of being a pi... |
| ispisys2 30216 | The property of being a pi... |
| inelpisys 30217 | Pi-systems are closed unde... |
| sigapisys 30218 | All sigma-algebras are pi-... |
| isldsys 30219 | The property of being a la... |
| pwldsys 30220 | The power set of the unive... |
| unelldsys 30221 | Lambda-systems are closed ... |
| sigaldsys 30222 | All sigma-algebras are lam... |
| ldsysgenld 30223 | The intersection of all la... |
| sigapildsyslem 30224 | Lemma for ~ sigapildsys . ... |
| sigapildsys 30225 | Sigma-algebra are exactly ... |
| ldgenpisyslem1 30226 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem2 30227 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem3 30228 | Lemma for ~ ldgenpisys . ... |
| ldgenpisys 30229 | The lambda system ` E ` ge... |
| dynkin 30230 | Dynkin's lambda-pi theorem... |
| isros 30231 | The property of being a ri... |
| rossspw 30232 | A ring of sets is a collec... |
| 0elros 30233 | A ring of sets contains th... |
| unelros 30234 | A ring of sets is closed u... |
| difelros 30235 | A ring of sets is closed u... |
| inelros 30236 | A ring of sets is closed u... |
| fiunelros 30237 | A ring of sets is closed u... |
| issros 30238 | The property of being a se... |
| srossspw 30239 | A semi-ring of sets is a c... |
| 0elsros 30240 | A semi-ring of sets contai... |
| inelsros 30241 | A semi-ring of sets is clo... |
| diffiunisros 30242 | In semiring of sets, compl... |
| rossros 30243 | Rings of sets are semi-rin... |
| brsiga 30246 | The Borel Algebra on real ... |
| brsigarn 30247 | The Borel Algebra is a sig... |
| brsigasspwrn 30248 | The Borel Algebra is a set... |
| unibrsiga 30249 | The union of the Borel Alg... |
| cldssbrsiga 30250 | A Borel Algebra contains a... |
| sxval 30253 | Value of the product sigma... |
| sxsiga 30254 | A product sigma-algebra is... |
| sxsigon 30255 | A product sigma-algebra is... |
| sxuni 30256 | The base set of a product ... |
| elsx 30257 | The cartesian product of t... |
| measbase 30260 | The base set of a measure ... |
| measval 30261 | The value of the ` measure... |
| ismeas 30262 | The property of being a me... |
| isrnmeas 30263 | The property of being a me... |
| dmmeas 30264 | The domain of a measure is... |
| measbasedom 30265 | The base set of a measure ... |
| measfrge0 30266 | A measure is a function ov... |
| measfn 30267 | A measure is a function on... |
| measvxrge0 30268 | The values of a measure ar... |
| measvnul 30269 | The measure of the empty s... |
| measge0 30270 | A measure is nonnegative. ... |
| measle0 30271 | If the measure of a given ... |
| measvun 30272 | The measure of a countable... |
| measxun2 30273 | The measure the union of t... |
| measun 30274 | The measure the union of t... |
| measvunilem 30275 | Lemma for ~ measvuni . (C... |
| measvunilem0 30276 | Lemma for ~ measvuni . (C... |
| measvuni 30277 | The measure of a countable... |
| measssd 30278 | A measure is monotone with... |
| measunl 30279 | A measure is sub-additive ... |
| measiuns 30280 | The measure of the union o... |
| measiun 30281 | A measure is sub-additive.... |
| meascnbl 30282 | A measure is continuous fr... |
| measinblem 30283 | Lemma for ~ measinb . (Co... |
| measinb 30284 | Building a measure restric... |
| measres 30285 | Building a measure restric... |
| measinb2 30286 | Building a measure restric... |
| measdivcstOLD 30287 | Division of a measure by a... |
| measdivcst 30288 | Division of a measure by a... |
| cntmeas 30289 | The Counting measure is a ... |
| pwcntmeas 30290 | The counting measure is a ... |
| cntnevol 30291 | Counting and Lebesgue meas... |
| voliune 30292 | The Lebesgue measure funct... |
| volfiniune 30293 | The Lebesgue measure funct... |
| volmeas 30294 | The Lebesgue measure is a ... |
| ddeval1 30297 | Value of the delta measure... |
| ddeval0 30298 | Value of the delta measure... |
| ddemeas 30299 | The Dirac delta measure is... |
| relae 30303 | 'almost everywhere' is a r... |
| brae 30304 | 'almost everywhere' relati... |
| braew 30305 | 'almost everywhere' relati... |
| truae 30306 | A truth holds almost every... |
| aean 30307 | A conjunction holds almost... |
| faeval 30309 | Value of the 'almost every... |
| relfae 30310 | The 'almost everywhere' bu... |
| brfae 30311 | 'almost everywhere' relati... |
| ismbfm 30314 | The predicate " ` F ` is a... |
| elunirnmbfm 30315 | The property of being a me... |
| mbfmfun 30316 | A measurable function is a... |
| mbfmf 30317 | A measurable function as a... |
| isanmbfm 30318 | The predicate to be a meas... |
| mbfmcnvima 30319 | The preimage by a measurab... |
| mbfmbfm 30320 | A measurable function to a... |
| mbfmcst 30321 | A constant function is mea... |
| 1stmbfm 30322 | The first projection map i... |
| 2ndmbfm 30323 | The second projection map ... |
| imambfm 30324 | If the sigma-algebra in th... |
| cnmbfm 30325 | A continuous function is m... |
| mbfmco 30326 | The composition of two mea... |
| mbfmco2 30327 | The pair building of two m... |
| mbfmvolf 30328 | Measurable functions with ... |
| elmbfmvol2 30329 | Measurable functions with ... |
| mbfmcnt 30330 | All functions are measurab... |
| br2base 30331 | The base set for the gener... |
| dya2ub 30332 | An upper bound for a dyadi... |
| sxbrsigalem0 30333 | The closed half-spaces of ... |
| sxbrsigalem3 30334 | The sigma-algebra generate... |
| dya2iocival 30335 | The function ` I ` returns... |
| dya2iocress 30336 | Dyadic intervals are subse... |
| dya2iocbrsiga 30337 | Dyadic intervals are Borel... |
| dya2icobrsiga 30338 | Dyadic intervals are Borel... |
| dya2icoseg 30339 | For any point and any clos... |
| dya2icoseg2 30340 | For any point and any open... |
| dya2iocrfn 30341 | The function returning dya... |
| dya2iocct 30342 | The dyadic rectangle set i... |
| dya2iocnrect 30343 | For any point of an open r... |
| dya2iocnei 30344 | For any point of an open s... |
| dya2iocuni 30345 | Every open set of ` ( RR X... |
| dya2iocucvr 30346 | The dyadic rectangular set... |
| sxbrsigalem1 30347 | The Borel algebra on ` ( R... |
| sxbrsigalem2 30348 | The sigma-algebra generate... |
| sxbrsigalem4 30349 | The Borel algebra on ` ( R... |
| sxbrsigalem5 30350 | First direction for ~ sxbr... |
| sxbrsigalem6 30351 | First direction for ~ sxbr... |
| sxbrsiga 30352 | The product sigma-algebra ... |
| omsval 30355 | Value of the function mapp... |
| omsfval 30356 | Value of the outer measure... |
| omscl 30357 | A closure lemma for the co... |
| omsf 30358 | A constructed outer measur... |
| oms0 30359 | A constructed outer measur... |
| omsmon 30360 | A constructed outer measur... |
| omssubaddlem 30361 | For any small margin ` E `... |
| omssubadd 30362 | A constructed outer measur... |
| carsgval 30365 | Value of the Caratheodory ... |
| carsgcl 30366 | Closure of the Caratheodor... |
| elcarsg 30367 | Property of being a Catath... |
| baselcarsg 30368 | The universe set, ` O ` , ... |
| 0elcarsg 30369 | The empty set is Caratheod... |
| carsguni 30370 | The union of all Caratheod... |
| elcarsgss 30371 | Caratheodory measurable se... |
| difelcarsg 30372 | The Caratheodory measurabl... |
| inelcarsg 30373 | The Caratheodory measurabl... |
| unelcarsg 30374 | The Caratheodory-measurabl... |
| difelcarsg2 30375 | The Caratheodory-measurabl... |
| carsgmon 30376 | Utility lemma: Apply mono... |
| carsgsigalem 30377 | Lemma for the following th... |
| fiunelcarsg 30378 | The Caratheodory measurabl... |
| carsgclctunlem1 30379 | Lemma for ~ carsgclctun . ... |
| carsggect 30380 | The outer measure is count... |
| carsgclctunlem2 30381 | Lemma for ~ carsgclctun . ... |
| carsgclctunlem3 30382 | Lemma for ~ carsgclctun . ... |
| carsgclctun 30383 | The Caratheodory measurabl... |
| carsgsiga 30384 | The Caratheodory measurabl... |
| omsmeas 30385 | The restriction of a const... |
| pmeasmono 30386 | This theorem's hypotheses ... |
| pmeasadd 30387 | A premeasure on a ring of ... |
| itgeq12dv 30388 | Equality theorem for an in... |
| sitgval 30394 | Value of the simple functi... |
| issibf 30395 | The predicate " ` F ` is a... |
| sibf0 30396 | The constant zero function... |
| sibfmbl 30397 | A simple function is measu... |
| sibff 30398 | A simple function is a fun... |
| sibfrn 30399 | A simple function has fini... |
| sibfima 30400 | Any preimage of a singleto... |
| sibfinima 30401 | The measure of the interse... |
| sibfof 30402 | Applying function operatio... |
| sitgfval 30403 | Value of the Bochner integ... |
| sitgclg 30404 | Closure of the Bochner int... |
| sitgclbn 30405 | Closure of the Bochner int... |
| sitgclcn 30406 | Closure of the Bochner int... |
| sitgclre 30407 | Closure of the Bochner int... |
| sitg0 30408 | The integral of the consta... |
| sitgf 30409 | The integral for simple fu... |
| sitgaddlemb 30410 | Lemma for * sitgadd . (Co... |
| sitmval 30411 | Value of the simple functi... |
| sitmfval 30412 | Value of the integral dist... |
| sitmcl 30413 | Closure of the integral di... |
| sitmf 30414 | The integral metric as a f... |
| oddpwdc 30416 | Lemma for ~ eulerpart . T... |
| oddpwdcv 30417 | Lemma for ~ eulerpart : va... |
| eulerpartlemsv1 30418 | Lemma for ~ eulerpart . V... |
| eulerpartlemelr 30419 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv2 30420 | Lemma for ~ eulerpart . V... |
| eulerpartlemsf 30421 | Lemma for ~ eulerpart . (... |
| eulerpartlems 30422 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv3 30423 | Lemma for ~ eulerpart . V... |
| eulerpartlemgc 30424 | Lemma for ~ eulerpart . (... |
| eulerpartleme 30425 | Lemma for ~ eulerpart . (... |
| eulerpartlemv 30426 | Lemma for ~ eulerpart . (... |
| eulerpartlemo 30427 | Lemma for ~ eulerpart : ` ... |
| eulerpartlemd 30428 | Lemma for ~ eulerpart : ` ... |
| eulerpartlem1 30429 | Lemma for ~ eulerpart . (... |
| eulerpartlemb 30430 | Lemma for ~ eulerpart . T... |
| eulerpartlemt0 30431 | Lemma for ~ eulerpart . (... |
| eulerpartlemf 30432 | Lemma for ~ eulerpart : O... |
| eulerpartlemt 30433 | Lemma for ~ eulerpart . (... |
| eulerpartgbij 30434 | Lemma for ~ eulerpart : T... |
| eulerpartlemgv 30435 | Lemma for ~ eulerpart : va... |
| eulerpartlemr 30436 | Lemma for ~ eulerpart . (... |
| eulerpartlemmf 30437 | Lemma for ~ eulerpart . (... |
| eulerpartlemgvv 30438 | Lemma for ~ eulerpart : va... |
| eulerpartlemgu 30439 | Lemma for ~ eulerpart : R... |
| eulerpartlemgh 30440 | Lemma for ~ eulerpart : T... |
| eulerpartlemgf 30441 | Lemma for ~ eulerpart : I... |
| eulerpartlemgs2 30442 | Lemma for ~ eulerpart : T... |
| eulerpartlemn 30443 | Lemma for ~ eulerpart . (... |
| eulerpart 30444 | Euler's theorem on partiti... |
| subiwrd 30447 | Lemma for ~ sseqp1 . (Con... |
| subiwrdlen 30448 | Length of a subword of an ... |
| iwrdsplit 30449 | Lemma for ~ sseqp1 . (Con... |
| sseqval 30450 | Value of the strong sequen... |
| sseqfv1 30451 | Value of the strong sequen... |
| sseqfn 30452 | A strong recursive sequenc... |
| sseqmw 30453 | Lemma for ~ sseqf amd ~ ss... |
| sseqf 30454 | A strong recursive sequenc... |
| sseqfres 30455 | The first elements in the ... |
| sseqfv2 30456 | Value of the strong sequen... |
| sseqp1 30457 | Value of the strong sequen... |
| fiblem 30460 | Lemma for ~ fib0 , ~ fib1 ... |
| fib0 30461 | Value of the Fibonacci seq... |
| fib1 30462 | Value of the Fibonacci seq... |
| fibp1 30463 | Value of the Fibonacci seq... |
| fib2 30464 | Value of the Fibonacci seq... |
| fib3 30465 | Value of the Fibonacci seq... |
| fib4 30466 | Value of the Fibonacci seq... |
| fib5 30467 | Value of the Fibonacci seq... |
| fib6 30468 | Value of the Fibonacci seq... |
| elprob 30471 | The property of being a pr... |
| domprobmeas 30472 | A probability measure is a... |
| domprobsiga 30473 | The domain of a probabilit... |
| probtot 30474 | The probability of the uni... |
| prob01 30475 | A probability is an elemen... |
| probnul 30476 | The probability of the emp... |
| unveldomd 30477 | The universe is an element... |
| unveldom 30478 | The universe is an element... |
| nuleldmp 30479 | The empty set is an elemen... |
| probcun 30480 | The probability of the uni... |
| probun 30481 | The probability of the uni... |
| probdif 30482 | The probability of the dif... |
| probinc 30483 | A probability law is incre... |
| probdsb 30484 | The probability of the com... |
| probmeasd 30485 | A probability measure is a... |
| probvalrnd 30486 | The value of a probability... |
| probtotrnd 30487 | The probability of the uni... |
| totprobd 30488 | Law of total probability, ... |
| totprob 30489 | Law of total probability. ... |
| probfinmeasbOLD 30490 | Build a probability measur... |
| probfinmeasb 30491 | Build a probability measur... |
| probmeasb 30492 | Build a probability from a... |
| cndprobval 30495 | The value of the condition... |
| cndprobin 30496 | An identity linking condit... |
| cndprob01 30497 | The conditional probabilit... |
| cndprobtot 30498 | The conditional probabilit... |
| cndprobnul 30499 | The conditional probabilit... |
| cndprobprob 30500 | The conditional probabilit... |
| bayesth 30501 | Bayes Theorem. (Contribut... |
| rrvmbfm 30504 | A real-valued random varia... |
| isrrvv 30505 | Elementhood to the set of ... |
| rrvvf 30506 | A real-valued random varia... |
| rrvfn 30507 | A real-valued random varia... |
| rrvdm 30508 | The domain of a random var... |
| rrvrnss 30509 | The range of a random vari... |
| rrvf2 30510 | A real-valued random varia... |
| rrvdmss 30511 | The domain of a random var... |
| rrvfinvima 30512 | For a real-value random va... |
| 0rrv 30513 | The constant function equa... |
| rrvadd 30514 | The sum of two random vari... |
| rrvmulc 30515 | A random variable multipli... |
| rrvsum 30516 | An indexed sum of random v... |
| orvcval 30519 | Value of the preimage mapp... |
| orvcval2 30520 | Another way to express the... |
| elorvc 30521 | Elementhood of a preimage.... |
| orvcval4 30522 | The value of the preimage ... |
| orvcoel 30523 | If the relation produces o... |
| orvccel 30524 | If the relation produces c... |
| elorrvc 30525 | Elementhood of a preimage ... |
| orrvcval4 30526 | The value of the preimage ... |
| orrvcoel 30527 | If the relation produces o... |
| orrvccel 30528 | If the relation produces c... |
| orvcgteel 30529 | Preimage maps produced by ... |
| orvcelval 30530 | Preimage maps produced by ... |
| orvcelel 30531 | Preimage maps produced by ... |
| dstrvval 30532 | The value of the distribut... |
| dstrvprob 30533 | The distribution of a rand... |
| orvclteel 30534 | Preimage maps produced by ... |
| dstfrvel 30535 | Elementhood of preimage ma... |
| dstfrvunirn 30536 | The limit of all preimage ... |
| orvclteinc 30537 | Preimage maps produced by ... |
| dstfrvinc 30538 | A cumulative distribution ... |
| dstfrvclim1 30539 | The limit of the cumulativ... |
| coinfliplem 30540 | Division in the extended r... |
| coinflipprob 30541 | The ` P ` we defined for c... |
| coinflipspace 30542 | The space of our coin-flip... |
| coinflipuniv 30543 | The universe of our coin-f... |
| coinfliprv 30544 | The ` X ` we defined for c... |
| coinflippv 30545 | The probability of heads i... |
| coinflippvt 30546 | The probability of tails i... |
| ballotlemoex 30547 | ` O ` is a set. (Contribu... |
| ballotlem1 30548 | The size of the universe i... |
| ballotlemelo 30549 | Elementhood in ` O ` . (C... |
| ballotlem2 30550 | The probability that the f... |
| ballotlemfval 30551 | The value of F. (Contribut... |
| ballotlemfelz 30552 | ` ( F `` C ) ` has values ... |
| ballotlemfp1 30553 | If the ` J ` th ballot is ... |
| ballotlemfc0 30554 | ` F ` takes value 0 betwee... |
| ballotlemfcc 30555 | ` F ` takes value 0 betwee... |
| ballotlemfmpn 30556 | ` ( F `` C ) ` finishes co... |
| ballotlemfval0 30557 | ` ( F `` C ) ` always star... |
| ballotleme 30558 | Elements of ` E ` . (Cont... |
| ballotlemodife 30559 | Elements of ` ( O \ E ) ` ... |
| ballotlem4 30560 | If the first pick is a vot... |
| ballotlem5 30561 | If A is not ahead througho... |
| ballotlemi 30562 | Value of ` I ` for a given... |
| ballotlemiex 30563 | Properties of ` ( I `` C )... |
| ballotlemi1 30564 | The first tie cannot be re... |
| ballotlemii 30565 | The first tie cannot be re... |
| ballotlemsup 30566 | The set of zeroes of ` F `... |
| ballotlemimin 30567 | ` ( I `` C ) ` is the firs... |
| ballotlemic 30568 | If the first vote is for B... |
| ballotlem1c 30569 | If the first vote is for A... |
| ballotlemsval 30570 | Value of ` S ` . (Contrib... |
| ballotlemsv 30571 | Value of ` S ` evaluated a... |
| ballotlemsgt1 30572 | ` S ` maps values less tha... |
| ballotlemsdom 30573 | Domain of ` S ` for a give... |
| ballotlemsel1i 30574 | The range ` ( 1 ... ( I ``... |
| ballotlemsf1o 30575 | The defined ` S ` is a bij... |
| ballotlemsi 30576 | The image by ` S ` of the ... |
| ballotlemsima 30577 | The image by ` S ` of an i... |
| ballotlemieq 30578 | If two countings share the... |
| ballotlemrval 30579 | Value of ` R ` . (Contrib... |
| ballotlemscr 30580 | The image of ` ( R `` C ) ... |
| ballotlemrv 30581 | Value of ` R ` evaluated a... |
| ballotlemrv1 30582 | Value of ` R ` before the ... |
| ballotlemrv2 30583 | Value of ` R ` after the t... |
| ballotlemro 30584 | Range of ` R ` is included... |
| ballotlemgval 30585 | Expand the value of ` .^ `... |
| ballotlemgun 30586 | A property of the defined ... |
| ballotlemfg 30587 | Express the value of ` ( F... |
| ballotlemfrc 30588 | Express the value of ` ( F... |
| ballotlemfrci 30589 | Reverse counting preserves... |
| ballotlemfrceq 30590 | Value of ` F ` for a rever... |
| ballotlemfrcn0 30591 | Value of ` F ` for a rever... |
| ballotlemrc 30592 | Range of ` R ` . (Contrib... |
| ballotlemirc 30593 | Applying ` R ` does not ch... |
| ballotlemrinv0 30594 | Lemma for ~ ballotlemrinv ... |
| ballotlemrinv 30595 | ` R ` is its own inverse :... |
| ballotlem1ri 30596 | When the vote on the first... |
| ballotlem7 30597 | ` R ` is a bijection betwe... |
| ballotlem8 30598 | There are as many counting... |
| ballotth 30599 | Bertrand's ballot problem ... |
| sgncl 30600 | Closure of the signum. (C... |
| sgnclre 30601 | Closure of the signum. (C... |
| sgnneg 30602 | Negation of the signum. (... |
| sgn3da 30603 | A conditional containing a... |
| sgnmul 30604 | Signum of a product. (Con... |
| sgnmulrp2 30605 | Multiplication by a positi... |
| sgnsub 30606 | Subtraction of a number of... |
| sgnnbi 30607 | Negative signum. (Contrib... |
| sgnpbi 30608 | Positive signum. (Contrib... |
| sgn0bi 30609 | Zero signum. (Contributed... |
| sgnsgn 30610 | Signum is idempotent. (Co... |
| sgnmulsgn 30611 | If two real numbers are of... |
| sgnmulsgp 30612 | If two real numbers are of... |
| fzssfzo 30613 | Condition for an integer i... |
| gsumncl 30614 | Closure of a group sum in ... |
| gsumnunsn 30615 | Closure of a group sum in ... |
| wrdfd 30616 | A word is a zero-based seq... |
| wrdres 30617 | Condition for the restrict... |
| wrdsplex 30618 | Existence of a split of a ... |
| ccatmulgnn0dir 30619 | Concatenation of words fol... |
| ofcccat 30620 | Letterwise operations on w... |
| ofcs1 30621 | Letterwise operations on a... |
| ofcs2 30622 | Letterwise operations on a... |
| plymul02 30623 | Product of a polynomial wi... |
| plymulx0 30624 | Coefficients of a polynomi... |
| plymulx 30625 | Coefficients of a polynomi... |
| plyrecld 30626 | Closure of a polynomial wi... |
| signsplypnf 30627 | The quotient of a polynomi... |
| signsply0 30628 | Lemma for the rule of sign... |
| signspval 30629 | The value of the skipping ... |
| signsw0glem 30630 | Neutral element property o... |
| signswbase 30631 | The base of ` W ` is the t... |
| signswplusg 30632 | The operation of ` W ` . ... |
| signsw0g 30633 | The neutral element of ` W... |
| signswmnd 30634 | ` W ` is a monoid structur... |
| signswrid 30635 | The zero-skipping operatio... |
| signswlid 30636 | The zero-skipping operatio... |
| signswn0 30637 | The zero-skipping operatio... |
| signswch 30638 | The zero-skipping operatio... |
| signslema 30639 | Computational part of sign... |
| signstfv 30640 | Value of the zero-skipping... |
| signstfval 30641 | Value of the zero-skipping... |
| signstcl 30642 | Closure of the zero skippi... |
| signstf 30643 | The zero skipping sign wor... |
| signstlen 30644 | Length of the zero skippin... |
| signstf0 30645 | Sign of a single letter wo... |
| signstfvn 30646 | Zero-skipping sign in a wo... |
| signsvtn0 30647 | If the last letter is non ... |
| signstfvp 30648 | Zero-skipping sign in a wo... |
| signstfvneq0 30649 | In case the first letter i... |
| signstfvcl 30650 | Closure of the zero skippi... |
| signstfvc 30651 | Zero-skipping sign in a wo... |
| signstres 30652 | Restriction of a zero skip... |
| signstfveq0a 30653 | Lemma for ~ signstfveq0 . ... |
| signstfveq0 30654 | In case the last letter is... |
| signsvvfval 30655 | The value of ` V ` , which... |
| signsvvf 30656 | ` V ` is a function. (Con... |
| signsvf0 30657 | There is no change of sign... |
| signsvf1 30658 | In a single-letter word, w... |
| signsvfn 30659 | Number of changes in a wor... |
| signsvtp 30660 | Adding a letter of the sam... |
| signsvtn 30661 | Adding a letter of a diffe... |
| signsvfpn 30662 | Adding a letter of the sam... |
| signsvfnn 30663 | Adding a letter of a diffe... |
| signlem0 30664 | Adding a zero as the highe... |
| signshf 30665 | ` H ` , corresponding to t... |
| signshwrd 30666 | ` H ` , corresponding to t... |
| signshlen 30667 | Length of ` H ` , correspo... |
| signshnz 30668 | ` H ` is not the empty wor... |
| efcld 30669 | Closure law for the expone... |
| iblidicc 30670 | The identity function is i... |
| rpsqrtcn 30671 | Continuity of the real pos... |
| divsqrtid 30672 | A real number divided by i... |
| cxpcncf1 30673 | The power function on comp... |
| efmul2picn 30674 | Multiplying by ` ( _i x. (... |
| fct2relem 30675 | Lemma for ~ ftc2re . (Con... |
| ftc2re 30676 | The Fundamental Theorem of... |
| fdvposlt 30677 | Functions with a positive ... |
| fdvneggt 30678 | Functions with a negative ... |
| fdvposle 30679 | Functions with a nonnegati... |
| fdvnegge 30680 | Functions with a non-posit... |
| prodfzo03 30681 | A product of three factors... |
| actfunsnf1o 30682 | The action ` F ` of extend... |
| actfunsnrndisj 30683 | The action ` F ` of extend... |
| itgexpif 30684 | The basis for the circle m... |
| fsum2dsub 30685 | Lemma for ~ breprexp - Re-... |
| reprval 30688 | Value of the representatio... |
| repr0 30689 | There is exactly one repre... |
| reprf 30690 | Members of the representat... |
| reprsum 30691 | Sums of values of the memb... |
| reprle 30692 | Upper bound to the terms i... |
| reprsuc 30693 | Express the representation... |
| reprfi 30694 | Bounded representations ar... |
| reprss 30695 | Representations with terms... |
| reprinrn 30696 | Representations with term ... |
| reprlt 30697 | There are no representatio... |
| hashreprin 30698 | Express a sum of represent... |
| reprgt 30699 | There are no representatio... |
| reprinfz1 30700 | For the representation of ... |
| reprfi2 30701 | Corollary of ~ reprinfz1 .... |
| reprfz1 30702 | Corollary of ~ reprinfz1 .... |
| hashrepr 30703 | Develop the number of repr... |
| reprpmtf1o 30704 | Transposing ` 0 ` and ` X ... |
| reprdifc 30705 | Express the representation... |
| chpvalz 30706 | Value of the second Chebys... |
| chtvalz 30707 | Value of the Chebyshev fun... |
| breprexplema 30708 | Lemma for ~ breprexp (indu... |
| breprexplemb 30709 | Lemma for ~ breprexp (clos... |
| breprexplemc 30710 | Lemma for ~ breprexp (indu... |
| breprexp 30711 | Express the ` S ` th power... |
| breprexpnat 30712 | Express the ` S ` th power... |
| vtsval 30715 | Value of the Vinogradov tr... |
| vtscl 30716 | Closure of the Vinogradov ... |
| vtsprod 30717 | Express the Vinogradov tri... |
| circlemeth 30718 | The Hardy, Littlewood and ... |
| circlemethnat 30719 | The Hardy, Littlewood and ... |
| circlevma 30720 | The Circle Method, where t... |
| circlemethhgt 30721 | The circle method, where t... |
| hgt750lemc 30725 | An upper bound to the summ... |
| hgt750lemd 30726 | An upper bound to the summ... |
| hgt749d 30727 | A deduction version of ~ a... |
| logdivsqrle 30728 | Conditions for ` ( ( log `... |
| hgt750lem 30729 | Lemma for ~ tgoldbachgtd .... |
| hgt750lem2 30730 | Decimal multiplication gal... |
| hgt750lemf 30731 | Lemma for the statement 7.... |
| hgt750lemg 30732 | Lemma for the statement 7.... |
| oddprm2 30733 | Two ways to write the set ... |
| hgt750lemb 30734 | An upper bound on the cont... |
| hgt750lema 30735 | An upper bound on the cont... |
| hgt750leme 30736 | An upper bound on the cont... |
| tgoldbachgnn 30737 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtde 30738 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtda 30739 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtd 30740 | Odd integers greater than ... |
| tgoldbachgt 30741 | Odd integers greater than ... |
| istrkg2d 30744 | Property of fulfilling dim... |
| axtglowdim2OLD 30745 | Lower dimension axiom for ... |
| axtgupdim2OLD 30746 | Upper dimension axiom for ... |
| afsval 30749 | Value of the AFS relation ... |
| brafs 30750 | Binary relation form of th... |
| tg5segofs 30751 | Rephrase ~ axtg5seg using ... |
| bnj170 30764 | ` /\ ` -manipulation. (Co... |
| bnj240 30765 | ` /\ ` -manipulation. (Co... |
| bnj248 30766 | ` /\ ` -manipulation. (Co... |
| bnj250 30767 | ` /\ ` -manipulation. (Co... |
| bnj251 30768 | ` /\ ` -manipulation. (Co... |
| bnj252 30769 | ` /\ ` -manipulation. (Co... |
| bnj253 30770 | ` /\ ` -manipulation. (Co... |
| bnj255 30771 | ` /\ ` -manipulation. (Co... |
| bnj256 30772 | ` /\ ` -manipulation. (Co... |
| bnj257 30773 | ` /\ ` -manipulation. (Co... |
| bnj258 30774 | ` /\ ` -manipulation. (Co... |
| bnj268 30775 | ` /\ ` -manipulation. (Co... |
| bnj290 30776 | ` /\ ` -manipulation. (Co... |
| bnj291 30777 | ` /\ ` -manipulation. (Co... |
| bnj312 30778 | ` /\ ` -manipulation. (Co... |
| bnj334 30779 | ` /\ ` -manipulation. (Co... |
| bnj345 30780 | ` /\ ` -manipulation. (Co... |
| bnj422 30781 | ` /\ ` -manipulation. (Co... |
| bnj432 30782 | ` /\ ` -manipulation. (Co... |
| bnj446 30783 | ` /\ ` -manipulation. (Co... |
| bnj23 30784 | First-order logic and set ... |
| bnj31 30785 | First-order logic and set ... |
| bnj62 30786 | First-order logic and set ... |
| bnj89 30787 | First-order logic and set ... |
| bnj90 30788 | First-order logic and set ... |
| bnj101 30789 | First-order logic and set ... |
| bnj105 30790 | First-order logic and set ... |
| bnj115 30791 | First-order logic and set ... |
| bnj132 30792 | First-order logic and set ... |
| bnj133 30793 | First-order logic and set ... |
| bnj142OLD 30794 | First-order logic and set ... |
| bnj145OLD 30795 | First-order logic and set ... |
| bnj156 30796 | First-order logic and set ... |
| bnj158 30797 | First-order logic and set ... |
| bnj168 30798 | First-order logic and set ... |
| bnj206 30799 | First-order logic and set ... |
| bnj216 30800 | First-order logic and set ... |
| bnj219 30801 | First-order logic and set ... |
| bnj226 30802 | First-order logic and set ... |
| bnj228 30803 | First-order logic and set ... |
| bnj519 30804 | First-order logic and set ... |
| bnj521 30805 | First-order logic and set ... |
| bnj524 30806 | First-order logic and set ... |
| bnj525 30807 | First-order logic and set ... |
| bnj534 30808 | First-order logic and set ... |
| bnj538 30809 | First-order logic and set ... |
| bnj538OLD 30810 | First-order logic and set ... |
| bnj529 30811 | First-order logic and set ... |
| bnj551 30812 | First-order logic and set ... |
| bnj563 30813 | First-order logic and set ... |
| bnj564 30814 | First-order logic and set ... |
| bnj593 30815 | First-order logic and set ... |
| bnj596 30816 | First-order logic and set ... |
| bnj610 30817 | Pass from equality ( ` x =... |
| bnj642 30818 | ` /\ ` -manipulation. (Co... |
| bnj643 30819 | ` /\ ` -manipulation. (Co... |
| bnj645 30820 | ` /\ ` -manipulation. (Co... |
| bnj658 30821 | ` /\ ` -manipulation. (Co... |
| bnj667 30822 | ` /\ ` -manipulation. (Co... |
| bnj705 30823 | ` /\ ` -manipulation. (Co... |
| bnj706 30824 | ` /\ ` -manipulation. (Co... |
| bnj707 30825 | ` /\ ` -manipulation. (Co... |
| bnj708 30826 | ` /\ ` -manipulation. (Co... |
| bnj721 30827 | ` /\ ` -manipulation. (Co... |
| bnj832 30828 | ` /\ ` -manipulation. (Co... |
| bnj835 30829 | ` /\ ` -manipulation. (Co... |
| bnj836 30830 | ` /\ ` -manipulation. (Co... |
| bnj837 30831 | ` /\ ` -manipulation. (Co... |
| bnj769 30832 | ` /\ ` -manipulation. (Co... |
| bnj770 30833 | ` /\ ` -manipulation. (Co... |
| bnj771 30834 | ` /\ ` -manipulation. (Co... |
| bnj887 30835 | ` /\ ` -manipulation. (Co... |
| bnj918 30836 | First-order logic and set ... |
| bnj919 30837 | First-order logic and set ... |
| bnj923 30838 | First-order logic and set ... |
| bnj927 30839 | First-order logic and set ... |
| bnj930 30840 | First-order logic and set ... |
| bnj931 30841 | First-order logic and set ... |
| bnj937 30842 | First-order logic and set ... |
| bnj941 30843 | First-order logic and set ... |
| bnj945 30844 | Technical lemma for ~ bnj6... |
| bnj946 30845 | First-order logic and set ... |
| bnj951 30846 | ` /\ ` -manipulation. (Co... |
| bnj956 30847 | First-order logic and set ... |
| bnj976 30848 | First-order logic and set ... |
| bnj982 30849 | First-order logic and set ... |
| bnj1019 30850 | First-order logic and set ... |
| bnj1023 30851 | First-order logic and set ... |
| bnj1095 30852 | First-order logic and set ... |
| bnj1096 30853 | First-order logic and set ... |
| bnj1098 30854 | First-order logic and set ... |
| bnj1101 30855 | First-order logic and set ... |
| bnj1113 30856 | First-order logic and set ... |
| bnj1109 30857 | First-order logic and set ... |
| bnj1131 30858 | First-order logic and set ... |
| bnj1138 30859 | First-order logic and set ... |
| bnj1142 30860 | First-order logic and set ... |
| bnj1143 30861 | First-order logic and set ... |
| bnj1146 30862 | First-order logic and set ... |
| bnj1149 30863 | First-order logic and set ... |
| bnj1185 30864 | First-order logic and set ... |
| bnj1196 30865 | First-order logic and set ... |
| bnj1198 30866 | First-order logic and set ... |
| bnj1209 30867 | First-order logic and set ... |
| bnj1211 30868 | First-order logic and set ... |
| bnj1213 30869 | First-order logic and set ... |
| bnj1212 30870 | First-order logic and set ... |
| bnj1219 30871 | First-order logic and set ... |
| bnj1224 30872 | First-order logic and set ... |
| bnj1230 30873 | First-order logic and set ... |
| bnj1232 30874 | First-order logic and set ... |
| bnj1235 30875 | First-order logic and set ... |
| bnj1239 30876 | First-order logic and set ... |
| bnj1238 30877 | First-order logic and set ... |
| bnj1241 30878 | First-order logic and set ... |
| bnj1247 30879 | First-order logic and set ... |
| bnj1254 30880 | First-order logic and set ... |
| bnj1262 30881 | First-order logic and set ... |
| bnj1266 30882 | First-order logic and set ... |
| bnj1265 30883 | First-order logic and set ... |
| bnj1275 30884 | First-order logic and set ... |
| bnj1276 30885 | First-order logic and set ... |
| bnj1292 30886 | First-order logic and set ... |
| bnj1293 30887 | First-order logic and set ... |
| bnj1294 30888 | First-order logic and set ... |
| bnj1299 30889 | First-order logic and set ... |
| bnj1304 30890 | First-order logic and set ... |
| bnj1316 30891 | First-order logic and set ... |
| bnj1317 30892 | First-order logic and set ... |
| bnj1322 30893 | First-order logic and set ... |
| bnj1340 30894 | First-order logic and set ... |
| bnj1345 30895 | First-order logic and set ... |
| bnj1350 30896 | First-order logic and set ... |
| bnj1351 30897 | First-order logic and set ... |
| bnj1352 30898 | First-order logic and set ... |
| bnj1361 30899 | First-order logic and set ... |
| bnj1366 30900 | First-order logic and set ... |
| bnj1379 30901 | First-order logic and set ... |
| bnj1383 30902 | First-order logic and set ... |
| bnj1385 30903 | First-order logic and set ... |
| bnj1386 30904 | First-order logic and set ... |
| bnj1397 30905 | First-order logic and set ... |
| bnj1400 30906 | First-order logic and set ... |
| bnj1405 30907 | First-order logic and set ... |
| bnj1422 30908 | First-order logic and set ... |
| bnj1424 30909 | First-order logic and set ... |
| bnj1436 30910 | First-order logic and set ... |
| bnj1441 30911 | First-order logic and set ... |
| bnj1454 30912 | First-order logic and set ... |
| bnj1459 30913 | First-order logic and set ... |
| bnj1464 30914 | Conversion of implicit sub... |
| bnj1465 30915 | First-order logic and set ... |
| bnj1468 30916 | Conversion of implicit sub... |
| bnj1476 30917 | First-order logic and set ... |
| bnj1502 30918 | First-order logic and set ... |
| bnj1503 30919 | First-order logic and set ... |
| bnj1517 30920 | First-order logic and set ... |
| bnj1521 30921 | First-order logic and set ... |
| bnj1533 30922 | First-order logic and set ... |
| bnj1534 30923 | First-order logic and set ... |
| bnj1536 30924 | First-order logic and set ... |
| bnj1538 30925 | First-order logic and set ... |
| bnj1541 30926 | First-order logic and set ... |
| bnj1542 30927 | First-order logic and set ... |
| bnj110 30928 | Well-founded induction res... |
| bnj157 30929 | Well-founded induction res... |
| bnj66 30930 | Technical lemma for ~ bnj6... |
| bnj91 30931 | First-order logic and set ... |
| bnj92 30932 | First-order logic and set ... |
| bnj93 30933 | Technical lemma for ~ bnj9... |
| bnj95 30934 | Technical lemma for ~ bnj1... |
| bnj96 30935 | Technical lemma for ~ bnj1... |
| bnj97 30936 | Technical lemma for ~ bnj1... |
| bnj98 30937 | Technical lemma for ~ bnj1... |
| bnj106 30938 | First-order logic and set ... |
| bnj118 30939 | First-order logic and set ... |
| bnj121 30940 | First-order logic and set ... |
| bnj124 30941 | Technical lemma for ~ bnj1... |
| bnj125 30942 | Technical lemma for ~ bnj1... |
| bnj126 30943 | Technical lemma for ~ bnj1... |
| bnj130 30944 | Technical lemma for ~ bnj1... |
| bnj149 30945 | Technical lemma for ~ bnj1... |
| bnj150 30946 | Technical lemma for ~ bnj1... |
| bnj151 30947 | Technical lemma for ~ bnj1... |
| bnj154 30948 | Technical lemma for ~ bnj1... |
| bnj155 30949 | Technical lemma for ~ bnj1... |
| bnj153 30950 | Technical lemma for ~ bnj8... |
| bnj207 30951 | Technical lemma for ~ bnj8... |
| bnj213 30952 | First-order logic and set ... |
| bnj222 30953 | Technical lemma for ~ bnj2... |
| bnj229 30954 | Technical lemma for ~ bnj5... |
| bnj517 30955 | Technical lemma for ~ bnj5... |
| bnj518 30956 | Technical lemma for ~ bnj8... |
| bnj523 30957 | Technical lemma for ~ bnj8... |
| bnj526 30958 | Technical lemma for ~ bnj8... |
| bnj528 30959 | Technical lemma for ~ bnj8... |
| bnj535 30960 | Technical lemma for ~ bnj8... |
| bnj539 30961 | Technical lemma for ~ bnj8... |
| bnj540 30962 | Technical lemma for ~ bnj8... |
| bnj543 30963 | Technical lemma for ~ bnj8... |
| bnj544 30964 | Technical lemma for ~ bnj8... |
| bnj545 30965 | Technical lemma for ~ bnj8... |
| bnj546 30966 | Technical lemma for ~ bnj8... |
| bnj548 30967 | Technical lemma for ~ bnj8... |
| bnj553 30968 | Technical lemma for ~ bnj8... |
| bnj554 30969 | Technical lemma for ~ bnj8... |
| bnj556 30970 | Technical lemma for ~ bnj8... |
| bnj557 30971 | Technical lemma for ~ bnj8... |
| bnj558 30972 | Technical lemma for ~ bnj8... |
| bnj561 30973 | Technical lemma for ~ bnj8... |
| bnj562 30974 | Technical lemma for ~ bnj8... |
| bnj570 30975 | Technical lemma for ~ bnj8... |
| bnj571 30976 | Technical lemma for ~ bnj8... |
| bnj605 30977 | Technical lemma. This lem... |
| bnj581 30978 | Technical lemma for ~ bnj5... |
| bnj589 30979 | Technical lemma for ~ bnj8... |
| bnj590 30980 | Technical lemma for ~ bnj8... |
| bnj591 30981 | Technical lemma for ~ bnj8... |
| bnj594 30982 | Technical lemma for ~ bnj8... |
| bnj580 30983 | Technical lemma for ~ bnj5... |
| bnj579 30984 | Technical lemma for ~ bnj8... |
| bnj602 30985 | Equality theorem for the `... |
| bnj607 30986 | Technical lemma for ~ bnj8... |
| bnj609 30987 | Technical lemma for ~ bnj8... |
| bnj611 30988 | Technical lemma for ~ bnj8... |
| bnj600 30989 | Technical lemma for ~ bnj8... |
| bnj601 30990 | Technical lemma for ~ bnj8... |
| bnj852 30991 | Technical lemma for ~ bnj6... |
| bnj864 30992 | Technical lemma for ~ bnj6... |
| bnj865 30993 | Technical lemma for ~ bnj6... |
| bnj873 30994 | Technical lemma for ~ bnj6... |
| bnj849 30995 | Technical lemma for ~ bnj6... |
| bnj882 30996 | Definition (using hypothes... |
| bnj18eq1 30997 | Equality theorem for trans... |
| bnj893 30998 | Property of ` _trCl ` . U... |
| bnj900 30999 | Technical lemma for ~ bnj6... |
| bnj906 31000 | Property of ` _trCl ` . (... |
| bnj908 31001 | Technical lemma for ~ bnj6... |
| bnj911 31002 | Technical lemma for ~ bnj6... |
| bnj916 31003 | Technical lemma for ~ bnj6... |
| bnj917 31004 | Technical lemma for ~ bnj6... |
| bnj934 31005 | Technical lemma for ~ bnj6... |
| bnj929 31006 | Technical lemma for ~ bnj6... |
| bnj938 31007 | Technical lemma for ~ bnj6... |
| bnj944 31008 | Technical lemma for ~ bnj6... |
| bnj953 31009 | Technical lemma for ~ bnj6... |
| bnj958 31010 | Technical lemma for ~ bnj6... |
| bnj1000 31011 | Technical lemma for ~ bnj8... |
| bnj965 31012 | Technical lemma for ~ bnj8... |
| bnj964 31013 | Technical lemma for ~ bnj6... |
| bnj966 31014 | Technical lemma for ~ bnj6... |
| bnj967 31015 | Technical lemma for ~ bnj6... |
| bnj969 31016 | Technical lemma for ~ bnj6... |
| bnj970 31017 | Technical lemma for ~ bnj6... |
| bnj910 31018 | Technical lemma for ~ bnj6... |
| bnj978 31019 | Technical lemma for ~ bnj6... |
| bnj981 31020 | Technical lemma for ~ bnj6... |
| bnj983 31021 | Technical lemma for ~ bnj6... |
| bnj984 31022 | Technical lemma for ~ bnj6... |
| bnj985 31023 | Technical lemma for ~ bnj6... |
| bnj986 31024 | Technical lemma for ~ bnj6... |
| bnj996 31025 | Technical lemma for ~ bnj6... |
| bnj998 31026 | Technical lemma for ~ bnj6... |
| bnj999 31027 | Technical lemma for ~ bnj6... |
| bnj1001 31028 | Technical lemma for ~ bnj6... |
| bnj1006 31029 | Technical lemma for ~ bnj6... |
| bnj1014 31030 | Technical lemma for ~ bnj6... |
| bnj1015 31031 | Technical lemma for ~ bnj6... |
| bnj1018 31032 | Technical lemma for ~ bnj6... |
| bnj1020 31033 | Technical lemma for ~ bnj6... |
| bnj1021 31034 | Technical lemma for ~ bnj6... |
| bnj907 31035 | Technical lemma for ~ bnj6... |
| bnj1029 31036 | Property of ` _trCl ` . (... |
| bnj1033 31037 | Technical lemma for ~ bnj6... |
| bnj1034 31038 | Technical lemma for ~ bnj6... |
| bnj1039 31039 | Technical lemma for ~ bnj6... |
| bnj1040 31040 | Technical lemma for ~ bnj6... |
| bnj1047 31041 | Technical lemma for ~ bnj6... |
| bnj1049 31042 | Technical lemma for ~ bnj6... |
| bnj1052 31043 | Technical lemma for ~ bnj6... |
| bnj1053 31044 | Technical lemma for ~ bnj6... |
| bnj1071 31045 | Technical lemma for ~ bnj6... |
| bnj1083 31046 | Technical lemma for ~ bnj6... |
| bnj1090 31047 | Technical lemma for ~ bnj6... |
| bnj1093 31048 | Technical lemma for ~ bnj6... |
| bnj1097 31049 | Technical lemma for ~ bnj6... |
| bnj1110 31050 | Technical lemma for ~ bnj6... |
| bnj1112 31051 | Technical lemma for ~ bnj6... |
| bnj1118 31052 | Technical lemma for ~ bnj6... |
| bnj1121 31053 | Technical lemma for ~ bnj6... |
| bnj1123 31054 | Technical lemma for ~ bnj6... |
| bnj1030 31055 | Technical lemma for ~ bnj6... |
| bnj1124 31056 | Property of ` _trCl ` . (... |
| bnj1133 31057 | Technical lemma for ~ bnj6... |
| bnj1128 31058 | Technical lemma for ~ bnj6... |
| bnj1127 31059 | Property of ` _trCl ` . (... |
| bnj1125 31060 | Property of ` _trCl ` . (... |
| bnj1145 31061 | Technical lemma for ~ bnj6... |
| bnj1147 31062 | Property of ` _trCl ` . (... |
| bnj1137 31063 | Property of ` _trCl ` . (... |
| bnj1148 31064 | Property of ` _pred ` . (... |
| bnj1136 31065 | Technical lemma for ~ bnj6... |
| bnj1152 31066 | Technical lemma for ~ bnj6... |
| bnj1154 31067 | Property of ` Fr ` . (Con... |
| bnj1171 31068 | Technical lemma for ~ bnj6... |
| bnj1172 31069 | Technical lemma for ~ bnj6... |
| bnj1173 31070 | Technical lemma for ~ bnj6... |
| bnj1174 31071 | Technical lemma for ~ bnj6... |
| bnj1175 31072 | Technical lemma for ~ bnj6... |
| bnj1176 31073 | Technical lemma for ~ bnj6... |
| bnj1177 31074 | Technical lemma for ~ bnj6... |
| bnj1186 31075 | Technical lemma for ~ bnj6... |
| bnj1190 31076 | Technical lemma for ~ bnj6... |
| bnj1189 31077 | Technical lemma for ~ bnj6... |
| bnj69 31078 | Existence of a minimal ele... |
| bnj1228 31079 | Existence of a minimal ele... |
| bnj1204 31080 | Well-founded induction. T... |
| bnj1234 31081 | Technical lemma for ~ bnj6... |
| bnj1245 31082 | Technical lemma for ~ bnj6... |
| bnj1256 31083 | Technical lemma for ~ bnj6... |
| bnj1259 31084 | Technical lemma for ~ bnj6... |
| bnj1253 31085 | Technical lemma for ~ bnj6... |
| bnj1279 31086 | Technical lemma for ~ bnj6... |
| bnj1286 31087 | Technical lemma for ~ bnj6... |
| bnj1280 31088 | Technical lemma for ~ bnj6... |
| bnj1296 31089 | Technical lemma for ~ bnj6... |
| bnj1309 31090 | Technical lemma for ~ bnj6... |
| bnj1307 31091 | Technical lemma for ~ bnj6... |
| bnj1311 31092 | Technical lemma for ~ bnj6... |
| bnj1318 31093 | Technical lemma for ~ bnj6... |
| bnj1326 31094 | Technical lemma for ~ bnj6... |
| bnj1321 31095 | Technical lemma for ~ bnj6... |
| bnj1364 31096 | Property of ` _FrSe ` . (... |
| bnj1371 31097 | Technical lemma for ~ bnj6... |
| bnj1373 31098 | Technical lemma for ~ bnj6... |
| bnj1374 31099 | Technical lemma for ~ bnj6... |
| bnj1384 31100 | Technical lemma for ~ bnj6... |
| bnj1388 31101 | Technical lemma for ~ bnj6... |
| bnj1398 31102 | Technical lemma for ~ bnj6... |
| bnj1413 31103 | Property of ` _trCl ` . (... |
| bnj1408 31104 | Technical lemma for ~ bnj1... |
| bnj1414 31105 | Property of ` _trCl ` . (... |
| bnj1415 31106 | Technical lemma for ~ bnj6... |
| bnj1416 31107 | Technical lemma for ~ bnj6... |
| bnj1418 31108 | Property of ` _pred ` . (... |
| bnj1417 31109 | Technical lemma for ~ bnj6... |
| bnj1421 31110 | Technical lemma for ~ bnj6... |
| bnj1444 31111 | Technical lemma for ~ bnj6... |
| bnj1445 31112 | Technical lemma for ~ bnj6... |
| bnj1446 31113 | Technical lemma for ~ bnj6... |
| bnj1447 31114 | Technical lemma for ~ bnj6... |
| bnj1448 31115 | Technical lemma for ~ bnj6... |
| bnj1449 31116 | Technical lemma for ~ bnj6... |
| bnj1442 31117 | Technical lemma for ~ bnj6... |
| bnj1450 31118 | Technical lemma for ~ bnj6... |
| bnj1423 31119 | Technical lemma for ~ bnj6... |
| bnj1452 31120 | Technical lemma for ~ bnj6... |
| bnj1466 31121 | Technical lemma for ~ bnj6... |
| bnj1467 31122 | Technical lemma for ~ bnj6... |
| bnj1463 31123 | Technical lemma for ~ bnj6... |
| bnj1489 31124 | Technical lemma for ~ bnj6... |
| bnj1491 31125 | Technical lemma for ~ bnj6... |
| bnj1312 31126 | Technical lemma for ~ bnj6... |
| bnj1493 31127 | Technical lemma for ~ bnj6... |
| bnj1497 31128 | Technical lemma for ~ bnj6... |
| bnj1498 31129 | Technical lemma for ~ bnj6... |
| bnj60 31130 | Well-founded recursion, pa... |
| bnj1514 31131 | Technical lemma for ~ bnj1... |
| bnj1518 31132 | Technical lemma for ~ bnj1... |
| bnj1519 31133 | Technical lemma for ~ bnj1... |
| bnj1520 31134 | Technical lemma for ~ bnj1... |
| bnj1501 31135 | Technical lemma for ~ bnj1... |
| bnj1500 31136 | Well-founded recursion, pa... |
| bnj1525 31137 | Technical lemma for ~ bnj1... |
| bnj1529 31138 | Technical lemma for ~ bnj1... |
| bnj1523 31139 | Technical lemma for ~ bnj1... |
| bnj1522 31140 | Well-founded recursion, pa... |
| quartfull 31147 | The quartic equation, writ... |
| deranglem 31148 | Lemma for derangements. (... |
| derangval 31149 | Define the derangement fun... |
| derangf 31150 | The derangement number is ... |
| derang0 31151 | The derangement number of ... |
| derangsn 31152 | The derangement number of ... |
| derangenlem 31153 | One half of ~ derangen . ... |
| derangen 31154 | The derangement number is ... |
| subfacval 31155 | The subfactorial is define... |
| derangen2 31156 | Write the derangement numb... |
| subfacf 31157 | The subfactorial is a func... |
| subfaclefac 31158 | The subfactorial is less t... |
| subfac0 31159 | The subfactorial at zero. ... |
| subfac1 31160 | The subfactorial at one. ... |
| subfacp1lem1 31161 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem2a 31162 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem2b 31163 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem3 31164 | Lemma for ~ subfacp1 . In... |
| subfacp1lem4 31165 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem5 31166 | Lemma for ~ subfacp1 . In... |
| subfacp1lem6 31167 | Lemma for ~ subfacp1 . By... |
| subfacp1 31168 | A two-term recurrence for ... |
| subfacval2 31169 | A closed-form expression f... |
| subfaclim 31170 | The subfactorial converges... |
| subfacval3 31171 | Another closed form expres... |
| derangfmla 31172 | The derangements formula, ... |
| erdszelem1 31173 | Lemma for ~ erdsze . (Con... |
| erdszelem2 31174 | Lemma for ~ erdsze . (Con... |
| erdszelem3 31175 | Lemma for ~ erdsze . (Con... |
| erdszelem4 31176 | Lemma for ~ erdsze . (Con... |
| erdszelem5 31177 | Lemma for ~ erdsze . (Con... |
| erdszelem6 31178 | Lemma for ~ erdsze . (Con... |
| erdszelem7 31179 | Lemma for ~ erdsze . (Con... |
| erdszelem8 31180 | Lemma for ~ erdsze . (Con... |
| erdszelem9 31181 | Lemma for ~ erdsze . (Con... |
| erdszelem10 31182 | Lemma for ~ erdsze . (Con... |
| erdszelem11 31183 | Lemma for ~ erdsze . (Con... |
| erdsze 31184 | The Erdős-Szekeres th... |
| erdsze2lem1 31185 | Lemma for ~ erdsze2 . (Co... |
| erdsze2lem2 31186 | Lemma for ~ erdsze2 . (Co... |
| erdsze2 31187 | Generalize the statement o... |
| kur14lem1 31188 | Lemma for ~ kur14 . (Cont... |
| kur14lem2 31189 | Lemma for ~ kur14 . Write... |
| kur14lem3 31190 | Lemma for ~ kur14 . A clo... |
| kur14lem4 31191 | Lemma for ~ kur14 . Compl... |
| kur14lem5 31192 | Lemma for ~ kur14 . Closu... |
| kur14lem6 31193 | Lemma for ~ kur14 . If ` ... |
| kur14lem7 31194 | Lemma for ~ kur14 : main p... |
| kur14lem8 31195 | Lemma for ~ kur14 . Show ... |
| kur14lem9 31196 | Lemma for ~ kur14 . Since... |
| kur14lem10 31197 | Lemma for ~ kur14 . Disch... |
| kur14 31198 | Kuratowski's closure-compl... |
| ispconn 31205 | The property of being a pa... |
| pconncn 31206 | The property of being a pa... |
| pconntop 31207 | A simply connected space i... |
| issconn 31208 | The property of being a si... |
| sconnpconn 31209 | A simply connected space i... |
| sconntop 31210 | A simply connected space i... |
| sconnpht 31211 | A closed path in a simply ... |
| cnpconn 31212 | An image of a path-connect... |
| pconnconn 31213 | A path-connected space is ... |
| txpconn 31214 | The topological product of... |
| ptpconn 31215 | The topological product of... |
| indispconn 31216 | The indiscrete topology (o... |
| connpconn 31217 | A connected and locally pa... |
| qtoppconn 31218 | A quotient of a path-conne... |
| pconnpi1 31219 | All fundamental groups in ... |
| sconnpht2 31220 | Any two paths in a simply ... |
| sconnpi1 31221 | A path-connected topologic... |
| txsconnlem 31222 | Lemma for ~ txsconn . (Co... |
| txsconn 31223 | The topological product of... |
| cvxpconn 31224 | A convex subset of the com... |
| cvxsconn 31225 | A convex subset of the com... |
| blsconn 31226 | An open ball in the comple... |
| cnllysconn 31227 | The topology of the comple... |
| resconn 31228 | A subset of ` RR ` is simp... |
| ioosconn 31229 | An open interval is simply... |
| iccsconn 31230 | A closed interval is simpl... |
| retopsconn 31231 | The real numbers are simpl... |
| iccllysconn 31232 | A closed interval is local... |
| rellysconn 31233 | The real numbers are local... |
| iisconn 31234 | The unit interval is simpl... |
| iillysconn 31235 | The unit interval is local... |
| iinllyconn 31236 | The unit interval is local... |
| fncvm 31239 | Lemma for covering maps. ... |
| cvmscbv 31240 | Change bound variables in ... |
| iscvm 31241 | The property of being a co... |
| cvmtop1 31242 | Reverse closure for a cove... |
| cvmtop2 31243 | Reverse closure for a cove... |
| cvmcn 31244 | A covering map is a contin... |
| cvmcov 31245 | Property of a covering map... |
| cvmsrcl 31246 | Reverse closure for an eve... |
| cvmsi 31247 | One direction of ~ cvmsval... |
| cvmsval 31248 | Elementhood in the set ` S... |
| cvmsss 31249 | An even covering is a subs... |
| cvmsn0 31250 | An even covering is nonemp... |
| cvmsuni 31251 | An even covering of ` U ` ... |
| cvmsdisj 31252 | An even covering of ` U ` ... |
| cvmshmeo 31253 | Every element of an even c... |
| cvmsf1o 31254 | ` F ` , localized to an el... |
| cvmscld 31255 | The sets of an even coveri... |
| cvmsss2 31256 | An open subset of an evenl... |
| cvmcov2 31257 | The covering map property ... |
| cvmseu 31258 | Every element in ` U. T ` ... |
| cvmsiota 31259 | Identify the unique elemen... |
| cvmopnlem 31260 | Lemma for ~ cvmopn . (Con... |
| cvmfolem 31261 | Lemma for ~ cvmfo . (Cont... |
| cvmopn 31262 | A covering map is an open ... |
| cvmliftmolem1 31263 | Lemma for ~ cvmliftmo . (... |
| cvmliftmolem2 31264 | Lemma for ~ cvmliftmo . (... |
| cvmliftmoi 31265 | A lift of a continuous fun... |
| cvmliftmo 31266 | A lift of a continuous fun... |
| cvmliftlem1 31267 | Lemma for ~ cvmlift . In ... |
| cvmliftlem2 31268 | Lemma for ~ cvmlift . ` W ... |
| cvmliftlem3 31269 | Lemma for ~ cvmlift . Sin... |
| cvmliftlem4 31270 | Lemma for ~ cvmlift . The... |
| cvmliftlem5 31271 | Lemma for ~ cvmlift . Def... |
| cvmliftlem6 31272 | Lemma for ~ cvmlift . Ind... |
| cvmliftlem7 31273 | Lemma for ~ cvmlift . Pro... |
| cvmliftlem8 31274 | Lemma for ~ cvmlift . The... |
| cvmliftlem9 31275 | Lemma for ~ cvmlift . The... |
| cvmliftlem10 31276 | Lemma for ~ cvmlift . The... |
| cvmliftlem11 31277 | Lemma for ~ cvmlift . (Co... |
| cvmliftlem13 31278 | Lemma for ~ cvmlift . The... |
| cvmliftlem14 31279 | Lemma for ~ cvmlift . Put... |
| cvmliftlem15 31280 | Lemma for ~ cvmlift . Dis... |
| cvmlift 31281 | One of the important prope... |
| cvmfo 31282 | A covering map is an onto ... |
| cvmliftiota 31283 | Write out a function ` H `... |
| cvmlift2lem1 31284 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9a 31285 | Lemma for ~ cvmlift2 and ~... |
| cvmlift2lem2 31286 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem3 31287 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem4 31288 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem5 31289 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem6 31290 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem7 31291 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem8 31292 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9 31293 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem10 31294 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem11 31295 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem12 31296 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem13 31297 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2 31298 | A two-dimensional version ... |
| cvmliftphtlem 31299 | Lemma for ~ cvmliftpht . ... |
| cvmliftpht 31300 | If ` G ` and ` H ` are pat... |
| cvmlift3lem1 31301 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem2 31302 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem3 31303 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem4 31304 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem5 31305 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem6 31306 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem7 31307 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem8 31308 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem9 31309 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3 31310 | A general version of ~ cvm... |
| snmlff 31311 | The function ` F ` from ~ ... |
| snmlfval 31312 | The function ` F ` from ~ ... |
| snmlval 31313 | The property " ` A ` is si... |
| snmlflim 31314 | If ` A ` is simply normal,... |
| mvtval 31397 | The set of variable typeco... |
| mrexval 31398 | The set of "raw expression... |
| mexval 31399 | The set of expressions, wh... |
| mexval2 31400 | The set of expressions, wh... |
| mdvval 31401 | The set of disjoint variab... |
| mvrsval 31402 | The set of variables in an... |
| mvrsfpw 31403 | The set of variables in an... |
| mrsubffval 31404 | The substitution of some v... |
| mrsubfval 31405 | The substitution of some v... |
| mrsubval 31406 | The substitution of some v... |
| mrsubcv 31407 | The value of a substituted... |
| mrsubvr 31408 | The value of a substituted... |
| mrsubff 31409 | A substitution is a functi... |
| mrsubrn 31410 | Although it is defined for... |
| mrsubff1 31411 | When restricted to complet... |
| mrsubff1o 31412 | When restricted to complet... |
| mrsub0 31413 | The value of the substitut... |
| mrsubf 31414 | A substitution is a functi... |
| mrsubccat 31415 | Substitution distributes o... |
| mrsubcn 31416 | A substitution does not ch... |
| elmrsubrn 31417 | Characterization of the su... |
| mrsubco 31418 | The composition of two sub... |
| mrsubvrs 31419 | The set of variables in a ... |
| msubffval 31420 | A substitution applied to ... |
| msubfval 31421 | A substitution applied to ... |
| msubval 31422 | A substitution applied to ... |
| msubrsub 31423 | A substitution applied to ... |
| msubty 31424 | The type of a substituted ... |
| elmsubrn 31425 | Characterization of substi... |
| msubrn 31426 | Although it is defined for... |
| msubff 31427 | A substitution is a functi... |
| msubco 31428 | The composition of two sub... |
| msubf 31429 | A substitution is a functi... |
| mvhfval 31430 | Value of the function mapp... |
| mvhval 31431 | Value of the function mapp... |
| mpstval 31432 | A pre-statement is an orde... |
| elmpst 31433 | Property of being a pre-st... |
| msrfval 31434 | Value of the reduct of a p... |
| msrval 31435 | Value of the reduct of a p... |
| mpstssv 31436 | A pre-statement is an orde... |
| mpst123 31437 | Decompose a pre-statement ... |
| mpstrcl 31438 | The elements of a pre-stat... |
| msrf 31439 | The reduct of a pre-statem... |
| msrrcl 31440 | If ` X ` and ` Y ` have th... |
| mstaval 31441 | Value of the set of statem... |
| msrid 31442 | The reduct of a statement ... |
| msrfo 31443 | The reduct of a pre-statem... |
| mstapst 31444 | A statement is a pre-state... |
| elmsta 31445 | Property of being a statem... |
| ismfs 31446 | A formal system is a tuple... |
| mfsdisj 31447 | The constants and variable... |
| mtyf2 31448 | The type function maps var... |
| mtyf 31449 | The type function maps var... |
| mvtss 31450 | The set of variable typeco... |
| maxsta 31451 | An axiom is a statement. ... |
| mvtinf 31452 | Each variable typecode has... |
| msubff1 31453 | When restricted to complet... |
| msubff1o 31454 | When restricted to complet... |
| mvhf 31455 | The function mapping varia... |
| mvhf1 31456 | The function mapping varia... |
| msubvrs 31457 | The set of variables in a ... |
| mclsrcl 31458 | Reverse closure for the cl... |
| mclsssvlem 31459 | Lemma for ~ mclsssv . (Co... |
| mclsval 31460 | The function mapping varia... |
| mclsssv 31461 | The closure of a set of ex... |
| ssmclslem 31462 | Lemma for ~ ssmcls . (Con... |
| vhmcls 31463 | All variable hypotheses ar... |
| ssmcls 31464 | The original expressions a... |
| ss2mcls 31465 | The closure is monotonic u... |
| mclsax 31466 | The closure is closed unde... |
| mclsind 31467 | Induction theorem for clos... |
| mppspstlem 31468 | Lemma for ~ mppspst . (Co... |
| mppsval 31469 | Definition of a provable p... |
| elmpps 31470 | Definition of a provable p... |
| mppspst 31471 | A provable pre-statement i... |
| mthmval 31472 | A theorem is a pre-stateme... |
| elmthm 31473 | A theorem is a pre-stateme... |
| mthmi 31474 | A statement whose reduct i... |
| mthmsta 31475 | A theorem is a pre-stateme... |
| mppsthm 31476 | A provable pre-statement i... |
| mthmblem 31477 | Lemma for ~ mthmb . (Cont... |
| mthmb 31478 | If two statements have the... |
| mthmpps 31479 | Given a theorem, there is ... |
| mclsppslem 31480 | The closure is closed unde... |
| mclspps 31481 | The closure is closed unde... |
| problem1 31558 | Practice problem 1. Clues... |
| problem2 31559 | Practice problem 2. Clues... |
| problem2OLD 31560 | Practice problem 2. Clues... |
| problem3 31561 | Practice problem 3. Clues... |
| problem4 31562 | Practice problem 4. Clues... |
| problem5 31563 | Practice problem 5. Clues... |
| quad3 31564 | Variant of quadratic equat... |
| climuzcnv 31565 | Utility lemma to convert b... |
| sinccvglem 31566 | ` ( ( sin `` x ) / x ) ~~>... |
| sinccvg 31567 | ` ( ( sin `` x ) / x ) ~~>... |
| circum 31568 | The circumference of a cir... |
| elfzm12 31569 | Membership in a curtailed ... |
| nn0seqcvg 31570 | A strictly-decreasing nonn... |
| lediv2aALT 31571 | Division of both sides of ... |
| abs2sqlei 31572 | The absolute values of two... |
| abs2sqlti 31573 | The absolute values of two... |
| abs2sqle 31574 | The absolute values of two... |
| abs2sqlt 31575 | The absolute values of two... |
| abs2difi 31576 | Difference of absolute val... |
| abs2difabsi 31577 | Absolute value of differen... |
| axextprim 31578 | ~ ax-ext without distinct ... |
| axrepprim 31579 | ~ ax-rep without distinct ... |
| axunprim 31580 | ~ ax-un without distinct v... |
| axpowprim 31581 | ~ ax-pow without distinct ... |
| axregprim 31582 | ~ ax-reg without distinct ... |
| axinfprim 31583 | ~ ax-inf without distinct ... |
| axacprim 31584 | ~ ax-ac without distinct v... |
| untelirr 31585 | We call a class "untanged"... |
| untuni 31586 | The union of a class is un... |
| untsucf 31587 | If a class is untangled, t... |
| unt0 31588 | The null set is untangled.... |
| untint 31589 | If there is an untangled e... |
| efrunt 31590 | If ` A ` is well-founded b... |
| untangtr 31591 | A transitive class is unta... |
| 3orel2 31592 | Partial elimination of a t... |
| 3orel3 31593 | Partial elimination of a t... |
| 3pm3.2ni 31594 | Triple negated disjunction... |
| 3jaodd 31595 | Double deduction form of ~... |
| 3orit 31596 | Closed form of ~ 3ori . (... |
| biimpexp 31597 | A biconditional in the ant... |
| 3orel13 31598 | Elimination of two disjunc... |
| nepss 31599 | Two classes are inequal if... |
| 3ccased 31600 | Triple disjunction form of... |
| dfso3 31601 | Expansion of the definitio... |
| brtpid1 31602 | A binary relation involvin... |
| brtpid2 31603 | A binary relation involvin... |
| brtpid3 31604 | A binary relation involvin... |
| ceqsrexv2 31605 | Alternate elimitation of a... |
| iota5f 31606 | A method for computing iot... |
| ceqsralv2 31607 | Alternate elimination of a... |
| dford5 31608 | A class is ordinal iff it ... |
| jath 31609 | Closed form of ~ ja . Pro... |
| sqdivzi 31610 | Distribution of square ove... |
| subdivcomb1 31611 | Bring a term in a subtract... |
| subdivcomb2 31612 | Bring a term in a subtract... |
| supfz 31613 | The supremum of a finite s... |
| inffz 31614 | The infimum of a finite se... |
| inffzOLD 31615 | The infimum of a finite se... |
| fz0n 31616 | The sequence ` ( 0 ... ( N... |
| shftvalg 31617 | Value of a sequence shifte... |
| divcnvlin 31618 | Limit of the ratio of two ... |
| climlec3 31619 | Comparison of a constant t... |
| logi 31620 | Calculate the logarithm of... |
| iexpire 31621 | ` _i ` raised to itself is... |
| bcneg1 31622 | The binomial coefficent ov... |
| bcm1nt 31623 | The proportion of one bion... |
| bcprod 31624 | A product identity for bin... |
| bccolsum 31625 | A column-sum rule for bino... |
| iprodefisumlem 31626 | Lemma for ~ iprodefisum . ... |
| iprodefisum 31627 | Applying the exponential f... |
| iprodgam 31628 | An infinite product versio... |
| faclimlem1 31629 | Lemma for ~ faclim . Clos... |
| faclimlem2 31630 | Lemma for ~ faclim . Show... |
| faclimlem3 31631 | Lemma for ~ faclim . Alge... |
| faclim 31632 | An infinite product expres... |
| iprodfac 31633 | An infinite product expres... |
| faclim2 31634 | Another factorial limit du... |
| pdivsq 31635 | Condition for a prime divi... |
| dvdspw 31636 | Exponentiation law for div... |
| gcd32 31637 | Swap the second and third ... |
| gcdabsorb 31638 | Absorption law for gcd. (... |
| brtp 31639 | A condition for a binary r... |
| dftr6 31640 | A potential definition of ... |
| coep 31641 | Composition with epsilon. ... |
| coepr 31642 | Composition with the conve... |
| dffr5 31643 | A quantifier free definiti... |
| dfso2 31644 | Quantifier free definition... |
| dfpo2 31645 | Quantifier free definition... |
| br8 31646 | Substitution for an eight-... |
| br6 31647 | Substitution for a six-pla... |
| br4 31648 | Substitution for a four-pl... |
| cnvco1 31649 | Another distributive law o... |
| cnvco2 31650 | Another distributive law o... |
| eldm3 31651 | Quantifier-free definition... |
| elrn3 31652 | Quantifier-free definition... |
| pocnv 31653 | The converse of a partial ... |
| socnv 31654 | The converse of a strict o... |
| sotrd 31655 | Transitivity law for stric... |
| sotr3 31656 | Transitivity law for stric... |
| soasym 31657 | Asymmetry law for strict o... |
| sotrine 31658 | Trichotomy law for strict ... |
| eqfunresadj 31659 | Law for adjoining an eleme... |
| eqfunressuc 31660 | Law for equality of restri... |
| funeldmb 31661 | If ` (/) ` is not part of ... |
| elintfv 31662 | Membership in an intersect... |
| funpsstri 31663 | A condition for subset tri... |
| fundmpss 31664 | If a class ` F ` is a prop... |
| fvresval 31665 | The value of a function at... |
| funsseq 31666 | Given two functions with e... |
| fununiq 31667 | The uniqueness condition o... |
| funbreq 31668 | An equality condition for ... |
| fprb 31669 | A condition for functionho... |
| br1steq 31670 | Uniqueness condition for t... |
| br2ndeq 31671 | Uniqueness condition for t... |
| br1steqg 31672 | Uniqueness condition for t... |
| br2ndeqg 31673 | Uniqueness condition for t... |
| br1steqgOLD 31674 | Obsolete version of ~ br1s... |
| br2ndeqgOLD 31675 | Obsolete version of ~ br2n... |
| dfdm5 31676 | Definition of domain in te... |
| dfrn5 31677 | Definition of range in ter... |
| opelco3 31678 | Alternate way of saying th... |
| elima4 31679 | Quantifier-free expression... |
| fv1stcnv 31680 | The value of the converse ... |
| fv2ndcnv 31681 | The value of the converse ... |
| imaindm 31682 | The image is unaffected by... |
| setinds 31683 | Principle of ` _E ` induct... |
| setinds2f 31684 | ` _E ` induction schema, u... |
| setinds2 31685 | ` _E ` induction schema, u... |
| elpotr 31686 | A class of transitive sets... |
| dford5reg 31687 | Given ~ ax-reg , an ordina... |
| dfon2lem1 31688 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem2 31689 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem3 31690 | Lemma for ~ dfon2 . All s... |
| dfon2lem4 31691 | Lemma for ~ dfon2 . If tw... |
| dfon2lem5 31692 | Lemma for ~ dfon2 . Two s... |
| dfon2lem6 31693 | Lemma for ~ dfon2 . A tra... |
| dfon2lem7 31694 | Lemma for ~ dfon2 . All e... |
| dfon2lem8 31695 | Lemma for ~ dfon2 . The i... |
| dfon2lem9 31696 | Lemma for ~ dfon2 . A cla... |
| dfon2 31697 | ` On ` consists of all set... |
| domep 31698 | The domain of the epsilon ... |
| rdgprc0 31699 | The value of the recursive... |
| rdgprc 31700 | The value of the recursive... |
| dfrdg2 31701 | Alternate definition of th... |
| dfrdg3 31702 | Generalization of ~ dfrdg2... |
| axextdfeq 31703 | A version of ~ ax-ext for ... |
| ax8dfeq 31704 | A version of ~ ax-8 for us... |
| axextdist 31705 | ~ ax-ext with distinctors ... |
| axext4dist 31706 | ~ axext4 with distinctors ... |
| 19.12b 31707 | Version of ~ 19.12vv with ... |
| exnel 31708 | There is always a set not ... |
| distel 31709 | Distinctors in terms of me... |
| axextndbi 31710 | ~ axextnd as a bicondition... |
| hbntg 31711 | A more general form of ~ h... |
| hbimtg 31712 | A more general and closed ... |
| hbaltg 31713 | A more general and closed ... |
| hbng 31714 | A more general form of ~ h... |
| hbimg 31715 | A more general form of ~ h... |
| tfisg 31716 | A closed form of ~ tfis . ... |
| dftrpred2 31719 | A definition of the transi... |
| trpredeq1 31720 | Equality theorem for trans... |
| trpredeq2 31721 | Equality theorem for trans... |
| trpredeq3 31722 | Equality theorem for trans... |
| trpredeq1d 31723 | Equality deduction for tra... |
| trpredeq2d 31724 | Equality deduction for tra... |
| trpredeq3d 31725 | Equality deduction for tra... |
| eltrpred 31726 | A class is a transitive pr... |
| trpredlem1 31727 | Technical lemma for transi... |
| trpredpred 31728 | Assuming it exists, the pr... |
| trpredss 31729 | The transitive predecessor... |
| trpredtr 31730 | The transitive predecessor... |
| trpredmintr 31731 | The transitive predecessor... |
| trpredelss 31732 | Given a transitive predece... |
| dftrpred3g 31733 | The transitive predecessor... |
| dftrpred4g 31734 | Another recursive expressi... |
| trpredpo 31735 | If ` R ` partially orders ... |
| trpred0 31736 | The class of transitive pr... |
| trpredex 31737 | The transitive predecessor... |
| trpredrec 31738 | If ` Y ` is an ` R ` , ` A... |
| frmin 31739 | Every (possibly proper) su... |
| frind 31740 | The principle of founded i... |
| frindi 31741 | The principle of founded i... |
| frinsg 31742 | Founded Induction Schema. ... |
| frins 31743 | Founded Induction Schema. ... |
| frins2fg 31744 | Founded Induction schema, ... |
| frins2f 31745 | Founded Induction schema, ... |
| frins2g 31746 | Founded Induction schema, ... |
| frins2 31747 | Founded Induction schema, ... |
| frins3 31748 | Founded Induction schema, ... |
| orderseqlem 31749 | Lemma for ~ poseq and ~ so... |
| poseq 31750 | A partial ordering of sequ... |
| soseq 31751 | A linear ordering of seque... |
| wsuceq123 31760 | Equality theorem for well-... |
| wsuceq1 31761 | Equality theorem for well-... |
| wsuceq2 31762 | Equality theorem for well-... |
| wsuceq3 31763 | Equality theorem for well-... |
| nfwsuc 31764 | Bound-variable hypothesis ... |
| wlimeq12 31765 | Equality theorem for the l... |
| wlimeq1 31766 | Equality theorem for the l... |
| wlimeq2 31767 | Equality theorem for the l... |
| nfwlim 31768 | Bound-variable hypothesis ... |
| elwlim 31769 | Membership in the limit cl... |
| elwlimOLD 31770 | Membership in the limit cl... |
| wzel 31771 | The zero of a well-founded... |
| wzelOLD 31772 | The zero of a well-founded... |
| wsuclem 31773 | Lemma for the supremum pro... |
| wsuclemOLD 31774 | Obsolete version of ~ wsuc... |
| wsucex 31775 | Existence theorem for well... |
| wsuccl 31776 | If ` X ` is a set with an ... |
| wsuclb 31777 | A well-founded successor i... |
| wlimss 31778 | The class of limit points ... |
| frr3g 31779 | Functions defined by found... |
| frrlem1 31780 | Lemma for founded recursio... |
| frrlem2 31781 | Lemma for founded recursio... |
| frrlem3 31782 | Lemma for founded recursio... |
| frrlem4 31783 | Lemma for founded recursio... |
| frrlem5 31784 | Lemma for founded recursio... |
| frrlem5b 31785 | Lemma for founded recursio... |
| frrlem5c 31786 | Lemma for founded recursio... |
| frrlem5d 31787 | Lemma for founded recursio... |
| frrlem5e 31788 | Lemma for founded recursio... |
| frrlem6 31789 | Lemma for founded recursio... |
| frrlem7 31790 | Lemma for founded recursio... |
| frrlem10 31791 | Lemma for founded recursio... |
| frrlem11 31792 | Lemma for founded recursio... |
| elno 31799 | Membership in the surreals... |
| sltval 31800 | The value of the surreal l... |
| bdayval 31801 | The value of the birthday ... |
| nofun 31802 | A surreal is a function. ... |
| nodmon 31803 | The domain of a surreal is... |
| norn 31804 | The range of a surreal is ... |
| nofnbday 31805 | A surreal is a function ov... |
| nodmord 31806 | The domain of a surreal ha... |
| elno2 31807 | An alternative condition f... |
| elno3 31808 | Another condition for memb... |
| sltval2 31809 | Alternate expression for s... |
| nofv 31810 | The function value of a su... |
| nosgnn0 31811 | ` (/) ` is not a surreal s... |
| nosgnn0i 31812 | If ` X ` is a surreal sign... |
| noreson 31813 | The restriction of a surre... |
| sltintdifex 31814 |
If ` A |
| sltres 31815 | If the restrictions of two... |
| noxp1o 31816 | The Cartesian product of a... |
| noseponlem 31817 | Lemma for ~ nosepon . Con... |
| nosepon 31818 | Given two unequal surreals... |
| noextend 31819 | Extending a surreal by one... |
| noextendseq 31820 | Extend a surreal by a sequ... |
| noextenddif 31821 | Calculate the place where ... |
| noextendlt 31822 | Extending a surreal with a... |
| noextendgt 31823 | Extending a surreal with a... |
| nolesgn2o 31824 | Given ` A ` less than or e... |
| nolesgn2ores 31825 | Given ` A ` less than or e... |
| sltsolem1 31826 | Lemma for ~ sltso . The s... |
| sltso 31827 | Surreal less than totally ... |
| bdayfo 31828 | The birthday function maps... |
| fvnobday 31829 | The value of a surreal at ... |
| nosepnelem 31830 | Lemma for ~ nosepne . (Co... |
| nosepne 31831 | The value of two non-equal... |
| nosep1o 31832 | If the value of a surreal ... |
| nosepdmlem 31833 | Lemma for ~ nosepdm . (Co... |
| nosepdm 31834 | The first place two surrea... |
| nosepeq 31835 | The values of two surreals... |
| nosepssdm 31836 | Given two non-equal surrea... |
| nodenselem4 31837 | Lemma for ~ nodense . Sho... |
| nodenselem5 31838 | Lemma for ~ nodense . If ... |
| nodenselem6 31839 | The restriction of a surre... |
| nodenselem7 31840 | Lemma for ~ nodense . ` A ... |
| nodenselem8 31841 | Lemma for ~ nodense . Giv... |
| nodense 31842 | Given two distinct surreal... |
| bdayimaon 31843 | Lemma for full-eta propert... |
| nolt02olem 31844 | Lemma for ~ nolt02o . If ... |
| nolt02o 31845 | Given ` A ` less than ` B ... |
| noresle 31846 | Restriction law for surrea... |
| nomaxmo 31847 | A class of surreals has at... |
| noprefixmo 31848 | In any class of surreals, ... |
| nosupno 31849 | The next several theorems ... |
| nosupdm 31850 | The domain of the surreal ... |
| nosupbday 31851 | Birthday bounding law for ... |
| nosupfv 31852 | The value of surreal supre... |
| nosupres 31853 | A restriction law for surr... |
| nosupbnd1lem1 31854 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1lem2 31855 | Lemma for ~ nosupbnd1 . W... |
| nosupbnd1lem3 31856 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem4 31857 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem5 31858 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem6 31859 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1 31860 | Bounding law from below fo... |
| nosupbnd2lem1 31861 | Bounding law from above wh... |
| nosupbnd2 31862 | Bounding law from above fo... |
| noetalem1 31863 | Lemma for ~ noeta . Estab... |
| noetalem2 31864 | Lemma for ~ noeta . ` Z ` ... |
| noetalem3 31865 | Lemma for ~ noeta . When ... |
| noetalem4 31866 | Lemma for ~ noeta . Bound... |
| noetalem5 31867 | Lemma for ~ noeta . The f... |
| noeta 31868 | The full-eta axiom for the... |
| sltirr 31871 | Surreal less than is irref... |
| slttr 31872 | Surreal less than is trans... |
| sltasym 31873 | Surreal less than is asymm... |
| sltlin 31874 | Surreal less than obeys tr... |
| slttrieq2 31875 | Trichotomy law for surreal... |
| slttrine 31876 | Trichotomy law for surreal... |
| slenlt 31877 | Surreal less than or equal... |
| sltnle 31878 | Surreal less than in terms... |
| sleloe 31879 | Surreal less than or equal... |
| sletri3 31880 | Trichotomy law for surreal... |
| sltletr 31881 | Surreal transitive law. (... |
| slelttr 31882 | Surreal transitive law. (... |
| sletr 31883 | Surreal transitive law. (... |
| slttrd 31884 | Surreal less than is trans... |
| sltletrd 31885 | Surreal less than is trans... |
| slelttrd 31886 | Surreal less than is trans... |
| sletrd 31887 | Surreal less than or equal... |
| bdayfun 31888 | The birthday function is a... |
| bdayfn 31889 | The birthday function is a... |
| bdaydm 31890 | The birthday function's do... |
| bdayrn 31891 | The birthday function's ra... |
| bdayelon 31892 | The value of the birthday ... |
| nocvxminlem 31893 | Lemma for ~ nocvxmin . Gi... |
| nocvxmin 31894 | Given a nonempty convex cl... |
| noprc 31895 | The surreal numbers are a ... |
| brsslt 31900 | Binary relation form of th... |
| ssltex1 31901 | The first argument of surr... |
| ssltex2 31902 | The second argument of sur... |
| ssltss1 31903 | The first argument of surr... |
| ssltss2 31904 | The second argument of sur... |
| ssltsep 31905 | The separation property of... |
| sssslt1 31906 | Relationship between surre... |
| sssslt2 31907 | Relationship between surre... |
| nulsslt 31908 | The empty set is less than... |
| nulssgt 31909 | The empty set is greater t... |
| conway 31910 | Conway's Simplicity Theore... |
| scutval 31911 | The value of the surreal c... |
| scutcut 31912 | Cut properties of the surr... |
| scutbday 31913 | The birthday of the surrea... |
| sslttr 31914 | Transitive law for surreal... |
| ssltun1 31915 | Union law for surreal set ... |
| ssltun2 31916 | Union law for surreal set ... |
| scutun12 31917 | Union law for surreal cuts... |
| dmscut 31918 | The domain of the surreal ... |
| scutf 31919 | Functionhood statement for... |
| etasslt 31920 | A restatement of ~ noeta u... |
| scutbdaybnd 31921 | An upper bound on the birt... |
| scutbdaylt 31922 | If a surreal lies in a gap... |
| slerec 31923 | A comparison law for surre... |
| sltrec 31924 | A comparison law for surre... |
| madeval 31935 | The value of the made by f... |
| madeval2 31936 | Alternative characterizati... |
| txpss3v 31985 | A tail Cartesian product i... |
| txprel 31986 | A tail Cartesian product i... |
| brtxp 31987 | Characterize a ternary rel... |
| brtxp2 31988 | The binary relation over a... |
| dfpprod2 31989 | Expanded definition of par... |
| pprodcnveq 31990 | A converse law for paralle... |
| pprodss4v 31991 | The parallel product is a ... |
| brpprod 31992 | Characterize a quaternary ... |
| brpprod3a 31993 | Condition for parallel pro... |
| brpprod3b 31994 | Condition for parallel pro... |
| relsset 31995 | The subset class is a bina... |
| brsset 31996 | For sets, the ` SSet ` bin... |
| idsset 31997 | ` _I ` is equal to the int... |
| eltrans 31998 | Membership in the class of... |
| dfon3 31999 | A quantifier-free definiti... |
| dfon4 32000 | Another quantifier-free de... |
| brtxpsd 32001 | Expansion of a common form... |
| brtxpsd2 32002 | Another common abbreviatio... |
| brtxpsd3 32003 | A third common abbreviatio... |
| relbigcup 32004 | The ` Bigcup ` relationshi... |
| brbigcup 32005 | Binary relation over ` Big... |
| dfbigcup2 32006 | ` Bigcup ` using maps-to n... |
| fobigcup 32007 | ` Bigcup ` maps the univer... |
| fnbigcup 32008 | ` Bigcup ` is a function o... |
| fvbigcup 32009 | For sets, ` Bigcup ` yield... |
| elfix 32010 | Membership in the fixpoint... |
| elfix2 32011 | Alternative membership in ... |
| dffix2 32012 | The fixpoints of a class i... |
| fixssdm 32013 | The fixpoints of a class a... |
| fixssrn 32014 | The fixpoints of a class a... |
| fixcnv 32015 | The fixpoints of a class a... |
| fixun 32016 | The fixpoint operator dist... |
| ellimits 32017 | Membership in the class of... |
| limitssson 32018 | The class of all limit ord... |
| dfom5b 32019 | A quantifier-free definiti... |
| sscoid 32020 | A condition for subset and... |
| dffun10 32021 | Another potential definiti... |
| elfuns 32022 | Membership in the class of... |
| elfunsg 32023 | Closed form of ~ elfuns . ... |
| brsingle 32024 | The binary relation form o... |
| elsingles 32025 | Membership in the class of... |
| fnsingle 32026 | The singleton relationship... |
| fvsingle 32027 | The value of the singleton... |
| dfsingles2 32028 | Alternate definition of th... |
| snelsingles 32029 | A singleton is a member of... |
| dfiota3 32030 | A definiton of iota using ... |
| dffv5 32031 | Another quantifier free de... |
| unisnif 32032 | Express union of singleton... |
| brimage 32033 | Binary relation form of th... |
| brimageg 32034 | Closed form of ~ brimage .... |
| funimage 32035 | ` Image A ` is a function.... |
| fnimage 32036 | ` Image R ` is a function ... |
| imageval 32037 | The image functor in maps-... |
| fvimage 32038 | Value of the image functor... |
| brcart 32039 | Binary relation form of th... |
| brdomain 32040 | Binary relation form of th... |
| brrange 32041 | Binary relation form of th... |
| brdomaing 32042 | Closed form of ~ brdomain ... |
| brrangeg 32043 | Closed form of ~ brrange .... |
| brimg 32044 | Binary relation form of th... |
| brapply 32045 | Binary relation form of th... |
| brcup 32046 | Binary relation form of th... |
| brcap 32047 | Binary relation form of th... |
| brsuccf 32048 | Binary relation form of th... |
| funpartlem 32049 | Lemma for ~ funpartfun . ... |
| funpartfun 32050 | The functional part of ` F... |
| funpartss 32051 | The functional part of ` F... |
| funpartfv 32052 | The function value of the ... |
| fullfunfnv 32053 | The full functional part o... |
| fullfunfv 32054 | The function value of the ... |
| brfullfun 32055 | A binary relation form con... |
| brrestrict 32056 | Binary relation form of th... |
| dfrecs2 32057 | A quantifier-free definiti... |
| dfrdg4 32058 | A quantifier-free definiti... |
| dfint3 32059 | Quantifier-free definition... |
| imagesset 32060 | The Image functor applied ... |
| brub 32061 | Binary relation form of th... |
| brlb 32062 | Binary relation form of th... |
| altopex 32067 | Alternative ordered pairs ... |
| altopthsn 32068 | Two alternate ordered pair... |
| altopeq12 32069 | Equality for alternate ord... |
| altopeq1 32070 | Equality for alternate ord... |
| altopeq2 32071 | Equality for alternate ord... |
| altopth1 32072 | Equality of the first memb... |
| altopth2 32073 | Equality of the second mem... |
| altopthg 32074 | Alternate ordered pair the... |
| altopthbg 32075 | Alternate ordered pair the... |
| altopth 32076 | The alternate ordered pair... |
| altopthb 32077 | Alternate ordered pair the... |
| altopthc 32078 | Alternate ordered pair the... |
| altopthd 32079 | Alternate ordered pair the... |
| altxpeq1 32080 | Equality for alternate Car... |
| altxpeq2 32081 | Equality for alternate Car... |
| elaltxp 32082 | Membership in alternate Ca... |
| altopelaltxp 32083 | Alternate ordered pair mem... |
| altxpsspw 32084 | An inclusion rule for alte... |
| altxpexg 32085 | The alternate Cartesian pr... |
| rankaltopb 32086 | Compute the rank of an alt... |
| nfaltop 32087 | Bound-variable hypothesis ... |
| sbcaltop 32088 | Distribution of class subs... |
| cgrrflx2d 32091 | Deduction form of ~ axcgrr... |
| cgrtr4d 32092 | Deduction form of ~ axcgrt... |
| cgrtr4and 32093 | Deduction form of ~ axcgrt... |
| cgrrflx 32094 | Reflexivity law for congru... |
| cgrrflxd 32095 | Deduction form of ~ cgrrfl... |
| cgrcomim 32096 | Congruence commutes on the... |
| cgrcom 32097 | Congruence commutes betwee... |
| cgrcomand 32098 | Deduction form of ~ cgrcom... |
| cgrtr 32099 | Transitivity law for congr... |
| cgrtrand 32100 | Deduction form of ~ cgrtr ... |
| cgrtr3 32101 | Transitivity law for congr... |
| cgrtr3and 32102 | Deduction form of ~ cgrtr3... |
| cgrcoml 32103 | Congruence commutes on the... |
| cgrcomr 32104 | Congruence commutes on the... |
| cgrcomlr 32105 | Congruence commutes on bot... |
| cgrcomland 32106 | Deduction form of ~ cgrcom... |
| cgrcomrand 32107 | Deduction form of ~ cgrcom... |
| cgrcomlrand 32108 | Deduction form of ~ cgrcom... |
| cgrtriv 32109 | Degenerate segments are co... |
| cgrid2 32110 | Identity law for congruenc... |
| cgrdegen 32111 | Two congruent segments are... |
| brofs 32112 | Binary relation form of th... |
| 5segofs 32113 | Rephrase ~ ax5seg using th... |
| ofscom 32114 | The outer five segment pre... |
| cgrextend 32115 | Link congruence over a pai... |
| cgrextendand 32116 | Deduction form of ~ cgrext... |
| segconeq 32117 | Two points that satsify th... |
| segconeu 32118 | Existential uniqueness ver... |
| btwntriv2 32119 | Betweenness always holds f... |
| btwncomim 32120 | Betweenness commutes. Imp... |
| btwncom 32121 | Betweenness commutes. (Co... |
| btwncomand 32122 | Deduction form of ~ btwnco... |
| btwntriv1 32123 | Betweenness always holds f... |
| btwnswapid 32124 | If you can swap the first ... |
| btwnswapid2 32125 | If you can swap arguments ... |
| btwnintr 32126 | Inner transitivity law for... |
| btwnexch3 32127 | Exchange the first endpoin... |
| btwnexch3and 32128 | Deduction form of ~ btwnex... |
| btwnouttr2 32129 | Outer transitivity law for... |
| btwnexch2 32130 | Exchange the outer point o... |
| btwnouttr 32131 | Outer transitivity law for... |
| btwnexch 32132 | Outer transitivity law for... |
| btwnexchand 32133 | Deduction form of ~ btwnex... |
| btwndiff 32134 | There is always a ` c ` di... |
| trisegint 32135 | A line segment between two... |
| funtransport 32138 | The ` TransportTo ` relati... |
| fvtransport 32139 | Calculate the value of the... |
| transportcl 32140 | Closure law for segment tr... |
| transportprops 32141 | Calculate the defining pro... |
| brifs 32150 | Binary relation form of th... |
| ifscgr 32151 | Inner five segment congrue... |
| cgrsub 32152 | Removing identical parts f... |
| brcgr3 32153 | Binary relation form of th... |
| cgr3permute3 32154 | Permutation law for three-... |
| cgr3permute1 32155 | Permutation law for three-... |
| cgr3permute2 32156 | Permutation law for three-... |
| cgr3permute4 32157 | Permutation law for three-... |
| cgr3permute5 32158 | Permutation law for three-... |
| cgr3tr4 32159 | Transitivity law for three... |
| cgr3com 32160 | Commutativity law for thre... |
| cgr3rflx 32161 | Identity law for three-pla... |
| cgrxfr 32162 | A line segment can be divi... |
| btwnxfr 32163 | A condition for extending ... |
| colinrel 32164 | Colinearity is a relations... |
| brcolinear2 32165 | Alternate colinearity bina... |
| brcolinear 32166 | The binary relation form o... |
| colinearex 32167 | The colinear predicate exi... |
| colineardim1 32168 | If ` A ` is colinear with ... |
| colinearperm1 32169 | Permutation law for coline... |
| colinearperm3 32170 | Permutation law for coline... |
| colinearperm2 32171 | Permutation law for coline... |
| colinearperm4 32172 | Permutation law for coline... |
| colinearperm5 32173 | Permutation law for coline... |
| colineartriv1 32174 | Trivial case of colinearit... |
| colineartriv2 32175 | Trivial case of colinearit... |
| btwncolinear1 32176 | Betweenness implies coline... |
| btwncolinear2 32177 | Betweenness implies coline... |
| btwncolinear3 32178 | Betweenness implies coline... |
| btwncolinear4 32179 | Betweenness implies coline... |
| btwncolinear5 32180 | Betweenness implies coline... |
| btwncolinear6 32181 | Betweenness implies coline... |
| colinearxfr 32182 | Transfer law for colineari... |
| lineext 32183 | Extend a line with a missi... |
| brofs2 32184 | Change some conditions for... |
| brifs2 32185 | Change some conditions for... |
| brfs 32186 | Binary relation form of th... |
| fscgr 32187 | Congruence law for the gen... |
| linecgr 32188 | Congruence rule for lines.... |
| linecgrand 32189 | Deduction form of ~ linecg... |
| lineid 32190 | Identity law for points on... |
| idinside 32191 | Law for finding a point in... |
| endofsegid 32192 | If ` A ` , ` B ` , and ` C... |
| endofsegidand 32193 | Deduction form of ~ endofs... |
| btwnconn1lem1 32194 | Lemma for ~ btwnconn1 . T... |
| btwnconn1lem2 32195 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem3 32196 | Lemma for ~ btwnconn1 . E... |
| btwnconn1lem4 32197 | Lemma for ~ btwnconn1 . A... |
| btwnconn1lem5 32198 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem6 32199 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem7 32200 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem8 32201 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem9 32202 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem10 32203 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem11 32204 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem12 32205 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem13 32206 | Lemma for ~ btwnconn1 . B... |
| btwnconn1lem14 32207 | Lemma for ~ btwnconn1 . F... |
| btwnconn1 32208 | Connectitivy law for betwe... |
| btwnconn2 32209 | Another connectivity law f... |
| btwnconn3 32210 | Inner connectivity law for... |
| midofsegid 32211 | If two points fall in the ... |
| segcon2 32212 | Generalization of ~ axsegc... |
| brsegle 32215 | Binary relation form of th... |
| brsegle2 32216 | Alternate characterization... |
| seglecgr12im 32217 | Substitution law for segme... |
| seglecgr12 32218 | Substitution law for segme... |
| seglerflx 32219 | Segment comparison is refl... |
| seglemin 32220 | Any segment is at least as... |
| segletr 32221 | Segment less than is trans... |
| segleantisym 32222 | Antisymmetry law for segme... |
| seglelin 32223 | Linearity law for segment ... |
| btwnsegle 32224 | If ` B ` falls between ` A... |
| colinbtwnle 32225 | Given three colinear point... |
| broutsideof 32228 | Binary relation form of ` ... |
| broutsideof2 32229 | Alternate form of ` Outsid... |
| outsidene1 32230 | Outsideness implies inequa... |
| outsidene2 32231 | Outsideness implies inequa... |
| btwnoutside 32232 | A principle linking outsid... |
| broutsideof3 32233 | Characterization of outsid... |
| outsideofrflx 32234 | Reflexitivity of outsidene... |
| outsideofcom 32235 | Commutitivity law for outs... |
| outsideoftr 32236 | Transitivity law for outsi... |
| outsideofeq 32237 | Uniqueness law for ` Outsi... |
| outsideofeu 32238 | Given a non-degenerate ray... |
| outsidele 32239 | Relate ` OutsideOf ` to ` ... |
| outsideofcol 32240 | Outside of implies colinea... |
| funray 32247 | Show that the ` Ray ` rela... |
| fvray 32248 | Calculate the value of the... |
| funline 32249 | Show that the ` Line ` rel... |
| linedegen 32250 | When ` Line ` is applied w... |
| fvline 32251 | Calculate the value of the... |
| liness 32252 | A line is a subset of the ... |
| fvline2 32253 | Alternate definition of a ... |
| lineunray 32254 | A line is composed of a po... |
| lineelsb2 32255 | If ` S ` lies on ` P Q ` ,... |
| linerflx1 32256 | Reflexivity law for line m... |
| linecom 32257 | Commutativity law for line... |
| linerflx2 32258 | Reflexivity law for line m... |
| ellines 32259 | Membership in the set of a... |
| linethru 32260 | If ` A ` is a line contain... |
| hilbert1.1 32261 | There is a line through an... |
| hilbert1.2 32262 | There is at most one line ... |
| linethrueu 32263 | There is a unique line goi... |
| lineintmo 32264 | Two distinct lines interse... |
| fwddifval 32269 | Calculate the value of the... |
| fwddifnval 32270 | The value of the forward d... |
| fwddifn0 32271 | The value of the n-iterate... |
| fwddifnp1 32272 | The value of the n-iterate... |
| rankung 32273 | The rank of the union of t... |
| ranksng 32274 | The rank of a singleton. ... |
| rankelg 32275 | The membership relation is... |
| rankpwg 32276 | The rank of a power set. ... |
| rank0 32277 | The rank of the empty set ... |
| rankeq1o 32278 | The only set with rank ` 1... |
| elhf 32281 | Membership in the heredita... |
| elhf2 32282 | Alternate form of membersh... |
| elhf2g 32283 | Hereditarily finiteness vi... |
| 0hf 32284 | The empty set is a heredit... |
| hfun 32285 | The union of two HF sets i... |
| hfsn 32286 | The singleton of an HF set... |
| hfadj 32287 | Adjoining one HF element t... |
| hfelhf 32288 | Any member of an HF set is... |
| hftr 32289 | The class of all hereditar... |
| hfext 32290 | Extensionality for HF sets... |
| hfuni 32291 | The union of an HF set is ... |
| hfpw 32292 | The power class of an HF s... |
| hfninf 32293 | ` _om ` is not hereditaril... |
| a1i14 32294 | Add two antecedents to a w... |
| a1i24 32295 | Add two antecedents to a w... |
| exp5d 32296 | An exportation inference. ... |
| exp5g 32297 | An exportation inference. ... |
| exp5k 32298 | An exportation inference. ... |
| exp56 32299 | An exportation inference. ... |
| exp58 32300 | An exportation inference. ... |
| exp510 32301 | An exportation inference. ... |
| exp511 32302 | An exportation inference. ... |
| exp512 32303 | An exportation inference. ... |
| 3com12d 32304 | Commutation in consequent.... |
| imp5p 32305 | A triple importation infer... |
| imp5q 32306 | A triple importation infer... |
| ecase13d 32307 | Deduction for elimination ... |
| subtr 32308 | Transitivity of implicit s... |
| subtr2 32309 | Transitivity of implicit s... |
| trer 32310 | A relation intersected wit... |
| elicc3 32311 | An equivalent membership c... |
| finminlem 32312 | A useful lemma about finit... |
| gtinf 32313 | Any number greater than an... |
| gtinfOLD 32314 | Any number greater than an... |
| opnrebl 32315 | A set is open in the stand... |
| opnrebl2 32316 | A set is open in the stand... |
| nn0prpwlem 32317 | Lemma for ~ nn0prpw . Use... |
| nn0prpw 32318 | Two nonnegative integers a... |
| topbnd 32319 | Two equivalent expressions... |
| opnbnd 32320 | A set is open iff it is di... |
| cldbnd 32321 | A set is closed iff it con... |
| ntruni 32322 | A union of interiors is a ... |
| clsun 32323 | A pairwise union of closur... |
| clsint2 32324 | The closure of an intersec... |
| opnregcld 32325 | A set is regularly closed ... |
| cldregopn 32326 | A set if regularly open if... |
| neiin 32327 | Two neighborhoods intersec... |
| hmeoclda 32328 | Homeomorphisms preserve cl... |
| hmeocldb 32329 | Homeomorphisms preserve cl... |
| ivthALT 32330 | An alternate proof of the ... |
| fnerel 32333 | Fineness is a relation. (... |
| isfne 32334 | The predicate " ` B ` is f... |
| isfne4 32335 | The predicate " ` B ` is f... |
| isfne4b 32336 | A condition for a topology... |
| isfne2 32337 | The predicate " ` B ` is f... |
| isfne3 32338 | The predicate " ` B ` is f... |
| fnebas 32339 | A finer cover covers the s... |
| fnetg 32340 | A finer cover generates a ... |
| fnessex 32341 | If ` B ` is finer than ` A... |
| fneuni 32342 | If ` B ` is finer than ` A... |
| fneint 32343 | If a cover is finer than a... |
| fness 32344 | A cover is finer than its ... |
| fneref 32345 | Reflexivity of the finenes... |
| fnetr 32346 | Transitivity of the finene... |
| fneval 32347 | Two covers are finer than ... |
| fneer 32348 | Fineness intersected with ... |
| topfne 32349 | Fineness for covers corres... |
| topfneec 32350 | A cover is equivalent to a... |
| topfneec2 32351 | A topology is precisely id... |
| fnessref 32352 | A cover is finer iff it ha... |
| refssfne 32353 | A cover is a refinement if... |
| neibastop1 32354 | A collection of neighborho... |
| neibastop2lem 32355 | Lemma for ~ neibastop2 . ... |
| neibastop2 32356 | In the topology generated ... |
| neibastop3 32357 | The topology generated by ... |
| topmtcl 32358 | The meet of a collection o... |
| topmeet 32359 | Two equivalent formulation... |
| topjoin 32360 | Two equivalent formulation... |
| fnemeet1 32361 | The meet of a collection o... |
| fnemeet2 32362 | The meet of equivalence cl... |
| fnejoin1 32363 | Join of equivalence classe... |
| fnejoin2 32364 | Join of equivalence classe... |
| fgmin 32365 | Minimality property of a g... |
| neifg 32366 | The neighborhood filter of... |
| tailfval 32367 | The tail function for a di... |
| tailval 32368 | The tail of an element in ... |
| eltail 32369 | An element of a tail. (Co... |
| tailf 32370 | The tail function of a dir... |
| tailini 32371 | A tail contains its initia... |
| tailfb 32372 | The collection of tails of... |
| filnetlem1 32373 | Lemma for ~ filnet . Chan... |
| filnetlem2 32374 | Lemma for ~ filnet . The ... |
| filnetlem3 32375 | Lemma for ~ filnet . (Con... |
| filnetlem4 32376 | Lemma for ~ filnet . (Con... |
| filnet 32377 | A filter has the same conv... |
| tb-ax1 32378 | The first of three axioms ... |
| tb-ax2 32379 | The second of three axioms... |
| tb-ax3 32380 | The third of three axioms ... |
| tbsyl 32381 | The weak syllogism from Ta... |
| re1ax2lem 32382 | Lemma for ~ re1ax2 . (Con... |
| re1ax2 32383 | ~ ax-2 rederived from the ... |
| naim1 32384 | Constructor theorem for ` ... |
| naim2 32385 | Constructor theorem for ` ... |
| naim1i 32386 | Constructor rule for ` -/\... |
| naim2i 32387 | Constructor rule for ` -/\... |
| naim12i 32388 | Constructor rule for ` -/\... |
| nabi1 32389 | Constructor theorem for ` ... |
| nabi2 32390 | Constructor theorem for ` ... |
| nabi1i 32391 | Constructor rule for ` -/\... |
| nabi2i 32392 | Constructor rule for ` -/\... |
| nabi12i 32393 | Constructor rule for ` -/\... |
| df3nandALT1 32396 | The double nand expressed ... |
| df3nandALT2 32397 | The double nand expressed ... |
| andnand1 32398 | Double and in terms of dou... |
| imnand2 32399 | An ` -> ` nand relation. ... |
| allt 32400 | For all sets, ` T. ` is tr... |
| alnof 32401 | For all sets, ` F. ` is no... |
| nalf 32402 | Not all sets hold ` F. ` a... |
| extt 32403 | There exists a set that ho... |
| nextnt 32404 | There does not exist a set... |
| nextf 32405 | There does not exist a set... |
| unnf 32406 | There does not exist exact... |
| unnt 32407 | There does not exist exact... |
| mont 32408 | There does not exist at mo... |
| mof 32409 | There exist at most one se... |
| meran1 32410 | A single axiom for proposi... |
| meran2 32411 | A single axiom for proposi... |
| meran3 32412 | A single axiom for proposi... |
| waj-ax 32413 | A single axiom for proposi... |
| lukshef-ax2 32414 | A single axiom for proposi... |
| arg-ax 32415 | ? (Contributed by Anthony... |
| negsym1 32416 | In the paper "On Variable ... |
| imsym1 32417 | A symmetry with ` -> ` . ... |
| bisym1 32418 | A symmetry with ` <-> ` . ... |
| consym1 32419 | A symmetry with ` /\ ` . ... |
| dissym1 32420 | A symmetry with ` \/ ` . ... |
| nandsym1 32421 | A symmetry with ` -/\ ` . ... |
| unisym1 32422 | A symmetry with ` A. ` . ... |
| exisym1 32423 | A symmetry with ` E. ` . ... |
| unqsym1 32424 | A symmetry with ` E! ` . ... |
| amosym1 32425 | A symmetry with ` E* ` . ... |
| subsym1 32426 | A symmetry with ` [ x / y ... |
| ontopbas 32427 | An ordinal number is a top... |
| onsstopbas 32428 | The class of ordinal numbe... |
| onpsstopbas 32429 | The class of ordinal numbe... |
| ontgval 32430 | The topology generated fro... |
| ontgsucval 32431 | The topology generated fro... |
| onsuctop 32432 | A successor ordinal number... |
| onsuctopon 32433 | One of the topologies on a... |
| ordtoplem 32434 | Membership of the class of... |
| ordtop 32435 | An ordinal is a topology i... |
| onsucconni 32436 | A successor ordinal number... |
| onsucconn 32437 | A successor ordinal number... |
| ordtopconn 32438 | An ordinal topology is con... |
| onintopssconn 32439 | An ordinal topology is con... |
| onsuct0 32440 | A successor ordinal number... |
| ordtopt0 32441 | An ordinal topology is T_0... |
| onsucsuccmpi 32442 | The successor of a success... |
| onsucsuccmp 32443 | The successor of a success... |
| limsucncmpi 32444 | The successor of a limit o... |
| limsucncmp 32445 | The successor of a limit o... |
| ordcmp 32446 | An ordinal topology is com... |
| ssoninhaus 32447 | The ordinal topologies ` 1... |
| onint1 32448 | The ordinal T_1 spaces are... |
| oninhaus 32449 | The ordinal Hausdorff spac... |
| fveleq 32450 | Please add description her... |
| findfvcl 32451 | Please add description her... |
| findreccl 32452 | Please add description her... |
| findabrcl 32453 | Please add description her... |
| nnssi2 32454 | Convert a theorem for real... |
| nnssi3 32455 | Convert a theorem for real... |
| nndivsub 32456 | Please add description her... |
| nndivlub 32457 | A factor of a positive int... |
| ee7.2aOLD 32460 | Lemma for Euclid's Element... |
| dnival 32461 | Value of the "distance to ... |
| dnicld1 32462 | Closure theorem for the "d... |
| dnicld2 32463 | Closure theorem for the "d... |
| dnif 32464 | The "distance to nearest i... |
| dnizeq0 32465 | The distance to nearest in... |
| dnizphlfeqhlf 32466 | The distance to nearest in... |
| rddif2 32467 | Variant of ~ rddif . (Con... |
| dnibndlem1 32468 | Lemma for ~ dnibnd . (Con... |
| dnibndlem2 32469 | Lemma for ~ dnibnd . (Con... |
| dnibndlem3 32470 | Lemma for ~ dnibnd . (Con... |
| dnibndlem4 32471 | Lemma for ~ dnibnd . (Con... |
| dnibndlem5 32472 | Lemma for ~ dnibnd . (Con... |
| dnibndlem6 32473 | Lemma for ~ dnibnd . (Con... |
| dnibndlem7 32474 | Lemma for ~ dnibnd . (Con... |
| dnibndlem8 32475 | Lemma for ~ dnibnd . (Con... |
| dnibndlem9 32476 | Lemma for ~ dnibnd . (Con... |
| dnibndlem10 32477 | Lemma for ~ dnibnd . (Con... |
| dnibndlem11 32478 | Lemma for ~ dnibnd . (Con... |
| dnibndlem12 32479 | Lemma for ~ dnibnd . (Con... |
| dnibndlem13 32480 | Lemma for ~ dnibnd . (Con... |
| dnibnd 32481 | The "distance to nearest i... |
| dnicn 32482 | The "distance to nearest i... |
| knoppcnlem1 32483 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem2 32484 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem3 32485 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem4 32486 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem5 32487 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem6 32488 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem7 32489 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem8 32490 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem9 32491 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem10 32492 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem11 32493 | Lemma for ~ knoppcn . (Co... |
| knoppcn 32494 | The continuous nowhere dif... |
| knoppcld 32495 | Closure theorem for Knopp'... |
| addgtge0d 32496 | Addition of positive and n... |
| unblimceq0lem 32497 | Lemma for ~ unblimceq0 . ... |
| unblimceq0 32498 | If ` F ` is unbounded near... |
| unbdqndv1 32499 | If the difference quotient... |
| unbdqndv2lem1 32500 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2lem2 32501 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2 32502 | Variant of ~ unbdqndv1 wit... |
| knoppndvlem1 32503 | Lemma for ~ knoppndv . (C... |
| knoppndvlem2 32504 | Lemma for ~ knoppndv . (C... |
| knoppndvlem3 32505 | Lemma for ~ knoppndv . (C... |
| knoppndvlem4 32506 | Lemma for ~ knoppndv . (C... |
| knoppndvlem5 32507 | Lemma for ~ knoppndv . (C... |
| knoppndvlem6 32508 | Lemma for ~ knoppndv . (C... |
| knoppndvlem7 32509 | Lemma for ~ knoppndv . (C... |
| knoppndvlem8 32510 | Lemma for ~ knoppndv . (C... |
| knoppndvlem9 32511 | Lemma for ~ knoppndv . (C... |
| knoppndvlem10 32512 | Lemma for ~ knoppndv . (C... |
| knoppndvlem11 32513 | Lemma for ~ knoppndv . (C... |
| knoppndvlem12 32514 | Lemma for ~ knoppndv . (C... |
| knoppndvlem13 32515 | Lemma for ~ knoppndv . (C... |
| knoppndvlem14 32516 | Lemma for ~ knoppndv . (C... |
| knoppndvlem15 32517 | Lemma for ~ knoppndv . (C... |
| knoppndvlem16 32518 | Lemma for ~ knoppndv . (C... |
| knoppndvlem17 32519 | Lemma for ~ knoppndv . (C... |
| knoppndvlem18 32520 | Lemma for ~ knoppndv . (C... |
| knoppndvlem19 32521 | Lemma for ~ knoppndv . (C... |
| knoppndvlem20 32522 | Lemma for ~ knoppndv . (C... |
| knoppndvlem21 32523 | Lemma for ~ knoppndv . (C... |
| knoppndvlem22 32524 | Lemma for ~ knoppndv . (C... |
| knoppndv 32525 | The continuous nowhere dif... |
| knoppf 32526 | Knopp's function is a func... |
| knoppcn2 32527 | Variant of ~ knoppcn with ... |
| cnndvlem1 32528 | Lemma for ~ cnndv . (Cont... |
| cnndvlem2 32529 | Lemma for ~ cnndv . (Cont... |
| cnndv 32530 | There exists a continuous ... |
| bj-mp2c 32531 | A double modus ponens infe... |
| bj-mp2d 32532 | A double modus ponens infe... |
| bj-0 32533 | A syntactic theorem. See ... |
| bj-1 32534 | In this proof, the use of ... |
| bj-a1k 32535 | Weakening of ~ ax-1 . Thi... |
| bj-jarri 32536 | Inference associated with ... |
| bj-jarrii 32537 | Inference associated with ... |
| bj-imim2ALT 32538 | More direct proof of ~ imi... |
| bj-imim21 32539 | The propositional function... |
| bj-imim21i 32540 | Inference associated with ... |
| bj-orim2 32541 | Proof of ~ orim2 from the ... |
| bj-curry 32542 | A non-intuitionistic posit... |
| bj-peirce 32543 | Proof of ~ peirce from min... |
| bj-currypeirce 32544 | Curry's axiom (a non-intui... |
| bj-peircecurry 32545 | Peirce's axiom ~ peirce im... |
| pm4.81ALT 32546 | Alternate proof of ~ pm4.8... |
| bj-con4iALT 32547 | Alternate proof of ~ con4i... |
| bj-con2com 32548 | A commuted form of the con... |
| bj-con2comi 32549 | Inference associated with ... |
| bj-pm2.01i 32550 | Inference associated with ... |
| bj-nimn 32551 | If a formula is true, then... |
| bj-nimni 32552 | Inference associated with ... |
| bj-peircei 32553 | Inference associated with ... |
| bj-looinvi 32554 | Inference associated with ... |
| bj-looinvii 32555 | Inference associated with ... |
| bj-jaoi1 32556 | Shortens ~ orfa2 (58>53), ... |
| bj-jaoi2 32557 | Shortens ~ consensus (110>... |
| bj-dfbi4 32558 | Alternate definition of th... |
| bj-dfbi5 32559 | Alternate definition of th... |
| bj-dfbi6 32560 | Alternate definition of th... |
| bj-bijust0 32561 | The general statement that... |
| bj-consensus 32562 | Version of ~ consensus exp... |
| bj-consensusALT 32563 | Alternate proof of ~ bj-co... |
| bj-dfifc2 32564 | This should be the alterna... |
| bj-df-ifc 32565 | The definition of "ifc" if... |
| bj-ififc 32566 | A theorem linking ` if- ` ... |
| bj-imbi12 32567 | Uncurried (imported) form ... |
| bj-biorfi 32568 | This should be labeled "bi... |
| bj-falor 32569 | Dual of ~ truan (which has... |
| bj-falor2 32570 | Dual of ~ truan . (Contri... |
| bj-bibibi 32571 | A property of the bicondit... |
| bj-imn3ani 32572 | Duplication of ~ bnj1224 .... |
| bj-andnotim 32573 | Two ways of expressing a c... |
| bj-bi3ant 32574 | This used to be in the mai... |
| bj-bisym 32575 | This used to be in the mai... |
| bj-axdd2 32576 | This implication, proved u... |
| bj-axd2d 32577 | This implication, proved u... |
| bj-axtd 32578 | This implication, proved f... |
| bj-gl4lem 32579 | Lemma for ~ bj-gl4 . Note... |
| bj-gl4 32580 | In a normal modal logic, t... |
| bj-axc4 32581 | Over minimal calculus, the... |
| prvlem1 32586 | An elementary property of ... |
| prvlem2 32587 | An elementary property of ... |
| bj-babygodel 32588 | See the section header com... |
| bj-babylob 32589 | See the section header com... |
| bj-godellob 32590 | Proof of Gödel's theo... |
| bj-genr 32591 | Generalization rule on the... |
| bj-genl 32592 | Generalization rule on the... |
| bj-genan 32593 | Generalization rule on a c... |
| bj-2alim 32594 | Closed form of ~ 2alimi . ... |
| bj-2exim 32595 | Closed form of ~ 2eximi . ... |
| bj-alanim 32596 | Closed form of ~ alanimi .... |
| bj-2albi 32597 | Closed form of ~ 2albii . ... |
| bj-notalbii 32598 | Equivalence of universal q... |
| bj-2exbi 32599 | Closed form of ~ 2exbii . ... |
| bj-3exbi 32600 | Closed form of ~ 3exbii . ... |
| bj-sylgt2 32601 | Uncurried (imported) form ... |
| bj-exlimh 32602 | Closed form of close to ~ ... |
| bj-exlimh2 32603 | Uncurried (imported) form ... |
| bj-alrimhi 32604 | An inference associated wi... |
| bj-alexim 32605 | Closed form of ~ aleximi (... |
| bj-nexdh 32606 | Closed form of ~ nexdh (ac... |
| bj-nexdh2 32607 | Uncurried (imported) form ... |
| bj-hbxfrbi 32608 | Closed form of ~ hbxfrbi .... |
| bj-exlime 32609 | Variant of ~ exlimih where... |
| bj-exnalimn 32610 | A transformation of quanti... |
| bj-exalim 32611 | Distributing quantifiers o... |
| bj-exalimi 32612 | An inference for distribut... |
| bj-exalims 32613 | Distributing quantifiers o... |
| bj-exalimsi 32614 | An inference for distribut... |
| bj-ax12ig 32615 | A lemma used to prove a we... |
| bj-ax12i 32616 | A weakening of ~ bj-ax12ig... |
| bj-ax12wlem 32617 | A lemma used to prove a we... |
| bj-ssbjust 32618 | Justification theorem for ... |
| bj-ssbim 32621 | Distribute substitution ov... |
| bj-ssbbi 32622 | Biconditional property for... |
| bj-ssbimi 32623 | Distribute substitution ov... |
| bj-ssbbii 32624 | Biconditional property for... |
| bj-alsb 32625 | If a proposition is true f... |
| bj-sbex 32626 | If a proposition is true f... |
| bj-ssbeq 32627 | Substitution in an equalit... |
| bj-ssb0 32628 | Substitution for a variabl... |
| bj-ssbequ 32629 | Equality property for subs... |
| bj-ssblem1 32630 | A lemma for the definiens ... |
| bj-ssblem2 32631 | An instance of ~ ax-11 pro... |
| bj-ssb1a 32632 | One direction of a simplif... |
| bj-ssb1 32633 | A simplified definition of... |
| bj-ax12 32634 | A weaker form of ~ ax-12 a... |
| bj-ax12ssb 32635 | The axiom ~ bj-ax12 expres... |
| bj-modal5e 32636 | Dual statement of ~ hbe1 (... |
| bj-19.41al 32637 | Special case of ~ 19.41 pr... |
| bj-equsexval 32638 | Special case of ~ equsexv ... |
| bj-sb56 32639 | Proof of ~ sb56 from Tarsk... |
| bj-dfssb2 32640 | An alternate definition of... |
| bj-ssbn 32641 | The result of a substituti... |
| bj-ssbft 32642 | See ~ sbft . This proof i... |
| bj-ssbequ2 32643 | Note that ~ ax-12 is used ... |
| bj-ssbequ1 32644 | This uses ~ ax-12 with a d... |
| bj-ssbid2 32645 | A special case of ~ bj-ssb... |
| bj-ssbid2ALT 32646 | Alternate proof of ~ bj-ss... |
| bj-ssbid1 32647 | A special case of ~ bj-ssb... |
| bj-ssbid1ALT 32648 | Alternate proof of ~ bj-ss... |
| bj-ssbssblem 32649 | Composition of two substit... |
| bj-ssbcom3lem 32650 | Lemma for bj-ssbcom3 when ... |
| bj-ax6elem1 32651 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6elem2 32652 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6e 32653 | Proof of ~ ax6e (hence ~ a... |
| bj-extru 32654 | There exists a variable su... |
| bj-alequexv 32655 | Version of ~ bj-alequex wi... |
| bj-spimvwt 32656 | Closed form of ~ spimvw . ... |
| bj-spimevw 32657 | Existential introduction, ... |
| bj-spnfw 32658 | Theorem close to a closed ... |
| bj-cbvexiw 32659 | Change bound variable. Th... |
| bj-cbvexivw 32660 | Change bound variable. Th... |
| bj-modald 32661 | A short form of the axiom ... |
| bj-denot 32662 | A weakening of ~ ax-6 and ... |
| bj-eqs 32663 | A lemma for substitutions,... |
| bj-cbvexw 32664 | Change bound variable. Th... |
| bj-ax12w 32665 | The general statement that... |
| bj-elequ2g 32666 | A form of ~ elequ2 with a ... |
| bj-ax89 32667 | A theorem which could be u... |
| bj-elequ12 32668 | An identity law for the no... |
| bj-cleljusti 32669 | One direction of ~ cleljus... |
| bj-alcomexcom 32670 | Commutation of universal q... |
| bj-hbalt 32671 | Closed form of ~ hbal . W... |
| axc11n11 32672 | Proof of ~ axc11n from { ~... |
| axc11n11r 32673 | Proof of ~ axc11n from { ~... |
| bj-axc16g16 32674 | Proof of ~ axc16g from { ~... |
| bj-ax12v3 32675 | A weak version of ~ ax-12 ... |
| bj-ax12v3ALT 32676 | Alternate proof of ~ bj-ax... |
| bj-sb 32677 | A weak variant of ~ sbid2 ... |
| bj-modalbe 32678 | The predicate-calculus ver... |
| bj-spst 32679 | Closed form of ~ sps . On... |
| bj-19.21bit 32680 | Closed form of ~ 19.21bi .... |
| bj-19.23bit 32681 | Closed form of ~ 19.23bi .... |
| bj-nexrt 32682 | Closed form of ~ nexr . C... |
| bj-alrim 32683 | Closed form of ~ alrimi . ... |
| bj-alrim2 32684 | Uncurried (imported) form ... |
| bj-nfdt0 32685 | A theorem close to a close... |
| bj-nfdt 32686 | Closed form of ~ nf5d and ... |
| bj-nexdt 32687 | Closed form of ~ nexd . (... |
| bj-nexdvt 32688 | Closed form of ~ nexdv . ... |
| bj-19.3t 32689 | Closed form of ~ 19.3 . (... |
| bj-alexbiex 32690 | Adding a second quantifier... |
| bj-exexbiex 32691 | Adding a second quantifier... |
| bj-alalbial 32692 | Adding a second quantifier... |
| bj-exalbial 32693 | Adding a second quantifier... |
| bj-19.9htbi 32694 | Strengthening ~ 19.9ht by ... |
| bj-hbntbi 32695 | Strengthening ~ hbnt by re... |
| bj-biexal1 32696 | A general FOL biconditiona... |
| bj-biexal2 32697 | A general FOL biconditiona... |
| bj-biexal3 32698 | A general FOL biconditiona... |
| bj-bialal 32699 | A general FOL biconditiona... |
| bj-biexex 32700 | A general FOL biconditiona... |
| bj-hbext 32701 | Closed form of ~ hbex . (... |
| bj-nfalt 32702 | Closed form of ~ nfal . (... |
| bj-nfext 32703 | Closed form of ~ nfex . (... |
| bj-eeanvw 32704 | Version of ~ eeanv with a ... |
| bj-modal4e 32705 | Dual statement of ~ hba1 (... |
| bj-modalb 32706 | A short form of the axiom ... |
| bj-axc10 32707 | Alternate (shorter) proof ... |
| bj-alequex 32708 | A fol lemma. See ~ bj-ale... |
| bj-spimt2 32709 | A step in the proof of ~ s... |
| bj-cbv3ta 32710 | Closed form of ~ cbv3 . (... |
| bj-cbv3tb 32711 | Closed form of ~ cbv3 . (... |
| bj-hbsb3t 32712 | A theorem close to a close... |
| bj-hbsb3 32713 | Shorter proof of ~ hbsb3 .... |
| bj-nfs1t 32714 | A theorem close to a close... |
| bj-nfs1t2 32715 | A theorem close to a close... |
| bj-nfs1 32716 | Shorter proof of ~ nfs1 (t... |
| bj-axc10v 32717 | Version of ~ axc10 with a ... |
| bj-spimtv 32718 | Version of ~ spimt with a ... |
| bj-spimedv 32719 | Version of ~ spimed with a... |
| bj-spimev 32720 | Version of ~ spime with a ... |
| bj-spimvv 32721 | Version of ~ spimv and ~ s... |
| bj-spimevv 32722 | Version of ~ spimev with a... |
| bj-spvv 32723 | Version of ~ spv with a dv... |
| bj-speiv 32724 | Version of ~ spei with a d... |
| bj-chvarv 32725 | Version of ~ chvar with a ... |
| bj-chvarvv 32726 | Version of ~ chvarv with a... |
| bj-cbv3v2 32727 | Version of ~ cbv3 with two... |
| bj-cbv3hv2 32728 | Version of ~ cbv3h with tw... |
| bj-cbv1v 32729 | Version of ~ cbv1 with a d... |
| bj-cbv1hv 32730 | Version of ~ cbv1h with a ... |
| bj-cbv2hv 32731 | Version of ~ cbv2h with a ... |
| bj-cbv2v 32732 | Version of ~ cbv2 with a d... |
| bj-cbvalvv 32733 | Version of ~ cbvalv with a... |
| bj-cbvexvv 32734 | Version of ~ cbvexv with a... |
| bj-cbvaldv 32735 | Version of ~ cbvald with a... |
| bj-cbvexdv 32736 | Version of ~ cbvexd with a... |
| bj-cbval2v 32737 | Version of ~ cbval2 with a... |
| bj-cbvex2v 32738 | Version of ~ cbvex2 with a... |
| bj-cbval2vv 32739 | Version of ~ cbval2v with ... |
| bj-cbvex2vv 32740 | Version of ~ cbvex2v with ... |
| bj-cbvaldvav 32741 | Version of ~ cbvaldva with... |
| bj-cbvexdvav 32742 | Version of ~ cbvexdva with... |
| bj-cbvex4vv 32743 | Version of ~ cbvex4v with ... |
| bj-equsalhv 32744 | Version of ~ equsalh with ... |
| bj-axc11nv 32745 | Version of ~ axc11n with a... |
| bj-aecomsv 32746 | Version of ~ aecoms with a... |
| bj-axc11v 32747 | Version of ~ axc11 with a ... |
| bj-dral1v 32748 | Version of ~ dral1 with a ... |
| bj-drex1v 32749 | Version of ~ drex1 with a ... |
| bj-drnf1v 32750 | Version of ~ drnf1 with a ... |
| bj-drnf2v 32751 | Version of ~ drnf2 with a ... |
| bj-equs45fv 32752 | Version of ~ equs45f with ... |
| bj-sb2v 32753 | Version of ~ sb2 with a dv... |
| bj-stdpc4v 32754 | Version of ~ stdpc4 with a... |
| bj-2stdpc4v 32755 | Version of ~ 2stdpc4 with ... |
| bj-sb3v 32756 | Version of ~ sb3 with a dv... |
| bj-sb4v 32757 | Version of ~ sb4 with a dv... |
| bj-hbs1 32758 | Version of ~ hbsb2 with a ... |
| bj-nfs1v 32759 | Version of ~ nfsb2 with a ... |
| bj-hbsb2av 32760 | Version of ~ hbsb2a with a... |
| bj-hbsb3v 32761 | Version of ~ hbsb3 with a ... |
| bj-equsb1v 32762 | Version of ~ equsb1 with a... |
| bj-sbftv 32763 | Version of ~ sbft with a d... |
| bj-sbfv 32764 | Version of ~ sbf with a dv... |
| bj-sbfvv 32765 | Version of ~ sbf with two ... |
| bj-sbtv 32766 | Version of ~ sbt with a dv... |
| bj-sb6 32767 | Remove dependency on ~ ax-... |
| bj-sb5 32768 | Remove dependency on ~ ax-... |
| bj-axext3 32769 | Remove dependency on ~ ax-... |
| bj-axext4 32770 | Remove dependency on ~ ax-... |
| bj-hbab1 32771 | Remove dependency on ~ ax-... |
| bj-nfsab1 32772 | Remove dependency on ~ ax-... |
| bj-abeq2 32773 | Remove dependency on ~ ax-... |
| bj-abeq1 32774 | Remove dependency on ~ ax-... |
| bj-abbi 32775 | Remove dependency on ~ ax-... |
| bj-abbi2i 32776 | Remove dependency on ~ ax-... |
| bj-abbii 32777 | Remove dependency on ~ ax-... |
| bj-abbid 32778 | Remove dependency on ~ ax-... |
| bj-abbidv 32779 | Remove dependency on ~ ax-... |
| bj-abbi2dv 32780 | Remove dependency on ~ ax-... |
| bj-abbi1dv 32781 | Remove dependency on ~ ax-... |
| bj-abid2 32782 | Remove dependency on ~ ax-... |
| bj-clabel 32783 | Remove dependency on ~ ax-... |
| bj-sbab 32784 | Remove dependency on ~ ax-... |
| bj-nfab1 32785 | Remove dependency on ~ ax-... |
| bj-vjust 32786 | Remove dependency on ~ ax-... |
| bj-cdeqab 32787 | Remove dependency on ~ ax-... |
| bj-axrep1 32788 | Remove dependency on ~ ax-... |
| bj-axrep2 32789 | Remove dependency on ~ ax-... |
| bj-axrep3 32790 | Remove dependency on ~ ax-... |
| bj-axrep4 32791 | Remove dependency on ~ ax-... |
| bj-axrep5 32792 | Remove dependency on ~ ax-... |
| bj-axsep 32793 | Remove dependency on ~ ax-... |
| bj-nalset 32794 | Remove dependency on ~ ax-... |
| bj-zfpow 32795 | Remove dependency on ~ ax-... |
| bj-el 32796 | Remove dependency on ~ ax-... |
| bj-dtru 32797 | Remove dependency on ~ ax-... |
| bj-axc16b 32798 | Remove dependency on ~ ax-... |
| bj-eunex 32799 | Remove dependency on ~ ax-... |
| bj-dtrucor 32800 | Remove dependency on ~ ax-... |
| bj-dtrucor2v 32801 | Version of ~ dtrucor2 with... |
| bj-dvdemo1 32802 | Remove dependency on ~ ax-... |
| bj-dvdemo2 32803 | Remove dependency on ~ ax-... |
| bj-sb3b 32804 | Simplified definition of s... |
| bj-hbaeb2 32805 | Biconditional version of a... |
| bj-hbaeb 32806 | Biconditional version of ~... |
| bj-hbnaeb 32807 | Biconditional version of ~... |
| bj-dvv 32808 | A special instance of ~ bj... |
| bj-equsal1t 32809 | Duplication of ~ wl-equsal... |
| bj-equsal1ti 32810 | Inference associated with ... |
| bj-equsal1 32811 | One direction of ~ equsal ... |
| bj-equsal2 32812 | One direction of ~ equsal ... |
| bj-equsal 32813 | Shorter proof of ~ equsal ... |
| stdpc5t 32814 | Closed form of ~ stdpc5 . ... |
| bj-stdpc5 32815 | More direct proof of ~ std... |
| 2stdpc5 32816 | A double ~ stdpc5 (one dir... |
| bj-19.21t 32817 | Proof of ~ 19.21t from ~ s... |
| exlimii 32818 | Inference associated with ... |
| ax11-pm 32819 | Proof of ~ ax-11 similar t... |
| ax6er 32820 | Commuted form of ~ ax6e . ... |
| exlimiieq1 32821 | Inferring a theorem when i... |
| exlimiieq2 32822 | Inferring a theorem when i... |
| ax11-pm2 32823 | Proof of ~ ax-11 from the ... |
| bj-sbsb 32824 | Biconditional showing two ... |
| bj-dfsb2 32825 | Alternate (dual) definitio... |
| bj-sbf3 32826 | Substitution has no effect... |
| bj-sbf4 32827 | Substitution has no effect... |
| bj-sbnf 32828 | Move non-free predicate in... |
| bj-eu3f 32829 | Version of ~ eu3v where th... |
| bj-eumo0 32830 | Existential uniqueness imp... |
| bj-sbidmOLD 32831 | Obsolete proof of ~ sbidm ... |
| bj-mo3OLD 32832 | Obsolete proof of ~ mo3 te... |
| bj-syl66ib 32833 | A mixed syllogism inferenc... |
| bj-dvelimdv 32834 | Deduction form of ~ dvelim... |
| bj-dvelimdv1 32835 | Curried (exported) form of... |
| bj-dvelimv 32836 | A version of ~ dvelim usin... |
| bj-nfeel2 32837 | Non-freeness in an equalit... |
| bj-axc14nf 32838 | Proof of a version of ~ ax... |
| bj-axc14 32839 | Alternate proof of ~ axc14... |
| eliminable1 32840 | A theorem used to prove th... |
| eliminable2a 32841 | A theorem used to prove th... |
| eliminable2b 32842 | A theorem used to prove th... |
| eliminable2c 32843 | A theorem used to prove th... |
| eliminable3a 32844 | A theorem used to prove th... |
| eliminable3b 32845 | A theorem used to prove th... |
| bj-termab 32846 | Every class can be written... |
| bj-cleljustab 32847 | An instance of ~ df-clel w... |
| bj-clelsb3 32848 | Remove dependency on ~ ax-... |
| bj-hblem 32849 | Remove dependency on ~ ax-... |
| bj-nfcjust 32850 | Remove dependency on ~ ax-... |
| bj-nfcrii 32851 | Remove dependency on ~ ax-... |
| bj-nfcri 32852 | Remove dependency on ~ ax-... |
| bj-nfnfc 32853 | Remove dependency on ~ ax-... |
| bj-vexwt 32854 | Closed form of ~ bj-vexw .... |
| bj-vexw 32855 | If ` ph ` is a theorem, th... |
| bj-vexwvt 32856 | Closed form of ~ bj-vexwv ... |
| bj-vexwv 32857 | Version of ~ bj-vexw with ... |
| bj-denotes 32858 | This would be the justific... |
| bj-issetwt 32859 | Closed form of ~ bj-issetw... |
| bj-issetw 32860 | The closest one can get to... |
| bj-elissetv 32861 | Version of ~ bj-elisset wi... |
| bj-elisset 32862 | Remove from ~ elisset depe... |
| bj-issetiv 32863 | Version of ~ bj-isseti wit... |
| bj-isseti 32864 | Remove from ~ isseti depen... |
| bj-ralvw 32865 | A weak version of ~ ralv n... |
| bj-rexvwv 32866 | A weak version of ~ rexv n... |
| bj-rababwv 32867 | A weak version of ~ rabab ... |
| bj-ralcom4 32868 | Remove from ~ ralcom4 depe... |
| bj-rexcom4 32869 | Remove from ~ rexcom4 depe... |
| bj-rexcom4a 32870 | Remove from ~ rexcom4a dep... |
| bj-rexcom4bv 32871 | Version of ~ bj-rexcom4b w... |
| bj-rexcom4b 32872 | Remove from ~ rexcom4b dep... |
| bj-ceqsalt0 32873 | The FOL content of ~ ceqsa... |
| bj-ceqsalt1 32874 | The FOL content of ~ ceqsa... |
| bj-ceqsalt 32875 | Remove from ~ ceqsalt depe... |
| bj-ceqsaltv 32876 | Version of ~ bj-ceqsalt wi... |
| bj-ceqsalg0 32877 | The FOL content of ~ ceqsa... |
| bj-ceqsalg 32878 | Remove from ~ ceqsalg depe... |
| bj-ceqsalgALT 32879 | Alternate proof of ~ bj-ce... |
| bj-ceqsalgv 32880 | Version of ~ bj-ceqsalg wi... |
| bj-ceqsalgvALT 32881 | Alternate proof of ~ bj-ce... |
| bj-ceqsal 32882 | Remove from ~ ceqsal depen... |
| bj-ceqsalv 32883 | Remove from ~ ceqsalv depe... |
| bj-spcimdv 32884 | Remove from ~ spcimdv depe... |
| bj-spcimdvv 32885 | Remove from ~ spcimdv depe... |
| bj-nfcsym 32886 | The class-form not-free pr... |
| bj-ax8 32887 | Proof of ~ ax-8 from ~ df-... |
| bj-df-clel 32888 | Candidate definition for ~... |
| bj-dfclel 32889 | Characterization of the el... |
| bj-ax9 32890 | Proof of ~ ax-9 from Tarsk... |
| bj-ax9-2 32891 | Proof of ~ ax-9 from Tarsk... |
| bj-cleqhyp 32892 | The hypothesis of ~ bj-df-... |
| bj-df-cleq 32893 | Candidate definition for ~... |
| bj-dfcleq 32894 | Proof of class extensional... |
| bj-sbeqALT 32895 | Substitution in an equalit... |
| bj-sbeq 32896 | Distribute proper substitu... |
| bj-sbceqgALT 32897 | Distribute proper substitu... |
| bj-csbsnlem 32898 | Lemma for ~ bj-csbsn (in t... |
| bj-csbsn 32899 | Substitution in a singleto... |
| bj-sbel1 32900 | Version of ~ sbcel1g when ... |
| bj-abv 32901 | The class of sets verifyin... |
| bj-ab0 32902 | The class of sets verifyin... |
| bj-abf 32903 | Shorter proof of ~ abf (wh... |
| bj-csbprc 32904 | More direct proof of ~ csb... |
| bj-exlimmpi 32905 | Lemma for ~ bj-vtoclg1f1 (... |
| bj-exlimmpbi 32906 | Lemma for theorems of the ... |
| bj-exlimmpbir 32907 | Lemma for theorems of the ... |
| bj-vtoclf 32908 | Remove dependency on ~ ax-... |
| bj-vtocl 32909 | Remove dependency on ~ ax-... |
| bj-vtoclg1f1 32910 | The FOL content of ~ vtocl... |
| bj-vtoclg1f 32911 | Reprove ~ vtoclg1f from ~ ... |
| bj-vtoclg1fv 32912 | Version of ~ bj-vtoclg1f w... |
| bj-rabbida2 32913 | Version of ~ rabbidva2 wit... |
| bj-rabbida 32914 | Version of ~ rabbidva with... |
| bj-rabbid 32915 | Version of ~ rabbidv with ... |
| bj-rabeqd 32916 | Deduction form of ~ rabeq ... |
| bj-rabeqbid 32917 | Version of ~ rabeqbidv wit... |
| bj-rabeqbida 32918 | Version of ~ rabeqbidva wi... |
| bj-seex 32919 | Version of ~ seex with a d... |
| bj-nfcf 32920 | Version of ~ df-nfc with a... |
| bj-axsep2 32921 | Remove dependency on ~ ax-... |
| bj-unrab 32922 | Generalization of ~ unrab ... |
| bj-inrab 32923 | Generalization of ~ inrab ... |
| bj-inrab2 32924 | Shorter proof of ~ inrab .... |
| bj-inrab3 32925 | Generalization of ~ dfrab3... |
| bj-rabtr 32926 | Restricted class abstracti... |
| bj-rabtrALT 32927 | Alternate proof of ~ bj-ra... |
| bj-rabtrALTALT 32928 | Alternate proof of ~ bj-ra... |
| bj-rabtrAUTO 32929 | Proof of ~ bj-rabtr found ... |
| bj-ru0 32932 | The FOL part of Russell's ... |
| bj-ru1 32933 | A version of Russell's par... |
| bj-ru 32934 | Remove dependency on ~ ax-... |
| bj-n0i 32935 | Inference associated with ... |
| bj-disjcsn 32936 | A class is disjoint from i... |
| bj-disjsn01 32937 | Disjointness of the single... |
| bj-1ex 32938 | ` 1o ` is a set. (Contrib... |
| bj-2ex 32939 | ` 2o ` is a set. (Contrib... |
| bj-0nel1 32940 | The empty set does not bel... |
| bj-1nel0 32941 | ` 1o ` does not belong to ... |
| bj-xpimasn 32942 | The image of a singleton, ... |
| bj-xpima1sn 32943 | The image of a singleton b... |
| bj-xpima1snALT 32944 | Alternate proof of ~ bj-xp... |
| bj-xpima2sn 32945 | The image of a singleton b... |
| bj-xpnzex 32946 | If the first factor of a p... |
| bj-xpexg2 32947 | Curried (exported) form of... |
| bj-xpnzexb 32948 | If the first factor of a p... |
| bj-cleq 32949 | Substitution property for ... |
| bj-sels 32950 | If a class is a set, then ... |
| bj-snsetex 32951 | The class of sets "whose s... |
| bj-clex 32952 | Sethood of certain classes... |
| bj-sngleq 32955 | Substitution property for ... |
| bj-elsngl 32956 | Characterization of the el... |
| bj-snglc 32957 | Characterization of the el... |
| bj-snglss 32958 | The singletonization of a ... |
| bj-0nelsngl 32959 | The empty set is not a mem... |
| bj-snglinv 32960 | Inverse of singletonizatio... |
| bj-snglex 32961 | A class is a set if and on... |
| bj-tageq 32964 | Substitution property for ... |
| bj-eltag 32965 | Characterization of the el... |
| bj-0eltag 32966 | The empty set belongs to t... |
| bj-tagn0 32967 | The tagging of a class is ... |
| bj-tagss 32968 | The tagging of a class is ... |
| bj-snglsstag 32969 | The singletonization is in... |
| bj-sngltagi 32970 | The singletonization is in... |
| bj-sngltag 32971 | The singletonization and t... |
| bj-tagci 32972 | Characterization of the el... |
| bj-tagcg 32973 | Characterization of the el... |
| bj-taginv 32974 | Inverse of tagging. (Cont... |
| bj-tagex 32975 | A class is a set if and on... |
| bj-xtageq 32976 | The products of a given cl... |
| bj-xtagex 32977 | The product of a set and t... |
| bj-projeq 32980 | Substitution property for ... |
| bj-projeq2 32981 | Substitution property for ... |
| bj-projun 32982 | The class projection on a ... |
| bj-projex 32983 | Sethood of the class proje... |
| bj-projval 32984 | Value of the class project... |
| bj-1upleq 32987 | Substitution property for ... |
| bj-pr1eq 32990 | Substitution property for ... |
| bj-pr1un 32991 | The first projection prese... |
| bj-pr1val 32992 | Value of the first project... |
| bj-pr11val 32993 | Value of the first project... |
| bj-pr1ex 32994 | Sethood of the first proje... |
| bj-1uplth 32995 | The characteristic propert... |
| bj-1uplex 32996 | A monuple is a set if and ... |
| bj-1upln0 32997 | A monuple is nonempty. (C... |
| bj-2upleq 33000 | Substitution property for ... |
| bj-pr21val 33001 | Value of the first project... |
| bj-pr2eq 33004 | Substitution property for ... |
| bj-pr2un 33005 | The second projection pres... |
| bj-pr2val 33006 | Value of the second projec... |
| bj-pr22val 33007 | Value of the second projec... |
| bj-pr2ex 33008 | Sethood of the second proj... |
| bj-2uplth 33009 | The characteristic propert... |
| bj-2uplex 33010 | A couple is a set if and o... |
| bj-2upln0 33011 | A couple is nonempty. (Co... |
| bj-2upln1upl 33012 | A couple is never equal to... |
| bj-disj2r 33013 | Relative version of ~ ssdi... |
| bj-sscon 33014 | Contraposition law for rel... |
| bj-vjust2 33015 | Justification theorem for ... |
| bj-df-v 33016 | Alternate definition of th... |
| bj-df-nul 33017 | Alternate definition of th... |
| bj-nul 33018 | Two formulations of the ax... |
| bj-nuliota 33019 | Definition of the empty se... |
| bj-nuliotaALT 33020 | Alternate proof of ~ bj-nu... |
| bj-vtoclgfALT 33021 | Alternate proof of ~ vtocl... |
| bj-pwcfsdom 33022 | Remove hypothesis from ~ p... |
| bj-grur1 33023 | Remove hypothesis from ~ g... |
| bj-evaleq 33024 | Equality theorem for the `... |
| bj-evalfun 33025 | The evaluation at a class ... |
| bj-evalfn 33026 | The evaluation at a class ... |
| bj-evalval 33027 | Value of the evaluation at... |
| bj-evalid 33028 | The evaluation at a set of... |
| bj-ndxarg 33029 | Proof of ~ ndxarg from ~ b... |
| bj-ndxid 33030 | Proof of ~ ndxid from ~ nd... |
| bj-evalidval 33031 | Closed general form of ~ s... |
| bj-rest00 33034 | An elementwise intersectio... |
| bj-restsn 33035 | An elementwise intersectio... |
| bj-restsnss 33036 | Special case of ~ bj-rests... |
| bj-restsnss2 33037 | Special case of ~ bj-rests... |
| bj-restsn0 33038 | An elementwise intersectio... |
| bj-restsn10 33039 | Special case of ~ bj-rests... |
| bj-restsnid 33040 | The elementwise intersecti... |
| bj-rest10 33041 | An elementwise intersectio... |
| bj-rest10b 33042 | Alternate version of ~ bj-... |
| bj-restn0 33043 | An elementwise intersectio... |
| bj-restn0b 33044 | Alternate version of ~ bj-... |
| bj-restpw 33045 | The elementwise intersecti... |
| bj-rest0 33046 | An elementwise intersectio... |
| bj-restb 33047 | An elementwise intersectio... |
| bj-restv 33048 | An elementwise intersectio... |
| bj-resta 33049 | An elementwise intersectio... |
| bj-restuni 33050 | The union of an elementwis... |
| bj-restuni2 33051 | The union of an elementwis... |
| bj-restreg 33052 | A reformulation of the axi... |
| bj-intss 33053 | A nonempty intersection of... |
| bj-raldifsn 33054 | All elements in a set sati... |
| bj-0int 33055 | If ` A ` is a collection o... |
| bj-mooreset 33056 | A Moore collection is a se... |
| bj-ismoore 33059 | Characterization of Moore ... |
| bj-ismoorec 33060 | Characterization of Moore ... |
| bj-ismoored0 33061 | Necessary condition to be ... |
| bj-ismoored 33062 | Necessary condition to be ... |
| bj-ismoored2 33063 | Necessary condition to be ... |
| bj-ismooredr 33064 | Sufficient condition to be... |
| bj-ismooredr2 33065 | Sufficient condition to be... |
| bj-discrmoore 33066 | The discrete Moore collect... |
| bj-0nmoore 33067 | The empty set is not a Moo... |
| bj-snmoore 33068 | A singleton is a Moore col... |
| bj-0nelmpt 33069 | The empty set is not an el... |
| bj-mptval 33070 | Value of a function given ... |
| bj-dfmpt2a 33071 | An equivalent definition o... |
| bj-mpt2mptALT 33072 | Alternate proof of ~ mpt2m... |
| bj-elid 33085 | Characterization of the el... |
| bj-elid2 33086 | Characterization of the el... |
| bj-elid3 33087 | Characterization of the el... |
| bj-diagval 33090 | Value of the diagonal. (C... |
| bj-eldiag 33091 | Characterization of the el... |
| bj-eldiag2 33092 | Characterization of the el... |
| bj-inftyexpiinv 33095 | Utility theorem for the in... |
| bj-inftyexpiinj 33096 | Injectivity of the paramet... |
| bj-inftyexpidisj 33097 | An element of the circle a... |
| bj-ccinftydisj 33100 | The circle at infinity is ... |
| bj-elccinfty 33101 | A lemma for infinite exten... |
| bj-ccssccbar 33104 | Complex numbers are extend... |
| bj-ccinftyssccbar 33105 | Infinite extended complex ... |
| bj-pinftyccb 33108 | The class ` pinfty ` is an... |
| bj-pinftynrr 33109 | The extended complex numbe... |
| bj-minftyccb 33112 | The class ` minfty ` is an... |
| bj-minftynrr 33113 | The extended complex numbe... |
| bj-pinftynminfty 33114 | The extended complex numbe... |
| bj-rrhatsscchat 33123 | The real projective line i... |
| bj-cmnssmnd 33136 | Commutative monoids are mo... |
| bj-cmnssmndel 33137 | Commutative monoids are mo... |
| bj-ablssgrp 33138 | Abelian groups are groups.... |
| bj-ablssgrpel 33139 | Abelian groups are groups ... |
| bj-ablsscmn 33140 | Abelian groups are commuta... |
| bj-ablsscmnel 33141 | Abelian groups are commuta... |
| bj-modssabl 33142 | (The additive groups of) m... |
| bj-vecssmod 33143 | Vector spaces are modules.... |
| bj-vecssmodel 33144 | Vector spaces are modules ... |
| bj-finsumval0 33147 | Value of a finite sum. (C... |
| bj-rrvecssvec 33150 | Real vector spaces are vec... |
| bj-rrvecssvecel 33151 | Real vector spaces are vec... |
| bj-rrvecsscmn 33152 | (The additive groups of) r... |
| bj-rrvecsscmnel 33153 | (The additive groups of) r... |
| bj-subcom 33154 | A consequence of commutati... |
| bj-ldiv 33155 | Left-division. (Contribut... |
| bj-rdiv 33156 | Right-division. (Contribu... |
| bj-mdiv 33157 | A division law. (Contribu... |
| bj-lineq 33158 | Solution of a (scalar) lin... |
| bj-lineqi 33159 | Solution of a (scalar) lin... |
| bj-bary1lem 33160 | A lemma for barycentric co... |
| bj-bary1lem1 33161 | Existence and uniqueness (... |
| bj-bary1 33162 | Barycentric coordinates in... |
| taupilem3 33165 | Lemma for tau-related theo... |
| taupilemrplb 33166 | A set of positive reals ha... |
| taupilem1 33167 | Lemma for ~ taupi . A pos... |
| taupilem2 33168 | Lemma for ~ taupi . The s... |
| taupi 33169 | Relationship between ` _ta... |
| dfgcd3 33170 | Alternate definition of th... |
| csbdif 33171 | Distribution of class subs... |
| csbpredg 33172 | Move class substitution in... |
| csbwrecsg 33173 | Move class substitution in... |
| csbrecsg 33174 | Move class substitution in... |
| csbrdgg 33175 | Move class substitution in... |
| csboprabg 33176 | Move class substitution in... |
| csbmpt22g 33177 | Move class substitution in... |
| mpnanrd 33178 | Eliminate the right side o... |
| con1bii2 33179 | A contraposition inference... |
| con2bii2 33180 | A contraposition inference... |
| vtoclefex 33181 | Implicit substitution of a... |
| rnmptsn 33182 | The range of a function ma... |
| f1omptsnlem 33183 | This is the core of the pr... |
| f1omptsn 33184 | A function mapping to sing... |
| mptsnunlem 33185 | This is the core of the pr... |
| mptsnun 33186 | A class ` B ` is equal to ... |
| dissneqlem 33187 | This is the core of the pr... |
| dissneq 33188 | Any topology that contains... |
| exlimim 33189 | Closed form of ~ exlimimd ... |
| exlimimd 33190 | Existential elimination ru... |
| exlimimdd 33191 | Existential elimination ru... |
| exellim 33192 | Closed form of ~ exellimdd... |
| exellimddv 33193 | Eliminate an antecedent wh... |
| topdifinfindis 33194 | Part of Exercise 3 of [Mun... |
| topdifinffinlem 33195 | This is the core of the pr... |
| topdifinffin 33196 | Part of Exercise 3 of [Mun... |
| topdifinf 33197 | Part of Exercise 3 of [Mun... |
| topdifinfeq 33198 | Two different ways of defi... |
| icorempt2 33199 | Closed-below, open-above i... |
| icoreresf 33200 | Closed-below, open-above i... |
| icoreval 33201 | Value of the closed-below,... |
| icoreelrnab 33202 | Elementhood in the set of ... |
| isbasisrelowllem1 33203 | Lemma for ~ isbasisrelowl ... |
| isbasisrelowllem2 33204 | Lemma for ~ isbasisrelowl ... |
| icoreclin 33205 | The set of closed-below, o... |
| isbasisrelowl 33206 | The set of all closed-belo... |
| icoreunrn 33207 | The union of all closed-be... |
| istoprelowl 33208 | The set of all closed-belo... |
| icoreelrn 33209 | A class abstraction which ... |
| iooelexlt 33210 | An element of an open inte... |
| relowlssretop 33211 | The lower limit topology o... |
| relowlpssretop 33212 | The lower limit topology o... |
| sucneqond 33213 | Inequality of an ordinal s... |
| sucneqoni 33214 | Inequality of an ordinal s... |
| onsucuni3 33215 | If an ordinal number has a... |
| 1oequni2o 33216 | The ordinal number ` 1o ` ... |
| rdgsucuni 33217 | If an ordinal number has a... |
| rdgeqoa 33218 | If a recursive function wi... |
| elxp8 33219 | Membership in a Cartesian ... |
| dffinxpf 33222 | This theorem is the same a... |
| finxpeq1 33223 | Equality theorem for Carte... |
| finxpeq2 33224 | Equality theorem for Carte... |
| csbfinxpg 33225 | Distribute proper substitu... |
| finxpreclem1 33226 | Lemma for ` ^^ ` recursion... |
| finxpreclem2 33227 | Lemma for ` ^^ ` recursion... |
| finxp0 33228 | The value of Cartesian exp... |
| finxp1o 33229 | The value of Cartesian exp... |
| finxpreclem3 33230 | Lemma for ` ^^ ` recursion... |
| finxpreclem4 33231 | Lemma for ` ^^ ` recursion... |
| finxpreclem5 33232 | Lemma for ` ^^ ` recursion... |
| finxpreclem6 33233 | Lemma for ` ^^ ` recursion... |
| finxpsuclem 33234 | Lemma for ~ finxpsuc . (C... |
| finxpsuc 33235 | The value of Cartesian exp... |
| finxp2o 33236 | The value of Cartesian exp... |
| finxp3o 33237 | The value of Cartesian exp... |
| finxpnom 33238 | Cartesian exponentiation w... |
| finxp00 33239 | Cartesian exponentiation o... |
| wl-section-prop 33240 | Intuitionistic logic is no... |
| wl-section-boot 33244 | In this section, I provide... |
| wl-imim1i 33245 | Inference adding common co... |
| wl-syl 33246 | An inference version of th... |
| wl-syl5 33247 | A syllogism rule of infere... |
| wl-pm2.18d 33248 | Deduction based on reducti... |
| wl-con4i 33249 | Inference rule. Copy of ~... |
| wl-pm2.24i 33250 | Inference rule. Copy of ~... |
| wl-a1i 33251 | Inference rule. Copy of ~... |
| wl-mpi 33252 | A nested modus ponens infe... |
| wl-imim2i 33253 | Inference adding common an... |
| wl-syl6 33254 | A syllogism rule of infere... |
| wl-ax3 33255 | ~ ax-3 proved from Lukasie... |
| wl-ax1 33256 | ~ ax-1 proved from Lukasie... |
| wl-pm2.27 33257 | This theorem, called "Asse... |
| wl-com12 33258 | Inference that swaps (comm... |
| wl-pm2.21 33259 | From a wff and its negatio... |
| wl-con1i 33260 | A contraposition inference... |
| wl-ja 33261 | Inference joining the ante... |
| wl-imim2 33262 | A closed form of syllogism... |
| wl-a1d 33263 | Deduction introducing an e... |
| wl-ax2 33264 | ~ ax-2 proved from Lukasie... |
| wl-id 33265 | Principle of identity. Th... |
| wl-notnotr 33266 | Converse of double negatio... |
| wl-pm2.04 33267 | Swap antecedents. Theorem... |
| wl-section-impchain 33268 | An implication like ` ( ps... |
| wl-impchain-mp-x 33269 | This series of theorems pr... |
| wl-impchain-mp-0 33270 | This theorem is the start ... |
| wl-impchain-mp-1 33271 | This theorem is in fact a ... |
| wl-impchain-mp-2 33272 | This theorem is in fact a ... |
| wl-impchain-com-1.x 33273 | It is often convenient to ... |
| wl-impchain-com-1.1 33274 | A degenerate form of antec... |
| wl-impchain-com-1.2 33275 | This theorem is in fact a ... |
| wl-impchain-com-1.3 33276 | This theorem is in fact a ... |
| wl-impchain-com-1.4 33277 | This theorem is in fact a ... |
| wl-impchain-com-n.m 33278 | This series of theorems al... |
| wl-impchain-com-2.3 33279 | This theorem is in fact a ... |
| wl-impchain-com-2.4 33280 | This theorem is in fact a ... |
| wl-impchain-com-3.2.1 33281 | This theorem is in fact a ... |
| wl-impchain-a1-x 33282 | If an implication chain is... |
| wl-impchain-a1-1 33283 | Inference rule, a copy of ... |
| wl-impchain-a1-2 33284 | Inference rule, a copy of ... |
| wl-impchain-a1-3 33285 | Inference rule, a copy of ... |
| wl-ax13lem1 33287 | A version of ~ ax-wl-13v w... |
| wl-jarri 33288 | Dropping a nested antecede... |
| wl-jarli 33289 | Dropping a nested conseque... |
| wl-mps 33290 | Replacing a nested consequ... |
| wl-syls1 33291 | Replacing a nested consequ... |
| wl-syls2 33292 | Replacing a nested anteced... |
| wl-embant 33293 | A true wff can always be a... |
| wl-orel12 33294 | In a conjunctive normal fo... |
| wl-cases2-dnf 33295 | A particular instance of ~... |
| wl-dfnan2 33296 | An alternative definition ... |
| wl-nancom 33297 | The 'nand' operator commut... |
| wl-nannan 33298 | Lemma for handling nested ... |
| wl-nannot 33299 | Show equivalence between n... |
| wl-nanbi1 33300 | Introduce a right anti-con... |
| wl-nanbi2 33301 | Introduce a left anti-conj... |
| wl-naev 33302 | If some set variables can ... |
| wl-hbae1 33303 | This specialization of ~ h... |
| wl-naevhba1v 33304 | An instance of ~ hbn1w app... |
| wl-hbnaev 33305 | Any variable is free in ` ... |
| wl-spae 33306 | Prove an instance of ~ sp ... |
| wl-cbv3vv 33307 | Avoiding ~ ax-11 . (Contr... |
| wl-speqv 33308 | Under the assumption ` -. ... |
| wl-19.8eqv 33309 | Under the assumption ` -. ... |
| wl-19.2reqv 33310 | Under the assumption ` -. ... |
| wl-dveeq12 33311 | The current form of ~ ax-1... |
| wl-nfalv 33312 | If ` x ` is not present in... |
| wl-nfimf1 33313 | An antecedent is irrelevan... |
| wl-nfnbi 33314 | Being free does not depend... |
| wl-nfae1 33315 | Unlike ~ nfae , this speci... |
| wl-nfnae1 33316 | Unlike ~ nfnae , this spec... |
| wl-aetr 33317 | A transitive law for varia... |
| wl-dral1d 33318 | A version of ~ dral1 with ... |
| wl-cbvalnaed 33319 | ~ wl-cbvalnae with a conte... |
| wl-cbvalnae 33320 | A more general version of ... |
| wl-exeq 33321 | The semantics of ` E. x y ... |
| wl-aleq 33322 | The semantics of ` A. x y ... |
| wl-nfeqfb 33323 | Extend ~ nfeqf to an equiv... |
| wl-nfs1t 33324 | If ` y ` is not free in ` ... |
| wl-equsald 33325 | Deduction version of ~ equ... |
| wl-equsal 33326 | A useful equivalence relat... |
| wl-equsal1t 33327 | The expression ` x = y ` i... |
| wl-equsalcom 33328 | This simple equivalence ea... |
| wl-equsal1i 33329 | The antecedent ` x = y ` i... |
| wl-sb6rft 33330 | A specialization of ~ wl-e... |
| wl-sbrimt 33331 | Substitution with a variab... |
| wl-sblimt 33332 | Substitution with a variab... |
| wl-sb8t 33333 | Substitution of variable i... |
| wl-sb8et 33334 | Substitution of variable i... |
| wl-sbhbt 33335 | Closed form of ~ sbhb . C... |
| wl-sbnf1 33336 | Two ways expressing that `... |
| wl-equsb3 33337 | ~ equsb3 with a distinctor... |
| wl-equsb4 33338 | Substitution applied to an... |
| wl-sb6nae 33339 | Version of ~ sb6 suitable ... |
| wl-sb5nae 33340 | Version of ~ sb5 suitable ... |
| wl-2sb6d 33341 | Version of ~ 2sb6 with a c... |
| wl-sbcom2d-lem1 33342 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d-lem2 33343 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d 33344 | Version of ~ sbcom2 with a... |
| wl-sbalnae 33345 | A theorem used in eliminat... |
| wl-sbal1 33346 | A theorem used in eliminat... |
| wl-sbal2 33347 | Move quantifier in and out... |
| wl-lem-exsb 33348 | This theorem provides a ba... |
| wl-lem-nexmo 33349 | This theorem provides a ba... |
| wl-lem-moexsb 33350 | The antecedent ` A. x ( ph... |
| wl-alanbii 33351 | This theorem extends ~ ala... |
| wl-mo2df 33352 | Version of ~ mo2 with a co... |
| wl-mo2tf 33353 | Closed form of ~ mo2 with ... |
| wl-eudf 33354 | Version of ~ df-eu with a ... |
| wl-eutf 33355 | Closed form of ~ df-eu wit... |
| wl-euequ1f 33356 | ~ euequ1 proved with a dis... |
| wl-mo2t 33357 | Closed form of ~ mo2 . (C... |
| wl-mo3t 33358 | Closed form of ~ mo3 . (C... |
| wl-sb8eut 33359 | Substitution of variable i... |
| wl-sb8mot 33360 | Substitution of variable i... |
| wl-ax11-lem1 33362 | A transitive law for varia... |
| wl-ax11-lem2 33363 | Lemma. (Contributed by Wo... |
| wl-ax11-lem3 33364 | Lemma. (Contributed by Wo... |
| wl-ax11-lem4 33365 | Lemma. (Contributed by Wo... |
| wl-ax11-lem5 33366 | Lemma. (Contributed by Wo... |
| wl-ax11-lem6 33367 | Lemma. (Contributed by Wo... |
| wl-ax11-lem7 33368 | Lemma. (Contributed by Wo... |
| wl-ax11-lem8 33369 | Lemma. (Contributed by Wo... |
| wl-ax11-lem9 33370 | The easy part when ` x ` c... |
| wl-ax11-lem10 33371 | We now have prepared every... |
| wl-sbcom3 33372 | Substituting ` y ` for ` x... |
| wel-wl 33374 | Redefine ` e. ` in a set c... |
| wel2-wl 33376 | Redefine ` e. ` in a set c... |
| wl-ax8clv1 33378 | Lifting the distinct varia... |
| wl-clelv2-just 33379 | Show that the definition ~... |
| wl-ax8clv2 33381 | Axiom ~ ax-wl-8cl carries ... |
| rabiun 33382 | Abstraction restricted to ... |
| iundif1 33383 | Indexed union of class dif... |
| imadifss 33384 | The difference of images i... |
| cureq 33385 | Equality theorem for curry... |
| unceq 33386 | Equality theorem for uncur... |
| curf 33387 | Functional property of cur... |
| uncf 33388 | Functional property of unc... |
| curfv 33389 | Value of currying. (Contr... |
| uncov 33390 | Value of uncurrying. (Con... |
| curunc 33391 | Currying of uncurrying. (... |
| unccur 33392 | Uncurrying of currying. (... |
| phpreu 33393 | Theorem related to pigeonh... |
| finixpnum 33394 | A finite Cartesian product... |
| fin2solem 33395 | Lemma for ~ fin2so . (Con... |
| fin2so 33396 | Any totally ordered Tarski... |
| ltflcei 33397 | Theorem to move the floor ... |
| leceifl 33398 | Theorem to move the floor ... |
| sin2h 33399 | Half-angle rule for sine. ... |
| cos2h 33400 | Half-angle rule for cosine... |
| tan2h 33401 | Half-angle rule for tangen... |
| pigt3 33402 | ` _pi ` is greater than 3.... |
| lindsdom 33403 | A linearly independent set... |
| lindsenlbs 33404 | A maximal linearly indepen... |
| matunitlindflem1 33405 | One direction of ~ matunit... |
| matunitlindflem2 33406 | One direction of ~ matunit... |
| matunitlindf 33407 | A matrix over a field is i... |
| ptrest 33408 | Expressing a restriction o... |
| ptrecube 33409 | Any point in an open set o... |
| poimirlem1 33410 | Lemma for ~ poimir - the v... |
| poimirlem2 33411 | Lemma for ~ poimir - conse... |
| poimirlem3 33412 | Lemma for ~ poimir to add ... |
| poimirlem4 33413 | Lemma for ~ poimir connect... |
| poimirlem5 33414 | Lemma for ~ poimir to esta... |
| poimirlem6 33415 | Lemma for ~ poimir establi... |
| poimirlem7 33416 | Lemma for ~ poimir , simil... |
| poimirlem8 33417 | Lemma for ~ poimir , estab... |
| poimirlem9 33418 | Lemma for ~ poimir , estab... |
| poimirlem10 33419 | Lemma for ~ poimir establi... |
| poimirlem11 33420 | Lemma for ~ poimir connect... |
| poimirlem12 33421 | Lemma for ~ poimir connect... |
| poimirlem13 33422 | Lemma for ~ poimir - for a... |
| poimirlem14 33423 | Lemma for ~ poimir - for a... |
| poimirlem15 33424 | Lemma for ~ poimir , that ... |
| poimirlem16 33425 | Lemma for ~ poimir establi... |
| poimirlem17 33426 | Lemma for ~ poimir establi... |
| poimirlem18 33427 | Lemma for ~ poimir stating... |
| poimirlem19 33428 | Lemma for ~ poimir establi... |
| poimirlem20 33429 | Lemma for ~ poimir establi... |
| poimirlem21 33430 | Lemma for ~ poimir stating... |
| poimirlem22 33431 | Lemma for ~ poimir , that ... |
| poimirlem23 33432 | Lemma for ~ poimir , two w... |
| poimirlem24 33433 | Lemma for ~ poimir , two w... |
| poimirlem25 33434 | Lemma for ~ poimir stating... |
| poimirlem26 33435 | Lemma for ~ poimir showing... |
| poimirlem27 33436 | Lemma for ~ poimir showing... |
| poimirlem28 33437 | Lemma for ~ poimir , a var... |
| poimirlem29 33438 | Lemma for ~ poimir connect... |
| poimirlem30 33439 | Lemma for ~ poimir combini... |
| poimirlem31 33440 | Lemma for ~ poimir , assig... |
| poimirlem32 33441 | Lemma for ~ poimir , combi... |
| poimir 33442 | Poincare-Miranda theorem. ... |
| broucube 33443 | Brouwer - or as Kulpa call... |
| heicant 33444 | Heine-Cantor theorem: a co... |
| opnmbllem0 33445 | Lemma for ~ ismblfin ; cou... |
| mblfinlem1 33446 | Lemma for ~ ismblfin , ord... |
| mblfinlem2 33447 | Lemma for ~ ismblfin , eff... |
| mblfinlem3 33448 | The difference between two... |
| mblfinlem4 33449 | Backward direction of ~ is... |
| ismblfin 33450 | Measurability in terms of ... |
| ovoliunnfl 33451 | ~ ovoliun is incompatible ... |
| ex-ovoliunnfl 33452 | Demonstration of ~ ovoliun... |
| voliunnfl 33453 | ~ voliun is incompatible w... |
| volsupnfl 33454 | ~ volsup is incompatible w... |
| 0mbf 33455 | The empty function is meas... |
| mbfresfi 33456 | Measurability of a piecewi... |
| mbfposadd 33457 | If the sum of two measurab... |
| cnambfre 33458 | A real-valued, a.e. contin... |
| dvtanlem 33459 | Lemma for ~ dvtan - the do... |
| dvtan 33460 | Derivative of tangent. (C... |
| itg2addnclem 33461 | An alternate expression fo... |
| itg2addnclem2 33462 | Lemma for ~ itg2addnc . T... |
| itg2addnclem3 33463 | Lemma incomprehensible in ... |
| itg2addnc 33464 | Alternate proof of ~ itg2a... |
| itg2gt0cn 33465 | ~ itg2gt0 holds on functio... |
| ibladdnclem 33466 | Lemma for ~ ibladdnc ; cf ... |
| ibladdnc 33467 | Choice-free analogue of ~ ... |
| itgaddnclem1 33468 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnclem2 33469 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnc 33470 | Choice-free analogue of ~ ... |
| iblsubnc 33471 | Choice-free analogue of ~ ... |
| itgsubnc 33472 | Choice-free analogue of ~ ... |
| iblabsnclem 33473 | Lemma for ~ iblabsnc ; cf.... |
| iblabsnc 33474 | Choice-free analogue of ~ ... |
| iblmulc2nc 33475 | Choice-free analogue of ~ ... |
| itgmulc2nclem1 33476 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nclem2 33477 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nc 33478 | Choice-free analogue of ~ ... |
| itgabsnc 33479 | Choice-free analogue of ~ ... |
| bddiblnc 33480 | Choice-free proof of ~ bdd... |
| cnicciblnc 33481 | Choice-free proof of ~ cni... |
| itggt0cn 33482 | ~ itggt0 holds for continu... |
| ftc1cnnclem 33483 | Lemma for ~ ftc1cnnc ; cf.... |
| ftc1cnnc 33484 | Choice-free proof of ~ ftc... |
| ftc1anclem1 33485 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem2 33486 | Lemma for ~ ftc1anc - rest... |
| ftc1anclem3 33487 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem4 33488 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem5 33489 | Lemma for ~ ftc1anc , the ... |
| ftc1anclem6 33490 | Lemma for ~ ftc1anc - cons... |
| ftc1anclem7 33491 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem8 33492 | Lemma for ~ ftc1anc . (Co... |
| ftc1anc 33493 | ~ ftc1a holds for function... |
| ftc2nc 33494 | Choice-free proof of ~ ftc... |
| asindmre 33495 | Real part of domain of dif... |
| dvasin 33496 | Derivative of arcsine. (C... |
| dvacos 33497 | Derivative of arccosine. ... |
| dvreasin 33498 | Real derivative of arcsine... |
| dvreacos 33499 | Real derivative of arccosi... |
| areacirclem1 33500 | Antiderivative of cross-se... |
| areacirclem2 33501 | Endpoint-inclusive continu... |
| areacirclem3 33502 | Integrability of cross-sec... |
| areacirclem4 33503 | Endpoint-inclusive continu... |
| areacirclem5 33504 | Finding the cross-section ... |
| areacirc 33505 | The area of a circle of ra... |
| anim12da 33506 | Conjoin antecedents and co... |
| unirep 33507 | Define a quantity whose de... |
| cover2 33508 | Two ways of expressing the... |
| cover2g 33509 | Two ways of expressing the... |
| brabg2 33510 | Relation by a binary relat... |
| opelopab3 33511 | Ordered pair membership in... |
| cocanfo 33512 | Cancellation of a surjecti... |
| brresi 33513 | Restriction of a binary re... |
| fnopabeqd 33514 | Equality deduction for fun... |
| fvopabf4g 33515 | Function value of an opera... |
| eqfnun 33516 | Two functions on ` A u. B ... |
| fnopabco 33517 | Composition of a function ... |
| opropabco 33518 | Composition of an operator... |
| f1opr 33519 | Condition for an operation... |
| cocnv 33520 | Composition with a functio... |
| f1ocan1fv 33521 | Cancel a composition by a ... |
| f1ocan2fv 33522 | Cancel a composition by th... |
| inixp 33523 | Intersection of Cartesian ... |
| upixp 33524 | Universal property of the ... |
| abrexdom 33525 | An indexed set is dominate... |
| abrexdom2 33526 | An indexed set is dominate... |
| ac6gf 33527 | Axiom of Choice. (Contrib... |
| indexa 33528 | If for every element of an... |
| indexdom 33529 | If for every element of an... |
| frinfm 33530 | A subset of a well-founded... |
| welb 33531 | A nonempty subset of a wel... |
| supex2g 33532 | Existence of supremum. (C... |
| supclt 33533 | Closure of supremum. (Con... |
| supubt 33534 | Upper bound property of su... |
| filbcmb 33535 | Combine a finite set of lo... |
| rdr 33536 | Two ways of expressing the... |
| fzmul 33537 | Membership of a product in... |
| sdclem2 33538 | Lemma for ~ sdc . (Contri... |
| sdclem1 33539 | Lemma for ~ sdc . (Contri... |
| sdc 33540 | Strong dependent choice. ... |
| fdc 33541 | Finite version of dependen... |
| fdc1 33542 | Variant of ~ fdc with no s... |
| seqpo 33543 | Two ways to say that a seq... |
| incsequz 33544 | An increasing sequence of ... |
| incsequz2 33545 | An increasing sequence of ... |
| nnubfi 33546 | A bounded above set of pos... |
| nninfnub 33547 | An infinite set of positiv... |
| subspopn 33548 | An open set is open in the... |
| neificl 33549 | Neighborhoods are closed u... |
| lpss2 33550 | Limit points of a subset a... |
| metf1o 33551 | Use a bijection with a met... |
| blssp 33552 | A ball in the subspace met... |
| mettrifi 33553 | Generalized triangle inequ... |
| lmclim2 33554 | A sequence in a metric spa... |
| geomcau 33555 | If the distance between co... |
| caures 33556 | The restriction of a Cauch... |
| caushft 33557 | A shifted Cauchy sequence ... |
| constcncf 33558 | A constant function is a c... |
| idcncf 33559 | The identity function is a... |
| sub1cncf 33560 | Subtracting a constant is ... |
| sub2cncf 33561 | Subtraction from a constan... |
| cnres2 33562 | The restriction of a conti... |
| cnresima 33563 | A continuous function is c... |
| cncfres 33564 | A continuous function on c... |
| istotbnd 33568 | The predicate "is a totall... |
| istotbnd2 33569 | The predicate "is a totall... |
| istotbnd3 33570 | A metric space is totally ... |
| totbndmet 33571 | The predicate "totally bou... |
| 0totbnd 33572 | The metric (there is only ... |
| sstotbnd2 33573 | Condition for a subset of ... |
| sstotbnd 33574 | Condition for a subset of ... |
| sstotbnd3 33575 | Use a net that is not nece... |
| totbndss 33576 | A subset of a totally boun... |
| equivtotbnd 33577 | If the metric ` M ` is "st... |
| isbnd 33579 | The predicate "is a bounde... |
| bndmet 33580 | A bounded metric space is ... |
| isbndx 33581 | A "bounded extended metric... |
| isbnd2 33582 | The predicate "is a bounde... |
| isbnd3 33583 | A metric space is bounded ... |
| isbnd3b 33584 | A metric space is bounded ... |
| bndss 33585 | A subset of a bounded metr... |
| blbnd 33586 | A ball is bounded. (Contr... |
| ssbnd 33587 | A subset of a metric space... |
| totbndbnd 33588 | A totally bounded metric s... |
| equivbnd 33589 | If the metric ` M ` is "st... |
| bnd2lem 33590 | Lemma for ~ equivbnd2 and ... |
| equivbnd2 33591 | If balls are totally bound... |
| prdsbnd 33592 | The product metric over fi... |
| prdstotbnd 33593 | The product metric over fi... |
| prdsbnd2 33594 | If balls are totally bound... |
| cntotbnd 33595 | A subset of the complex nu... |
| cnpwstotbnd 33596 | A subset of ` A ^ I ` , wh... |
| ismtyval 33599 | The set of isometries betw... |
| isismty 33600 | The condition "is an isome... |
| ismtycnv 33601 | The inverse of an isometry... |
| ismtyima 33602 | The image of a ball under ... |
| ismtyhmeolem 33603 | Lemma for ~ ismtyhmeo . (... |
| ismtyhmeo 33604 | An isometry is a homeomorp... |
| ismtybndlem 33605 | Lemma for ~ ismtybnd . (C... |
| ismtybnd 33606 | Isometries preserve bounde... |
| ismtyres 33607 | A restriction of an isomet... |
| heibor1lem 33608 | Lemma for ~ heibor1 . A c... |
| heibor1 33609 | One half of ~ heibor , tha... |
| heiborlem1 33610 | Lemma for ~ heibor . We w... |
| heiborlem2 33611 | Lemma for ~ heibor . Subs... |
| heiborlem3 33612 | Lemma for ~ heibor . Usin... |
| heiborlem4 33613 | Lemma for ~ heibor . Usin... |
| heiborlem5 33614 | Lemma for ~ heibor . The ... |
| heiborlem6 33615 | Lemma for ~ heibor . Sinc... |
| heiborlem7 33616 | Lemma for ~ heibor . Sinc... |
| heiborlem8 33617 | Lemma for ~ heibor . The ... |
| heiborlem9 33618 | Lemma for ~ heibor . Disc... |
| heiborlem10 33619 | Lemma for ~ heibor . The ... |
| heibor 33620 | Generalized Heine-Borel Th... |
| bfplem1 33621 | Lemma for ~ bfp . The seq... |
| bfplem2 33622 | Lemma for ~ bfp . Using t... |
| bfp 33623 | Banach fixed point theorem... |
| rrnval 33626 | The n-dimensional Euclidea... |
| rrnmval 33627 | The value of the Euclidean... |
| rrnmet 33628 | Euclidean space is a metri... |
| rrndstprj1 33629 | The distance between two p... |
| rrndstprj2 33630 | Bound on the distance betw... |
| rrncmslem 33631 | Lemma for ~ rrncms . (Con... |
| rrncms 33632 | Euclidean space is complet... |
| repwsmet 33633 | The supremum metric on ` R... |
| rrnequiv 33634 | The supremum metric on ` R... |
| rrntotbnd 33635 | A set in Euclidean space i... |
| rrnheibor 33636 | Heine-Borel theorem for Eu... |
| ismrer1 33637 | An isometry between ` RR `... |
| reheibor 33638 | Heine-Borel theorem for re... |
| iccbnd 33639 | A closed interval in ` RR ... |
| icccmpALT 33640 | A closed interval in ` RR ... |
| isass 33645 | The predicate "is an assoc... |
| isexid 33646 | The predicate ` G ` has a ... |
| ismgmOLD 33649 | Obsolete version of ~ ismg... |
| clmgmOLD 33650 | Obsolete version of ~ mgmc... |
| opidonOLD 33651 | Obsolete version of ~ mndp... |
| rngopidOLD 33652 | Obsolete version of ~ mndp... |
| opidon2OLD 33653 | Obsolete version of ~ mndp... |
| isexid2 33654 | If ` G e. ( Magma i^i ExId... |
| exidu1 33655 | Unicity of the left and ri... |
| idrval 33656 | The value of the identity ... |
| iorlid 33657 | A magma right and left ide... |
| cmpidelt 33658 | A magma right and left ide... |
| smgrpismgmOLD 33661 | Obsolete version of ~ sgrp... |
| issmgrpOLD 33662 | Obsolete version of ~ issg... |
| smgrpmgm 33663 | A semi-group is a magma. ... |
| smgrpassOLD 33664 | Obsolete version of ~ sgrp... |
| mndoissmgrpOLD 33667 | Obsolete version of ~ mnds... |
| mndoisexid 33668 | A monoid has an identity e... |
| mndoismgmOLD 33669 | Obsolete version of ~ mndm... |
| mndomgmid 33670 | A monoid is a magma with a... |
| ismndo 33671 | The predicate "is a monoid... |
| ismndo1 33672 | The predicate "is a monoid... |
| ismndo2 33673 | The predicate "is a monoid... |
| grpomndo 33674 | A group is a monoid. (Con... |
| exidcl 33675 | Closure of the binary oper... |
| exidreslem 33676 | Lemma for ~ exidres and ~ ... |
| exidres 33677 | The restriction of a binar... |
| exidresid 33678 | The restriction of a binar... |
| ablo4pnp 33679 | A commutative/associative ... |
| grpoeqdivid 33680 | Two group elements are equ... |
| grposnOLD 33681 | The group operation for th... |
| elghomlem1OLD 33684 | Obsolete as of 15-Mar-2020... |
| elghomlem2OLD 33685 | Obsolete as of 15-Mar-2020... |
| elghomOLD 33686 | Obsolete version of ~ isgh... |
| ghomlinOLD 33687 | Obsolete version of ~ ghml... |
| ghomidOLD 33688 | Obsolete version of ~ ghmi... |
| ghomf 33689 | Mapping property of a grou... |
| ghomco 33690 | The composition of two gro... |
| ghomdiv 33691 | Group homomorphisms preser... |
| grpokerinj 33692 | A group homomorphism is in... |
| relrngo 33695 | The class of all unital ri... |
| isrngo 33696 | The predicate "is a (unita... |
| isrngod 33697 | Conditions that determine ... |
| rngoi 33698 | The properties of a unital... |
| rngosm 33699 | Functionality of the multi... |
| rngocl 33700 | Closure of the multiplicat... |
| rngoid 33701 | The multiplication operati... |
| rngoideu 33702 | The unit element of a ring... |
| rngodi 33703 | Distributive law for the m... |
| rngodir 33704 | Distributive law for the m... |
| rngoass 33705 | Associative law for the mu... |
| rngo2 33706 | A ring element plus itself... |
| rngoablo 33707 | A ring's addition operatio... |
| rngoablo2 33708 | In a unital ring the addit... |
| rngogrpo 33709 | A ring's addition operatio... |
| rngone0 33710 | The base set of a ring is ... |
| rngogcl 33711 | Closure law for the additi... |
| rngocom 33712 | The addition operation of ... |
| rngoaass 33713 | The addition operation of ... |
| rngoa32 33714 | The addition operation of ... |
| rngoa4 33715 | Rearrangement of 4 terms i... |
| rngorcan 33716 | Right cancellation law for... |
| rngolcan 33717 | Left cancellation law for ... |
| rngo0cl 33718 | A ring has an additive ide... |
| rngo0rid 33719 | The additive identity of a... |
| rngo0lid 33720 | The additive identity of a... |
| rngolz 33721 | The zero of a unital ring ... |
| rngorz 33722 | The zero of a unital ring ... |
| rngosn3 33723 | Obsolete as of 25-Jan-2020... |
| rngosn4 33724 | Obsolete as of 25-Jan-2020... |
| rngosn6 33725 | Obsolete as of 25-Jan-2020... |
| rngonegcl 33726 | A ring is closed under neg... |
| rngoaddneg1 33727 | Adding the negative in a r... |
| rngoaddneg2 33728 | Adding the negative in a r... |
| rngosub 33729 | Subtraction in a ring, in ... |
| rngmgmbs4 33730 | The range of an internal o... |
| rngodm1dm2 33731 | In a unital ring the domai... |
| rngorn1 33732 | In a unital ring the range... |
| rngorn1eq 33733 | In a unital ring the range... |
| rngomndo 33734 | In a unital ring the multi... |
| rngoidmlem 33735 | The unit of a ring is an i... |
| rngolidm 33736 | The unit of a ring is an i... |
| rngoridm 33737 | The unit of a ring is an i... |
| rngo1cl 33738 | The unit of a ring belongs... |
| rngoueqz 33739 | Obsolete as of 23-Jan-2020... |
| rngonegmn1l 33740 | Negation in a ring is the ... |
| rngonegmn1r 33741 | Negation in a ring is the ... |
| rngoneglmul 33742 | Negation of a product in a... |
| rngonegrmul 33743 | Negation of a product in a... |
| rngosubdi 33744 | Ring multiplication distri... |
| rngosubdir 33745 | Ring multiplication distri... |
| zerdivemp1x 33746 | In a unitary ring a left i... |
| isdivrngo 33749 | The predicate "is a divisi... |
| drngoi 33750 | The properties of a divisi... |
| gidsn 33751 | Obsolete as of 23-Jan-2020... |
| zrdivrng 33752 | The zero ring is not a div... |
| dvrunz 33753 | In a division ring the uni... |
| isgrpda 33754 | Properties that determine ... |
| isdrngo1 33755 | The predicate "is a divisi... |
| divrngcl 33756 | The product of two nonzero... |
| isdrngo2 33757 | A division ring is a ring ... |
| isdrngo3 33758 | A division ring is a ring ... |
| rngohomval 33763 | The set of ring homomorphi... |
| isrngohom 33764 | The predicate "is a ring h... |
| rngohomf 33765 | A ring homomorphism is a f... |
| rngohomcl 33766 | Closure law for a ring hom... |
| rngohom1 33767 | A ring homomorphism preser... |
| rngohomadd 33768 | Ring homomorphisms preserv... |
| rngohommul 33769 | Ring homomorphisms preserv... |
| rngogrphom 33770 | A ring homomorphism is a g... |
| rngohom0 33771 | A ring homomorphism preser... |
| rngohomsub 33772 | Ring homomorphisms preserv... |
| rngohomco 33773 | The composition of two rin... |
| rngokerinj 33774 | A ring homomorphism is inj... |
| rngoisoval 33776 | The set of ring isomorphis... |
| isrngoiso 33777 | The predicate "is a ring i... |
| rngoiso1o 33778 | A ring isomorphism is a bi... |
| rngoisohom 33779 | A ring isomorphism is a ri... |
| rngoisocnv 33780 | The inverse of a ring isom... |
| rngoisoco 33781 | The composition of two rin... |
| isriscg 33783 | The ring isomorphism relat... |
| isrisc 33784 | The ring isomorphism relat... |
| risc 33785 | The ring isomorphism relat... |
| risci 33786 | Determine that two rings a... |
| riscer 33787 | Ring isomorphism is an equ... |
| iscom2 33794 | A device to add commutativ... |
| iscrngo 33795 | The predicate "is a commut... |
| iscrngo2 33796 | The predicate "is a commut... |
| iscringd 33797 | Conditions that determine ... |
| flddivrng 33798 | A field is a division ring... |
| crngorngo 33799 | A commutative ring is a ri... |
| crngocom 33800 | The multiplication operati... |
| crngm23 33801 | Commutative/associative la... |
| crngm4 33802 | Commutative/associative la... |
| fldcrng 33803 | A field is a commutative r... |
| isfld2 33804 | The predicate "is a field"... |
| crngohomfo 33805 | The image of a homomorphis... |
| idlval 33812 | The class of ideals of a r... |
| isidl 33813 | The predicate "is an ideal... |
| isidlc 33814 | The predicate "is an ideal... |
| idlss 33815 | An ideal of ` R ` is a sub... |
| idlcl 33816 | An element of an ideal is ... |
| idl0cl 33817 | An ideal contains ` 0 ` . ... |
| idladdcl 33818 | An ideal is closed under a... |
| idllmulcl 33819 | An ideal is closed under m... |
| idlrmulcl 33820 | An ideal is closed under m... |
| idlnegcl 33821 | An ideal is closed under n... |
| idlsubcl 33822 | An ideal is closed under s... |
| rngoidl 33823 | A ring ` R ` is an ` R ` i... |
| 0idl 33824 | The set containing only ` ... |
| 1idl 33825 | Two ways of expressing the... |
| 0rngo 33826 | In a ring, ` 0 = 1 ` iff t... |
| divrngidl 33827 | The only ideals in a divis... |
| intidl 33828 | The intersection of a none... |
| inidl 33829 | The intersection of two id... |
| unichnidl 33830 | The union of a nonempty ch... |
| keridl 33831 | The kernel of a ring homom... |
| pridlval 33832 | The class of prime ideals ... |
| ispridl 33833 | The predicate "is a prime ... |
| pridlidl 33834 | A prime ideal is an ideal.... |
| pridlnr 33835 | A prime ideal is a proper ... |
| pridl 33836 | The main property of a pri... |
| ispridl2 33837 | A condition that shows an ... |
| maxidlval 33838 | The set of maximal ideals ... |
| ismaxidl 33839 | The predicate "is a maxima... |
| maxidlidl 33840 | A maximal ideal is an idea... |
| maxidlnr 33841 | A maximal ideal is proper.... |
| maxidlmax 33842 | A maximal ideal is a maxim... |
| maxidln1 33843 | One is not contained in an... |
| maxidln0 33844 | A ring with a maximal idea... |
| isprrngo 33849 | The predicate "is a prime ... |
| prrngorngo 33850 | A prime ring is a ring. (... |
| smprngopr 33851 | A simple ring (one whose o... |
| divrngpr 33852 | A division ring is a prime... |
| isdmn 33853 | The predicate "is a domain... |
| isdmn2 33854 | The predicate "is a domain... |
| dmncrng 33855 | A domain is a commutative ... |
| dmnrngo 33856 | A domain is a ring. (Cont... |
| flddmn 33857 | A field is a domain. (Con... |
| igenval 33860 | The ideal generated by a s... |
| igenss 33861 | A set is a subset of the i... |
| igenidl 33862 | The ideal generated by a s... |
| igenmin 33863 | The ideal generated by a s... |
| igenidl2 33864 | The ideal generated by an ... |
| igenval2 33865 | The ideal generated by a s... |
| prnc 33866 | A principal ideal (an idea... |
| isfldidl 33867 | Determine if a ring is a f... |
| isfldidl2 33868 | Determine if a ring is a f... |
| ispridlc 33869 | The predicate "is a prime ... |
| pridlc 33870 | Property of a prime ideal ... |
| pridlc2 33871 | Property of a prime ideal ... |
| pridlc3 33872 | Property of a prime ideal ... |
| isdmn3 33873 | The predicate "is a domain... |
| dmnnzd 33874 | A domain has no zero-divis... |
| dmncan1 33875 | Cancellation law for domai... |
| dmncan2 33876 | Cancellation law for domai... |
| efald2 33877 | A proof by contradiction. ... |
| notbinot1 33878 | Simplification rule of neg... |
| bicontr 33879 | Biimplication of its own n... |
| impor 33880 | An equivalent formula for ... |
| orfa 33881 | The falsum ` F. ` can be r... |
| notbinot2 33882 | Commutation rule between n... |
| biimpor 33883 | A rewriting rule for biimp... |
| unitresl 33884 | A lemma for Conjunctive No... |
| unitresr 33885 | A lemma for Conjunctive No... |
| orfa1 33886 | Add a contradicting disjun... |
| orfa2 33887 | Remove a contradicting dis... |
| bifald 33888 | Infer the equivalence to a... |
| orsild 33889 | A lemma for not-or-not eli... |
| orsird 33890 | A lemma for not-or-not eli... |
| orcomdd 33891 | Commutativity of logic dis... |
| cnf1dd 33892 | A lemma for Conjunctive No... |
| cnf2dd 33893 | A lemma for Conjunctive No... |
| cnfn1dd 33894 | A lemma for Conjunctive No... |
| cnfn2dd 33895 | A lemma for Conjunctive No... |
| or32dd 33896 | A rearrangement of disjunc... |
| notornotel1 33897 | A lemma for not-or-not eli... |
| notornotel2 33898 | A lemma for not-or-not eli... |
| contrd 33899 | A proof by contradiction, ... |
| an12i 33900 | An inference from commutin... |
| exmid2 33901 | An excluded middle law. (... |
| selconj 33902 | An inference for selecting... |
| truconj 33903 | Add true as a conjunct. (... |
| orel 33904 | An inference for disjuncti... |
| negel 33905 | An inference for negation ... |
| botel 33906 | An inference for bottom el... |
| tradd 33907 | Add top ad a conjunct. (C... |
| sbtru 33908 | Substitution does not chan... |
| sbfal 33909 | Substitution does not chan... |
| sbcani 33910 | Distribution of class subs... |
| sbcori 33911 | Distribution of class subs... |
| sbcimi 33912 | Distribution of class subs... |
| sbceqi 33913 | Distribution of class subs... |
| sbcni 33914 | Move class substitution in... |
| sbali 33915 | Discard class substitution... |
| sbexi 33916 | Discard class substitution... |
| sbcalf 33917 | Move universal quantifier ... |
| sbcexf 33918 | Move existential quantifie... |
| sbcalfi 33919 | Move universal quantifier ... |
| sbcexfi 33920 | Move existential quantifie... |
| csbvargi 33921 | The proper substitution of... |
| csbconstgi 33922 | The proper substitution of... |
| spsbcdi 33923 | A lemma for eliminating a ... |
| alrimii 33924 | A lemma for introducing a ... |
| spesbcdi 33925 | A lemma for introducing an... |
| exlimddvf 33926 | A lemma for eliminating an... |
| exlimddvfi 33927 | A lemma for eliminating an... |
| sbceq1ddi 33928 | A lemma for eliminating in... |
| sbccom2lem 33929 | Lemma for ~ sbccom2 . (Co... |
| sbccom2 33930 | Commutative law for double... |
| sbccom2f 33931 | Commutative law for double... |
| sbccom2fi 33932 | Commutative law for double... |
| sbcgfi 33933 | Substitution for a variabl... |
| csbcom2fi 33934 | Commutative law for double... |
| csbgfi 33935 | Substitution for a variabl... |
| fald 33936 | Refutation of falsity, in ... |
| tsim1 33937 | A Tseitin axiom for logica... |
| tsim2 33938 | A Tseitin axiom for logica... |
| tsim3 33939 | A Tseitin axiom for logica... |
| tsbi1 33940 | A Tseitin axiom for logica... |
| tsbi2 33941 | A Tseitin axiom for logica... |
| tsbi3 33942 | A Tseitin axiom for logica... |
| tsbi4 33943 | A Tseitin axiom for logica... |
| tsxo1 33944 | A Tseitin axiom for logica... |
| tsxo2 33945 | A Tseitin axiom for logica... |
| tsxo3 33946 | A Tseitin axiom for logica... |
| tsxo4 33947 | A Tseitin axiom for logica... |
| tsan1 33948 | A Tseitin axiom for logica... |
| tsan2 33949 | A Tseitin axiom for logica... |
| tsan3 33950 | A Tseitin axiom for logica... |
| tsna1 33951 | A Tseitin axiom for logica... |
| tsna2 33952 | A Tseitin axiom for logica... |
| tsna3 33953 | A Tseitin axiom for logica... |
| tsor1 33954 | A Tseitin axiom for logica... |
| tsor2 33955 | A Tseitin axiom for logica... |
| tsor3 33956 | A Tseitin axiom for logica... |
| ts3an1 33957 | A Tseitin axiom for triple... |
| ts3an2 33958 | A Tseitin axiom for triple... |
| ts3an3 33959 | A Tseitin axiom for triple... |
| ts3or1 33960 | A Tseitin axiom for triple... |
| ts3or2 33961 | A Tseitin axiom for triple... |
| ts3or3 33962 | A Tseitin axiom for triple... |
| iuneq2f 33963 | Equality deduction for ind... |
| abeq12 33964 | Equality deduction for cla... |
| rabeq12f 33965 | Equality deduction for res... |
| csbeq12 33966 | Equality deduction for sub... |
| nfbii2 33967 | Equality deduction for not... |
| sbeqi 33968 | Equality deduction for sub... |
| ralbi12f 33969 | Equality deduction for res... |
| oprabbi 33970 | Equality deduction for cla... |
| mpt2bi123f 33971 | Equality deduction for map... |
| iuneq12f 33972 | Equality deduction for ind... |
| iineq12f 33973 | Equality deduction for ind... |
| opabbi 33974 | Equality deduction for cla... |
| mptbi12f 33975 | Equality deduction for map... |
| scottexf 33976 | A version of ~ scottex wit... |
| scott0f 33977 | A version of ~ scott0 with... |
| scottn0f 33978 | A version of ~ scott0f wit... |
| ac6s3f 33979 | Generalization of the Axio... |
| ac6s6 33980 | Generalization of the Axio... |
| ac6s6f 33981 | Generalization of the Axio... |
| elv 33983 | New way ( ~ elv , ~ el2v t... |
| el2v 33984 | New way ( ~ elv , ~ el2v t... |
| el2v1 33985 | New way ( ~ elv , ~ el2v t... |
| el2v2 33986 | New way ( ~ elv , ~ el2v t... |
| el3v 33987 | New way ( ~ elv , ~ el2v t... |
| el3v1 33988 | New way ( ~ elv , ~ el2v t... |
| el3v2 33989 | New way ( ~ elv , ~ el2v t... |
| el3v3 33990 | New way ( ~ elv , ~ el2v t... |
| el3v12 33991 | New way ( ~ elv , ~ el2v t... |
| el3v13 33992 | New way ( ~ elv , ~ el2v t... |
| el3v23 33993 | New way ( ~ elv , ~ el2v t... |
| biancom 33994 | Commuting conjunction in a... |
| biancomd 33995 | Commuting conjunction in a... |
| anbi1ci 33996 | Introduce a left and the s... |
| anbi1cd 33997 | Introduce a left and the s... |
| an2anr 33998 | Double commutation in conj... |
| anan 33999 | Multiple commutations in c... |
| triantru3 34000 | A wff is equivalent to its... |
| eqeltr 34001 | Substitution of equal clas... |
| eqelb 34002 | Substitution of equal clas... |
| eqeqan1d 34003 | Implication of introducing... |
| eqeqan2d 34004 | Implication of introducing... |
| ineqcom 34005 | Two ways of saying that tw... |
| ineqcomi 34006 | Disjointness inference (wh... |
| inres2 34007 | Two ways of expressing the... |
| ssbr 34008 | Subclass theorem for binar... |
| coideq 34009 | Equality theorem for compo... |
| ralanid 34010 | Cancellation law for restr... |
| nexmo 34011 | If there is no case where ... |
| 3albii 34012 | Inference adding three uni... |
| 3ralbii 34013 | Inference adding three res... |
| rabbii 34014 | Equivalent wff's correspon... |
| rabbieq 34015 | Equivalent wff's correspon... |
| rabimbieq 34016 | Restricted equivalent wff'... |
| abeqin 34017 | Intersection with class ab... |
| abeqinbi 34018 | Intersection with class ab... |
| rabeqel 34019 | Class element of a restric... |
| eqrelf 34020 | The equality connective be... |
| releleccnv 34021 | Elementhood in a converse ... |
| releccnveq 34022 | Equality of converse ` R `... |
| opelvvdif 34023 | Negated elementhood of ord... |
| vvdifopab 34024 | Ordered-pair class abstrac... |
| brvdif 34025 | Binary relation with unive... |
| brvdif2 34026 | Binary relation with unive... |
| brvvdif 34027 | Binary relation with the c... |
| brvbrvvdif 34028 | Binary relation with the c... |
| brcnvep 34029 | The converse of the binary... |
| elecALTV 34030 | Elementhood in the ` R ` -... |
| opelresALTV 34031 | Ordered pair elementhood i... |
| brresALTV 34032 | Binary relation on a restr... |
| brcnvepres 34033 | Restricted converse epsilo... |
| brinxp2ALTV 34034 | Intersection with cross pr... |
| brres2 34035 | Binary relation on a restr... |
| eldmres 34036 | Elementhood in the domain ... |
| eldm4 34037 | Elementhood in a domain. ... |
| eldmres2 34038 | Elementhood in the domain ... |
| eceq1i 34039 | Equality theorem for ` C `... |
| eceq2i 34040 | Equality theorem for the `... |
| eceq2d 34041 | Equality theorem for the `... |
| elecres 34042 | Elementhood in the restric... |
| ecres 34043 | Restricted coset of ` B ` ... |
| ecres2 34044 | The restricted coset of ` ... |
| eccnvepres 34045 | Restricted converse epsilo... |
| eleccnvep 34046 | Elementhood in the convers... |
| eccnvep 34047 | The converse epsilon coset... |
| extep 34048 | Property of epsilon relati... |
| eccnvepres2 34049 | The restricted converse ep... |
| eccnvepres3 34050 | Condition for a restricted... |
| eldmqsres 34051 | Elementhood in a restricte... |
| eldmqsres2 34052 | Elementhood in a restricte... |
| qsss1 34053 | Subclass theorem for quoti... |
| qseq1i 34054 | Equality theorem for quoti... |
| qseq1d 34055 | Equality theorem for quoti... |
| qseq2i 34056 | Equality theorem for quoti... |
| qseq2d 34057 | Equality theorem for quoti... |
| qseq12 34058 | Equality theorem for quoti... |
| brinxprnres 34059 | Binary relation on a restr... |
| inxprnres 34060 | Restriction of a class as ... |
| dfres4 34061 | Alternate definition of th... |
| exan3 34062 | Equivalent expressions wit... |
| exanres 34063 | Equivalent expressions wit... |
| exanres3 34064 | Equivalent expressions wit... |
| exanres2 34065 | Equivalent expressions wit... |
| cnvepres 34066 | Restricted converse epsilo... |
| ssrel3 34067 | Subclass relation in anoth... |
| eqrel2 34068 | Equality of relations. (C... |
| relinxp 34069 | Intersection with a Cartes... |
| rncnv 34070 | Range of converse is the d... |
| dfdm6 34071 | Alternate definition of do... |
| dfrn6 34072 | Alternate definition of ra... |
| rncnvepres 34073 | The range of the restricte... |
| dmecd 34074 | Equality of the coset of `... |
| dmec2d 34075 | Equality of the coset of `... |
| inxpssres 34076 | Intersection with a Cartes... |
| brid 34077 | Property of the identity b... |
| ideq2 34078 | For sets, the identity bin... |
| idresssidinxp 34079 | Condition for the identity... |
| idreseqidinxp 34080 | Condition for the identity... |
| extid 34081 | Property of identity relat... |
| inxpss 34082 | Two ways to say that an in... |
| idinxpss 34083 | Two ways to say that an in... |
| inxpss3 34084 | Two ways to say that an in... |
| inxpss2 34085 | Two ways to say that inter... |
| inxpssidinxp 34086 | Two ways to say that inter... |
| idinxpssinxp 34087 | Two ways to say that inter... |
| idinxpres 34088 | The intersection of the id... |
| idinxpssinxp2 34089 | Identity intersection with... |
| idinxpssinxp3 34090 | Identity intersection with... |
| idinxpssinxp4 34091 | Identity intersection with... |
| relcnveq3 34092 | Two ways of saying a relat... |
| relcnveq 34093 | Two ways of saying a relat... |
| relcnveq2 34094 | Two ways of saying a relat... |
| relcnveq4 34095 | Two ways of saying a relat... |
| qsresid 34096 | Simplification of a specia... |
| nel02 34097 | The empty set has no eleme... |
| n0elqs 34098 | Two ways of expressing tha... |
| n0elqs2 34099 | Two ways of expressing tha... |
| ecex2 34100 | Condition for a coset to b... |
| uniqsALTV 34101 | The union of a quotient se... |
| rnresequniqs 34102 | The range of a restriction... |
| n0el2 34103 | Two ways of expressing tha... |
| cnvepresex 34104 | Sethood condition for the ... |
| inex2ALTV 34105 | Sethood condition for the ... |
| inex3 34106 | More general sethood condi... |
| inxpex 34107 | Sethood condition for the ... |
| eqres 34108 | Converting a class constan... |
| opidORIG 34109 | Please delete when ~ opidg... |
| opideq 34110 | Equality conditions for or... |
| opelinxp 34111 | Ordered pair element in an... |
| iss2 34112 | A subclass of the identity... |
| eldmcnv 34113 | Elementhood in a domain of... |
| dfrel5 34114 | Alternate definition of th... |
| dfrel6 34115 | Alternate definition of th... |
| cnvresrn 34116 | Converse restricted to ran... |
| ecin0 34117 | Two ways of saying that th... |
| ecinn0 34118 | Two ways of saying that th... |
| ineleq 34119 | Lemma for ~ inecmo . (Con... |
| inecmo 34120 | Lemma for ~~? dfeldisj5 (v... |
| inecmo2 34121 | Lemma for ~~? dfeldisj5 , ... |
| ineccnvmo 34122 | Lemma for ~ ineccnvmo2 . ... |
| alrmomo 34123 | Lemma for ~ ineccnvmo2 . ... |
| alrmomo2 34124 | Lemma for ~ inecmo3 . (Co... |
| ineccnvmo2 34125 | Lemma for ~~? dffunsALTV5 ... |
| inecmo3 34126 | Lemma for ~~? dfdisjs5 , ~... |
| motr 34127 | Lemma for ~~? trcoss . (C... |
| bropabid 34128 | Lemma for ~~? inxptxp . (... |
| inxp2 34129 | Intersection with a Cartes... |
| opabssi 34130 | Lemma for ~ opabf . (Cont... |
| opabf 34131 | A class abstraction of a c... |
| ec0 34132 | The empty-coset of a class... |
| 0qs 34133 | Quotient set with the empt... |
| xrnss3v 34135 | A range Cartesian product ... |
| xrnrel 34136 | A range Cartesian product ... |
| prtlem60 34137 | Lemma for ~ prter3 . (Con... |
| bicomdd 34138 | Commute two sides of a bic... |
| jca2r 34139 | Inference conjoining the c... |
| jca3 34140 | Inference conjoining the c... |
| prtlem70 34141 | Lemma for ~ prter3 : a rea... |
| ibdr 34142 | Reverse of ~ ibd . (Contr... |
| pm5.31r 34143 | Variant of ~ pm5.31 . (Co... |
| prtlem100 34144 | Lemma for ~ prter3 . (Con... |
| prtlem5 34145 | Lemma for ~ prter1 , ~ prt... |
| prtlem80 34146 | Lemma for ~ prter2 . (Con... |
| brabsb2 34147 | A closed form of ~ brabsb ... |
| eqbrrdv2 34148 | Other version of ~ eqbrrdi... |
| prtlem9 34149 | Lemma for ~ prter3 . (Con... |
| prtlem10 34150 | Lemma for ~ prter3 . (Con... |
| prtlem11 34151 | Lemma for ~ prter2 . (Con... |
| prtlem12 34152 | Lemma for ~ prtex and ~ pr... |
| prtlem13 34153 | Lemma for ~ prter1 , ~ prt... |
| prtlem16 34154 | Lemma for ~ prtex , ~ prte... |
| prtlem400 34155 | Lemma for ~ prter2 and als... |
| erprt 34158 | The quotient set of an equ... |
| prtlem14 34159 | Lemma for ~ prter1 , ~ prt... |
| prtlem15 34160 | Lemma for ~ prter1 and ~ p... |
| prtlem17 34161 | Lemma for ~ prter2 . (Con... |
| prtlem18 34162 | Lemma for ~ prter2 . (Con... |
| prtlem19 34163 | Lemma for ~ prter2 . (Con... |
| prter1 34164 | Every partition generates ... |
| prtex 34165 | The equivalence relation g... |
| prter2 34166 | The quotient set of the eq... |
| prter3 34167 | For every partition there ... |
| axc5 34178 | This theorem repeats ~ sp ... |
| ax4fromc4 34179 | Rederivation of axiom ~ ax... |
| ax10fromc7 34180 | Rederivation of axiom ~ ax... |
| ax6fromc10 34181 | Rederivation of axiom ~ ax... |
| hba1-o 34182 | The setvar ` x ` is not fr... |
| axc4i-o 34183 | Inference version of ~ ax-... |
| equid1 34184 | Proof of ~ equid from our ... |
| equcomi1 34185 | Proof of ~ equcomi from ~ ... |
| aecom-o 34186 | Commutation law for identi... |
| aecoms-o 34187 | A commutation rule for ide... |
| hbae-o 34188 | All variables are effectiv... |
| dral1-o 34189 | Formula-building lemma for... |
| ax12fromc15 34190 | Rederivation of axiom ~ ax... |
| ax13fromc9 34191 | Derive ~ ax-13 from ~ ax-c... |
| ax5ALT 34192 | Axiom to quantify a variab... |
| sps-o 34193 | Generalization of antecede... |
| hbequid 34194 | Bound-variable hypothesis ... |
| nfequid-o 34195 | Bound-variable hypothesis ... |
| axc5c7 34196 | Proof of a single axiom th... |
| axc5c7toc5 34197 | Rederivation of ~ ax-c5 fr... |
| axc5c7toc7 34198 | Rederivation of ~ ax-c7 fr... |
| axc711 34199 | Proof of a single axiom th... |
| nfa1-o 34200 | ` x ` is not free in ` A. ... |
| axc711toc7 34201 | Rederivation of ~ ax-c7 fr... |
| axc711to11 34202 | Rederivation of ~ ax-11 fr... |
| axc5c711 34203 | Proof of a single axiom th... |
| axc5c711toc5 34204 | Rederivation of ~ ax-c5 fr... |
| axc5c711toc7 34205 | Rederivation of ~ ax-c7 fr... |
| axc5c711to11 34206 | Rederivation of ~ ax-11 fr... |
| equidqe 34207 | ~ equid with existential q... |
| axc5sp1 34208 | A special case of ~ ax-c5 ... |
| equidq 34209 | ~ equid with universal qua... |
| equid1ALT 34210 | Alternate proof of ~ equid... |
| axc11nfromc11 34211 | Rederivation of ~ ax-c11n ... |
| naecoms-o 34212 | A commutation rule for dis... |
| hbnae-o 34213 | All variables are effectiv... |
| dvelimf-o 34214 | Proof of ~ dvelimh that us... |
| dral2-o 34215 | Formula-building lemma for... |
| aev-o 34216 | A "distinctor elimination"... |
| ax5eq 34217 | Theorem to add distinct qu... |
| dveeq2-o 34218 | Quantifier introduction wh... |
| axc16g-o 34219 | A generalization of axiom ... |
| dveeq1-o 34220 | Quantifier introduction wh... |
| dveeq1-o16 34221 | Version of ~ dveeq1 using ... |
| ax5el 34222 | Theorem to add distinct qu... |
| axc11n-16 34223 | This theorem shows that, g... |
| dveel2ALT 34224 | Alternate proof of ~ dveel... |
| ax12f 34225 | Basis step for constructin... |
| ax12eq 34226 | Basis step for constructin... |
| ax12el 34227 | Basis step for constructin... |
| ax12indn 34228 | Induction step for constru... |
| ax12indi 34229 | Induction step for constru... |
| ax12indalem 34230 | Lemma for ~ ax12inda2 and ... |
| ax12inda2ALT 34231 | Alternate proof of ~ ax12i... |
| ax12inda2 34232 | Induction step for constru... |
| ax12inda 34233 | Induction step for constru... |
| ax12v2-o 34234 | Rederivation of ~ ax-c15 f... |
| ax12a2-o 34235 | Derive ~ ax-c15 from a hyp... |
| axc11-o 34236 | Show that ~ ax-c11 can be ... |
| fsumshftd 34237 | Index shift of a finite su... |
| fsumshftdOLD 34238 | Obsolete version of ~ fsum... |
| riotaclbgBAD 34240 | Closure of restricted iota... |
| riotaclbBAD 34241 | Closure of restricted iota... |
| riotasvd 34242 | Deduction version of ~ rio... |
| riotasv2d 34243 | Value of description binde... |
| riotasv2s 34244 | The value of description b... |
| riotasv 34245 | Value of description binde... |
| riotasv3d 34246 | A property ` ch ` holding ... |
| elimhyps 34247 | A version of ~ elimhyp usi... |
| dedths 34248 | A version of weak deductio... |
| renegclALT 34249 | Closure law for negative o... |
| elimhyps2 34250 | Generalization of ~ elimhy... |
| dedths2 34251 | Generalization of ~ dedths... |
| 19.9alt 34252 | Version of ~ 19.9t for uni... |
| nfcxfrdf 34253 | A utility lemma to transfe... |
| nfded 34254 | A deduction theorem that c... |
| nfded2 34255 | A deduction theorem that c... |
| nfunidALT2 34256 | Deduction version of ~ nfu... |
| nfunidALT 34257 | Deduction version of ~ nfu... |
| nfopdALT 34258 | Deduction version of bound... |
| cnaddcom 34259 | Recover the commutative la... |
| toycom 34260 | Show the commutative law f... |
| lshpset 34265 | The set of all hyperplanes... |
| islshp 34266 | The predicate "is a hyperp... |
| islshpsm 34267 | Hyperplane properties expr... |
| lshplss 34268 | A hyperplane is a subspace... |
| lshpne 34269 | A hyperplane is not equal ... |
| lshpnel 34270 | A hyperplane's generating ... |
| lshpnelb 34271 | The subspace sum of a hype... |
| lshpnel2N 34272 | Condition that determines ... |
| lshpne0 34273 | The member of the span in ... |
| lshpdisj 34274 | A hyperplane and the span ... |
| lshpcmp 34275 | If two hyperplanes are com... |
| lshpinN 34276 | The intersection of two di... |
| lsatset 34277 | The set of all 1-dim subsp... |
| islsat 34278 | The predicate "is a 1-dim ... |
| lsatlspsn2 34279 | The span of a nonzero sing... |
| lsatlspsn 34280 | The span of a nonzero sing... |
| islsati 34281 | A 1-dim subspace (atom) (o... |
| lsateln0 34282 | A 1-dim subspace (atom) (o... |
| lsatlss 34283 | The set of 1-dim subspaces... |
| lsatlssel 34284 | An atom is a subspace. (C... |
| lsatssv 34285 | An atom is a set of vector... |
| lsatn0 34286 | A 1-dim subspace (atom) of... |
| lsatspn0 34287 | The span of a vector is an... |
| lsator0sp 34288 | The span of a vector is ei... |
| lsatssn0 34289 | A subspace (or any class) ... |
| lsatcmp 34290 | If two atoms are comparabl... |
| lsatcmp2 34291 | If an atom is included in ... |
| lsatel 34292 | A nonzero vector in an ato... |
| lsatelbN 34293 | A nonzero vector in an ato... |
| lsat2el 34294 | Two atoms sharing a nonzer... |
| lsmsat 34295 | Convert comparison of atom... |
| lsatfixedN 34296 | Show equality with the spa... |
| lsmsatcv 34297 | Subspace sum has the cover... |
| lssatomic 34298 | The lattice of subspaces i... |
| lssats 34299 | The lattice of subspaces i... |
| lpssat 34300 | Two subspaces in a proper ... |
| lrelat 34301 | Subspaces are relatively a... |
| lssatle 34302 | The ordering of two subspa... |
| lssat 34303 | Two subspaces in a proper ... |
| islshpat 34304 | Hyperplane properties expr... |
| lcvfbr 34307 | The covers relation for a ... |
| lcvbr 34308 | The covers relation for a ... |
| lcvbr2 34309 | The covers relation for a ... |
| lcvbr3 34310 | The covers relation for a ... |
| lcvpss 34311 | The covers relation implie... |
| lcvnbtwn 34312 | The covers relation implie... |
| lcvntr 34313 | The covers relation is not... |
| lcvnbtwn2 34314 | The covers relation implie... |
| lcvnbtwn3 34315 | The covers relation implie... |
| lsmcv2 34316 | Subspace sum has the cover... |
| lcvat 34317 | If a subspace covers anoth... |
| lsatcv0 34318 | An atom covers the zero su... |
| lsatcveq0 34319 | A subspace covered by an a... |
| lsat0cv 34320 | A subspace is an atom iff ... |
| lcvexchlem1 34321 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem2 34322 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem3 34323 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem4 34324 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem5 34325 | Lemma for ~ lcvexch . (Co... |
| lcvexch 34326 | Subspaces satisfy the exch... |
| lcvp 34327 | Covering property of Defin... |
| lcv1 34328 | Covering property of a sub... |
| lcv2 34329 | Covering property of a sub... |
| lsatexch 34330 | The atom exchange property... |
| lsatnle 34331 | The meet of a subspace and... |
| lsatnem0 34332 | The meet of distinct atoms... |
| lsatexch1 34333 | The atom exch1ange propert... |
| lsatcv0eq 34334 | If the sum of two atoms co... |
| lsatcv1 34335 | Two atoms covering the zer... |
| lsatcvatlem 34336 | Lemma for ~ lsatcvat . (C... |
| lsatcvat 34337 | A nonzero subspace less th... |
| lsatcvat2 34338 | A subspace covered by the ... |
| lsatcvat3 34339 | A condition implying that ... |
| islshpcv 34340 | Hyperplane properties expr... |
| l1cvpat 34341 | A subspace covered by the ... |
| l1cvat 34342 | Create an atom under an el... |
| lshpat 34343 | Create an atom under a hyp... |
| lflset 34346 | The set of linear function... |
| islfl 34347 | The predicate "is a linear... |
| lfli 34348 | Property of a linear funct... |
| islfld 34349 | Properties that determine ... |
| lflf 34350 | A linear functional is a f... |
| lflcl 34351 | A linear functional value ... |
| lfl0 34352 | A linear functional is zer... |
| lfladd 34353 | Property of a linear funct... |
| lflsub 34354 | Property of a linear funct... |
| lflmul 34355 | Property of a linear funct... |
| lfl0f 34356 | The zero function is a fun... |
| lfl1 34357 | A nonzero functional has a... |
| lfladdcl 34358 | Closure of addition of two... |
| lfladdcom 34359 | Commutativity of functiona... |
| lfladdass 34360 | Associativity of functiona... |
| lfladd0l 34361 | Functional addition with t... |
| lflnegcl 34362 | Closure of the negative of... |
| lflnegl 34363 | A functional plus its nega... |
| lflvscl 34364 | Closure of a scalar produc... |
| lflvsdi1 34365 | Distributive law for (righ... |
| lflvsdi2 34366 | Reverse distributive law f... |
| lflvsdi2a 34367 | Reverse distributive law f... |
| lflvsass 34368 | Associative law for (right... |
| lfl0sc 34369 | The (right vector space) s... |
| lflsc0N 34370 | The scalar product with th... |
| lfl1sc 34371 | The (right vector space) s... |
| lkrfval 34374 | The kernel of a functional... |
| lkrval 34375 | Value of the kernel of a f... |
| ellkr 34376 | Membership in the kernel o... |
| lkrval2 34377 | Value of the kernel of a f... |
| ellkr2 34378 | Membership in the kernel o... |
| lkrcl 34379 | A member of the kernel of ... |
| lkrf0 34380 | The value of a functional ... |
| lkr0f 34381 | The kernel of the zero fun... |
| lkrlss 34382 | The kernel of a linear fun... |
| lkrssv 34383 | The kernel of a linear fun... |
| lkrsc 34384 | The kernel of a nonzero sc... |
| lkrscss 34385 | The kernel of a scalar pro... |
| eqlkr 34386 | Two functionals with the s... |
| eqlkr2 34387 | Two functionals with the s... |
| eqlkr3 34388 | Two functionals with the s... |
| lkrlsp 34389 | The subspace sum of a kern... |
| lkrlsp2 34390 | The subspace sum of a kern... |
| lkrlsp3 34391 | The subspace sum of a kern... |
| lkrshp 34392 | The kernel of a nonzero fu... |
| lkrshp3 34393 | The kernels of nonzero fun... |
| lkrshpor 34394 | The kernel of a functional... |
| lkrshp4 34395 | A kernel is a hyperplane i... |
| lshpsmreu 34396 | Lemma for ~ lshpkrex . Sh... |
| lshpkrlem1 34397 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem2 34398 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem3 34399 | Lemma for ~ lshpkrex . De... |
| lshpkrlem4 34400 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem5 34401 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem6 34402 | Lemma for ~ lshpkrex . Sh... |
| lshpkrcl 34403 | The set ` G ` defined by h... |
| lshpkr 34404 | The kernel of functional `... |
| lshpkrex 34405 | There exists a functional ... |
| lshpset2N 34406 | The set of all hyperplanes... |
| islshpkrN 34407 | The predicate "is a hyperp... |
| lfl1dim 34408 | Equivalent expressions for... |
| lfl1dim2N 34409 | Equivalent expressions for... |
| ldualset 34412 | Define the (left) dual of ... |
| ldualvbase 34413 | The vectors of a dual spac... |
| ldualelvbase 34414 | Utility theorem for conver... |
| ldualfvadd 34415 | Vector addition in the dua... |
| ldualvadd 34416 | Vector addition in the dua... |
| ldualvaddcl 34417 | The value of vector additi... |
| ldualvaddval 34418 | The value of the value of ... |
| ldualsca 34419 | The ring of scalars of the... |
| ldualsbase 34420 | Base set of scalar ring fo... |
| ldualsaddN 34421 | Scalar addition for the du... |
| ldualsmul 34422 | Scalar multiplication for ... |
| ldualfvs 34423 | Scalar product operation f... |
| ldualvs 34424 | Scalar product operation v... |
| ldualvsval 34425 | Value of scalar product op... |
| ldualvscl 34426 | The scalar product operati... |
| ldualvaddcom 34427 | Commutative law for vector... |
| ldualvsass 34428 | Associative law for scalar... |
| ldualvsass2 34429 | Associative law for scalar... |
| ldualvsdi1 34430 | Distributive law for scala... |
| ldualvsdi2 34431 | Reverse distributive law f... |
| ldualgrplem 34432 | Lemma for ~ ldualgrp . (C... |
| ldualgrp 34433 | The dual of a vector space... |
| ldual0 34434 | The zero scalar of the dua... |
| ldual1 34435 | The unit scalar of the dua... |
| ldualneg 34436 | The negative of a scalar o... |
| ldual0v 34437 | The zero vector of the dua... |
| ldual0vcl 34438 | The dual zero vector is a ... |
| lduallmodlem 34439 | Lemma for ~ lduallmod . (... |
| lduallmod 34440 | The dual of a left module ... |
| lduallvec 34441 | The dual of a left vector ... |
| ldualvsub 34442 | The value of vector subtra... |
| ldualvsubcl 34443 | Closure of vector subtract... |
| ldualvsubval 34444 | The value of the value of ... |
| ldualssvscl 34445 | Closure of scalar product ... |
| ldualssvsubcl 34446 | Closure of vector subtract... |
| ldual0vs 34447 | Scalar zero times a functi... |
| lkr0f2 34448 | The kernel of the zero fun... |
| lduallkr3 34449 | The kernels of nonzero fun... |
| lkrpssN 34450 | Proper subset relation bet... |
| lkrin 34451 | Intersection of the kernel... |
| eqlkr4 34452 | Two functionals with the s... |
| ldual1dim 34453 | Equivalent expressions for... |
| ldualkrsc 34454 | The kernel of a nonzero sc... |
| lkrss 34455 | The kernel of a scalar pro... |
| lkrss2N 34456 | Two functionals with kerne... |
| lkreqN 34457 | Proportional functionals h... |
| lkrlspeqN 34458 | Condition for colinear fun... |
| isopos 34467 | The predicate "is an ortho... |
| opposet 34468 | Every orthoposet is a pose... |
| oposlem 34469 | Lemma for orthoposet prope... |
| op01dm 34470 | Conditions necessary for z... |
| op0cl 34471 | An orthoposet has a zero e... |
| op1cl 34472 | An orthoposet has a unit e... |
| op0le 34473 | Orthoposet zero is less th... |
| ople0 34474 | An element less than or eq... |
| opnlen0 34475 | An element not less than a... |
| lub0N 34476 | The least upper bound of t... |
| opltn0 34477 | A lattice element greater ... |
| ople1 34478 | Any element is less than t... |
| op1le 34479 | If the orthoposet unit is ... |
| glb0N 34480 | The greatest lower bound o... |
| opoccl 34481 | Closure of orthocomplement... |
| opococ 34482 | Double negative law for or... |
| opcon3b 34483 | Contraposition law for ort... |
| opcon2b 34484 | Orthocomplement contraposi... |
| opcon1b 34485 | Orthocomplement contraposi... |
| oplecon3 34486 | Contraposition law for ort... |
| oplecon3b 34487 | Contraposition law for ort... |
| oplecon1b 34488 | Contraposition law for str... |
| opoc1 34489 | Orthocomplement of orthopo... |
| opoc0 34490 | Orthocomplement of orthopo... |
| opltcon3b 34491 | Contraposition law for str... |
| opltcon1b 34492 | Contraposition law for str... |
| opltcon2b 34493 | Contraposition law for str... |
| opexmid 34494 | Law of excluded middle for... |
| opnoncon 34495 | Law of contradiction for o... |
| riotaocN 34496 | The orthocomplement of the... |
| cmtfvalN 34497 | Value of commutes relation... |
| cmtvalN 34498 | Equivalence for commutes r... |
| isolat 34499 | The predicate "is an ortho... |
| ollat 34500 | An ortholattice is a latti... |
| olop 34501 | An ortholattice is an orth... |
| olposN 34502 | An ortholattice is a poset... |
| isolatiN 34503 | Properties that determine ... |
| oldmm1 34504 | De Morgan's law for meet i... |
| oldmm2 34505 | De Morgan's law for meet i... |
| oldmm3N 34506 | De Morgan's law for meet i... |
| oldmm4 34507 | De Morgan's law for meet i... |
| oldmj1 34508 | De Morgan's law for join i... |
| oldmj2 34509 | De Morgan's law for join i... |
| oldmj3 34510 | De Morgan's law for join i... |
| oldmj4 34511 | De Morgan's law for join i... |
| olj01 34512 | An ortholattice element jo... |
| olj02 34513 | An ortholattice element jo... |
| olm11 34514 | The meet of an ortholattic... |
| olm12 34515 | The meet of an ortholattic... |
| latmassOLD 34516 | Ortholattice meet is assoc... |
| latm12 34517 | A rearrangement of lattice... |
| latm32 34518 | A rearrangement of lattice... |
| latmrot 34519 | Rotate lattice meet of 3 c... |
| latm4 34520 | Rearrangement of lattice m... |
| latmmdiN 34521 | Lattice meet distributes o... |
| latmmdir 34522 | Lattice meet distributes o... |
| olm01 34523 | Meet with lattice zero is ... |
| olm02 34524 | Meet with lattice zero is ... |
| isoml 34525 | The predicate "is an ortho... |
| isomliN 34526 | Properties that determine ... |
| omlol 34527 | An orthomodular lattice is... |
| omlop 34528 | An orthomodular lattice is... |
| omllat 34529 | An orthomodular lattice is... |
| omllaw 34530 | The orthomodular law. (Co... |
| omllaw2N 34531 | Variation of orthomodular ... |
| omllaw3 34532 | Orthomodular law equivalen... |
| omllaw4 34533 | Orthomodular law equivalen... |
| omllaw5N 34534 | The orthomodular law. Rem... |
| cmtcomlemN 34535 | Lemma for ~ cmtcomN . ( ~... |
| cmtcomN 34536 | Commutation is symmetric. ... |
| cmt2N 34537 | Commutation with orthocomp... |
| cmt3N 34538 | Commutation with orthocomp... |
| cmt4N 34539 | Commutation with orthocomp... |
| cmtbr2N 34540 | Alternate definition of th... |
| cmtbr3N 34541 | Alternate definition for t... |
| cmtbr4N 34542 | Alternate definition for t... |
| lecmtN 34543 | Ordered elements commute. ... |
| cmtidN 34544 | Any element commutes with ... |
| omlfh1N 34545 | Foulis-Holland Theorem, pa... |
| omlfh3N 34546 | Foulis-Holland Theorem, pa... |
| omlmod1i2N 34547 | Analogue of modular law ~ ... |
| omlspjN 34548 | Contraction of a Sasaki pr... |
| cvrfval 34555 | Value of covers relation "... |
| cvrval 34556 | Binary relation expressing... |
| cvrlt 34557 | The covers relation implie... |
| cvrnbtwn 34558 | There is no element betwee... |
| ncvr1 34559 | No element covers the latt... |
| cvrletrN 34560 | Property of an element abo... |
| cvrval2 34561 | Binary relation expressing... |
| cvrnbtwn2 34562 | The covers relation implie... |
| cvrnbtwn3 34563 | The covers relation implie... |
| cvrcon3b 34564 | Contraposition law for the... |
| cvrle 34565 | The covers relation implie... |
| cvrnbtwn4 34566 | The covers relation implie... |
| cvrnle 34567 | The covers relation implie... |
| cvrne 34568 | The covers relation implie... |
| cvrnrefN 34569 | The covers relation is not... |
| cvrcmp 34570 | If two lattice elements th... |
| cvrcmp2 34571 | If two lattice elements co... |
| pats 34572 | The set of atoms in a pose... |
| isat 34573 | The predicate "is an atom"... |
| isat2 34574 | The predicate "is an atom"... |
| atcvr0 34575 | An atom covers zero. ( ~ ... |
| atbase 34576 | An atom is a member of the... |
| atssbase 34577 | The set of atoms is a subs... |
| 0ltat 34578 | An atom is greater than ze... |
| leatb 34579 | A poset element less than ... |
| leat 34580 | A poset element less than ... |
| leat2 34581 | A nonzero poset element le... |
| leat3 34582 | A poset element less than ... |
| meetat 34583 | The meet of any element wi... |
| meetat2 34584 | The meet of any element wi... |
| isatl 34586 | The predicate "is an atomi... |
| atllat 34587 | An atomic lattice is a lat... |
| atlpos 34588 | An atomic lattice is a pos... |
| atl0dm 34589 | Condition necessary for ze... |
| atl0cl 34590 | An atomic lattice has a ze... |
| atl0le 34591 | Orthoposet zero is less th... |
| atlle0 34592 | An element less than or eq... |
| atlltn0 34593 | A lattice element greater ... |
| isat3 34594 | The predicate "is an atom"... |
| atn0 34595 | An atom is not zero. ( ~ ... |
| atnle0 34596 | An atom is not less than o... |
| atlen0 34597 | A lattice element is nonze... |
| atcmp 34598 | If two atoms are comparabl... |
| atncmp 34599 | Frequently-used variation ... |
| atnlt 34600 | Two atoms cannot satisfy t... |
| atcvreq0 34601 | An element covered by an a... |
| atncvrN 34602 | Two atoms cannot satisfy t... |
| atlex 34603 | Every nonzero element of a... |
| atnle 34604 | Two ways of expressing "an... |
| atnem0 34605 | The meet of distinct atoms... |
| atlatmstc 34606 | An atomic, complete, ortho... |
| atlatle 34607 | The ordering of two Hilber... |
| atlrelat1 34608 | An atomistic lattice with ... |
| iscvlat 34610 | The predicate "is an atomi... |
| iscvlat2N 34611 | The predicate "is an atomi... |
| cvlatl 34612 | An atomic lattice with the... |
| cvllat 34613 | An atomic lattice with the... |
| cvlposN 34614 | An atomic lattice with the... |
| cvlexch1 34615 | An atomic covering lattice... |
| cvlexch2 34616 | An atomic covering lattice... |
| cvlexchb1 34617 | An atomic covering lattice... |
| cvlexchb2 34618 | An atomic covering lattice... |
| cvlexch3 34619 | An atomic covering lattice... |
| cvlexch4N 34620 | An atomic covering lattice... |
| cvlatexchb1 34621 | A version of ~ cvlexchb1 f... |
| cvlatexchb2 34622 | A version of ~ cvlexchb2 f... |
| cvlatexch1 34623 | Atom exchange property. (... |
| cvlatexch2 34624 | Atom exchange property. (... |
| cvlatexch3 34625 | Atom exchange property. (... |
| cvlcvr1 34626 | The covering property. Pr... |
| cvlcvrp 34627 | A Hilbert lattice satisfie... |
| cvlatcvr1 34628 | An atom is covered by its ... |
| cvlatcvr2 34629 | An atom is covered by its ... |
| cvlsupr2 34630 | Two equivalent ways of exp... |
| cvlsupr3 34631 | Two equivalent ways of exp... |
| cvlsupr4 34632 | Consequence of superpositi... |
| cvlsupr5 34633 | Consequence of superpositi... |
| cvlsupr6 34634 | Consequence of superpositi... |
| cvlsupr7 34635 | Consequence of superpositi... |
| cvlsupr8 34636 | Consequence of superpositi... |
| ishlat1 34639 | The predicate "is a Hilber... |
| ishlat2 34640 | The predicate "is a Hilber... |
| ishlat3N 34641 | The predicate "is a Hilber... |
| ishlatiN 34642 | Properties that determine ... |
| hlomcmcv 34643 | A Hilbert lattice is ortho... |
| hloml 34644 | A Hilbert lattice is ortho... |
| hlclat 34645 | A Hilbert lattice is compl... |
| hlcvl 34646 | A Hilbert lattice is an at... |
| hlatl 34647 | A Hilbert lattice is atomi... |
| hlol 34648 | A Hilbert lattice is an or... |
| hlop 34649 | A Hilbert lattice is an or... |
| hllat 34650 | A Hilbert lattice is a lat... |
| hlomcmat 34651 | A Hilbert lattice is ortho... |
| hlpos 34652 | A Hilbert lattice is a pos... |
| hlatjcl 34653 | Closure of join operation.... |
| hlatjcom 34654 | Commutatitivity of join op... |
| hlatjidm 34655 | Idempotence of join operat... |
| hlatjass 34656 | Lattice join is associativ... |
| hlatj12 34657 | Swap 1st and 2nd members o... |
| hlatj32 34658 | Swap 2nd and 3rd members o... |
| hlatjrot 34659 | Rotate lattice join of 3 c... |
| hlatj4 34660 | Rearrangement of lattice j... |
| hlatlej1 34661 | A join's first argument is... |
| hlatlej2 34662 | A join's second argument i... |
| glbconN 34663 | De Morgan's law for GLB an... |
| glbconxN 34664 | De Morgan's law for GLB an... |
| atnlej1 34665 | If an atom is not less tha... |
| atnlej2 34666 | If an atom is not less tha... |
| hlsuprexch 34667 | A Hilbert lattice has the ... |
| hlexch1 34668 | A Hilbert lattice has the ... |
| hlexch2 34669 | A Hilbert lattice has the ... |
| hlexchb1 34670 | A Hilbert lattice has the ... |
| hlexchb2 34671 | A Hilbert lattice has the ... |
| hlsupr 34672 | A Hilbert lattice has the ... |
| hlsupr2 34673 | A Hilbert lattice has the ... |
| hlhgt4 34674 | A Hilbert lattice has a he... |
| hlhgt2 34675 | A Hilbert lattice has a he... |
| hl0lt1N 34676 | Lattice 0 is less than lat... |
| hlexch3 34677 | A Hilbert lattice has the ... |
| hlexch4N 34678 | A Hilbert lattice has the ... |
| hlatexchb1 34679 | A version of ~ hlexchb1 fo... |
| hlatexchb2 34680 | A version of ~ hlexchb2 fo... |
| hlatexch1 34681 | Atom exchange property. (... |
| hlatexch2 34682 | Atom exchange property. (... |
| hlatmstcOLDN 34683 | An atomic, complete, ortho... |
| hlatle 34684 | The ordering of two Hilber... |
| hlateq 34685 | The equality of two Hilber... |
| hlrelat1 34686 | An atomistic lattice with ... |
| hlrelat5N 34687 | An atomistic lattice with ... |
| hlrelat 34688 | A Hilbert lattice is relat... |
| hlrelat2 34689 | A consequence of relative ... |
| exatleN 34690 | A condition for an atom to... |
| hl2at 34691 | A Hilbert lattice has at l... |
| atex 34692 | At least one atom exists. ... |
| intnatN 34693 | If the intersection with a... |
| 2llnne2N 34694 | Condition implying that tw... |
| 2llnneN 34695 | Condition implying that tw... |
| cvr1 34696 | A Hilbert lattice has the ... |
| cvr2N 34697 | Less-than and covers equiv... |
| hlrelat3 34698 | The Hilbert lattice is rel... |
| cvrval3 34699 | Binary relation expressing... |
| cvrval4N 34700 | Binary relation expressing... |
| cvrval5 34701 | Binary relation expressing... |
| cvrp 34702 | A Hilbert lattice satisfie... |
| atcvr1 34703 | An atom is covered by its ... |
| atcvr2 34704 | An atom is covered by its ... |
| cvrexchlem 34705 | Lemma for ~ cvrexch . ( ~... |
| cvrexch 34706 | A Hilbert lattice satisfie... |
| cvratlem 34707 | Lemma for ~ cvrat . ( ~ a... |
| cvrat 34708 | A nonzero Hilbert lattice ... |
| ltltncvr 34709 | A chained strong ordering ... |
| ltcvrntr 34710 | Non-transitive condition f... |
| cvrntr 34711 | The covers relation is not... |
| atcvr0eq 34712 | The covers relation is not... |
| lnnat 34713 | A line (the join of two di... |
| atcvrj0 34714 | Two atoms covering the zer... |
| cvrat2 34715 | A Hilbert lattice element ... |
| atcvrneN 34716 | Inequality derived from at... |
| atcvrj1 34717 | Condition for an atom to b... |
| atcvrj2b 34718 | Condition for an atom to b... |
| atcvrj2 34719 | Condition for an atom to b... |
| atleneN 34720 | Inequality derived from at... |
| atltcvr 34721 | An equivalence of less-tha... |
| atle 34722 | Any nonzero element has an... |
| atlt 34723 | Two atoms are unequal iff ... |
| atlelt 34724 | Transfer less-than relatio... |
| 2atlt 34725 | Given an atom less than an... |
| atexchcvrN 34726 | Atom exchange property. V... |
| atexchltN 34727 | Atom exchange property. V... |
| cvrat3 34728 | A condition implying that ... |
| cvrat4 34729 | A condition implying exist... |
| cvrat42 34730 | Commuted version of ~ cvra... |
| 2atjm 34731 | The meet of a line (expres... |
| atbtwn 34732 | Property of a 3rd atom ` R... |
| atbtwnexOLDN 34733 | There exists a 3rd atom ` ... |
| atbtwnex 34734 | Given atoms ` P ` in ` X `... |
| 3noncolr2 34735 | Two ways to express 3 non-... |
| 3noncolr1N 34736 | Two ways to express 3 non-... |
| hlatcon3 34737 | Atom exchange combined wit... |
| hlatcon2 34738 | Atom exchange combined wit... |
| 4noncolr3 34739 | A way to express 4 non-col... |
| 4noncolr2 34740 | A way to express 4 non-col... |
| 4noncolr1 34741 | A way to express 4 non-col... |
| athgt 34742 | A Hilbert lattice, whose h... |
| 3dim0 34743 | There exists a 3-dimension... |
| 3dimlem1 34744 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem2 34745 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3a 34746 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem3 34747 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3OLDN 34748 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4a 34749 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem4 34750 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4OLDN 34751 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1lem5 34752 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1 34753 | Construct a 3-dimensional ... |
| 3dim2 34754 | Construct 2 new layers on ... |
| 3dim3 34755 | Construct a new layer on t... |
| 2dim 34756 | Generate a height-3 elemen... |
| 1dimN 34757 | An atom is covered by a he... |
| 1cvrco 34758 | The orthocomplement of an ... |
| 1cvratex 34759 | There exists an atom less ... |
| 1cvratlt 34760 | An atom less than or equal... |
| 1cvrjat 34761 | An element covered by the ... |
| 1cvrat 34762 | Create an atom under an el... |
| ps-1 34763 | The join of two atoms ` R ... |
| ps-2 34764 | Lattice analogue for the p... |
| 2atjlej 34765 | Two atoms are different if... |
| hlatexch3N 34766 | Rearrange join of atoms in... |
| hlatexch4 34767 | Exchange 2 atoms. (Contri... |
| ps-2b 34768 | Variation of projective ge... |
| 3atlem1 34769 | Lemma for ~ 3at . (Contri... |
| 3atlem2 34770 | Lemma for ~ 3at . (Contri... |
| 3atlem3 34771 | Lemma for ~ 3at . (Contri... |
| 3atlem4 34772 | Lemma for ~ 3at . (Contri... |
| 3atlem5 34773 | Lemma for ~ 3at . (Contri... |
| 3atlem6 34774 | Lemma for ~ 3at . (Contri... |
| 3atlem7 34775 | Lemma for ~ 3at . (Contri... |
| 3at 34776 | Any three non-colinear ato... |
| llnset 34791 | The set of lattice lines i... |
| islln 34792 | The predicate "is a lattic... |
| islln4 34793 | The predicate "is a lattic... |
| llni 34794 | Condition implying a latti... |
| llnbase 34795 | A lattice line is a lattic... |
| islln3 34796 | The predicate "is a lattic... |
| islln2 34797 | The predicate "is a lattic... |
| llni2 34798 | The join of two different ... |
| llnnleat 34799 | An atom cannot majorize a ... |
| llnneat 34800 | A lattice line is not an a... |
| 2atneat 34801 | The join of two distinct a... |
| llnn0 34802 | A lattice line is nonzero.... |
| islln2a 34803 | The predicate "is a lattic... |
| llnle 34804 | Any element greater than 0... |
| atcvrlln2 34805 | An atom under a line is co... |
| atcvrlln 34806 | An element covering an ato... |
| llnexatN 34807 | Given an atom on a line, t... |
| llncmp 34808 | If two lattice lines are c... |
| llnnlt 34809 | Two lattice lines cannot s... |
| 2llnmat 34810 | Two intersecting lines int... |
| 2at0mat0 34811 | Special case of ~ 2atmat0 ... |
| 2atmat0 34812 | The meet of two unequal li... |
| 2atm 34813 | An atom majorized by two d... |
| ps-2c 34814 | Variation of projective ge... |
| lplnset 34815 | The set of lattice planes ... |
| islpln 34816 | The predicate "is a lattic... |
| islpln4 34817 | The predicate "is a lattic... |
| lplni 34818 | Condition implying a latti... |
| islpln3 34819 | The predicate "is a lattic... |
| lplnbase 34820 | A lattice plane is a latti... |
| islpln5 34821 | The predicate "is a lattic... |
| islpln2 34822 | The predicate "is a lattic... |
| lplni2 34823 | The join of 3 different at... |
| lvolex3N 34824 | There is an atom outside o... |
| llnmlplnN 34825 | The intersection of a line... |
| lplnle 34826 | Any element greater than 0... |
| lplnnle2at 34827 | A lattice line (or atom) c... |
| lplnnleat 34828 | A lattice plane cannot maj... |
| lplnnlelln 34829 | A lattice plane is not les... |
| 2atnelpln 34830 | The join of two atoms is n... |
| lplnneat 34831 | No lattice plane is an ato... |
| lplnnelln 34832 | No lattice plane is a latt... |
| lplnn0N 34833 | A lattice plane is nonzero... |
| islpln2a 34834 | The predicate "is a lattic... |
| islpln2ah 34835 | The predicate "is a lattic... |
| lplnriaN 34836 | Property of a lattice plan... |
| lplnribN 34837 | Property of a lattice plan... |
| lplnric 34838 | Property of a lattice plan... |
| lplnri1 34839 | Property of a lattice plan... |
| lplnri2N 34840 | Property of a lattice plan... |
| lplnri3N 34841 | Property of a lattice plan... |
| lplnllnneN 34842 | Two lattice lines defined ... |
| llncvrlpln2 34843 | A lattice line under a lat... |
| llncvrlpln 34844 | An element covering a latt... |
| 2lplnmN 34845 | If the join of two lattice... |
| 2llnmj 34846 | The meet of two lattice li... |
| 2atmat 34847 | The meet of two intersecti... |
| lplncmp 34848 | If two lattice planes are ... |
| lplnexatN 34849 | Given a lattice line on a ... |
| lplnexllnN 34850 | Given an atom on a lattice... |
| lplnnlt 34851 | Two lattice planes cannot ... |
| 2llnjaN 34852 | The join of two different ... |
| 2llnjN 34853 | The join of two different ... |
| 2llnm2N 34854 | The meet of two different ... |
| 2llnm3N 34855 | Two lattice lines in a lat... |
| 2llnm4 34856 | Two lattice lines that maj... |
| 2llnmeqat 34857 | An atom equals the interse... |
| lvolset 34858 | The set of 3-dim lattice v... |
| islvol 34859 | The predicate "is a 3-dim ... |
| islvol4 34860 | The predicate "is a 3-dim ... |
| lvoli 34861 | Condition implying a 3-dim... |
| islvol3 34862 | The predicate "is a 3-dim ... |
| lvoli3 34863 | Condition implying a 3-dim... |
| lvolbase 34864 | A 3-dim lattice volume is ... |
| islvol5 34865 | The predicate "is a 3-dim ... |
| islvol2 34866 | The predicate "is a 3-dim ... |
| lvoli2 34867 | The join of 4 different at... |
| lvolnle3at 34868 | A lattice plane (or lattic... |
| lvolnleat 34869 | An atom cannot majorize a ... |
| lvolnlelln 34870 | A lattice line cannot majo... |
| lvolnlelpln 34871 | A lattice plane cannot maj... |
| 3atnelvolN 34872 | The join of 3 atoms is not... |
| 2atnelvolN 34873 | The join of two atoms is n... |
| lvolneatN 34874 | No lattice volume is an at... |
| lvolnelln 34875 | No lattice volume is a lat... |
| lvolnelpln 34876 | No lattice volume is a lat... |
| lvoln0N 34877 | A lattice volume is nonzer... |
| islvol2aN 34878 | The predicate "is a lattic... |
| 4atlem0a 34879 | Lemma for ~ 4at . (Contri... |
| 4atlem0ae 34880 | Lemma for ~ 4at . (Contri... |
| 4atlem0be 34881 | Lemma for ~ 4at . (Contri... |
| 4atlem3 34882 | Lemma for ~ 4at . Break i... |
| 4atlem3a 34883 | Lemma for ~ 4at . Break i... |
| 4atlem3b 34884 | Lemma for ~ 4at . Break i... |
| 4atlem4a 34885 | Lemma for ~ 4at . Frequen... |
| 4atlem4b 34886 | Lemma for ~ 4at . Frequen... |
| 4atlem4c 34887 | Lemma for ~ 4at . Frequen... |
| 4atlem4d 34888 | Lemma for ~ 4at . Frequen... |
| 4atlem9 34889 | Lemma for ~ 4at . Substit... |
| 4atlem10a 34890 | Lemma for ~ 4at . Substit... |
| 4atlem10b 34891 | Lemma for ~ 4at . Substit... |
| 4atlem10 34892 | Lemma for ~ 4at . Combine... |
| 4atlem11a 34893 | Lemma for ~ 4at . Substit... |
| 4atlem11b 34894 | Lemma for ~ 4at . Substit... |
| 4atlem11 34895 | Lemma for ~ 4at . Combine... |
| 4atlem12a 34896 | Lemma for ~ 4at . Substit... |
| 4atlem12b 34897 | Lemma for ~ 4at . Substit... |
| 4atlem12 34898 | Lemma for ~ 4at . Combine... |
| 4at 34899 | Four atoms determine a lat... |
| 4at2 34900 | Four atoms determine a lat... |
| lplncvrlvol2 34901 | A lattice line under a lat... |
| lplncvrlvol 34902 | An element covering a latt... |
| lvolcmp 34903 | If two lattice planes are ... |
| lvolnltN 34904 | Two lattice volumes cannot... |
| 2lplnja 34905 | The join of two different ... |
| 2lplnj 34906 | The join of two different ... |
| 2lplnm2N 34907 | The meet of two different ... |
| 2lplnmj 34908 | The meet of two lattice pl... |
| dalemkehl 34909 | Lemma for ~ dath . Freque... |
| dalemkelat 34910 | Lemma for ~ dath . Freque... |
| dalemkeop 34911 | Lemma for ~ dath . Freque... |
| dalempea 34912 | Lemma for ~ dath . Freque... |
| dalemqea 34913 | Lemma for ~ dath . Freque... |
| dalemrea 34914 | Lemma for ~ dath . Freque... |
| dalemsea 34915 | Lemma for ~ dath . Freque... |
| dalemtea 34916 | Lemma for ~ dath . Freque... |
| dalemuea 34917 | Lemma for ~ dath . Freque... |
| dalemyeo 34918 | Lemma for ~ dath . Freque... |
| dalemzeo 34919 | Lemma for ~ dath . Freque... |
| dalemclpjs 34920 | Lemma for ~ dath . Freque... |
| dalemclqjt 34921 | Lemma for ~ dath . Freque... |
| dalemclrju 34922 | Lemma for ~ dath . Freque... |
| dalem-clpjq 34923 | Lemma for ~ dath . Freque... |
| dalemceb 34924 | Lemma for ~ dath . Freque... |
| dalempeb 34925 | Lemma for ~ dath . Freque... |
| dalemqeb 34926 | Lemma for ~ dath . Freque... |
| dalemreb 34927 | Lemma for ~ dath . Freque... |
| dalemseb 34928 | Lemma for ~ dath . Freque... |
| dalemteb 34929 | Lemma for ~ dath . Freque... |
| dalemueb 34930 | Lemma for ~ dath . Freque... |
| dalempjqeb 34931 | Lemma for ~ dath . Freque... |
| dalemsjteb 34932 | Lemma for ~ dath . Freque... |
| dalemtjueb 34933 | Lemma for ~ dath . Freque... |
| dalemqrprot 34934 | Lemma for ~ dath . Freque... |
| dalemyeb 34935 | Lemma for ~ dath . Freque... |
| dalemcnes 34936 | Lemma for ~ dath . Freque... |
| dalempnes 34937 | Lemma for ~ dath . Freque... |
| dalemqnet 34938 | Lemma for ~ dath . Freque... |
| dalempjsen 34939 | Lemma for ~ dath . Freque... |
| dalemply 34940 | Lemma for ~ dath . Freque... |
| dalemsly 34941 | Lemma for ~ dath . Freque... |
| dalemswapyz 34942 | Lemma for ~ dath . Swap t... |
| dalemrot 34943 | Lemma for ~ dath . Rotate... |
| dalemrotyz 34944 | Lemma for ~ dath . Rotate... |
| dalem1 34945 | Lemma for ~ dath . Show t... |
| dalemcea 34946 | Lemma for ~ dath . Freque... |
| dalem2 34947 | Lemma for ~ dath . Show t... |
| dalemdea 34948 | Lemma for ~ dath . Freque... |
| dalemeea 34949 | Lemma for ~ dath . Freque... |
| dalem3 34950 | Lemma for ~ dalemdnee . (... |
| dalem4 34951 | Lemma for ~ dalemdnee . (... |
| dalemdnee 34952 | Lemma for ~ dath . Axis o... |
| dalem5 34953 | Lemma for ~ dath . Atom `... |
| dalem6 34954 | Lemma for ~ dath . Analog... |
| dalem7 34955 | Lemma for ~ dath . Analog... |
| dalem8 34956 | Lemma for ~ dath . Plane ... |
| dalem-cly 34957 | Lemma for ~ dalem9 . Cent... |
| dalem9 34958 | Lemma for ~ dath . Since ... |
| dalem10 34959 | Lemma for ~ dath . Atom `... |
| dalem11 34960 | Lemma for ~ dath . Analog... |
| dalem12 34961 | Lemma for ~ dath . Analog... |
| dalem13 34962 | Lemma for ~ dalem14 . (Co... |
| dalem14 34963 | Lemma for ~ dath . Planes... |
| dalem15 34964 | Lemma for ~ dath . The ax... |
| dalem16 34965 | Lemma for ~ dath . The at... |
| dalem17 34966 | Lemma for ~ dath . When p... |
| dalem18 34967 | Lemma for ~ dath . Show t... |
| dalem19 34968 | Lemma for ~ dath . Show t... |
| dalemccea 34969 | Lemma for ~ dath . Freque... |
| dalemddea 34970 | Lemma for ~ dath . Freque... |
| dalem-ccly 34971 | Lemma for ~ dath . Freque... |
| dalem-ddly 34972 | Lemma for ~ dath . Freque... |
| dalemccnedd 34973 | Lemma for ~ dath . Freque... |
| dalemclccjdd 34974 | Lemma for ~ dath . Freque... |
| dalemcceb 34975 | Lemma for ~ dath . Freque... |
| dalemswapyzps 34976 | Lemma for ~ dath . Swap t... |
| dalemrotps 34977 | Lemma for ~ dath . Rotate... |
| dalemcjden 34978 | Lemma for ~ dath . Show t... |
| dalem20 34979 | Lemma for ~ dath . Show t... |
| dalem21 34980 | Lemma for ~ dath . Show t... |
| dalem22 34981 | Lemma for ~ dath . Show t... |
| dalem23 34982 | Lemma for ~ dath . Show t... |
| dalem24 34983 | Lemma for ~ dath . Show t... |
| dalem25 34984 | Lemma for ~ dath . Show t... |
| dalem27 34985 | Lemma for ~ dath . Show t... |
| dalem28 34986 | Lemma for ~ dath . Lemma ... |
| dalem29 34987 | Lemma for ~ dath . Analog... |
| dalem30 34988 | Lemma for ~ dath . Analog... |
| dalem31N 34989 | Lemma for ~ dath . Analog... |
| dalem32 34990 | Lemma for ~ dath . Analog... |
| dalem33 34991 | Lemma for ~ dath . Analog... |
| dalem34 34992 | Lemma for ~ dath . Analog... |
| dalem35 34993 | Lemma for ~ dath . Analog... |
| dalem36 34994 | Lemma for ~ dath . Analog... |
| dalem37 34995 | Lemma for ~ dath . Analog... |
| dalem38 34996 | Lemma for ~ dath . Plane ... |
| dalem39 34997 | Lemma for ~ dath . Auxili... |
| dalem40 34998 | Lemma for ~ dath . Analog... |
| dalem41 34999 | Lemma for ~ dath . (Contr... |
| dalem42 35000 | Lemma for ~ dath . Auxili... |
| dalem43 35001 | Lemma for ~ dath . Planes... |
| dalem44 35002 | Lemma for ~ dath . Dummy ... |
| dalem45 35003 | Lemma for ~ dath . Dummy ... |
| dalem46 35004 | Lemma for ~ dath . Analog... |
| dalem47 35005 | Lemma for ~ dath . Analog... |
| dalem48 35006 | Lemma for ~ dath . Analog... |
| dalem49 35007 | Lemma for ~ dath . Analog... |
| dalem50 35008 | Lemma for ~ dath . Analog... |
| dalem51 35009 | Lemma for ~ dath . Constr... |
| dalem52 35010 | Lemma for ~ dath . Lines ... |
| dalem53 35011 | Lemma for ~ dath . The au... |
| dalem54 35012 | Lemma for ~ dath . Line `... |
| dalem55 35013 | Lemma for ~ dath . Lines ... |
| dalem56 35014 | Lemma for ~ dath . Analog... |
| dalem57 35015 | Lemma for ~ dath . Axis o... |
| dalem58 35016 | Lemma for ~ dath . Analog... |
| dalem59 35017 | Lemma for ~ dath . Analog... |
| dalem60 35018 | Lemma for ~ dath . ` B ` i... |
| dalem61 35019 | Lemma for ~ dath . Show t... |
| dalem62 35020 | Lemma for ~ dath . Elimin... |
| dalem63 35021 | Lemma for ~ dath . Combin... |
| dath 35022 | Desargues' Theorem of proj... |
| dath2 35023 | Version of Desargues' Theo... |
| lineset 35024 | The set of lines in a Hilb... |
| isline 35025 | The predicate "is a line".... |
| islinei 35026 | Condition implying "is a l... |
| pointsetN 35027 | The set of points in a Hil... |
| ispointN 35028 | The predicate "is a point"... |
| atpointN 35029 | The singleton of an atom i... |
| psubspset 35030 | The set of projective subs... |
| ispsubsp 35031 | The predicate "is a projec... |
| ispsubsp2 35032 | The predicate "is a projec... |
| psubspi 35033 | Property of a projective s... |
| psubspi2N 35034 | Property of a projective s... |
| 0psubN 35035 | The empty set is a project... |
| snatpsubN 35036 | The singleton of an atom i... |
| pointpsubN 35037 | A point (singleton of an a... |
| linepsubN 35038 | A line is a projective sub... |
| atpsubN 35039 | The set of all atoms is a ... |
| psubssat 35040 | A projective subspace cons... |
| psubatN 35041 | A member of a projective s... |
| pmapfval 35042 | The projective map of a Hi... |
| pmapval 35043 | Value of the projective ma... |
| elpmap 35044 | Member of a projective map... |
| pmapssat 35045 | The projective map of a Hi... |
| pmapssbaN 35046 | A weakening of ~ pmapssat ... |
| pmaple 35047 | The projective map of a Hi... |
| pmap11 35048 | The projective map of a Hi... |
| pmapat 35049 | The projective map of an a... |
| elpmapat 35050 | Member of the projective m... |
| pmap0 35051 | Value of the projective ma... |
| pmapeq0 35052 | A projective map value is ... |
| pmap1N 35053 | Value of the projective ma... |
| pmapsub 35054 | The projective map of a Hi... |
| pmapglbx 35055 | The projective map of the ... |
| pmapglb 35056 | The projective map of the ... |
| pmapglb2N 35057 | The projective map of the ... |
| pmapglb2xN 35058 | The projective map of the ... |
| pmapmeet 35059 | The projective map of a me... |
| isline2 35060 | Definition of line in term... |
| linepmap 35061 | A line described with a pr... |
| isline3 35062 | Definition of line in term... |
| isline4N 35063 | Definition of line in term... |
| lneq2at 35064 | A line equals the join of ... |
| lnatexN 35065 | There is an atom in a line... |
| lnjatN 35066 | Given an atom in a line, t... |
| lncvrelatN 35067 | A lattice element covered ... |
| lncvrat 35068 | A line covers the atoms it... |
| lncmp 35069 | If two lines are comparabl... |
| 2lnat 35070 | Two intersecting lines int... |
| 2atm2atN 35071 | Two joins with a common at... |
| 2llnma1b 35072 | Generalization of ~ 2llnma... |
| 2llnma1 35073 | Two different intersecting... |
| 2llnma3r 35074 | Two different intersecting... |
| 2llnma2 35075 | Two different intersecting... |
| 2llnma2rN 35076 | Two different intersecting... |
| cdlema1N 35077 | A condition for required f... |
| cdlema2N 35078 | A condition for required f... |
| cdlemblem 35079 | Lemma for ~ cdlemb . (Con... |
| cdlemb 35080 | Given two atoms not less t... |
| paddfval 35083 | Projective subspace sum op... |
| paddval 35084 | Projective subspace sum op... |
| elpadd 35085 | Member of a projective sub... |
| elpaddn0 35086 | Member of projective subsp... |
| paddvaln0N 35087 | Projective subspace sum op... |
| elpaddri 35088 | Condition implying members... |
| elpaddatriN 35089 | Condition implying members... |
| elpaddat 35090 | Membership in a projective... |
| elpaddatiN 35091 | Consequence of membership ... |
| elpadd2at 35092 | Membership in a projective... |
| elpadd2at2 35093 | Membership in a projective... |
| paddunssN 35094 | Projective subspace sum in... |
| elpadd0 35095 | Member of projective subsp... |
| paddval0 35096 | Projective subspace sum wi... |
| padd01 35097 | Projective subspace sum wi... |
| padd02 35098 | Projective subspace sum wi... |
| paddcom 35099 | Projective subspace sum co... |
| paddssat 35100 | A projective subspace sum ... |
| sspadd1 35101 | A projective subspace sum ... |
| sspadd2 35102 | A projective subspace sum ... |
| paddss1 35103 | Subset law for projective ... |
| paddss2 35104 | Subset law for projective ... |
| paddss12 35105 | Subset law for projective ... |
| paddasslem1 35106 | Lemma for ~ paddass . (Co... |
| paddasslem2 35107 | Lemma for ~ paddass . (Co... |
| paddasslem3 35108 | Lemma for ~ paddass . Res... |
| paddasslem4 35109 | Lemma for ~ paddass . Com... |
| paddasslem5 35110 | Lemma for ~ paddass . Sho... |
| paddasslem6 35111 | Lemma for ~ paddass . (Co... |
| paddasslem7 35112 | Lemma for ~ paddass . Com... |
| paddasslem8 35113 | Lemma for ~ paddass . (Co... |
| paddasslem9 35114 | Lemma for ~ paddass . Com... |
| paddasslem10 35115 | Lemma for ~ paddass . Use... |
| paddasslem11 35116 | Lemma for ~ paddass . The... |
| paddasslem12 35117 | Lemma for ~ paddass . The... |
| paddasslem13 35118 | Lemma for ~ paddass . The... |
| paddasslem14 35119 | Lemma for ~ paddass . Rem... |
| paddasslem15 35120 | Lemma for ~ paddass . Use... |
| paddasslem16 35121 | Lemma for ~ paddass . Use... |
| paddasslem17 35122 | Lemma for ~ paddass . The... |
| paddasslem18 35123 | Lemma for ~ paddass . Com... |
| paddass 35124 | Projective subspace sum is... |
| padd12N 35125 | Commutative/associative la... |
| padd4N 35126 | Rearrangement of 4 terms i... |
| paddidm 35127 | Projective subspace sum is... |
| paddclN 35128 | The projective sum of two ... |
| paddssw1 35129 | Subset law for projective ... |
| paddssw2 35130 | Subset law for projective ... |
| paddss 35131 | Subset law for projective ... |
| pmodlem1 35132 | Lemma for ~ pmod1i . (Con... |
| pmodlem2 35133 | Lemma for ~ pmod1i . (Con... |
| pmod1i 35134 | The modular law holds in a... |
| pmod2iN 35135 | Dual of the modular law. ... |
| pmodN 35136 | The modular law for projec... |
| pmodl42N 35137 | Lemma derived from modular... |
| pmapjoin 35138 | The projective map of the ... |
| pmapjat1 35139 | The projective map of the ... |
| pmapjat2 35140 | The projective map of the ... |
| pmapjlln1 35141 | The projective map of the ... |
| hlmod1i 35142 | A version of the modular l... |
| atmod1i1 35143 | Version of modular law ~ p... |
| atmod1i1m 35144 | Version of modular law ~ p... |
| atmod1i2 35145 | Version of modular law ~ p... |
| llnmod1i2 35146 | Version of modular law ~ p... |
| atmod2i1 35147 | Version of modular law ~ p... |
| atmod2i2 35148 | Version of modular law ~ p... |
| llnmod2i2 35149 | Version of modular law ~ p... |
| atmod3i1 35150 | Version of modular law tha... |
| atmod3i2 35151 | Version of modular law tha... |
| atmod4i1 35152 | Version of modular law tha... |
| atmod4i2 35153 | Version of modular law tha... |
| llnexchb2lem 35154 | Lemma for ~ llnexchb2 . (... |
| llnexchb2 35155 | Line exchange property (co... |
| llnexch2N 35156 | Line exchange property (co... |
| dalawlem1 35157 | Lemma for ~ dalaw . Speci... |
| dalawlem2 35158 | Lemma for ~ dalaw . Utili... |
| dalawlem3 35159 | Lemma for ~ dalaw . First... |
| dalawlem4 35160 | Lemma for ~ dalaw . Secon... |
| dalawlem5 35161 | Lemma for ~ dalaw . Speci... |
| dalawlem6 35162 | Lemma for ~ dalaw . First... |
| dalawlem7 35163 | Lemma for ~ dalaw . Secon... |
| dalawlem8 35164 | Lemma for ~ dalaw . Speci... |
| dalawlem9 35165 | Lemma for ~ dalaw . Speci... |
| dalawlem10 35166 | Lemma for ~ dalaw . Combi... |
| dalawlem11 35167 | Lemma for ~ dalaw . First... |
| dalawlem12 35168 | Lemma for ~ dalaw . Secon... |
| dalawlem13 35169 | Lemma for ~ dalaw . Speci... |
| dalawlem14 35170 | Lemma for ~ dalaw . Combi... |
| dalawlem15 35171 | Lemma for ~ dalaw . Swap ... |
| dalaw 35172 | Desargues' law, derived fr... |
| pclfvalN 35175 | The projective subspace cl... |
| pclvalN 35176 | Value of the projective su... |
| pclclN 35177 | Closure of the projective ... |
| elpclN 35178 | Membership in the projecti... |
| elpcliN 35179 | Implication of membership ... |
| pclssN 35180 | Ordering is preserved by s... |
| pclssidN 35181 | A set of atoms is included... |
| pclidN 35182 | The projective subspace cl... |
| pclbtwnN 35183 | A projective subspace sand... |
| pclunN 35184 | The projective subspace cl... |
| pclun2N 35185 | The projective subspace cl... |
| pclfinN 35186 | The projective subspace cl... |
| pclcmpatN 35187 | The set of projective subs... |
| polfvalN 35190 | The projective subspace po... |
| polvalN 35191 | Value of the projective su... |
| polval2N 35192 | Alternate expression for v... |
| polsubN 35193 | The polarity of a set of a... |
| polssatN 35194 | The polarity of a set of a... |
| pol0N 35195 | The polarity of the empty ... |
| pol1N 35196 | The polarity of the whole ... |
| 2pol0N 35197 | The closed subspace closur... |
| polpmapN 35198 | The polarity of a projecti... |
| 2polpmapN 35199 | Double polarity of a proje... |
| 2polvalN 35200 | Value of double polarity. ... |
| 2polssN 35201 | A set of atoms is a subset... |
| 3polN 35202 | Triple polarity cancels to... |
| polcon3N 35203 | Contraposition law for pol... |
| 2polcon4bN 35204 | Contraposition law for pol... |
| polcon2N 35205 | Contraposition law for pol... |
| polcon2bN 35206 | Contraposition law for pol... |
| pclss2polN 35207 | The projective subspace cl... |
| pcl0N 35208 | The projective subspace cl... |
| pcl0bN 35209 | The projective subspace cl... |
| pmaplubN 35210 | The LUB of a projective ma... |
| sspmaplubN 35211 | A set of atoms is a subset... |
| 2pmaplubN 35212 | Double projective map of a... |
| paddunN 35213 | The closure of the project... |
| poldmj1N 35214 | De Morgan's law for polari... |
| pmapj2N 35215 | The projective map of the ... |
| pmapocjN 35216 | The projective map of the ... |
| polatN 35217 | The polarity of the single... |
| 2polatN 35218 | Double polarity of the sin... |
| pnonsingN 35219 | The intersection of a set ... |
| psubclsetN 35222 | The set of closed projecti... |
| ispsubclN 35223 | The predicate "is a closed... |
| psubcliN 35224 | Property of a closed proje... |
| psubcli2N 35225 | Property of a closed proje... |
| psubclsubN 35226 | A closed projective subspa... |
| psubclssatN 35227 | A closed projective subspa... |
| pmapidclN 35228 | Projective map of the LUB ... |
| 0psubclN 35229 | The empty set is a closed ... |
| 1psubclN 35230 | The set of all atoms is a ... |
| atpsubclN 35231 | A point (singleton of an a... |
| pmapsubclN 35232 | A projective map value is ... |
| ispsubcl2N 35233 | Alternate predicate for "i... |
| psubclinN 35234 | The intersection of two cl... |
| paddatclN 35235 | The projective sum of a cl... |
| pclfinclN 35236 | The projective subspace cl... |
| linepsubclN 35237 | A line is a closed project... |
| polsubclN 35238 | A polarity is a closed pro... |
| poml4N 35239 | Orthomodular law for proje... |
| poml5N 35240 | Orthomodular law for proje... |
| poml6N 35241 | Orthomodular law for proje... |
| osumcllem1N 35242 | Lemma for ~ osumclN . (Co... |
| osumcllem2N 35243 | Lemma for ~ osumclN . (Co... |
| osumcllem3N 35244 | Lemma for ~ osumclN . (Co... |
| osumcllem4N 35245 | Lemma for ~ osumclN . (Co... |
| osumcllem5N 35246 | Lemma for ~ osumclN . (Co... |
| osumcllem6N 35247 | Lemma for ~ osumclN . Use... |
| osumcllem7N 35248 | Lemma for ~ osumclN . (Co... |
| osumcllem8N 35249 | Lemma for ~ osumclN . (Co... |
| osumcllem9N 35250 | Lemma for ~ osumclN . (Co... |
| osumcllem10N 35251 | Lemma for ~ osumclN . Con... |
| osumcllem11N 35252 | Lemma for ~ osumclN . (Co... |
| osumclN 35253 | Closure of orthogonal sum.... |
| pmapojoinN 35254 | For orthogonal elements, p... |
| pexmidN 35255 | Excluded middle law for cl... |
| pexmidlem1N 35256 | Lemma for ~ pexmidN . Hol... |
| pexmidlem2N 35257 | Lemma for ~ pexmidN . (Co... |
| pexmidlem3N 35258 | Lemma for ~ pexmidN . Use... |
| pexmidlem4N 35259 | Lemma for ~ pexmidN . (Co... |
| pexmidlem5N 35260 | Lemma for ~ pexmidN . (Co... |
| pexmidlem6N 35261 | Lemma for ~ pexmidN . (Co... |
| pexmidlem7N 35262 | Lemma for ~ pexmidN . Con... |
| pexmidlem8N 35263 | Lemma for ~ pexmidN . The... |
| pexmidALTN 35264 | Excluded middle law for cl... |
| pl42lem1N 35265 | Lemma for ~ pl42N . (Cont... |
| pl42lem2N 35266 | Lemma for ~ pl42N . (Cont... |
| pl42lem3N 35267 | Lemma for ~ pl42N . (Cont... |
| pl42lem4N 35268 | Lemma for ~ pl42N . (Cont... |
| pl42N 35269 | Law holding in a Hilbert l... |
| watfvalN 35278 | The W atoms function. (Co... |
| watvalN 35279 | Value of the W atoms funct... |
| iswatN 35280 | The predicate "is a W atom... |
| lhpset 35281 | The set of co-atoms (latti... |
| islhp 35282 | The predicate "is a co-ato... |
| islhp2 35283 | The predicate "is a co-ato... |
| lhpbase 35284 | A co-atom is a member of t... |
| lhp1cvr 35285 | The lattice unit covers a ... |
| lhplt 35286 | An atom under a co-atom is... |
| lhp2lt 35287 | The join of two atoms unde... |
| lhpexlt 35288 | There exists an atom less ... |
| lhp0lt 35289 | A co-atom is greater than ... |
| lhpn0 35290 | A co-atom is nonzero. TOD... |
| lhpexle 35291 | There exists an atom under... |
| lhpexnle 35292 | There exists an atom not u... |
| lhpexle1lem 35293 | Lemma for ~ lhpexle1 and o... |
| lhpexle1 35294 | There exists an atom under... |
| lhpexle2lem 35295 | Lemma for ~ lhpexle2 . (C... |
| lhpexle2 35296 | There exists atom under a ... |
| lhpexle3lem 35297 | There exists atom under a ... |
| lhpexle3 35298 | There exists atom under a ... |
| lhpex2leN 35299 | There exist at least two d... |
| lhpoc 35300 | The orthocomplement of a c... |
| lhpoc2N 35301 | The orthocomplement of an ... |
| lhpocnle 35302 | The orthocomplement of a c... |
| lhpocat 35303 | The orthocomplement of a c... |
| lhpocnel 35304 | The orthocomplement of a c... |
| lhpocnel2 35305 | The orthocomplement of a c... |
| lhpjat1 35306 | The join of a co-atom (hyp... |
| lhpjat2 35307 | The join of a co-atom (hyp... |
| lhpj1 35308 | The join of a co-atom (hyp... |
| lhpmcvr 35309 | The meet of a lattice hype... |
| lhpmcvr2 35310 | Alternate way to express t... |
| lhpmcvr3 35311 | Specialization of ~ lhpmcv... |
| lhpmcvr4N 35312 | Specialization of ~ lhpmcv... |
| lhpmcvr5N 35313 | Specialization of ~ lhpmcv... |
| lhpmcvr6N 35314 | Specialization of ~ lhpmcv... |
| lhpm0atN 35315 | If the meet of a lattice h... |
| lhpmat 35316 | An element covered by the ... |
| lhpmatb 35317 | An element covered by the ... |
| lhp2at0 35318 | Join and meet with differe... |
| lhp2atnle 35319 | Inequality for 2 different... |
| lhp2atne 35320 | Inequality for joins with ... |
| lhp2at0nle 35321 | Inequality for 2 different... |
| lhp2at0ne 35322 | Inequality for joins with ... |
| lhpelim 35323 | Eliminate an atom not unde... |
| lhpmod2i2 35324 | Modular law for hyperplane... |
| lhpmod6i1 35325 | Modular law for hyperplane... |
| lhprelat3N 35326 | The Hilbert lattice is rel... |
| cdlemb2 35327 | Given two atoms not under ... |
| lhple 35328 | Property of a lattice elem... |
| lhpat 35329 | Create an atom under a co-... |
| lhpat4N 35330 | Property of an atom under ... |
| lhpat2 35331 | Create an atom under a co-... |
| lhpat3 35332 | There is only one atom und... |
| 4atexlemk 35333 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemw 35334 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempw 35335 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemp 35336 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemq 35337 | Lemma for ~ 4atexlem7 . (... |
| 4atexlems 35338 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemt 35339 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemutvt 35340 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempnq 35341 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnslpq 35342 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkl 35343 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkc 35344 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemwb 35345 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempsb 35346 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemqtb 35347 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempns 35348 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemswapqr 35349 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemu 35350 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemv 35351 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemunv 35352 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemtlw 35353 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemntlpq 35354 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemc 35355 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnclw 35356 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex2 35357 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemcnd 35358 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex4 35359 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemex6 35360 | Lemma for ~ 4atexlem7 . (... |
| 4atexlem7 35361 | Whenever there are at leas... |
| 4atex 35362 | Whenever there are at leas... |
| 4atex2 35363 | More general version of ~ ... |
| 4atex2-0aOLDN 35364 | Same as ~ 4atex2 except th... |
| 4atex2-0bOLDN 35365 | Same as ~ 4atex2 except th... |
| 4atex2-0cOLDN 35366 | Same as ~ 4atex2 except th... |
| 4atex3 35367 | More general version of ~ ... |
| lautset 35368 | The set of lattice automor... |
| islaut 35369 | The predictate "is a latti... |
| lautle 35370 | Less-than or equal propert... |
| laut1o 35371 | A lattice automorphism is ... |
| laut11 35372 | One-to-one property of a l... |
| lautcl 35373 | A lattice automorphism val... |
| lautcnvclN 35374 | Reverse closure of a latti... |
| lautcnvle 35375 | Less-than or equal propert... |
| lautcnv 35376 | The converse of a lattice ... |
| lautlt 35377 | Less-than property of a la... |
| lautcvr 35378 | Covering property of a lat... |
| lautj 35379 | Meet property of a lattice... |
| lautm 35380 | Meet property of a lattice... |
| lauteq 35381 | A lattice automorphism arg... |
| idlaut 35382 | The identity function is a... |
| lautco 35383 | The composition of two lat... |
| pautsetN 35384 | The set of projective auto... |
| ispautN 35385 | The predictate "is a proje... |
| ldilfset 35394 | The mapping from fiducial ... |
| ldilset 35395 | The set of lattice dilatio... |
| isldil 35396 | The predicate "is a lattic... |
| ldillaut 35397 | A lattice dilation is an a... |
| ldil1o 35398 | A lattice dilation is a on... |
| ldilval 35399 | Value of a lattice dilatio... |
| idldil 35400 | The identity function is a... |
| ldilcnv 35401 | The converse of a lattice ... |
| ldilco 35402 | The composition of two lat... |
| ltrnfset 35403 | The set of all lattice tra... |
| ltrnset 35404 | The set of lattice transla... |
| isltrn 35405 | The predicate "is a lattic... |
| isltrn2N 35406 | The predicate "is a lattic... |
| ltrnu 35407 | Uniqueness property of a l... |
| ltrnldil 35408 | A lattice translation is a... |
| ltrnlaut 35409 | A lattice translation is a... |
| ltrn1o 35410 | A lattice translation is a... |
| ltrncl 35411 | Closure of a lattice trans... |
| ltrn11 35412 | One-to-one property of a l... |
| ltrncnvnid 35413 | If a translation is differ... |
| ltrncoidN 35414 | Two translations are equal... |
| ltrnle 35415 | Less-than or equal propert... |
| ltrncnvleN 35416 | Less-than or equal propert... |
| ltrnm 35417 | Lattice translation of a m... |
| ltrnj 35418 | Lattice translation of a m... |
| ltrncvr 35419 | Covering property of a lat... |
| ltrnval1 35420 | Value of a lattice transla... |
| ltrnid 35421 | A lattice translation is t... |
| ltrnnid 35422 | If a lattice translation i... |
| ltrnatb 35423 | The lattice translation of... |
| ltrncnvatb 35424 | The converse of the lattic... |
| ltrnel 35425 | The lattice translation of... |
| ltrnat 35426 | The lattice translation of... |
| ltrncnvat 35427 | The converse of the lattic... |
| ltrncnvel 35428 | The converse of the lattic... |
| ltrncoelN 35429 | Composition of lattice tra... |
| ltrncoat 35430 | Composition of lattice tra... |
| ltrncoval 35431 | Two ways to express value ... |
| ltrncnv 35432 | The converse of a lattice ... |
| ltrn11at 35433 | Frequently used one-to-one... |
| ltrneq2 35434 | The equality of two transl... |
| ltrneq 35435 | The equality of two transl... |
| idltrn 35436 | The identity function is a... |
| ltrnmw 35437 | Property of lattice transl... |
| ltrnmwOLD 35438 | Property of lattice transl... |
| dilfsetN 35439 | The mapping from fiducial ... |
| dilsetN 35440 | The set of dilations for a... |
| isdilN 35441 | The predicate "is a dilati... |
| trnfsetN 35442 | The mapping from fiducial ... |
| trnsetN 35443 | The set of translations fo... |
| istrnN 35444 | The predicate "is a transl... |
| trlfset 35447 | The set of all traces of l... |
| trlset 35448 | The set of traces of latti... |
| trlval 35449 | The value of the trace of ... |
| trlval2 35450 | The value of the trace of ... |
| trlcl 35451 | Closure of the trace of a ... |
| trlcnv 35452 | The trace of the converse ... |
| trljat1 35453 | The value of a translation... |
| trljat2 35454 | The value of a translation... |
| trljat3 35455 | The value of a translation... |
| trlat 35456 | If an atom differs from it... |
| trl0 35457 | If an atom not under the f... |
| trlator0 35458 | The trace of a lattice tra... |
| trlatn0 35459 | The trace of a lattice tra... |
| trlnidat 35460 | The trace of a lattice tra... |
| ltrnnidn 35461 | If a lattice translation i... |
| ltrnideq 35462 | Property of the identity l... |
| trlid0 35463 | The trace of the identity ... |
| trlnidatb 35464 | A lattice translation is n... |
| trlid0b 35465 | A lattice translation is t... |
| trlnid 35466 | Different translations wit... |
| ltrn2ateq 35467 | Property of the equality o... |
| ltrnateq 35468 | If any atom (under ` W ` )... |
| ltrnatneq 35469 | If any atom (under ` W ` )... |
| ltrnatlw 35470 | If the value of an atom eq... |
| trlle 35471 | The trace of a lattice tra... |
| trlne 35472 | The trace of a lattice tra... |
| trlnle 35473 | The atom not under the fid... |
| trlval3 35474 | The value of the trace of ... |
| trlval4 35475 | The value of the trace of ... |
| trlval5 35476 | The value of the trace of ... |
| arglem1N 35477 | Lemma for Desargues' law. ... |
| cdlemc1 35478 | Part of proof of Lemma C i... |
| cdlemc2 35479 | Part of proof of Lemma C i... |
| cdlemc3 35480 | Part of proof of Lemma C i... |
| cdlemc4 35481 | Part of proof of Lemma C i... |
| cdlemc5 35482 | Lemma for ~ cdlemc . (Con... |
| cdlemc6 35483 | Lemma for ~ cdlemc . (Con... |
| cdlemc 35484 | Lemma C in [Crawley] p. 11... |
| cdlemd1 35485 | Part of proof of Lemma D i... |
| cdlemd2 35486 | Part of proof of Lemma D i... |
| cdlemd3 35487 | Part of proof of Lemma D i... |
| cdlemd4 35488 | Part of proof of Lemma D i... |
| cdlemd5 35489 | Part of proof of Lemma D i... |
| cdlemd6 35490 | Part of proof of Lemma D i... |
| cdlemd7 35491 | Part of proof of Lemma D i... |
| cdlemd8 35492 | Part of proof of Lemma D i... |
| cdlemd9 35493 | Part of proof of Lemma D i... |
| cdlemd 35494 | If two translations agree ... |
| ltrneq3 35495 | Two translations agree at ... |
| cdleme00a 35496 | Part of proof of Lemma E i... |
| cdleme0aa 35497 | Part of proof of Lemma E i... |
| cdleme0a 35498 | Part of proof of Lemma E i... |
| cdleme0b 35499 | Part of proof of Lemma E i... |
| cdleme0c 35500 | Part of proof of Lemma E i... |
| cdleme0cp 35501 | Part of proof of Lemma E i... |
| cdleme0cq 35502 | Part of proof of Lemma E i... |
| cdleme0dN 35503 | Part of proof of Lemma E i... |
| cdleme0e 35504 | Part of proof of Lemma E i... |
| cdleme0fN 35505 | Part of proof of Lemma E i... |
| cdleme0gN 35506 | Part of proof of Lemma E i... |
| cdlemeulpq 35507 | Part of proof of Lemma E i... |
| cdleme01N 35508 | Part of proof of Lemma E i... |
| cdleme02N 35509 | Part of proof of Lemma E i... |
| cdleme0ex1N 35510 | Part of proof of Lemma E i... |
| cdleme0ex2N 35511 | Part of proof of Lemma E i... |
| cdleme0moN 35512 | Part of proof of Lemma E i... |
| cdleme1b 35513 | Part of proof of Lemma E i... |
| cdleme1 35514 | Part of proof of Lemma E i... |
| cdleme2 35515 | Part of proof of Lemma E i... |
| cdleme3b 35516 | Part of proof of Lemma E i... |
| cdleme3c 35517 | Part of proof of Lemma E i... |
| cdleme3d 35518 | Part of proof of Lemma E i... |
| cdleme3e 35519 | Part of proof of Lemma E i... |
| cdleme3fN 35520 | Part of proof of Lemma E i... |
| cdleme3g 35521 | Part of proof of Lemma E i... |
| cdleme3h 35522 | Part of proof of Lemma E i... |
| cdleme3fa 35523 | Part of proof of Lemma E i... |
| cdleme3 35524 | Part of proof of Lemma E i... |
| cdleme4 35525 | Part of proof of Lemma E i... |
| cdleme4a 35526 | Part of proof of Lemma E i... |
| cdleme5 35527 | Part of proof of Lemma E i... |
| cdleme6 35528 | Part of proof of Lemma E i... |
| cdleme7aa 35529 | Part of proof of Lemma E i... |
| cdleme7a 35530 | Part of proof of Lemma E i... |
| cdleme7b 35531 | Part of proof of Lemma E i... |
| cdleme7c 35532 | Part of proof of Lemma E i... |
| cdleme7d 35533 | Part of proof of Lemma E i... |
| cdleme7e 35534 | Part of proof of Lemma E i... |
| cdleme7ga 35535 | Part of proof of Lemma E i... |
| cdleme7 35536 | Part of proof of Lemma E i... |
| cdleme8 35537 | Part of proof of Lemma E i... |
| cdleme9a 35538 | Part of proof of Lemma E i... |
| cdleme9b 35539 | Utility lemma for Lemma E ... |
| cdleme9 35540 | Part of proof of Lemma E i... |
| cdleme10 35541 | Part of proof of Lemma E i... |
| cdleme8tN 35542 | Part of proof of Lemma E i... |
| cdleme9taN 35543 | Part of proof of Lemma E i... |
| cdleme9tN 35544 | Part of proof of Lemma E i... |
| cdleme10tN 35545 | Part of proof of Lemma E i... |
| cdleme16aN 35546 | Part of proof of Lemma E i... |
| cdleme11a 35547 | Part of proof of Lemma E i... |
| cdleme11c 35548 | Part of proof of Lemma E i... |
| cdleme11dN 35549 | Part of proof of Lemma E i... |
| cdleme11e 35550 | Part of proof of Lemma E i... |
| cdleme11fN 35551 | Part of proof of Lemma E i... |
| cdleme11g 35552 | Part of proof of Lemma E i... |
| cdleme11h 35553 | Part of proof of Lemma E i... |
| cdleme11j 35554 | Part of proof of Lemma E i... |
| cdleme11k 35555 | Part of proof of Lemma E i... |
| cdleme11l 35556 | Part of proof of Lemma E i... |
| cdleme11 35557 | Part of proof of Lemma E i... |
| cdleme12 35558 | Part of proof of Lemma E i... |
| cdleme13 35559 | Part of proof of Lemma E i... |
| cdleme14 35560 | Part of proof of Lemma E i... |
| cdleme15a 35561 | Part of proof of Lemma E i... |
| cdleme15b 35562 | Part of proof of Lemma E i... |
| cdleme15c 35563 | Part of proof of Lemma E i... |
| cdleme15d 35564 | Part of proof of Lemma E i... |
| cdleme15 35565 | Part of proof of Lemma E i... |
| cdleme16b 35566 | Part of proof of Lemma E i... |
| cdleme16c 35567 | Part of proof of Lemma E i... |
| cdleme16d 35568 | Part of proof of Lemma E i... |
| cdleme16e 35569 | Part of proof of Lemma E i... |
| cdleme16f 35570 | Part of proof of Lemma E i... |
| cdleme16g 35571 | Part of proof of Lemma E i... |
| cdleme16 35572 | Part of proof of Lemma E i... |
| cdleme17a 35573 | Part of proof of Lemma E i... |
| cdleme17b 35574 | Lemma leading to ~ cdleme1... |
| cdleme17c 35575 | Part of proof of Lemma E i... |
| cdleme17d1 35576 | Part of proof of Lemma E i... |
| cdleme0nex 35577 | Part of proof of Lemma E i... |
| cdleme18a 35578 | Part of proof of Lemma E i... |
| cdleme18b 35579 | Part of proof of Lemma E i... |
| cdleme18c 35580 | Part of proof of Lemma E i... |
| cdleme22gb 35581 | Utility lemma for Lemma E ... |
| cdleme18d 35582 | Part of proof of Lemma E i... |
| cdlemesner 35583 | Part of proof of Lemma E i... |
| cdlemedb 35584 | Part of proof of Lemma E i... |
| cdlemeda 35585 | Part of proof of Lemma E i... |
| cdlemednpq 35586 | Part of proof of Lemma E i... |
| cdlemednuN 35587 | Part of proof of Lemma E i... |
| cdleme20zN 35588 | Part of proof of Lemma E i... |
| cdleme20y 35589 | Part of proof of Lemma E i... |
| cdleme20yOLD 35590 | Part of proof of Lemma E i... |
| cdleme19a 35591 | Part of proof of Lemma E i... |
| cdleme19b 35592 | Part of proof of Lemma E i... |
| cdleme19c 35593 | Part of proof of Lemma E i... |
| cdleme19d 35594 | Part of proof of Lemma E i... |
| cdleme19e 35595 | Part of proof of Lemma E i... |
| cdleme19f 35596 | Part of proof of Lemma E i... |
| cdleme20aN 35597 | Part of proof of Lemma E i... |
| cdleme20bN 35598 | Part of proof of Lemma E i... |
| cdleme20c 35599 | Part of proof of Lemma E i... |
| cdleme20d 35600 | Part of proof of Lemma E i... |
| cdleme20e 35601 | Part of proof of Lemma E i... |
| cdleme20f 35602 | Part of proof of Lemma E i... |
| cdleme20g 35603 | Part of proof of Lemma E i... |
| cdleme20h 35604 | Part of proof of Lemma E i... |
| cdleme20i 35605 | Part of proof of Lemma E i... |
| cdleme20j 35606 | Part of proof of Lemma E i... |
| cdleme20k 35607 | Part of proof of Lemma E i... |
| cdleme20l1 35608 | Part of proof of Lemma E i... |
| cdleme20l2 35609 | Part of proof of Lemma E i... |
| cdleme20l 35610 | Part of proof of Lemma E i... |
| cdleme20m 35611 | Part of proof of Lemma E i... |
| cdleme20 35612 | Combine ~ cdleme19f and ~ ... |
| cdleme21a 35613 | Part of proof of Lemma E i... |
| cdleme21b 35614 | Part of proof of Lemma E i... |
| cdleme21c 35615 | Part of proof of Lemma E i... |
| cdleme21at 35616 | Part of proof of Lemma E i... |
| cdleme21ct 35617 | Part of proof of Lemma E i... |
| cdleme21d 35618 | Part of proof of Lemma E i... |
| cdleme21e 35619 | Part of proof of Lemma E i... |
| cdleme21f 35620 | Part of proof of Lemma E i... |
| cdleme21g 35621 | Part of proof of Lemma E i... |
| cdleme21h 35622 | Part of proof of Lemma E i... |
| cdleme21i 35623 | Part of proof of Lemma E i... |
| cdleme21j 35624 | Combine ~ cdleme20 and ~ c... |
| cdleme21 35625 | Part of proof of Lemma E i... |
| cdleme21k 35626 | Eliminate ` S =/= T ` cond... |
| cdleme22aa 35627 | Part of proof of Lemma E i... |
| cdleme22a 35628 | Part of proof of Lemma E i... |
| cdleme22b 35629 | Part of proof of Lemma E i... |
| cdleme22cN 35630 | Part of proof of Lemma E i... |
| cdleme22d 35631 | Part of proof of Lemma E i... |
| cdleme22e 35632 | Part of proof of Lemma E i... |
| cdleme22eALTN 35633 | Part of proof of Lemma E i... |
| cdleme22f 35634 | Part of proof of Lemma E i... |
| cdleme22f2 35635 | Part of proof of Lemma E i... |
| cdleme22g 35636 | Part of proof of Lemma E i... |
| cdleme23a 35637 | Part of proof of Lemma E i... |
| cdleme23b 35638 | Part of proof of Lemma E i... |
| cdleme23c 35639 | Part of proof of Lemma E i... |
| cdleme24 35640 | Quantified version of ~ cd... |
| cdleme25a 35641 | Lemma for ~ cdleme25b . (... |
| cdleme25b 35642 | Transform ~ cdleme24 . TO... |
| cdleme25c 35643 | Transform ~ cdleme25b . (... |
| cdleme25dN 35644 | Transform ~ cdleme25c . (... |
| cdleme25cl 35645 | Show closure of the unique... |
| cdleme25cv 35646 | Change bound variables in ... |
| cdleme26e 35647 | Part of proof of Lemma E i... |
| cdleme26ee 35648 | Part of proof of Lemma E i... |
| cdleme26eALTN 35649 | Part of proof of Lemma E i... |
| cdleme26fALTN 35650 | Part of proof of Lemma E i... |
| cdleme26f 35651 | Part of proof of Lemma E i... |
| cdleme26f2ALTN 35652 | Part of proof of Lemma E i... |
| cdleme26f2 35653 | Part of proof of Lemma E i... |
| cdleme27cl 35654 | Part of proof of Lemma E i... |
| cdleme27a 35655 | Part of proof of Lemma E i... |
| cdleme27b 35656 | Lemma for ~ cdleme27N . (... |
| cdleme27N 35657 | Part of proof of Lemma E i... |
| cdleme28a 35658 | Lemma for ~ cdleme25b . T... |
| cdleme28b 35659 | Lemma for ~ cdleme25b . T... |
| cdleme28c 35660 | Part of proof of Lemma E i... |
| cdleme28 35661 | Quantified version of ~ cd... |
| cdleme29ex 35662 | Lemma for ~ cdleme29b . (... |
| cdleme29b 35663 | Transform ~ cdleme28 . (C... |
| cdleme29c 35664 | Transform ~ cdleme28b . (... |
| cdleme29cl 35665 | Show closure of the unique... |
| cdleme30a 35666 | Part of proof of Lemma E i... |
| cdleme31so 35667 | Part of proof of Lemma E i... |
| cdleme31sn 35668 | Part of proof of Lemma E i... |
| cdleme31sn1 35669 | Part of proof of Lemma E i... |
| cdleme31se 35670 | Part of proof of Lemma D i... |
| cdleme31se2 35671 | Part of proof of Lemma D i... |
| cdleme31sc 35672 | Part of proof of Lemma E i... |
| cdleme31sde 35673 | Part of proof of Lemma D i... |
| cdleme31snd 35674 | Part of proof of Lemma D i... |
| cdleme31sdnN 35675 | Part of proof of Lemma E i... |
| cdleme31sn1c 35676 | Part of proof of Lemma E i... |
| cdleme31sn2 35677 | Part of proof of Lemma E i... |
| cdleme31fv 35678 | Part of proof of Lemma E i... |
| cdleme31fv1 35679 | Part of proof of Lemma E i... |
| cdleme31fv1s 35680 | Part of proof of Lemma E i... |
| cdleme31fv2 35681 | Part of proof of Lemma E i... |
| cdleme31id 35682 | Part of proof of Lemma E i... |
| cdlemefrs29pre00 35683 | ***START OF VALUE AT ATOM ... |
| cdlemefrs29bpre0 35684 | TODO fix comment. (Contri... |
| cdlemefrs29bpre1 35685 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29cpre1 35686 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29clN 35687 | TODO: NOT USED? Show clo... |
| cdlemefrs32fva 35688 | Part of proof of Lemma E i... |
| cdlemefrs32fva1 35689 | Part of proof of Lemma E i... |
| cdlemefr29exN 35690 | Lemma for ~ cdlemefs29bpre... |
| cdlemefr27cl 35691 | Part of proof of Lemma E i... |
| cdlemefr32sn2aw 35692 | Show that ` [_ R / s ]_ N ... |
| cdlemefr32snb 35693 | Show closure of ` [_ R / s... |
| cdlemefr29bpre0N 35694 | TODO fix comment. (Contri... |
| cdlemefr29clN 35695 | Show closure of the unique... |
| cdleme43frv1snN 35696 | Value of ` [_ R / s ]_ N `... |
| cdlemefr32fvaN 35697 | Part of proof of Lemma E i... |
| cdlemefr32fva1 35698 | Part of proof of Lemma E i... |
| cdlemefr31fv1 35699 | Value of ` ( F `` R ) ` wh... |
| cdlemefs29pre00N 35700 | FIX COMMENT. TODO: see if ... |
| cdlemefs27cl 35701 | Part of proof of Lemma E i... |
| cdlemefs32sn1aw 35702 | Show that ` [_ R / s ]_ N ... |
| cdlemefs32snb 35703 | Show closure of ` [_ R / s... |
| cdlemefs29bpre0N 35704 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29bpre1N 35705 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29cpre1N 35706 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29clN 35707 | Show closure of the unique... |
| cdleme43fsv1snlem 35708 | Value of ` [_ R / s ]_ N `... |
| cdleme43fsv1sn 35709 | Value of ` [_ R / s ]_ N `... |
| cdlemefs32fvaN 35710 | Part of proof of Lemma E i... |
| cdlemefs32fva1 35711 | Part of proof of Lemma E i... |
| cdlemefs31fv1 35712 | Value of ` ( F `` R ) ` wh... |
| cdlemefr44 35713 | Value of f(r) when r is an... |
| cdlemefs44 35714 | Value of f_s(r) when r is ... |
| cdlemefr45 35715 | Value of f(r) when r is an... |
| cdlemefr45e 35716 | Explicit expansion of ~ cd... |
| cdlemefs45 35717 | Value of f_s(r) when r is ... |
| cdlemefs45ee 35718 | Explicit expansion of ~ cd... |
| cdlemefs45eN 35719 | Explicit expansion of ~ cd... |
| cdleme32sn1awN 35720 | Show that ` [_ R / s ]_ N ... |
| cdleme41sn3a 35721 | Show that ` [_ R / s ]_ N ... |
| cdleme32sn2awN 35722 | Show that ` [_ R / s ]_ N ... |
| cdleme32snaw 35723 | Show that ` [_ R / s ]_ N ... |
| cdleme32snb 35724 | Show closure of ` [_ R / s... |
| cdleme32fva 35725 | Part of proof of Lemma D i... |
| cdleme32fva1 35726 | Part of proof of Lemma D i... |
| cdleme32fvaw 35727 | Show that ` ( F `` R ) ` i... |
| cdleme32fvcl 35728 | Part of proof of Lemma D i... |
| cdleme32a 35729 | Part of proof of Lemma D i... |
| cdleme32b 35730 | Part of proof of Lemma D i... |
| cdleme32c 35731 | Part of proof of Lemma D i... |
| cdleme32d 35732 | Part of proof of Lemma D i... |
| cdleme32e 35733 | Part of proof of Lemma D i... |
| cdleme32f 35734 | Part of proof of Lemma D i... |
| cdleme32le 35735 | Part of proof of Lemma D i... |
| cdleme35a 35736 | Part of proof of Lemma E i... |
| cdleme35fnpq 35737 | Part of proof of Lemma E i... |
| cdleme35b 35738 | Part of proof of Lemma E i... |
| cdleme35c 35739 | Part of proof of Lemma E i... |
| cdleme35d 35740 | Part of proof of Lemma E i... |
| cdleme35e 35741 | Part of proof of Lemma E i... |
| cdleme35f 35742 | Part of proof of Lemma E i... |
| cdleme35g 35743 | Part of proof of Lemma E i... |
| cdleme35h 35744 | Part of proof of Lemma E i... |
| cdleme35h2 35745 | Part of proof of Lemma E i... |
| cdleme35sn2aw 35746 | Part of proof of Lemma E i... |
| cdleme35sn3a 35747 | Part of proof of Lemma E i... |
| cdleme36a 35748 | Part of proof of Lemma E i... |
| cdleme36m 35749 | Part of proof of Lemma E i... |
| cdleme37m 35750 | Part of proof of Lemma E i... |
| cdleme38m 35751 | Part of proof of Lemma E i... |
| cdleme38n 35752 | Part of proof of Lemma E i... |
| cdleme39a 35753 | Part of proof of Lemma E i... |
| cdleme39n 35754 | Part of proof of Lemma E i... |
| cdleme40m 35755 | Part of proof of Lemma E i... |
| cdleme40n 35756 | Part of proof of Lemma E i... |
| cdleme40v 35757 | Part of proof of Lemma E i... |
| cdleme40w 35758 | Part of proof of Lemma E i... |
| cdleme42a 35759 | Part of proof of Lemma E i... |
| cdleme42c 35760 | Part of proof of Lemma E i... |
| cdleme42d 35761 | Part of proof of Lemma E i... |
| cdleme41sn3aw 35762 | Part of proof of Lemma E i... |
| cdleme41sn4aw 35763 | Part of proof of Lemma E i... |
| cdleme41snaw 35764 | Part of proof of Lemma E i... |
| cdleme41fva11 35765 | Part of proof of Lemma E i... |
| cdleme42b 35766 | Part of proof of Lemma E i... |
| cdleme42e 35767 | Part of proof of Lemma E i... |
| cdleme42f 35768 | Part of proof of Lemma E i... |
| cdleme42g 35769 | Part of proof of Lemma E i... |
| cdleme42h 35770 | Part of proof of Lemma E i... |
| cdleme42i 35771 | Part of proof of Lemma E i... |
| cdleme42k 35772 | Part of proof of Lemma E i... |
| cdleme42ke 35773 | Part of proof of Lemma E i... |
| cdleme42keg 35774 | Part of proof of Lemma E i... |
| cdleme42mN 35775 | Part of proof of Lemma E i... |
| cdleme42mgN 35776 | Part of proof of Lemma E i... |
| cdleme43aN 35777 | Part of proof of Lemma E i... |
| cdleme43bN 35778 | Lemma for Lemma E in [Craw... |
| cdleme43cN 35779 | Part of proof of Lemma E i... |
| cdleme43dN 35780 | Part of proof of Lemma E i... |
| cdleme46f2g2 35781 | Conversion for ` G ` to re... |
| cdleme46f2g1 35782 | Conversion for ` G ` to re... |
| cdleme17d2 35783 | Part of proof of Lemma E i... |
| cdleme17d3 35784 | TODO: FIX COMMENT. (Contr... |
| cdleme17d4 35785 | TODO: FIX COMMENT. (Contr... |
| cdleme17d 35786 | Part of proof of Lemma E i... |
| cdleme48fv 35787 | Part of proof of Lemma D i... |
| cdleme48fvg 35788 | Remove ` P =/= Q ` conditi... |
| cdleme46fvaw 35789 | Show that ` ( F `` R ) ` i... |
| cdleme48bw 35790 | TODO: fix comment. TODO: ... |
| cdleme48b 35791 | TODO: fix comment. (Contr... |
| cdleme46frvlpq 35792 | Show that ` ( F `` S ) ` i... |
| cdleme46fsvlpq 35793 | Show that ` ( F `` R ) ` i... |
| cdlemeg46fvcl 35794 | TODO: fix comment. (Contr... |
| cdleme4gfv 35795 | Part of proof of Lemma D i... |
| cdlemeg47b 35796 | TODO: FIX COMMENT. (Contr... |
| cdlemeg47rv 35797 | Value of g_s(r) when r is ... |
| cdlemeg47rv2 35798 | Value of g_s(r) when r is ... |
| cdlemeg49le 35799 | Part of proof of Lemma D i... |
| cdlemeg46bOLDN 35800 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46c 35801 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46rvOLDN 35802 | Value of g_s(r) when r is ... |
| cdlemeg46rv2OLDN 35803 | Value of g_s(r) when r is ... |
| cdlemeg46fvaw 35804 | Show that ` ( F `` R ) ` i... |
| cdlemeg46nlpq 35805 | Show that ` ( G `` S ) ` i... |
| cdlemeg46ngfr 35806 | TODO FIX COMMENT g(f(s))=s... |
| cdlemeg46nfgr 35807 | TODO FIX COMMENT f(g(s))=s... |
| cdlemeg46sfg 35808 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fjgN 35809 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46rjgN 35810 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46fjv 35811 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fsfv 35812 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46frv 35813 | TODO FIX COMMENT. (f(r) ` ... |
| cdlemeg46v1v2 35814 | TODO FIX COMMENT v_1 = v_2... |
| cdlemeg46vrg 35815 | TODO FIX COMMENT v_1 ` <_ ... |
| cdlemeg46rgv 35816 | TODO FIX COMMENT r ` <_ ` ... |
| cdlemeg46req 35817 | TODO FIX COMMENT r = (v_1 ... |
| cdlemeg46gfv 35818 | TODO FIX COMMENT p. 115 pe... |
| cdlemeg46gfr 35819 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gfre 35820 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gf 35821 | TODO FIX COMMENT Eliminate... |
| cdlemeg46fgN 35822 | TODO FIX COMMENT p. 116 pe... |
| cdleme48d 35823 | TODO: fix comment. (Contr... |
| cdleme48gfv1 35824 | TODO: fix comment. (Contr... |
| cdleme48gfv 35825 | TODO: fix comment. (Contr... |
| cdleme48fgv 35826 | TODO: fix comment. (Contr... |
| cdlemeg49lebilem 35827 | Part of proof of Lemma D i... |
| cdleme50lebi 35828 | Part of proof of Lemma D i... |
| cdleme50eq 35829 | Part of proof of Lemma D i... |
| cdleme50f 35830 | Part of proof of Lemma D i... |
| cdleme50f1 35831 | Part of proof of Lemma D i... |
| cdleme50rnlem 35832 | Part of proof of Lemma D i... |
| cdleme50rn 35833 | Part of proof of Lemma D i... |
| cdleme50f1o 35834 | Part of proof of Lemma D i... |
| cdleme50laut 35835 | Part of proof of Lemma D i... |
| cdleme50ldil 35836 | Part of proof of Lemma D i... |
| cdleme50trn1 35837 | Part of proof that ` F ` i... |
| cdleme50trn2a 35838 | Part of proof that ` F ` i... |
| cdleme50trn2 35839 | Part of proof that ` F ` i... |
| cdleme50trn12 35840 | Part of proof that ` F ` i... |
| cdleme50trn3 35841 | Part of proof that ` F ` i... |
| cdleme50trn123 35842 | Part of proof that ` F ` i... |
| cdleme51finvfvN 35843 | Part of proof of Lemma E i... |
| cdleme51finvN 35844 | Part of proof of Lemma E i... |
| cdleme50ltrn 35845 | Part of proof of Lemma E i... |
| cdleme51finvtrN 35846 | Part of proof of Lemma E i... |
| cdleme50ex 35847 | Part of Lemma E in [Crawle... |
| cdleme 35848 | Lemma E in [Crawley] p. 11... |
| cdlemf1 35849 | Part of Lemma F in [Crawle... |
| cdlemf2 35850 | Part of Lemma F in [Crawle... |
| cdlemf 35851 | Lemma F in [Crawley] p. 11... |
| cdlemfnid 35852 | ~ cdlemf with additional c... |
| cdlemftr3 35853 | Special case of ~ cdlemf s... |
| cdlemftr2 35854 | Special case of ~ cdlemf s... |
| cdlemftr1 35855 | Part of proof of Lemma G o... |
| cdlemftr0 35856 | Special case of ~ cdlemf s... |
| trlord 35857 | The ordering of two Hilber... |
| cdlemg1a 35858 | Shorter expression for ` G... |
| cdlemg1b2 35859 | This theorem can be used t... |
| cdlemg1idlemN 35860 | Lemma for ~ cdlemg1idN . ... |
| cdlemg1fvawlemN 35861 | Lemma for ~ ltrniotafvawN ... |
| cdlemg1ltrnlem 35862 | Lemma for ~ ltrniotacl . ... |
| cdlemg1finvtrlemN 35863 | Lemma for ~ ltrniotacnvN .... |
| cdlemg1bOLDN 35864 | This theorem can be used t... |
| cdlemg1idN 35865 | Version of ~ cdleme31id wi... |
| ltrniotafvawN 35866 | Version of ~ cdleme46fvaw ... |
| ltrniotacl 35867 | Version of ~ cdleme50ltrn ... |
| ltrniotacnvN 35868 | Version of ~ cdleme51finvt... |
| ltrniotaval 35869 | Value of the unique transl... |
| ltrniotacnvval 35870 | Converse value of the uniq... |
| ltrniotaidvalN 35871 | Value of the unique transl... |
| ltrniotavalbN 35872 | Value of the unique transl... |
| cdlemeiota 35873 | A translation is uniquely ... |
| cdlemg1ci2 35874 | Any function of the form o... |
| cdlemg1cN 35875 | Any translation belongs to... |
| cdlemg1cex 35876 | Any translation is one of ... |
| cdlemg2cN 35877 | Any translation belongs to... |
| cdlemg2dN 35878 | This theorem can be used t... |
| cdlemg2cex 35879 | Any translation is one of ... |
| cdlemg2ce 35880 | Utility theorem to elimina... |
| cdlemg2jlemOLDN 35881 | Part of proof of Lemma E i... |
| cdlemg2fvlem 35882 | Lemma for ~ cdlemg2fv . (... |
| cdlemg2klem 35883 | ~ cdleme42keg with simpler... |
| cdlemg2idN 35884 | Version of ~ cdleme31id wi... |
| cdlemg3a 35885 | Part of proof of Lemma G i... |
| cdlemg2jOLDN 35886 | TODO: Replace this with ~... |
| cdlemg2fv 35887 | Value of a translation in ... |
| cdlemg2fv2 35888 | Value of a translation in ... |
| cdlemg2k 35889 | ~ cdleme42keg with simpler... |
| cdlemg2kq 35890 | ~ cdlemg2k with ` P ` and ... |
| cdlemg2l 35891 | TODO: FIX COMMENT. (Contr... |
| cdlemg2m 35892 | TODO: FIX COMMENT. (Contr... |
| cdlemg5 35893 | TODO: Is there a simpler ... |
| cdlemb3 35894 | Given two atoms not under ... |
| cdlemg7fvbwN 35895 | Properties of a translatio... |
| cdlemg4a 35896 | TODO: FIX COMMENT If fg(p... |
| cdlemg4b1 35897 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b2 35898 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b12 35899 | TODO: FIX COMMENT. (Contr... |
| cdlemg4c 35900 | TODO: FIX COMMENT. (Contr... |
| cdlemg4d 35901 | TODO: FIX COMMENT. (Contr... |
| cdlemg4e 35902 | TODO: FIX COMMENT. (Contr... |
| cdlemg4f 35903 | TODO: FIX COMMENT. (Contr... |
| cdlemg4g 35904 | TODO: FIX COMMENT. (Contr... |
| cdlemg4 35905 | TODO: FIX COMMENT. (Contr... |
| cdlemg6a 35906 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6b 35907 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6c 35908 | TODO: FIX COMMENT. (Contr... |
| cdlemg6d 35909 | TODO: FIX COMMENT. (Contr... |
| cdlemg6e 35910 | TODO: FIX COMMENT. (Contr... |
| cdlemg6 35911 | TODO: FIX COMMENT. (Contr... |
| cdlemg7fvN 35912 | Value of a translation com... |
| cdlemg7aN 35913 | TODO: FIX COMMENT. (Contr... |
| cdlemg7N 35914 | TODO: FIX COMMENT. (Contr... |
| cdlemg8a 35915 | TODO: FIX COMMENT. (Contr... |
| cdlemg8b 35916 | TODO: FIX COMMENT. (Contr... |
| cdlemg8c 35917 | TODO: FIX COMMENT. (Contr... |
| cdlemg8d 35918 | TODO: FIX COMMENT. (Contr... |
| cdlemg8 35919 | TODO: FIX COMMENT. (Contr... |
| cdlemg9a 35920 | TODO: FIX COMMENT. (Contr... |
| cdlemg9b 35921 | The triples ` <. P , ( F `... |
| cdlemg9 35922 | The triples ` <. P , ( F `... |
| cdlemg10b 35923 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10bALTN 35924 | TODO: FIX COMMENT. TODO: ... |
| cdlemg11a 35925 | TODO: FIX COMMENT. (Contr... |
| cdlemg11aq 35926 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10c 35927 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10a 35928 | TODO: FIX COMMENT. (Contr... |
| cdlemg10 35929 | TODO: FIX COMMENT. (Contr... |
| cdlemg11b 35930 | TODO: FIX COMMENT. (Contr... |
| cdlemg12a 35931 | TODO: FIX COMMENT. (Contr... |
| cdlemg12b 35932 | The triples ` <. P , ( F `... |
| cdlemg12c 35933 | The triples ` <. P , ( F `... |
| cdlemg12d 35934 | TODO: FIX COMMENT. (Contr... |
| cdlemg12e 35935 | TODO: FIX COMMENT. (Contr... |
| cdlemg12f 35936 | TODO: FIX COMMENT. (Contr... |
| cdlemg12g 35937 | TODO: FIX COMMENT. TODO: ... |
| cdlemg12 35938 | TODO: FIX COMMENT. (Contr... |
| cdlemg13a 35939 | TODO: FIX COMMENT. (Contr... |
| cdlemg13 35940 | TODO: FIX COMMENT. (Contr... |
| cdlemg14f 35941 | TODO: FIX COMMENT. (Contr... |
| cdlemg14g 35942 | TODO: FIX COMMENT. (Contr... |
| cdlemg15a 35943 | Eliminate the ` ( F `` P )... |
| cdlemg15 35944 | Eliminate the ` ( (... |
| cdlemg16 35945 | Part of proof of Lemma G o... |
| cdlemg16ALTN 35946 | This version of ~ cdlemg16... |
| cdlemg16z 35947 | Eliminate ` ( ( F `... |
| cdlemg16zz 35948 | Eliminate ` P =/= Q ` from... |
| cdlemg17a 35949 | TODO: FIX COMMENT. (Contr... |
| cdlemg17b 35950 | Part of proof of Lemma G i... |
| cdlemg17dN 35951 | TODO: fix comment. (Contr... |
| cdlemg17dALTN 35952 | Same as ~ cdlemg17dN with ... |
| cdlemg17e 35953 | TODO: fix comment. (Contr... |
| cdlemg17f 35954 | TODO: fix comment. (Contr... |
| cdlemg17g 35955 | TODO: fix comment. (Contr... |
| cdlemg17h 35956 | TODO: fix comment. (Contr... |
| cdlemg17i 35957 | TODO: fix comment. (Contr... |
| cdlemg17ir 35958 | TODO: fix comment. (Contr... |
| cdlemg17j 35959 | TODO: fix comment. (Contr... |
| cdlemg17pq 35960 | Utility theorem for swappi... |
| cdlemg17bq 35961 | ~ cdlemg17b with ` P ` and... |
| cdlemg17iqN 35962 | ~ cdlemg17i with ` P ` and... |
| cdlemg17irq 35963 | ~ cdlemg17ir with ` P ` an... |
| cdlemg17jq 35964 | ~ cdlemg17j with ` P ` and... |
| cdlemg17 35965 | Part of Lemma G of [Crawle... |
| cdlemg18a 35966 | Show two lines are differe... |
| cdlemg18b 35967 | Lemma for ~ cdlemg18c . T... |
| cdlemg18c 35968 | Show two lines intersect a... |
| cdlemg18d 35969 | Show two lines intersect a... |
| cdlemg18 35970 | Show two lines intersect a... |
| cdlemg19a 35971 | Show two lines intersect a... |
| cdlemg19 35972 | Show two lines intersect a... |
| cdlemg20 35973 | Show two lines intersect a... |
| cdlemg21 35974 | Version of cdlemg19 with `... |
| cdlemg22 35975 | ~ cdlemg21 with ` ( F `` P... |
| cdlemg24 35976 | Combine ~ cdlemg16z and ~ ... |
| cdlemg37 35977 | Use ~ cdlemg8 to eliminate... |
| cdlemg25zz 35978 | ~ cdlemg16zz restated for ... |
| cdlemg26zz 35979 | ~ cdlemg16zz restated for ... |
| cdlemg27a 35980 | For use with case when ` (... |
| cdlemg28a 35981 | Part of proof of Lemma G o... |
| cdlemg31b0N 35982 | TODO: Fix comment. (Cont... |
| cdlemg31b0a 35983 | TODO: Fix comment. (Cont... |
| cdlemg27b 35984 | TODO: Fix comment. (Cont... |
| cdlemg31a 35985 | TODO: fix comment. (Contr... |
| cdlemg31b 35986 | TODO: fix comment. (Contr... |
| cdlemg31c 35987 | Show that when ` N ` is an... |
| cdlemg31d 35988 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33b0 35989 | TODO: Fix comment. (Cont... |
| cdlemg33c0 35990 | TODO: Fix comment. (Cont... |
| cdlemg28b 35991 | Part of proof of Lemma G o... |
| cdlemg28 35992 | Part of proof of Lemma G o... |
| cdlemg29 35993 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33a 35994 | TODO: Fix comment. (Cont... |
| cdlemg33b 35995 | TODO: Fix comment. (Cont... |
| cdlemg33c 35996 | TODO: Fix comment. (Cont... |
| cdlemg33d 35997 | TODO: Fix comment. (Cont... |
| cdlemg33e 35998 | TODO: Fix comment. (Cont... |
| cdlemg33 35999 | Combine ~ cdlemg33b , ~ cd... |
| cdlemg34 36000 | Use cdlemg33 to eliminate ... |
| cdlemg35 36001 | TODO: Fix comment. TODO:... |
| cdlemg36 36002 | Use cdlemg35 to eliminate ... |
| cdlemg38 36003 | Use ~ cdlemg37 to eliminat... |
| cdlemg39 36004 | Eliminate ` =/= ` conditio... |
| cdlemg40 36005 | Eliminate ` P =/= Q ` cond... |
| cdlemg41 36006 | Convert ~ cdlemg40 to func... |
| ltrnco 36007 | The composition of two tra... |
| trlcocnv 36008 | Swap the arguments of the ... |
| trlcoabs 36009 | Absorption into a composit... |
| trlcoabs2N 36010 | Absorption of the trace of... |
| trlcoat 36011 | The trace of a composition... |
| trlcocnvat 36012 | Commonly used special case... |
| trlconid 36013 | The composition of two dif... |
| trlcolem 36014 | Lemma for ~ trlco . (Cont... |
| trlco 36015 | The trace of a composition... |
| trlcone 36016 | If two translations have d... |
| cdlemg42 36017 | Part of proof of Lemma G o... |
| cdlemg43 36018 | Part of proof of Lemma G o... |
| cdlemg44a 36019 | Part of proof of Lemma G o... |
| cdlemg44b 36020 | Eliminate ` ( F `` P ) =/=... |
| cdlemg44 36021 | Part of proof of Lemma G o... |
| cdlemg47a 36022 | TODO: fix comment. TODO: ... |
| cdlemg46 36023 | Part of proof of Lemma G o... |
| cdlemg47 36024 | Part of proof of Lemma G o... |
| cdlemg48 36025 | Elmininate ` h ` from ~ cd... |
| ltrncom 36026 | Composition is commutative... |
| ltrnco4 36027 | Rearrange a composition of... |
| trljco 36028 | Trace joined with trace of... |
| trljco2 36029 | Trace joined with trace of... |
| tgrpfset 36032 | The translation group maps... |
| tgrpset 36033 | The translation group for ... |
| tgrpbase 36034 | The base set of the transl... |
| tgrpopr 36035 | The group operation of the... |
| tgrpov 36036 | The group operation value ... |
| tgrpgrplem 36037 | Lemma for ~ tgrpgrp . (Co... |
| tgrpgrp 36038 | The translation group is a... |
| tgrpabl 36039 | The translation group is a... |
| tendofset 36046 | The set of all trace-prese... |
| tendoset 36047 | The set of trace-preservin... |
| istendo 36048 | The predicate "is a trace-... |
| tendotp 36049 | Trace-preserving property ... |
| istendod 36050 | Deduce the predicate "is a... |
| tendof 36051 | Functionality of a trace-p... |
| tendoeq1 36052 | Condition determining equa... |
| tendovalco 36053 | Value of composition of tr... |
| tendocoval 36054 | Value of composition of en... |
| tendocl 36055 | Closure of a trace-preserv... |
| tendoco2 36056 | Distribution of compositio... |
| tendoidcl 36057 | The identity is a trace-pr... |
| tendo1mul 36058 | Multiplicative identity mu... |
| tendo1mulr 36059 | Multiplicative identity mu... |
| tendococl 36060 | The composition of two tra... |
| tendoid 36061 | The identity value of a tr... |
| tendoeq2 36062 | Condition determining equa... |
| tendoplcbv 36063 | Define sum operation for t... |
| tendopl 36064 | Value of endomorphism sum ... |
| tendopl2 36065 | Value of result of endomor... |
| tendoplcl2 36066 | Value of result of endomor... |
| tendoplco2 36067 | Value of result of endomor... |
| tendopltp 36068 | Trace-preserving property ... |
| tendoplcl 36069 | Endomorphism sum is a trac... |
| tendoplcom 36070 | The endomorphism sum opera... |
| tendoplass 36071 | The endomorphism sum opera... |
| tendodi1 36072 | Endomorphism composition d... |
| tendodi2 36073 | Endomorphism composition d... |
| tendo0cbv 36074 | Define additive identity f... |
| tendo02 36075 | Value of additive identity... |
| tendo0co2 36076 | The additive identity trac... |
| tendo0tp 36077 | Trace-preserving property ... |
| tendo0cl 36078 | The additive identity is a... |
| tendo0pl 36079 | Property of the additive i... |
| tendo0plr 36080 | Property of the additive i... |
| tendoicbv 36081 | Define inverse function fo... |
| tendoi 36082 | Value of inverse endomorph... |
| tendoi2 36083 | Value of additive inverse ... |
| tendoicl 36084 | Closure of the additive in... |
| tendoipl 36085 | Property of the additive i... |
| tendoipl2 36086 | Property of the additive i... |
| erngfset 36087 | The division rings on trac... |
| erngset 36088 | The division ring on trace... |
| erngbase 36089 | The base set of the divisi... |
| erngfplus 36090 | Ring addition operation. ... |
| erngplus 36091 | Ring addition operation. ... |
| erngplus2 36092 | Ring addition operation. ... |
| erngfmul 36093 | Ring multiplication operat... |
| erngmul 36094 | Ring addition operation. ... |
| erngfset-rN 36095 | The division rings on trac... |
| erngset-rN 36096 | The division ring on trace... |
| erngbase-rN 36097 | The base set of the divisi... |
| erngfplus-rN 36098 | Ring addition operation. ... |
| erngplus-rN 36099 | Ring addition operation. ... |
| erngplus2-rN 36100 | Ring addition operation. ... |
| erngfmul-rN 36101 | Ring multiplication operat... |
| erngmul-rN 36102 | Ring addition operation. ... |
| cdlemh1 36103 | Part of proof of Lemma H o... |
| cdlemh2 36104 | Part of proof of Lemma H o... |
| cdlemh 36105 | Lemma H of [Crawley] p. 11... |
| cdlemi1 36106 | Part of proof of Lemma I o... |
| cdlemi2 36107 | Part of proof of Lemma I o... |
| cdlemi 36108 | Lemma I of [Crawley] p. 11... |
| cdlemj1 36109 | Part of proof of Lemma J o... |
| cdlemj2 36110 | Part of proof of Lemma J o... |
| cdlemj3 36111 | Part of proof of Lemma J o... |
| tendocan 36112 | Cancellation law: if the v... |
| tendoid0 36113 | A trace-preserving endomor... |
| tendo0mul 36114 | Additive identity multipli... |
| tendo0mulr 36115 | Additive identity multipli... |
| tendo1ne0 36116 | The identity (unity) is no... |
| tendoconid 36117 | The composition (product) ... |
| tendotr 36118 | The trace of the value of ... |
| cdlemk1 36119 | Part of proof of Lemma K o... |
| cdlemk2 36120 | Part of proof of Lemma K o... |
| cdlemk3 36121 | Part of proof of Lemma K o... |
| cdlemk4 36122 | Part of proof of Lemma K o... |
| cdlemk5a 36123 | Part of proof of Lemma K o... |
| cdlemk5 36124 | Part of proof of Lemma K o... |
| cdlemk6 36125 | Part of proof of Lemma K o... |
| cdlemk8 36126 | Part of proof of Lemma K o... |
| cdlemk9 36127 | Part of proof of Lemma K o... |
| cdlemk9bN 36128 | Part of proof of Lemma K o... |
| cdlemki 36129 | Part of proof of Lemma K o... |
| cdlemkvcl 36130 | Part of proof of Lemma K o... |
| cdlemk10 36131 | Part of proof of Lemma K o... |
| cdlemksv 36132 | Part of proof of Lemma K o... |
| cdlemksel 36133 | Part of proof of Lemma K o... |
| cdlemksat 36134 | Part of proof of Lemma K o... |
| cdlemksv2 36135 | Part of proof of Lemma K o... |
| cdlemk7 36136 | Part of proof of Lemma K o... |
| cdlemk11 36137 | Part of proof of Lemma K o... |
| cdlemk12 36138 | Part of proof of Lemma K o... |
| cdlemkoatnle 36139 | Utility lemma. (Contribut... |
| cdlemk13 36140 | Part of proof of Lemma K o... |
| cdlemkole 36141 | Utility lemma. (Contribut... |
| cdlemk14 36142 | Part of proof of Lemma K o... |
| cdlemk15 36143 | Part of proof of Lemma K o... |
| cdlemk16a 36144 | Part of proof of Lemma K o... |
| cdlemk16 36145 | Part of proof of Lemma K o... |
| cdlemk17 36146 | Part of proof of Lemma K o... |
| cdlemk1u 36147 | Part of proof of Lemma K o... |
| cdlemk5auN 36148 | Part of proof of Lemma K o... |
| cdlemk5u 36149 | Part of proof of Lemma K o... |
| cdlemk6u 36150 | Part of proof of Lemma K o... |
| cdlemkj 36151 | Part of proof of Lemma K o... |
| cdlemkuvN 36152 | Part of proof of Lemma K o... |
| cdlemkuel 36153 | Part of proof of Lemma K o... |
| cdlemkuat 36154 | Part of proof of Lemma K o... |
| cdlemkuv2 36155 | Part of proof of Lemma K o... |
| cdlemk18 36156 | Part of proof of Lemma K o... |
| cdlemk19 36157 | Part of proof of Lemma K o... |
| cdlemk7u 36158 | Part of proof of Lemma K o... |
| cdlemk11u 36159 | Part of proof of Lemma K o... |
| cdlemk12u 36160 | Part of proof of Lemma K o... |
| cdlemk21N 36161 | Part of proof of Lemma K o... |
| cdlemk20 36162 | Part of proof of Lemma K o... |
| cdlemkoatnle-2N 36163 | Utility lemma. (Contribut... |
| cdlemk13-2N 36164 | Part of proof of Lemma K o... |
| cdlemkole-2N 36165 | Utility lemma. (Contribut... |
| cdlemk14-2N 36166 | Part of proof of Lemma K o... |
| cdlemk15-2N 36167 | Part of proof of Lemma K o... |
| cdlemk16-2N 36168 | Part of proof of Lemma K o... |
| cdlemk17-2N 36169 | Part of proof of Lemma K o... |
| cdlemkj-2N 36170 | Part of proof of Lemma K o... |
| cdlemkuv-2N 36171 | Part of proof of Lemma K o... |
| cdlemkuel-2N 36172 | Part of proof of Lemma K o... |
| cdlemkuv2-2 36173 | Part of proof of Lemma K o... |
| cdlemk18-2N 36174 | Part of proof of Lemma K o... |
| cdlemk19-2N 36175 | Part of proof of Lemma K o... |
| cdlemk7u-2N 36176 | Part of proof of Lemma K o... |
| cdlemk11u-2N 36177 | Part of proof of Lemma K o... |
| cdlemk12u-2N 36178 | Part of proof of Lemma K o... |
| cdlemk21-2N 36179 | Part of proof of Lemma K o... |
| cdlemk20-2N 36180 | Part of proof of Lemma K o... |
| cdlemk22 36181 | Part of proof of Lemma K o... |
| cdlemk30 36182 | Part of proof of Lemma K o... |
| cdlemkuu 36183 | Convert between function a... |
| cdlemk31 36184 | Part of proof of Lemma K o... |
| cdlemk32 36185 | Part of proof of Lemma K o... |
| cdlemkuel-3 36186 | Part of proof of Lemma K o... |
| cdlemkuv2-3N 36187 | Part of proof of Lemma K o... |
| cdlemk18-3N 36188 | Part of proof of Lemma K o... |
| cdlemk22-3 36189 | Part of proof of Lemma K o... |
| cdlemk23-3 36190 | Part of proof of Lemma K o... |
| cdlemk24-3 36191 | Part of proof of Lemma K o... |
| cdlemk25-3 36192 | Part of proof of Lemma K o... |
| cdlemk26b-3 36193 | Part of proof of Lemma K o... |
| cdlemk26-3 36194 | Part of proof of Lemma K o... |
| cdlemk27-3 36195 | Part of proof of Lemma K o... |
| cdlemk28-3 36196 | Part of proof of Lemma K o... |
| cdlemk33N 36197 | Part of proof of Lemma K o... |
| cdlemk34 36198 | Part of proof of Lemma K o... |
| cdlemk29-3 36199 | Part of proof of Lemma K o... |
| cdlemk35 36200 | Part of proof of Lemma K o... |
| cdlemk36 36201 | Part of proof of Lemma K o... |
| cdlemk37 36202 | Part of proof of Lemma K o... |
| cdlemk38 36203 | Part of proof of Lemma K o... |
| cdlemk39 36204 | Part of proof of Lemma K o... |
| cdlemk40 36205 | TODO: fix comment. (Contr... |
| cdlemk40t 36206 | TODO: fix comment. (Contr... |
| cdlemk40f 36207 | TODO: fix comment. (Contr... |
| cdlemk41 36208 | Part of proof of Lemma K o... |
| cdlemkfid1N 36209 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid1 36210 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid2N 36211 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid2 36212 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid3N 36213 | TODO: is this useful or sh... |
| cdlemky 36214 | Part of proof of Lemma K o... |
| cdlemkyu 36215 | Convert between function a... |
| cdlemkyuu 36216 | ~ cdlemkyu with some hypot... |
| cdlemk11ta 36217 | Part of proof of Lemma K o... |
| cdlemk19ylem 36218 | Lemma for ~ cdlemk19y . (... |
| cdlemk11tb 36219 | Part of proof of Lemma K o... |
| cdlemk19y 36220 | ~ cdlemk19 with simpler hy... |
| cdlemkid3N 36221 | Lemma for ~ cdlemkid . (C... |
| cdlemkid4 36222 | Lemma for ~ cdlemkid . (C... |
| cdlemkid5 36223 | Lemma for ~ cdlemkid . (C... |
| cdlemkid 36224 | The value of the tau funct... |
| cdlemk35s 36225 | Substitution version of ~ ... |
| cdlemk35s-id 36226 | Substitution version of ~ ... |
| cdlemk39s 36227 | Substitution version of ~ ... |
| cdlemk39s-id 36228 | Substitution version of ~ ... |
| cdlemk42 36229 | Part of proof of Lemma K o... |
| cdlemk19xlem 36230 | Lemma for ~ cdlemk19x . (... |
| cdlemk19x 36231 | ~ cdlemk19 with simpler hy... |
| cdlemk42yN 36232 | Part of proof of Lemma K o... |
| cdlemk11tc 36233 | Part of proof of Lemma K o... |
| cdlemk11t 36234 | Part of proof of Lemma K o... |
| cdlemk45 36235 | Part of proof of Lemma K o... |
| cdlemk46 36236 | Part of proof of Lemma K o... |
| cdlemk47 36237 | Part of proof of Lemma K o... |
| cdlemk48 36238 | Part of proof of Lemma K o... |
| cdlemk49 36239 | Part of proof of Lemma K o... |
| cdlemk50 36240 | Part of proof of Lemma K o... |
| cdlemk51 36241 | Part of proof of Lemma K o... |
| cdlemk52 36242 | Part of proof of Lemma K o... |
| cdlemk53a 36243 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53b 36244 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53 36245 | Part of proof of Lemma K o... |
| cdlemk54 36246 | Part of proof of Lemma K o... |
| cdlemk55a 36247 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55b 36248 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55 36249 | Part of proof of Lemma K o... |
| cdlemkyyN 36250 | Part of proof of Lemma K o... |
| cdlemk43N 36251 | Part of proof of Lemma K o... |
| cdlemk35u 36252 | Substitution version of ~ ... |
| cdlemk55u1 36253 | Lemma for ~ cdlemk55u . (... |
| cdlemk55u 36254 | Part of proof of Lemma K o... |
| cdlemk39u1 36255 | Lemma for ~ cdlemk39u . (... |
| cdlemk39u 36256 | Part of proof of Lemma K o... |
| cdlemk19u1 36257 | ~ cdlemk19 with simpler hy... |
| cdlemk19u 36258 | Part of Lemma K of [Crawle... |
| cdlemk56 36259 | Part of Lemma K of [Crawle... |
| cdlemk19w 36260 | Use a fixed element to eli... |
| cdlemk56w 36261 | Use a fixed element to eli... |
| cdlemk 36262 | Lemma K of [Crawley] p. 11... |
| tendoex 36263 | Generalization of Lemma K ... |
| cdleml1N 36264 | Part of proof of Lemma L o... |
| cdleml2N 36265 | Part of proof of Lemma L o... |
| cdleml3N 36266 | Part of proof of Lemma L o... |
| cdleml4N 36267 | Part of proof of Lemma L o... |
| cdleml5N 36268 | Part of proof of Lemma L o... |
| cdleml6 36269 | Part of proof of Lemma L o... |
| cdleml7 36270 | Part of proof of Lemma L o... |
| cdleml8 36271 | Part of proof of Lemma L o... |
| cdleml9 36272 | Part of proof of Lemma L o... |
| dva1dim 36273 | Two expressions for the 1-... |
| dvhb1dimN 36274 | Two expressions for the 1-... |
| erng1lem 36275 | Value of the endomorphism ... |
| erngdvlem1 36276 | Lemma for ~ eringring . (... |
| erngdvlem2N 36277 | Lemma for ~ eringring . (... |
| erngdvlem3 36278 | Lemma for ~ eringring . (... |
| erngdvlem4 36279 | Lemma for ~ erngdv . (Con... |
| eringring 36280 | An endomorphism ring is a ... |
| erngdv 36281 | An endomorphism ring is a ... |
| erng0g 36282 | The division ring zero of ... |
| erng1r 36283 | The division ring unit of ... |
| erngdvlem1-rN 36284 | Lemma for ~ eringring . (... |
| erngdvlem2-rN 36285 | Lemma for ~ eringring . (... |
| erngdvlem3-rN 36286 | Lemma for ~ eringring . (... |
| erngdvlem4-rN 36287 | Lemma for ~ erngdv . (Con... |
| erngring-rN 36288 | An endomorphism ring is a ... |
| erngdv-rN 36289 | An endomorphism ring is a ... |
| dvafset 36292 | The constructed partial ve... |
| dvaset 36293 | The constructed partial ve... |
| dvasca 36294 | The ring base set of the c... |
| dvabase 36295 | The ring base set of the c... |
| dvafplusg 36296 | Ring addition operation fo... |
| dvaplusg 36297 | Ring addition operation fo... |
| dvaplusgv 36298 | Ring addition operation fo... |
| dvafmulr 36299 | Ring multiplication operat... |
| dvamulr 36300 | Ring multiplication operat... |
| dvavbase 36301 | The vectors (vector base s... |
| dvafvadd 36302 | The vector sum operation f... |
| dvavadd 36303 | Ring addition operation fo... |
| dvafvsca 36304 | Ring addition operation fo... |
| dvavsca 36305 | Ring addition operation fo... |
| tendospid 36306 | Identity property of endom... |
| tendospcl 36307 | Closure of endomorphism sc... |
| tendospass 36308 | Associative law for endomo... |
| tendospdi1 36309 | Forward distributive law f... |
| tendocnv 36310 | Converse of a trace-preser... |
| tendospdi2 36311 | Reverse distributive law f... |
| tendospcanN 36312 | Cancellation law for trace... |
| dvaabl 36313 | The constructed partial ve... |
| dvalveclem 36314 | Lemma for ~ dvalvec . (Co... |
| dvalvec 36315 | The constructed partial ve... |
| dva0g 36316 | The zero vector of partial... |
| diaffval 36319 | The partial isomorphism A ... |
| diafval 36320 | The partial isomorphism A ... |
| diaval 36321 | The partial isomorphism A ... |
| diaelval 36322 | Member of the partial isom... |
| diafn 36323 | Functionality and domain o... |
| diadm 36324 | Domain of the partial isom... |
| diaeldm 36325 | Member of domain of the pa... |
| diadmclN 36326 | A member of domain of the ... |
| diadmleN 36327 | A member of domain of the ... |
| dian0 36328 | The value of the partial i... |
| dia0eldmN 36329 | The lattice zero belongs t... |
| dia1eldmN 36330 | The fiducial hyperplane (t... |
| diass 36331 | The value of the partial i... |
| diael 36332 | A member of the value of t... |
| diatrl 36333 | Trace of a member of the p... |
| diaelrnN 36334 | Any value of the partial i... |
| dialss 36335 | The value of partial isomo... |
| diaord 36336 | The partial isomorphism A ... |
| dia11N 36337 | The partial isomorphism A ... |
| diaf11N 36338 | The partial isomorphism A ... |
| diaclN 36339 | Closure of partial isomorp... |
| diacnvclN 36340 | Closure of partial isomorp... |
| dia0 36341 | The value of the partial i... |
| dia1N 36342 | The value of the partial i... |
| dia1elN 36343 | The largest subspace in th... |
| diaglbN 36344 | Partial isomorphism A of a... |
| diameetN 36345 | Partial isomorphism A of a... |
| diainN 36346 | Inverse partial isomorphis... |
| diaintclN 36347 | The intersection of partia... |
| diasslssN 36348 | The partial isomorphism A ... |
| diassdvaN 36349 | The partial isomorphism A ... |
| dia1dim 36350 | Two expressions for the 1-... |
| dia1dim2 36351 | Two expressions for a 1-di... |
| dia1dimid 36352 | A vector (translation) bel... |
| dia2dimlem1 36353 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem2 36354 | Lemma for ~ dia2dim . Def... |
| dia2dimlem3 36355 | Lemma for ~ dia2dim . Def... |
| dia2dimlem4 36356 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem5 36357 | Lemma for ~ dia2dim . The... |
| dia2dimlem6 36358 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem7 36359 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem8 36360 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem9 36361 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem10 36362 | Lemma for ~ dia2dim . Con... |
| dia2dimlem11 36363 | Lemma for ~ dia2dim . Con... |
| dia2dimlem12 36364 | Lemma for ~ dia2dim . Obt... |
| dia2dimlem13 36365 | Lemma for ~ dia2dim . Eli... |
| dia2dim 36366 | A two-dimensional subspace... |
| dvhfset 36369 | The constructed full vecto... |
| dvhset 36370 | The constructed full vecto... |
| dvhsca 36371 | The ring of scalars of the... |
| dvhbase 36372 | The ring base set of the c... |
| dvhfplusr 36373 | Ring addition operation fo... |
| dvhfmulr 36374 | Ring multiplication operat... |
| dvhmulr 36375 | Ring multiplication operat... |
| dvhvbase 36376 | The vectors (vector base s... |
| dvhelvbasei 36377 | Vector membership in the c... |
| dvhvaddcbv 36378 | Change bound variables to ... |
| dvhvaddval 36379 | The vector sum operation f... |
| dvhfvadd 36380 | The vector sum operation f... |
| dvhvadd 36381 | The vector sum operation f... |
| dvhopvadd 36382 | The vector sum operation f... |
| dvhopvadd2 36383 | The vector sum operation f... |
| dvhvaddcl 36384 | Closure of the vector sum ... |
| dvhvaddcomN 36385 | Commutativity of vector su... |
| dvhvaddass 36386 | Associativity of vector su... |
| dvhvscacbv 36387 | Change bound variables to ... |
| dvhvscaval 36388 | The scalar product operati... |
| dvhfvsca 36389 | Scalar product operation f... |
| dvhvsca 36390 | Scalar product operation f... |
| dvhopvsca 36391 | Scalar product operation f... |
| dvhvscacl 36392 | Closure of the scalar prod... |
| tendoinvcl 36393 | Closure of multiplicative ... |
| tendolinv 36394 | Left multiplicative invers... |
| tendorinv 36395 | Right multiplicative inver... |
| dvhgrp 36396 | The full vector space ` U ... |
| dvhlveclem 36397 | Lemma for ~ dvhlvec . TOD... |
| dvhlvec 36398 | The full vector space ` U ... |
| dvhlmod 36399 | The full vector space ` U ... |
| dvh0g 36400 | The zero vector of vector ... |
| dvheveccl 36401 | Properties of a unit vecto... |
| dvhopclN 36402 | Closure of a ` DVecH ` vec... |
| dvhopaddN 36403 | Sum of ` DVecH ` vectors e... |
| dvhopspN 36404 | Scalar product of ` DVecH ... |
| dvhopN 36405 | Decompose a ` DVecH ` vect... |
| dvhopellsm 36406 | Ordered pair membership in... |
| cdlemm10N 36407 | The image of the map ` G `... |
| docaffvalN 36410 | Subspace orthocomplement f... |
| docafvalN 36411 | Subspace orthocomplement f... |
| docavalN 36412 | Subspace orthocomplement f... |
| docaclN 36413 | Closure of subspace orthoc... |
| diaocN 36414 | Value of partial isomorphi... |
| doca2N 36415 | Double orthocomplement of ... |
| doca3N 36416 | Double orthocomplement of ... |
| dvadiaN 36417 | Any closed subspace is a m... |
| diarnN 36418 | Partial isomorphism A maps... |
| diaf1oN 36419 | The partial isomorphism A ... |
| djaffvalN 36422 | Subspace join for ` DVecA ... |
| djafvalN 36423 | Subspace join for ` DVecA ... |
| djavalN 36424 | Subspace join for ` DVecA ... |
| djaclN 36425 | Closure of subspace join f... |
| djajN 36426 | Transfer lattice join to `... |
| dibffval 36429 | The partial isomorphism B ... |
| dibfval 36430 | The partial isomorphism B ... |
| dibval 36431 | The partial isomorphism B ... |
| dibopelvalN 36432 | Member of the partial isom... |
| dibval2 36433 | Value of the partial isomo... |
| dibopelval2 36434 | Member of the partial isom... |
| dibval3N 36435 | Value of the partial isomo... |
| dibelval3 36436 | Member of the partial isom... |
| dibopelval3 36437 | Member of the partial isom... |
| dibelval1st 36438 | Membership in value of the... |
| dibelval1st1 36439 | Membership in value of the... |
| dibelval1st2N 36440 | Membership in value of the... |
| dibelval2nd 36441 | Membership in value of the... |
| dibn0 36442 | The value of the partial i... |
| dibfna 36443 | Functionality and domain o... |
| dibdiadm 36444 | Domain of the partial isom... |
| dibfnN 36445 | Functionality and domain o... |
| dibdmN 36446 | Domain of the partial isom... |
| dibeldmN 36447 | Member of domain of the pa... |
| dibord 36448 | The isomorphism B for a la... |
| dib11N 36449 | The isomorphism B for a la... |
| dibf11N 36450 | The partial isomorphism A ... |
| dibclN 36451 | Closure of partial isomorp... |
| dibvalrel 36452 | The value of partial isomo... |
| dib0 36453 | The value of partial isomo... |
| dib1dim 36454 | Two expressions for the 1-... |
| dibglbN 36455 | Partial isomorphism B of a... |
| dibintclN 36456 | The intersection of partia... |
| dib1dim2 36457 | Two expressions for a 1-di... |
| dibss 36458 | The partial isomorphism B ... |
| diblss 36459 | The value of partial isomo... |
| diblsmopel 36460 | Membership in subspace sum... |
| dicffval 36463 | The partial isomorphism C ... |
| dicfval 36464 | The partial isomorphism C ... |
| dicval 36465 | The partial isomorphism C ... |
| dicopelval 36466 | Membership in value of the... |
| dicelvalN 36467 | Membership in value of the... |
| dicval2 36468 | The partial isomorphism C ... |
| dicelval3 36469 | Member of the partial isom... |
| dicopelval2 36470 | Membership in value of the... |
| dicelval2N 36471 | Membership in value of the... |
| dicfnN 36472 | Functionality and domain o... |
| dicdmN 36473 | Domain of the partial isom... |
| dicvalrelN 36474 | The value of partial isomo... |
| dicssdvh 36475 | The partial isomorphism C ... |
| dicelval1sta 36476 | Membership in value of the... |
| dicelval1stN 36477 | Membership in value of the... |
| dicelval2nd 36478 | Membership in value of the... |
| dicvaddcl 36479 | Membership in value of the... |
| dicvscacl 36480 | Membership in value of the... |
| dicn0 36481 | The value of the partial i... |
| diclss 36482 | The value of partial isomo... |
| diclspsn 36483 | The value of isomorphism C... |
| cdlemn2 36484 | Part of proof of Lemma N o... |
| cdlemn2a 36485 | Part of proof of Lemma N o... |
| cdlemn3 36486 | Part of proof of Lemma N o... |
| cdlemn4 36487 | Part of proof of Lemma N o... |
| cdlemn4a 36488 | Part of proof of Lemma N o... |
| cdlemn5pre 36489 | Part of proof of Lemma N o... |
| cdlemn5 36490 | Part of proof of Lemma N o... |
| cdlemn6 36491 | Part of proof of Lemma N o... |
| cdlemn7 36492 | Part of proof of Lemma N o... |
| cdlemn8 36493 | Part of proof of Lemma N o... |
| cdlemn9 36494 | Part of proof of Lemma N o... |
| cdlemn10 36495 | Part of proof of Lemma N o... |
| cdlemn11a 36496 | Part of proof of Lemma N o... |
| cdlemn11b 36497 | Part of proof of Lemma N o... |
| cdlemn11c 36498 | Part of proof of Lemma N o... |
| cdlemn11pre 36499 | Part of proof of Lemma N o... |
| cdlemn11 36500 | Part of proof of Lemma N o... |
| cdlemn 36501 | Lemma N of [Crawley] p. 12... |
| dihordlem6 36502 | Part of proof of Lemma N o... |
| dihordlem7 36503 | Part of proof of Lemma N o... |
| dihordlem7b 36504 | Part of proof of Lemma N o... |
| dihjustlem 36505 | Part of proof after Lemma ... |
| dihjust 36506 | Part of proof after Lemma ... |
| dihord1 36507 | Part of proof after Lemma ... |
| dihord2a 36508 | Part of proof after Lemma ... |
| dihord2b 36509 | Part of proof after Lemma ... |
| dihord2cN 36510 | Part of proof after Lemma ... |
| dihord11b 36511 | Part of proof after Lemma ... |
| dihord10 36512 | Part of proof after Lemma ... |
| dihord11c 36513 | Part of proof after Lemma ... |
| dihord2pre 36514 | Part of proof after Lemma ... |
| dihord2pre2 36515 | Part of proof after Lemma ... |
| dihord2 36516 | Part of proof after Lemma ... |
| dihffval 36519 | The isomorphism H for a la... |
| dihfval 36520 | Isomorphism H for a lattic... |
| dihval 36521 | Value of isomorphism H for... |
| dihvalc 36522 | Value of isomorphism H for... |
| dihlsscpre 36523 | Closure of isomorphism H f... |
| dihvalcqpre 36524 | Value of isomorphism H for... |
| dihvalcq 36525 | Value of isomorphism H for... |
| dihvalb 36526 | Value of isomorphism H for... |
| dihopelvalbN 36527 | Ordered pair member of the... |
| dihvalcqat 36528 | Value of isomorphism H for... |
| dih1dimb 36529 | Two expressions for a 1-di... |
| dih1dimb2 36530 | Isomorphism H at an atom u... |
| dih1dimc 36531 | Isomorphism H at an atom n... |
| dib2dim 36532 | Extend ~ dia2dim to partia... |
| dih2dimb 36533 | Extend ~ dib2dim to isomor... |
| dih2dimbALTN 36534 | Extend ~ dia2dim to isomor... |
| dihopelvalcqat 36535 | Ordered pair member of the... |
| dihvalcq2 36536 | Value of isomorphism H for... |
| dihopelvalcpre 36537 | Member of value of isomorp... |
| dihopelvalc 36538 | Member of value of isomorp... |
| dihlss 36539 | The value of isomorphism H... |
| dihss 36540 | The value of isomorphism H... |
| dihssxp 36541 | An isomorphism H value is ... |
| dihopcl 36542 | Closure of an ordered pair... |
| xihopellsmN 36543 | Ordered pair membership in... |
| dihopellsm 36544 | Ordered pair membership in... |
| dihord6apre 36545 | Part of proof that isomorp... |
| dihord3 36546 | The isomorphism H for a la... |
| dihord4 36547 | The isomorphism H for a la... |
| dihord5b 36548 | Part of proof that isomorp... |
| dihord6b 36549 | Part of proof that isomorp... |
| dihord6a 36550 | Part of proof that isomorp... |
| dihord5apre 36551 | Part of proof that isomorp... |
| dihord5a 36552 | Part of proof that isomorp... |
| dihord 36553 | The isomorphism H is order... |
| dih11 36554 | The isomorphism H is one-t... |
| dihf11lem 36555 | Functionality of the isomo... |
| dihf11 36556 | The isomorphism H for a la... |
| dihfn 36557 | Functionality and domain o... |
| dihdm 36558 | Domain of isomorphism H. (... |
| dihcl 36559 | Closure of isomorphism H. ... |
| dihcnvcl 36560 | Closure of isomorphism H c... |
| dihcnvid1 36561 | The converse isomorphism o... |
| dihcnvid2 36562 | The isomorphism of a conve... |
| dihcnvord 36563 | Ordering property for conv... |
| dihcnv11 36564 | The converse of isomorphis... |
| dihsslss 36565 | The isomorphism H maps to ... |
| dihrnlss 36566 | The isomorphism H maps to ... |
| dihrnss 36567 | The isomorphism H maps to ... |
| dihvalrel 36568 | The value of isomorphism H... |
| dih0 36569 | The value of isomorphism H... |
| dih0bN 36570 | A lattice element is zero ... |
| dih0vbN 36571 | A vector is zero iff its s... |
| dih0cnv 36572 | The isomorphism H converse... |
| dih0rn 36573 | The zero subspace belongs ... |
| dih0sb 36574 | A subspace is zero iff the... |
| dih1 36575 | The value of isomorphism H... |
| dih1rn 36576 | The full vector space belo... |
| dih1cnv 36577 | The isomorphism H converse... |
| dihwN 36578 | Value of isomorphism H at ... |
| dihmeetlem1N 36579 | Isomorphism H of a conjunc... |
| dihglblem5apreN 36580 | A conjunction property of ... |
| dihglblem5aN 36581 | A conjunction property of ... |
| dihglblem2aN 36582 | Lemma for isomorphism H of... |
| dihglblem2N 36583 | The GLB of a set of lattic... |
| dihglblem3N 36584 | Isomorphism H of a lattice... |
| dihglblem3aN 36585 | Isomorphism H of a lattice... |
| dihglblem4 36586 | Isomorphism H of a lattice... |
| dihglblem5 36587 | Isomorphism H of a lattice... |
| dihmeetlem2N 36588 | Isomorphism H of a conjunc... |
| dihglbcpreN 36589 | Isomorphism H of a lattice... |
| dihglbcN 36590 | Isomorphism H of a lattice... |
| dihmeetcN 36591 | Isomorphism H of a lattice... |
| dihmeetbN 36592 | Isomorphism H of a lattice... |
| dihmeetbclemN 36593 | Lemma for isomorphism H of... |
| dihmeetlem3N 36594 | Lemma for isomorphism H of... |
| dihmeetlem4preN 36595 | Lemma for isomorphism H of... |
| dihmeetlem4N 36596 | Lemma for isomorphism H of... |
| dihmeetlem5 36597 | Part of proof that isomorp... |
| dihmeetlem6 36598 | Lemma for isomorphism H of... |
| dihmeetlem7N 36599 | Lemma for isomorphism H of... |
| dihjatc1 36600 | Lemma for isomorphism H of... |
| dihjatc2N 36601 | Isomorphism H of join with... |
| dihjatc3 36602 | Isomorphism H of join with... |
| dihmeetlem8N 36603 | Lemma for isomorphism H of... |
| dihmeetlem9N 36604 | Lemma for isomorphism H of... |
| dihmeetlem10N 36605 | Lemma for isomorphism H of... |
| dihmeetlem11N 36606 | Lemma for isomorphism H of... |
| dihmeetlem12N 36607 | Lemma for isomorphism H of... |
| dihmeetlem13N 36608 | Lemma for isomorphism H of... |
| dihmeetlem14N 36609 | Lemma for isomorphism H of... |
| dihmeetlem15N 36610 | Lemma for isomorphism H of... |
| dihmeetlem16N 36611 | Lemma for isomorphism H of... |
| dihmeetlem17N 36612 | Lemma for isomorphism H of... |
| dihmeetlem18N 36613 | Lemma for isomorphism H of... |
| dihmeetlem19N 36614 | Lemma for isomorphism H of... |
| dihmeetlem20N 36615 | Lemma for isomorphism H of... |
| dihmeetALTN 36616 | Isomorphism H of a lattice... |
| dih1dimatlem0 36617 | Lemma for ~ dih1dimat . (... |
| dih1dimatlem 36618 | Lemma for ~ dih1dimat . (... |
| dih1dimat 36619 | Any 1-dimensional subspace... |
| dihlsprn 36620 | The span of a vector belon... |
| dihlspsnssN 36621 | A subspace included in a 1... |
| dihlspsnat 36622 | The inverse isomorphism H ... |
| dihatlat 36623 | The isomorphism H of an at... |
| dihat 36624 | There exists at least one ... |
| dihpN 36625 | The value of isomorphism H... |
| dihlatat 36626 | The reverse isomorphism H ... |
| dihatexv 36627 | There is a nonzero vector ... |
| dihatexv2 36628 | There is a nonzero vector ... |
| dihglblem6 36629 | Isomorphism H of a lattice... |
| dihglb 36630 | Isomorphism H of a lattice... |
| dihglb2 36631 | Isomorphism H of a lattice... |
| dihmeet 36632 | Isomorphism H of a lattice... |
| dihintcl 36633 | The intersection of closed... |
| dihmeetcl 36634 | Closure of closed subspace... |
| dihmeet2 36635 | Reverse isomorphism H of a... |
| dochffval 36638 | Subspace orthocomplement f... |
| dochfval 36639 | Subspace orthocomplement f... |
| dochval 36640 | Subspace orthocomplement f... |
| dochval2 36641 | Subspace orthocomplement f... |
| dochcl 36642 | Closure of subspace orthoc... |
| dochlss 36643 | A subspace orthocomplement... |
| dochssv 36644 | A subspace orthocomplement... |
| dochfN 36645 | Domain and codomain of the... |
| dochvalr 36646 | Orthocomplement of a close... |
| doch0 36647 | Orthocomplement of the zer... |
| doch1 36648 | Orthocomplement of the uni... |
| dochoc0 36649 | The zero subspace is close... |
| dochoc1 36650 | The unit subspace (all vec... |
| dochvalr2 36651 | Orthocomplement of a close... |
| dochvalr3 36652 | Orthocomplement of a close... |
| doch2val2 36653 | Double orthocomplement for... |
| dochss 36654 | Subset law for orthocomple... |
| dochocss 36655 | Double negative law for or... |
| dochoc 36656 | Double negative law for or... |
| dochsscl 36657 | If a set of vectors is inc... |
| dochoccl 36658 | A set of vectors is closed... |
| dochord 36659 | Ordering law for orthocomp... |
| dochord2N 36660 | Ordering law for orthocomp... |
| dochord3 36661 | Ordering law for orthocomp... |
| doch11 36662 | Orthocomplement is one-to-... |
| dochsordN 36663 | Strict ordering law for or... |
| dochn0nv 36664 | An orthocomplement is nonz... |
| dihoml4c 36665 | Version of ~ dihoml4 with ... |
| dihoml4 36666 | Orthomodular law for const... |
| dochspss 36667 | The span of a set of vecto... |
| dochocsp 36668 | The span of an orthocomple... |
| dochspocN 36669 | The span of an orthocomple... |
| dochocsn 36670 | The double orthocomplement... |
| dochsncom 36671 | Swap vectors in an orthoco... |
| dochsat 36672 | The double orthocomplement... |
| dochshpncl 36673 | If a hyperplane is not clo... |
| dochlkr 36674 | Equivalent conditions for ... |
| dochkrshp 36675 | The closure of a kernel is... |
| dochkrshp2 36676 | Properties of the closure ... |
| dochkrshp3 36677 | Properties of the closure ... |
| dochkrshp4 36678 | Properties of the closure ... |
| dochdmj1 36679 | De Morgan-like law for sub... |
| dochnoncon 36680 | Law of noncontradiction. ... |
| dochnel2 36681 | A nonzero member of a subs... |
| dochnel 36682 | A nonzero vector doesn't b... |
| djhffval 36685 | Subspace join for ` DVecH ... |
| djhfval 36686 | Subspace join for ` DVecH ... |
| djhval 36687 | Subspace join for ` DVecH ... |
| djhval2 36688 | Value of subspace join for... |
| djhcl 36689 | Closure of subspace join f... |
| djhlj 36690 | Transfer lattice join to `... |
| djhljjN 36691 | Lattice join in terms of `... |
| djhjlj 36692 | ` DVecH ` vector space clo... |
| djhj 36693 | ` DVecH ` vector space clo... |
| djhcom 36694 | Subspace join commutes. (... |
| djhspss 36695 | Subspace span of union is ... |
| djhsumss 36696 | Subspace sum is a subset o... |
| dihsumssj 36697 | The subspace sum of two is... |
| djhunssN 36698 | Subspace union is a subset... |
| dochdmm1 36699 | De Morgan-like law for clo... |
| djhexmid 36700 | Excluded middle property o... |
| djh01 36701 | Closed subspace join with ... |
| djh02 36702 | Closed subspace join with ... |
| djhlsmcl 36703 | A closed subspace sum equa... |
| djhcvat42 36704 | A covering property. ( ~ ... |
| dihjatb 36705 | Isomorphism H of lattice j... |
| dihjatc 36706 | Isomorphism H of lattice j... |
| dihjatcclem1 36707 | Lemma for isomorphism H of... |
| dihjatcclem2 36708 | Lemma for isomorphism H of... |
| dihjatcclem3 36709 | Lemma for ~ dihjatcc . (C... |
| dihjatcclem4 36710 | Lemma for isomorphism H of... |
| dihjatcc 36711 | Isomorphism H of lattice j... |
| dihjat 36712 | Isomorphism H of lattice j... |
| dihprrnlem1N 36713 | Lemma for ~ dihprrn , show... |
| dihprrnlem2 36714 | Lemma for ~ dihprrn . (Co... |
| dihprrn 36715 | The span of a vector pair ... |
| djhlsmat 36716 | The sum of two subspace at... |
| dihjat1lem 36717 | Subspace sum of a closed s... |
| dihjat1 36718 | Subspace sum of a closed s... |
| dihsmsprn 36719 | Subspace sum of a closed s... |
| dihjat2 36720 | The subspace sum of a clos... |
| dihjat3 36721 | Isomorphism H of lattice j... |
| dihjat4 36722 | Transfer the subspace sum ... |
| dihjat6 36723 | Transfer the subspace sum ... |
| dihsmsnrn 36724 | The subspace sum of two si... |
| dihsmatrn 36725 | The subspace sum of a clos... |
| dihjat5N 36726 | Transfer lattice join with... |
| dvh4dimat 36727 | There is an atom that is o... |
| dvh3dimatN 36728 | There is an atom that is o... |
| dvh2dimatN 36729 | Given an atom, there exist... |
| dvh1dimat 36730 | There exists an atom. (Co... |
| dvh1dim 36731 | There exists a nonzero vec... |
| dvh4dimlem 36732 | Lemma for ~ dvh4dimN . (C... |
| dvhdimlem 36733 | Lemma for ~ dvh2dim and ~ ... |
| dvh2dim 36734 | There is a vector that is ... |
| dvh3dim 36735 | There is a vector that is ... |
| dvh4dimN 36736 | There is a vector that is ... |
| dvh3dim2 36737 | There is a vector that is ... |
| dvh3dim3N 36738 | There is a vector that is ... |
| dochsnnz 36739 | The orthocomplement of a s... |
| dochsatshp 36740 | The orthocomplement of a s... |
| dochsatshpb 36741 | The orthocomplement of a s... |
| dochsnshp 36742 | The orthocomplement of a n... |
| dochshpsat 36743 | A hyperplane is closed iff... |
| dochkrsat 36744 | The orthocomplement of a k... |
| dochkrsat2 36745 | The orthocomplement of a k... |
| dochsat0 36746 | The orthocomplement of a k... |
| dochkrsm 36747 | The subspace sum of a clos... |
| dochexmidat 36748 | Special case of excluded m... |
| dochexmidlem1 36749 | Lemma for ~ dochexmid . H... |
| dochexmidlem2 36750 | Lemma for ~ dochexmid . (... |
| dochexmidlem3 36751 | Lemma for ~ dochexmid . U... |
| dochexmidlem4 36752 | Lemma for ~ dochexmid . (... |
| dochexmidlem5 36753 | Lemma for ~ dochexmid . (... |
| dochexmidlem6 36754 | Lemma for ~ dochexmid . (... |
| dochexmidlem7 36755 | Lemma for ~ dochexmid . C... |
| dochexmidlem8 36756 | Lemma for ~ dochexmid . T... |
| dochexmid 36757 | Excluded middle law for cl... |
| dochsnkrlem1 36758 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem2 36759 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem3 36760 | Lemma for ~ dochsnkr . (C... |
| dochsnkr 36761 | A (closed) kernel expresse... |
| dochsnkr2 36762 | Kernel of the explicit fun... |
| dochsnkr2cl 36763 | The ` X ` determining func... |
| dochflcl 36764 | Closure of the explicit fu... |
| dochfl1 36765 | The value of the explicit ... |
| dochfln0 36766 | The value of a functional ... |
| dochkr1 36767 | A nonzero functional has a... |
| dochkr1OLDN 36768 | A nonzero functional has a... |
| lpolsetN 36771 | The set of polarities of a... |
| islpolN 36772 | The predicate "is a polari... |
| islpoldN 36773 | Properties that determine ... |
| lpolfN 36774 | Functionality of a polarit... |
| lpolvN 36775 | The polarity of the whole ... |
| lpolconN 36776 | Contraposition property of... |
| lpolsatN 36777 | The polarity of an atomic ... |
| lpolpolsatN 36778 | Property of a polarity. (... |
| dochpolN 36779 | The subspace orthocompleme... |
| lcfl1lem 36780 | Property of a functional w... |
| lcfl1 36781 | Property of a functional w... |
| lcfl2 36782 | Property of a functional w... |
| lcfl3 36783 | Property of a functional w... |
| lcfl4N 36784 | Property of a functional w... |
| lcfl5 36785 | Property of a functional w... |
| lcfl5a 36786 | Property of a functional w... |
| lcfl6lem 36787 | Lemma for ~ lcfl6 . A fun... |
| lcfl7lem 36788 | Lemma for ~ lcfl7N . If t... |
| lcfl6 36789 | Property of a functional w... |
| lcfl7N 36790 | Property of a functional w... |
| lcfl8 36791 | Property of a functional w... |
| lcfl8a 36792 | Property of a functional w... |
| lcfl8b 36793 | Property of a nonzero func... |
| lcfl9a 36794 | Property implying that a f... |
| lclkrlem1 36795 | The set of functionals hav... |
| lclkrlem2a 36796 | Lemma for ~ lclkr . Use ~... |
| lclkrlem2b 36797 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2c 36798 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2d 36799 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2e 36800 | Lemma for ~ lclkr . The k... |
| lclkrlem2f 36801 | Lemma for ~ lclkr . Const... |
| lclkrlem2g 36802 | Lemma for ~ lclkr . Compa... |
| lclkrlem2h 36803 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2i 36804 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2j 36805 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2k 36806 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2l 36807 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2m 36808 | Lemma for ~ lclkr . Const... |
| lclkrlem2n 36809 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2o 36810 | Lemma for ~ lclkr . When ... |
| lclkrlem2p 36811 | Lemma for ~ lclkr . When ... |
| lclkrlem2q 36812 | Lemma for ~ lclkr . The s... |
| lclkrlem2r 36813 | Lemma for ~ lclkr . When ... |
| lclkrlem2s 36814 | Lemma for ~ lclkr . Thus,... |
| lclkrlem2t 36815 | Lemma for ~ lclkr . We el... |
| lclkrlem2u 36816 | Lemma for ~ lclkr . ~ lclk... |
| lclkrlem2v 36817 | Lemma for ~ lclkr . When ... |
| lclkrlem2w 36818 | Lemma for ~ lclkr . This ... |
| lclkrlem2x 36819 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2y 36820 | Lemma for ~ lclkr . Resta... |
| lclkrlem2 36821 | The set of functionals hav... |
| lclkr 36822 | The set of functionals wit... |
| lcfls1lem 36823 | Property of a functional w... |
| lcfls1N 36824 | Property of a functional w... |
| lcfls1c 36825 | Property of a functional w... |
| lclkrslem1 36826 | The set of functionals hav... |
| lclkrslem2 36827 | The set of functionals hav... |
| lclkrs 36828 | The set of functionals hav... |
| lclkrs2 36829 | The set of functionals wit... |
| lcfrvalsnN 36830 | Reconstruction from the du... |
| lcfrlem1 36831 | Lemma for ~ lcfr . Note t... |
| lcfrlem2 36832 | Lemma for ~ lcfr . (Contr... |
| lcfrlem3 36833 | Lemma for ~ lcfr . (Contr... |
| lcfrlem4 36834 | Lemma for ~ lcfr . (Contr... |
| lcfrlem5 36835 | Lemma for ~ lcfr . The se... |
| lcfrlem6 36836 | Lemma for ~ lcfr . Closur... |
| lcfrlem7 36837 | Lemma for ~ lcfr . Closur... |
| lcfrlem8 36838 | Lemma for ~ lcf1o and ~ lc... |
| lcfrlem9 36839 | Lemma for ~ lcf1o . (This... |
| lcf1o 36840 | Define a function ` J ` th... |
| lcfrlem10 36841 | Lemma for ~ lcfr . (Contr... |
| lcfrlem11 36842 | Lemma for ~ lcfr . (Contr... |
| lcfrlem12N 36843 | Lemma for ~ lcfr . (Contr... |
| lcfrlem13 36844 | Lemma for ~ lcfr . (Contr... |
| lcfrlem14 36845 | Lemma for ~ lcfr . (Contr... |
| lcfrlem15 36846 | Lemma for ~ lcfr . (Contr... |
| lcfrlem16 36847 | Lemma for ~ lcfr . (Contr... |
| lcfrlem17 36848 | Lemma for ~ lcfr . Condit... |
| lcfrlem18 36849 | Lemma for ~ lcfr . (Contr... |
| lcfrlem19 36850 | Lemma for ~ lcfr . (Contr... |
| lcfrlem20 36851 | Lemma for ~ lcfr . (Contr... |
| lcfrlem21 36852 | Lemma for ~ lcfr . (Contr... |
| lcfrlem22 36853 | Lemma for ~ lcfr . (Contr... |
| lcfrlem23 36854 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem24 36855 | Lemma for ~ lcfr . (Contr... |
| lcfrlem25 36856 | Lemma for ~ lcfr . Specia... |
| lcfrlem26 36857 | Lemma for ~ lcfr . Specia... |
| lcfrlem27 36858 | Lemma for ~ lcfr . Specia... |
| lcfrlem28 36859 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem29 36860 | Lemma for ~ lcfr . (Contr... |
| lcfrlem30 36861 | Lemma for ~ lcfr . (Contr... |
| lcfrlem31 36862 | Lemma for ~ lcfr . (Contr... |
| lcfrlem32 36863 | Lemma for ~ lcfr . (Contr... |
| lcfrlem33 36864 | Lemma for ~ lcfr . (Contr... |
| lcfrlem34 36865 | Lemma for ~ lcfr . (Contr... |
| lcfrlem35 36866 | Lemma for ~ lcfr . (Contr... |
| lcfrlem36 36867 | Lemma for ~ lcfr . (Contr... |
| lcfrlem37 36868 | Lemma for ~ lcfr . (Contr... |
| lcfrlem38 36869 | Lemma for ~ lcfr . Combin... |
| lcfrlem39 36870 | Lemma for ~ lcfr . Elimin... |
| lcfrlem40 36871 | Lemma for ~ lcfr . Elimin... |
| lcfrlem41 36872 | Lemma for ~ lcfr . Elimin... |
| lcfrlem42 36873 | Lemma for ~ lcfr . Elimin... |
| lcfr 36874 | Reconstruction of a subspa... |
| lcdfval 36877 | Dual vector space of funct... |
| lcdval 36878 | Dual vector space of funct... |
| lcdval2 36879 | Dual vector space of funct... |
| lcdlvec 36880 | The dual vector space of f... |
| lcdlmod 36881 | The dual vector space of f... |
| lcdvbase 36882 | Vector base set of a dual ... |
| lcdvbasess 36883 | The vector base set of the... |
| lcdvbaselfl 36884 | A vector in the base set o... |
| lcdvbasecl 36885 | Closure of the value of a ... |
| lcdvadd 36886 | Vector addition for the cl... |
| lcdvaddval 36887 | The value of the value of ... |
| lcdsca 36888 | The ring of scalars of the... |
| lcdsbase 36889 | Base set of scalar ring fo... |
| lcdsadd 36890 | Scalar addition for the cl... |
| lcdsmul 36891 | Scalar multiplication for ... |
| lcdvs 36892 | Scalar product for the clo... |
| lcdvsval 36893 | Value of scalar product op... |
| lcdvscl 36894 | The scalar product operati... |
| lcdlssvscl 36895 | Closure of scalar product ... |
| lcdvsass 36896 | Associative law for scalar... |
| lcd0 36897 | The zero scalar of the clo... |
| lcd1 36898 | The unit scalar of the clo... |
| lcdneg 36899 | The unit scalar of the clo... |
| lcd0v 36900 | The zero functional in the... |
| lcd0v2 36901 | The zero functional in the... |
| lcd0vvalN 36902 | Value of the zero function... |
| lcd0vcl 36903 | Closure of the zero functi... |
| lcd0vs 36904 | A scalar zero times a func... |
| lcdvs0N 36905 | A scalar times the zero fu... |
| lcdvsub 36906 | The value of vector subtra... |
| lcdvsubval 36907 | The value of the value of ... |
| lcdlss 36908 | Subspaces of a dual vector... |
| lcdlss2N 36909 | Subspaces of a dual vector... |
| lcdlsp 36910 | Span in the set of functio... |
| lcdlkreqN 36911 | Colinear functionals have ... |
| lcdlkreq2N 36912 | Colinear functionals have ... |
| mapdffval 36915 | Projectivity from vector s... |
| mapdfval 36916 | Projectivity from vector s... |
| mapdval 36917 | Value of projectivity from... |
| mapdvalc 36918 | Value of projectivity from... |
| mapdval2N 36919 | Value of projectivity from... |
| mapdval3N 36920 | Value of projectivity from... |
| mapdval4N 36921 | Value of projectivity from... |
| mapdval5N 36922 | Value of projectivity from... |
| mapdordlem1a 36923 | Lemma for ~ mapdord . (Co... |
| mapdordlem1bN 36924 | Lemma for ~ mapdord . (Co... |
| mapdordlem1 36925 | Lemma for ~ mapdord . (Co... |
| mapdordlem2 36926 | Lemma for ~ mapdord . Ord... |
| mapdord 36927 | Ordering property of the m... |
| mapd11 36928 | The map defined by ~ df-ma... |
| mapddlssN 36929 | The mapping of a subspace ... |
| mapdsn 36930 | Value of the map defined b... |
| mapdsn2 36931 | Value of the map defined b... |
| mapdsn3 36932 | Value of the map defined b... |
| mapd1dim2lem1N 36933 | Value of the map defined b... |
| mapdrvallem2 36934 | Lemma for ~ mapdrval . TO... |
| mapdrvallem3 36935 | Lemma for ~ mapdrval . (C... |
| mapdrval 36936 | Given a dual subspace ` R ... |
| mapd1o 36937 | The map defined by ~ df-ma... |
| mapdrn 36938 | Range of the map defined b... |
| mapdunirnN 36939 | Union of the range of the ... |
| mapdrn2 36940 | Range of the map defined b... |
| mapdcnvcl 36941 | Closure of the converse of... |
| mapdcl 36942 | Closure the value of the m... |
| mapdcnvid1N 36943 | Converse of the value of t... |
| mapdsord 36944 | Strong ordering property o... |
| mapdcl2 36945 | The mapping of a subspace ... |
| mapdcnvid2 36946 | Value of the converse of t... |
| mapdcnvordN 36947 | Ordering property of the c... |
| mapdcnv11N 36948 | The converse of the map de... |
| mapdcv 36949 | Covering property of the c... |
| mapdincl 36950 | Closure of dual subspace i... |
| mapdin 36951 | Subspace intersection is p... |
| mapdlsmcl 36952 | Closure of dual subspace s... |
| mapdlsm 36953 | Subspace sum is preserved ... |
| mapd0 36954 | Projectivity map of the ze... |
| mapdcnvatN 36955 | Atoms are preserved by the... |
| mapdat 36956 | Atoms are preserved by the... |
| mapdspex 36957 | The map of a span equals t... |
| mapdn0 36958 | Transfer nonzero property ... |
| mapdncol 36959 | Transfer non-colinearity f... |
| mapdindp 36960 | Transfer (part of) vector ... |
| mapdpglem1 36961 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2 36962 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2a 36963 | Lemma for ~ mapdpg . (Con... |
| mapdpglem3 36964 | Lemma for ~ mapdpg . Baer... |
| mapdpglem4N 36965 | Lemma for ~ mapdpg . (Con... |
| mapdpglem5N 36966 | Lemma for ~ mapdpg . (Con... |
| mapdpglem6 36967 | Lemma for ~ mapdpg . Baer... |
| mapdpglem8 36968 | Lemma for ~ mapdpg . Baer... |
| mapdpglem9 36969 | Lemma for ~ mapdpg . Baer... |
| mapdpglem10 36970 | Lemma for ~ mapdpg . Baer... |
| mapdpglem11 36971 | Lemma for ~ mapdpg . (Con... |
| mapdpglem12 36972 | Lemma for ~ mapdpg . TODO... |
| mapdpglem13 36973 | Lemma for ~ mapdpg . (Con... |
| mapdpglem14 36974 | Lemma for ~ mapdpg . (Con... |
| mapdpglem15 36975 | Lemma for ~ mapdpg . (Con... |
| mapdpglem16 36976 | Lemma for ~ mapdpg . Baer... |
| mapdpglem17N 36977 | Lemma for ~ mapdpg . Baer... |
| mapdpglem18 36978 | Lemma for ~ mapdpg . Baer... |
| mapdpglem19 36979 | Lemma for ~ mapdpg . Baer... |
| mapdpglem20 36980 | Lemma for ~ mapdpg . Baer... |
| mapdpglem21 36981 | Lemma for ~ mapdpg . (Con... |
| mapdpglem22 36982 | Lemma for ~ mapdpg . Baer... |
| mapdpglem23 36983 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30a 36984 | Lemma for ~ mapdpg . (Con... |
| mapdpglem30b 36985 | Lemma for ~ mapdpg . (Con... |
| mapdpglem25 36986 | Lemma for ~ mapdpg . Baer... |
| mapdpglem26 36987 | Lemma for ~ mapdpg . Baer... |
| mapdpglem27 36988 | Lemma for ~ mapdpg . Baer... |
| mapdpglem29 36989 | Lemma for ~ mapdpg . Baer... |
| mapdpglem28 36990 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30 36991 | Lemma for ~ mapdpg . Baer... |
| mapdpglem31 36992 | Lemma for ~ mapdpg . Baer... |
| mapdpglem24 36993 | Lemma for ~ mapdpg . Exis... |
| mapdpglem32 36994 | Lemma for ~ mapdpg . Uniq... |
| mapdpg 36995 | Part 1 of proof of the fir... |
| baerlem3lem1 36996 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem1 36997 | Lemma for ~ baerlem5a . (... |
| baerlem5blem1 36998 | Lemma for ~ baerlem5b . (... |
| baerlem3lem2 36999 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem2 37000 | Lemma for ~ baerlem5a . (... |
| baerlem5blem2 37001 | Lemma for ~ baerlem5b . (... |
| baerlem3 37002 | An equality that holds whe... |
| baerlem5a 37003 | An equality that holds whe... |
| baerlem5b 37004 | An equality that holds whe... |
| baerlem5amN 37005 | An equality that holds whe... |
| baerlem5bmN 37006 | An equality that holds whe... |
| baerlem5abmN 37007 | An equality that holds whe... |
| mapdindp0 37008 | Vector independence lemma.... |
| mapdindp1 37009 | Vector independence lemma.... |
| mapdindp2 37010 | Vector independence lemma.... |
| mapdindp3 37011 | Vector independence lemma.... |
| mapdindp4 37012 | Vector independence lemma.... |
| mapdhval 37013 | Lemmma for ~~? mapdh . (C... |
| mapdhval0 37014 | Lemmma for ~~? mapdh . (C... |
| mapdhval2 37015 | Lemmma for ~~? mapdh . (C... |
| mapdhcl 37016 | Lemmma for ~~? mapdh . (C... |
| mapdheq 37017 | Lemmma for ~~? mapdh . Th... |
| mapdheq2 37018 | Lemmma for ~~? mapdh . On... |
| mapdheq2biN 37019 | Lemmma for ~~? mapdh . Pa... |
| mapdheq4lem 37020 | Lemma for ~ mapdheq4 . Pa... |
| mapdheq4 37021 | Lemma for ~~? mapdh . Par... |
| mapdh6lem1N 37022 | Lemma for ~ mapdh6N . Par... |
| mapdh6lem2N 37023 | Lemma for ~ mapdh6N . Par... |
| mapdh6aN 37024 | Lemma for ~ mapdh6N . Par... |
| mapdh6b0N 37025 | Lemmma for ~ mapdh6N . (C... |
| mapdh6bN 37026 | Lemmma for ~ mapdh6N . (C... |
| mapdh6cN 37027 | Lemmma for ~ mapdh6N . (C... |
| mapdh6dN 37028 | Lemmma for ~ mapdh6N . (C... |
| mapdh6eN 37029 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6fN 37030 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6gN 37031 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6hN 37032 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6iN 37033 | Lemmma for ~ mapdh6N . El... |
| mapdh6jN 37034 | Lemmma for ~ mapdh6N . El... |
| mapdh6kN 37035 | Lemmma for ~ mapdh6N . El... |
| mapdh6N 37036 | Part (6) of [Baer] p. 47 l... |
| mapdh7eN 37037 | Part (7) of [Baer] p. 48 l... |
| mapdh7cN 37038 | Part (7) of [Baer] p. 48 l... |
| mapdh7dN 37039 | Part (7) of [Baer] p. 48 l... |
| mapdh7fN 37040 | Part (7) of [Baer] p. 48 l... |
| mapdh75e 37041 | Part (7) of [Baer] p. 48 l... |
| mapdh75cN 37042 | Part (7) of [Baer] p. 48 l... |
| mapdh75d 37043 | Part (7) of [Baer] p. 48 l... |
| mapdh75fN 37044 | Part (7) of [Baer] p. 48 l... |
| hvmapffval 37047 | Map from nonzero vectors t... |
| hvmapfval 37048 | Map from nonzero vectors t... |
| hvmapval 37049 | Value of map from nonzero ... |
| hvmapvalvalN 37050 | Value of value of map (i.e... |
| hvmapidN 37051 | The value of the vector to... |
| hvmap1o 37052 | The vector to functional m... |
| hvmapclN 37053 | Closure of the vector to f... |
| hvmap1o2 37054 | The vector to functional m... |
| hvmapcl2 37055 | Closure of the vector to f... |
| hvmaplfl 37056 | The vector to functional m... |
| hvmaplkr 37057 | Kernel of the vector to fu... |
| mapdhvmap 37058 | Relationship between ` map... |
| lspindp5 37059 | Obtain an independent vect... |
| hdmaplem1 37060 | Lemma to convert a frequen... |
| hdmaplem2N 37061 | Lemma to convert a frequen... |
| hdmaplem3 37062 | Lemma to convert a frequen... |
| hdmaplem4 37063 | Lemma to convert a frequen... |
| mapdh8a 37064 | Part of Part (8) in [Baer]... |
| mapdh8aa 37065 | Part of Part (8) in [Baer]... |
| mapdh8ab 37066 | Part of Part (8) in [Baer]... |
| mapdh8ac 37067 | Part of Part (8) in [Baer]... |
| mapdh8ad 37068 | Part of Part (8) in [Baer]... |
| mapdh8b 37069 | Part of Part (8) in [Baer]... |
| mapdh8c 37070 | Part of Part (8) in [Baer]... |
| mapdh8d0N 37071 | Part of Part (8) in [Baer]... |
| mapdh8d 37072 | Part of Part (8) in [Baer]... |
| mapdh8e 37073 | Part of Part (8) in [Baer]... |
| mapdh8fN 37074 | Part of Part (8) in [Baer]... |
| mapdh8g 37075 | Part of Part (8) in [Baer]... |
| mapdh8i 37076 | Part of Part (8) in [Baer]... |
| mapdh8j 37077 | Part of Part (8) in [Baer]... |
| mapdh8 37078 | Part (8) in [Baer] p. 48. ... |
| mapdh9a 37079 | Lemma for part (9) in [Bae... |
| mapdh9aOLDN 37080 | Lemma for part (9) in [Bae... |
| hdmap1ffval 37085 | Preliminary map from vecto... |
| hdmap1fval 37086 | Preliminary map from vecto... |
| hdmap1vallem 37087 | Value of preliminary map f... |
| hdmap1val 37088 | Value of preliminary map f... |
| hdmap1val0 37089 | Value of preliminary map f... |
| hdmap1val2 37090 | Value of preliminary map f... |
| hdmap1eq 37091 | The defining equation for ... |
| hdmap1cbv 37092 | Frequently used lemma to c... |
| hdmap1valc 37093 | Connect the value of the p... |
| hdmap1cl 37094 | Convert closure theorem ~ ... |
| hdmap1eq2 37095 | Convert ~ mapdheq2 to use ... |
| hdmap1eq4N 37096 | Convert ~ mapdheq4 to use ... |
| hdmap1l6lem1 37097 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6lem2 37098 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6a 37099 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6b0N 37100 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6b 37101 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6c 37102 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6d 37103 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6e 37104 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6f 37105 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6g 37106 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6h 37107 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6i 37108 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6j 37109 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6k 37110 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6 37111 | Part (6) of [Baer] p. 47 l... |
| hdmap1p6N 37112 | (Convert ~ mapdh6N to use ... |
| hdmap1eulem 37113 | Lemma for ~ hdmap1eu . TO... |
| hdmap1eulemOLDN 37114 | Lemma for ~ hdmap1euOLDN .... |
| hdmap1eu 37115 | Convert ~ mapdh9a to use t... |
| hdmap1euOLDN 37116 | Convert ~ mapdh9aOLDN to u... |
| hdmap1neglem1N 37117 | Lemma for ~ hdmapneg . TO... |
| hdmapffval 37118 | Map from vectors to functi... |
| hdmapfval 37119 | Map from vectors to functi... |
| hdmapval 37120 | Value of map from vectors ... |
| hdmapfnN 37121 | Functionality of map from ... |
| hdmapcl 37122 | Closure of map from vector... |
| hdmapval2lem 37123 | Lemma for ~ hdmapval2 . (... |
| hdmapval2 37124 | Value of map from vectors ... |
| hdmapval0 37125 | Value of map from vectors ... |
| hdmapeveclem 37126 | Lemma for ~ hdmapevec . T... |
| hdmapevec 37127 | Value of map from vectors ... |
| hdmapevec2 37128 | The inner product of the r... |
| hdmapval3lemN 37129 | Value of map from vectors ... |
| hdmapval3N 37130 | Value of map from vectors ... |
| hdmap10lem 37131 | Lemma for ~ hdmap10 . (Co... |
| hdmap10 37132 | Part 10 in [Baer] p. 48 li... |
| hdmap11lem1 37133 | Lemma for ~ hdmapadd . (C... |
| hdmap11lem2 37134 | Lemma for ~ hdmapadd . (C... |
| hdmapadd 37135 | Part 11 in [Baer] p. 48 li... |
| hdmapeq0 37136 | Part of proof of part 12 i... |
| hdmapnzcl 37137 | Nonzero vector closure of ... |
| hdmapneg 37138 | Part of proof of part 12 i... |
| hdmapsub 37139 | Part of proof of part 12 i... |
| hdmap11 37140 | Part of proof of part 12 i... |
| hdmaprnlem1N 37141 | Part of proof of part 12 i... |
| hdmaprnlem3N 37142 | Part of proof of part 12 i... |
| hdmaprnlem3uN 37143 | Part of proof of part 12 i... |
| hdmaprnlem4tN 37144 | Lemma for ~ hdmaprnN . TO... |
| hdmaprnlem4N 37145 | Part of proof of part 12 i... |
| hdmaprnlem6N 37146 | Part of proof of part 12 i... |
| hdmaprnlem7N 37147 | Part of proof of part 12 i... |
| hdmaprnlem8N 37148 | Part of proof of part 12 i... |
| hdmaprnlem9N 37149 | Part of proof of part 12 i... |
| hdmaprnlem3eN 37150 | Lemma for ~ hdmaprnN . (C... |
| hdmaprnlem10N 37151 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem11N 37152 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem15N 37153 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem16N 37154 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem17N 37155 | Lemma for ~ hdmaprnN . In... |
| hdmaprnN 37156 | Part of proof of part 12 i... |
| hdmapf1oN 37157 | Part 12 in [Baer] p. 49. ... |
| hdmap14lem1a 37158 | Prior to part 14 in [Baer]... |
| hdmap14lem2a 37159 | Prior to part 14 in [Baer]... |
| hdmap14lem1 37160 | Prior to part 14 in [Baer]... |
| hdmap14lem2N 37161 | Prior to part 14 in [Baer]... |
| hdmap14lem3 37162 | Prior to part 14 in [Baer]... |
| hdmap14lem4a 37163 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem4 37164 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem6 37165 | Case where ` F ` is zero. ... |
| hdmap14lem7 37166 | Combine cases of ` F ` . ... |
| hdmap14lem8 37167 | Part of proof of part 14 i... |
| hdmap14lem9 37168 | Part of proof of part 14 i... |
| hdmap14lem10 37169 | Part of proof of part 14 i... |
| hdmap14lem11 37170 | Part of proof of part 14 i... |
| hdmap14lem12 37171 | Lemma for proof of part 14... |
| hdmap14lem13 37172 | Lemma for proof of part 14... |
| hdmap14lem14 37173 | Part of proof of part 14 i... |
| hdmap14lem15 37174 | Part of proof of part 14 i... |
| hgmapffval 37177 | Map from the scalar divisi... |
| hgmapfval 37178 | Map from the scalar divisi... |
| hgmapval 37179 | Value of map from the scal... |
| hgmapfnN 37180 | Functionality of scalar si... |
| hgmapcl 37181 | Closure of scalar sigma ma... |
| hgmapdcl 37182 | Closure of the vector spac... |
| hgmapvs 37183 | Part 15 of [Baer] p. 50 li... |
| hgmapval0 37184 | Value of the scalar sigma ... |
| hgmapval1 37185 | Value of the scalar sigma ... |
| hgmapadd 37186 | Part 15 of [Baer] p. 50 li... |
| hgmapmul 37187 | Part 15 of [Baer] p. 50 li... |
| hgmaprnlem1N 37188 | Lemma for ~ hgmaprnN . (C... |
| hgmaprnlem2N 37189 | Lemma for ~ hgmaprnN . Pa... |
| hgmaprnlem3N 37190 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem4N 37191 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem5N 37192 | Lemma for ~ hgmaprnN . El... |
| hgmaprnN 37193 | Part of proof of part 16 i... |
| hgmap11 37194 | The scalar sigma map is on... |
| hgmapf1oN 37195 | The scalar sigma map is a ... |
| hgmapeq0 37196 | The scalar sigma map is ze... |
| hdmapipcl 37197 | The inner product (Hermiti... |
| hdmapln1 37198 | Linearity property that wi... |
| hdmaplna1 37199 | Additive property of first... |
| hdmaplns1 37200 | Subtraction property of fi... |
| hdmaplnm1 37201 | Multiplicative property of... |
| hdmaplna2 37202 | Additive property of secon... |
| hdmapglnm2 37203 | g-linear property of secon... |
| hdmapgln2 37204 | g-linear property that wil... |
| hdmaplkr 37205 | Kernel of the vector to du... |
| hdmapellkr 37206 | Membership in the kernel (... |
| hdmapip0 37207 | Zero property that will be... |
| hdmapip1 37208 | Construct a proportional v... |
| hdmapip0com 37209 | Commutation property of Ba... |
| hdmapinvlem1 37210 | Line 27 in [Baer] p. 110. ... |
| hdmapinvlem2 37211 | Line 28 in [Baer] p. 110, ... |
| hdmapinvlem3 37212 | Line 30 in [Baer] p. 110, ... |
| hdmapinvlem4 37213 | Part 1.1 of Proposition 1 ... |
| hdmapglem5 37214 | Part 1.2 in [Baer] p. 110 ... |
| hgmapvvlem1 37215 | Involution property of sca... |
| hgmapvvlem2 37216 | Lemma for ~ hgmapvv . Eli... |
| hgmapvvlem3 37217 | Lemma for ~ hgmapvv . Eli... |
| hgmapvv 37218 | Value of a double involuti... |
| hdmapglem7a 37219 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7b 37220 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7 37221 | Lemma for ~ hdmapg . Line... |
| hdmapg 37222 | Apply the scalar sigma fun... |
| hdmapoc 37223 | Express our constructed or... |
| hlhilset 37226 | The final Hilbert space co... |
| hlhilsca 37227 | The scalar of the final co... |
| hlhilbase 37228 | The base set of the final ... |
| hlhilplus 37229 | The vector addition for th... |
| hlhilslem 37230 | Lemma for ~ hlhilsbase2 . ... |
| hlhilsbase 37231 | The scalar base set of the... |
| hlhilsplus 37232 | Scalar addition for the fi... |
| hlhilsmul 37233 | Scalar multiplication for ... |
| hlhilsbase2 37234 | The scalar base set of the... |
| hlhilsplus2 37235 | Scalar addition for the fi... |
| hlhilsmul2 37236 | Scalar multiplication for ... |
| hlhils0 37237 | The scalar ring zero for t... |
| hlhils1N 37238 | The scalar ring unity for ... |
| hlhilvsca 37239 | The scalar product for the... |
| hlhilip 37240 | Inner product operation fo... |
| hlhilipval 37241 | Value of inner product ope... |
| hlhilnvl 37242 | The involution operation o... |
| hlhillvec 37243 | The final constructed Hilb... |
| hlhildrng 37244 | The star division ring for... |
| hlhilsrnglem 37245 | Lemma for ~ hlhilsrng . (... |
| hlhilsrng 37246 | The star division ring for... |
| hlhil0 37247 | The zero vector for the fi... |
| hlhillsm 37248 | The vector sum operation f... |
| hlhilocv 37249 | The orthocomplement for th... |
| hlhillcs 37250 | The closed subspaces of th... |
| hlhilphllem 37251 | Lemma for ~ hlhil . (Cont... |
| hlhilhillem 37252 | Lemma for ~ hlhil . (Cont... |
| hlathil 37253 | Construction of a Hilbert ... |
| rntrclfvOAI 37254 | The range of the transitiv... |
| moxfr 37255 | Transfer at-most-one betwe... |
| imaiinfv 37256 | Indexed intersection of an... |
| elrfi 37257 | Elementhood in a set of re... |
| elrfirn 37258 | Elementhood in a set of re... |
| elrfirn2 37259 | Elementhood in a set of re... |
| cmpfiiin 37260 | In a compact topology, a s... |
| ismrcd1 37261 | Any function from the subs... |
| ismrcd2 37262 | Second half of ~ ismrcd1 .... |
| istopclsd 37263 | A closure function which s... |
| ismrc 37264 | A function is a Moore clos... |
| isnacs 37267 | Expand definition of Noeth... |
| nacsfg 37268 | In a Noetherian-type closu... |
| isnacs2 37269 | Express Noetherian-type cl... |
| mrefg2 37270 | Slight variation on finite... |
| mrefg3 37271 | Slight variation on finite... |
| nacsacs 37272 | A closure system of Noethe... |
| isnacs3 37273 | A choice-free order equiva... |
| incssnn0 37274 | Transitivity induction of ... |
| nacsfix 37275 | An increasing sequence of ... |
| constmap 37276 | A constant (represented wi... |
| mapco2g 37277 | Renaming indexes in a tupl... |
| mapco2 37278 | Post-composition (renaming... |
| mapfzcons 37279 | Extending a one-based mapp... |
| mapfzcons1 37280 | Recover prefix mapping fro... |
| mapfzcons1cl 37281 | A nonempty mapping has a p... |
| mapfzcons2 37282 | Recover added element from... |
| mptfcl 37283 | Interpret range of a maps-... |
| mzpclval 37288 | Substitution lemma for ` m... |
| elmzpcl 37289 | Double substitution lemma ... |
| mzpclall 37290 | The set of all functions w... |
| mzpcln0 37291 | Corrolary of ~ mzpclall : ... |
| mzpcl1 37292 | Defining property 1 of a p... |
| mzpcl2 37293 | Defining property 2 of a p... |
| mzpcl34 37294 | Defining properties 3 and ... |
| mzpval 37295 | Value of the ` mzPoly ` fu... |
| dmmzp 37296 | ` mzPoly ` is defined for ... |
| mzpincl 37297 | Polynomial closedness is a... |
| mzpconst 37298 | Constant functions are pol... |
| mzpf 37299 | A polynomial function is a... |
| mzpproj 37300 | A projection function is p... |
| mzpadd 37301 | The pointwise sum of two p... |
| mzpmul 37302 | The pointwise product of t... |
| mzpconstmpt 37303 | A constant function expres... |
| mzpaddmpt 37304 | Sum of polynomial function... |
| mzpmulmpt 37305 | Product of polynomial func... |
| mzpsubmpt 37306 | The difference of two poly... |
| mzpnegmpt 37307 | Negation of a polynomial f... |
| mzpexpmpt 37308 | Raise a polynomial functio... |
| mzpindd 37309 | "Structural" induction to ... |
| mzpmfp 37310 | Relationship between multi... |
| mzpsubst 37311 | Substituting polynomials f... |
| mzprename 37312 | Simplified version of ~ mz... |
| mzpresrename 37313 | A polynomial is a polynomi... |
| mzpcompact2lem 37314 | Lemma for ~ mzpcompact2 . ... |
| mzpcompact2 37315 | Polynomials are finitary o... |
| coeq0i 37316 | ~ coeq0 but without explic... |
| fzsplit1nn0 37317 | Split a finite 1-based set... |
| eldiophb 37320 | Initial expression of Diop... |
| eldioph 37321 | Condition for a set to be ... |
| diophrw 37322 | Renaming and adding unused... |
| eldioph2lem1 37323 | Lemma for ~ eldioph2 . Co... |
| eldioph2lem2 37324 | Lemma for ~ eldioph2 . Co... |
| eldioph2 37325 | Construct a Diophantine se... |
| eldioph2b 37326 | While Diophantine sets wer... |
| eldiophelnn0 37327 | Remove antecedent on ` B `... |
| eldioph3b 37328 | Define Diophantine sets in... |
| eldioph3 37329 | Inference version of ~ eld... |
| ellz1 37330 | Membership in a lower set ... |
| lzunuz 37331 | The union of a lower set o... |
| fz1eqin 37332 | Express a one-based finite... |
| lzenom 37333 | Lower integers are countab... |
| elmapresaun 37334 | ~ fresaun transposed to ma... |
| elmapresaunres2 37335 | ~ fresaunres2 transposed t... |
| diophin 37336 | If two sets are Diophantin... |
| diophun 37337 | If two sets are Diophantin... |
| eldiophss 37338 | Diophantine sets are sets ... |
| diophrex 37339 | Projecting a Diophantine s... |
| eq0rabdioph 37340 | This is the first of a num... |
| eqrabdioph 37341 | Diophantine set builder fo... |
| 0dioph 37342 | The null set is Diophantin... |
| vdioph 37343 | The "universal" set (as la... |
| anrabdioph 37344 | Diophantine set builder fo... |
| orrabdioph 37345 | Diophantine set builder fo... |
| 3anrabdioph 37346 | Diophantine set builder fo... |
| 3orrabdioph 37347 | Diophantine set builder fo... |
| 2sbcrex 37348 | Exchange an existential qu... |
| sbcrexgOLD 37349 | Interchange class substitu... |
| 2sbcrexOLD 37350 | Exchange an existential qu... |
| sbc2rex 37351 | Exchange a substitution wi... |
| sbc2rexgOLD 37352 | Exchange a substitution wi... |
| sbc4rex 37353 | Exchange a substitution wi... |
| sbc4rexgOLD 37354 | Exchange a substitution wi... |
| sbcrot3 37355 | Rotate a sequence of three... |
| sbcrot5 37356 | Rotate a sequence of five ... |
| sbccomieg 37357 | Commute two explicit subst... |
| rexrabdioph 37358 | Diophantine set builder fo... |
| rexfrabdioph 37359 | Diophantine set builder fo... |
| 2rexfrabdioph 37360 | Diophantine set builder fo... |
| 3rexfrabdioph 37361 | Diophantine set builder fo... |
| 4rexfrabdioph 37362 | Diophantine set builder fo... |
| 6rexfrabdioph 37363 | Diophantine set builder fo... |
| 7rexfrabdioph 37364 | Diophantine set builder fo... |
| rabdiophlem1 37365 | Lemma for arithmetic dioph... |
| rabdiophlem2 37366 | Lemma for arithmetic dioph... |
| elnn0rabdioph 37367 | Diophantine set builder fo... |
| rexzrexnn0 37368 | Rewrite a quantification o... |
| lerabdioph 37369 | Diophantine set builder fo... |
| eluzrabdioph 37370 | Diophantine set builder fo... |
| elnnrabdioph 37371 | Diophantine set builder fo... |
| ltrabdioph 37372 | Diophantine set builder fo... |
| nerabdioph 37373 | Diophantine set builder fo... |
| dvdsrabdioph 37374 | Divisibility is a Diophant... |
| eldioph4b 37375 | Membership in ` Dioph ` ex... |
| eldioph4i 37376 | Forward-only version of ~ ... |
| diophren 37377 | Change variables in a Diop... |
| rabrenfdioph 37378 | Change variable numbers in... |
| rabren3dioph 37379 | Change variable numbers in... |
| fphpd 37380 | Pigeonhole principle expre... |
| fphpdo 37381 | Pigeonhole principle for s... |
| ctbnfien 37382 | An infinite subset of a co... |
| fiphp3d 37383 | Infinite pigeonhole princi... |
| rencldnfilem 37384 | Lemma for ~ rencldnfi . (... |
| rencldnfi 37385 | A set of real numbers whic... |
| irrapxlem1 37386 | Lemma for ~ irrapx1 . Div... |
| irrapxlem2 37387 | Lemma for ~ irrapx1 . Two... |
| irrapxlem3 37388 | Lemma for ~ irrapx1 . By ... |
| irrapxlem4 37389 | Lemma for ~ irrapx1 . Eli... |
| irrapxlem5 37390 | Lemma for ~ irrapx1 . Swi... |
| irrapxlem6 37391 | Lemma for ~ irrapx1 . Exp... |
| irrapx1 37392 | Dirichlet's approximation ... |
| pellexlem1 37393 | Lemma for ~ pellex . Arit... |
| pellexlem2 37394 | Lemma for ~ pellex . Arit... |
| pellexlem3 37395 | Lemma for ~ pellex . To e... |
| pellexlem4 37396 | Lemma for ~ pellex . Invo... |
| pellexlem5 37397 | Lemma for ~ pellex . Invo... |
| pellexlem6 37398 | Lemma for ~ pellex . Doin... |
| pellex 37399 | Every Pell equation has a ... |
| pell1qrval 37410 | Value of the set of first-... |
| elpell1qr 37411 | Membership in a first-quad... |
| pell14qrval 37412 | Value of the set of positi... |
| elpell14qr 37413 | Membership in the set of p... |
| pell1234qrval 37414 | Value of the set of genera... |
| elpell1234qr 37415 | Membership in the set of g... |
| pell1234qrre 37416 | General Pell solutions are... |
| pell1234qrne0 37417 | No solution to a Pell equa... |
| pell1234qrreccl 37418 | General solutions of the P... |
| pell1234qrmulcl 37419 | General solutions of the P... |
| pell14qrss1234 37420 | A positive Pell solution i... |
| pell14qrre 37421 | A positive Pell solution i... |
| pell14qrne0 37422 | A positive Pell solution i... |
| pell14qrgt0 37423 | A positive Pell solution i... |
| pell14qrrp 37424 | A positive Pell solution i... |
| pell1234qrdich 37425 | A general Pell solution is... |
| elpell14qr2 37426 | A number is a positive Pel... |
| pell14qrmulcl 37427 | Positive Pell solutions ar... |
| pell14qrreccl 37428 | Positive Pell solutions ar... |
| pell14qrdivcl 37429 | Positive Pell solutions ar... |
| pell14qrexpclnn0 37430 | Lemma for ~ pell14qrexpcl ... |
| pell14qrexpcl 37431 | Positive Pell solutions ar... |
| pell1qrss14 37432 | First-quadrant Pell soluti... |
| pell14qrdich 37433 | A positive Pell solution i... |
| pell1qrge1 37434 | A Pell solution in the fir... |
| pell1qr1 37435 | 1 is a Pell solution and i... |
| elpell1qr2 37436 | The first quadrant solutio... |
| pell1qrgaplem 37437 | Lemma for ~ pell1qrgap . ... |
| pell1qrgap 37438 | First-quadrant Pell soluti... |
| pell14qrgap 37439 | Positive Pell solutions ar... |
| pell14qrgapw 37440 | Positive Pell solutions ar... |
| pellqrexplicit 37441 | Condition for a calculated... |
| infmrgelbi 37442 | Any lower bound of a nonem... |
| pellqrex 37443 | There is a nontrivial solu... |
| pellfundval 37444 | Value of the fundamental s... |
| pellfundre 37445 | The fundamental solution o... |
| pellfundge 37446 | Lower bound on the fundame... |
| pellfundgt1 37447 | Weak lower bound on the Pe... |
| pellfundlb 37448 | A nontrivial first quadran... |
| pellfundglb 37449 | If a real is larger than t... |
| pellfundex 37450 | The fundamental solution a... |
| pellfund14gap 37451 | There are no solutions bet... |
| pellfundrp 37452 | The fundamental Pell solut... |
| pellfundne1 37453 | The fundamental Pell solut... |
| reglogcl 37454 | General logarithm is a rea... |
| reglogltb 37455 | General logarithm preserve... |
| reglogleb 37456 | General logarithm preserve... |
| reglogmul 37457 | Multiplication law for gen... |
| reglogexp 37458 | Power law for general log.... |
| reglogbas 37459 | General log of the base is... |
| reglog1 37460 | General log of 1 is 0. (C... |
| reglogexpbas 37461 | General log of a power of ... |
| pellfund14 37462 | Every positive Pell soluti... |
| pellfund14b 37463 | The positive Pell solution... |
| rmxfval 37468 | Value of the X sequence. ... |
| rmyfval 37469 | Value of the Y sequence. ... |
| rmspecsqrtnq 37470 | The discriminant used to d... |
| rmspecsqrtnqOLD 37471 | Obsolete version of ~ rmsp... |
| rmspecnonsq 37472 | The discriminant used to d... |
| qirropth 37473 | This lemma implements the ... |
| rmspecfund 37474 | The base of exponent used ... |
| rmxyelqirr 37475 | The solutions used to cons... |
| rmxypairf1o 37476 | The function used to extra... |
| rmxyelxp 37477 | Lemma for ~ frmx and ~ frm... |
| frmx 37478 | The X sequence is a nonneg... |
| frmy 37479 | The Y sequence is an integ... |
| rmxyval 37480 | Main definition of the X a... |
| rmspecpos 37481 | The discriminant used to d... |
| rmxycomplete 37482 | The X and Y sequences take... |
| rmxynorm 37483 | The X and Y sequences defi... |
| rmbaserp 37484 | The base of exponentiation... |
| rmxyneg 37485 | Negation law for X and Y s... |
| rmxyadd 37486 | Addition formula for X and... |
| rmxy1 37487 | Value of the X and Y seque... |
| rmxy0 37488 | Value of the X and Y seque... |
| rmxneg 37489 | Negation law (even functio... |
| rmx0 37490 | Value of X sequence at 0. ... |
| rmx1 37491 | Value of X sequence at 1. ... |
| rmxadd 37492 | Addition formula for X seq... |
| rmyneg 37493 | Negation formula for Y seq... |
| rmy0 37494 | Value of Y sequence at 0. ... |
| rmy1 37495 | Value of Y sequence at 1. ... |
| rmyadd 37496 | Addition formula for Y seq... |
| rmxp1 37497 | Special addition-of-1 form... |
| rmyp1 37498 | Special addition of 1 form... |
| rmxm1 37499 | Subtraction of 1 formula f... |
| rmym1 37500 | Subtraction of 1 formula f... |
| rmxluc 37501 | The X sequence is a Lucas ... |
| rmyluc 37502 | The Y sequence is a Lucas ... |
| rmyluc2 37503 | Lucas sequence property of... |
| rmxdbl 37504 | "Double-angle formula" for... |
| rmydbl 37505 | "Double-angle formula" for... |
| monotuz 37506 | A function defined on an u... |
| monotoddzzfi 37507 | A function which is odd an... |
| monotoddzz 37508 | A function (given implicit... |
| oddcomabszz 37509 | An odd function which take... |
| 2nn0ind 37510 | Induction on nonnegative i... |
| zindbi 37511 | Inductively transfer a pro... |
| expmordi 37512 | Mantissa ordering relation... |
| rpexpmord 37513 | Mantissa ordering relation... |
| rmxypos 37514 | For all nonnegative indice... |
| ltrmynn0 37515 | The Y-sequence is strictly... |
| ltrmxnn0 37516 | The X-sequence is strictly... |
| lermxnn0 37517 | The X-sequence is monotoni... |
| rmxnn 37518 | The X-sequence is defined ... |
| ltrmy 37519 | The Y-sequence is strictly... |
| rmyeq0 37520 | Y is zero only at zero. (... |
| rmyeq 37521 | Y is one-to-one. (Contrib... |
| lermy 37522 | Y is monotonic (non-strict... |
| rmynn 37523 | ` rmY ` is positive for po... |
| rmynn0 37524 | ` rmY ` is nonnegative for... |
| rmyabs 37525 | ` rmY ` commutes with ` ab... |
| jm2.24nn 37526 | X(n) is strictly greater t... |
| jm2.17a 37527 | First half of lemma 2.17 o... |
| jm2.17b 37528 | Weak form of the second ha... |
| jm2.17c 37529 | Second half of lemma 2.17 ... |
| jm2.24 37530 | Lemma 2.24 of [JonesMatija... |
| rmygeid 37531 | Y(n) increases faster than... |
| congtr 37532 | A wff of the form ` A || (... |
| congadd 37533 | If two pairs of numbers ar... |
| congmul 37534 | If two pairs of numbers ar... |
| congsym 37535 | Congruence mod ` A ` is a ... |
| congneg 37536 | If two integers are congru... |
| congsub 37537 | If two pairs of numbers ar... |
| congid 37538 | Every integer is congruent... |
| mzpcong 37539 | Polynomials commute with c... |
| congrep 37540 | Every integer is congruent... |
| congabseq 37541 | If two integers are congru... |
| acongid 37542 | A wff like that in this th... |
| acongsym 37543 | Symmetry of alternating co... |
| acongneg2 37544 | Negate right side of alter... |
| acongtr 37545 | Transitivity of alternatin... |
| acongeq12d 37546 | Substitution deduction for... |
| acongrep 37547 | Every integer is alternati... |
| fzmaxdif 37548 | Bound on the difference be... |
| fzneg 37549 | Reflection of a finite ran... |
| acongeq 37550 | Two numbers in the fundame... |
| dvdsacongtr 37551 | Alternating congruence pas... |
| coprmdvdsb 37552 | Multiplication by a coprim... |
| modabsdifz 37553 | Divisibility in terms of m... |
| dvdsabsmod0 37554 | Divisibility in terms of m... |
| jm2.18 37555 | Theorem 2.18 of [JonesMati... |
| jm2.19lem1 37556 | Lemma for ~ jm2.19 . X an... |
| jm2.19lem2 37557 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem3 37558 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem4 37559 | Lemma for ~ jm2.19 . Exte... |
| jm2.19 37560 | Lemma 2.19 of [JonesMatija... |
| jm2.21 37561 | Lemma for ~ jm2.20nn . Ex... |
| jm2.22 37562 | Lemma for ~ jm2.20nn . Ap... |
| jm2.23 37563 | Lemma for ~ jm2.20nn . Tr... |
| jm2.20nn 37564 | Lemma 2.20 of [JonesMatija... |
| jm2.25lem1 37565 | Lemma for ~ jm2.26 . (Con... |
| jm2.25 37566 | Lemma for ~ jm2.26 . Rema... |
| jm2.26a 37567 | Lemma for ~ jm2.26 . Reve... |
| jm2.26lem3 37568 | Lemma for ~ jm2.26 . Use ... |
| jm2.26 37569 | Lemma 2.26 of [JonesMatija... |
| jm2.15nn0 37570 | Lemma 2.15 of [JonesMatija... |
| jm2.16nn0 37571 | Lemma 2.16 of [JonesMatija... |
| jm2.27a 37572 | Lemma for ~ jm2.27 . Reve... |
| jm2.27b 37573 | Lemma for ~ jm2.27 . Expa... |
| jm2.27c 37574 | Lemma for ~ jm2.27 . Forw... |
| jm2.27 37575 | Lemma 2.27 of [JonesMatija... |
| jm2.27dlem1 37576 | Lemma for ~ rmydioph . Su... |
| jm2.27dlem2 37577 | Lemma for ~ rmydioph . Th... |
| jm2.27dlem3 37578 | Lemma for ~ rmydioph . In... |
| jm2.27dlem4 37579 | Lemma for ~ rmydioph . In... |
| jm2.27dlem5 37580 | Lemma for ~ rmydioph . Us... |
| rmydioph 37581 | ~ jm2.27 restated in terms... |
| rmxdiophlem 37582 | X can be expressed in term... |
| rmxdioph 37583 | X is a Diophantine functio... |
| jm3.1lem1 37584 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem2 37585 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem3 37586 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1 37587 | Diophantine expression for... |
| expdiophlem1 37588 | Lemma for ~ expdioph . Fu... |
| expdiophlem2 37589 | Lemma for ~ expdioph . Ex... |
| expdioph 37590 | The exponential function i... |
| setindtr 37591 | Epsilon induction for sets... |
| setindtrs 37592 | Epsilon induction scheme w... |
| dford3lem1 37593 | Lemma for ~ dford3 . (Con... |
| dford3lem2 37594 | Lemma for ~ dford3 . (Con... |
| dford3 37595 | Ordinals are precisely the... |
| dford4 37596 | ~ dford3 expressed in prim... |
| wopprc 37597 | Unrelated: Wiener pairs t... |
| rpnnen3lem 37598 | Lemma for ~ rpnnen3 . (Co... |
| rpnnen3 37599 | Dedekind cut injection of ... |
| axac10 37600 | Characterization of choice... |
| harinf 37601 | The Hartogs number of an i... |
| wdom2d2 37602 | Deduction for weak dominan... |
| ttac 37603 | Tarski's theorem about cho... |
| pw2f1ocnv 37604 | Define a bijection between... |
| pw2f1o2 37605 | Define a bijection between... |
| pw2f1o2val 37606 | Function value of the ~ pw... |
| pw2f1o2val2 37607 | Membership in a mapped set... |
| soeq12d 37608 | Equality deduction for tot... |
| freq12d 37609 | Equality deduction for fou... |
| weeq12d 37610 | Equality deduction for wel... |
| limsuc2 37611 | Limit ordinals in the sens... |
| wepwsolem 37612 | Transfer an ordering on ch... |
| wepwso 37613 | A well-ordering induces a ... |
| dnnumch1 37614 | Define an enumeration of a... |
| dnnumch2 37615 | Define an enumeration (wea... |
| dnnumch3lem 37616 | Value of the ordinal injec... |
| dnnumch3 37617 | Define an injection from a... |
| dnwech 37618 | Define a well-ordering fro... |
| fnwe2val 37619 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem1 37620 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem2 37621 | Lemma for ~ fnwe2 . An el... |
| fnwe2lem3 37622 | Lemma for ~ fnwe2 . Trich... |
| fnwe2 37623 | A well-ordering can be con... |
| aomclem1 37624 | Lemma for ~ dfac11 . This... |
| aomclem2 37625 | Lemma for ~ dfac11 . Succ... |
| aomclem3 37626 | Lemma for ~ dfac11 . Succ... |
| aomclem4 37627 | Lemma for ~ dfac11 . Limi... |
| aomclem5 37628 | Lemma for ~ dfac11 . Comb... |
| aomclem6 37629 | Lemma for ~ dfac11 . Tran... |
| aomclem7 37630 | Lemma for ~ dfac11 . ` ( R... |
| aomclem8 37631 | Lemma for ~ dfac11 . Perf... |
| dfac11 37632 | The right-hand side of thi... |
| kelac1 37633 | Kelley's choice, basic for... |
| kelac2lem 37634 | Lemma for ~ kelac2 and ~ d... |
| kelac2 37635 | Kelley's choice, most comm... |
| dfac21 37636 | Tychonoff's theorem is a c... |
| islmodfg 37639 | Property of a finitely gen... |
| islssfg 37640 | Property of a finitely gen... |
| islssfg2 37641 | Property of a finitely gen... |
| islssfgi 37642 | Finitely spanned subspaces... |
| fglmod 37643 | Finitely generated left mo... |
| lsmfgcl 37644 | The sum of two finitely ge... |
| islnm 37647 | Property of being a Noethe... |
| islnm2 37648 | Property of being a Noethe... |
| lnmlmod 37649 | A Noetherian left module i... |
| lnmlssfg 37650 | A submodule of Noetherian ... |
| lnmlsslnm 37651 | All submodules of a Noethe... |
| lnmfg 37652 | A Noetherian left module i... |
| kercvrlsm 37653 | The domain of a linear fun... |
| lmhmfgima 37654 | A homomorphism maps finite... |
| lnmepi 37655 | Epimorphic images of Noeth... |
| lmhmfgsplit 37656 | If the kernel and range of... |
| lmhmlnmsplit 37657 | If the kernel and range of... |
| lnmlmic 37658 | Noetherian is an invariant... |
| pwssplit4 37659 | Splitting for structure po... |
| filnm 37660 | Finite left modules are No... |
| pwslnmlem0 37661 | Zeroeth powers are Noether... |
| pwslnmlem1 37662 | First powers are Noetheria... |
| pwslnmlem2 37663 | A sum of powers is Noether... |
| pwslnm 37664 | Finite powers of Noetheria... |
| unxpwdom3 37665 | Weaker version of ~ unxpwd... |
| pwfi2f1o 37666 | The ~ pw2f1o bijection rel... |
| pwfi2en 37667 | Finitely supported indicat... |
| frlmpwfi 37668 | Formal linear combinations... |
| gicabl 37669 | Being Abelian is a group i... |
| imasgim 37670 | A relabeling of the elemen... |
| isnumbasgrplem1 37671 | A set which is equipollent... |
| harn0 37672 | The Hartogs number of a se... |
| numinfctb 37673 | A numerable infinite set c... |
| isnumbasgrplem2 37674 | If the (to be thought of a... |
| isnumbasgrplem3 37675 | Every nonempty numerable s... |
| isnumbasabl 37676 | A set is numerable iff it ... |
| isnumbasgrp 37677 | A set is numerable iff it ... |
| dfacbasgrp 37678 | A choice equivalent in abs... |
| islnr 37681 | Property of a left-Noether... |
| lnrring 37682 | Left-Noetherian rings are ... |
| lnrlnm 37683 | Left-Noetherian rings have... |
| islnr2 37684 | Property of being a left-N... |
| islnr3 37685 | Relate left-Noetherian rin... |
| lnr2i 37686 | Given an ideal in a left-N... |
| lpirlnr 37687 | Left principal ideal rings... |
| lnrfrlm 37688 | Finite-dimensional free mo... |
| lnrfg 37689 | Finitely-generated modules... |
| lnrfgtr 37690 | A submodule of a finitely ... |
| hbtlem1 37693 | Value of the leading coeff... |
| hbtlem2 37694 | Leading coefficient ideals... |
| hbtlem7 37695 | Functionality of leading c... |
| hbtlem4 37696 | The leading ideal function... |
| hbtlem3 37697 | The leading ideal function... |
| hbtlem5 37698 | The leading ideal function... |
| hbtlem6 37699 | There is a finite set of p... |
| hbt 37700 | The Hilbert Basis Theorem ... |
| dgrsub2 37705 | Subtracting two polynomial... |
| elmnc 37706 | Property of a monic polyno... |
| mncply 37707 | A monic polynomial is a po... |
| mnccoe 37708 | A monic polynomial has lea... |
| mncn0 37709 | A monic polynomial is not ... |
| dgraaval 37714 | Value of the degree functi... |
| dgraalem 37715 | Properties of the degree o... |
| dgraacl 37716 | Closure of the degree func... |
| dgraaf 37717 | Degree function on algebra... |
| dgraaub 37718 | Upper bound on degree of a... |
| dgraa0p 37719 | A rational polynomial of d... |
| mpaaeu 37720 | An algebraic number has ex... |
| mpaaval 37721 | Value of the minimal polyn... |
| mpaalem 37722 | Properties of the minimal ... |
| mpaacl 37723 | Minimal polynomial is a po... |
| mpaadgr 37724 | Minimal polynomial has deg... |
| mpaaroot 37725 | The minimal polynomial of ... |
| mpaamn 37726 | Minimal polynomial is moni... |
| itgoval 37731 | Value of the integral-over... |
| aaitgo 37732 | The standard algebraic num... |
| itgoss 37733 | An integral element is int... |
| itgocn 37734 | All integral elements are ... |
| cnsrexpcl 37735 | Exponentiation is closed i... |
| fsumcnsrcl 37736 | Finite sums are closed in ... |
| cnsrplycl 37737 | Polynomials are closed in ... |
| rgspnval 37738 | Value of the ring-span of ... |
| rgspncl 37739 | The ring-span of a set is ... |
| rgspnssid 37740 | The ring-span of a set con... |
| rgspnmin 37741 | The ring-span is contained... |
| rgspnid 37742 | The span of a subring is i... |
| rngunsnply 37743 | Adjoining one element to a... |
| flcidc 37744 | Finite linear combinations... |
| algstr 37747 | Lemma to shorten proofs of... |
| algbase 37748 | The base set of a construc... |
| algaddg 37749 | The additive operation of ... |
| algmulr 37750 | The multiplicative operati... |
| algsca 37751 | The set of scalars of a co... |
| algvsca 37752 | The scalar product operati... |
| mendval 37753 | Value of the module endomo... |
| mendbas 37754 | Base set of the module end... |
| mendplusgfval 37755 | Addition in the module end... |
| mendplusg 37756 | A specific addition in the... |
| mendmulrfval 37757 | Multiplication in the modu... |
| mendmulr 37758 | A specific multiplication ... |
| mendsca 37759 | The module endomorphism al... |
| mendvscafval 37760 | Scalar multiplication in t... |
| mendvsca 37761 | A specific scalar multipli... |
| mendring 37762 | The module endomorphism al... |
| mendlmod 37763 | The module endomorphism al... |
| mendassa 37764 | The module endomorphism al... |
| issdrg 37767 | Property of a division sub... |
| issdrg2 37768 | Property of a division sub... |
| acsfn1p 37769 | Construction of a closure ... |
| subrgacs 37770 | Closure property of subrin... |
| sdrgacs 37771 | Closure property of divisi... |
| cntzsdrg 37772 | Centralizers in division r... |
| idomrootle 37773 | No element of an integral ... |
| idomodle 37774 | Limit on the number of ` N... |
| fiuneneq 37775 | Two finite sets of equal s... |
| idomsubgmo 37776 | The units of an integral d... |
| proot1mul 37777 | Any primitive ` N ` -th ro... |
| proot1hash 37778 | If an integral domain has ... |
| proot1ex 37779 | The complex field has prim... |
| isdomn3 37782 | Nonzero elements form a mu... |
| mon1pid 37783 | Monicity and degree of the... |
| mon1psubm 37784 | Monic polynomials are a mu... |
| deg1mhm 37785 | Homomorphic property of th... |
| cytpfn 37786 | Functionality of the cyclo... |
| cytpval 37787 | Substitutions for the Nth ... |
| fgraphopab 37788 | Express a function as a su... |
| fgraphxp 37789 | Express a function as a su... |
| hausgraph 37790 | The graph of a continuous ... |
| ioounsn 37795 | The closure of the upper e... |
| iocunico 37796 | Split an open interval int... |
| iocinico 37797 | The intersection of two se... |
| iocmbl 37798 | An open-below, closed-abov... |
| cnioobibld 37799 | A bounded, continuous func... |
| itgpowd 37800 | The integral of a monomial... |
| arearect 37801 | The area of a rectangle wh... |
| areaquad 37802 | The area of a quadrilatera... |
| ifpan123g 37803 | Conjunction of conditional... |
| ifpan23 37804 | Conjunction of conditional... |
| ifpdfor2 37805 | Define or in terms of cond... |
| ifporcor 37806 | Corollary of commutation o... |
| ifpdfan2 37807 | Define and with conditiona... |
| ifpancor 37808 | Corollary of commutation o... |
| ifpdfor 37809 | Define or in terms of cond... |
| ifpdfan 37810 | Define and with conditiona... |
| ifpbi2 37811 | Equivalence theorem for co... |
| ifpbi3 37812 | Equivalence theorem for co... |
| ifpim1 37813 | Restate implication as con... |
| ifpnot 37814 | Restate negated wff as con... |
| ifpid2 37815 | Restate wff as conditional... |
| ifpim2 37816 | Restate implication as con... |
| ifpbi23 37817 | Equivalence theorem for co... |
| ifpdfbi 37818 | Define biimplication as co... |
| ifpbiidcor 37819 | Restatement of ~ biid . (... |
| ifpbicor 37820 | Corollary of commutation o... |
| ifpxorcor 37821 | Corollary of commutation o... |
| ifpbi1 37822 | Equivalence theorem for co... |
| ifpnot23 37823 | Negation of conditional lo... |
| ifpnotnotb 37824 | Factor conditional logic o... |
| ifpnorcor 37825 | Corollary of commutation o... |
| ifpnancor 37826 | Corollary of commutation o... |
| ifpnot23b 37827 | Negation of conditional lo... |
| ifpbiidcor2 37828 | Restatement of ~ biid . (... |
| ifpnot23c 37829 | Negation of conditional lo... |
| ifpnot23d 37830 | Negation of conditional lo... |
| ifpdfnan 37831 | Define nand as conditional... |
| ifpdfxor 37832 | Define xor as conditional ... |
| ifpbi12 37833 | Equivalence theorem for co... |
| ifpbi13 37834 | Equivalence theorem for co... |
| ifpbi123 37835 | Equivalence theorem for co... |
| ifpidg 37836 | Restate wff as conditional... |
| ifpid3g 37837 | Restate wff as conditional... |
| ifpid2g 37838 | Restate wff as conditional... |
| ifpid1g 37839 | Restate wff as conditional... |
| ifpim23g 37840 | Restate implication as con... |
| ifpim3 37841 | Restate implication as con... |
| ifpnim1 37842 | Restate negated implicatio... |
| ifpim4 37843 | Restate implication as con... |
| ifpnim2 37844 | Restate negated implicatio... |
| ifpim123g 37845 | Implication of conditional... |
| ifpim1g 37846 | Implication of conditional... |
| ifp1bi 37847 | Substitute the first eleme... |
| ifpbi1b 37848 | When the first variable is... |
| ifpimimb 37849 | Factor conditional logic o... |
| ifpororb 37850 | Factor conditional logic o... |
| ifpananb 37851 | Factor conditional logic o... |
| ifpnannanb 37852 | Factor conditional logic o... |
| ifpor123g 37853 | Disjunction of conditional... |
| ifpimim 37854 | Consequnce of implication.... |
| ifpbibib 37855 | Factor conditional logic o... |
| ifpxorxorb 37856 | Factor conditional logic o... |
| rp-fakeimass 37857 | A special case where impli... |
| rp-fakeanorass 37858 | A special case where a mix... |
| rp-fakeoranass 37859 | A special case where a mix... |
| rp-fakenanass 37860 | A special case where nand ... |
| rp-fakeinunass 37861 | A special case where a mix... |
| rp-fakeuninass 37862 | A special case where a mix... |
| rp-isfinite5 37863 | A set is said to be finite... |
| rp-isfinite6 37864 | A set is said to be finite... |
| pwelg 37865 | The powerclass is an eleme... |
| pwinfig 37866 | The powerclass of an infin... |
| pwinfi2 37867 | The powerclass of an infin... |
| pwinfi3 37868 | The powerclass of an infin... |
| pwinfi 37869 | The powerclass of an infin... |
| fipjust 37870 | A definition of the finite... |
| cllem0 37871 | The class of all sets with... |
| superficl 37872 | The class of all supersets... |
| superuncl 37873 | The class of all supersets... |
| ssficl 37874 | The class of all subsets o... |
| ssuncl 37875 | The class of all subsets o... |
| ssdifcl 37876 | The class of all subsets o... |
| sssymdifcl 37877 | The class of all subsets o... |
| fiinfi 37878 | If two classes have the fi... |
| rababg 37879 | Condition when restricted ... |
| elintabg 37880 | Two ways of saying a set i... |
| elinintab 37881 | Two ways of saying a set i... |
| elmapintrab 37882 | Two ways to say a set is a... |
| elinintrab 37883 | Two ways of saying a set i... |
| inintabss 37884 | Upper bound on intersectio... |
| inintabd 37885 | Value of the intersection ... |
| xpinintabd 37886 | Value of the intersection ... |
| relintabex 37887 | If the intersection of a c... |
| elcnvcnvintab 37888 | Two ways of saying a set i... |
| relintab 37889 | Value of the intersection ... |
| nonrel 37890 | A non-relation is equal to... |
| elnonrel 37891 | Only an ordered pair where... |
| cnvssb 37892 | Subclass theorem for conve... |
| relnonrel 37893 | The non-relation part of a... |
| cnvnonrel 37894 | The converse of the non-re... |
| brnonrel 37895 | A non-relation cannot rela... |
| dmnonrel 37896 | The domain of the non-rela... |
| rnnonrel 37897 | The range of the non-relat... |
| resnonrel 37898 | A restriction of the non-r... |
| imanonrel 37899 | An image under the non-rel... |
| cononrel1 37900 | Composition with the non-r... |
| cononrel2 37901 | Composition with the non-r... |
| elmapintab 37902 | Two ways to say a set is a... |
| fvnonrel 37903 | The function value of any ... |
| elinlem 37904 | Two ways to say a set is a... |
| elcnvcnvlem 37905 | Two ways to say a set is a... |
| cnvcnvintabd 37906 | Value of the relationship ... |
| elcnvlem 37907 | Two ways to say a set is a... |
| elcnvintab 37908 | Two ways of saying a set i... |
| cnvintabd 37909 | Value of the converse of t... |
| undmrnresiss 37910 | Two ways of saying the ide... |
| reflexg 37911 | Two ways of saying a relat... |
| cnvssco 37912 | A condition weaker than re... |
| refimssco 37913 | Reflexive relations are su... |
| cleq2lem 37914 | Equality implies bijection... |
| cbvcllem 37915 | Change of bound variable i... |
| intabssd 37916 | When for each element ` y ... |
| clublem 37917 | If a superset ` Y ` of ` X... |
| clss2lem 37918 | The closure of a property ... |
| dfid7 37919 | Definition of identity rel... |
| mptrcllem 37920 | Show two versions of a clo... |
| cotrintab 37921 | The intersection of a clas... |
| rclexi 37922 | The reflexive closure of a... |
| rtrclexlem 37923 | Existence of relation impl... |
| rtrclex 37924 | The reflexive-transitive c... |
| trclubgNEW 37925 | If a relation exists then ... |
| trclubNEW 37926 | If a relation exists then ... |
| trclexi 37927 | The transitive closure of ... |
| rtrclexi 37928 | The reflexive-transitive c... |
| clrellem 37929 | When the property ` ps ` h... |
| clcnvlem 37930 | When ` A ` , an upper boun... |
| cnvtrucl0 37931 | The converse of the trivia... |
| cnvrcl0 37932 | The converse of the reflex... |
| cnvtrcl0 37933 | The converse of the transi... |
| dmtrcl 37934 | The domain of the transiti... |
| rntrcl 37935 | The range of the transitiv... |
| dfrtrcl5 37936 | Definition of reflexive-tr... |
| trcleq2lemRP 37937 | Equality implies bijection... |
| al3im 37938 | Version of ~ ax-4 for a ne... |
| intima0 37939 | Two ways of expressing the... |
| elimaint 37940 | Element of image of inters... |
| csbcog 37941 | Distribute proper substitu... |
| cnviun 37942 | Converse of indexed union.... |
| imaiun1 37943 | The image of an indexed un... |
| coiun1 37944 | Composition with an indexe... |
| elintima 37945 | Element of intersection of... |
| intimass 37946 | The image under the inters... |
| intimass2 37947 | The image under the inters... |
| intimag 37948 | Requirement for the image ... |
| intimasn 37949 | Two ways to express the im... |
| intimasn2 37950 | Two ways to express the im... |
| ss2iundf 37951 | Subclass theorem for index... |
| ss2iundv 37952 | Subclass theorem for index... |
| cbviuneq12df 37953 | Rule used to change the bo... |
| cbviuneq12dv 37954 | Rule used to change the bo... |
| conrel1d 37955 | Deduction about compositio... |
| conrel2d 37956 | Deduction about compositio... |
| trrelind 37957 | The intersection of transi... |
| xpintrreld 37958 | The intersection of a tran... |
| restrreld 37959 | The restriction of a trans... |
| trrelsuperreldg 37960 | Concrete construction of a... |
| trficl 37961 | The class of all transitiv... |
| cnvtrrel 37962 | The converse of a transiti... |
| trrelsuperrel2dg 37963 | Concrete construction of a... |
| dfrcl2 37966 | Reflexive closure of a rel... |
| dfrcl3 37967 | Reflexive closure of a rel... |
| dfrcl4 37968 | Reflexive closure of a rel... |
| relexp2 37969 | A set operated on by the r... |
| relexpnul 37970 | If the domain and range of... |
| eliunov2 37971 | Membership in the indexed ... |
| eltrclrec 37972 | Membership in the indexed ... |
| elrtrclrec 37973 | Membership in the indexed ... |
| briunov2 37974 | Two classes related by the... |
| brmptiunrelexpd 37975 | If two elements are connec... |
| fvmptiunrelexplb0d 37976 | If the indexed union range... |
| fvmptiunrelexplb0da 37977 | If the indexed union range... |
| fvmptiunrelexplb1d 37978 | If the indexed union range... |
| brfvid 37979 | If two elements are connec... |
| brfvidRP 37980 | If two elements are connec... |
| fvilbd 37981 | A set is a subset of its i... |
| fvilbdRP 37982 | A set is a subset of its i... |
| brfvrcld 37983 | If two elements are connec... |
| brfvrcld2 37984 | If two elements are connec... |
| fvrcllb0d 37985 | A restriction of the ident... |
| fvrcllb0da 37986 | A restriction of the ident... |
| fvrcllb1d 37987 | A set is a subset of its i... |
| brtrclrec 37988 | Two classes related by the... |
| brrtrclrec 37989 | Two classes related by the... |
| briunov2uz 37990 | Two classes related by the... |
| eliunov2uz 37991 | Membership in the indexed ... |
| ov2ssiunov2 37992 | Any particular operator va... |
| relexp0eq 37993 | The zeroth power of relati... |
| iunrelexp0 37994 | Simplification of zeroth p... |
| relexpxpnnidm 37995 | Any positive power of a cr... |
| relexpiidm 37996 | Any power of any restricti... |
| relexpss1d 37997 | The relational power of a ... |
| comptiunov2i 37998 | The composition two indexe... |
| corclrcl 37999 | The reflexive closure is i... |
| iunrelexpmin1 38000 | The indexed union of relat... |
| relexpmulnn 38001 | With exponents limited to ... |
| relexpmulg 38002 | With ordered exponents, th... |
| trclrelexplem 38003 | The union of relational po... |
| iunrelexpmin2 38004 | The indexed union of relat... |
| relexp01min 38005 | With exponents limited to ... |
| relexp1idm 38006 | Repeated raising a relatio... |
| relexp0idm 38007 | Repeated raising a relatio... |
| relexp0a 38008 | Absorbtion law for zeroth ... |
| relexpxpmin 38009 | The composition of powers ... |
| relexpaddss 38010 | The composition of two pow... |
| iunrelexpuztr 38011 | The indexed union of relat... |
| dftrcl3 38012 | Transitive closure of a re... |
| brfvtrcld 38013 | If two elements are connec... |
| fvtrcllb1d 38014 | A set is a subset of its i... |
| trclfvcom 38015 | The transitive closure of ... |
| cnvtrclfv 38016 | The converse of the transi... |
| cotrcltrcl 38017 | The transitive closure is ... |
| trclimalb2 38018 | Lower bound for image unde... |
| brtrclfv2 38019 | Two ways to indicate two e... |
| trclfvdecomr 38020 | The transitive closure of ... |
| trclfvdecoml 38021 | The transitive closure of ... |
| dmtrclfvRP 38022 | The domain of the transiti... |
| rntrclfvRP 38023 | The range of the transitiv... |
| rntrclfv 38024 | The range of the transitiv... |
| dfrtrcl3 38025 | Reflexive-transitive closu... |
| brfvrtrcld 38026 | If two elements are connec... |
| fvrtrcllb0d 38027 | A restriction of the ident... |
| fvrtrcllb0da 38028 | A restriction of the ident... |
| fvrtrcllb1d 38029 | A set is a subset of its i... |
| dfrtrcl4 38030 | Reflexive-transitive closu... |
| corcltrcl 38031 | The composition of the ref... |
| cortrcltrcl 38032 | Composition with the refle... |
| corclrtrcl 38033 | Composition with the refle... |
| cotrclrcl 38034 | The composition of the ref... |
| cortrclrcl 38035 | Composition with the refle... |
| cotrclrtrcl 38036 | Composition with the refle... |
| cortrclrtrcl 38037 | The reflexive-transitive c... |
| frege77d 38038 | If the images of both ` { ... |
| frege81d 38039 | If the image of ` U ` is a... |
| frege83d 38040 | If the image of the union ... |
| frege96d 38041 | If ` C ` follows ` A ` in ... |
| frege87d 38042 | If the images of both ` { ... |
| frege91d 38043 | If ` B ` follows ` A ` in ... |
| frege97d 38044 | If ` A ` contains all elem... |
| frege98d 38045 | If ` C ` follows ` A ` and... |
| frege102d 38046 | If either ` A ` and ` C ` ... |
| frege106d 38047 | If ` B ` follows ` A ` in ... |
| frege108d 38048 | If either ` A ` and ` C ` ... |
| frege109d 38049 | If ` A ` contains all elem... |
| frege114d 38050 | If either ` R ` relates ` ... |
| frege111d 38051 | If either ` A ` and ` C ` ... |
| frege122d 38052 | If ` F ` is a function, ` ... |
| frege124d 38053 | If ` F ` is a function, ` ... |
| frege126d 38054 | If ` F ` is a function, ` ... |
| frege129d 38055 | If ` F ` is a function and... |
| frege131d 38056 | If ` F ` is a function and... |
| frege133d 38057 | If ` F ` is a function and... |
| dfxor4 38058 | Express exclusive-or in te... |
| dfxor5 38059 | Express exclusive-or in te... |
| df3or2 38060 | Express triple-or in terms... |
| df3an2 38061 | Express triple-and in term... |
| nev 38062 | Express that not every set... |
| ndisj 38063 | Express that an intersecti... |
| 0pssin 38064 | Express that an intersecti... |
| rp-imass 38065 | If the ` R ` -image of a c... |
| dfhe2 38068 | The property of relation `... |
| dfhe3 38069 | The property of relation `... |
| heeq12 38070 | Equality law for relations... |
| heeq1 38071 | Equality law for relations... |
| heeq2 38072 | Equality law for relations... |
| sbcheg 38073 | Distribute proper substitu... |
| hess 38074 | Subclass law for relations... |
| xphe 38075 | Any Cartesian product is h... |
| 0he 38076 | The empty relation is here... |
| 0heALT 38077 | The empty relation is here... |
| he0 38078 | Any relation is hereditary... |
| unhe1 38079 | The union of two relations... |
| snhesn 38080 | Any singleton is hereditar... |
| idhe 38081 | The identity relation is h... |
| psshepw 38082 | The relation between sets ... |
| sshepw 38083 | The relation between sets ... |
| rp-simp2-frege 38086 | Simplification of triple c... |
| rp-simp2 38087 | Simplification of triple c... |
| rp-frege3g 38088 | Add antecedent to ~ ax-fre... |
| frege3 38089 | Add antecedent to ~ ax-fre... |
| rp-misc1-frege 38090 | Double-use of ~ ax-frege2 ... |
| rp-frege24 38091 | Introducing an embedded an... |
| rp-frege4g 38092 | Deduction related to distr... |
| frege4 38093 | Special case of closed for... |
| frege5 38094 | A closed form of ~ syl . ... |
| rp-7frege 38095 | Distribute antecedent and ... |
| rp-4frege 38096 | Elimination of a nested an... |
| rp-6frege 38097 | Elimination of a nested an... |
| rp-8frege 38098 | Eliminate antecedent when ... |
| rp-frege25 38099 | Closed form for ~ a1dd . ... |
| frege6 38100 | A closed form of ~ imim2d ... |
| axfrege8 38101 | Swap antecedents. Identic... |
| frege7 38102 | A closed form of ~ syl6 . ... |
| frege26 38104 | Identical to ~ idd . Prop... |
| frege27 38105 | We cannot (at the same tim... |
| frege9 38106 | Closed form of ~ syl with ... |
| frege12 38107 | A closed form of ~ com23 .... |
| frege11 38108 | Elimination of a nested an... |
| frege24 38109 | Closed form for ~ a1d . D... |
| frege16 38110 | A closed form of ~ com34 .... |
| frege25 38111 | Closed form for ~ a1dd . ... |
| frege18 38112 | Closed form of a syllogism... |
| frege22 38113 | A closed form of ~ com45 .... |
| frege10 38114 | Result commuting anteceden... |
| frege17 38115 | A closed form of ~ com3l .... |
| frege13 38116 | A closed form of ~ com3r .... |
| frege14 38117 | Closed form of a deduction... |
| frege19 38118 | A closed form of ~ syl6 . ... |
| frege23 38119 | Syllogism followed by rota... |
| frege15 38120 | A closed form of ~ com4r .... |
| frege21 38121 | Replace antecedent in ante... |
| frege20 38122 | A closed form of ~ syl8 . ... |
| axfrege28 38123 | Contraposition. Identical... |
| frege29 38125 | Closed form of ~ con3d . ... |
| frege30 38126 | Commuted, closed form of ~... |
| axfrege31 38127 | Identical to ~ notnotr . ... |
| frege32 38129 | Deduce ~ con1 from ~ con3 ... |
| frege33 38130 | If ` ph ` or ` ps ` takes ... |
| frege34 38131 | If as a conseqence of the ... |
| frege35 38132 | Commuted, closed form of ~... |
| frege36 38133 | The case in which ` ps ` i... |
| frege37 38134 | If ` ch ` is a necessary c... |
| frege38 38135 | Identical to ~ pm2.21 . P... |
| frege39 38136 | Syllogism between ~ pm2.18... |
| frege40 38137 | Anything implies ~ pm2.18 ... |
| axfrege41 38138 | Identical to ~ notnot . A... |
| frege42 38140 | Not not ~ id . Propositio... |
| frege43 38141 | If there is a choice only ... |
| frege44 38142 | Similar to a commuted ~ pm... |
| frege45 38143 | Deduce ~ pm2.6 from ~ con1... |
| frege46 38144 | If ` ps ` holds when ` ph ... |
| frege47 38145 | Deduce consequence follows... |
| frege48 38146 | Closed form of syllogism w... |
| frege49 38147 | Closed form of deduction w... |
| frege50 38148 | Closed form of ~ jaoi . P... |
| frege51 38149 | Compare with ~ jaod . Pro... |
| axfrege52a 38150 | Justification for ~ ax-fre... |
| frege52aid 38152 | The case when the content ... |
| frege53aid 38153 | Specialization of ~ frege5... |
| frege53a 38154 | Lemma for ~ frege55a . Pr... |
| axfrege54a 38155 | Justification for ~ ax-fre... |
| frege54cor0a 38157 | Synonym for logical equiva... |
| frege54cor1a 38158 | Reflexive equality. (Cont... |
| frege55aid 38159 | Lemma for ~ frege57aid . ... |
| frege55lem1a 38160 | Necessary deduction regard... |
| frege55lem2a 38161 | Core proof of Proposition ... |
| frege55a 38162 | Proposition 55 of [Frege18... |
| frege55cor1a 38163 | Proposition 55 of [Frege18... |
| frege56aid 38164 | Lemma for ~ frege57aid . ... |
| frege56a 38165 | Proposition 56 of [Frege18... |
| frege57aid 38166 | This is the all imporant f... |
| frege57a 38167 | Analogue of ~ frege57aid .... |
| axfrege58a 38168 | Identical to ~ anifp . Ju... |
| frege58acor 38170 | Lemma for ~ frege59a . (C... |
| frege59a 38171 | A kind of Aristotelian inf... |
| frege60a 38172 | Swap antecedents of ~ ax-f... |
| frege61a 38173 | Lemma for ~ frege65a . Pr... |
| frege62a 38174 | A kind of Aristotelian inf... |
| frege63a 38175 | Proposition 63 of [Frege18... |
| frege64a 38176 | Lemma for ~ frege65a . Pr... |
| frege65a 38177 | A kind of Aristotelian inf... |
| frege66a 38178 | Swap antecedents of ~ freg... |
| frege67a 38179 | Lemma for ~ frege68a . Pr... |
| frege68a 38180 | Combination of applying a ... |
| axfrege52c 38181 | Justification for ~ ax-fre... |
| frege52b 38183 | The case when the content ... |
| frege53b 38184 | Lemma for frege102 (via ~ ... |
| axfrege54c 38185 | Reflexive equality of clas... |
| frege54b 38187 | Reflexive equality of sets... |
| frege54cor1b 38188 | Reflexive equality. (Cont... |
| frege55lem1b 38189 | Necessary deduction regard... |
| frege55lem2b 38190 | Lemma for ~ frege55b . Co... |
| frege55b 38191 | Lemma for ~ frege57b . Pr... |
| frege56b 38192 | Lemma for ~ frege57b . Pr... |
| frege57b 38193 | Analogue of ~ frege57aid .... |
| axfrege58b 38194 | If ` A. x ph ` is affirmed... |
| frege58bid 38196 | If ` A. x ph ` is affirmed... |
| frege58bcor 38197 | Lemma for ~ frege59b . (C... |
| frege59b 38198 | A kind of Aristotelian inf... |
| frege60b 38199 | Swap antecedents of ~ ax-f... |
| frege61b 38200 | Lemma for ~ frege65b . Pr... |
| frege62b 38201 | A kind of Aristotelian inf... |
| frege63b 38202 | Lemma for ~ frege91 . Pro... |
| frege64b 38203 | Lemma for ~ frege65b . Pr... |
| frege65b 38204 | A kind of Aristotelian inf... |
| frege66b 38205 | Swap antecedents of ~ freg... |
| frege67b 38206 | Lemma for ~ frege68b . Pr... |
| frege68b 38207 | Combination of applying a ... |
| frege53c 38208 | Proposition 53 of [Frege18... |
| frege54cor1c 38209 | Reflexive equality. (Cont... |
| frege55lem1c 38210 | Necessary deduction regard... |
| frege55lem2c 38211 | Core proof of Proposition ... |
| frege55c 38212 | Proposition 55 of [Frege18... |
| frege56c 38213 | Lemma for ~ frege57c . Pr... |
| frege57c 38214 | Swap order of implication ... |
| frege58c 38215 | Principle related to ~ sp ... |
| frege59c 38216 | A kind of Aristotelian inf... |
| frege60c 38217 | Swap antecedents of ~ freg... |
| frege61c 38218 | Lemma for ~ frege65c . Pr... |
| frege62c 38219 | A kind of Aristotelian inf... |
| frege63c 38220 | Analogue of ~ frege63b . ... |
| frege64c 38221 | Lemma for ~ frege65c . Pr... |
| frege65c 38222 | A kind of Aristotelian inf... |
| frege66c 38223 | Swap antecedents of ~ freg... |
| frege67c 38224 | Lemma for ~ frege68c . Pr... |
| frege68c 38225 | Combination of applying a ... |
| dffrege69 38226 | If from the proposition th... |
| frege70 38227 | Lemma for ~ frege72 . Pro... |
| frege71 38228 | Lemma for ~ frege72 . Pro... |
| frege72 38229 | If property ` A ` is hered... |
| frege73 38230 | Lemma for ~ frege87 . Pro... |
| frege74 38231 | If ` X ` has a property ` ... |
| frege75 38232 | If from the proposition th... |
| dffrege76 38233 | If from the two propositio... |
| frege77 38234 | If ` Y ` follows ` X ` in ... |
| frege78 38235 | Commuted form of of ~ freg... |
| frege79 38236 | Distributed form of ~ freg... |
| frege80 38237 | Add additional condition t... |
| frege81 38238 | If ` X ` has a property ` ... |
| frege82 38239 | Closed-form deduction base... |
| frege83 38240 | Apply commuted form of ~ f... |
| frege84 38241 | Commuted form of ~ frege81... |
| frege85 38242 | Commuted form of ~ frege77... |
| frege86 38243 | Conclusion about element o... |
| frege87 38244 | If ` Z ` is a result of an... |
| frege88 38245 | Commuted form of ~ frege87... |
| frege89 38246 | One direction of ~ dffrege... |
| frege90 38247 | Add antecedent to ~ frege8... |
| frege91 38248 | Every result of an applica... |
| frege92 38249 | Inference from ~ frege91 .... |
| frege93 38250 | Necessary condition for tw... |
| frege94 38251 | Looking one past a pair re... |
| frege95 38252 | Looking one past a pair re... |
| frege96 38253 | Every result of an applica... |
| frege97 38254 | The property of following ... |
| frege98 38255 | If ` Y ` follows ` X ` and... |
| dffrege99 38256 | If ` Z ` is identical with... |
| frege100 38257 | One direction of ~ dffrege... |
| frege101 38258 | Lemma for ~ frege102 . Pr... |
| frege102 38259 | If ` Z ` belongs to the ` ... |
| frege103 38260 | Proposition 103 of [Frege1... |
| frege104 38261 | Proposition 104 of [Frege1... |
| frege105 38262 | Proposition 105 of [Frege1... |
| frege106 38263 | Whatever follows ` X ` in ... |
| frege107 38264 | Proposition 107 of [Frege1... |
| frege108 38265 | If ` Y ` belongs to the ` ... |
| frege109 38266 | The property of belonging ... |
| frege110 38267 | Proposition 110 of [Frege1... |
| frege111 38268 | If ` Y ` belongs to the ` ... |
| frege112 38269 | Identity implies belonging... |
| frege113 38270 | Proposition 113 of [Frege1... |
| frege114 38271 | If ` X ` belongs to the ` ... |
| dffrege115 38272 | If from the the circumstan... |
| frege116 38273 | One direction of ~ dffrege... |
| frege117 38274 | Lemma for ~ frege118 . Pr... |
| frege118 38275 | Simplified application of ... |
| frege119 38276 | Lemma for ~ frege120 . Pr... |
| frege120 38277 | Simplified application of ... |
| frege121 38278 | Lemma for ~ frege122 . Pr... |
| frege122 38279 | If ` X ` is a result of an... |
| frege123 38280 | Lemma for ~ frege124 . Pr... |
| frege124 38281 | If ` X ` is a result of an... |
| frege125 38282 | Lemma for ~ frege126 . Pr... |
| frege126 38283 | If ` M ` follows ` Y ` in ... |
| frege127 38284 | Communte antecedents of ~ ... |
| frege128 38285 | Lemma for ~ frege129 . Pr... |
| frege129 38286 | If the procedure ` R ` is ... |
| frege130 38287 | Lemma for ~ frege131 . Pr... |
| frege131 38288 | If the procedure ` R ` is ... |
| frege132 38289 | Lemma for ~ frege133 . Pr... |
| frege133 38290 | If the procedure ` R ` is ... |
| enrelmap 38291 | The set of all possible re... |
| enrelmapr 38292 | The set of all possible re... |
| enmappw 38293 | The set of all mappings fr... |
| enmappwid 38294 | The set of all mappings fr... |
| rfovd 38295 | Value of the operator, ` (... |
| rfovfvd 38296 | Value of the operator, ` (... |
| rfovfvfvd 38297 | Value of the operator, ` (... |
| rfovcnvf1od 38298 | Properties of the operator... |
| rfovcnvd 38299 | Value of the converse of t... |
| rfovf1od 38300 | The value of the operator,... |
| rfovcnvfvd 38301 | Value of the converse of t... |
| fsovd 38302 | Value of the operator, ` (... |
| fsovrfovd 38303 | The operator which gives a... |
| fsovfvd 38304 | Value of the operator, ` (... |
| fsovfvfvd 38305 | Value of the operator, ` (... |
| fsovfd 38306 | The operator, ` ( A O B ) ... |
| fsovcnvlem 38307 | The ` O ` operator, which ... |
| fsovcnvd 38308 | The value of the converse ... |
| fsovcnvfvd 38309 | The value of the converse ... |
| fsovf1od 38310 | The value of ` ( A O B ) `... |
| dssmapfvd 38311 | Value of the duality opera... |
| dssmapfv2d 38312 | Value of the duality opera... |
| dssmapfv3d 38313 | Value of the duality opera... |
| dssmapnvod 38314 | For any base set ` B ` the... |
| dssmapf1od 38315 | For any base set ` B ` the... |
| dssmap2d 38316 | For any base set ` B ` the... |
| sscon34b 38317 | Relative complementation r... |
| rcompleq 38318 | Two subclasses are equal i... |
| or3or 38319 | Decompose disjunction into... |
| andi3or 38320 | Distribute over triple dis... |
| uneqsn 38321 | If a union of classes is e... |
| df3o2 38322 | Ordinal 3 is the triplet c... |
| df3o3 38323 | Ordinal 3 , fully expanded... |
| brfvimex 38324 | If a binary relation holds... |
| brovmptimex 38325 | If a binary relation holds... |
| brovmptimex1 38326 | If a binary relation holds... |
| brovmptimex2 38327 | If a binary relation holds... |
| brcoffn 38328 | Conditions allowing the de... |
| brcofffn 38329 | Conditions allowing the de... |
| brco2f1o 38330 | Conditions allowing the de... |
| brco3f1o 38331 | Conditions allowing the de... |
| ntrclsbex 38332 | If (pseudo-)interior and (... |
| ntrclsrcomplex 38333 | The relative complement of... |
| neik0imk0p 38334 | Kuratowski's K0 axiom impl... |
| ntrk2imkb 38335 | If an interior function is... |
| ntrkbimka 38336 | If the interiors of disjoi... |
| ntrk0kbimka 38337 | If the interiors of disjoi... |
| clsk3nimkb 38338 | If the base set is not emp... |
| clsk1indlem0 38339 | The ansatz closure functio... |
| clsk1indlem2 38340 | The ansatz closure functio... |
| clsk1indlem3 38341 | The ansatz closure functio... |
| clsk1indlem4 38342 | The ansatz closure functio... |
| clsk1indlem1 38343 | The ansatz closure functio... |
| clsk1independent 38344 | For generalized closure fu... |
| neik0pk1imk0 38345 | Kuratowski's K0' and K1 ax... |
| isotone1 38346 | Two different ways to say ... |
| isotone2 38347 | Two different ways to say ... |
| ntrk1k3eqk13 38348 | An interior function is bo... |
| ntrclsf1o 38349 | If (pseudo-)interior and (... |
| ntrclsnvobr 38350 | If (pseudo-)interior and (... |
| ntrclsiex 38351 | If (pseudo-)interior and (... |
| ntrclskex 38352 | If (pseudo-)interior and (... |
| ntrclsfv1 38353 | If (pseudo-)interior and (... |
| ntrclsfv2 38354 | If (pseudo-)interior and (... |
| ntrclselnel1 38355 | If (pseudo-)interior and (... |
| ntrclselnel2 38356 | If (pseudo-)interior and (... |
| ntrclsfv 38357 | The value of the interior ... |
| ntrclsfveq1 38358 | If interior and closure fu... |
| ntrclsfveq2 38359 | If interior and closure fu... |
| ntrclsfveq 38360 | If interior and closure fu... |
| ntrclsss 38361 | If interior and closure fu... |
| ntrclsneine0lem 38362 | If (pseudo-)interior and (... |
| ntrclsneine0 38363 | If (pseudo-)interior and (... |
| ntrclscls00 38364 | If (pseudo-)interior and (... |
| ntrclsiso 38365 | If (pseudo-)interior and (... |
| ntrclsk2 38366 | An interior function is co... |
| ntrclskb 38367 | The interiors of disjoint ... |
| ntrclsk3 38368 | The intersection of interi... |
| ntrclsk13 38369 | The interior of the inters... |
| ntrclsk4 38370 | Idempotence of the interio... |
| ntrneibex 38371 | If (pseudo-)interior and (... |
| ntrneircomplex 38372 | The relative complement of... |
| ntrneif1o 38373 | If (pseudo-)interior and (... |
| ntrneiiex 38374 | If (pseudo-)interior and (... |
| ntrneinex 38375 | If (pseudo-)interior and (... |
| ntrneicnv 38376 | If (pseudo-)interior and (... |
| ntrneifv1 38377 | If (pseudo-)interior and (... |
| ntrneifv2 38378 | If (pseudo-)interior and (... |
| ntrneiel 38379 | If (pseudo-)interior and (... |
| ntrneifv3 38380 | The value of the neighbors... |
| ntrneineine0lem 38381 | If (pseudo-)interior and (... |
| ntrneineine1lem 38382 | If (pseudo-)interior and (... |
| ntrneifv4 38383 | The value of the interior ... |
| ntrneiel2 38384 | Membership in iterated int... |
| ntrneineine0 38385 | If (pseudo-)interior and (... |
| ntrneineine1 38386 | If (pseudo-)interior and (... |
| ntrneicls00 38387 | If (pseudo-)interior and (... |
| ntrneicls11 38388 | If (pseudo-)interior and (... |
| ntrneiiso 38389 | If (pseudo-)interior and (... |
| ntrneik2 38390 | An interior function is co... |
| ntrneix2 38391 | An interior (closure) func... |
| ntrneikb 38392 | The interiors of disjoint ... |
| ntrneixb 38393 | The interiors (closures) o... |
| ntrneik3 38394 | The intersection of interi... |
| ntrneix3 38395 | The closure of the union o... |
| ntrneik13 38396 | The interior of the inters... |
| ntrneix13 38397 | The closure of the union o... |
| ntrneik4w 38398 | Idempotence of the interio... |
| ntrneik4 38399 | Idempotence of the interio... |
| clsneibex 38400 | If (pseudo-)closure and (p... |
| clsneircomplex 38401 | The relative complement of... |
| clsneif1o 38402 | If a (pseudo-)closure func... |
| clsneicnv 38403 | If a (pseudo-)closure func... |
| clsneikex 38404 | If closure and neighborhoo... |
| clsneinex 38405 | If closure and neighborhoo... |
| clsneiel1 38406 | If a (pseudo-)closure func... |
| clsneiel2 38407 | If a (pseudo-)closure func... |
| clsneifv3 38408 | Value of the neighborhoods... |
| clsneifv4 38409 | Value of the the closure (... |
| neicvgbex 38410 | If (pseudo-)neighborhood a... |
| neicvgrcomplex 38411 | The relative complement of... |
| neicvgf1o 38412 | If neighborhood and conver... |
| neicvgnvo 38413 | If neighborhood and conver... |
| neicvgnvor 38414 | If neighborhood and conver... |
| neicvgmex 38415 | If the neighborhoods and c... |
| neicvgnex 38416 | If the neighborhoods and c... |
| neicvgel1 38417 | A subset being an element ... |
| neicvgel2 38418 | The complement of a subset... |
| neicvgfv 38419 | The value of the neighborh... |
| ntrrn 38420 | The range of the interior ... |
| ntrf 38421 | The interior function of a... |
| ntrf2 38422 | The interior function is a... |
| ntrelmap 38423 | The interior function is a... |
| clsf2 38424 | The closure function is a ... |
| clselmap 38425 | The closure function is a ... |
| dssmapntrcls 38426 | The interior and closure o... |
| dssmapclsntr 38427 | The closure and interior o... |
| gneispa 38428 | Each point ` p ` of the ne... |
| gneispb 38429 | Given a neighborhood ` N `... |
| gneispace2 38430 | The predicate that ` F ` i... |
| gneispace3 38431 | The predicate that ` F ` i... |
| gneispace 38432 | The predicate that ` F ` i... |
| gneispacef 38433 | A generic neighborhood spa... |
| gneispacef2 38434 | A generic neighborhood spa... |
| gneispacefun 38435 | A generic neighborhood spa... |
| gneispacern 38436 | A generic neighborhood spa... |
| gneispacern2 38437 | A generic neighborhood spa... |
| gneispace0nelrn 38438 | A generic neighborhood spa... |
| gneispace0nelrn2 38439 | A generic neighborhood spa... |
| gneispace0nelrn3 38440 | A generic neighborhood spa... |
| gneispaceel 38441 | Every neighborhood of a po... |
| gneispaceel2 38442 | Every neighborhood of a po... |
| gneispacess 38443 | All supersets of a neighbo... |
| gneispacess2 38444 | All supersets of a neighbo... |
| k0004lem1 38445 | Application of ~ ssin to r... |
| k0004lem2 38446 | A mapping with a particula... |
| k0004lem3 38447 | When the value of a mappin... |
| k0004val 38448 | The topological simplex of... |
| k0004ss1 38449 | The topological simplex of... |
| k0004ss2 38450 | The topological simplex of... |
| k0004ss3 38451 | The topological simplex of... |
| k0004val0 38452 | The topological simplex of... |
| inductionexd 38453 | Simple induction example. ... |
| wwlemuld 38454 | Natural deduction form of ... |
| leeq1d 38455 | Specialization of ~ breq1d... |
| leeq2d 38456 | Specialization of ~ breq2d... |
| absmulrposd 38457 | Specialization of absmuld ... |
| imadisjld 38458 | Natural dduction form of o... |
| imadisjlnd 38459 | Natural deduction form of ... |
| wnefimgd 38460 | The image of a mapping fro... |
| fco2d 38461 | Natural deduction form of ... |
| fvco3d 38462 | Natural deduction form of ... |
| wfximgfd 38463 | The value of a function on... |
| extoimad 38464 | If |f(x)| <= C for all x t... |
| imo72b2lem0 38465 | Lemma for ~ imo72b2 . (Co... |
| suprleubrd 38466 | Natural deduction form of ... |
| imo72b2lem2 38467 | Lemma for ~ imo72b2 . (Co... |
| syldbl2 38468 | Stacked hypotheseis implie... |
| funfvima2d 38469 | A function's value in a pr... |
| suprlubrd 38470 | Natural deduction form of ... |
| imo72b2lem1 38471 | Lemma for ~ imo72b2 . (Co... |
| lemuldiv3d 38472 | 'Less than or equal to' re... |
| lemuldiv4d 38473 | 'Less than or equal to' re... |
| rspcdvinvd 38474 | If something is true for a... |
| imo72b2 38475 | IMO 1972 B2. (14th Intern... |
| int-addcomd 38476 | AdditionCommutativity gene... |
| int-addassocd 38477 | AdditionAssociativity gene... |
| int-addsimpd 38478 | AdditionSimplification gen... |
| int-mulcomd 38479 | MultiplicationCommutativit... |
| int-mulassocd 38480 | MultiplicationAssociativit... |
| int-mulsimpd 38481 | MultiplicationSimplificati... |
| int-leftdistd 38482 | AdditionMultiplicationLeft... |
| int-rightdistd 38483 | AdditionMultiplicationRigh... |
| int-sqdefd 38484 | SquareDefinition generator... |
| int-mul11d 38485 | First MultiplicationOne ge... |
| int-mul12d 38486 | Second MultiplicationOne g... |
| int-add01d 38487 | First AdditionZero generat... |
| int-add02d 38488 | Second AdditionZero genera... |
| int-sqgeq0d 38489 | SquareGEQZero generator ru... |
| int-eqprincd 38490 | PrincipleOfEquality genera... |
| int-eqtransd 38491 | EqualityTransitivity gener... |
| int-eqmvtd 38492 | EquMoveTerm generator rule... |
| int-eqineqd 38493 | EquivalenceImpliesDoubleIn... |
| int-ineqmvtd 38494 | IneqMoveTerm generator rul... |
| int-ineq1stprincd 38495 | FirstPrincipleOfInequality... |
| int-ineq2ndprincd 38496 | SecondPrincipleOfInequalit... |
| int-ineqtransd 38497 | InequalityTransitivity gen... |
| unitadd 38498 | Theorem used in conjunctio... |
| gsumws3 38499 | Valuation of a length 3 wo... |
| gsumws4 38500 | Valuation of a length 4 wo... |
| amgm2d 38501 | Arithmetic-geometric mean ... |
| amgm3d 38502 | Arithmetic-geometric mean ... |
| amgm4d 38503 | Arithmetic-geometric mean ... |
| nanorxor 38504 | 'nand' is equivalent to th... |
| undisjrab 38505 | Union of two disjoint rest... |
| iso0 38506 | The empty set is an ` R , ... |
| ssrecnpr 38507 | ` RR ` is a subset of both... |
| seff 38508 | Let set ` S ` be the real ... |
| sblpnf 38509 | The infinity ball in the a... |
| prmunb2 38510 | The primes are unbounded. ... |
| dvgrat 38511 | Ratio test for divergence ... |
| cvgdvgrat 38512 | Ratio test for convergence... |
| radcnvrat 38513 | Let ` L ` be the limit, if... |
| reldvds 38514 | The divides relation is in... |
| nznngen 38515 | All positive integers in t... |
| nzss 38516 | The set of multiples of _m... |
| nzin 38517 | The intersection of the se... |
| nzprmdif 38518 | Subtract one prime's multi... |
| hashnzfz 38519 | Special case of ~ hashdvds... |
| hashnzfz2 38520 | Special case of ~ hashnzfz... |
| hashnzfzclim 38521 | As the upper bound ` K ` o... |
| caofcan 38522 | Transfer a cancellation la... |
| ofsubid 38523 | Function analogue of ~ sub... |
| ofmul12 38524 | Function analogue of ~ mul... |
| ofdivrec 38525 | Function analogue of ~ div... |
| ofdivcan4 38526 | Function analogue of ~ div... |
| ofdivdiv2 38527 | Function analogue of ~ div... |
| lhe4.4ex1a 38528 | Example of the Fundamental... |
| dvsconst 38529 | Derivative of a constant f... |
| dvsid 38530 | Derivative of the identity... |
| dvsef 38531 | Derivative of the exponent... |
| expgrowthi 38532 | Exponential growth and dec... |
| dvconstbi 38533 | The derivative of a functi... |
| expgrowth 38534 | Exponential growth and dec... |
| bccval 38537 | Value of the generalized b... |
| bcccl 38538 | Closure of the generalized... |
| bcc0 38539 | The generalized binomial c... |
| bccp1k 38540 | Generalized binomial coeff... |
| bccm1k 38541 | Generalized binomial coeff... |
| bccn0 38542 | Generalized binomial coeff... |
| bccn1 38543 | Generalized binomial coeff... |
| bccbc 38544 | The binomial coefficient a... |
| uzmptshftfval 38545 | When ` F ` is a maps-to fu... |
| dvradcnv2 38546 | The radius of convergence ... |
| binomcxplemwb 38547 | Lemma for ~ binomcxp . Th... |
| binomcxplemnn0 38548 | Lemma for ~ binomcxp . Wh... |
| binomcxplemrat 38549 | Lemma for ~ binomcxp . As... |
| binomcxplemfrat 38550 | Lemma for ~ binomcxp . ~ b... |
| binomcxplemradcnv 38551 | Lemma for ~ binomcxp . By... |
| binomcxplemdvbinom 38552 | Lemma for ~ binomcxp . By... |
| binomcxplemcvg 38553 | Lemma for ~ binomcxp . Th... |
| binomcxplemdvsum 38554 | Lemma for ~ binomcxp . Th... |
| binomcxplemnotnn0 38555 | Lemma for ~ binomcxp . Wh... |
| binomcxp 38556 | Generalize the binomial th... |
| pm10.12 38557 | Theorem *10.12 in [Whitehe... |
| pm10.14 38558 | Theorem *10.14 in [Whitehe... |
| pm10.251 38559 | Theorem *10.251 in [Whiteh... |
| pm10.252 38560 | Theorem *10.252 in [Whiteh... |
| pm10.253 38561 | Theorem *10.253 in [Whiteh... |
| albitr 38562 | Theorem *10.301 in [Whiteh... |
| pm10.42 38563 | Theorem *10.42 in [Whitehe... |
| pm10.52 38564 | Theorem *10.52 in [Whitehe... |
| pm10.53 38565 | Theorem *10.53 in [Whitehe... |
| pm10.541 38566 | Theorem *10.541 in [Whiteh... |
| pm10.542 38567 | Theorem *10.542 in [Whiteh... |
| pm10.55 38568 | Theorem *10.55 in [Whitehe... |
| pm10.56 38569 | Theorem *10.56 in [Whitehe... |
| pm10.57 38570 | Theorem *10.57 in [Whitehe... |
| 2alanimi 38571 | Removes two universal quan... |
| 2al2imi 38572 | Removes two universal quan... |
| pm11.11 38573 | Theorem *11.11 in [Whitehe... |
| pm11.12 38574 | Theorem *11.12 in [Whitehe... |
| 19.21vv 38575 | Compare Theorem *11.3 in [... |
| 2alim 38576 | Theorem *11.32 in [Whitehe... |
| 2albi 38577 | Theorem *11.33 in [Whitehe... |
| 2exim 38578 | Theorem *11.34 in [Whitehe... |
| 2exbi 38579 | Theorem *11.341 in [Whiteh... |
| spsbce-2 38580 | Theorem *11.36 in [Whitehe... |
| 19.33-2 38581 | Theorem *11.421 in [Whiteh... |
| 19.36vv 38582 | Theorem *11.43 in [Whitehe... |
| 19.31vv 38583 | Theorem *11.44 in [Whitehe... |
| 19.37vv 38584 | Theorem *11.46 in [Whitehe... |
| 19.28vv 38585 | Theorem *11.47 in [Whitehe... |
| pm11.52 38586 | Theorem *11.52 in [Whitehe... |
| 2exanali 38587 | Theorem *11.521 in [Whiteh... |
| aaanv 38588 | Theorem *11.56 in [Whitehe... |
| pm11.57 38589 | Theorem *11.57 in [Whitehe... |
| pm11.58 38590 | Theorem *11.58 in [Whitehe... |
| pm11.59 38591 | Theorem *11.59 in [Whitehe... |
| pm11.6 38592 | Theorem *11.6 in [Whitehea... |
| pm11.61 38593 | Theorem *11.61 in [Whitehe... |
| pm11.62 38594 | Theorem *11.62 in [Whitehe... |
| pm11.63 38595 | Theorem *11.63 in [Whitehe... |
| pm11.7 38596 | Theorem *11.7 in [Whitehea... |
| pm11.71 38597 | Theorem *11.71 in [Whitehe... |
| sbeqal1 38598 | If ` x = y ` always implie... |
| sbeqal1i 38599 | Suppose you know ` x = y `... |
| sbeqal2i 38600 | If ` x = y ` implies ` x =... |
| sbeqalbi 38601 | When both ` x ` and ` z ` ... |
| axc5c4c711 38602 | Proof of a theorem that ca... |
| axc5c4c711toc5 38603 | Rederivation of ~ sp from ... |
| axc5c4c711toc4 38604 | Rederivation of ~ axc4 fro... |
| axc5c4c711toc7 38605 | Rederivation of ~ axc7 fro... |
| axc5c4c711to11 38606 | Rederivation of ~ ax-11 fr... |
| axc11next 38607 | This theorem shows that, g... |
| pm13.13a 38608 | One result of theorem *13.... |
| pm13.13b 38609 | Theorem *13.13 in [Whitehe... |
| pm13.14 38610 | Theorem *13.14 in [Whitehe... |
| pm13.192 38611 | Theorem *13.192 in [Whiteh... |
| pm13.193 38612 | Theorem *13.193 in [Whiteh... |
| pm13.194 38613 | Theorem *13.194 in [Whiteh... |
| pm13.195 38614 | Theorem *13.195 in [Whiteh... |
| pm13.196a 38615 | Theorem *13.196 in [Whiteh... |
| 2sbc6g 38616 | Theorem *13.21 in [Whitehe... |
| 2sbc5g 38617 | Theorem *13.22 in [Whitehe... |
| iotain 38618 | Equivalence between two di... |
| iotaexeu 38619 | The iota class exists. Th... |
| iotasbc 38620 | Definition *14.01 in [Whit... |
| iotasbc2 38621 | Theorem *14.111 in [Whiteh... |
| pm14.12 38622 | Theorem *14.12 in [Whitehe... |
| pm14.122a 38623 | Theorem *14.122 in [Whiteh... |
| pm14.122b 38624 | Theorem *14.122 in [Whiteh... |
| pm14.122c 38625 | Theorem *14.122 in [Whiteh... |
| pm14.123a 38626 | Theorem *14.123 in [Whiteh... |
| pm14.123b 38627 | Theorem *14.123 in [Whiteh... |
| pm14.123c 38628 | Theorem *14.123 in [Whiteh... |
| pm14.18 38629 | Theorem *14.18 in [Whitehe... |
| iotaequ 38630 | Theorem *14.2 in [Whitehea... |
| iotavalb 38631 | Theorem *14.202 in [Whiteh... |
| iotasbc5 38632 | Theorem *14.205 in [Whiteh... |
| pm14.24 38633 | Theorem *14.24 in [Whitehe... |
| iotavalsb 38634 | Theorem *14.242 in [Whiteh... |
| sbiota1 38635 | Theorem *14.25 in [Whitehe... |
| sbaniota 38636 | Theorem *14.26 in [Whitehe... |
| eubi 38637 | Theorem *14.271 in [Whiteh... |
| iotasbcq 38638 | Theorem *14.272 in [Whiteh... |
| elnev 38639 | Any set that contains one ... |
| rusbcALT 38640 | A version of Russell's par... |
| compel 38641 | Equivalence between two wa... |
| compeq 38642 | Equality between two ways ... |
| compne 38643 | The complement of ` A ` is... |
| compneOLD 38644 | Obsolete proof of ~ compne... |
| compab 38645 | Two ways of saying "the co... |
| conss34OLD 38646 | Obsolete proof of ~ compls... |
| conss2 38647 | Contrapositive law for sub... |
| conss1 38648 | Contrapositive law for sub... |
| ralbidar 38649 | More general form of ~ ral... |
| rexbidar 38650 | More general form of ~ rex... |
| dropab1 38651 | Theorem to aid use of the ... |
| dropab2 38652 | Theorem to aid use of the ... |
| ipo0 38653 | If the identity relation p... |
| ifr0 38654 | A class that is founded by... |
| ordpss 38655 | ~ ordelpss with an anteced... |
| fvsb 38656 | Explicit substitution of a... |
| fveqsb 38657 | Implicit substitution of a... |
| xpexb 38658 | A Cartesian product exists... |
| trelpss 38659 | An element of a transitive... |
| addcomgi 38660 | Generalization of commutat... |
| addrval 38670 | Value of the operation of ... |
| subrval 38671 | Value of the operation of ... |
| mulvval 38672 | Value of the operation of ... |
| addrfv 38673 | Vector addition at a value... |
| subrfv 38674 | Vector subtraction at a va... |
| mulvfv 38675 | Scalar multiplication at a... |
| addrfn 38676 | Vector addition produces a... |
| subrfn 38677 | Vector subtraction produce... |
| mulvfn 38678 | Scalar multiplication prod... |
| addrcom 38679 | Vector addition is commuta... |
| idiALT 38683 | Placeholder for ~ idi . T... |
| exbir 38684 | Exportation implication al... |
| 3impexpbicom 38685 | Version of ~ 3impexp where... |
| 3impexpbicomi 38686 | Inference associated with ... |
| bi1imp 38687 | Importation inference simi... |
| bi2imp 38688 | Importation inference simi... |
| bi3impb 38689 | Similar to ~ 3impb with im... |
| bi3impa 38690 | Similar to ~ 3impa with im... |
| bi23impib 38691 | ~ 3impib with the inner im... |
| bi13impib 38692 | ~ 3impib with the outer im... |
| bi123impib 38693 | ~ 3impib with the implicat... |
| bi13impia 38694 | ~ 3impia with the outer im... |
| bi123impia 38695 | ~ 3impia with the implicat... |
| bi33imp12 38696 | ~ 3imp with innermost impl... |
| bi23imp13 38697 | ~ 3imp with middle implica... |
| bi13imp23 38698 | ~ 3imp with outermost impl... |
| bi13imp2 38699 | Similar to ~ 3imp except t... |
| bi12imp3 38700 | Similar to ~ 3imp except a... |
| bi23imp1 38701 | Similar to ~ 3imp except a... |
| bi123imp0 38702 | Similar to ~ 3imp except a... |
| 4animp1 38703 | A single hypothesis unific... |
| 4an31 38704 | A rearrangement of conjunc... |
| 4an4132 38705 | A rearrangement of conjunc... |
| expcomdg 38706 | Biconditional form of ~ ex... |
| iidn3 38707 | ~ idn3 without virtual ded... |
| ee222 38708 | ~ e222 without virtual ded... |
| ee3bir 38709 | Right-biconditional form o... |
| ee13 38710 | ~ e13 without virtual dedu... |
| ee121 38711 | ~ e121 without virtual ded... |
| ee122 38712 | ~ e122 without virtual ded... |
| ee333 38713 | ~ e333 without virtual ded... |
| ee323 38714 | ~ e323 without virtual ded... |
| 3ornot23 38715 | If the second and third di... |
| orbi1r 38716 | ~ orbi1 with order of disj... |
| 3orbi123 38717 | ~ pm4.39 with a 3-conjunct... |
| syl5imp 38718 | Closed form of ~ syl5 . D... |
| impexpd 38719 | The following User's Proof... |
| com3rgbi 38720 | The following User's Proof... |
| impexpdcom 38721 | The following User's Proof... |
| ee1111 38722 | Non-virtual deduction form... |
| pm2.43bgbi 38723 | Logical equivalence of a 2... |
| pm2.43cbi 38724 | Logical equivalence of a 3... |
| ee233 38725 | Non-virtual deduction form... |
| imbi13 38726 | Join three logical equival... |
| ee33 38727 | Non-virtual deduction form... |
| con5 38728 | Biconditional contrapositi... |
| con5i 38729 | Inference form of ~ con5 .... |
| exlimexi 38730 | Inference similar to Theor... |
| sb5ALT 38731 | Equivalence for substituti... |
| eexinst01 38732 | ~ exinst01 without virtual... |
| eexinst11 38733 | ~ exinst11 without virtual... |
| vk15.4j 38734 | Excercise 4j of Unit 15 of... |
| notnotrALT 38735 | Converse of double negatio... |
| con3ALT2 38736 | Contraposition. Alternate... |
| ssralv2 38737 | Quantification restricted ... |
| sbc3or 38738 | ~ sbcor with a 3-disjuncts... |
| sbcangOLD 38739 | Distribution of class subs... |
| sbcorgOLD 38740 | Distribution of class subs... |
| sbcbiiOLD 38741 | Formula-building inference... |
| sbc3orgOLD 38742 | ~ sbcorgOLD with a 3-disju... |
| alrim3con13v 38743 | Closed form of ~ alrimi wi... |
| rspsbc2 38744 | ~ rspsbc with two quantify... |
| sbcoreleleq 38745 | Substitution of a setvar v... |
| tratrb 38746 | If a class is transitive a... |
| ordelordALT 38747 | An element of an ordinal c... |
| sbcim2g 38748 | Distribution of class subs... |
| sbcbi 38749 | Implication form of ~ sbcb... |
| trsbc 38750 | Formula-building inference... |
| truniALT 38751 | The union of a class of tr... |
| sbcalgOLD 38752 | Move universal quantifier ... |
| sbcexgOLD 38753 | Move existential quantifie... |
| sbcel12gOLD 38754 | Distribute proper substitu... |
| sbcel2gOLD 38755 | Move proper substitution i... |
| sbcssOLD 38756 | Distribute proper substitu... |
| onfrALTlem5 38757 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem4 38758 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem3 38759 | Lemma for ~ onfrALT . (Co... |
| ggen31 38760 | ~ gen31 without virtual de... |
| onfrALTlem2 38761 | Lemma for ~ onfrALT . (Co... |
| cbvexsv 38762 | A theorem pertaining to th... |
| onfrALTlem1 38763 | Lemma for ~ onfrALT . (Co... |
| onfrALT 38764 | The epsilon relation is fo... |
| csbeq2gOLD 38765 | Formula-building implicati... |
| 19.41rg 38766 | Closed form of right-to-le... |
| opelopab4 38767 | Ordered pair membership in... |
| 2pm13.193 38768 | ~ pm13.193 for two variabl... |
| hbntal 38769 | A closed form of ~ hbn . ~... |
| hbimpg 38770 | A closed form of ~ hbim . ... |
| hbalg 38771 | Closed form of ~ hbal . D... |
| hbexg 38772 | Closed form of ~ nfex . D... |
| ax6e2eq 38773 | Alternate form of ~ ax6e f... |
| ax6e2nd 38774 | If at least two sets exist... |
| ax6e2ndeq 38775 | "At least two sets exist" ... |
| 2sb5nd 38776 | Equivalence for double sub... |
| 2uasbanh 38777 | Distribute the unabbreviat... |
| 2uasban 38778 | Distribute the unabbreviat... |
| e2ebind 38779 | Absorption of an existenti... |
| elpwgded 38780 | ~ elpwgdedVD in convention... |
| trelded 38781 | Deduction form of ~ trel .... |
| jaoded 38782 | Deduction form of ~ jao . ... |
| sbtT 38783 | A substitution into a theo... |
| not12an2impnot1 38784 | If a double conjunction is... |
| in1 38787 | Inference form of ~ df-vd1... |
| iin1 38788 | ~ in1 without virtual dedu... |
| dfvd1ir 38789 | Inference form of ~ df-vd1... |
| idn1 38790 | Virtual deduction identity... |
| dfvd1imp 38791 | Left-to-right part of defi... |
| dfvd1impr 38792 | Right-to-left part of defi... |
| dfvd2 38795 | Definition of a 2-hypothes... |
| dfvd2an 38798 | Definition of a 2-hypothes... |
| dfvd2ani 38799 | Inference form of ~ dfvd2a... |
| dfvd2anir 38800 | Right-to-left inference fo... |
| dfvd2i 38801 | Inference form of ~ dfvd2 ... |
| dfvd2ir 38802 | Right-to-left inference fo... |
| dfvd3 38807 | Definition of a 3-hypothes... |
| dfvd3i 38808 | Inference form of ~ dfvd3 ... |
| dfvd3ir 38809 | Right-to-left inference fo... |
| dfvd3an 38810 | Definition of a 3-hypothes... |
| dfvd3ani 38811 | Inference form of ~ dfvd3a... |
| dfvd3anir 38812 | Right-to-left inference fo... |
| vd01 38822 | A virtual hypothesis virtu... |
| vd02 38823 | Two virtual hypotheses vir... |
| vd03 38824 | A theorem is virtually inf... |
| vd12 38825 | A virtual deduction with 1... |
| vd13 38826 | A virtual deduction with 1... |
| vd23 38827 | A virtual deduction with 2... |
| dfvd2imp 38828 | The virtual deduction form... |
| dfvd2impr 38829 | A 2-antecedent nested impl... |
| in2 38830 | The virtual deduction intr... |
| int2 38831 | The virtual deduction intr... |
| iin2 38832 | ~ in2 without virtual dedu... |
| in2an 38833 | The virtual deduction intr... |
| in3 38834 | The virtual deduction intr... |
| iin3 38835 | ~ in3 without virtual dedu... |
| in3an 38836 | The virtual deduction intr... |
| int3 38837 | The virtual deduction intr... |
| idn2 38838 | Virtual deduction identity... |
| iden2 38839 | Virtual deduction identity... |
| idn3 38840 | Virtual deduction identity... |
| gen11 38841 | Virtual deduction generali... |
| gen11nv 38842 | Virtual deduction generali... |
| gen12 38843 | Virtual deduction generali... |
| gen21 38844 | Virtual deduction generali... |
| gen21nv 38845 | Virtual deduction form of ... |
| gen31 38846 | Virtual deduction generali... |
| gen22 38847 | Virtual deduction generali... |
| ggen22 38848 | ~ gen22 without virtual de... |
| exinst 38849 | Existential Instantiation.... |
| exinst01 38850 | Existential Instantiation.... |
| exinst11 38851 | Existential Instantiation.... |
| e1a 38852 | A Virtual deduction elimin... |
| el1 38853 | A Virtual deduction elimin... |
| e1bi 38854 | Biconditional form of ~ e1... |
| e1bir 38855 | Right biconditional form o... |
| e2 38856 | A virtual deduction elimin... |
| e2bi 38857 | Biconditional form of ~ e2... |
| e2bir 38858 | Right biconditional form o... |
| ee223 38859 | ~ e223 without virtual ded... |
| e223 38860 | A virtual deduction elimin... |
| e222 38861 | A virtual deduction elimin... |
| e220 38862 | A virtual deduction elimin... |
| ee220 38863 | ~ e220 without virtual ded... |
| e202 38864 | A virtual deduction elimin... |
| ee202 38865 | ~ e202 without virtual ded... |
| e022 38866 | A virtual deduction elimin... |
| ee022 38867 | ~ e022 without virtual ded... |
| e002 38868 | A virtual deduction elimin... |
| ee002 38869 | ~ e002 without virtual ded... |
| e020 38870 | A virtual deduction elimin... |
| ee020 38871 | ~ e020 without virtual ded... |
| e200 38872 | A virtual deduction elimin... |
| ee200 38873 | ~ e200 without virtual ded... |
| e221 38874 | A virtual deduction elimin... |
| ee221 38875 | ~ e221 without virtual ded... |
| e212 38876 | A virtual deduction elimin... |
| ee212 38877 | ~ e212 without virtual ded... |
| e122 38878 | A virtual deduction elimin... |
| e112 38879 | A virtual deduction elimin... |
| ee112 38880 | ~ e112 without virtual ded... |
| e121 38881 | A virtual deduction elimin... |
| e211 38882 | A virtual deduction elimin... |
| ee211 38883 | ~ e211 without virtual ded... |
| e210 38884 | A virtual deduction elimin... |
| ee210 38885 | ~ e210 without virtual ded... |
| e201 38886 | A virtual deduction elimin... |
| ee201 38887 | ~ e201 without virtual ded... |
| e120 38888 | A virtual deduction elimin... |
| ee120 38889 | Virtual deduction rule ~ e... |
| e021 38890 | A virtual deduction elimin... |
| ee021 38891 | ~ e021 without virtual ded... |
| e012 38892 | A virtual deduction elimin... |
| ee012 38893 | ~ e012 without virtual ded... |
| e102 38894 | A virtual deduction elimin... |
| ee102 38895 | ~ e102 without virtual ded... |
| e22 38896 | A virtual deduction elimin... |
| e22an 38897 | Conjunction form of ~ e22 ... |
| ee22an 38898 | ~ e22an without virtual de... |
| e111 38899 | A virtual deduction elimin... |
| e1111 38900 | A virtual deduction elimin... |
| e110 38901 | A virtual deduction elimin... |
| ee110 38902 | ~ e110 without virtual ded... |
| e101 38903 | A virtual deduction elimin... |
| ee101 38904 | ~ e101 without virtual ded... |
| e011 38905 | A virtual deduction elimin... |
| ee011 38906 | ~ e011 without virtual ded... |
| e100 38907 | A virtual deduction elimin... |
| ee100 38908 | ~ e100 without virtual ded... |
| e010 38909 | A virtual deduction elimin... |
| ee010 38910 | ~ e010 without virtual ded... |
| e001 38911 | A virtual deduction elimin... |
| ee001 38912 | ~ e001 without virtual ded... |
| e11 38913 | A virtual deduction elimin... |
| e11an 38914 | Conjunction form of ~ e11 ... |
| ee11an 38915 | ~ e11an without virtual de... |
| e01 38916 | A virtual deduction elimin... |
| e01an 38917 | Conjunction form of ~ e01 ... |
| ee01an 38918 | ~ e01an without virtual de... |
| e10 38919 | A virtual deduction elimin... |
| e10an 38920 | Conjunction form of ~ e10 ... |
| ee10an 38921 | ~ e10an without virtual de... |
| e02 38922 | A virtual deduction elimin... |
| e02an 38923 | Conjunction form of ~ e02 ... |
| ee02an 38924 | ~ e02an without virtual de... |
| eel021old 38925 | ~ el021old without virtual... |
| el021old 38926 | A virtual deduction elimin... |
| eel132 38927 | ~ syl2an with antecedents ... |
| eel000cT 38928 | An elimination deduction. ... |
| eel0TT 38929 | An elimination deduction. ... |
| eelT00 38930 | An elimination deduction. ... |
| eelTTT 38931 | An elimination deduction. ... |
| eelT11 38932 | An elimination deduction. ... |
| eelT1 38933 | Syllogism inference combin... |
| eelT12 38934 | An elimination deduction. ... |
| eelTT1 38935 | An elimination deduction. ... |
| eelT01 38936 | An elimination deduction. ... |
| eel0T1 38937 | An elimination deduction. ... |
| eel12131 38938 | An elimination deduction. ... |
| eel2131 38939 | ~ syl2an with antecedents ... |
| eel3132 38940 | ~ syl2an with antecedents ... |
| eel0321old 38941 | ~ el0321old without virtua... |
| el0321old 38942 | A virtual deduction elimin... |
| eel2122old 38943 | ~ el2122old without virtua... |
| el2122old 38944 | A virtual deduction elimin... |
| eel0001 38945 | An elimination deduction. ... |
| eel0000 38946 | Elimination rule similar t... |
| eel1111 38947 | Four-hypothesis eliminatio... |
| eel00001 38948 | An elimination deduction. ... |
| eel00000 38949 | Elimination rule similar ~... |
| eel11111 38950 | Five-hypothesis eliminatio... |
| e12 38951 | A virtual deduction elimin... |
| e12an 38952 | Conjunction form of ~ e12 ... |
| el12 38953 | Virtual deduction form of ... |
| e20 38954 | A virtual deduction elimin... |
| e20an 38955 | Conjunction form of ~ e20 ... |
| ee20an 38956 | ~ e20an without virtual de... |
| e21 38957 | A virtual deduction elimin... |
| e21an 38958 | Conjunction form of ~ e21 ... |
| ee21an 38959 | ~ e21an without virtual de... |
| e333 38960 | A virtual deduction elimin... |
| e33 38961 | A virtual deduction elimin... |
| e33an 38962 | Conjunction form of ~ e33 ... |
| ee33an 38963 | ~ e33an without virtual de... |
| e3 38964 | Meta-connective form of ~ ... |
| e3bi 38965 | Biconditional form of ~ e3... |
| e3bir 38966 | Right biconditional form o... |
| e03 38967 | A virtual deduction elimin... |
| ee03 38968 | ~ e03 without virtual dedu... |
| e03an 38969 | Conjunction form of ~ e03 ... |
| ee03an 38970 | Conjunction form of ~ ee03... |
| e30 38971 | A virtual deduction elimin... |
| ee30 38972 | ~ e30 without virtual dedu... |
| e30an 38973 | A virtual deduction elimin... |
| ee30an 38974 | Conjunction form of ~ ee30... |
| e13 38975 | A virtual deduction elimin... |
| e13an 38976 | A virtual deduction elimin... |
| ee13an 38977 | ~ e13an without virtual de... |
| e31 38978 | A virtual deduction elimin... |
| ee31 38979 | ~ e31 without virtual dedu... |
| e31an 38980 | A virtual deduction elimin... |
| ee31an 38981 | ~ e31an without virtual de... |
| e23 38982 | A virtual deduction elimin... |
| e23an 38983 | A virtual deduction elimin... |
| ee23an 38984 | ~ e23an without virtual de... |
| e32 38985 | A virtual deduction elimin... |
| ee32 38986 | ~ e32 without virtual dedu... |
| e32an 38987 | A virtual deduction elimin... |
| ee32an 38988 | ~ e33an without virtual de... |
| e123 38989 | A virtual deduction elimin... |
| ee123 38990 | ~ e123 without virtual ded... |
| el123 38991 | A virtual deduction elimin... |
| e233 38992 | A virtual deduction elimin... |
| e323 38993 | A virtual deduction elimin... |
| e000 38994 | A virtual deduction elimin... |
| e00 38995 | Elimination rule identical... |
| e00an 38996 | Elimination rule identical... |
| eel00cT 38997 | An elimination deduction. ... |
| eelTT 38998 | An elimination deduction. ... |
| e0a 38999 | Elimination rule identical... |
| eelT 39000 | An elimination deduction. ... |
| eel0cT 39001 | An elimination deduction. ... |
| eelT0 39002 | An elimination deduction. ... |
| e0bi 39003 | Elimination rule identical... |
| e0bir 39004 | Elimination rule identical... |
| uun0.1 39005 | Convention notation form o... |
| un0.1 39006 | ` T. ` is the constant tru... |
| uunT1 39007 | A deduction unionizing a n... |
| uunT1p1 39008 | A deduction unionizing a n... |
| uunT21 39009 | A deduction unionizing a n... |
| uun121 39010 | A deduction unionizing a n... |
| uun121p1 39011 | A deduction unionizing a n... |
| uun132 39012 | A deduction unionizing a n... |
| uun132p1 39013 | A deduction unionizing a n... |
| anabss7p1 39014 | A deduction unionizing a n... |
| un10 39015 | A unionizing deduction. (... |
| un01 39016 | A unionizing deduction. (... |
| un2122 39017 | A deduction unionizing a n... |
| uun2131 39018 | A deduction unionizing a n... |
| uun2131p1 39019 | A deduction unionizing a n... |
| uunTT1 39020 | A deduction unionizing a n... |
| uunTT1p1 39021 | A deduction unionizing a n... |
| uunTT1p2 39022 | A deduction unionizing a n... |
| uunT11 39023 | A deduction unionizing a n... |
| uunT11p1 39024 | A deduction unionizing a n... |
| uunT11p2 39025 | A deduction unionizing a n... |
| uunT12 39026 | A deduction unionizing a n... |
| uunT12p1 39027 | A deduction unionizing a n... |
| uunT12p2 39028 | A deduction unionizing a n... |
| uunT12p3 39029 | A deduction unionizing a n... |
| uunT12p4 39030 | A deduction unionizing a n... |
| uunT12p5 39031 | A deduction unionizing a n... |
| uun111 39032 | A deduction unionizing a n... |
| 3anidm12p1 39033 | A deduction unionizing a n... |
| 3anidm12p2 39034 | A deduction unionizing a n... |
| uun123 39035 | A deduction unionizing a n... |
| uun123p1 39036 | A deduction unionizing a n... |
| uun123p2 39037 | A deduction unionizing a n... |
| uun123p3 39038 | A deduction unionizing a n... |
| uun123p4 39039 | A deduction unionizing a n... |
| uun2221 39040 | A deduction unionizing a n... |
| uun2221p1 39041 | A deduction unionizing a n... |
| uun2221p2 39042 | A deduction unionizing a n... |
| 3impdirp1 39043 | A deduction unionizing a n... |
| 3impcombi 39044 | A 1-hypothesis proposition... |
| trsspwALT 39045 | Virtual deduction proof of... |
| trsspwALT2 39046 | Virtual deduction proof of... |
| trsspwALT3 39047 | Short predicate calculus p... |
| sspwtr 39048 | Virtual deduction proof of... |
| sspwtrALT 39049 | Virtual deduction proof of... |
| csbabgOLD 39050 | Move substitution into a c... |
| csbunigOLD 39051 | Distribute proper substitu... |
| csbfv12gALTOLD 39052 | Move class substitution in... |
| csbxpgOLD 39053 | Distribute proper substitu... |
| csbingOLD 39054 | Distribute proper substitu... |
| csbresgOLD 39055 | Distribute proper substitu... |
| csbrngOLD 39056 | Distribute proper substitu... |
| csbima12gALTOLD 39057 | Move class substitution in... |
| sspwtrALT2 39058 | Short predicate calculus p... |
| pwtrVD 39059 | Virtual deduction proof of... |
| pwtrrVD 39060 | Virtual deduction proof of... |
| suctrALT 39061 | The successor of a transit... |
| snssiALTVD 39062 | Virtual deduction proof of... |
| snssiALT 39063 | If a class is an element o... |
| snsslVD 39064 | Virtual deduction proof of... |
| snssl 39065 | If a singleton is a subcla... |
| snelpwrVD 39066 | Virtual deduction proof of... |
| unipwrVD 39067 | Virtual deduction proof of... |
| unipwr 39068 | A class is a subclass of t... |
| sstrALT2VD 39069 | Virtual deduction proof of... |
| sstrALT2 39070 | Virtual deduction proof of... |
| suctrALT2VD 39071 | Virtual deduction proof of... |
| suctrALT2 39072 | Virtual deduction proof of... |
| elex2VD 39073 | Virtual deduction proof of... |
| elex22VD 39074 | Virtual deduction proof of... |
| eqsbc3rVD 39075 | Virtual deduction proof of... |
| zfregs2VD 39076 | Virtual deduction proof of... |
| tpid3gVD 39077 | Virtual deduction proof of... |
| en3lplem1VD 39078 | Virtual deduction proof of... |
| en3lplem2VD 39079 | Virtual deduction proof of... |
| en3lpVD 39080 | Virtual deduction proof of... |
| simplbi2VD 39081 | Virtual deduction proof of... |
| 3ornot23VD 39082 | Virtual deduction proof of... |
| orbi1rVD 39083 | Virtual deduction proof of... |
| bitr3VD 39084 | Virtual deduction proof of... |
| 3orbi123VD 39085 | Virtual deduction proof of... |
| sbc3orgVD 39086 | Virtual deduction proof of... |
| 19.21a3con13vVD 39087 | Virtual deduction proof of... |
| exbirVD 39088 | Virtual deduction proof of... |
| exbiriVD 39089 | Virtual deduction proof of... |
| rspsbc2VD 39090 | Virtual deduction proof of... |
| 3impexpVD 39091 | Virtual deduction proof of... |
| 3impexpbicomVD 39092 | Virtual deduction proof of... |
| 3impexpbicomiVD 39093 | Virtual deduction proof of... |
| sbcel1gvOLD 39094 | Class substitution into a ... |
| sbcoreleleqVD 39095 | Virtual deduction proof of... |
| hbra2VD 39096 | Virtual deduction proof of... |
| tratrbVD 39097 | Virtual deduction proof of... |
| al2imVD 39098 | Virtual deduction proof of... |
| syl5impVD 39099 | Virtual deduction proof of... |
| idiVD 39100 | Virtual deduction proof of... |
| ancomstVD 39101 | Closed form of ~ ancoms . ... |
| ssralv2VD 39102 | Quantification restricted ... |
| ordelordALTVD 39103 | An element of an ordinal c... |
| equncomVD 39104 | If a class equals the unio... |
| equncomiVD 39105 | Inference form of ~ equnco... |
| sucidALTVD 39106 | A set belongs to its succe... |
| sucidALT 39107 | A set belongs to its succe... |
| sucidVD 39108 | A set belongs to its succe... |
| imbi12VD 39109 | Implication form of ~ imbi... |
| imbi13VD 39110 | Join three logical equival... |
| sbcim2gVD 39111 | Distribution of class subs... |
| sbcbiVD 39112 | Implication form of ~ sbcb... |
| trsbcVD 39113 | Formula-building inference... |
| truniALTVD 39114 | The union of a class of tr... |
| ee33VD 39115 | Non-virtual deduction form... |
| trintALTVD 39116 | The intersection of a clas... |
| trintALT 39117 | The intersection of a clas... |
| undif3VD 39118 | The first equality of Exer... |
| sbcssgVD 39119 | Virtual deduction proof of... |
| csbingVD 39120 | Virtual deduction proof of... |
| onfrALTlem5VD 39121 | Virtual deduction proof of... |
| onfrALTlem4VD 39122 | Virtual deduction proof of... |
| onfrALTlem3VD 39123 | Virtual deduction proof of... |
| simplbi2comtVD 39124 | Virtual deduction proof of... |
| onfrALTlem2VD 39125 | Virtual deduction proof of... |
| onfrALTlem1VD 39126 | Virtual deduction proof of... |
| onfrALTVD 39127 | Virtual deduction proof of... |
| csbeq2gVD 39128 | Virtual deduction proof of... |
| csbsngVD 39129 | Virtual deduction proof of... |
| csbxpgVD 39130 | Virtual deduction proof of... |
| csbresgVD 39131 | Virtual deduction proof of... |
| csbrngVD 39132 | Virtual deduction proof of... |
| csbima12gALTVD 39133 | Virtual deduction proof of... |
| csbunigVD 39134 | Virtual deduction proof of... |
| csbfv12gALTVD 39135 | Virtual deduction proof of... |
| con5VD 39136 | Virtual deduction proof of... |
| relopabVD 39137 | Virtual deduction proof of... |
| 19.41rgVD 39138 | Virtual deduction proof of... |
| 2pm13.193VD 39139 | Virtual deduction proof of... |
| hbimpgVD 39140 | Virtual deduction proof of... |
| hbalgVD 39141 | Virtual deduction proof of... |
| hbexgVD 39142 | Virtual deduction proof of... |
| ax6e2eqVD 39143 | The following User's Proof... |
| ax6e2ndVD 39144 | The following User's Proof... |
| ax6e2ndeqVD 39145 | The following User's Proof... |
| 2sb5ndVD 39146 | The following User's Proof... |
| 2uasbanhVD 39147 | The following User's Proof... |
| e2ebindVD 39148 | The following User's Proof... |
| sb5ALTVD 39149 | The following User's Proof... |
| vk15.4jVD 39150 | The following User's Proof... |
| notnotrALTVD 39151 | The following User's Proof... |
| con3ALTVD 39152 | The following User's Proof... |
| elpwgdedVD 39153 | Membership in a power clas... |
| sspwimp 39154 | If a class is a subclass o... |
| sspwimpVD 39155 | The following User's Proof... |
| sspwimpcf 39156 | If a class is a subclass o... |
| sspwimpcfVD 39157 | The following User's Proof... |
| suctrALTcf 39158 | The sucessor of a transiti... |
| suctrALTcfVD 39159 | The following User's Proof... |
| suctrALT3 39160 | The successor of a transit... |
| sspwimpALT 39161 | If a class is a subclass o... |
| unisnALT 39162 | A set equals the union of ... |
| notnotrALT2 39163 | Converse of double negatio... |
| sspwimpALT2 39164 | If a class is a subclass o... |
| e2ebindALT 39165 | Absorption of an existenti... |
| ax6e2ndALT 39166 | If at least two sets exist... |
| ax6e2ndeqALT 39167 | "At least two sets exist" ... |
| 2sb5ndALT 39168 | Equivalence for double sub... |
| chordthmALT 39169 | The intersecting chords th... |
| isosctrlem1ALT 39170 | Lemma for ~ isosctr . Thi... |
| iunconnlem2 39171 | The indexed union of conne... |
| iunconnALT 39172 | The indexed union of conne... |
| sineq0ALT 39173 | A complex number whose sin... |
| evth2f 39174 | A version of ~ evth2 using... |
| elunif 39175 | A version of ~ eluni using... |
| rzalf 39176 | A version of ~ rzal using ... |
| fvelrnbf 39177 | A version of ~ fvelrnb usi... |
| rfcnpre1 39178 | If F is a continuous funct... |
| ubelsupr 39179 | If U belongs to A and U is... |
| fsumcnf 39180 | A finite sum of functions ... |
| mulltgt0 39181 | The product of a negative ... |
| rspcegf 39182 | A version of ~ rspcev usin... |
| rabexgf 39183 | A version of ~ rabexg usin... |
| fcnre 39184 | A function continuous with... |
| sumsnd 39185 | A sum of a singleton is th... |
| evthf 39186 | A version of ~ evth using ... |
| cnfex 39187 | The class of continuous fu... |
| fnchoice 39188 | For a finite set, a choice... |
| refsumcn 39189 | A finite sum of continuous... |
| rfcnpre2 39190 | If ` F ` is a continuous f... |
| cncmpmax 39191 | When the hypothesis for th... |
| rfcnpre3 39192 | If F is a continuous funct... |
| rfcnpre4 39193 | If F is a continuous funct... |
| sumpair 39194 | Sum of two distinct comple... |
| rfcnnnub 39195 | Given a real continuous fu... |
| refsum2cnlem1 39196 | This is the core Lemma for... |
| refsum2cn 39197 | The sum of two continuus r... |
| elunnel2 39198 | A member of a union that i... |
| adantlllr 39199 | Deduction adding a conjunc... |
| 3adantlr3 39200 | Deduction adding a conjunc... |
| nnxrd 39201 | A natural number is an ext... |
| 3adantll2 39202 | Deduction adding a conjunc... |
| 3adantll3 39203 | Deduction adding a conjunc... |
| ssnel 39204 | If not element of a set, t... |
| jcn 39205 | Inference joining the cons... |
| elabrexg 39206 | Elementhood in an image se... |
| ifeq123d 39207 | Equality deduction for con... |
| sncldre 39208 | A singleton is closed w.r.... |
| n0p 39209 | A polynomial with a nonzer... |
| pm2.65ni 39210 | Inference rule for proof b... |
| pwssfi 39211 | Every element of the power... |
| iuneq2df 39212 | Equality deduction for ind... |
| nnfoctb 39213 | There exists a mapping fro... |
| ssinss1d 39214 | Intersection preserves sub... |
| 0un 39215 | The union of the empty set... |
| elpwinss 39216 | An element of the powerset... |
| unidmex 39217 | If ` F ` is a set, then ` ... |
| ndisj2 39218 | A non disjointness conditi... |
| zenom 39219 | The set of integer numbers... |
| rexsngf 39220 | Restricted existential qua... |
| uzwo4 39221 | Well-ordering principle: a... |
| unisn0 39222 | The union of the singleton... |
| ssin0 39223 | If two classes are disjoin... |
| inabs3 39224 | Absorption law for interse... |
| pwpwuni 39225 | Relationship between power... |
| disjiun2 39226 | In a disjoint collection, ... |
| 0pwfi 39227 | The empty set is in any po... |
| ssinss2d 39228 | Intersection preserves sub... |
| zct 39229 | The set of integer numbers... |
| iunxsngf2 39230 | A singleton index picks ou... |
| pwfin0 39231 | A finite set always belong... |
| uzct 39232 | An upper integer set is co... |
| iunxsnf 39233 | A singleton index picks ou... |
| fiiuncl 39234 | If a set is closed under t... |
| iunp1 39235 | The addition of the next s... |
| fiunicl 39236 | If a set is closed under t... |
| ixpeq2d 39237 | Equality theorem for infin... |
| disjxp1 39238 | The sets of a cartesian pr... |
| disjsnxp 39239 | The sets in the cartesian ... |
| eliind 39240 | Membership in indexed inte... |
| rspcef 39241 | Restricted existential spe... |
| inn0f 39242 | A non-empty intersection. ... |
| ixpssmapc 39243 | An infinite Cartesian prod... |
| inn0 39244 | A non-empty intersection. ... |
| elintd 39245 | Membership in class inters... |
| eqneltri 39246 | If a class is not an eleme... |
| ssdf 39247 | A sufficient condition for... |
| brneqtrd 39248 | Substitution of equal clas... |
| ssnct 39249 | A set containing an uncoun... |
| ssuniint 39250 | Sufficient condition for b... |
| elintdv 39251 | Membership in class inters... |
| ssd 39252 | A sufficient condition for... |
| ralimralim 39253 | Introducing any antecedent... |
| snelmap 39254 | Membership of the element ... |
| dfcleqf 39255 | Equality connective betwee... |
| xrnmnfpnf 39256 | An extended real that is n... |
| nelrnmpt 39257 | Non-membership in the rang... |
| snn0d 39258 | The singleton of a set is ... |
| iuneq1i 39259 | Equality theorem for index... |
| nssrex 39260 | Negation of subclass relat... |
| nelpr2 39261 | If a class is not an eleme... |
| nelpr1 39262 | If a class is not an eleme... |
| iunssf 39263 | Subset theorem for an inde... |
| ssinc 39264 | Inclusion relation for a m... |
| ssdec 39265 | Inclusion relation for a m... |
| elixpconstg 39266 | Membership in an infinite ... |
| iineq1d 39267 | Equality theorem for index... |
| metpsmet 39268 | A metric is a pseudometric... |
| ixpssixp 39269 | Subclass theorem for infin... |
| ballss3 39270 | A sufficient condition for... |
| iunssd 39271 | Subset theorem for an inde... |
| iunincfi 39272 | Given a sequence of increa... |
| nsstr 39273 | If it's not a subclass, it... |
| rabbida 39274 | Equivalent wff's yield equ... |
| rexanuz3 39275 | Combine two different uppe... |
| rabeqd 39276 | Equality theorem for restr... |
| cbvmpt22 39277 | Rule to change the second ... |
| cbvmpt21 39278 | Rule to change the first b... |
| eliuniin 39279 | Indexed union of indexed i... |
| ssabf 39280 | Subclass of a class abstra... |
| uniexd 39281 | Deduction version of the Z... |
| pwexd 39282 | Deduction version of the p... |
| pssnssi 39283 | A proper subclass does not... |
| rabidim2 39284 | Membership in a restricted... |
| xpexd 39285 | The Cartesian product of t... |
| eluni2f 39286 | Membership in class union.... |
| eliin2f 39287 | Membership in indexed inte... |
| nssd 39288 | Negation of subclass relat... |
| iineq12dv 39289 | Equality deduction for ind... |
| supxrcld 39290 | The supremum of an arbitra... |
| elrestd 39291 | A sufficient condition for... |
| eliuniincex 39292 | Counterexample to show tha... |
| eliincex 39293 | Counterexample to show tha... |
| eliinid 39294 | Membership in an indexed i... |
| abssf 39295 | Class abstraction in a sub... |
| fexd 39296 | If the domain of a mapping... |
| supxrubd 39297 | A member of a set of exten... |
| ssrabf 39298 | Subclass of a restricted c... |
| eliin2 39299 | Membership in indexed inte... |
| ssrab2f 39300 | Subclass relation for a re... |
| restuni3 39301 | The underlying set of a su... |
| rabssf 39302 | Restricted class abstracti... |
| eliuniin2 39303 | Indexed union of indexed i... |
| restuni4 39304 | The underlying set of a su... |
| restuni6 39305 | The underlying set of a su... |
| restuni5 39306 | The underlying set of a su... |
| unirestss 39307 | The union of an elementwis... |
| ne0d 39308 | If a set has elements, the... |
| iniin1 39309 | Indexed intersection of in... |
| iniin2 39310 | Indexed intersection of in... |
| cbvrabv2 39311 | A more general version of ... |
| iinssiin 39312 | Subset implication for an ... |
| eliind2 39313 | Membership in indexed inte... |
| iinssd 39314 | Subset implication for an ... |
| ralrimia 39315 | Inference from Theorem 19.... |
| tpid2g 39316 | Closed theorem form of ~ t... |
| rabbida2 39317 | Equivalent wff's yield equ... |
| iinexd 39318 | The existence of an indexe... |
| rabexf 39319 | Separation Scheme in terms... |
| rabbida3 39320 | Equivalent wff's yield equ... |
| resexd 39321 | The restriction of a set i... |
| tpid1g 39322 | Closed theorem form of ~ t... |
| fnexd 39323 | If the domain of a functio... |
| r19.36vf 39324 | Restricted quantifier vers... |
| raleqd 39325 | Equality deduction for res... |
| ralimda 39326 | Deduction quantifying both... |
| iinssf 39327 | Subset implication for an ... |
| iinssdf 39328 | Subset implication for an ... |
| ifcli 39329 | Membership (closure) of a ... |
| resabs2i 39330 | Absorption law for restric... |
| ssdf2 39331 | A sufficient condition for... |
| rabssd 39332 | Restricted class abstracti... |
| ssrind 39333 | Add right intersection to ... |
| rexnegd 39334 | Minus a real number. (Con... |
| rexlimd3 39335 | * Inference from Theorem 1... |
| resabs1i 39336 | Absorption law for restric... |
| nel1nelin 39337 | Membership in an intersect... |
| nel2nelin 39338 | Membership in an intersect... |
| rexlimdva2 39339 | Inference from Theorem 19.... |
| nel1nelini 39340 | Membership in an intersect... |
| nel2nelini 39341 | Membership in an intersect... |
| eliunid 39342 | Membership in indexed unio... |
| reximddv3 39343 | Deduction from Theorem 19.... |
| reximdd 39344 | Deduction from Theorem 19.... |
| unfid 39345 | The union of two finite se... |
| unima 39346 | Image of a union. (Contri... |
| feq1dd 39347 | Equality deduction for fun... |
| fnresdmss 39348 | A function does not change... |
| fmptsnxp 39349 | Maps-to notation and cross... |
| fvmpt2bd 39350 | Value of a function given ... |
| rnmptfi 39351 | The range of a function wi... |
| fresin2 39352 | Restriction of a function ... |
| rnmptc 39353 | Range of a constant functi... |
| ffi 39354 | A function with finite dom... |
| suprnmpt 39355 | An explicit bound for the ... |
| rnffi 39356 | The range of a function wi... |
| mptelpm 39357 | A function in maps-to nota... |
| rnmptpr 39358 | Range of a function define... |
| resmpti 39359 | Restriction of the mapping... |
| founiiun 39360 | Union expressed as an inde... |
| f1oeq2d 39361 | Equality deduction for one... |
| rnresun 39362 | Distribution law for range... |
| f1oeq1d 39363 | Equality deduction for one... |
| dffo3f 39364 | An onto mapping expressed ... |
| rnresss 39365 | The range of a restriction... |
| elrnmptd 39366 | The range of a function in... |
| elrnmptf 39367 | The range of a function in... |
| rnmptssrn 39368 | Inclusion relation for two... |
| disjf1 39369 | A 1 to 1 mapping built fro... |
| rnsnf 39370 | The range of a function wh... |
| wessf1ornlem 39371 | Given a function ` F ` on ... |
| wessf1orn 39372 | Given a function ` F ` on ... |
| foelrnf 39373 | Property of a surjective f... |
| nelrnres 39374 | If ` A ` is not in the ran... |
| disjrnmpt2 39375 | Disjointness of the range ... |
| elrnmpt1sf 39376 | Elementhood in an image se... |
| founiiun0 39377 | Union expressed as an inde... |
| disjf1o 39378 | A bijection built from dis... |
| fompt 39379 | Express being onto for a m... |
| disjinfi 39380 | Only a finite number of di... |
| fvovco 39381 | Value of the composition o... |
| ssnnf1octb 39382 | There exists a bijection b... |
| mapdm0OLD 39383 | The empty set is the only ... |
| nnf1oxpnn 39384 | There is a bijection betwe... |
| rnmptssd 39385 | The range of an operation ... |
| projf1o 39386 | A biijection from a set to... |
| fvmap 39387 | Function value for a membe... |
| mapsnd 39388 | The value of set exponenti... |
| fvixp2 39389 | Projection of a factor of ... |
| fidmfisupp 39390 | A function with a finite d... |
| mapsnend 39391 | Set exponentiation to a si... |
| choicefi 39392 | For a finite set, a choice... |
| mpct 39393 | The exponentiation of a co... |
| cnmetcoval 39394 | Value of the distance func... |
| fcomptss 39395 | Express composition of two... |
| elmapsnd 39396 | Membership in a set expone... |
| mapss2 39397 | Subset inheritance for set... |
| fsneq 39398 | Equality condition for two... |
| difmap 39399 | Difference of two sets exp... |
| unirnmap 39400 | Given a subset of a set ex... |
| inmap 39401 | Intersection of two sets e... |
| fcoss 39402 | Composition of two mapping... |
| fsneqrn 39403 | Equality condition for two... |
| difmapsn 39404 | Difference of two sets exp... |
| mapssbi 39405 | Subset inheritance for set... |
| unirnmapsn 39406 | Equality theorem for a sub... |
| iunmapss 39407 | The indexed union of set e... |
| ssmapsn 39408 | A subset ` C ` of a set ex... |
| iunmapsn 39409 | The indexed union of set e... |
| absfico 39410 | Mapping domain and codomai... |
| icof 39411 | The set of left-closed rig... |
| rnmpt0 39412 | The range of a function in... |
| rnmptn0 39413 | The range of a function in... |
| elpmrn 39414 | The range of a partial fun... |
| imaexi 39415 | The image of a set is a se... |
| axccdom 39416 | Relax the constraint on ax... |
| dmmptdf 39417 | The domain of the mapping ... |
| elpmi2 39418 | The domain of a partial fu... |
| dmrelrnrel 39419 | A relation preserving func... |
| fdmd 39420 | The domain of a mapping. ... |
| fco3 39421 | Functionality of a composi... |
| dmexd 39422 | The domain of a set is a s... |
| fvcod 39423 | Value of a function compos... |
| fcod 39424 | Composition of two mapping... |
| freld 39425 | A mapping is a relation. ... |
| frnd 39426 | The range of a mapping. (... |
| elrnmpt2id 39427 | Membership in the range of... |
| fvmptelrn 39428 | A function's value belongs... |
| axccd 39429 | An alternative version of ... |
| axccd2 39430 | An alternative version of ... |
| funimassd 39431 | Sufficient condition for t... |
| fimassd 39432 | The image of a class is a ... |
| feqresmptf 39433 | Express a restricted funct... |
| fnmptd 39434 | The maps-to notation defin... |
| elrnmpt1d 39435 | Elementhood in an image se... |
| dmresss 39436 | The domain of a restrictio... |
| mptima 39437 | Image of a function in map... |
| dmmptssf 39438 | The domain of a mapping is... |
| dmmptdf2 39439 | The domain of the mapping ... |
| dmuz 39440 | Domain of the upper intege... |
| fndmd 39441 | The domain of a function. ... |
| fmptd2f 39442 | Domain and codomain of the... |
| mpteq1df 39443 | An equality theorem for th... |
| mptexf 39444 | If the domain of a functio... |
| fvmptd2 39445 | Deduction version of ~ fvm... |
| fvmpt4 39446 | Value of a function given ... |
| fvmptd3 39447 | Deduction version of ~ fvm... |
| fmptf 39448 | Functionality of the mappi... |
| resimass 39449 | The image of a restriction... |
| mptssid 39450 | The mapping operation expr... |
| mptfnd 39451 | The maps-to notation defin... |
| mpteq12da 39452 | An equality inference for ... |
| rnmptlb 39453 | Boundness below of the ran... |
| elpreimad 39454 | Membership in the preimage... |
| rnmptbddlem 39455 | Boundness of the range of ... |
| rnmptbdd 39456 | Boundness of the range of ... |
| mptima2 39457 | Image of a function in map... |
| fvelimad 39458 | Function value in an image... |
| fnfvimad 39459 | A function's value belongs... |
| fmptd2 39460 | Domain and codomain of the... |
| funimaeq 39461 | Membership relation for th... |
| rnmptssf 39462 | The range of an operation ... |
| rnmptbd2lem 39463 | Boundness below of the ran... |
| rnmptbd2 39464 | Boundness below of the ran... |
| infnsuprnmpt 39465 | The indexed infimum of rea... |
| suprclrnmpt 39466 | Closure of the indexed sup... |
| suprubrnmpt2 39467 | A member of a nonempty ind... |
| suprubrnmpt 39468 | A member of a nonempty ind... |
| rnmptssdf 39469 | The range of an operation ... |
| rnmptbdlem 39470 | Boundness above of the ran... |
| rnmptbd 39471 | Boundness above of the ran... |
| rnmptss2 39472 | The range of an operation ... |
| elmptima 39473 | The image of a function in... |
| ralrnmpt3 39474 | A restricted quantifier ov... |
| fvelima2 39475 | Function value in an image... |
| funresd 39476 | A restriction of a functio... |
| rnmptssbi 39477 | The range of an operation ... |
| fnfvima2 39478 | Given an element of the pr... |
| fnfvelrnd 39479 | A function's value belongs... |
| imass2d 39480 | Subset theorem for image. ... |
| imassmpt 39481 | Membership relation for th... |
| fnssresd 39482 | Restriction of a function ... |
| fpmd 39483 | A total function is a part... |
| fconst7 39484 | An alternative way to expr... |
| sub2times 39485 | Subtracting from a number,... |
| xrltled 39486 | 'Less than' implies 'less ... |
| abssubrp 39487 | The distance of two distin... |
| elfzfzo 39488 | Relationship between membe... |
| oddfl 39489 | Odd number representation ... |
| abscosbd 39490 | Bound for the absolute val... |
| mul13d 39491 | Commutative/associative la... |
| negpilt0 39492 | Negative ` _pi ` is negati... |
| dstregt0 39493 | A complex number ` A ` tha... |
| subadd4b 39494 | Rearrangement of 4 terms i... |
| xrlttri5d 39495 | Not equal and not larger i... |
| neglt 39496 | The negative of a positive... |
| zltlesub 39497 | If an integer ` N ` is sma... |
| divlt0gt0d 39498 | The ratio of a negative nu... |
| subsub23d 39499 | Swap subtrahend and result... |
| 2timesgt 39500 | Double of a positive real ... |
| reopn 39501 | The reals are open with re... |
| elfzop1le2 39502 | A member in a half-open in... |
| sub31 39503 | Swap the first and third t... |
| nnne1ge2 39504 | A positive integer which i... |
| lefldiveq 39505 | A closed enough, smaller r... |
| negsubdi3d 39506 | Distribution of negative o... |
| ltdiv2dd 39507 | Division of a positive num... |
| absnpncand 39508 | Triangular inequality, com... |
| abssinbd 39509 | Bound for the absolute val... |
| halffl 39510 | Floor of ` ( 1 / 2 ) ` . ... |
| monoords 39511 | Ordering relation for a st... |
| hashssle 39512 | The size of a subset of a ... |
| lttri5d 39513 | Not equal and not larger i... |
| fzisoeu 39514 | A finite ordered set has a... |
| lt3addmuld 39515 | If three real numbers are ... |
| absnpncan2d 39516 | Triangular inequality, com... |
| fperiodmullem 39517 | A function with period T i... |
| fperiodmul 39518 | A function with period T i... |
| upbdrech 39519 | Choice of an upper bound f... |
| lt4addmuld 39520 | If four real numbers are l... |
| absnpncan3d 39521 | Triangular inequality, com... |
| upbdrech2 39522 | Choice of an upper bound f... |
| ssfiunibd 39523 | A finite union of bounded ... |
| fz1ssfz0 39524 | Subset relationship for fi... |
| fzdifsuc2 39525 | Remove a successor from th... |
| fzsscn 39526 | A finite sequence of integ... |
| divcan8d 39527 | A cancellation law for div... |
| dmmcand 39528 | Cancellation law for divis... |
| fzssre 39529 | A finite sequence of integ... |
| elfzelzd 39530 | A member of a finite set o... |
| bccld 39531 | A binomial coefficient, in... |
| leadd12dd 39532 | Addition to both sides of ... |
| fzssnn0 39533 | A finite set of sequential... |
| xreqle 39534 | Equality implies 'less tha... |
| xaddid2d 39535 | ` 0 ` is a left identity f... |
| xadd0ge 39536 | A number is less than or e... |
| elfzolem1 39537 | A member in a half-open in... |
| xrgtned 39538 | 'Greater than' implies not... |
| xrleneltd 39539 | 'Less than or equal to' an... |
| xaddcomd 39540 | The extended real addition... |
| supxrre3 39541 | The supremum of a nonempty... |
| uzfissfz 39542 | For any finite subset of t... |
| xleadd2d 39543 | Addition of extended reals... |
| suprltrp 39544 | The supremum of a nonempty... |
| xleadd1d 39545 | Addition of extended reals... |
| xreqled 39546 | Equality implies 'less tha... |
| xrgepnfd 39547 | An extended real greater o... |
| xrge0nemnfd 39548 | A nonnegative extended rea... |
| supxrgere 39549 | If a real number can be ap... |
| iuneqfzuzlem 39550 | Lemma for ~ iuneqfzuz : he... |
| iuneqfzuz 39551 | If two unions indexed by u... |
| xle2addd 39552 | Adding both side of two in... |
| supxrgelem 39553 | If an extended real number... |
| supxrge 39554 | If an extended real number... |
| suplesup 39555 | If any element of ` A ` ca... |
| infxrglb 39556 | The infimum of a set of ex... |
| xadd0ge2 39557 | A number is less than or e... |
| nepnfltpnf 39558 | An extended real that is n... |
| ltadd12dd 39559 | Addition to both sides of ... |
| nemnftgtmnft 39560 | An extended real that is n... |
| xrgtso 39561 | 'Greater than' is a strict... |
| rpex 39562 | The positive reals form a ... |
| xrge0ge0 39563 | A nonnegative extended rea... |
| xrssre 39564 | A subset of extended reals... |
| ssuzfz 39565 | A finite subset of the upp... |
| absfun 39566 | The absolute value is a fu... |
| infrpge 39567 | The infimum of a non empty... |
| xrlexaddrp 39568 | If an extended real number... |
| supsubc 39569 | The supremum function dist... |
| xralrple2 39570 | Show that ` A ` is less th... |
| nnuzdisj 39571 | The first ` N ` elements o... |
| ltdivgt1 39572 | Divsion by a number greate... |
| xrltned 39573 | 'Less than' implies not eq... |
| nnsplit 39574 | Express the set of positiv... |
| divdiv3d 39575 | Division into a fraction. ... |
| abslt2sqd 39576 | Comparison of the square o... |
| qenom 39577 | The set of rational number... |
| qct 39578 | The set of rational number... |
| xrltnled 39579 | 'Less than' in terms of 'l... |
| lenlteq 39580 | 'less than or equal to' bu... |
| xrred 39581 | An extended real that is n... |
| rr2sscn2 39582 | ` RR^ 2 ` is a subset of C... |
| infxr 39583 | The infimum of a set of ex... |
| infxrunb2 39584 | The infimum of an unbounde... |
| infxrbnd2 39585 | The infimum of a bounded-b... |
| infleinflem1 39586 | Lemma for ~ infleinf , cas... |
| infleinflem2 39587 | Lemma for ~ infleinf , whe... |
| infleinf 39588 | If any element of ` B ` ca... |
| xralrple4 39589 | Show that ` A ` is less th... |
| xralrple3 39590 | Show that ` A ` is less th... |
| eluzelzd 39591 | A member of an upper set o... |
| suplesup2 39592 | If any element of ` A ` is... |
| recnnltrp 39593 | ` N ` is a natural number ... |
| fiminre2 39594 | A nonempty finite set of r... |
| nnn0 39595 | The set of positive intege... |
| fzct 39596 | A finite set of sequential... |
| rpgtrecnn 39597 | Any positive real number i... |
| fzossuz 39598 | A half-open integer interv... |
| fzossz 39599 | A half-open integer interv... |
| infrefilb 39600 | The infimum of a finite se... |
| infxrrefi 39601 | The real and extended real... |
| xrralrecnnle 39602 | Show that ` A ` is less th... |
| fzoct 39603 | A finite set of sequential... |
| frexr 39604 | A function taking real val... |
| nnrecrp 39605 | The reciprocal of a positi... |
| qred 39606 | A rational number is a rea... |
| reclt0d 39607 | The reciprocal of a negati... |
| lt0neg1dd 39608 | If a number is negative, i... |
| mnfled 39609 | Minus infinity is less tha... |
| xrleidd 39610 | 'Less than or equal to' is... |
| negelrpd 39611 | The negation of a negative... |
| infxrcld 39612 | The infimum of an arbitrar... |
| xrralrecnnge 39613 | Show that ` A ` is less th... |
| reclt0 39614 | The reciprocal of a negati... |
| ltmulneg 39615 | Multiplying by a negative ... |
| allbutfi 39616 | For all but finitely many.... |
| ltdiv23neg 39617 | Swap denominator with othe... |
| xreqnltd 39618 | A consequence of trichotom... |
| mnfnre2 39619 | Minus infinity is not a re... |
| uzssre 39620 | An upper set of integers i... |
| zssxr 39621 | The integers are a subset ... |
| fisupclrnmpt 39622 | A nonempty finite indexed ... |
| supxrunb3 39623 | The supremum of an unbound... |
| elfzod 39624 | Membership in a half-open ... |
| fimaxre4 39625 | A nonempty finite set of r... |
| ren0 39626 | The set of reals is nonemp... |
| eluzelz2 39627 | A member of an upper set o... |
| pnfnre2 39628 | Plus infinity is not a rea... |
| resabs2d 39629 | Absorption law for restric... |
| uzid2 39630 | Membership of the least me... |
| uzidd 39631 | Membership of the least me... |
| supxrleubrnmpt 39632 | The supremum of a nonempty... |
| uzssre2 39633 | An upper set of integers i... |
| uzssd 39634 | Subset relationship for tw... |
| eluzd 39635 | Membership in an upper set... |
| elfzd 39636 | Membership in a finite set... |
| infxrlbrnmpt2 39637 | A member of a nonempty ind... |
| xrre4 39638 | An extended real is real i... |
| uz0 39639 | The upper integers functio... |
| eluzelz2d 39640 | A member of an upper set o... |
| infleinf2 39641 | If any element in ` B ` is... |
| unb2ltle 39642 | "Unbounded below" expresse... |
| uzidd2 39643 | Membership of the least me... |
| uzssd2 39644 | Subset relationship for tw... |
| rexabslelem 39645 | An indexed set of absolute... |
| rexabsle 39646 | An indexed set of absolute... |
| allbutfiinf 39647 | Given a "for all but finit... |
| supxrrernmpt 39648 | The real and extended real... |
| suprleubrnmpt 39649 | The supremum of a nonempty... |
| infrnmptle 39650 | An indexed infimum of exte... |
| infxrunb3 39651 | The infimum of an unbounde... |
| uzn0d 39652 | The upper integers are all... |
| uzssd3 39653 | Subset relationship for tw... |
| rexabsle2 39654 | An indexed set of absolute... |
| infxrunb3rnmpt 39655 | The infimum of an unbounde... |
| supxrre3rnmpt 39656 | The indexed supremum of a ... |
| uzublem 39657 | A set of reals, indexed by... |
| uzub 39658 | A set of reals, indexed by... |
| ssrexr 39659 | A subset of the reals is a... |
| supxrmnf2 39660 | Removing minus infinity fr... |
| supxrcli 39661 | The supremum of an arbitra... |
| uzid3 39662 | Membership of the least me... |
| infxrlesupxr 39663 | The supremum of a nonempty... |
| xnegeqd 39664 | Equality of two extended n... |
| xnegrecl 39665 | The extended real negative... |
| xnegnegi 39666 | Extended real version of ~... |
| xnegeqi 39667 | Equality of two extended n... |
| nfxnegd 39668 | Deduction version of ~ nfx... |
| xnegnegd 39669 | Extended real version of ~... |
| uzred 39670 | An upper integer is a real... |
| xnegcli 39671 | Closure of extended real n... |
| supminfrnmpt 39672 | The indexed supremum of a ... |
| ceilged 39673 | The ceiling of a real numb... |
| infxrpnf 39674 | Adding plus infinity to a ... |
| infxrrnmptcl 39675 | The infimum of an arbitrar... |
| leneg2d 39676 | Negative of one side of 'l... |
| supxrltinfxr 39677 | The supremum of the empty ... |
| max1d 39678 | A number is less than or e... |
| ceilcld 39679 | Closure of the ceiling fun... |
| supxrleubrnmptf 39680 | The supremum of a nonempty... |
| nleltd 39681 | 'Not less than or equal to... |
| zxrd 39682 | An integer is an extended ... |
| infxrgelbrnmpt 39683 | The infimum of an indexed ... |
| rphalfltd 39684 | Half of a positive real is... |
| uzssz2 39685 | An upper set of integers i... |
| 1xr 39686 | ` 1 ` is an extended real ... |
| leneg3d 39687 | Negative of one side of 'l... |
| max2d 39688 | A number is less than or e... |
| uzn0bi 39689 | The upper integers functio... |
| xnegrecl2 39690 | If the extended real negat... |
| nfxneg 39691 | Bound-variable hypothesis ... |
| uzxrd 39692 | An upper integer is an ext... |
| infxrpnf2 39693 | Removing plus infinity fro... |
| supminfxr 39694 | The extended real suprema ... |
| infrpgernmpt 39695 | The infimum of a non empty... |
| xnegre 39696 | An extended real is real i... |
| xnegrecl2d 39697 | If the extended real negat... |
| uzxr 39698 | An upper integer is an ext... |
| supminfxr2 39699 | The extended real suprema ... |
| xnegred 39700 | An extended real is real i... |
| supminfxrrnmpt 39701 | The indexed supremum of a ... |
| min1d 39702 | The minimum of two numbers... |
| min2d 39703 | The minimum of two numbers... |
| pnfged 39704 | Plus infinity is an upper ... |
| xrnpnfmnf 39705 | An extended real that is n... |
| uzsscn 39706 | An upper set of integers i... |
| absimnre 39707 | The absolute value of the ... |
| uzsscn2 39708 | An upper set of integers i... |
| xrtgcntopre 39709 | The standard topologies on... |
| absimlere 39710 | The absolute value of the ... |
| rpssxr 39711 | The positive reals are a s... |
| gtnelioc 39712 | A real number larger than ... |
| ioossioc 39713 | An open interval is a subs... |
| ioondisj2 39714 | A condition for two open i... |
| ioondisj1 39715 | A condition for two open i... |
| ioosscn 39716 | An open interval is a set ... |
| ioogtlb 39717 | An element of a closed int... |
| evthiccabs 39718 | Extreme Value Theorem on y... |
| ltnelicc 39719 | A real number smaller than... |
| eliood 39720 | Membership in an open real... |
| iooabslt 39721 | An upper bound for the dis... |
| gtnelicc 39722 | A real number greater than... |
| iooinlbub 39723 | An open interval has empty... |
| iocgtlb 39724 | An element of a left open ... |
| iocleub 39725 | An element of a left open ... |
| eliccd 39726 | Membership in a closed rea... |
| iccssred 39727 | A closed real interval is ... |
| eliccre 39728 | A member of a closed inter... |
| eliooshift 39729 | Element of an open interva... |
| eliocd 39730 | Membership in a left open,... |
| snunioo2 39731 | The closure of one end of ... |
| icoltub 39732 | An element of a left close... |
| tgiooss 39733 | The restriction of the com... |
| eliocre 39734 | A member of a left open, r... |
| iooltub 39735 | An element of an open inte... |
| ioontr 39736 | The interior of an interva... |
| eliccxr 39737 | A member of a closed inter... |
| snunioo1 39738 | The closure of one end of ... |
| lbioc 39739 | An left open right closed ... |
| ioomidp 39740 | The midpoint is an element... |
| iccdifioo 39741 | If the open inverval is re... |
| iccdifprioo 39742 | An open interval is the cl... |
| ioossioobi 39743 | Biconditional form of ~ io... |
| iccshift 39744 | A closed interval shifted ... |
| iccsuble 39745 | An upper bound to the dist... |
| iocopn 39746 | A left open right closed i... |
| eliccelioc 39747 | Membership in a closed int... |
| iooshift 39748 | An open interval shifted b... |
| iccintsng 39749 | Intersection of two adiace... |
| icoiccdif 39750 | Left closed, right open in... |
| icoopn 39751 | A left closed right open i... |
| icoub 39752 | A left-closed, right-open ... |
| eliccxrd 39753 | Membership in a closed rea... |
| pnfel0pnf 39754 | ` +oo ` is a nonnegative e... |
| ge0nemnf2 39755 | A nonnegative extended rea... |
| eliccnelico 39756 | An element of a closed int... |
| eliccelicod 39757 | A member of a closed inter... |
| ge0xrre 39758 | A nonnegative extended rea... |
| ge0lere 39759 | A nonnegative extended Rea... |
| elicores 39760 | Membership in a left-close... |
| inficc 39761 | The infimum of a nonempty ... |
| qinioo 39762 | The rational numbers are d... |
| lenelioc 39763 | A real number smaller than... |
| ioonct 39764 | C non empty open interval ... |
| xrgtnelicc 39765 | A real number greater than... |
| iccdificc 39766 | The difference of two clos... |
| iocnct 39767 | A non empty left-open, rig... |
| iccnct 39768 | A closed interval, with mo... |
| iooiinicc 39769 | A closed interval expresse... |
| iccgelbd 39770 | An element of a closed int... |
| iooltubd 39771 | An element of an open inte... |
| icoltubd 39772 | An element of a left close... |
| qelioo 39773 | The rational numbers are d... |
| tgqioo2 39774 | Every open set of reals is... |
| iccleubd 39775 | An element of a closed int... |
| elioored 39776 | A member of an open interv... |
| ioogtlbd 39777 | An element of a closed int... |
| ioofun 39778 | ` (,) ` is a function. (C... |
| icomnfinre 39779 | A left-closed, right-open,... |
| sqrlearg 39780 | The square compared with i... |
| ressiocsup 39781 | If the supremum belongs to... |
| ressioosup 39782 | If the supremum does not b... |
| iooiinioc 39783 | A left-open, right-closed ... |
| ressiooinf 39784 | If the infimum does not be... |
| icogelbd 39785 | An element of a left close... |
| iocleubd 39786 | An element of a left open ... |
| uzinico 39787 | An upper interval of integ... |
| preimaiocmnf 39788 | Preimage of a right-closed... |
| uzinico2 39789 | An upper interval of integ... |
| uzinico3 39790 | An upper interval of integ... |
| icossico2 39791 | Condition for a closed-bel... |
| dmico 39792 | The domain of the closed-b... |
| ndmico 39793 | The closed-below, open-abo... |
| uzubioo 39794 | The upper integers are unb... |
| uzubico 39795 | The upper integers are unb... |
| uzubioo2 39796 | The upper integers are unb... |
| uzubico2 39797 | The upper integers are unb... |
| iocgtlbd 39798 | An element of a left open ... |
| xrtgioo2 39799 | The topology on the extend... |
| tgioo4 39800 | The standard topology on t... |
| fsumclf 39801 | Closure of a finite sum of... |
| fsummulc1f 39802 | Closure of a finite sum of... |
| fsumnncl 39803 | Closure of a non empty, fi... |
| fsumsplit1 39804 | Separate out a term in a f... |
| fsumge0cl 39805 | The finite sum of nonnegat... |
| fsumf1of 39806 | Re-index a finite sum usin... |
| fsumiunss 39807 | Sum over a disjoint indexe... |
| fsumreclf 39808 | Closure of a finite sum of... |
| fsumlessf 39809 | A shorter sum of nonnegati... |
| fsumsupp0 39810 | Finite sum of function val... |
| fsumsermpt 39811 | A finite sum expressed in ... |
| fmul01 39812 | Multiplying a finite numbe... |
| fmulcl 39813 | If ' Y ' is closed under t... |
| fmuldfeqlem1 39814 | induction step for the pro... |
| fmuldfeq 39815 | X and Z are two equivalent... |
| fmul01lt1lem1 39816 | Given a finite multiplicat... |
| fmul01lt1lem2 39817 | Given a finite multiplicat... |
| fmul01lt1 39818 | Given a finite multiplicat... |
| cncfmptss 39819 | A continuous complex funct... |
| rrpsscn 39820 | The positive reals are a s... |
| mulc1cncfg 39821 | A version of ~ mulc1cncf u... |
| infrglb 39822 | The infimum of a nonempty ... |
| expcnfg 39823 | If ` F ` is a complex cont... |
| prodeq2ad 39824 | Equality deduction for pro... |
| fprodsplit1 39825 | Separate out a term in a f... |
| fprodexp 39826 | Positive integer exponenti... |
| fprodabs2 39827 | The absolute value of a fi... |
| fprod0 39828 | A finite product with a ze... |
| mccllem 39829 | * Induction step for ~ mcc... |
| mccl 39830 | A multinomial coefficient,... |
| fprodcnlem 39831 | A finite product of functi... |
| fprodcn 39832 | A finite product of functi... |
| clim1fr1 39833 | A class of sequences of fr... |
| isumneg 39834 | Negation of a converging s... |
| climrec 39835 | Limit of the reciprocal of... |
| climmulf 39836 | A version of ~ climmul usi... |
| climexp 39837 | The limit of natural power... |
| climinf 39838 | A bounded monotonic non in... |
| climsuselem1 39839 | The subsequence index ` I ... |
| climsuse 39840 | A subsequence ` G ` of a c... |
| climrecf 39841 | A version of ~ climrec usi... |
| climneg 39842 | Complex limit of the negat... |
| climinff 39843 | A version of ~ climinf usi... |
| climdivf 39844 | Limit of the ratio of two ... |
| climreeq 39845 | If ` F ` is a real functio... |
| ellimciota 39846 | An explicit value for the ... |
| climaddf 39847 | A version of ~ climadd usi... |
| mullimc 39848 | Limit of the product of tw... |
| ellimcabssub0 39849 | An equivalent condition fo... |
| limcdm0 39850 | If a function has empty do... |
| islptre 39851 | An equivalence condition f... |
| limccog 39852 | Limit of the composition o... |
| limciccioolb 39853 | The limit of a function at... |
| climf 39854 | Express the predicate: Th... |
| mullimcf 39855 | Limit of the multiplicatio... |
| constlimc 39856 | Limit of constant function... |
| rexlim2d 39857 | Inference removing two res... |
| idlimc 39858 | Limit of the identity func... |
| divcnvg 39859 | The sequence of reciprocal... |
| limcperiod 39860 | If ` F ` is a periodic fun... |
| limcrecl 39861 | If ` F ` is a real-valued ... |
| sumnnodd 39862 | A series indexed by ` NN `... |
| lptioo2 39863 | The upper bound of an open... |
| lptioo1 39864 | The lower bound of an open... |
| elprn1 39865 | A member of an unordered p... |
| elprn2 39866 | A member of an unordered p... |
| limcmptdm 39867 | The domain of a map-to fun... |
| clim2f 39868 | Express the predicate: Th... |
| limcicciooub 39869 | The limit of a function at... |
| ltmod 39870 | A sufficient condition for... |
| islpcn 39871 | A characterization for a l... |
| lptre2pt 39872 | If a set in the real line ... |
| limsupre 39873 | If a sequence is bounded, ... |
| limcresiooub 39874 | The left limit doesn't cha... |
| limcresioolb 39875 | The right limit doesn't ch... |
| limcleqr 39876 | If the left and the right ... |
| lptioo2cn 39877 | The upper bound of an open... |
| lptioo1cn 39878 | The lower bound of an open... |
| neglimc 39879 | Limit of the negative func... |
| addlimc 39880 | Sum of two limits. (Contr... |
| 0ellimcdiv 39881 | If the numerator converges... |
| clim2cf 39882 | Express the predicate ` F ... |
| limclner 39883 | For a limit point, both fr... |
| sublimc 39884 | Subtraction of two limits.... |
| reclimc 39885 | Limit of the reciprocal of... |
| clim0cf 39886 | Express the predicate ` F ... |
| limclr 39887 | For a limit point, both fr... |
| divlimc 39888 | Limit of the quotient of t... |
| expfac 39889 | Factorial grows faster tha... |
| climconstmpt 39890 | A constant sequence conver... |
| climresmpt 39891 | A function restricted to u... |
| climsubmpt 39892 | Limit of the difference of... |
| climsubc2mpt 39893 | Limit of the difference of... |
| climsubc1mpt 39894 | Limit of the difference of... |
| fnlimfv 39895 | The value of the limit fun... |
| climreclf 39896 | The limit of a convergent ... |
| climeldmeq 39897 | Two functions that are eve... |
| climf2 39898 | Express the predicate: Th... |
| fnlimcnv 39899 | The sequence of function v... |
| climeldmeqmpt 39900 | Two functions that are eve... |
| climfveq 39901 | Two functions that are eve... |
| clim2f2 39902 | Express the predicate: Th... |
| climfveqmpt 39903 | Two functions that are eve... |
| climd 39904 | Express the predicate: Th... |
| clim2d 39905 | The limit of complex numbe... |
| fnlimfvre 39906 | The limit function of real... |
| allbutfifvre 39907 | Given a sequence of real-v... |
| climleltrp 39908 | The limit of complex numbe... |
| fnlimfvre2 39909 | The limit function of real... |
| fnlimf 39910 | The limit function of real... |
| fnlimabslt 39911 | A sequence of function val... |
| climfveqf 39912 | Two functions that are eve... |
| climmptf 39913 | Exhibit a function ` G ` w... |
| climfveqmpt3 39914 | Two functions that are eve... |
| climeldmeqf 39915 | Two functions that are eve... |
| climreclmpt 39916 | The limit of B convergent ... |
| limsupref 39917 | If a sequence is bounded, ... |
| limsupbnd1f 39918 | If a sequence is eventuall... |
| climbddf 39919 | A converging sequence of c... |
| climeqf 39920 | Two functions that are eve... |
| climeldmeqmpt3 39921 | Two functions that are eve... |
| limsupcld 39922 | Closure of the superior li... |
| climfv 39923 | The limit of a convergent ... |
| limsupval3 39924 | The superior limit of an i... |
| climfveqmpt2 39925 | Two functions that are eve... |
| limsup0 39926 | The superior limit of the ... |
| climeldmeqmpt2 39927 | Two functions that are eve... |
| limsupresre 39928 | The supremum limit of a fu... |
| climeqmpt 39929 | Two functions that are eve... |
| climfvd 39930 | The limit of a convergent ... |
| limsuplesup 39931 | An upper bound for the sup... |
| limsupresico 39932 | The superior limit doesn't... |
| limsuppnfdlem 39933 | If the restriction of a fu... |
| limsuppnfd 39934 | If the restriction of a fu... |
| limsupresuz 39935 | If the real part of the do... |
| limsupub 39936 | If the limsup is not ` +oo... |
| limsupres 39937 | The superior limit of a re... |
| climinf2lem 39938 | A convergent, non-increasi... |
| climinf2 39939 | A convergent, non-increasi... |
| limsupvaluz 39940 | The superior limit, when t... |
| limsupresuz2 39941 | If the domain of a functio... |
| limsuppnflem 39942 | If the restriction of a fu... |
| limsuppnf 39943 | If the restriction of a fu... |
| limsupubuzlem 39944 | If the limsup is not ` +oo... |
| limsupubuz 39945 | For a real-valued function... |
| climinf2mpt 39946 | A bounded below, monotonic... |
| climinfmpt 39947 | A bounded below, monotonic... |
| climinf3 39948 | A convergent, non-increasi... |
| limsupvaluzmpt 39949 | The superior limit, when t... |
| limsupequzmpt2 39950 | Two functions that are eve... |
| limsupubuzmpt 39951 | If the limsup is not ` +oo... |
| limsupmnflem 39952 | The superior limit of a fu... |
| limsupmnf 39953 | The superior limit of a fu... |
| limsupequzlem 39954 | Two functions that are eve... |
| limsupequz 39955 | Two functions that are eve... |
| limsupre2lem 39956 | Given a function on the ex... |
| limsupre2 39957 | Given a function on the ex... |
| limsupmnfuzlem 39958 | The superior limit of a fu... |
| limsupmnfuz 39959 | The superior limit of a fu... |
| limsupequzmptlem 39960 | Two functions that are eve... |
| limsupequzmpt 39961 | Two functions that are eve... |
| limsupre2mpt 39962 | Given a function on the ex... |
| limsupequzmptf 39963 | Two functions that are eve... |
| limsupre3lem 39964 | Given a function on the ex... |
| limsupre3 39965 | Given a function on the ex... |
| limsupre3mpt 39966 | Given a function on the ex... |
| limsupre3uzlem 39967 | Given a function on the ex... |
| limsupre3uz 39968 | Given a function on the ex... |
| limsupreuz 39969 | Given a function on the re... |
| limsupvaluz2 39970 | The superior limit, when t... |
| limsupreuzmpt 39971 | Given a function on the re... |
| supcnvlimsup 39972 | If a function on a set of ... |
| supcnvlimsupmpt 39973 | If a function on a set of ... |
| 0cnv 39974 | If (/) is a complex number... |
| climuzlem 39975 | Express the predicate: Th... |
| climuz 39976 | Express the predicate: Th... |
| lmbr3v 39977 | Express the binary relatio... |
| climisp 39978 | If a sequence converges to... |
| lmbr3 39979 | Express the binary relatio... |
| climrescn 39980 | A sequence converging w.r.... |
| climxrrelem 39981 | If a seqence ranging over ... |
| climxrre 39982 | If a sequence ranging over... |
| limsuplt2 39985 | The defining property of t... |
| liminfgord 39986 | Ordering property of the i... |
| limsupvald 39987 | The superior limit of a se... |
| limsupresicompt 39988 | The superior limit doesn't... |
| limsupcli 39989 | Closure of the superior li... |
| liminfgf 39990 | Closure of the inferior li... |
| liminfval 39991 | The inferior limit of a se... |
| climlimsup 39992 | A sequence of real numbers... |
| limsupge 39993 | The defining property of t... |
| liminfgval 39994 | Value of the inferior limi... |
| liminfcl 39995 | Closure of the inferior li... |
| liminfvald 39996 | The inferior limit of a se... |
| liminfval5 39997 | The inferior limit of an i... |
| limsupresxr 39998 | The superior limit of a fu... |
| liminfresxr 39999 | The inferior limit of a fu... |
| liminfval2 40000 | The superior limit, relati... |
| climlimsupcex 40001 | Counterexample for ~ climl... |
| liminfcld 40002 | Closure of the inferior li... |
| liminfresico 40003 | The inferior limit doesn't... |
| limsup10exlem 40004 | The range of the given fun... |
| limsup10ex 40005 | The superior limit of a fu... |
| liminf10ex 40006 | The inferior limit of a fu... |
| liminflelimsuplem 40007 | The superior limit is grea... |
| liminflelimsup 40008 | The superior limit is grea... |
| limsupgtlem 40009 | For any positive real, the... |
| limsupgt 40010 | Given a sequence of real n... |
| liminfresre 40011 | The inferior limit of a fu... |
| liminfresicompt 40012 | The inferior limit doesn't... |
| liminfltlimsupex 40013 | An example where the ` lim... |
| liminfgelimsup 40014 | The inferior limit is grea... |
| liminfvalxr 40015 | Alternate definition of ` ... |
| liminfresuz 40016 | If the real part of the do... |
| liminflelimsupuz 40017 | The superior limit is grea... |
| liminfvalxrmpt 40018 | Alternate definition of ` ... |
| liminfresuz2 40019 | If the domain of a functio... |
| liminfgelimsupuz 40020 | The inferior limit is grea... |
| liminfval4 40021 | Alternate definition of ` ... |
| liminfval3 40022 | Alternate definition of ` ... |
| liminfequzmpt2 40023 | Two functions that are eve... |
| liminfvaluz 40024 | Alternate definition of ` ... |
| liminf0 40025 | The inferior limit of the ... |
| limsupval4 40026 | Alternate definition of ` ... |
| liminfvaluz2 40027 | Alternate definition of ` ... |
| liminfvaluz3 40028 | Alternate definition of ` ... |
| liminflelimsupcex 40029 | A counterexample for ~ lim... |
| limsupvaluz3 40030 | Alternate definition of ` ... |
| liminfvaluz4 40031 | Alternate definition of ` ... |
| limsupvaluz4 40032 | Alternate definition of ` ... |
| climliminflimsupd 40033 | If a sequence of real numb... |
| liminfreuzlem 40034 | Given a function on the re... |
| liminfreuz 40035 | Given a function on the re... |
| liminfltlem 40036 | Given a sequence of real n... |
| liminflt 40037 | Given a sequence of real n... |
| climliminf 40038 | A sequence of real numbers... |
| liminflimsupclim 40039 | A sequence of real numbers... |
| climliminflimsup 40040 | A sequence of real numbers... |
| climliminflimsup2 40041 | A sequence of real numbers... |
| climliminflimsup3 40042 | A sequence of real numbers... |
| climliminflimsup4 40043 | A sequence of real numbers... |
| xlimrel 40046 | The limit on extended real... |
| xlimres 40047 | A function converges iff i... |
| xlimcl 40048 | The limit of a sequence of... |
| rexlimddv2 40049 | Restricted existential eli... |
| xlimclim 40050 | Given a sequence of reals,... |
| xlimconst 40051 | A constant sequence conver... |
| climxlim 40052 | A converging sequence in t... |
| xlimbr 40053 | Express the binary relatio... |
| fuzxrpmcn 40054 | A function mapping from an... |
| cnrefiisplem 40055 | Lemma for ~ cnrefiisp (som... |
| cnrefiisp 40056 | A non-real, complex number... |
| xlimxrre 40057 | If a sequence ranging over... |
| xlimmnfvlem1 40058 | The "only if" part of the ... |
| xlimmnfvlem2 40059 | The "if" part of the bicon... |
| xlimmnfv 40060 | A function converges to mi... |
| xlimconst2 40061 | A sequence that eventually... |
| xlimpnfvlem1 40062 | The "only if" part of the ... |
| xlimpnfvlem2 40063 | The "if" part of the bicon... |
| xlimpnfv 40064 | A function converges to pl... |
| xlimclim2lem 40065 | Lemma for ~ xlimclim2 . H... |
| xlimclim2 40066 | Given a sequence of extend... |
| xlimmnf 40067 | A function converges to mi... |
| xlimpnf 40068 | A function converges to pl... |
| xlimmnfmpt 40069 | A function converges to pl... |
| xlimpnfmpt 40070 | A function converges to pl... |
| climxlim2lem 40071 | In this lemma for ~ climxl... |
| climxlim2 40072 | A sequence of extended rea... |
| dfxlim2v 40073 | An alternative definition ... |
| dfxlim2 40074 | An alternative definition ... |
| coseq0 40075 | A complex number whose cos... |
| sinmulcos 40076 | Multiplication formula for... |
| coskpi2 40077 | The cosine of an integer m... |
| cosnegpi 40078 | The cosine of negative ` _... |
| sinaover2ne0 40079 | If ` A ` in ` ( 0 , 2 _pi ... |
| cosknegpi 40080 | The cosine of an integer m... |
| mulcncff 40081 | The multiplication of two ... |
| subcncf 40082 | The addition of two contin... |
| cncfmptssg 40083 | A continuous complex funct... |
| constcncfg 40084 | A constant function is a c... |
| idcncfg 40085 | The identity function is a... |
| addcncf 40086 | The addition of two contin... |
| cncfshift 40087 | A periodic continuous func... |
| resincncf 40088 | ` sin ` restricted to real... |
| addccncf2 40089 | Adding a constant is a con... |
| 0cnf 40090 | The empty set is a continu... |
| fsumcncf 40091 | The finite sum of continuo... |
| cncfperiod 40092 | A periodic continuous func... |
| subcncff 40093 | The subtraction of two con... |
| negcncfg 40094 | The opposite of a continuo... |
| cnfdmsn 40095 | A function with a singleto... |
| cncfcompt 40096 | Composition of continuous ... |
| addcncff 40097 | The addition of two contin... |
| ioccncflimc 40098 | Limit at the upper bound, ... |
| cncfuni 40099 | A function is continuous i... |
| icccncfext 40100 | A continuous function on a... |
| cncficcgt0 40101 | A the absolute value of a ... |
| icocncflimc 40102 | Limit at the lower bound, ... |
| cncfdmsn 40103 | A complex function with a ... |
| divcncff 40104 | The quotient of two contin... |
| cncfshiftioo 40105 | A periodic continuous func... |
| cncfiooicclem1 40106 | A continuous function ` F ... |
| cncfiooicc 40107 | A continuous function ` F ... |
| cncfiooiccre 40108 | A continuous function ` F ... |
| cncfioobdlem 40109 | ` G ` actually extends ` F... |
| cncfioobd 40110 | A continuous function ` F ... |
| jumpncnp 40111 | Jump discontinuity or disc... |
| cncfcompt2 40112 | Composition of continuous ... |
| cxpcncf2 40113 | The complex power function... |
| fprodcncf 40114 | The finite product of cont... |
| add1cncf 40115 | Addition to a constant is ... |
| add2cncf 40116 | Addition to a constant is ... |
| sub1cncfd 40117 | Subtracting a constant is ... |
| sub2cncfd 40118 | Subtraction from a constan... |
| fprodsub2cncf 40119 | ` F ` is continuous. (Con... |
| fprodadd2cncf 40120 | ` F ` is continuous. (Con... |
| fprodsubrecnncnvlem 40121 | The sequence ` S ` of fini... |
| fprodsubrecnncnv 40122 | The sequence ` S ` of fini... |
| fprodaddrecnncnvlem 40123 | The sequence ` S ` of fini... |
| fprodaddrecnncnv 40124 | The sequence ` S ` of fini... |
| dvsinexp 40125 | The derivative of sin^N . ... |
| dvcosre 40126 | The real derivative of the... |
| dvsinax 40127 | Derivative exercise: the d... |
| dvsubf 40128 | The subtraction rule for e... |
| dvmptconst 40129 | Function-builder for deriv... |
| dvcnre 40130 | From compex differentiatio... |
| dvmptidg 40131 | Function-builder for deriv... |
| dvresntr 40132 | Function-builder for deriv... |
| fperdvper 40133 | The derivative of a period... |
| dvmptresicc 40134 | Derivative of a function r... |
| dvasinbx 40135 | Derivative exercise: the d... |
| dvresioo 40136 | Restriction of a derivativ... |
| dvdivf 40137 | The quotient rule for ever... |
| dvdivbd 40138 | A sufficient condition for... |
| dvsubcncf 40139 | A sufficient condition for... |
| dvmulcncf 40140 | A sufficient condition for... |
| dvcosax 40141 | Derivative exercise: the d... |
| dvdivcncf 40142 | A sufficient condition for... |
| dvbdfbdioolem1 40143 | Given a function with boun... |
| dvbdfbdioolem2 40144 | A function on an open inte... |
| dvbdfbdioo 40145 | A function on an open inte... |
| ioodvbdlimc1lem1 40146 | If ` F ` has bounded deriv... |
| ioodvbdlimc1lem2 40147 | Limit at the lower bound o... |
| ioodvbdlimc1 40148 | A real function with bound... |
| ioodvbdlimc2lem 40149 | Limit at the upper bound o... |
| ioodvbdlimc2 40150 | A real function with bound... |
| dvdmsscn 40151 | ` X ` is a subset of ` CC ... |
| dvmptmulf 40152 | Function-builder for deriv... |
| dvnmptdivc 40153 | Function-builder for itera... |
| dvdsn1add 40154 | If ` K ` divides ` N ` but... |
| dvxpaek 40155 | Derivative of the polynomi... |
| dvnmptconst 40156 | The ` N ` -th derivative o... |
| dvnxpaek 40157 | The ` n ` -th derivative o... |
| dvnmul 40158 | Function-builder for the `... |
| dvmptfprodlem 40159 | Induction step for ~ dvmpt... |
| dvmptfprod 40160 | Function-builder for deriv... |
| dvnprodlem1 40161 | ` D ` is bijective. (Cont... |
| dvnprodlem2 40162 | Induction step for ~ dvnpr... |
| dvnprodlem3 40163 | The multinomial formula fo... |
| dvnprod 40164 | The multinomial formula fo... |
| itgsin0pilem1 40165 | Calculation of the integra... |
| ibliccsinexp 40166 | sin^n on a closed interval... |
| itgsin0pi 40167 | Calculation of the integra... |
| iblioosinexp 40168 | sin^n on an open integral ... |
| itgsinexplem1 40169 | Integration by parts is ap... |
| itgsinexp 40170 | A recursive formula for th... |
| iblconstmpt 40171 | A constant function is int... |
| itgeq1d 40172 | Equality theorem for an in... |
| mbf0 40173 | The empty set is a measura... |
| mbfres2cn 40174 | Measurability of a piecewi... |
| vol0 40175 | The measure of the empty s... |
| ditgeqiooicc 40176 | A function ` F ` on an ope... |
| volge0 40177 | The volume of a set is alw... |
| cnbdibl 40178 | A continuous bounded funct... |
| snmbl 40179 | A singleton is measurable.... |
| ditgeq3d 40180 | Equality theorem for the d... |
| iblempty 40181 | The empty function is inte... |
| iblsplit 40182 | The union of two integrabl... |
| volsn 40183 | A singleton has 0 Lebesgue... |
| itgvol0 40184 | If the domani is negligibl... |
| itgcoscmulx 40185 | Exercise: the integral of ... |
| iblsplitf 40186 | A version of ~ iblsplit us... |
| ibliooicc 40187 | If a function is integrabl... |
| volioc 40188 | The measure of left open, ... |
| iblspltprt 40189 | If a function is integrabl... |
| itgsincmulx 40190 | Exercise: the integral of ... |
| itgsubsticclem 40191 | lemma for ~ itgsubsticc . ... |
| itgsubsticc 40192 | Integration by u-substitut... |
| itgioocnicc 40193 | The integral of a piecewis... |
| iblcncfioo 40194 | A continuous function ` F ... |
| itgspltprt 40195 | The ` S. ` integral splits... |
| itgiccshift 40196 | The integral of a function... |
| itgperiod 40197 | The integral of a periodic... |
| itgsbtaddcnst 40198 | Integral substitution, add... |
| itgeq2d 40199 | Equality theorem for an in... |
| volico 40200 | The measure of left closed... |
| sublevolico 40201 | The Lebesgue measure of a ... |
| dmvolss 40202 | Lebesgue measurable sets a... |
| ismbl3 40203 | The predicate " ` A ` is L... |
| volioof 40204 | The function that assigns ... |
| ovolsplit 40205 | The Lebesgue outer measure... |
| fvvolioof 40206 | The function value of the ... |
| volioore 40207 | The measure of an open int... |
| fvvolicof 40208 | The function value of the ... |
| voliooico 40209 | An open interval and a lef... |
| ismbl4 40210 | The predicate " ` A ` is L... |
| volioofmpt 40211 | ` ( ( vol o. (,) ) o. F ) ... |
| volicoff 40212 | ` ( ( vol o. [,) ) o. F ) ... |
| voliooicof 40213 | The Lebesgue measure of op... |
| volicofmpt 40214 | ` ( ( vol o. [,) ) o. F ) ... |
| volicc 40215 | The Lebesgue measure of a ... |
| voliccico 40216 | A closed interval and a le... |
| mbfdmssre 40217 | The domain of a measurable... |
| stoweidlem1 40218 | Lemma for ~ stoweid . Thi... |
| stoweidlem2 40219 | lemma for ~ stoweid : here... |
| stoweidlem3 40220 | Lemma for ~ stoweid : if `... |
| stoweidlem4 40221 | Lemma for ~ stoweid : a cl... |
| stoweidlem5 40222 | There exists a δ as ... |
| stoweidlem6 40223 | Lemma for ~ stoweid : two ... |
| stoweidlem7 40224 | This lemma is used to prov... |
| stoweidlem8 40225 | Lemma for ~ stoweid : two ... |
| stoweidlem9 40226 | Lemma for ~ stoweid : here... |
| stoweidlem10 40227 | Lemma for ~ stoweid . Thi... |
| stoweidlem11 40228 | This lemma is used to prov... |
| stoweidlem12 40229 | Lemma for ~ stoweid . Thi... |
| stoweidlem13 40230 | Lemma for ~ stoweid . Thi... |
| stoweidlem14 40231 | There exists a ` k ` as in... |
| stoweidlem15 40232 | This lemma is used to prov... |
| stoweidlem16 40233 | Lemma for ~ stoweid . The... |
| stoweidlem17 40234 | This lemma proves that the... |
| stoweidlem18 40235 | This theorem proves Lemma ... |
| stoweidlem19 40236 | If a set of real functions... |
| stoweidlem20 40237 | If a set A of real functio... |
| stoweidlem21 40238 | Once the Stone Weierstrass... |
| stoweidlem22 40239 | If a set of real functions... |
| stoweidlem23 40240 | This lemma is used to prov... |
| stoweidlem24 40241 | This lemma proves that for... |
| stoweidlem25 40242 | This lemma proves that for... |
| stoweidlem26 40243 | This lemma is used to prov... |
| stoweidlem27 40244 | This lemma is used to prov... |
| stoweidlem28 40245 | There exists a δ as ... |
| stoweidlem29 40246 | When the hypothesis for th... |
| stoweidlem30 40247 | This lemma is used to prov... |
| stoweidlem31 40248 | This lemma is used to prov... |
| stoweidlem32 40249 | If a set A of real functio... |
| stoweidlem33 40250 | If a set of real functions... |
| stoweidlem34 40251 | This lemma proves that for... |
| stoweidlem35 40252 | This lemma is used to prov... |
| stoweidlem36 40253 | This lemma is used to prov... |
| stoweidlem37 40254 | This lemma is used to prov... |
| stoweidlem38 40255 | This lemma is used to prov... |
| stoweidlem39 40256 | This lemma is used to prov... |
| stoweidlem40 40257 | This lemma proves that q_n... |
| stoweidlem41 40258 | This lemma is used to prov... |
| stoweidlem42 40259 | This lemma is used to prov... |
| stoweidlem43 40260 | This lemma is used to prov... |
| stoweidlem44 40261 | This lemma is used to prov... |
| stoweidlem45 40262 | This lemma proves that, gi... |
| stoweidlem46 40263 | This lemma proves that set... |
| stoweidlem47 40264 | Subtracting a constant fro... |
| stoweidlem48 40265 | This lemma is used to prov... |
| stoweidlem49 40266 | There exists a function q_... |
| stoweidlem50 40267 | This lemma proves that set... |
| stoweidlem51 40268 | There exists a function x ... |
| stoweidlem52 40269 | There exists a neighborood... |
| stoweidlem53 40270 | This lemma is used to prov... |
| stoweidlem54 40271 | There exists a function ` ... |
| stoweidlem55 40272 | This lemma proves the exis... |
| stoweidlem56 40273 | This theorem proves Lemma ... |
| stoweidlem57 40274 | There exists a function x ... |
| stoweidlem58 40275 | This theorem proves Lemma ... |
| stoweidlem59 40276 | This lemma proves that the... |
| stoweidlem60 40277 | This lemma proves that the... |
| stoweidlem61 40278 | This lemma proves that the... |
| stoweidlem62 40279 | This theorem proves the St... |
| stoweid 40280 | This theorem proves the St... |
| stowei 40281 | This theorem proves the St... |
| wallispilem1 40282 | ` I ` is monotone: increas... |
| wallispilem2 40283 | A first set of properties ... |
| wallispilem3 40284 | I maps to real values. (C... |
| wallispilem4 40285 | ` F ` maps to explicit exp... |
| wallispilem5 40286 | The sequence ` H ` converg... |
| wallispi 40287 | Wallis' formula for π :... |
| wallispi2lem1 40288 | An intermediate step betwe... |
| wallispi2lem2 40289 | Two expressions are proven... |
| wallispi2 40290 | An alternative version of ... |
| stirlinglem1 40291 | A simple limit of fraction... |
| stirlinglem2 40292 | ` A ` maps to positive rea... |
| stirlinglem3 40293 | Long but simple algebraic ... |
| stirlinglem4 40294 | Algebraic manipulation of ... |
| stirlinglem5 40295 | If ` T ` is between ` 0 ` ... |
| stirlinglem6 40296 | A series that converges to... |
| stirlinglem7 40297 | Algebraic manipulation of ... |
| stirlinglem8 40298 | If ` A ` converges to ` C ... |
| stirlinglem9 40299 | ` ( ( B `` N ) - ( B `` ( ... |
| stirlinglem10 40300 | A bound for any B(N)-B(N +... |
| stirlinglem11 40301 | ` B ` is decreasing. (Con... |
| stirlinglem12 40302 | The sequence ` B ` is boun... |
| stirlinglem13 40303 | ` B ` is decreasing and ha... |
| stirlinglem14 40304 | The sequence ` A ` converg... |
| stirlinglem15 40305 | The Stirling's formula is ... |
| stirling 40306 | Stirling's approximation f... |
| stirlingr 40307 | Stirling's approximation f... |
| dirkerval 40308 | The N_th Dirichlet Kernel.... |
| dirker2re 40309 | The Dirchlet Kernel value ... |
| dirkerdenne0 40310 | The Dirchlet Kernel denomi... |
| dirkerval2 40311 | The N_th Dirichlet Kernel ... |
| dirkerre 40312 | The Dirichlet Kernel at an... |
| dirkerper 40313 | the Dirichlet Kernel has p... |
| dirkerf 40314 | For any natural number ` N... |
| dirkertrigeqlem1 40315 | Sum of an even number of a... |
| dirkertrigeqlem2 40316 | Trigonomic equality lemma ... |
| dirkertrigeqlem3 40317 | Trigonometric equality lem... |
| dirkertrigeq 40318 | Trigonometric equality for... |
| dirkeritg 40319 | The definite integral of t... |
| dirkercncflem1 40320 | If ` Y ` is a multiple of ... |
| dirkercncflem2 40321 | Lemma used to prove that t... |
| dirkercncflem3 40322 | The Dirichlet Kernel is co... |
| dirkercncflem4 40323 | The Dirichlet Kernel is co... |
| dirkercncf 40324 | For any natural number ` N... |
| fourierdlem1 40325 | A partition interval is a ... |
| fourierdlem2 40326 | Membership in a partition.... |
| fourierdlem3 40327 | Membership in a partition.... |
| fourierdlem4 40328 | ` E ` is a function that m... |
| fourierdlem5 40329 | ` S ` is a function. (Con... |
| fourierdlem6 40330 | ` X ` is in the periodic p... |
| fourierdlem7 40331 | The difference between a p... |
| fourierdlem8 40332 | A partition interval is a ... |
| fourierdlem9 40333 | ` H ` is a complex functio... |
| fourierdlem10 40334 | Condition on the bounds of... |
| fourierdlem11 40335 | If there is a partition, t... |
| fourierdlem12 40336 | A point of a partition is ... |
| fourierdlem13 40337 | Value of ` V ` in terms of... |
| fourierdlem14 40338 | Given the partition ` V ` ... |
| fourierdlem15 40339 | The range of the partition... |
| fourierdlem16 40340 | The coefficients of the fo... |
| fourierdlem17 40341 | The defined ` L ` is actua... |
| fourierdlem18 40342 | The function ` S ` is cont... |
| fourierdlem19 40343 | If two elements of ` D ` h... |
| fourierdlem20 40344 | Every interval in the part... |
| fourierdlem21 40345 | The coefficients of the fo... |
| fourierdlem22 40346 | The coefficients of the fo... |
| fourierdlem23 40347 | If ` F ` is continuous and... |
| fourierdlem24 40348 | A sufficient condition for... |
| fourierdlem25 40349 | If ` C ` is not in the ran... |
| fourierdlem26 40350 | Periodic image of a point ... |
| fourierdlem27 40351 | A partition open interval ... |
| fourierdlem28 40352 | Derivative of ` ( F `` ( X... |
| fourierdlem29 40353 | Explicit function value fo... |
| fourierdlem30 40354 | Sum of three small pieces ... |
| fourierdlem31 40355 | If ` A ` is finite and for... |
| fourierdlem32 40356 | Limit of a continuous func... |
| fourierdlem33 40357 | Limit of a continuous func... |
| fourierdlem34 40358 | A partition is one to one.... |
| fourierdlem35 40359 | There is a single point in... |
| fourierdlem36 40360 | ` F ` is an isomorphism. ... |
| fourierdlem37 40361 | ` I ` is a function that m... |
| fourierdlem38 40362 | The function ` F ` is cont... |
| fourierdlem39 40363 | Integration by parts of ... |
| fourierdlem40 40364 | ` H ` is a continuous func... |
| fourierdlem41 40365 | Lemma used to prove that e... |
| fourierdlem42 40366 | The set of points in a mov... |
| fourierdlem43 40367 | ` K ` is a real function. ... |
| fourierdlem44 40368 | A condition for having ` (... |
| fourierdlem46 40369 | The function ` F ` has a l... |
| fourierdlem47 40370 | For ` r ` large enough, th... |
| fourierdlem48 40371 | The given periodic functio... |
| fourierdlem49 40372 | The given periodic functio... |
| fourierdlem50 40373 | Continuity of ` O ` and it... |
| fourierdlem51 40374 | ` X ` is in the periodic p... |
| fourierdlem52 40375 | d16:d17,d18:jca |- ( ph ->... |
| fourierdlem53 40376 | The limit of ` F ( s ) ` a... |
| fourierdlem54 40377 | Given a partition ` Q ` an... |
| fourierdlem55 40378 | ` U ` is a real function. ... |
| fourierdlem56 40379 | Derivative of the ` K ` fu... |
| fourierdlem57 40380 | The derivative of ` O ` . ... |
| fourierdlem58 40381 | The derivative of ` K ` is... |
| fourierdlem59 40382 | The derivative of ` H ` is... |
| fourierdlem60 40383 | Given a differentiable fun... |
| fourierdlem61 40384 | Given a differentiable fun... |
| fourierdlem62 40385 | The function ` K ` is cont... |
| fourierdlem63 40386 | The upper bound of interva... |
| fourierdlem64 40387 | The partition ` V ` is fin... |
| fourierdlem65 40388 | The distance of two adjace... |
| fourierdlem66 40389 | Value of the ` G ` functio... |
| fourierdlem67 40390 | ` G ` is a function. (Con... |
| fourierdlem68 40391 | The derivative of ` O ` is... |
| fourierdlem69 40392 | A piecewise continuous fun... |
| fourierdlem70 40393 | A piecewise continuous fun... |
| fourierdlem71 40394 | A periodic piecewise conti... |
| fourierdlem72 40395 | The derivative of ` O ` is... |
| fourierdlem73 40396 | A version of the Riemann L... |
| fourierdlem74 40397 | Given a piecewise smooth f... |
| fourierdlem75 40398 | Given a piecewise smooth f... |
| fourierdlem76 40399 | Continuity of ` O ` and it... |
| fourierdlem77 40400 | If ` H ` is bounded, then ... |
| fourierdlem78 40401 | ` G ` is continuous when r... |
| fourierdlem79 40402 | ` E ` projects every inter... |
| fourierdlem80 40403 | The derivative of ` O ` is... |
| fourierdlem81 40404 | The integral of a piecewis... |
| fourierdlem82 40405 | Integral by substitution, ... |
| fourierdlem83 40406 | The fourier partial sum fo... |
| fourierdlem84 40407 | If ` F ` is piecewise coni... |
| fourierdlem85 40408 | Limit of the function ` G ... |
| fourierdlem86 40409 | Continuity of ` O ` and it... |
| fourierdlem87 40410 | The integral of ` G ` goes... |
| fourierdlem88 40411 | Given a piecewise continuo... |
| fourierdlem89 40412 | Given a piecewise continuo... |
| fourierdlem90 40413 | Given a piecewise continuo... |
| fourierdlem91 40414 | Given a piecewise continuo... |
| fourierdlem92 40415 | The integral of a piecewis... |
| fourierdlem93 40416 | Integral by substitution (... |
| fourierdlem94 40417 | For a piecewise smooth fun... |
| fourierdlem95 40418 | Algebraic manipulation of ... |
| fourierdlem96 40419 | limit for ` F ` at the low... |
| fourierdlem97 40420 | ` F ` is continuous on the... |
| fourierdlem98 40421 | ` F ` is continuous on the... |
| fourierdlem99 40422 | limit for ` F ` at the upp... |
| fourierdlem100 40423 | A piecewise continuous fun... |
| fourierdlem101 40424 | Integral by substitution f... |
| fourierdlem102 40425 | For a piecewise smooth fun... |
| fourierdlem103 40426 | The half lower part of the... |
| fourierdlem104 40427 | The half upper part of the... |
| fourierdlem105 40428 | A piecewise continuous fun... |
| fourierdlem106 40429 | For a piecewise smooth fun... |
| fourierdlem107 40430 | The integral of a piecewis... |
| fourierdlem108 40431 | The integral of a piecewis... |
| fourierdlem109 40432 | The integral of a piecewis... |
| fourierdlem110 40433 | The integral of a piecewis... |
| fourierdlem111 40434 | The fourier partial sum fo... |
| fourierdlem112 40435 | Here abbreviations (local ... |
| fourierdlem113 40436 | Fourier series convergence... |
| fourierdlem114 40437 | Fourier series convergence... |
| fourierdlem115 40438 | Fourier serier convergence... |
| fourierd 40439 | Fourier series convergence... |
| fourierclimd 40440 | Fourier series convergence... |
| fourierclim 40441 | Fourier series convergence... |
| fourier 40442 | Fourier series convergence... |
| fouriercnp 40443 | If ` F ` is continuous at ... |
| fourier2 40444 | Fourier series convergence... |
| sqwvfoura 40445 | Fourier coefficients for t... |
| sqwvfourb 40446 | Fourier series ` B ` coeff... |
| fourierswlem 40447 | The Fourier series for the... |
| fouriersw 40448 | Fourier series convergence... |
| fouriercn 40449 | If the derivative of ` F `... |
| elaa2lem 40450 | Elementhood in the set of ... |
| elaa2 40451 | Elementhood in the set of ... |
| etransclem1 40452 | ` H ` is a function. (Con... |
| etransclem2 40453 | Derivative of ` G ` . (Co... |
| etransclem3 40454 | The given ` if ` term is a... |
| etransclem4 40455 | ` F ` expressed as a finit... |
| etransclem5 40456 | A change of bound variable... |
| etransclem6 40457 | A change of bound variable... |
| etransclem7 40458 | The given product is an in... |
| etransclem8 40459 | ` F ` is a function. (Con... |
| etransclem9 40460 | If ` K ` divides ` N ` but... |
| etransclem10 40461 | The given ` if ` term is a... |
| etransclem11 40462 | A change of bound variable... |
| etransclem12 40463 | ` C ` applied to ` N ` . ... |
| etransclem13 40464 | ` F ` applied to ` Y ` . ... |
| etransclem14 40465 | Value of the term ` T ` , ... |
| etransclem15 40466 | Value of the term ` T ` , ... |
| etransclem16 40467 | Every element in the range... |
| etransclem17 40468 | The ` N ` -th derivative o... |
| etransclem18 40469 | The given function is inte... |
| etransclem19 40470 | The ` N ` -th derivative o... |
| etransclem20 40471 | ` H ` is smooth. (Contrib... |
| etransclem21 40472 | The ` N ` -th derivative o... |
| etransclem22 40473 | The ` N ` -th derivative o... |
| etransclem23 40474 | This is the claim proof in... |
| etransclem24 40475 | ` P ` divides the I -th de... |
| etransclem25 40476 | ` P ` factorial divides th... |
| etransclem26 40477 | Every term in the sum of t... |
| etransclem27 40478 | The ` N ` -th derivative o... |
| etransclem28 40479 | ` ( P - 1 ) ` factorial di... |
| etransclem29 40480 | The ` N ` -th derivative o... |
| etransclem30 40481 | The ` N ` -th derivative o... |
| etransclem31 40482 | The ` N ` -th derivative o... |
| etransclem32 40483 | This is the proof for the ... |
| etransclem33 40484 | ` F ` is smooth. (Contrib... |
| etransclem34 40485 | The ` N ` -th derivative o... |
| etransclem35 40486 | ` P ` does not divide the ... |
| etransclem36 40487 | The ` N ` -th derivative o... |
| etransclem37 40488 | ` ( P - 1 ) ` factorial di... |
| etransclem38 40489 | ` P ` divides the I -th de... |
| etransclem39 40490 | ` G ` is a function. (Con... |
| etransclem40 40491 | The ` N ` -th derivative o... |
| etransclem41 40492 | ` P ` does not divide the ... |
| etransclem42 40493 | The ` N ` -th derivative o... |
| etransclem43 40494 | ` G ` is a continuous func... |
| etransclem44 40495 | The given finite sum is no... |
| etransclem45 40496 | ` K ` is an integer. (Con... |
| etransclem46 40497 | This is the proof for equa... |
| etransclem47 40498 | ` _e ` is transcendental. ... |
| etransclem48 40499 | ` _e ` is transcendental. ... |
| etransc 40500 | ` _e ` is transcendental. ... |
| rrxtopn 40501 | The topology of the genera... |
| rrxngp 40502 | Generalized Euclidean real... |
| rrxbasefi 40503 | The base of the generalize... |
| rrxtps 40504 | Generalized Euclidean real... |
| rrxdsfi 40505 | The distance over generali... |
| rrxtopnfi 40506 | The topology of the n-dime... |
| rrxmetfi 40507 | Euclidean space is a metri... |
| rrxtopon 40508 | The topology on Generalize... |
| rrxtop 40509 | The topology on Generalize... |
| rrndistlt 40510 | Given two points in the sp... |
| rrxtoponfi 40511 | The topology on n-dimensio... |
| rrxunitopnfi 40512 | The base set of the standa... |
| rrxtopn0 40513 | The topology of the zero-d... |
| qndenserrnbllem 40514 | n-dimensional rational num... |
| qndenserrnbl 40515 | n-dimensional rational num... |
| rrxtopn0b 40516 | The topology of the zero-d... |
| qndenserrnopnlem 40517 | n-dimensional rational num... |
| qndenserrnopn 40518 | n-dimensional rational num... |
| qndenserrn 40519 | n-dimensional rational num... |
| rrxsnicc 40520 | A multidimensional singlet... |
| rrnprjdstle 40521 | The distance between two p... |
| rrndsmet 40522 | ` D ` is a metric for the ... |
| rrndsxmet 40523 | ` D ` is an extended metri... |
| ioorrnopnlem 40524 | The a point in an indexed ... |
| ioorrnopn 40525 | The indexed product of ope... |
| ioorrnopnxrlem 40526 | Given a point ` F ` that b... |
| ioorrnopnxr 40527 | The indexed product of ope... |
| issal 40534 | Express the predicate " ` ... |
| pwsal 40535 | The power set of a given s... |
| salunicl 40536 | SAlg sigma-algebra is clos... |
| saluncl 40537 | The union of two sets in a... |
| prsal 40538 | The pair of the empty set ... |
| saldifcl 40539 | The complement of an eleme... |
| 0sal 40540 | The empty set belongs to e... |
| salgenval 40541 | The sigma-algebra generate... |
| saliuncl 40542 | SAlg sigma-algebra is clos... |
| salincl 40543 | The intersection of two se... |
| saluni 40544 | A set is an element of any... |
| saliincl 40545 | SAlg sigma-algebra is clos... |
| saldifcl2 40546 | The difference of two elem... |
| intsaluni 40547 | The union of an arbitrary ... |
| intsal 40548 | The arbitrary intersection... |
| salgenn0 40549 | The set used in the defini... |
| salgencl 40550 | ` SalGen ` actually genera... |
| issald 40551 | Sufficient condition to pr... |
| salexct 40552 | An example of non trivial ... |
| sssalgen 40553 | A set is a subset of the s... |
| salgenss 40554 | The sigma-algebra generate... |
| salgenuni 40555 | The base set of the sigma-... |
| issalgend 40556 | One side of ~ dfsalgen2 . ... |
| salexct2 40557 | An example of a subset tha... |
| unisalgen 40558 | The union of a set belongs... |
| dfsalgen2 40559 | Alternate characterization... |
| salexct3 40560 | An example of a sigma-alge... |
| salgencntex 40561 | This counterexample shows ... |
| salgensscntex 40562 | This counterexample shows ... |
| issalnnd 40563 | Sufficient condition to pr... |
| dmvolsal 40564 | Lebesgue measurable sets f... |
| saldifcld 40565 | The complement of an eleme... |
| saluncld 40566 | The union of two sets in a... |
| salgencld 40567 | ` SalGen ` actually genera... |
| 0sald 40568 | The empty set belongs to e... |
| iooborel 40569 | An open interval is a Bore... |
| salincld 40570 | The intersection of two se... |
| salunid 40571 | A set is an element of any... |
| unisalgen2 40572 | The union of a set belongs... |
| bor1sal 40573 | The Borel sigma-algebra on... |
| iocborel 40574 | A left-open, right-closed ... |
| subsaliuncllem 40575 | A subspace sigma-algebra i... |
| subsaliuncl 40576 | A subspace sigma-algebra i... |
| subsalsal 40577 | A subspace sigma-algebra i... |
| subsaluni 40578 | A set belongs to the subsp... |
| sge0rnre 40581 | When ` sum^ ` is applied t... |
| fge0icoicc 40582 | If ` F ` maps to nonnegati... |
| sge0val 40583 | The value of the sum of no... |
| fge0npnf 40584 | If ` F ` maps to nonnegati... |
| sge0rnn0 40585 | The range used in the defi... |
| sge0vald 40586 | The value of the sum of no... |
| fge0iccico 40587 | A range of nonnegative ext... |
| gsumge0cl 40588 | Closure of group sum, for ... |
| sge0reval 40589 | Value of the sum of nonneg... |
| sge0pnfval 40590 | If a term in the sum of no... |
| fge0iccre 40591 | A range of nonnegative ext... |
| sge0z 40592 | Any nonnegative extended s... |
| sge00 40593 | The sum of nonnegative ext... |
| fsumlesge0 40594 | Every finite subsum of non... |
| sge0revalmpt 40595 | Value of the sum of nonneg... |
| sge0sn 40596 | A sum of a nonnegative ext... |
| sge0tsms 40597 | ` sum^ ` applied to a nonn... |
| sge0cl 40598 | The arbitrary sum of nonne... |
| sge0f1o 40599 | Re-index a nonnegative ext... |
| sge0snmpt 40600 | A sum of a nonnegative ext... |
| sge0ge0 40601 | The sum of nonnegative ext... |
| sge0xrcl 40602 | The arbitrary sum of nonne... |
| sge0repnf 40603 | The of nonnegative extende... |
| sge0fsum 40604 | The arbitrary sum of a fin... |
| sge0rern 40605 | If the sum of nonnegative ... |
| sge0supre 40606 | If the arbitrary sum of no... |
| sge0fsummpt 40607 | The arbitrary sum of a fin... |
| sge0sup 40608 | The arbitrary sum of nonne... |
| sge0less 40609 | A shorter sum of nonnegati... |
| sge0rnbnd 40610 | The range used in the defi... |
| sge0pr 40611 | Sum of a pair of nonnegati... |
| sge0gerp 40612 | The arbitrary sum of nonne... |
| sge0pnffigt 40613 | If the sum of nonnegative ... |
| sge0ssre 40614 | If a sum of nonnegative ex... |
| sge0lefi 40615 | A sum of nonnegative exten... |
| sge0lessmpt 40616 | A shorter sum of nonnegati... |
| sge0ltfirp 40617 | If the sum of nonnegative ... |
| sge0prle 40618 | The sum of a pair of nonne... |
| sge0gerpmpt 40619 | The arbitrary sum of nonne... |
| sge0resrnlem 40620 | The sum of nonnegative ext... |
| sge0resrn 40621 | The sum of nonnegative ext... |
| sge0ssrempt 40622 | If a sum of nonnegative ex... |
| sge0resplit 40623 | ` sum^ ` splits into two p... |
| sge0le 40624 | If all of the terms of sum... |
| sge0ltfirpmpt 40625 | If the extended sum of non... |
| sge0split 40626 | Split a sum of nonnegative... |
| sge0lempt 40627 | If all of the terms of sum... |
| sge0splitmpt 40628 | Split a sum of nonnegative... |
| sge0ss 40629 | Change the index set to a ... |
| sge0iunmptlemfi 40630 | Sum of nonnegative extende... |
| sge0p1 40631 | The addition of the next t... |
| sge0iunmptlemre 40632 | Sum of nonnegative extende... |
| sge0fodjrnlem 40633 | Re-index a nonnegative ext... |
| sge0fodjrn 40634 | Re-index a nonnegative ext... |
| sge0iunmpt 40635 | Sum of nonnegative extende... |
| sge0iun 40636 | Sum of nonnegative extende... |
| sge0nemnf 40637 | The generalized sum of non... |
| sge0rpcpnf 40638 | The sum of an infinite num... |
| sge0rernmpt 40639 | If the sum of nonnegative ... |
| sge0lefimpt 40640 | A sum of nonnegative exten... |
| nn0ssge0 40641 | Nonnegative integers are n... |
| sge0clmpt 40642 | The generalized sum of non... |
| sge0ltfirpmpt2 40643 | If the extended sum of non... |
| sge0isum 40644 | If a series of nonnegative... |
| sge0xrclmpt 40645 | The generalized sum of non... |
| sge0xp 40646 | Combine two generalized su... |
| sge0isummpt 40647 | If a series of nonnegative... |
| sge0ad2en 40648 | The value of the infinite ... |
| sge0isummpt2 40649 | If a series of nonnegative... |
| sge0xaddlem1 40650 | The extended addition of t... |
| sge0xaddlem2 40651 | The extended addition of t... |
| sge0xadd 40652 | The extended addition of t... |
| sge0fsummptf 40653 | The generalized sum of a f... |
| sge0snmptf 40654 | A sum of a nonnegative ext... |
| sge0ge0mpt 40655 | The sum of nonnegative ext... |
| sge0repnfmpt 40656 | The of nonnegative extende... |
| sge0pnffigtmpt 40657 | If the generalized sum of ... |
| sge0splitsn 40658 | Separate out a term in a g... |
| sge0pnffsumgt 40659 | If the sum of nonnegative ... |
| sge0gtfsumgt 40660 | If the generalized sum of ... |
| sge0uzfsumgt 40661 | If a real number is smalle... |
| sge0pnfmpt 40662 | If a term in the sum of no... |
| sge0seq 40663 | A series of nonnegative re... |
| sge0reuz 40664 | Value of the generalized s... |
| sge0reuzb 40665 | Value of the generalized s... |
| ismea 40668 | Express the predicate " ` ... |
| dmmeasal 40669 | The domain of a measure is... |
| meaf 40670 | A measure is a function th... |
| mea0 40671 | The measure of the empty s... |
| nnfoctbdjlem 40672 | There exists a mapping fro... |
| nnfoctbdj 40673 | There exists a mapping fro... |
| meadjuni 40674 | The measure of the disjoin... |
| meacl 40675 | The measure of a set is a ... |
| iundjiunlem 40676 | The sets in the sequence `... |
| iundjiun 40677 | Given a sequence ` E ` of ... |
| meaxrcl 40678 | The measure of a set is an... |
| meadjun 40679 | The measure of the union o... |
| meassle 40680 | The measure of a set is la... |
| meaunle 40681 | The measure of the union o... |
| meadjiunlem 40682 | The sum of nonnegative ext... |
| meadjiun 40683 | The measure of the disjoin... |
| ismeannd 40684 | Sufficient condition to pr... |
| meaiunlelem 40685 | The measure of the union o... |
| meaiunle 40686 | The measure of the union o... |
| psmeasurelem 40687 | ` M ` applied to a disjoin... |
| psmeasure 40688 | Point supported measure, R... |
| voliunsge0lem 40689 | The Lebesgue measure funct... |
| voliunsge0 40690 | The Lebesgue measure funct... |
| volmea 40691 | The Lebeasgue measure on t... |
| meage0 40692 | If the measure of a measur... |
| meadjunre 40693 | The measure of the union o... |
| meassre 40694 | If the measure of a measur... |
| meale0eq0 40695 | A measure that is smaller ... |
| meadif 40696 | The measure of the differe... |
| meaiuninclem 40697 | Measures are continuous fr... |
| meaiuninc 40698 | Measures are continuous fr... |
| meaiuninc2 40699 | Measures are continuous fr... |
| meaiininclem 40700 | Measures are continuous fr... |
| meaiininc 40701 | Measures are continuous fr... |
| meaiininc2 40702 | Measures are continuous fr... |
| caragenval 40707 | The sigma-algebra generate... |
| isome 40708 | Express the predicate " ` ... |
| caragenel 40709 | Membership in the Caratheo... |
| omef 40710 | An outer measure is a func... |
| ome0 40711 | The outer measure of the e... |
| omessle 40712 | The outer measure of a set... |
| omedm 40713 | The domain of an outer mea... |
| caragensplit 40714 | If ` E ` is in the set gen... |
| caragenelss 40715 | An element of the Caratheo... |
| carageneld 40716 | Membership in the Caratheo... |
| omecl 40717 | The outer measure of a set... |
| caragenss 40718 | The sigma-algebra generate... |
| omeunile 40719 | The outer measure of the u... |
| caragen0 40720 | The empty set belongs to a... |
| omexrcl 40721 | The outer measure of a set... |
| caragenunidm 40722 | The base set of an outer m... |
| caragensspw 40723 | The sigma-algebra generate... |
| omessre 40724 | If the outer measure of a ... |
| caragenuni 40725 | The base set of the sigma-... |
| caragenuncllem 40726 | The Caratheodory's constru... |
| caragenuncl 40727 | The Caratheodory's constru... |
| caragendifcl 40728 | The Caratheodory's constru... |
| caragenfiiuncl 40729 | The Caratheodory's constru... |
| omeunle 40730 | The outer measure of the u... |
| omeiunle 40731 | The outer measure of the i... |
| omelesplit 40732 | The outer measure of a set... |
| omeiunltfirp 40733 | If the outer measure of a ... |
| omeiunlempt 40734 | The outer measure of the i... |
| carageniuncllem1 40735 | The outer measure of ` A i... |
| carageniuncllem2 40736 | The Caratheodory's constru... |
| carageniuncl 40737 | The Caratheodory's constru... |
| caragenunicl 40738 | The Caratheodory's constru... |
| caragensal 40739 | Caratheodory's method gene... |
| caratheodorylem1 40740 | Lemma used to prove that C... |
| caratheodorylem2 40741 | Caratheodory's constructio... |
| caratheodory 40742 | Caratheodory's constructio... |
| 0ome 40743 | The map that assigns 0 to ... |
| isomenndlem 40744 | ` O ` is sub-additive w.r.... |
| isomennd 40745 | Sufficient condition to pr... |
| caragenel2d 40746 | Membership in the Caratheo... |
| omege0 40747 | If the outer measure of a ... |
| omess0 40748 | If the outer measure of a ... |
| caragencmpl 40749 | A measure built with the C... |
| vonval 40754 | Value of the Lebesgue meas... |
| ovnval 40755 | Value of the Lebesgue oute... |
| elhoi 40756 | Membership in a multidimen... |
| icoresmbl 40757 | A closed-below, open-above... |
| hoissre 40758 | The projection of a half-o... |
| ovnval2 40759 | Value of the Lebesgue oute... |
| volicorecl 40760 | The Lebesgue measure of a ... |
| hoiprodcl 40761 | The pre-measure of half-op... |
| hoicvr 40762 | ` I ` is a countable set o... |
| hoissrrn 40763 | A half-open interval is a ... |
| ovn0val 40764 | The Lebesgue outer measure... |
| ovnn0val 40765 | The value of a (multidimen... |
| ovnval2b 40766 | Value of the Lebesgue oute... |
| volicorescl 40767 | The Lebesgue measure of a ... |
| ovnprodcl 40768 | The product used in the de... |
| hoiprodcl2 40769 | The pre-measure of half-op... |
| hoicvrrex 40770 | Any subset of the multidim... |
| ovnsupge0 40771 | The set used in the defini... |
| ovnlecvr 40772 | Given a subset of multidim... |
| ovnpnfelsup 40773 | ` +oo ` is an element of t... |
| ovnsslelem 40774 | The (multidimensional, non... |
| ovnssle 40775 | The (multidimensional) Leb... |
| ovnlerp 40776 | The Lebesgue outer measure... |
| ovnf 40777 | The Lebesgue outer measure... |
| ovncvrrp 40778 | The Lebesgue outer measure... |
| ovn0lem 40779 | For any finite dimension, ... |
| ovn0 40780 | For any finite dimension, ... |
| ovncl 40781 | The Lebesgue outer measure... |
| ovn02 40782 | For the zero-dimensional s... |
| ovnxrcl 40783 | The Lebesgue outer measure... |
| ovnsubaddlem1 40784 | The Lebesgue outer measure... |
| ovnsubaddlem2 40785 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd 40786 | ` ( voln* `` X ) ` is suba... |
| ovnome 40787 | ` ( voln* `` X ) ` is an o... |
| vonmea 40788 | ` ( voln `` X ) ` is a mea... |
| volicon0 40789 | The measure of a nonempty ... |
| hsphoif 40790 | ` H ` is a function (that ... |
| hoidmvval 40791 | The dimensional volume of ... |
| hoissrrn2 40792 | A half-open interval is a ... |
| hsphoival 40793 | ` H ` is a function (that ... |
| hoiprodcl3 40794 | The pre-measure of half-op... |
| volicore 40795 | The Lebesgue measure of a ... |
| hoidmvcl 40796 | The dimensional volume of ... |
| hoidmv0val 40797 | The dimensional volume of ... |
| hoidmvn0val 40798 | The dimensional volume of ... |
| hsphoidmvle2 40799 | The dimensional volume of ... |
| hsphoidmvle 40800 | The dimensional volume of ... |
| hoidmvval0 40801 | The dimensional volume of ... |
| hoiprodp1 40802 | The dimensional volume of ... |
| sge0hsphoire 40803 | If the generalized sum of ... |
| hoidmvval0b 40804 | The dimensional volume of ... |
| hoidmv1lelem1 40805 | The supremum of ` U ` belo... |
| hoidmv1lelem2 40806 | This is the contradiction ... |
| hoidmv1lelem3 40807 | The dimensional volume of ... |
| hoidmv1le 40808 | The dimensional volume of ... |
| hoidmvlelem1 40809 | The supremum of ` U ` belo... |
| hoidmvlelem2 40810 | This is the contradiction ... |
| hoidmvlelem3 40811 | This is the contradiction ... |
| hoidmvlelem4 40812 | The dimensional volume of ... |
| hoidmvlelem5 40813 | The dimensional volume of ... |
| hoidmvle 40814 | The dimensional volume of ... |
| ovnhoilem1 40815 | The Lebesgue outer measure... |
| ovnhoilem2 40816 | The Lebesgue outer measure... |
| ovnhoi 40817 | The Lebesgue outer measure... |
| dmovn 40818 | The domain of the Lebesgue... |
| hoicoto2 40819 | The half-open interval exp... |
| dmvon 40820 | Lebesgue measurable n-dime... |
| hoi2toco 40821 | The half-open interval exp... |
| hoidifhspval 40822 | ` D ` is a function that r... |
| hspval 40823 | The value of the half-spac... |
| ovnlecvr2 40824 | Given a subset of multidim... |
| ovncvr2 40825 | ` B ` and ` T ` are the le... |
| dmovnsal 40826 | The domain of the Lebesgue... |
| unidmovn 40827 | Base set of the n-dimensio... |
| rrnmbl 40828 | The set of n-dimensional R... |
| hoidifhspval2 40829 | ` D ` is a function that r... |
| hspdifhsp 40830 | A n-dimensional half-open ... |
| unidmvon 40831 | Base set of the n-dimensio... |
| hoidifhspf 40832 | ` D ` is a function that r... |
| hoidifhspval3 40833 | ` D ` is a function that r... |
| hoidifhspdmvle 40834 | The dimensional volume of ... |
| voncmpl 40835 | The Lebesgue measure is co... |
| hoiqssbllem1 40836 | The center of the n-dimens... |
| hoiqssbllem2 40837 | The center of the n-dimens... |
| hoiqssbllem3 40838 | A n-dimensional ball conta... |
| hoiqssbl 40839 | A n-dimensional ball conta... |
| hspmbllem1 40840 | Any half-space of the n-di... |
| hspmbllem2 40841 | Any half-space of the n-di... |
| hspmbllem3 40842 | Any half-space of the n-di... |
| hspmbl 40843 | Any half-space of the n-di... |
| hoimbllem 40844 | Any n-dimensional half-ope... |
| hoimbl 40845 | Any n-dimensional half-ope... |
| opnvonmbllem1 40846 | The half-open interval exp... |
| opnvonmbllem2 40847 | An open subset of the n-di... |
| opnvonmbl 40848 | An open subset of the n-di... |
| opnssborel 40849 | Open sets of a generalized... |
| borelmbl 40850 | All Borel subsets of the n... |
| volicorege0 40851 | The Lebesgue measure of a ... |
| isvonmbl 40852 | The predicate " ` A ` is m... |
| mblvon 40853 | The n-dimensional Lebesgue... |
| vonmblss 40854 | n-dimensional Lebesgue mea... |
| volico2 40855 | The measure of left closed... |
| vonmblss2 40856 | n-dimensional Lebesgue mea... |
| ovolval2lem 40857 | The value of the Lebesgue ... |
| ovolval2 40858 | The value of the Lebesgue ... |
| ovnsubadd2lem 40859 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd2 40860 | ` ( voln* `` X ) ` is suba... |
| ovolval3 40861 | The value of the Lebesgue ... |
| ovnsplit 40862 | The n-dimensional Lebesgue... |
| ovolval4lem1 40863 | |- ( ( ph /\ n e. A ) -> ... |
| ovolval4lem2 40864 | The value of the Lebesgue ... |
| ovolval4 40865 | The value of the Lebesgue ... |
| ovolval5lem1 40866 | |- ( ph -> ( sum^ ` ( n e.... |
| ovolval5lem2 40867 | |- ( ( ph /\ n e. NN ) ->... |
| ovolval5lem3 40868 | The value of the Lebesgue ... |
| ovolval5 40869 | The value of the Lebesgue ... |
| ovnovollem1 40870 | if ` F ` is a cover of ` B... |
| ovnovollem2 40871 | if ` I ` is a cover of ` (... |
| ovnovollem3 40872 | The 1-dimensional Lebesgue... |
| ovnovol 40873 | The 1-dimensional Lebesgue... |
| vonvolmbllem 40874 | If a subset ` B ` of real ... |
| vonvolmbl 40875 | A subset of Real numbers i... |
| vonvol 40876 | The 1-dimensional Lebesgue... |
| vonvolmbl2 40877 | A subset ` X ` of the spac... |
| vonvol2 40878 | The 1-dimensional Lebesgue... |
| hoimbl2 40879 | Any n-dimensional half-ope... |
| voncl 40880 | The Lebesgue measure of a ... |
| vonhoi 40881 | The Lebesgue outer measure... |
| vonxrcl 40882 | The Lebesgue measure of a ... |
| ioosshoi 40883 | A n-dimensional open inter... |
| vonn0hoi 40884 | The Lebesgue outer measure... |
| von0val 40885 | The Lebesgue measure (for ... |
| vonhoire 40886 | The Lebesgue measure of a ... |
| iinhoiicclem 40887 | A n-dimensional closed int... |
| iinhoiicc 40888 | A n-dimensional closed int... |
| iunhoiioolem 40889 | A n-dimensional open inter... |
| iunhoiioo 40890 | A n-dimensional open inter... |
| ioovonmbl 40891 | Any n-dimensional open int... |
| iccvonmbllem 40892 | Any n-dimensional closed i... |
| iccvonmbl 40893 | Any n-dimensional closed i... |
| vonioolem1 40894 | The sequence of the measur... |
| vonioolem2 40895 | The n-dimensional Lebesgue... |
| vonioo 40896 | The n-dimensional Lebesgue... |
| vonicclem1 40897 | The sequence of the measur... |
| vonicclem2 40898 | The n-dimensional Lebesgue... |
| vonicc 40899 | The n-dimensional Lebesgue... |
| snvonmbl 40900 | A n-dimensional singleton ... |
| vonn0ioo 40901 | The n-dimensional Lebesgue... |
| vonn0icc 40902 | The n-dimensional Lebesgue... |
| ctvonmbl 40903 | Any n-dimensional countabl... |
| vonn0ioo2 40904 | The n-dimensional Lebesgue... |
| vonsn 40905 | The n-dimensional Lebesgue... |
| vonn0icc2 40906 | The n-dimensional Lebesgue... |
| vonct 40907 | The n-dimensional Lebesgue... |
| vitali2 40908 | There are non-measurable s... |
| pimltmnf2 40911 | Given a real-valued functi... |
| preimagelt 40912 | The preimage of a right-op... |
| preimalegt 40913 | The preimage of a left-ope... |
| pimconstlt0 40914 | Given a constant function,... |
| pimconstlt1 40915 | Given a constant function,... |
| pimltpnf 40916 | Given a real-valued functi... |
| pimgtpnf2 40917 | Given a real-valued functi... |
| salpreimagelt 40918 | If all the preimages of le... |
| pimrecltpos 40919 | The preimage of an unbound... |
| salpreimalegt 40920 | If all the preimages of ri... |
| pimiooltgt 40921 | The preimage of an open in... |
| preimaicomnf 40922 | Preimage of an open interv... |
| pimltpnf2 40923 | Given a real-valued functi... |
| pimgtmnf2 40924 | Given a real-valued functi... |
| pimdecfgtioc 40925 | Given a non-increasing fun... |
| pimincfltioc 40926 | Given a non decreasing fun... |
| pimdecfgtioo 40927 | Given a non decreasing fun... |
| pimincfltioo 40928 | Given a non decreasing fun... |
| preimaioomnf 40929 | Preimage of an open interv... |
| preimageiingt 40930 | A preimage of a left-close... |
| preimaleiinlt 40931 | A preimage of a left-open,... |
| pimgtmnf 40932 | Given a real-valued functi... |
| pimrecltneg 40933 | The preimage of an unbound... |
| salpreimagtge 40934 | If all the preimages of le... |
| salpreimaltle 40935 | If all the preimages of ri... |
| issmflem 40936 | The predicate " ` F ` is a... |
| issmf 40937 | The predicate " ` F ` is a... |
| salpreimalelt 40938 | If all the preimages of ri... |
| salpreimagtlt 40939 | If all the preimages of le... |
| smfpreimalt 40940 | Given a function measurabl... |
| smff 40941 | A function measurable w.r.... |
| smfdmss 40942 | The domain of a function m... |
| issmff 40943 | The predicate " ` F ` is a... |
| issmfd 40944 | A sufficient condition for... |
| smfpreimaltf 40945 | Given a function measurabl... |
| issmfdf 40946 | A sufficient condition for... |
| sssmf 40947 | The restriction of a sigma... |
| mbfresmf 40948 | A Real valued, measurable ... |
| cnfsmf 40949 | A continuous function is m... |
| incsmflem 40950 | A non decreasing function ... |
| incsmf 40951 | A real-valued, non-decreas... |
| smfsssmf 40952 | If a function is measurabl... |
| issmflelem 40953 | The predicate " ` F ` is a... |
| issmfle 40954 | The predicate " ` F ` is a... |
| smfpimltmpt 40955 | Given a function measurabl... |
| smfpimltxr 40956 | Given a function measurabl... |
| issmfdmpt 40957 | A sufficient condition for... |
| smfconst 40958 | Given a sigma-algebra over... |
| sssmfmpt 40959 | The restriction of a sigma... |
| cnfrrnsmf 40960 | A function, continuous fro... |
| smfid 40961 | The identity function is B... |
| bormflebmf 40962 | A Borel measurable functio... |
| smfpreimale 40963 | Given a function measurabl... |
| issmfgtlem 40964 | The predicate " ` F ` is a... |
| issmfgt 40965 | The predicate " ` F ` is a... |
| issmfled 40966 | A sufficient condition for... |
| smfpimltxrmpt 40967 | Given a function measurabl... |
| smfmbfcex 40968 | A constant function, with ... |
| issmfgtd 40969 | A sufficient condition for... |
| smfpreimagt 40970 | Given a function measurabl... |
| smfaddlem1 40971 | Given the sum of two funct... |
| smfaddlem2 40972 | The sum of two sigma-measu... |
| smfadd 40973 | The sum of two sigma-measu... |
| decsmflem 40974 | A non-increasing function ... |
| decsmf 40975 | A real-valued, non-increas... |
| smfpreimagtf 40976 | Given a function measurabl... |
| issmfgelem 40977 | The predicate " ` F ` is a... |
| issmfge 40978 | The predicate " ` F ` is a... |
| smflimlem1 40979 | Lemma for the proof that t... |
| smflimlem2 40980 | Lemma for the proof that t... |
| smflimlem3 40981 | The limit of sigma-measura... |
| smflimlem4 40982 | Lemma for the proof that t... |
| smflimlem5 40983 | Lemma for the proof that t... |
| smflimlem6 40984 | Lemma for the proof that t... |
| smflim 40985 | The limit of sigma-measura... |
| nsssmfmbflem 40986 | The sigma-measurable funct... |
| nsssmfmbf 40987 | The sigma-measurable funct... |
| smfpimgtxr 40988 | Given a function measurabl... |
| smfpimgtmpt 40989 | Given a function measurabl... |
| smfpreimage 40990 | Given a function measurabl... |
| mbfpsssmf 40991 | Real valued, measurable fu... |
| smfpimgtxrmpt 40992 | Given a function measurabl... |
| smfpimioompt 40993 | Given a function measurabl... |
| smfpimioo 40994 | Given a function measurabl... |
| smfresal 40995 | Given a sigma-measurable f... |
| smfrec 40996 | The reciprocal of a sigma-... |
| smfres 40997 | The restriction of sigma-m... |
| smfmullem1 40998 | The multiplication of two ... |
| smfmullem2 40999 | The multiplication of two ... |
| smfmullem3 41000 | The multiplication of two ... |
| smfmullem4 41001 | The multiplication of two ... |
| smfmul 41002 | The multiplication of two ... |
| smfmulc1 41003 | A sigma-measurable functio... |
| smfdiv 41004 | The fraction of two sigma-... |
| smfpimbor1lem1 41005 | Every open set belongs to ... |
| smfpimbor1lem2 41006 | Given a sigma-measurable f... |
| smfpimbor1 41007 | Given a sigma-measurable f... |
| smf2id 41008 | Twice the identity functio... |
| smfco 41009 | The composition of a Borel... |
| smfneg 41010 | The negative of a sigma-me... |
| smffmpt 41011 | A function measurable w.r.... |
| smflim2 41012 | The limit of a sequence of... |
| smfpimcclem 41013 | Lemma for ~ smfpimcc given... |
| smfpimcc 41014 | Given a countable set of s... |
| issmfle2d 41015 | A sufficient condition for... |
| smflimmpt 41016 | The limit of a sequence of... |
| smfsuplem1 41017 | The supremum of a countabl... |
| smfsuplem2 41018 | The supremum of a countabl... |
| smfsuplem3 41019 | The supremum of a countabl... |
| smfsup 41020 | The supremum of a countabl... |
| smfsupmpt 41021 | The supremum of a countabl... |
| smfsupxr 41022 | The supremum of a countabl... |
| smfinflem 41023 | The infimum of a countable... |
| smfinf 41024 | The infimum of a countable... |
| smfinfmpt 41025 | The infimum of a countable... |
| smflimsuplem1 41026 | If ` H ` converges, the ` ... |
| smflimsuplem2 41027 | The superior limit of a se... |
| smflimsuplem3 41028 | The limit of the ` ( H `` ... |
| smflimsuplem4 41029 | If ` H ` converges, the ` ... |
| smflimsuplem5 41030 | ` H ` converges to the sup... |
| smflimsuplem6 41031 | The superior limit of a se... |
| smflimsuplem7 41032 | The superior limit of a se... |
| smflimsuplem8 41033 | The superior limit of a se... |
| smflimsup 41034 | The superior limit of a se... |
| smflimsupmpt 41035 | The superior limit of a se... |
| smfliminflem 41036 | The inferior limit of a co... |
| smfliminf 41037 | The inferior limit of a co... |
| smfliminfmpt 41038 | The inferior limit of a co... |
| sigarval 41039 | Define the signed area by ... |
| sigarim 41040 | Signed area takes value in... |
| sigarac 41041 | Signed area is anticommuta... |
| sigaraf 41042 | Signed area is additive by... |
| sigarmf 41043 | Signed area is additive (w... |
| sigaras 41044 | Signed area is additive by... |
| sigarms 41045 | Signed area is additive (w... |
| sigarls 41046 | Signed area is linear by t... |
| sigarid 41047 | Signed area of a flat para... |
| sigarexp 41048 | Expand the signed area for... |
| sigarperm 41049 | Signed area ` ( A - C ) G ... |
| sigardiv 41050 | If signed area between vec... |
| sigarimcd 41051 | Signed area takes value in... |
| sigariz 41052 | If signed area is zero, th... |
| sigarcol 41053 | Given three points ` A ` ,... |
| sharhght 41054 | Let ` A B C ` be a triangl... |
| sigaradd 41055 | Subtracting (double) area ... |
| cevathlem1 41056 | Ceva's theorem first lemma... |
| cevathlem2 41057 | Ceva's theorem second lemm... |
| cevath 41058 | Ceva's theorem. Let ` A B... |
| hirstL-ax3 41059 | The third axiom of a syste... |
| ax3h 41060 | Recovery of ~ ax-3 from ~ ... |
| aibandbiaiffaiffb 41061 | A closed form showing (a i... |
| aibandbiaiaiffb 41062 | A closed form showing (a i... |
| notatnand 41063 | Do not use. Use intnanr i... |
| aistia 41064 | Given a is equivalent to `... |
| aisfina 41065 | Given a is equivalent to `... |
| bothtbothsame 41066 | Given both a, b are equiva... |
| bothfbothsame 41067 | Given both a, b are equiva... |
| aiffbbtat 41068 | Given a is equivalent to b... |
| aisbbisfaisf 41069 | Given a is equivalent to b... |
| axorbtnotaiffb 41070 | Given a is exclusive to b,... |
| aiffnbandciffatnotciffb 41071 | Given a is equivalent to (... |
| axorbciffatcxorb 41072 | Given a is equivalent to (... |
| aibnbna 41073 | Given a implies b, (not b)... |
| aibnbaif 41074 | Given a implies b, not b, ... |
| aiffbtbat 41075 | Given a is equivalent to b... |
| astbstanbst 41076 | Given a is equivalent to T... |
| aistbistaandb 41077 | Given a is equivalent to T... |
| aisbnaxb 41078 | Given a is equivalent to b... |
| atbiffatnnb 41079 | If a implies b, then a imp... |
| bisaiaisb 41080 | Application of bicom1 with... |
| atbiffatnnbalt 41081 | If a implies b, then a imp... |
| abnotbtaxb 41082 | Assuming a, not b, there e... |
| abnotataxb 41083 | Assuming not a, b, there e... |
| conimpf 41084 | Assuming a, not b, and a i... |
| conimpfalt 41085 | Assuming a, not b, and a i... |
| aistbisfiaxb 41086 | Given a is equivalent to T... |
| aisfbistiaxb 41087 | Given a is equivalent to F... |
| aifftbifffaibif 41088 | Given a is equivalent to T... |
| aifftbifffaibifff 41089 | Given a is equivalent to T... |
| atnaiana 41090 | Given a, it is not the cas... |
| ainaiaandna 41091 | Given a, a implies it is n... |
| abcdta 41092 | Given (((a and b) and c) a... |
| abcdtb 41093 | Given (((a and b) and c) a... |
| abcdtc 41094 | Given (((a and b) and c) a... |
| abcdtd 41095 | Given (((a and b) and c) a... |
| abciffcbatnabciffncba 41096 | Operands in a biconditiona... |
| abciffcbatnabciffncbai 41097 | Operands in a biconditiona... |
| nabctnabc 41098 | not ( a -> ( b /\ c ) ) we... |
| jabtaib 41099 | For when pm3.4 lacks a pm3... |
| onenotinotbothi 41100 | From one negated implicati... |
| twonotinotbothi 41101 | From these two negated imp... |
| clifte 41102 | show d is the same as an i... |
| cliftet 41103 | show d is the same as an i... |
| clifteta 41104 | show d is the same as an i... |
| cliftetb 41105 | show d is the same as an i... |
| confun 41106 | Given the hypotheses there... |
| confun2 41107 | Confun simplified to two p... |
| confun3 41108 | Confun's more complex form... |
| confun4 41109 | An attempt at derivative. ... |
| confun5 41110 | An attempt at derivative. ... |
| plcofph 41111 | Given, a,b and a "definiti... |
| pldofph 41112 | Given, a,b c, d, "definiti... |
| plvcofph 41113 | Given, a,b,d, and "definit... |
| plvcofphax 41114 | Given, a,b,d, and "definit... |
| plvofpos 41115 | rh is derivable because ON... |
| mdandyv0 41116 | Given the equivalences set... |
| mdandyv1 41117 | Given the equivalences set... |
| mdandyv2 41118 | Given the equivalences set... |
| mdandyv3 41119 | Given the equivalences set... |
| mdandyv4 41120 | Given the equivalences set... |
| mdandyv5 41121 | Given the equivalences set... |
| mdandyv6 41122 | Given the equivalences set... |
| mdandyv7 41123 | Given the equivalences set... |
| mdandyv8 41124 | Given the equivalences set... |
| mdandyv9 41125 | Given the equivalences set... |
| mdandyv10 41126 | Given the equivalences set... |
| mdandyv11 41127 | Given the equivalences set... |
| mdandyv12 41128 | Given the equivalences set... |
| mdandyv13 41129 | Given the equivalences set... |
| mdandyv14 41130 | Given the equivalences set... |
| mdandyv15 41131 | Given the equivalences set... |
| mdandyvr0 41132 | Given the equivalences set... |
| mdandyvr1 41133 | Given the equivalences set... |
| mdandyvr2 41134 | Given the equivalences set... |
| mdandyvr3 41135 | Given the equivalences set... |
| mdandyvr4 41136 | Given the equivalences set... |
| mdandyvr5 41137 | Given the equivalences set... |
| mdandyvr6 41138 | Given the equivalences set... |
| mdandyvr7 41139 | Given the equivalences set... |
| mdandyvr8 41140 | Given the equivalences set... |
| mdandyvr9 41141 | Given the equivalences set... |
| mdandyvr10 41142 | Given the equivalences set... |
| mdandyvr11 41143 | Given the equivalences set... |
| mdandyvr12 41144 | Given the equivalences set... |
| mdandyvr13 41145 | Given the equivalences set... |
| mdandyvr14 41146 | Given the equivalences set... |
| mdandyvr15 41147 | Given the equivalences set... |
| mdandyvrx0 41148 | Given the exclusivities se... |
| mdandyvrx1 41149 | Given the exclusivities se... |
| mdandyvrx2 41150 | Given the exclusivities se... |
| mdandyvrx3 41151 | Given the exclusivities se... |
| mdandyvrx4 41152 | Given the exclusivities se... |
| mdandyvrx5 41153 | Given the exclusivities se... |
| mdandyvrx6 41154 | Given the exclusivities se... |
| mdandyvrx7 41155 | Given the exclusivities se... |
| mdandyvrx8 41156 | Given the exclusivities se... |
| mdandyvrx9 41157 | Given the exclusivities se... |
| mdandyvrx10 41158 | Given the exclusivities se... |
| mdandyvrx11 41159 | Given the exclusivities se... |
| mdandyvrx12 41160 | Given the exclusivities se... |
| mdandyvrx13 41161 | Given the exclusivities se... |
| mdandyvrx14 41162 | Given the exclusivities se... |
| mdandyvrx15 41163 | Given the exclusivities se... |
| H15NH16TH15IH16 41164 | Given 15 hypotheses and a ... |
| dandysum2p2e4 41165 | CONTRADICTION PRO... |
| mdandysum2p2e4 41166 | CONTRADICTION PROVED AT 1 ... |
| r19.32 41167 | Theorem 19.32 of [Margaris... |
| rexsb 41168 | An equivalent expression f... |
| rexrsb 41169 | An equivalent expression f... |
| 2rexsb 41170 | An equivalent expression f... |
| 2rexrsb 41171 | An equivalent expression f... |
| cbvral2 41172 | Change bound variables of ... |
| cbvrex2 41173 | Change bound variables of ... |
| 2ralbiim 41174 | Split a biconditional and ... |
| raaan2 41175 | Rearrange restricted quant... |
| rmoimi 41176 | Restricted "at most one" i... |
| 2reu5a 41177 | Double restricted existent... |
| reuimrmo 41178 | Restricted uniqueness impl... |
| rmoanim 41179 | Introduction of a conjunct... |
| reuan 41180 | Introduction of a conjunct... |
| 2reurex 41181 | Double restricted quantifi... |
| 2reurmo 41182 | Double restricted quantifi... |
| 2reu2rex 41183 | Double restricted existent... |
| 2rmoswap 41184 | A condition allowing swap ... |
| 2rexreu 41185 | Double restricted existent... |
| 2reu1 41186 | Double restricted existent... |
| 2reu2 41187 | Double restricted existent... |
| 2reu3 41188 | Double restricted existent... |
| 2reu4a 41189 | Definition of double restr... |
| 2reu4 41190 | Definition of double restr... |
| 2reu7 41191 | Two equivalent expressions... |
| 2reu8 41192 | Two equivalent expressions... |
| ralbinrald 41199 | Elemination of a restricte... |
| nvelim 41200 | If a class is the universa... |
| alneu 41201 | If a statement holds for a... |
| eu2ndop1stv 41202 | If there is a unique secon... |
| eldmressn 41203 | Element of the domain of a... |
| fveqvfvv 41204 | If a function's value at a... |
| funresfunco 41205 | Composition of two functio... |
| fnresfnco 41206 | Composition of two functio... |
| funcoressn 41207 | A composition restricted t... |
| funressnfv 41208 | A restriction to a singlet... |
| dfateq12d 41209 | Equality deduction for "de... |
| nfdfat 41210 | Bound-variable hypothesis ... |
| dfdfat2 41211 | Alternate definition of th... |
| dfafv2 41212 | Alternative definition of ... |
| afveq12d 41213 | Equality deduction for fun... |
| afveq1 41214 | Equality theorem for funct... |
| afveq2 41215 | Equality theorem for funct... |
| nfafv 41216 | Bound-variable hypothesis ... |
| csbafv12g 41217 | Move class substitution in... |
| afvfundmfveq 41218 | If a class is a function r... |
| afvnfundmuv 41219 | If a set is not in the dom... |
| ndmafv 41220 | The value of a class outsi... |
| afvvdm 41221 | If the function value of a... |
| nfunsnafv 41222 | If the restriction of a cl... |
| afvvfunressn 41223 | If the function value of a... |
| afvprc 41224 | A function's value at a pr... |
| afvvv 41225 | If a function's value at a... |
| afvpcfv0 41226 | If the value of the altern... |
| afvnufveq 41227 | The value of the alternati... |
| afvvfveq 41228 | The value of the alternati... |
| afv0fv0 41229 | If the value of the altern... |
| afvfvn0fveq 41230 | If the function's value at... |
| afv0nbfvbi 41231 | The function's value at an... |
| afvfv0bi 41232 | The function's value at an... |
| afveu 41233 | The value of a function at... |
| fnbrafvb 41234 | Equivalence of function va... |
| fnopafvb 41235 | Equivalence of function va... |
| funbrafvb 41236 | Equivalence of function va... |
| funopafvb 41237 | Equivalence of function va... |
| funbrafv 41238 | The second argument of a b... |
| funbrafv2b 41239 | Function value in terms of... |
| dfafn5a 41240 | Representation of a functi... |
| dfafn5b 41241 | Representation of a functi... |
| fnrnafv 41242 | The range of a function ex... |
| afvelrnb 41243 | A member of a function's r... |
| afvelrnb0 41244 | A member of a function's r... |
| dfaimafn 41245 | Alternate definition of th... |
| dfaimafn2 41246 | Alternate definition of th... |
| afvelima 41247 | Function value in an image... |
| afvelrn 41248 | A function's value belongs... |
| fnafvelrn 41249 | A function's value belongs... |
| fafvelrn 41250 | A function's value belongs... |
| ffnafv 41251 | A function maps to a class... |
| afvres 41252 | The value of a restricted ... |
| tz6.12-afv 41253 | Function value. Theorem 6... |
| tz6.12-1-afv 41254 | Function value (Theorem 6.... |
| dmfcoafv 41255 | Domains of a function comp... |
| afvco2 41256 | Value of a function compos... |
| rlimdmafv 41257 | Two ways to express that a... |
| aoveq123d 41258 | Equality deduction for ope... |
| nfaov 41259 | Bound-variable hypothesis ... |
| csbaovg 41260 | Move class substitution in... |
| aovfundmoveq 41261 | If a class is a function r... |
| aovnfundmuv 41262 | If an ordered pair is not ... |
| ndmaov 41263 | The value of an operation ... |
| ndmaovg 41264 | The value of an operation ... |
| aovvdm 41265 | If the operation value of ... |
| nfunsnaov 41266 | If the restriction of a cl... |
| aovvfunressn 41267 | If the operation value of ... |
| aovprc 41268 | The value of an operation ... |
| aovrcl 41269 | Reverse closure for an ope... |
| aovpcov0 41270 | If the alternative value o... |
| aovnuoveq 41271 | The alternative value of t... |
| aovvoveq 41272 | The alternative value of t... |
| aov0ov0 41273 | If the alternative value o... |
| aovovn0oveq 41274 | If the operation's value a... |
| aov0nbovbi 41275 | The operation's value on a... |
| aovov0bi 41276 | The operation's value on a... |
| rspceaov 41277 | A frequently used special ... |
| fnotaovb 41278 | Equivalence of operation v... |
| ffnaov 41279 | An operation maps to a cla... |
| faovcl 41280 | Closure law for an operati... |
| aovmpt4g 41281 | Value of a function given ... |
| aoprssdm 41282 | Domain of closure of an op... |
| ndmaovcl 41283 | The "closure" of an operat... |
| ndmaovrcl 41284 | Reverse closure law, in co... |
| ndmaovcom 41285 | Any operation is commutati... |
| ndmaovass 41286 | Any operation is associati... |
| ndmaovdistr 41287 | Any operation is distribut... |
| dfnelbr2 41290 | Alternate definition of th... |
| nelbr 41291 | The binary relation of a s... |
| nelbrim 41292 | If a set is related to ano... |
| nelbrnel 41293 | A set is related to anothe... |
| nelbrnelim 41294 | If a set is related to ano... |
| ralralimp 41295 | Selecting one of two alter... |
| elprneb 41296 | An element of a proper uno... |
| opidg 41297 | The ordered pair ` <. A , ... |
| otiunsndisjX 41298 | The union of singletons co... |
| fvifeq 41299 | Equality of function value... |
| rnfdmpr 41300 | The range of a one-to-one ... |
| imarnf1pr 41301 | The image of the range of ... |
| funop1 41302 | A function is an ordered p... |
| fun2dmnopgexmpl 41303 | A function with a domain c... |
| opabresex0d 41304 | A collection of ordered pa... |
| opabbrfex0d 41305 | A collection of ordered pa... |
| opabresexd 41306 | A collection of ordered pa... |
| opabbrfexd 41307 | A collection of ordered pa... |
| leltletr 41308 | Transitive law, weaker for... |
| cnambpcma 41309 | ((a-b)+c)-a = c-a holds fo... |
| cnapbmcpd 41310 | ((a+b)-c)+d = ((a+d)+b)-c ... |
| leaddsuble 41311 | Addition and subtraction o... |
| 2leaddle2 41312 | If two real numbers are le... |
| ltnltne 41313 | Variant of trichotomy law ... |
| p1lep2 41314 | A real number increasd by ... |
| ltsubsubaddltsub 41315 | If the result of subtracti... |
| zm1nn 41316 | An integer minus 1 is posi... |
| nn0resubcl 41317 | Closure law for subtractio... |
| zgeltp1eq 41318 | If an integer is between a... |
| 1t10e1p1e11 41319 | 11 is 1 times 10 to the po... |
| 1t10e1p1e11OLD 41320 | Obsolete version of ~ 1t10... |
| deccarry 41321 | Add 1 to a 2 digit number ... |
| eluzge0nn0 41322 | If an integer is greater t... |
| nltle2tri 41323 | Negated extended trichotom... |
| ssfz12 41324 | Subset relationship for fi... |
| elfz2z 41325 | Membership of an integer i... |
| 2elfz3nn0 41326 | If there are two elements ... |
| fz0addcom 41327 | The addition of two member... |
| 2elfz2melfz 41328 | If the sum of two integers... |
| fz0addge0 41329 | The sum of two integers in... |
| elfzlble 41330 | Membership of an integer i... |
| elfzelfzlble 41331 | Membership of an element o... |
| fzopred 41332 | Join a predecessor to the ... |
| fzopredsuc 41333 | Join a predecessor and a s... |
| 1fzopredsuc 41334 | Join 0 and a successor to ... |
| el1fzopredsuc 41335 | An element of an open inte... |
| subsubelfzo0 41336 | Subtracting a difference f... |
| fzoopth 41337 | A half-open integer range ... |
| 2ffzoeq 41338 | Two functions over a half-... |
| m1mod0mod1 41339 | An integer decreased by 1 ... |
| elmod2 41340 | An integer modulo 2 is eit... |
| smonoord 41341 | Ordering relation for a st... |
| fsummsndifre 41342 | A finite sum with one of i... |
| fsumsplitsndif 41343 | Separate out a term in a f... |
| fsummmodsndifre 41344 | A finite sum of summands m... |
| fsummmodsnunz 41345 | A finite sum of summands m... |
| setsidel 41346 | The injected slot is an el... |
| setsnidel 41347 | The injected slot is an el... |
| setsv 41348 | The value of the structure... |
| iccpval 41351 | Partition consisting of a ... |
| iccpart 41352 | A special partition. Corr... |
| iccpartimp 41353 | Implications for a class b... |
| iccpartres 41354 | The restriction of a parti... |
| iccpartxr 41355 | If there is a partition, t... |
| iccpartgtprec 41356 | If there is a partition, t... |
| iccpartipre 41357 | If there is a partition, t... |
| iccpartiltu 41358 | If there is a partition, t... |
| iccpartigtl 41359 | If there is a partition, t... |
| iccpartlt 41360 | If there is a partition, t... |
| iccpartltu 41361 | If there is a partition, t... |
| iccpartgtl 41362 | If there is a partition, t... |
| iccpartgt 41363 | If there is a partition, t... |
| iccpartleu 41364 | If there is a partition, t... |
| iccpartgel 41365 | If there is a partition, t... |
| iccpartrn 41366 | If there is a partition, t... |
| iccpartf 41367 | The range of the partition... |
| iccpartel 41368 | If there is a partition, t... |
| iccelpart 41369 | An element of any partitio... |
| iccpartiun 41370 | A half opened interval of ... |
| icceuelpartlem 41371 | Lemma for ~ icceuelpart . ... |
| icceuelpart 41372 | An element of a partitione... |
| iccpartdisj 41373 | The segments of a partitio... |
| iccpartnel 41374 | A point of a partition is ... |
| fargshiftfv 41375 | If a class is a function, ... |
| fargshiftf 41376 | If a class is a function, ... |
| fargshiftf1 41377 | If a function is 1-1, then... |
| fargshiftfo 41378 | If a function is onto, the... |
| fargshiftfva 41379 | The values of a shifted fu... |
| lswn0 41380 | The last symbol of a not e... |
| pfxval 41383 | Value of a prefix. (Contr... |
| pfx00 41384 | A zero length prefix. (Co... |
| pfx0 41385 | A prefix of an empty set i... |
| pfxcl 41386 | Closure of the prefix extr... |
| pfxmpt 41387 | Value of the prefix extrac... |
| pfxres 41388 | Value of the prefix extrac... |
| pfxf 41389 | A prefix of a word is a fu... |
| pfxfn 41390 | Value of the prefix extrac... |
| pfxlen 41391 | Length of a prefix. Could... |
| pfxid 41392 | A word is a prefix of itse... |
| pfxrn 41393 | The range of a prefix of a... |
| pfxn0 41394 | A prefix consisting of at ... |
| pfxnd 41395 | The value of the prefix ex... |
| pfxlen0 41396 | Length of a prefix of a wo... |
| addlenrevpfx 41397 | The sum of the lengths of ... |
| addlenpfx 41398 | The sum of the lengths of ... |
| pfxfv 41399 | A symbol in a prefix of a ... |
| pfxfv0 41400 | The first symbol in a pref... |
| pfxtrcfv 41401 | A symbol in a word truncat... |
| pfxtrcfv0 41402 | The first symbol in a word... |
| pfxfvlsw 41403 | The last symbol in a (not ... |
| pfxeq 41404 | The prefixes of two words ... |
| pfxtrcfvl 41405 | The last symbol in a word ... |
| pfxsuffeqwrdeq 41406 | Two words are equal if and... |
| pfxsuff1eqwrdeq 41407 | Two (nonempty) words are e... |
| disjwrdpfx 41408 | Sets of words are disjoint... |
| ccatpfx 41409 | Joining a prefix with an a... |
| pfxccat1 41410 | Recover the left half of a... |
| pfx1 41411 | A prefix of length 1. (Co... |
| pfx2 41412 | A prefix of length 2. (Co... |
| pfxswrd 41413 | A prefix of a subword. Co... |
| swrdpfx 41414 | A subword of a prefix. Co... |
| pfxpfx 41415 | A prefix of a prefix. Cou... |
| pfxpfxid 41416 | A prefix of a prefix with ... |
| pfxcctswrd 41417 | The concatenation of the p... |
| lenpfxcctswrd 41418 | The length of the concaten... |
| lenrevpfxcctswrd 41419 | The length of the concaten... |
| pfxlswccat 41420 | Reconstruct a nonempty wor... |
| ccats1pfxeq 41421 | The last symbol of a word ... |
| ccats1pfxeqrex 41422 | There exists a symbol such... |
| pfxccatin12lem1 41423 | Lemma 1 for ~ pfxccatin12 ... |
| pfxccatin12lem2 41424 | Lemma 2 for ~ pfxccatin12 ... |
| pfxccatin12 41425 | The subword of a concatena... |
| pfxccat3 41426 | The subword of a concatena... |
| pfxccatpfx1 41427 | A prefix of a concatenatio... |
| pfxccatpfx2 41428 | A prefix of a concatenatio... |
| pfxccat3a 41429 | A prefix of a concatenatio... |
| pfxccatid 41430 | A prefix of a concatenatio... |
| ccats1pfxeqbi 41431 | A word is a prefix of a wo... |
| pfxccatin12d 41432 | The subword of a concatena... |
| reuccatpfxs1lem 41433 | Lemma for ~ reuccatpfxs1 .... |
| reuccatpfxs1 41434 | There is a unique word hav... |
| splvalpfx 41435 | Value of the substring rep... |
| repswpfx 41436 | A prefix of a repeated sym... |
| cshword2 41437 | Perform a cyclical shift f... |
| pfxco 41438 | Mapping of words commutes ... |
| fmtno 41441 | The ` N ` th Fermat number... |
| fmtnoge3 41442 | Each Fermat number is grea... |
| fmtnonn 41443 | Each Fermat number is a po... |
| fmtnom1nn 41444 | A Fermat number minus one ... |
| fmtnoodd 41445 | Each Fermat number is odd.... |
| fmtnorn 41446 | A Fermat number is a funct... |
| fmtnof1 41447 | The enumeration of the Fer... |
| fmtnoinf 41448 | The set of Fermat numbers ... |
| fmtnorec1 41449 | The first recurrence relat... |
| sqrtpwpw2p 41450 | The floor of the square ro... |
| fmtnosqrt 41451 | The floor of the square ro... |
| fmtno0 41452 | The ` 0 ` th Fermat number... |
| fmtno1 41453 | The ` 1 ` st Fermat number... |
| fmtnorec2lem 41454 | Lemma for ~ fmtnorec2 (ind... |
| fmtnorec2 41455 | The second recurrence rela... |
| fmtnodvds 41456 | Any Fermat number divides ... |
| goldbachthlem1 41457 | Lemma 1 for ~ goldbachth .... |
| goldbachthlem2 41458 | Lemma 2 for ~ goldbachth .... |
| goldbachth 41459 | Goldbach's theorem: Two d... |
| fmtnorec3 41460 | The third recurrence relat... |
| fmtnorec4 41461 | The fourth recurrence rela... |
| fmtno2 41462 | The ` 2 ` nd Fermat number... |
| fmtno3 41463 | The ` 3 ` rd Fermat number... |
| fmtno4 41464 | The ` 4 ` th Fermat number... |
| fmtno5lem1 41465 | Lemma 1 for ~ fmtno5 . (C... |
| fmtno5lem2 41466 | Lemma 2 for ~ fmtno5 . (C... |
| fmtno5lem3 41467 | Lemma 3 for ~ fmtno5 . (C... |
| fmtno5lem4 41468 | Lemma 4 for ~ fmtno5 . (C... |
| fmtno5 41469 | The ` 5 ` th Fermat number... |
| fmtno0prm 41470 | The ` 0 ` th Fermat number... |
| fmtno1prm 41471 | The ` 1 ` st Fermat number... |
| fmtno2prm 41472 | The ` 2 ` nd Fermat number... |
| 257prm 41473 | 257 is a prime number (the... |
| fmtno3prm 41474 | The ` 3 ` rd Fermat number... |
| odz2prm2pw 41475 | Any power of two is coprim... |
| fmtnoprmfac1lem 41476 | Lemma for ~ fmtnoprmfac1 :... |
| fmtnoprmfac1 41477 | Divisor of Fermat number (... |
| fmtnoprmfac2lem1 41478 | Lemma for ~ fmtnoprmfac2 .... |
| fmtnoprmfac2 41479 | Divisor of Fermat number (... |
| fmtnofac2lem 41480 | Lemma for ~ fmtnofac2 (Ind... |
| fmtnofac2 41481 | Divisor of Fermat number (... |
| fmtnofac1 41482 | Divisor of Fermat number (... |
| fmtno4sqrt 41483 | The floor of the square ro... |
| fmtno4prmfac 41484 | If P was a (prime) factor ... |
| fmtno4prmfac193 41485 | If P was a (prime) factor ... |
| fmtno4nprmfac193 41486 | 193 is not a (prime) facto... |
| fmtno4prm 41487 | The ` 4 `-th Fermat number... |
| 65537prm 41488 | 65537 is a prime number (t... |
| fmtnofz04prm 41489 | The first five Fermat numb... |
| fmtnole4prm 41490 | The first five Fermat numb... |
| fmtno5faclem1 41491 | Lemma 1 for ~ fmtno5fac . ... |
| fmtno5faclem2 41492 | Lemma 2 for ~ fmtno5fac . ... |
| fmtno5faclem3 41493 | Lemma 3 for ~ fmtno5fac . ... |
| fmtno5fac 41494 | The factorisation of the `... |
| fmtno5nprm 41495 | The ` 5 ` th Fermat number... |
| prmdvdsfmtnof1lem1 41496 | Lemma 1 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof1lem2 41497 | Lemma 2 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof 41498 | The mapping of a Fermat nu... |
| prmdvdsfmtnof1 41499 | The mapping of a Fermat nu... |
| prminf2 41500 | The set of prime numbers i... |
| pwdif 41501 | The difference of two numb... |
| pwm1geoserALT 41502 | The n-th power of a number... |
| 2pwp1prm 41503 | For every prime number of ... |
| 2pwp1prmfmtno 41504 | Every prime number of the ... |
| m2prm 41505 | The second Mersenne number... |
| m3prm 41506 | The third Mersenne number ... |
| 2exp5 41507 | Two to the fifth power is ... |
| flsqrt 41508 | A condition equivalent to ... |
| flsqrt5 41509 | The floor of the square ro... |
| 3ndvds4 41510 | 3 does not divide 4. (Con... |
| 139prmALT 41511 | 139 is a prime number. In... |
| 31prm 41512 | 31 is a prime number. In ... |
| m5prm 41513 | The fifth Mersenne number ... |
| 2exp7 41514 | Two to the seventh power i... |
| 127prm 41515 | 127 is a prime number. (C... |
| m7prm 41516 | The seventh Mersenne numbe... |
| 2exp11 41517 | Two to the eleventh power ... |
| m11nprm 41518 | The eleventh Mersenne numb... |
| mod42tp1mod8 41519 | If a number is ` 3 ` modul... |
| sfprmdvdsmersenne 41520 | If ` Q ` is a safe prime (... |
| sgprmdvdsmersenne 41521 | If ` P ` is a Sophie Germa... |
| lighneallem1 41522 | Lemma 1 for ~ lighneal . ... |
| lighneallem2 41523 | Lemma 2 for ~ lighneal . ... |
| lighneallem3 41524 | Lemma 3 for ~ lighneal . ... |
| lighneallem4a 41525 | Lemma 1 for ~ lighneallem4... |
| lighneallem4b 41526 | Lemma 2 for ~ lighneallem4... |
| lighneallem4 41527 | Lemma 3 for ~ lighneal . ... |
| lighneal 41528 | If a power of a prime ` P ... |
| modexp2m1d 41529 | The square of an integer w... |
| proththdlem 41530 | Lemma for ~ proththd . (C... |
| proththd 41531 | Proth's theorem (1878). I... |
| 5tcu2e40 41532 | 5 times the cube of 2 is 4... |
| 3exp4mod41 41533 | 3 to the fourth power is -... |
| 41prothprmlem1 41534 | Lemma 1 for ~ 41prothprm .... |
| 41prothprmlem2 41535 | Lemma 2 for ~ 41prothprm .... |
| 41prothprm 41536 | 41 is a _Proth prime_. (C... |
| iseven 41541 | The predicate "is an even ... |
| isodd 41542 | The predicate "is an odd n... |
| evenz 41543 | An even number is an integ... |
| oddz 41544 | An odd number is an intege... |
| evendiv2z 41545 | The result of dividing an ... |
| oddp1div2z 41546 | The result of dividing an ... |
| oddm1div2z 41547 | The result of dividing an ... |
| isodd2 41548 | The predicate "is an odd n... |
| dfodd2 41549 | Alternate definition for o... |
| dfodd6 41550 | Alternate definition for o... |
| dfeven4 41551 | Alternate definition for e... |
| evenm1odd 41552 | The predecessor of an even... |
| evenp1odd 41553 | The successor of an even n... |
| oddp1eveni 41554 | The successor of an odd nu... |
| oddm1eveni 41555 | The predecessor of an odd ... |
| evennodd 41556 | An even number is not an o... |
| oddneven 41557 | An odd number is not an ev... |
| enege 41558 | The negative of an even nu... |
| onego 41559 | The negative of an odd num... |
| m1expevenALTV 41560 | Exponentiation of -1 by an... |
| m1expoddALTV 41561 | Exponentiation of -1 by an... |
| dfeven2 41562 | Alternate definition for e... |
| dfodd3 41563 | Alternate definition for o... |
| iseven2 41564 | The predicate "is an even ... |
| isodd3 41565 | The predicate "is an odd n... |
| 2dvdseven 41566 | 2 divides an even number. ... |
| 2ndvdsodd 41567 | 2 does not divide an odd n... |
| 2dvdsoddp1 41568 | 2 divides an odd number in... |
| 2dvdsoddm1 41569 | 2 divides an odd number de... |
| dfeven3 41570 | Alternate definition for e... |
| dfodd4 41571 | Alternate definition for o... |
| dfodd5 41572 | Alternate definition for o... |
| zefldiv2ALTV 41573 | The floor of an even numbe... |
| zofldiv2ALTV 41574 | The floor of an odd numer ... |
| oddflALTV 41575 | Odd number representation ... |
| iseven5 41576 | The predicate "is an even ... |
| isodd7 41577 | The predicate "is an odd n... |
| dfeven5 41578 | Alternate definition for e... |
| dfodd7 41579 | Alternate definition for o... |
| zneoALTV 41580 | No even integer equals an ... |
| zeoALTV 41581 | An integer is even or odd.... |
| zeo2ALTV 41582 | An integer is even or odd ... |
| nneoALTV 41583 | A positive integer is even... |
| nneoiALTV 41584 | A positive integer is even... |
| odd2np1ALTV 41585 | An integer is odd iff it i... |
| oddm1evenALTV 41586 | An integer is odd iff its ... |
| oddp1evenALTV 41587 | An integer is odd iff its ... |
| oexpnegALTV 41588 | The exponential of the neg... |
| oexpnegnz 41589 | The exponential of the neg... |
| bits0ALTV 41590 | Value of the zeroth bit. ... |
| bits0eALTV 41591 | The zeroth bit of an even ... |
| bits0oALTV 41592 | The zeroth bit of an odd n... |
| divgcdoddALTV 41593 | Either ` A / ( A gcd B ) `... |
| opoeALTV 41594 | The sum of two odds is eve... |
| opeoALTV 41595 | The sum of an odd and an e... |
| omoeALTV 41596 | The difference of two odds... |
| omeoALTV 41597 | The difference of an odd a... |
| oddprmALTV 41598 | A prime not equal to ` 2 `... |
| 0evenALTV 41599 | 0 is an even number. (Con... |
| 0noddALTV 41600 | 0 is not an odd number. (... |
| 1oddALTV 41601 | 1 is an odd number. (Cont... |
| 1nevenALTV 41602 | 1 is not an even number. ... |
| 2evenALTV 41603 | 2 is an even number. (Con... |
| 2noddALTV 41604 | 2 is not an odd number. (... |
| nn0o1gt2ALTV 41605 | An odd nonnegative integer... |
| nnoALTV 41606 | An alternate characterizat... |
| nn0oALTV 41607 | An alternate characterizat... |
| nn0e 41608 | An alternate characterizat... |
| nn0onn0exALTV 41609 | For each odd nonnegative i... |
| nn0enn0exALTV 41610 | For each even nonnegative ... |
| nnpw2evenALTV 41611 | 2 to the power of a positi... |
| epoo 41612 | The sum of an even and an ... |
| emoo 41613 | The difference of an even ... |
| epee 41614 | The sum of two even number... |
| emee 41615 | The difference of two even... |
| evensumeven 41616 | If a summand is even, the ... |
| 3odd 41617 | 3 is an odd number. (Cont... |
| 4even 41618 | 4 is an even number. (Con... |
| 5odd 41619 | 5 is an odd number. (Cont... |
| 6even 41620 | 6 is an even number. (Con... |
| 7odd 41621 | 7 is an odd number. (Cont... |
| 8even 41622 | 8 is an even number. (Con... |
| evenprm2 41623 | A prime number is even iff... |
| oddprmne2 41624 | Every prime number not bei... |
| oddprmuzge3 41625 | A prime number which is od... |
| evenltle 41626 | If an even number is great... |
| odd2prm2 41627 | If an odd number is the su... |
| even3prm2 41628 | If an even number is the s... |
| mogoldbblem 41629 | Lemma for ~ mogoldbb . (C... |
| perfectALTVlem1 41630 | Lemma for ~ perfectALTV . ... |
| perfectALTVlem2 41631 | Lemma for ~ perfectALTV . ... |
| perfectALTV 41632 | The Euclid-Euler theorem, ... |
| isgbe 41639 | The predicate "is an even ... |
| isgbow 41640 | The predicate "is a weak o... |
| isgbo 41641 | The predicate "is an odd G... |
| gbeeven 41642 | An even Goldbach number is... |
| gbowodd 41643 | A weak odd Goldbach number... |
| gbogbow 41644 | A (strong) odd Goldbach nu... |
| gboodd 41645 | An odd Goldbach number is ... |
| gbepos 41646 | Any even Goldbach number i... |
| gbowpos 41647 | Any weak odd Goldbach numb... |
| gbopos 41648 | Any odd Goldbach number is... |
| gbegt5 41649 | Any even Goldbach number i... |
| gbowgt5 41650 | Any weak odd Goldbach numb... |
| gbowge7 41651 | Any weak odd Goldbach numb... |
| gboge9 41652 | Any odd Goldbach number is... |
| gbege6 41653 | Any even Goldbach number i... |
| gbpart6 41654 | The Goldbach partition of ... |
| gbpart7 41655 | The (weak) Goldbach partit... |
| gbpart8 41656 | The Goldbach partition of ... |
| gbpart9 41657 | The (strong) Goldbach part... |
| gbpart11 41658 | The (strong) Goldbach part... |
| 6gbe 41659 | 6 is an even Goldbach numb... |
| 7gbow 41660 | 7 is a weak odd Goldbach n... |
| 8gbe 41661 | 8 is an even Goldbach numb... |
| 9gbo 41662 | 9 is an odd Goldbach numbe... |
| 11gbo 41663 | 11 is an odd Goldbach numb... |
| stgoldbwt 41664 | If the strong ternary Gold... |
| sbgoldbwt 41665 | If the strong binary Goldb... |
| sbgoldbst 41666 | If the strong binary Goldb... |
| sbgoldbaltlem1 41667 | Lemma 1 for ~ sbgoldbalt :... |
| sbgoldbaltlem2 41668 | Lemma 2 for ~ sbgoldbalt :... |
| sbgoldbalt 41669 | An alternate (related to t... |
| sbgoldbb 41670 | If the strong binary Goldb... |
| sgoldbeven3prm 41671 | If the binary Goldbach con... |
| sbgoldbm 41672 | If the strong binary Goldb... |
| mogoldbb 41673 | If the modern version of t... |
| sbgoldbmb 41674 | The strong binary Goldbach... |
| sbgoldbo 41675 | If the strong binary Goldb... |
| nnsum3primes4 41676 | 4 is the sum of at most 3 ... |
| nnsum4primes4 41677 | 4 is the sum of at most 4 ... |
| nnsum3primesprm 41678 | Every prime is "the sum of... |
| nnsum4primesprm 41679 | Every prime is "the sum of... |
| nnsum3primesgbe 41680 | Any even Goldbach number i... |
| nnsum4primesgbe 41681 | Any even Goldbach number i... |
| nnsum3primesle9 41682 | Every integer greater than... |
| nnsum4primesle9 41683 | Every integer greater than... |
| nnsum4primesodd 41684 | If the (weak) ternary Gold... |
| nnsum4primesoddALTV 41685 | If the (strong) ternary Go... |
| evengpop3 41686 | If the (weak) ternary Gold... |
| evengpoap3 41687 | If the (strong) ternary Go... |
| nnsum4primeseven 41688 | If the (weak) ternary Gold... |
| nnsum4primesevenALTV 41689 | If the (strong) ternary Go... |
| wtgoldbnnsum4prm 41690 | If the (weak) ternary Gold... |
| stgoldbnnsum4prm 41691 | If the (strong) ternary Go... |
| bgoldbnnsum3prm 41692 | If the binary Goldbach con... |
| bgoldbtbndlem1 41693 | Lemma 1 for ~ bgoldbtbnd :... |
| bgoldbtbndlem2 41694 | Lemma 2 for ~ bgoldbtbnd .... |
| bgoldbtbndlem3 41695 | Lemma 3 for ~ bgoldbtbnd .... |
| bgoldbtbndlem4 41696 | Lemma 4 for ~ bgoldbtbnd .... |
| bgoldbtbnd 41697 | If the binary Goldbach con... |
| tgoldbachgtALTV 41700 | Variant of Thierry Arnoux'... |
| bgoldbachlt 41701 | The binary Goldbach conjec... |
| tgblthelfgott 41703 | The ternary Goldbach conje... |
| tgoldbachlt 41704 | The ternary Goldbach conje... |
| tgoldbach 41705 | The ternary Goldbach conje... |
| bgoldbachltOLD 41707 | Obsolete version of ~ bgol... |
| tgblthelfgottOLD 41709 | Obsolete version of ~ tgbl... |
| tgoldbachltOLD 41710 | Obsolete version of ~ tgol... |
| tgoldbachOLD 41712 | Obsolete version of ~ tgol... |
| 1hegrlfgr 41713 | A graph ` G ` with one hyp... |
| upwlksfval 41716 | The set of simple walks (i... |
| isupwlk 41717 | Properties of a pair of fu... |
| isupwlkg 41718 | Generalisation of ~ isupwl... |
| upwlkbprop 41719 | Basic properties of a simp... |
| upwlkwlk 41720 | A simple walk is a walk. ... |
| upgrwlkupwlk 41721 | In a pseudograph, a walk i... |
| upgrwlkupwlkb 41722 | In a pseudograph, the defi... |
| upgrisupwlkALT 41723 | Alternate proof of ~ upgri... |
| sprid 41724 | Two identical representati... |
| elsprel 41725 | An unordered pair is an el... |
| spr0nelg 41726 | The empty set is not an el... |
| sprval 41729 | The set of all unordered p... |
| sprvalpw 41730 | The set of all unordered p... |
| sprssspr 41731 | The set of all unordered p... |
| spr0el 41732 | The empty set is not an un... |
| sprvalpwn0 41733 | The set of all unordered p... |
| sprel 41734 | An element of the set of a... |
| prssspr 41735 | An element of a subset of ... |
| prelspr 41736 | An unordered pair of eleme... |
| prsprel 41737 | The elements of a pair fro... |
| prsssprel 41738 | The elements of a pair fro... |
| sprvalpwle2 41739 | The set of all unordered p... |
| sprsymrelfvlem 41740 | Lemma for ~ sprsymrelf and... |
| sprsymrelf1lem 41741 | Lemma for ~ sprsymrelf1 . ... |
| sprsymrelfolem1 41742 | Lemma 1 for ~ sprsymrelfo ... |
| sprsymrelfolem2 41743 | Lemma 2 for ~ sprsymrelfo ... |
| sprsymrelfv 41744 | The value of the function ... |
| sprsymrelf 41745 | The mapping ` F ` is a fun... |
| sprsymrelf1 41746 | The mapping ` F ` is a one... |
| sprsymrelfo 41747 | The mapping ` F ` is a fun... |
| sprsymrelf1o 41748 | The mapping ` F ` is a bij... |
| sprbisymrel 41749 | There is a bijection betwe... |
| sprsymrelen 41750 | The class ` P ` of subsets... |
| upgredgssspr 41751 | The set of edges of a pseu... |
| uspgropssxp 41752 | The set ` G ` of "simple p... |
| uspgrsprfv 41753 | The value of the function ... |
| uspgrsprf 41754 | The mapping ` F ` is a fun... |
| uspgrsprf1 41755 | The mapping ` F ` is a one... |
| uspgrsprfo 41756 | The mapping ` F ` is a fun... |
| uspgrsprf1o 41757 | The mapping ` F ` is a bij... |
| uspgrex 41758 | The class ` G ` of all "si... |
| uspgrbispr 41759 | There is a bijection betwe... |
| uspgrspren 41760 | The set ` G ` of the "simp... |
| uspgrymrelen 41761 | The set ` G ` of the "simp... |
| uspgrbisymrel 41762 | There is a bijection betwe... |
| uspgrbisymrelALT 41763 | Alternate proof of ~ uspgr... |
| ovn0dmfun 41764 | If a class operation value... |
| xpsnopab 41765 | A Cartesian product with a... |
| xpiun 41766 | A Cartesian product expres... |
| ovn0ssdmfun 41767 | If a class' operation valu... |
| fnxpdmdm 41768 | The domain of the domain o... |
| cnfldsrngbas 41769 | The base set of a subring ... |
| cnfldsrngadd 41770 | The group addition operati... |
| cnfldsrngmul 41771 | The ring multiplication op... |
| plusfreseq 41772 | If the empty set is not co... |
| mgmplusfreseq 41773 | If the empty set is not co... |
| 0mgm 41774 | A set with an empty base s... |
| mgmpropd 41775 | If two structures have the... |
| ismgmd 41776 | Deduce a magma from its pr... |
| mgmhmrcl 41781 | Reverse closure of a magma... |
| submgmrcl 41782 | Reverse closure for submag... |
| ismgmhm 41783 | Property of a magma homomo... |
| mgmhmf 41784 | A magma homomorphism is a ... |
| mgmhmpropd 41785 | Magma homomorphism depends... |
| mgmhmlin 41786 | A magma homomorphism prese... |
| mgmhmf1o 41787 | A magma homomorphism is bi... |
| idmgmhm 41788 | The identity homomorphism ... |
| issubmgm 41789 | Expand definition of a sub... |
| issubmgm2 41790 | Submagmas are subsets that... |
| rabsubmgmd 41791 | Deduction for proving that... |
| submgmss 41792 | Submagmas are subsets of t... |
| submgmid 41793 | Every magma is trivially a... |
| submgmcl 41794 | Submagmas are closed under... |
| submgmmgm 41795 | Submagmas are themselves m... |
| submgmbas 41796 | The base set of a submagma... |
| subsubmgm 41797 | A submagma of a submagma i... |
| resmgmhm 41798 | Restriction of a magma hom... |
| resmgmhm2 41799 | One direction of ~ resmgmh... |
| resmgmhm2b 41800 | Restriction of the codomai... |
| mgmhmco 41801 | The composition of magma h... |
| mgmhmima 41802 | The homomorphic image of a... |
| mgmhmeql 41803 | The equalizer of two magma... |
| submgmacs 41804 | Submagmas are an algebraic... |
| ismhm0 41805 | Property of a monoid homom... |
| mhmismgmhm 41806 | Each monoid homomorphism i... |
| opmpt2ismgm 41807 | A structure with a group a... |
| copissgrp 41808 | A structure with a constan... |
| copisnmnd 41809 | A structure with a constan... |
| 0nodd 41810 | 0 is not an odd integer. ... |
| 1odd 41811 | 1 is an odd integer. (Con... |
| 2nodd 41812 | 2 is not an odd integer. ... |
| oddibas 41813 | Lemma 1 for ~ oddinmgm : ... |
| oddiadd 41814 | Lemma 2 for ~ oddinmgm : ... |
| oddinmgm 41815 | The structure of all odd i... |
| nnsgrpmgm 41816 | The structure of positive ... |
| nnsgrp 41817 | The structure of positive ... |
| nnsgrpnmnd 41818 | The structure of positive ... |
| iscllaw 41825 | The predicate "is a closed... |
| iscomlaw 41826 | The predicate "is a commut... |
| clcllaw 41827 | Closure of a closed operat... |
| isasslaw 41828 | The predicate "is an assoc... |
| asslawass 41829 | Associativity of an associ... |
| mgmplusgiopALT 41830 | Slot 2 (group operation) o... |
| sgrpplusgaopALT 41831 | Slot 2 (group operation) o... |
| intopval 41838 | The internal (binary) oper... |
| intop 41839 | An internal (binary) opera... |
| clintopval 41840 | The closed (internal binar... |
| assintopval 41841 | The associative (closed in... |
| assintopmap 41842 | The associative (closed in... |
| isclintop 41843 | The predicate "is a closed... |
| clintop 41844 | A closed (internal binary)... |
| assintop 41845 | An associative (closed int... |
| isassintop 41846 | The predicate "is an assoc... |
| clintopcllaw 41847 | The closure law holds for ... |
| assintopcllaw 41848 | The closure low holds for ... |
| assintopasslaw 41849 | The associative low holds ... |
| assintopass 41850 | An associative (closed int... |
| ismgmALT 41859 | The predicate "is a magma.... |
| iscmgmALT 41860 | The predicate "is a commut... |
| issgrpALT 41861 | The predicate "is a semigr... |
| iscsgrpALT 41862 | The predicate "is a commut... |
| mgm2mgm 41863 | Equivalence of the two def... |
| sgrp2sgrp 41864 | Equivalence of the two def... |
| idfusubc0 41865 | The identity functor for a... |
| idfusubc 41866 | The identity functor for a... |
| inclfusubc 41867 | The "inclusion functor" fr... |
| lmod0rng 41868 | If the scalar ring of a mo... |
| nzrneg1ne0 41869 | The additive inverse of th... |
| 0ringdif 41870 | A zero ring is a ring whic... |
| 0ringbas 41871 | The base set of a zero rin... |
| 0ring1eq0 41872 | In a zero ring, a ring whi... |
| nrhmzr 41873 | There is no ring homomorph... |
| isrng 41876 | The predicate "is a non-un... |
| rngabl 41877 | A non-unital ring is an (a... |
| rngmgp 41878 | A non-unital ring is a sem... |
| ringrng 41879 | A unital ring is a (non-un... |
| ringssrng 41880 | The unital rings are (non-... |
| isringrng 41881 | The predicate "is a unital... |
| rngdir 41882 | Distributive law for the m... |
| rngcl 41883 | Closure of the multiplicat... |
| rnglz 41884 | The zero of a nonunital ri... |
| rnghmrcl 41889 | Reverse closure of a non-u... |
| rnghmfn 41890 | The mapping of two non-uni... |
| rnghmval 41891 | The set of the non-unital ... |
| isrnghm 41892 | A function is a non-unital... |
| isrnghmmul 41893 | A function is a non-unital... |
| rnghmmgmhm 41894 | A non-unital ring homomorp... |
| rnghmval2 41895 | The non-unital ring homomo... |
| isrngisom 41896 | An isomorphism of non-unit... |
| rngimrcl 41897 | Reverse closure for an iso... |
| rnghmghm 41898 | A non-unital ring homomorp... |
| rnghmf 41899 | A ring homomorphism is a f... |
| rnghmmul 41900 | A homomorphism of non-unit... |
| isrnghm2d 41901 | Demonstration of non-unita... |
| isrnghmd 41902 | Demonstration of non-unita... |
| rnghmf1o 41903 | A non-unital ring homomorp... |
| isrngim 41904 | An isomorphism of non-unit... |
| rngimf1o 41905 | An isomorphism of non-unit... |
| rngimrnghm 41906 | An isomorphism of non-unit... |
| rnghmco 41907 | The composition of non-uni... |
| idrnghm 41908 | The identity homomorphism ... |
| c0mgm 41909 | The constant mapping to ze... |
| c0mhm 41910 | The constant mapping to ze... |
| c0ghm 41911 | The constant mapping to ze... |
| c0rhm 41912 | The constant mapping to ze... |
| c0rnghm 41913 | The constant mapping to ze... |
| c0snmgmhm 41914 | The constant mapping to ze... |
| c0snmhm 41915 | The constant mapping to ze... |
| c0snghm 41916 | The constant mapping to ze... |
| zrrnghm 41917 | The constant mapping to ze... |
| rhmfn 41918 | The mapping of two rings t... |
| rhmval 41919 | The ring homomorphisms bet... |
| rhmisrnghm 41920 | Each unital ring homomorph... |
| lidldomn1 41921 | If a (left) ideal (which i... |
| lidlssbas 41922 | The base set of the restri... |
| lidlbas 41923 | A (left) ideal of a ring i... |
| lidlabl 41924 | A (left) ideal of a ring i... |
| lidlmmgm 41925 | The multiplicative group o... |
| lidlmsgrp 41926 | The multiplicative group o... |
| lidlrng 41927 | A (left) ideal of a ring i... |
| zlidlring 41928 | The zero (left) ideal of a... |
| uzlidlring 41929 | Only the zero (left) ideal... |
| lidldomnnring 41930 | A (left) ideal of a domain... |
| 0even 41931 | 0 is an even integer. (Co... |
| 1neven 41932 | 1 is not an even integer. ... |
| 2even 41933 | 2 is an even integer. (Co... |
| 2zlidl 41934 | The even integers are a (l... |
| 2zrng 41935 | The ring of integers restr... |
| 2zrngbas 41936 | The base set of R is the s... |
| 2zrngadd 41937 | The group addition operati... |
| 2zrng0 41938 | The additive identity of R... |
| 2zrngamgm 41939 | R is an (additive) magma. ... |
| 2zrngasgrp 41940 | R is an (additive) semigro... |
| 2zrngamnd 41941 | R is an (additive) monoid.... |
| 2zrngacmnd 41942 | R is a commutative (additi... |
| 2zrngagrp 41943 | R is an (additive) group. ... |
| 2zrngaabl 41944 | R is an (additive) abelian... |
| 2zrngmul 41945 | The ring multiplication op... |
| 2zrngmmgm 41946 | R is a (multiplicative) ma... |
| 2zrngmsgrp 41947 | R is a (multiplicative) se... |
| 2zrngALT 41948 | The ring of integers restr... |
| 2zrngnmlid 41949 | R has no multiplicative (l... |
| 2zrngnmrid 41950 | R has no multiplicative (r... |
| 2zrngnmlid2 41951 | R has no multiplicative (l... |
| 2zrngnring 41952 | R is not a unital ring. (... |
| cznrnglem 41953 | Lemma for ~ cznrng : The ... |
| cznabel 41954 | The ring constructed from ... |
| cznrng 41955 | The ring constructed from ... |
| cznnring 41956 | The ring constructed from ... |
| rngcvalALTV 41961 | Value of the category of n... |
| rngcval 41962 | Value of the category of n... |
| rnghmresfn 41963 | The class of non-unital ri... |
| rnghmresel 41964 | An element of the non-unit... |
| rngcbas 41965 | Set of objects of the cate... |
| rngchomfval 41966 | Set of arrows of the categ... |
| rngchom 41967 | Set of arrows of the categ... |
| elrngchom 41968 | A morphism of non-unital r... |
| rngchomfeqhom 41969 | The functionalized Hom-set... |
| rngccofval 41970 | Composition in the categor... |
| rngcco 41971 | Composition in the categor... |
| dfrngc2 41972 | Alternate definition of th... |
| rnghmsscmap2 41973 | The non-unital ring homomo... |
| rnghmsscmap 41974 | The non-unital ring homomo... |
| rnghmsubcsetclem1 41975 | Lemma 1 for ~ rnghmsubcset... |
| rnghmsubcsetclem2 41976 | Lemma 2 for ~ rnghmsubcset... |
| rnghmsubcsetc 41977 | The non-unital ring homomo... |
| rngccat 41978 | The category of non-unital... |
| rngcid 41979 | The identity arrow in the ... |
| rngcsect 41980 | A section in the category ... |
| rngcinv 41981 | An inverse in the category... |
| rngciso 41982 | An isomorphism in the cate... |
| rngcbasALTV 41983 | Set of objects of the cate... |
| rngchomfvalALTV 41984 | Set of arrows of the categ... |
| rngchomALTV 41985 | Set of arrows of the categ... |
| elrngchomALTV 41986 | A morphism of non-unital r... |
| rngccofvalALTV 41987 | Composition in the categor... |
| rngccoALTV 41988 | Composition in the categor... |
| rngccatidALTV 41989 | Lemma for ~ rngccatALTV . ... |
| rngccatALTV 41990 | The category of non-unital... |
| rngcidALTV 41991 | The identity arrow in the ... |
| rngcsectALTV 41992 | A section in the category ... |
| rngcinvALTV 41993 | An inverse in the category... |
| rngcisoALTV 41994 | An isomorphism in the cate... |
| rngchomffvalALTV 41995 | The value of the functiona... |
| rngchomrnghmresALTV 41996 | The value of the functiona... |
| rngcifuestrc 41997 | The "inclusion functor" fr... |
| funcrngcsetc 41998 | The "natural forgetful fun... |
| funcrngcsetcALT 41999 | Alternate proof of ~ funcr... |
| zrinitorngc 42000 | The zero ring is an initia... |
| zrtermorngc 42001 | The zero ring is a termina... |
| zrzeroorngc 42002 | The zero ring is a zero ob... |
| ringcvalALTV 42007 | Value of the category of r... |
| ringcval 42008 | Value of the category of u... |
| rhmresfn 42009 | The class of unital ring h... |
| rhmresel 42010 | An element of the unital r... |
| ringcbas 42011 | Set of objects of the cate... |
| ringchomfval 42012 | Set of arrows of the categ... |
| ringchom 42013 | Set of arrows of the categ... |
| elringchom 42014 | A morphism of unital rings... |
| ringchomfeqhom 42015 | The functionalized Hom-set... |
| ringccofval 42016 | Composition in the categor... |
| ringcco 42017 | Composition in the categor... |
| dfringc2 42018 | Alternate definition of th... |
| rhmsscmap2 42019 | The unital ring homomorphi... |
| rhmsscmap 42020 | The unital ring homomorphi... |
| rhmsubcsetclem1 42021 | Lemma 1 for ~ rhmsubcsetc ... |
| rhmsubcsetclem2 42022 | Lemma 2 for ~ rhmsubcsetc ... |
| rhmsubcsetc 42023 | The unital ring homomorphi... |
| ringccat 42024 | The category of unital rin... |
| ringcid 42025 | The identity arrow in the ... |
| rhmsscrnghm 42026 | The unital ring homomorphi... |
| rhmsubcrngclem1 42027 | Lemma 1 for ~ rhmsubcrngc ... |
| rhmsubcrngclem2 42028 | Lemma 2 for ~ rhmsubcrngc ... |
| rhmsubcrngc 42029 | The unital ring homomorphi... |
| rngcresringcat 42030 | The restriction of the cat... |
| ringcsect 42031 | A section in the category ... |
| ringcinv 42032 | An inverse in the category... |
| ringciso 42033 | An isomorphism in the cate... |
| ringcbasbas 42034 | An element of the base set... |
| funcringcsetc 42035 | The "natural forgetful fun... |
| funcringcsetcALTV2lem1 42036 | Lemma 1 for ~ funcringcset... |
| funcringcsetcALTV2lem2 42037 | Lemma 2 for ~ funcringcset... |
| funcringcsetcALTV2lem3 42038 | Lemma 3 for ~ funcringcset... |
| funcringcsetcALTV2lem4 42039 | Lemma 4 for ~ funcringcset... |
| funcringcsetcALTV2lem5 42040 | Lemma 5 for ~ funcringcset... |
| funcringcsetcALTV2lem6 42041 | Lemma 6 for ~ funcringcset... |
| funcringcsetcALTV2lem7 42042 | Lemma 7 for ~ funcringcset... |
| funcringcsetcALTV2lem8 42043 | Lemma 8 for ~ funcringcset... |
| funcringcsetcALTV2lem9 42044 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV2 42045 | The "natural forgetful fun... |
| ringcbasALTV 42046 | Set of objects of the cate... |
| ringchomfvalALTV 42047 | Set of arrows of the categ... |
| ringchomALTV 42048 | Set of arrows of the categ... |
| elringchomALTV 42049 | A morphism of rings is a f... |
| ringccofvalALTV 42050 | Composition in the categor... |
| ringccoALTV 42051 | Composition in the categor... |
| ringccatidALTV 42052 | Lemma for ~ ringccatALTV .... |
| ringccatALTV 42053 | The category of rings is a... |
| ringcidALTV 42054 | The identity arrow in the ... |
| ringcsectALTV 42055 | A section in the category ... |
| ringcinvALTV 42056 | An inverse in the category... |
| ringcisoALTV 42057 | An isomorphism in the cate... |
| ringcbasbasALTV 42058 | An element of the base set... |
| funcringcsetclem1ALTV 42059 | Lemma 1 for ~ funcringcset... |
| funcringcsetclem2ALTV 42060 | Lemma 2 for ~ funcringcset... |
| funcringcsetclem3ALTV 42061 | Lemma 3 for ~ funcringcset... |
| funcringcsetclem4ALTV 42062 | Lemma 4 for ~ funcringcset... |
| funcringcsetclem5ALTV 42063 | Lemma 5 for ~ funcringcset... |
| funcringcsetclem6ALTV 42064 | Lemma 6 for ~ funcringcset... |
| funcringcsetclem7ALTV 42065 | Lemma 7 for ~ funcringcset... |
| funcringcsetclem8ALTV 42066 | Lemma 8 for ~ funcringcset... |
| funcringcsetclem9ALTV 42067 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV 42068 | The "natural forgetful fun... |
| irinitoringc 42069 | The ring of integers is an... |
| zrtermoringc 42070 | The zero ring is a termina... |
| zrninitoringc 42071 | The zero ring is not an in... |
| nzerooringczr 42072 | There is no zero object in... |
| srhmsubclem1 42073 | Lemma 1 for ~ srhmsubc . ... |
| srhmsubclem2 42074 | Lemma 2 for ~ srhmsubc . ... |
| srhmsubclem3 42075 | Lemma 3 for ~ srhmsubc . ... |
| srhmsubc 42076 | According to ~ df-subc , t... |
| sringcat 42077 | The restriction of the cat... |
| crhmsubc 42078 | According to ~ df-subc , t... |
| cringcat 42079 | The restriction of the cat... |
| drhmsubc 42080 | According to ~ df-subc , t... |
| drngcat 42081 | The restriction of the cat... |
| fldcat 42082 | The restriction of the cat... |
| fldc 42083 | The restriction of the cat... |
| fldhmsubc 42084 | According to ~ df-subc , t... |
| rngcrescrhm 42085 | The category of non-unital... |
| rhmsubclem1 42086 | Lemma 1 for ~ rhmsubc . (... |
| rhmsubclem2 42087 | Lemma 2 for ~ rhmsubc . (... |
| rhmsubclem3 42088 | Lemma 3 for ~ rhmsubc . (... |
| rhmsubclem4 42089 | Lemma 4 for ~ rhmsubc . (... |
| rhmsubc 42090 | According to ~ df-subc , t... |
| rhmsubccat 42091 | The restriction of the cat... |
| srhmsubcALTVlem1 42092 | Lemma 1 for ~ srhmsubcALTV... |
| srhmsubcALTVlem2 42093 | Lemma 2 for ~ srhmsubcALTV... |
| srhmsubcALTV 42094 | According to ~ df-subc , t... |
| sringcatALTV 42095 | The restriction of the cat... |
| crhmsubcALTV 42096 | According to ~ df-subc , t... |
| cringcatALTV 42097 | The restriction of the cat... |
| drhmsubcALTV 42098 | According to ~ df-subc , t... |
| drngcatALTV 42099 | The restriction of the cat... |
| fldcatALTV 42100 | The restriction of the cat... |
| fldcALTV 42101 | The restriction of the cat... |
| fldhmsubcALTV 42102 | According to ~ df-subc , t... |
| rngcrescrhmALTV 42103 | The category of non-unital... |
| rhmsubcALTVlem1 42104 | Lemma 1 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem2 42105 | Lemma 2 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem3 42106 | Lemma 3 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem4 42107 | Lemma 4 for ~ rhmsubcALTV ... |
| rhmsubcALTV 42108 | According to ~ df-subc , t... |
| rhmsubcALTVcat 42109 | The restriction of the cat... |
| xpprsng 42110 | The Cartesian product of a... |
| opeliun2xp 42111 | Membership of an ordered p... |
| eliunxp2 42112 | Membership in a union of C... |
| mpt2mptx2 42113 | Express a two-argument fun... |
| cbvmpt2x2 42114 | Rule to change the bound v... |
| dmmpt2ssx2 42115 | The domain of a mapping is... |
| mpt2exxg2 42116 | Existence of an operation ... |
| ovmpt2rdxf 42117 | Value of an operation give... |
| ovmpt2rdx 42118 | Value of an operation give... |
| ovmpt2x2 42119 | The value of an operation ... |
| fdmdifeqresdif 42120 | The restriction of a condi... |
| offvalfv 42121 | The function operation exp... |
| ofaddmndmap 42122 | The function operation app... |
| mapsnop 42123 | A singleton of an ordered ... |
| mapprop 42124 | An unordered pair containi... |
| ztprmneprm 42125 | A prime is not an integer ... |
| 2t6m3t4e0 42126 | 2 times 6 minus 3 times 4 ... |
| ssnn0ssfz 42127 | For any finite subset of `... |
| nn0sumltlt 42128 | If the sum of two nonnegat... |
| bcpascm1 42129 | Pascal's rule for the bino... |
| altgsumbc 42130 | The sum of binomial coeffi... |
| altgsumbcALT 42131 | Alternate proof of ~ altgs... |
| zlmodzxzlmod 42132 | The ` ZZ `-module ` ZZ X. ... |
| zlmodzxzel 42133 | An element of the (base se... |
| zlmodzxz0 42134 | The ` 0 ` of the ` ZZ `-mo... |
| zlmodzxzscm 42135 | The scalar multiplication ... |
| zlmodzxzadd 42136 | The addition of the ` ZZ `... |
| zlmodzxzsubm 42137 | The subtraction of the ` Z... |
| zlmodzxzsub 42138 | The subtraction of the ` Z... |
| gsumpr 42139 | Group sum of a pair. (Con... |
| mgpsumunsn 42140 | Extract a summand/factor f... |
| mgpsumz 42141 | If the group sum for the m... |
| mgpsumn 42142 | If the group sum for the m... |
| gsumsplit2f 42143 | Split a group sum into two... |
| gsumdifsndf 42144 | Extract a summand from a f... |
| exple2lt6 42145 | A nonnegative integer to t... |
| pgrple2abl 42146 | Every symmetric group on a... |
| pgrpgt2nabl 42147 | Every symmetric group on a... |
| invginvrid 42148 | Identity for a multiplicat... |
| rmsupp0 42149 | The support of a mapping o... |
| domnmsuppn0 42150 | The support of a mapping o... |
| rmsuppss 42151 | The support of a mapping o... |
| mndpsuppss 42152 | The support of a mapping o... |
| scmsuppss 42153 | The support of a mapping o... |
| rmsuppfi 42154 | The support of a mapping o... |
| rmfsupp 42155 | A mapping of a multiplicat... |
| mndpsuppfi 42156 | The support of a mapping o... |
| mndpfsupp 42157 | A mapping of a scalar mult... |
| scmsuppfi 42158 | The support of a mapping o... |
| scmfsupp 42159 | A mapping of a scalar mult... |
| suppmptcfin 42160 | The support of a mapping w... |
| mptcfsupp 42161 | A mapping with value 0 exc... |
| fsuppmptdmf 42162 | A mapping with a finite do... |
| lmodvsmdi 42163 | Multiple distributive law ... |
| gsumlsscl 42164 | Closure of a group sum in ... |
| ascl0 42165 | The scalar 0 embedded into... |
| ascl1 42166 | The scalar 1 embedded into... |
| assaascl0 42167 | The scalar 0 embedded into... |
| assaascl1 42168 | The scalar 1 embedded into... |
| ply1vr1smo 42169 | The variable in a polynomi... |
| ply1ass23l 42170 | Associative identity with ... |
| ply1sclrmsm 42171 | The ring multiplication of... |
| coe1id 42172 | Coefficient vector of the ... |
| coe1sclmulval 42173 | The value of the coefficie... |
| ply1mulgsumlem1 42174 | Lemma 1 for ~ ply1mulgsum ... |
| ply1mulgsumlem2 42175 | Lemma 2 for ~ ply1mulgsum ... |
| ply1mulgsumlem3 42176 | Lemma 3 for ~ ply1mulgsum ... |
| ply1mulgsumlem4 42177 | Lemma 4 for ~ ply1mulgsum ... |
| ply1mulgsum 42178 | The product of two polynom... |
| evl1at0 42179 | Polynomial evaluation for ... |
| evl1at1 42180 | Polynomial evaluation for ... |
| linply1 42181 | A term of the form ` x - C... |
| lineval 42182 | A term of the form ` x - C... |
| zringsubgval 42183 | Subtraction in the ring of... |
| linevalexample 42184 | The polynomial ` x - 3 ` o... |
| dmatALTval 42189 | The algebra of ` N ` x ` N... |
| dmatALTbas 42190 | The base set of the algebr... |
| dmatALTbasel 42191 | An element of the base set... |
| dmatbas 42192 | The set of all ` N ` x ` N... |
| lincop 42197 | A linear combination as op... |
| lincval 42198 | The value of a linear comb... |
| dflinc2 42199 | Alternative definition of ... |
| lcoop 42200 | A linear combination as op... |
| lcoval 42201 | The value of a linear comb... |
| lincfsuppcl 42202 | A linear combination of ve... |
| linccl 42203 | A linear combination of ve... |
| lincval0 42204 | The value of an empty line... |
| lincvalsng 42205 | The linear combination ove... |
| lincvalsn 42206 | The linear combination ove... |
| lincvalpr 42207 | The linear combination ove... |
| lincval1 42208 | The linear combination ove... |
| lcosn0 42209 | Properties of a linear com... |
| lincvalsc0 42210 | The linear combination whe... |
| lcoc0 42211 | Properties of a linear com... |
| linc0scn0 42212 | If a set contains the zero... |
| lincdifsn 42213 | A vector is a linear combi... |
| linc1 42214 | A vector is a linear combi... |
| lincellss 42215 | A linear combination of a ... |
| lco0 42216 | The set of empty linear co... |
| lcoel0 42217 | The zero vector is always ... |
| lincsum 42218 | The sum of two linear comb... |
| lincscm 42219 | A linear combinations mult... |
| lincsumcl 42220 | The sum of two linear comb... |
| lincscmcl 42221 | The multiplication of a li... |
| lincsumscmcl 42222 | The sum of a linear combin... |
| lincolss 42223 | According to the statement... |
| ellcoellss 42224 | Every linear combination o... |
| lcoss 42225 | A set of vectors of a modu... |
| lspsslco 42226 | Lemma for ~ lspeqlco . (C... |
| lcosslsp 42227 | Lemma for ~ lspeqlco . (C... |
| lspeqlco 42228 | Equivalence of a _span_ of... |
| rellininds 42232 | The class defining the rel... |
| linindsv 42234 | The classes of the module ... |
| islininds 42235 | The property of being a li... |
| linindsi 42236 | The implications of being ... |
| linindslinci 42237 | The implications of being ... |
| islinindfis 42238 | The property of being a li... |
| islinindfiss 42239 | The property of being a li... |
| linindscl 42240 | A linearly independent set... |
| lindepsnlininds 42241 | A linearly dependent subse... |
| islindeps 42242 | The property of being a li... |
| lincext1 42243 | Property 1 of an extension... |
| lincext2 42244 | Property 2 of an extension... |
| lincext3 42245 | Property 3 of an extension... |
| lindslinindsimp1 42246 | Implication 1 for ~ lindsl... |
| lindslinindimp2lem1 42247 | Lemma 1 for ~ lindslininds... |
| lindslinindimp2lem2 42248 | Lemma 2 for ~ lindslininds... |
| lindslinindimp2lem3 42249 | Lemma 3 for ~ lindslininds... |
| lindslinindimp2lem4 42250 | Lemma 4 for ~ lindslininds... |
| lindslinindsimp2lem5 42251 | Lemma 5 for ~ lindslininds... |
| lindslinindsimp2 42252 | Implication 2 for ~ lindsl... |
| lindslininds 42253 | Equivalence of definitions... |
| linds0 42254 | The empty set is always a ... |
| el0ldep 42255 | A set containing the zero ... |
| el0ldepsnzr 42256 | A set containing the zero ... |
| lindsrng01 42257 | Any subset of a module is ... |
| lindszr 42258 | Any subset of a module ove... |
| snlindsntorlem 42259 | Lemma for ~ snlindsntor . ... |
| snlindsntor 42260 | A singleton is linearly in... |
| ldepsprlem 42261 | Lemma for ~ ldepspr . (Co... |
| ldepspr 42262 | If a vector is a scalar mu... |
| lincresunit3lem3 42263 | Lemma 3 for ~ lincresunit3... |
| lincresunitlem1 42264 | Lemma 1 for properties of ... |
| lincresunitlem2 42265 | Lemma for properties of a ... |
| lincresunit1 42266 | Property 1 of a specially ... |
| lincresunit2 42267 | Property 2 of a specially ... |
| lincresunit3lem1 42268 | Lemma 1 for ~ lincresunit3... |
| lincresunit3lem2 42269 | Lemma 2 for ~ lincresunit3... |
| lincresunit3 42270 | Property 3 of a specially ... |
| lincreslvec3 42271 | Property 3 of a specially ... |
| islindeps2 42272 | Conditions for being a lin... |
| islininds2 42273 | Implication of being a lin... |
| isldepslvec2 42274 | Alternative definition of ... |
| lindssnlvec 42275 | A singleton not containing... |
| lmod1lem1 42276 | Lemma 1 for ~ lmod1 . (Co... |
| lmod1lem2 42277 | Lemma 2 for ~ lmod1 . (Co... |
| lmod1lem3 42278 | Lemma 3 for ~ lmod1 . (Co... |
| lmod1lem4 42279 | Lemma 4 for ~ lmod1 . (Co... |
| lmod1lem5 42280 | Lemma 5 for ~ lmod1 . (Co... |
| lmod1 42281 | The (smallest) structure r... |
| lmod1zr 42282 | The (smallest) structure r... |
| lmod1zrnlvec 42283 | There is a (left) module (... |
| lmodn0 42284 | Left modules exist. (Cont... |
| zlmodzxzequa 42285 | Example of an equation wit... |
| zlmodzxznm 42286 | Example of a linearly depe... |
| zlmodzxzldeplem 42287 | A and B are not equal. (C... |
| zlmodzxzequap 42288 | Example of an equation wit... |
| zlmodzxzldeplem1 42289 | Lemma 1 for ~ zlmodzxzldep... |
| zlmodzxzldeplem2 42290 | Lemma 2 for ~ zlmodzxzldep... |
| zlmodzxzldeplem3 42291 | Lemma 3 for ~ zlmodzxzldep... |
| zlmodzxzldeplem4 42292 | Lemma 4 for ~ zlmodzxzldep... |
| zlmodzxzldep 42293 | { A , B } is a linearly de... |
| ldepsnlinclem1 42294 | Lemma 1 for ~ ldepsnlinc .... |
| ldepsnlinclem2 42295 | Lemma 2 for ~ ldepsnlinc .... |
| lvecpsslmod 42296 | The class of all (left) ve... |
| ldepsnlinc 42297 | The reverse implication of... |
| ldepslinc 42298 | For (left) vector spaces, ... |
| offval0 42299 | Value of an operation appl... |
| suppdm 42300 | If the range of a function... |
| eluz2cnn0n1 42301 | An integer greater than 1 ... |
| divge1b 42302 | The ratio of a real number... |
| divgt1b 42303 | The ratio of a real number... |
| ltsubaddb 42304 | Equivalence for the "less ... |
| ltsubsubb 42305 | Equivalence for the "less ... |
| ltsubadd2b 42306 | Equivalence for the "less ... |
| divsub1dir 42307 | Distribution of division o... |
| expnegico01 42308 | An integer greater than 1 ... |
| elfzolborelfzop1 42309 | An element of a half-open ... |
| pw2m1lepw2m1 42310 | 2 to the power of a positi... |
| zgtp1leeq 42311 | If an integer is between a... |
| flsubz 42312 | An integer can be moved in... |
| fldivmod 42313 | Expressing the floor of a ... |
| mod0mul 42314 | If an integer is 0 modulo ... |
| modn0mul 42315 | If an integer is not 0 mod... |
| m1modmmod 42316 | An integer decreased by 1 ... |
| difmodm1lt 42317 | The difference between an ... |
| nn0onn0ex 42318 | For each odd nonnegative i... |
| nn0enn0ex 42319 | For each even nonnegative ... |
| nneop 42320 | A positive integer is even... |
| nneom 42321 | A positive integer is even... |
| nn0eo 42322 | A nonnegative integer is e... |
| nnpw2even 42323 | 2 to the power of a positi... |
| zefldiv2 42324 | The floor of an even integ... |
| zofldiv2 42325 | The floor of an odd intege... |
| nn0ofldiv2 42326 | The floor of an odd nonneg... |
| flnn0div2ge 42327 | The floor of a positive in... |
| flnn0ohalf 42328 | The floor of the half of a... |
| logcxp0 42329 | Logarithm of a complex pow... |
| regt1loggt0 42330 | The natural logarithm for ... |
| fdivval 42333 | The quotient of two functi... |
| fdivmpt 42334 | The quotient of two functi... |
| fdivmptf 42335 | The quotient of two functi... |
| refdivmptf 42336 | The quotient of two functi... |
| fdivpm 42337 | The quotient of two functi... |
| refdivpm 42338 | The quotient of two functi... |
| fdivmptfv 42339 | The function value of a qu... |
| refdivmptfv 42340 | The function value of a qu... |
| bigoval 42343 | Set of functions of order ... |
| elbigofrcl 42344 | Reverse closure of the "bi... |
| elbigo 42345 | Properties of a function o... |
| elbigo2 42346 | Properties of a function o... |
| elbigo2r 42347 | Sufficient condition for a... |
| elbigof 42348 | A function of order G(x) i... |
| elbigodm 42349 | The domain of a function o... |
| elbigoimp 42350 | The defining property of a... |
| elbigolo1 42351 | A function (into the posit... |
| rege1logbrege0 42352 | The general logarithm, wit... |
| rege1logbzge0 42353 | The general logarithm, wit... |
| fllogbd 42354 | A real number is between t... |
| relogbmulbexp 42355 | The logarithm of the produ... |
| relogbdivb 42356 | The logarithm of the quoti... |
| logbge0b 42357 | The logarithm of a number ... |
| logblt1b 42358 | The logarithm of a number ... |
| fldivexpfllog2 42359 | The floor of a positive re... |
| nnlog2ge0lt1 42360 | A positive integer is 1 if... |
| logbpw2m1 42361 | The floor of the binary lo... |
| fllog2 42362 | The floor of the binary lo... |
| blenval 42365 | The binary length of an in... |
| blen0 42366 | The binary length of 0. (... |
| blenn0 42367 | The binary length of a "nu... |
| blenre 42368 | The binary length of a pos... |
| blennn 42369 | The binary length of a pos... |
| blennnelnn 42370 | The binary length of a pos... |
| blennn0elnn 42371 | The binary length of a non... |
| blenpw2 42372 | The binary length of a pow... |
| blenpw2m1 42373 | The binary length of a pow... |
| nnpw2blen 42374 | A positive integer is betw... |
| nnpw2blenfzo 42375 | A positive integer is betw... |
| nnpw2blenfzo2 42376 | A positive integer is eith... |
| nnpw2pmod 42377 | Every positive integer can... |
| blen1 42378 | The binary length of 1. (... |
| blen2 42379 | The binary length of 2. (... |
| nnpw2p 42380 | Every positive integer can... |
| nnpw2pb 42381 | A number is a positive int... |
| blen1b 42382 | The binary length of a non... |
| blennnt2 42383 | The binary length of a pos... |
| nnolog2flm1 42384 | The floor of the binary lo... |
| blennn0em1 42385 | The binary length of the h... |
| blennngt2o2 42386 | The binary length of an od... |
| blengt1fldiv2p1 42387 | The binary length of an in... |
| blennn0e2 42388 | The binary length of an ev... |
| digfval 42391 | Operation to obtain the ` ... |
| digval 42392 | The ` K ` th digit of a no... |
| digvalnn0 42393 | The ` K ` th digit of a no... |
| nn0digval 42394 | The ` K ` th digit of a no... |
| dignn0fr 42395 | The digits of the fraction... |
| dignn0ldlem 42396 | Lemma for ~ dignnld . (Co... |
| dignnld 42397 | The leading digits of a po... |
| dig2nn0ld 42398 | The leading digits of a po... |
| dig2nn1st 42399 | The first (relevant) digit... |
| dig0 42400 | All digits of 0 are 0. (C... |
| digexp 42401 | The ` K ` th digit of a po... |
| dig1 42402 | All but one digits of 1 ar... |
| 0dig1 42403 | The ` 0 ` th digit of 1 is... |
| 0dig2pr01 42404 | The integers 0 and 1 corre... |
| dig2nn0 42405 | A digit of a nonnegative i... |
| 0dig2nn0e 42406 | The last bit of an even in... |
| 0dig2nn0o 42407 | The last bit of an odd int... |
| dig2bits 42408 | The ` K ` th digit of a no... |
| dignn0flhalflem1 42409 | Lemma 1 for ~ dignn0flhalf... |
| dignn0flhalflem2 42410 | Lemma 2 for ~ dignn0flhalf... |
| dignn0ehalf 42411 | The digits of the half of ... |
| dignn0flhalf 42412 | The digits of the rounded ... |
| nn0sumshdiglemA 42413 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglemB 42414 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglem1 42415 | Lemma 1 for ~ nn0sumshdig ... |
| nn0sumshdiglem2 42416 | Lemma 2 for ~ nn0sumshdig ... |
| nn0sumshdig 42417 | A nonnegative integer can ... |
| nn0mulfsum 42418 | Trivial algorithm to calcu... |
| nn0mullong 42419 | Standard algorithm (also k... |
| nfintd 42420 | Bound-variable hypothesis ... |
| nfiund 42421 | Bound-variable hypothesis ... |
| iunord 42422 | The indexed union of a col... |
| iunordi 42423 | The indexed union of a col... |
| rspcdf 42424 | Restricted specialization,... |
| spd 42425 | Specialization deduction, ... |
| spcdvw 42426 | A version of ~ spcdv where... |
| tfis2d 42427 | Transfinite Induction Sche... |
| bnd2d 42428 | Deduction form of ~ bnd2 .... |
| dffun3f 42429 | Alternate definition of fu... |
| setrecseq 42432 | Equality theorem for set r... |
| nfsetrecs 42433 | Bound-variable hypothesis ... |
| setrec1lem1 42434 | Lemma for ~ setrec1 . Thi... |
| setrec1lem2 42435 | Lemma for ~ setrec1 . If ... |
| setrec1lem3 42436 | Lemma for ~ setrec1 . If ... |
| setrec1lem4 42437 | Lemma for ~ setrec1 . If ... |
| setrec1 42438 | This is the first of two f... |
| setrec2fun 42439 | This is the second of two ... |
| setrec2lem1 42440 | Lemma for ~ setrec2 . The... |
| setrec2lem2 42441 | Lemma for ~ setrec2 . The... |
| setrec2 42442 | This is the second of two ... |
| setrec2v 42443 | Version of ~ setrec2 with ... |
| elsetrecslem 42444 | Lemma for ~ elsetrecs . A... |
| elsetrecs 42445 | A set ` A ` is an element ... |
| vsetrec 42446 | Construct ` _V ` using set... |
| 0setrec 42447 | If a function sends the em... |
| onsetreclem1 42448 | Lemma for ~ onsetrec . (C... |
| onsetreclem2 42449 | Lemma for ~ onsetrec . (C... |
| onsetreclem3 42450 | Lemma for ~ onsetrec . (C... |
| onsetrec 42451 | Construct ` On ` using set... |
| elpglem1 42454 | Lemma for ~ elpg . (Contr... |
| elpglem2 42455 | Lemma for ~ elpg . (Contr... |
| elpglem3 42456 | Lemma for ~ elpg . (Contr... |
| elpg 42457 | Membership in the class of... |
| 19.8ad 42458 | If a wff is true, it is tr... |
| sbidd 42459 | An identity theorem for su... |
| sbidd-misc 42460 | An identity theorem for su... |
| gte-lte 42465 | Simple relationship betwee... |
| gt-lt 42466 | Simple relationship betwee... |
| gte-lteh 42467 | Relationship between ` <_ ... |
| gt-lth 42468 | Relationship between ` < `... |
| ex-gt 42469 | Simple example of ` > ` , ... |
| ex-gte 42470 | Simple example of ` >_ ` ,... |
| sinhval-named 42477 | Value of the named sinh fu... |
| coshval-named 42478 | Value of the named cosh fu... |
| tanhval-named 42479 | Value of the named tanh fu... |
| sinh-conventional 42480 | Conventional definition of... |
| sinhpcosh 42481 | Prove that ` ( sinh `` A )... |
| secval 42488 | Value of the secant functi... |
| cscval 42489 | Value of the cosecant func... |
| cotval 42490 | Value of the cotangent fun... |
| seccl 42491 | The closure of the secant ... |
| csccl 42492 | The closure of the cosecan... |
| cotcl 42493 | The closure of the cotange... |
| reseccl 42494 | The closure of the secant ... |
| recsccl 42495 | The closure of the cosecan... |
| recotcl 42496 | The closure of the cotange... |
| recsec 42497 | The reciprocal of secant i... |
| reccsc 42498 | The reciprocal of cosecant... |
| reccot 42499 | The reciprocal of cotangen... |
| rectan 42500 | The reciprocal of tangent ... |
| sec0 42501 | The value of the secant fu... |
| onetansqsecsq 42502 | Prove the tangent squared ... |
| cotsqcscsq 42503 | Prove the tangent squared ... |
| ifnmfalse 42504 | If A is not a member of B,... |
| logb2aval 42505 | Define the value of the ` ... |
| comraddi 42512 | Commute RHS addition. See... |
| mvlladdd 42513 | Move LHS left addition to ... |
| mvlraddi 42514 | Move LHS right addition to... |
| mvrladdd 42515 | Move RHS left addition to ... |
| mvrladdi 42516 | Move RHS left addition to ... |
| assraddsubd 42517 | Associate RHS addition-sub... |
| assraddsubi 42518 | Associate RHS addition-sub... |
| joinlmuladdmuli 42519 | Join AB+CB into (A+C) on L... |
| joinlmulsubmuld 42520 | Join AB-CB into (A-C) on L... |
| joinlmulsubmuli 42521 | Join AB-CB into (A-C) on L... |
| mvlrmuld 42522 | Move LHS right multiplicat... |
| mvlrmuli 42523 | Move LHS right multiplicat... |
| i2linesi 42524 | Solve for the intersection... |
| i2linesd 42525 | Solve for the intersection... |
| alimp-surprise 42526 | Demonstrate that when usin... |
| alimp-no-surprise 42527 | There is no "surprise" in ... |
| empty-surprise 42528 | Demonstrate that when usin... |
| empty-surprise2 42529 | "Prove" that false is true... |
| eximp-surprise 42530 | Show what implication insi... |
| eximp-surprise2 42531 | Show that "there exists" w... |
| alsconv 42536 | There is an equivalence be... |
| alsi1d 42537 | Deduction rule: Given "al... |
| alsi2d 42538 | Deduction rule: Given "al... |
| alsc1d 42539 | Deduction rule: Given "al... |
| alsc2d 42540 | Deduction rule: Given "al... |
| alscn0d 42541 | Deduction rule: Given "al... |
| alsi-no-surprise 42542 | Demonstrate that there is ... |
| 5m4e1 42543 | Prove that 5 - 4 = 1. (Co... |
| 2p2ne5 42544 | Prove that ` 2 + 2 =/= 5 `... |
| resolution 42545 | Resolution rule. This is ... |
| testable 42546 | In classical logic all wff... |
| aacllem 42547 | Lemma for other theorems a... |
| amgmwlem 42548 | Weighted version of ~ amgm... |
| amgmlemALT 42549 | Alternate proof of ~ amgml... |
| amgmw2d 42550 | Weighted arithmetic-geomet... |
| young2d 42551 | Young's inequality for ` n... |
| Copyright terms: Public domain | W3C validator |