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Type | Label | Description |
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Statement | ||
Theorem | nndivdvds 10201 | Strong form of dvdsval2 10198 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
Theorem | nndivides 10202* | Definition of the divides relation for positive integers. (Contributed by AV, 26-Jul-2021.) |
Theorem | dvdsdc 10203 | Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.) |
DECID | ||
Theorem | moddvds 10204 | Two ways to say (mod ), see also definition in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | dvds0lem 10205 | A lemma to assist theorems of with no antecedents. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvds1lem 10206* | A lemma to assist theorems of with one antecedent. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvds2lem 10207* | A lemma to assist theorems of with two antecedents. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | iddvds 10208 | An integer divides itself. Theorem 1.1(a) in [ApostolNT] p. 14 (reflexive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | 1dvds 10209 | 1 divides any integer. Theorem 1.1(f) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvds0 10210 | Any integer divides 0. Theorem 1.1(g) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | negdvdsb 10211 | An integer divides another iff its negation does. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsnegb 10212 | An integer divides another iff it divides its negation. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | absdvdsb 10213 | An integer divides another iff its absolute value does. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsabsb 10214 | An integer divides another iff it divides its absolute value. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | 0dvds 10215 | Only 0 is divisible by 0. Theorem 1.1(h) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | zdvdsdc 10216 | Divisibility of integers is decidable. (Contributed by Jim Kingdon, 17-Jan-2022.) |
DECID | ||
Theorem | dvdsmul1 10217 | An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsmul2 10218 | An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | iddvdsexp 10219 | An integer divides a positive integer power of itself. (Contributed by Paul Chapman, 26-Oct-2012.) |
Theorem | muldvds1 10220 | If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | muldvds2 10221 | If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdscmul 10222 | Multiplication by a constant maintains the divides relation. Theorem 1.1(d) in [ApostolNT] p. 14 (multiplication property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsmulc 10223 | Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdscmulr 10224 | Cancellation law for the divides relation. Theorem 1.1(e) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsmulcr 10225 | Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | summodnegmod 10226 | The sum of two integers modulo a positive integer equals zero iff the first of the two integers equals the negative of the other integer modulo the positive integer. (Contributed by AV, 25-Jul-2021.) |
Theorem | modmulconst 10227 | Constant multiplication in a modulo operation, see theorem 5.3 in [ApostolNT] p. 108. (Contributed by AV, 21-Jul-2021.) |
Theorem | dvds2ln 10228 | If an integer divides each of two other integers, it divides any linear combination of them. Theorem 1.1(c) in [ApostolNT] p. 14 (linearity property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvds2add 10229 | If an integer divides each of two other integers, it divides their sum. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvds2sub 10230 | If an integer divides each of two other integers, it divides their difference. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvds2subd 10231 | Natural deduction form of dvds2sub 10230. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Theorem | dvdstr 10232 | The divides relation is transitive. Theorem 1.1(b) in [ApostolNT] p. 14 (transitive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsmultr1 10233 | If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.) |
Theorem | dvdsmultr1d 10234 | Natural deduction form of dvdsmultr1 10233. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Theorem | dvdsmultr2 10235 | If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.) |
Theorem | ordvdsmul 10236 | If an integer divides either of two others, it divides their product. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
Theorem | dvdssub2 10237 | If an integer divides a difference, then it divides one term iff it divides the other. (Contributed by Mario Carneiro, 13-Jul-2014.) |
Theorem | dvdsadd 10238 | An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 13-Jul-2014.) |
Theorem | dvdsaddr 10239 | An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | dvdssub 10240 | An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | dvdssubr 10241 | An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | dvdsadd2b 10242 | Adding a multiple of the base does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
Theorem | dvdslelemd 10243 | Lemma for dvdsle 10244. (Contributed by Jim Kingdon, 8-Nov-2021.) |
Theorem | dvdsle 10244 | The divisors of a positive integer are bounded by it. The proof does not use . (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsleabs 10245 | The divisors of a nonzero integer are bounded by its absolute value. Theorem 1.1(i) in [ApostolNT] p. 14 (comparison property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.) |
Theorem | dvdsleabs2 10246 | Transfer divisibility to an order constraint on absolute values. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
Theorem | dvdsabseq 10247 | If two integers divide each other, they must be equal, up to a difference in sign. Theorem 1.1(j) in [ApostolNT] p. 14. (Contributed by Mario Carneiro, 30-May-2014.) (Revised by AV, 7-Aug-2021.) |
Theorem | dvdseq 10248 | If two nonnegative integers divide each other, they must be equal. (Contributed by Mario Carneiro, 30-May-2014.) (Proof shortened by AV, 7-Aug-2021.) |
Theorem | divconjdvds 10249 | If a nonzero integer divides another integer , the other integer divided by the nonzero integer (i.e. the divisor conjugate of to ) divides the other integer . Theorem 1.1(k) in [ApostolNT] p. 14. (Contributed by AV, 7-Aug-2021.) |
Theorem | dvdsdivcl 10250* | The complement of a divisor of is also a divisor of . (Contributed by Mario Carneiro, 2-Jul-2015.) (Proof shortened by AV, 9-Aug-2021.) |
Theorem | dvdsflip 10251* | An involution of the divisors of a number. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 13-May-2016.) |
Theorem | dvdsssfz1 10252* | The set of divisors of a number is a subset of a finite set. (Contributed by Mario Carneiro, 22-Sep-2014.) |
Theorem | dvds1 10253 | The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.) |
Theorem | alzdvds 10254* | Only 0 is divisible by all integers. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsext 10255* | Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Theorem | fzm1ndvds 10256 | No number between and divides . (Contributed by Mario Carneiro, 24-Jan-2015.) |
Theorem | fzo0dvdseq 10257 | Zero is the only one of the first nonnegative integers that is divisible by . (Contributed by Stefan O'Rear, 6-Sep-2015.) |
..^ | ||
Theorem | fzocongeq 10258 | Two different elements of a half-open range are not congruent mod its length. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
..^ ..^ | ||
Theorem | addmodlteqALT 10259 | Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. Shorter proof of addmodlteq 9400 based on the "divides" relation. (Contributed by AV, 14-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
..^ ..^ | ||
Theorem | dvdsfac 10260 | A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.) |
Theorem | dvdsexp 10261 | A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | dvdsmod 10262 | Any number whose mod base is divisible by a divisor of the base is also divisible by . This means that primes will also be relatively prime to the base when reduced for any base. (Contributed by Mario Carneiro, 13-Mar-2014.) |
Theorem | mulmoddvds 10263 | If an integer is divisible by a positive integer, the product of this integer with another integer modulo the positive integer is 0. (Contributed by Alexander van der Vekens, 30-Aug-2018.) |
Theorem | 3dvdsdec 10264 | A decimal number is divisible by three iff the sum of its two "digits" is divisible by three. The term "digits" in its narrow sense is only correct if and actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers and , especially if is itself a decimal number, e.g. ;. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 8-Sep-2021.) |
; | ||
Theorem | 3dvds2dec 10265 | A decimal number is divisible by three iff the sum of its three "digits" is divisible by three. The term "digits" in its narrow sense is only correct if , and actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers , and . (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.) |
;; | ||
The set of integers can be partitioned into the set of even numbers and the set of odd numbers, see zeo4 10269. Instead of defining new class variables Even and Odd to represent these sets, we use the idiom to say that " is even" (which implies , see evenelz 10266) and to say that " is odd" (under the assumption that ). The previously proven theorems about even and odd numbers, like zneo 8448, zeo 8452, zeo2 8453, etc. use different representations, which are equivalent with the representations using the divides relation, see evend2 10289 and oddp1d2 10290. The corresponding theorems are zeneo 10270, zeo3 10267 and zeo4 10269. | ||
Theorem | evenelz 10266 | An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 10200. (Contributed by AV, 22-Jun-2021.) |
Theorem | zeo3 10267 | An integer is even or odd. (Contributed by AV, 17-Jun-2021.) |
Theorem | zeoxor 10268 | An integer is even or odd but not both. (Contributed by Jim Kingdon, 10-Nov-2021.) |
Theorem | zeo4 10269 | An integer is even or odd but not both. (Contributed by AV, 17-Jun-2021.) |
Theorem | zeneo 10270 | No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. This variant of zneo 8448 follows immediately from the fact that a contradiction implies anything, see pm2.21i 607. (Contributed by AV, 22-Jun-2021.) |
Theorem | odd2np1lem 10271* | Lemma for odd2np1 10272. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | odd2np1 10272* | An integer is odd iff it is one plus twice another integer. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | even2n 10273* | An integer is even iff it is twice another integer. (Contributed by AV, 25-Jun-2020.) |
Theorem | oddm1even 10274 | An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) |
Theorem | oddp1even 10275 | An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) |
Theorem | oexpneg 10276 | The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.) |
Theorem | mod2eq0even 10277 | An integer is 0 modulo 2 iff it is even (i.e. divisible by 2), see example 2 in [ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.) |
Theorem | fz01or 10278 | An integer is in the integer range from zero to one iff it is either zero or one. (Contributed by Jim Kingdon, 11-Nov-2021.) |
Theorem | mod2eq1n2dvds 10279 | An integer is 1 modulo 2 iff it is odd (i.e. not divisible by 2), see example 3 in [ApostolNT] p. 107. (Contributed by AV, 24-May-2020.) |
Theorem | oddnn02np1 10280* | A nonnegative integer is odd iff it is one plus twice another nonnegative integer. (Contributed by AV, 19-Jun-2021.) |
Theorem | oddge22np1 10281* | An integer greater than one is odd iff it is one plus twice a positive integer. (Contributed by AV, 16-Aug-2021.) |
Theorem | evennn02n 10282* | A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.) |
Theorem | evennn2n 10283* | A positive integer is even iff it is twice another positive integer. (Contributed by AV, 12-Aug-2021.) |
Theorem | 2tp1odd 10284 | A number which is twice an integer increased by 1 is odd. (Contributed by AV, 16-Jul-2021.) |
Theorem | mulsucdiv2z 10285 | An integer multiplied with its successor divided by 2 yields an integer, i.e. an integer multiplied with its successor is even. (Contributed by AV, 19-Jul-2021.) |
Theorem | sqoddm1div8z 10286 | A squared odd number minus 1 divided by 8 is an integer. (Contributed by AV, 19-Jul-2021.) |
Theorem | 2teven 10287 | A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021.) |
Theorem | zeo5 10288 | An integer is either even or odd, version of zeo3 10267 avoiding the negation of the representation of an odd number. (Proposed by BJ, 21-Jun-2021.) (Contributed by AV, 26-Jun-2020.) |
Theorem | evend2 10289 | An integer is even iff its quotient with 2 is an integer. This is a representation of even numbers without using the divides relation, see zeo 8452 and zeo2 8453. (Contributed by AV, 22-Jun-2021.) |
Theorem | oddp1d2 10290 | An integer is odd iff its successor divided by 2 is an integer. This is a representation of odd numbers without using the divides relation, see zeo 8452 and zeo2 8453. (Contributed by AV, 22-Jun-2021.) |
Theorem | zob 10291 | Alternate characterizations of an odd number. (Contributed by AV, 7-Jun-2020.) |
Theorem | oddm1d2 10292 | An integer is odd iff its predecessor divided by 2 is an integer. This is another representation of odd numbers without using the divides relation. (Contributed by AV, 18-Jun-2021.) (Proof shortened by AV, 22-Jun-2021.) |
Theorem | ltoddhalfle 10293 | An integer is less than half of an odd number iff it is less than or equal to the half of the predecessor of the odd number (which is an even number). (Contributed by AV, 29-Jun-2021.) |
Theorem | halfleoddlt 10294 | An integer is greater than half of an odd number iff it is greater than or equal to the half of the odd number. (Contributed by AV, 1-Jul-2021.) |
Theorem | opoe 10295 | The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | omoe 10296 | The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | opeo 10297 | The sum of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | omeo 10298 | The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | m1expe 10299 | Exponentiation of -1 by an even power. Variant of m1expeven 9523. (Contributed by AV, 25-Jun-2021.) |
Theorem | m1expo 10300 | Exponentiation of -1 by an odd power. (Contributed by AV, 26-Jun-2021.) |
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