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Type | Label | Description |
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Statement | ||
Theorem | dvdsmul1 11701 | An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsmul2 11702 | An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | iddvdsexp 11703 | An integer divides a positive integer power of itself. (Contributed by Paul Chapman, 26-Oct-2012.) |
Theorem | muldvds1 11704 | If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | muldvds2 11705 | If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdscmul 11706 | 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 11707 | Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdscmulr 11708 | Cancellation law for the divides relation. Theorem 1.1(e) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsmulcr 11709 | Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | summodnegmod 11710 | 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 11711 | Constant multiplication in a modulo operation, see theorem 5.3 in [ApostolNT] p. 108. (Contributed by AV, 21-Jul-2021.) |
Theorem | dvds2ln 11712 | 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 11713 | If an integer divides each of two other integers, it divides their sum. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvds2sub 11714 | If an integer divides each of two other integers, it divides their difference. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvds2subd 11715 | Deduction form of dvds2sub 11714. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Theorem | dvdstr 11716 | 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 11717 | If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.) |
Theorem | dvdsmultr1d 11718 | Natural deduction form of dvdsmultr1 11717. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Theorem | dvdsmultr2 11719 | If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.) |
Theorem | ordvdsmul 11720 | 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 11721 | 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 11722 | 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 11723 | An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | dvdssub 11724 | An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | dvdssubr 11725 | An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | dvdsadd2b 11726 | Adding a multiple of the base does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
Theorem | dvdslelemd 11727 | Lemma for dvdsle 11728. (Contributed by Jim Kingdon, 8-Nov-2021.) |
Theorem | dvdsle 11728 | The divisors of a positive integer are bounded by it. The proof does not use . (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsleabs 11729 | 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 11730 | Transfer divisibility to an order constraint on absolute values. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
Theorem | dvdsabseq 11731 | 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 11732 | 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 11733 | 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 11734* | 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 11735* | 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 11736* | The set of divisors of a number is a subset of a finite set. (Contributed by Mario Carneiro, 22-Sep-2014.) |
Theorem | dvds1 11737 | The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.) |
Theorem | alzdvds 11738* | Only 0 is divisible by all integers. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | dvdsext 11739* | Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Theorem | fzm1ndvds 11740 | No number between and divides . (Contributed by Mario Carneiro, 24-Jan-2015.) |
Theorem | fzo0dvdseq 11741 | Zero is the only one of the first nonnegative integers that is divisible by . (Contributed by Stefan O'Rear, 6-Sep-2015.) |
..^ | ||
Theorem | fzocongeq 11742 | Two different elements of a half-open range are not congruent mod its length. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
..^ ..^ | ||
Theorem | addmodlteqALT 11743 | 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 10290 based on the "divides" relation. (Contributed by AV, 14-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
..^ ..^ | ||
Theorem | dvdsfac 11744 | A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.) |
Theorem | dvdsexp 11745 | A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | dvdsmod 11746 | 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 11747 | 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 11748 | 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 11749 | 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 11753. 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 11750) and to say that " is odd" (under the assumption that ). The previously proven theorems about even and odd numbers, like zneo 9259, zeo 9263, zeo2 9264, etc. use different representations, which are equivalent with the representations using the divides relation, see evend2 11772 and oddp1d2 11773. The corresponding theorems are zeneo 11754, zeo3 11751 and zeo4 11753. | ||
Theorem | evenelz 11750 | An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 11681. (Contributed by AV, 22-Jun-2021.) |
Theorem | zeo3 11751 | An integer is even or odd. (Contributed by AV, 17-Jun-2021.) |
Theorem | zeoxor 11752 | An integer is even or odd but not both. (Contributed by Jim Kingdon, 10-Nov-2021.) |
Theorem | zeo4 11753 | An integer is even or odd but not both. (Contributed by AV, 17-Jun-2021.) |
Theorem | zeneo 11754 | 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 9259 follows immediately from the fact that a contradiction implies anything, see pm2.21i 636. (Contributed by AV, 22-Jun-2021.) |
Theorem | odd2np1lem 11755* | Lemma for odd2np1 11756. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | odd2np1 11756* | 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 11757* | An integer is even iff it is twice another integer. (Contributed by AV, 25-Jun-2020.) |
Theorem | oddm1even 11758 | An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) |
Theorem | oddp1even 11759 | An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) |
Theorem | oexpneg 11760 | The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.) |
Theorem | mod2eq0even 11761 | 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 | mod2eq1n2dvds 11762 | 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 11763* | A nonnegative integer is odd iff it is one plus twice another nonnegative integer. (Contributed by AV, 19-Jun-2021.) |
Theorem | oddge22np1 11764* | An integer greater than one is odd iff it is one plus twice a positive integer. (Contributed by AV, 16-Aug-2021.) |
Theorem | evennn02n 11765* | A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.) |
Theorem | evennn2n 11766* | A positive integer is even iff it is twice another positive integer. (Contributed by AV, 12-Aug-2021.) |
Theorem | 2tp1odd 11767 | A number which is twice an integer increased by 1 is odd. (Contributed by AV, 16-Jul-2021.) |
Theorem | mulsucdiv2z 11768 | 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 11769 | A squared odd number minus 1 divided by 8 is an integer. (Contributed by AV, 19-Jul-2021.) |
Theorem | 2teven 11770 | A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021.) |
Theorem | zeo5 11771 | An integer is either even or odd, version of zeo3 11751 avoiding the negation of the representation of an odd number. (Proposed by BJ, 21-Jun-2021.) (Contributed by AV, 26-Jun-2020.) |
Theorem | evend2 11772 | 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 9263 and zeo2 9264. (Contributed by AV, 22-Jun-2021.) |
Theorem | oddp1d2 11773 | 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 9263 and zeo2 9264. (Contributed by AV, 22-Jun-2021.) |
Theorem | zob 11774 | Alternate characterizations of an odd number. (Contributed by AV, 7-Jun-2020.) |
Theorem | oddm1d2 11775 | 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 11776 | 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 11777 | 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 11778 | The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | omoe 11779 | The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | opeo 11780 | 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 11781 | 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 11782 | Exponentiation of -1 by an even power. Variant of m1expeven 10459. (Contributed by AV, 25-Jun-2021.) |
Theorem | m1expo 11783 | Exponentiation of -1 by an odd power. (Contributed by AV, 26-Jun-2021.) |
Theorem | m1exp1 11784 | Exponentiation of negative one is one iff the exponent is even. (Contributed by AV, 20-Jun-2021.) |
Theorem | nn0enne 11785 | A positive integer is an even nonnegative integer iff it is an even positive integer. (Contributed by AV, 30-May-2020.) |
Theorem | nn0ehalf 11786 | The half of an even nonnegative integer is a nonnegative integer. (Contributed by AV, 22-Jun-2020.) (Revised by AV, 28-Jun-2021.) |
Theorem | nnehalf 11787 | The half of an even positive integer is a positive integer. (Contributed by AV, 28-Jun-2021.) |
Theorem | nn0o1gt2 11788 | An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.) |
Theorem | nno 11789 | An alternate characterization of an odd integer greater than 1. (Contributed by AV, 2-Jun-2020.) |
Theorem | nn0o 11790 | An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Proof shortened by AV, 2-Jun-2020.) |
Theorem | nn0ob 11791 | Alternate characterizations of an odd nonnegative integer. (Contributed by AV, 4-Jun-2020.) |
Theorem | nn0oddm1d2 11792 | A positive integer is odd iff its predecessor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.) |
Theorem | nnoddm1d2 11793 | A positive integer is odd iff its successor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.) |
Theorem | z0even 11794 | 0 is even. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 23-Jun-2021.) |
Theorem | n2dvds1 11795 | 2 does not divide 1 (common case). That means 1 is odd. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | n2dvdsm1 11796 | 2 does not divide -1. That means -1 is odd. (Contributed by AV, 15-Aug-2021.) |
Theorem | z2even 11797 | 2 is even. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 23-Jun-2021.) |
Theorem | n2dvds3 11798 | 2 does not divide 3, i.e. 3 is an odd number. (Contributed by AV, 28-Feb-2021.) |
Theorem | z4even 11799 | 4 is an even number. (Contributed by AV, 23-Jul-2020.) (Revised by AV, 4-Jul-2021.) |
Theorem | 4dvdseven 11800 | An integer which is divisible by 4 is an even integer. (Contributed by AV, 4-Jul-2021.) |
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