P. 117 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 11601-11700 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	6p5e11OLD 11601	Obsolete proof of 6p5e11 11600 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
;

Theorem	6p6e12 11602	6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7p3e10 11603	7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem	7p3e10bOLD 11604	Obsolete proof of 7p3e10 11603 as of 6-Sep-2021. (Contributed by Stanislas Polu, 7-Apr-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
;

Theorem	7p4e11 11605	7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	7p4e11OLD 11606	Obsolete proof of 7p4e11 11605 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
;

Theorem	7p5e12 11607	7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7p6e13 11608	7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7p7e14 11609	7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p2e10 11610	8 + 2 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem	8p2e10bOLD 11611	Obsolete proof of 8p2e10 11610 as of 6-Sep-2021. (Contributed by Stanislas Polu, 7-Apr-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
;

Theorem	8p3e11 11612	8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	8p3e11OLD 11613	Obsolete proof of 8p3e11 11612 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
;

Theorem	8p4e12 11614	8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p5e13 11615	8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p6e14 11616	8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p7e15 11617	8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8p8e16 11618	8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p2e11 11619	9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	9p2e11OLD 11620	Obsolete proof of 9p2e11 11619 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
;

Theorem	9p3e12 11621	9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p4e13 11622	9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p5e14 11623	9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p6e15 11624	9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p7e16 11625	9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p8e17 11626	9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9p9e18 11627	9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	10p10e20 11628	10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ; ;

Theorem	10p10e20OLD 11629	Obsolete version of 10p10e20 11628 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
;

Theorem	10m1e9 11630	10 - 1 = 9. (Contributed by AV, 6-Sep-2021.)
;

Theorem	4t3lem 11631	Lemma for 4t3e12 11632 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)


Theorem	4t3e12 11632	4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	4t4e16 11633	4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	5t2e10 11634	5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.)
;

Theorem	5t3e15 11635	5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	5t3e15OLD 11636	Obsolete proof of 5t3e15 11635 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
;

Theorem	5t4e20 11637	5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	5t4e20OLD 11638	Obsolete proof of 5t4e20 11637 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
;

Theorem	5t5e25 11639	5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	5t5e25OLD 11640	Obsolete proof of 5t5e25 11639 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
;

Theorem	6t2e12 11641	6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t3e18 11642	6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t4e24 11643	6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	6t5e30 11644	6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6t5e30OLD 11645	Obsolete proof of 6t5e30 11644 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
;

Theorem	6t6e36 11646	6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	6t6e36OLD 11647	Obsolete proof of 6t6e36 11646 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
;

Theorem	7t2e14 11648	7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t3e21 11649	7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t4e28 11650	7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t5e35 11651	7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t6e42 11652	7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	7t7e49 11653	7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t2e16 11654	8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t3e24 11655	8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t4e32 11656	8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t5e40 11657	8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	8t5e40OLD 11658	Obsolete proof of 8t5e40 11657 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
;

Theorem	8t6e48 11659	8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	8t6e48OLD 11660	Obsolete proof of 8t6e48 11659 as of 6-Sep-2021. (Contributed by Mario Carneiro, 19-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
;

Theorem	8t7e56 11661	8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	8t8e64 11662	8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t2e18 11663	9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t3e27 11664	9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t4e36 11665	9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t5e45 11666	9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t6e54 11667	9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t7e63 11668	9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t8e72 11669	9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t9e81 11670	9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	9t11e99 11671	9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	9t11e99OLD 11672	Obsolete proof of 9t11e99 11671 as of 6-Sep-2021. (Contributed by AV, 14-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
; ;

Theorem	9lt10 11673	9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	8lt10 11674	8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	7lt10 11675	7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	6lt10 11676	6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	5lt10 11677	5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	4lt10 11678	4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	3lt10 11679	3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	2lt10 11680	2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	1lt10 11681	1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	decbin0 11682	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	decbin2 11683	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	decbin3 11684	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	halfthird 11685	Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.)


Theorem	5recm6rec 11686	One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.)
;

5.4.11 Upper sets of integers

Syntax	cuz 11687	Extend class notation with the upper integer function. Read " " as "the set of integers greater than or equal to ."


Definition	df-uz 11688*	Define a function whose value at is the semi-infinite set of contiguous integers starting at , which we will also call the upper integers starting at . Read " " as "the set of integers greater than or equal to ." See uzval 11689 for its value, uzssz 11707 for its relationship to , nnuz 11723 and nn0uz 11722 for its relationships to and , and eluz1 11691 and eluz2 11693 for its membership relations. (Contributed by NM, 5-Sep-2005.)


Theorem	uzval 11689*	The value of the upper integers function. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)


Theorem	uzf 11690	The domain and range of the upper integers function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 3-Nov-2013.)


Theorem	eluz1 11691	Membership in the upper set of integers starting at . (Contributed by NM, 5-Sep-2005.)
$/\$

Theorem	eluzel2 11692	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)


Theorem	eluz2 11693	Membership in an upper set of integers. We use the fact that a function's value (under our function value definition) is empty outside of its domain to show . (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
$/\$ $/\$

Theorem	eluzmn 11694	Membership in an earlier upper set of integers. (Contributed by Thierry Arnoux, 8-Oct-2018.)
$/\$

Theorem	eluz1i 11695	Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)
$/\$

Theorem	eluzuzle 11696	An integer in an upper set of integers is an element of an upper set of integers with a smaller bound. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
$/\$

Theorem	eluzelz 11697	A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.)


Theorem	eluzelre 11698	A member of an upper set of integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)


Theorem	eluzelcn 11699	A member of an upper set of integers is a complex number. (Contributed by Glauco Siliprandi, 29-Jun-2017.)


Theorem	eluzle 11700	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42551

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