P. 121 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12001-12100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dvdsadd 12001	An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 13-Jul-2014.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> M || ( M + N ) ) )
Theorem	dvdsaddr 12002	An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> M || ( N + M ) ) )
Theorem	dvdssub 12003	An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> M || ( M - N ) ) )
Theorem	dvdssubr 12004	An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> M || ( N - M ) ) )
Theorem	dvdsadd2b 12005	Adding a multiple of the base does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.)
|- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> ( A || B <-> A || ( C + B ) ) )
Theorem	dvdsaddre2b 12006	Adding a multiple of the base does not affect divisibility. Variant of dvdsadd2b 12005 only requiring B to be a real number (not necessarily an integer). (Contributed by AV, 19-Jul-2021.)
|- ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) -> ( A || B <-> A || ( C + B ) ) )
Theorem	fsumdvds 12007*	If every term in a sum is divisible by N, then so is the sum. (Contributed by Mario Carneiro, 17-Jan-2015.)
|- ( ph -> A e. Fin )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ k e. A ) -> B e. ZZ )   &    |- ( ( ph /\ k e. A ) -> N || B )   =>    |- ( ph -> N || sum_ k e. A B )
Theorem	dvdslelemd 12008	Lemma for dvdsle 12009. (Contributed by Jim Kingdon, 8-Nov-2021.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. NN )   &    |- ( ph -> K e. ZZ )   &    |- ( ph -> N < M )   =>    |- ( ph -> ( K x. M ) =/= N )
Theorem	dvdsle 12009	The divisors of a positive integer are bounded by it. The proof does not use /. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. NN ) -> ( M || N -> M <_ N ) )
Theorem	dvdsleabs 12010	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.)
|- ( ( M e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( M || N -> M <_ ( abs ` N ) ) )
Theorem	dvdsleabs2 12011	Transfer divisibility to an order constraint on absolute values. (Contributed by Stefan O'Rear, 24-Sep-2014.)
|- ( ( M e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( M || N -> ( abs ` M ) <_ ( abs ` N ) ) )
Theorem	dvdsabseq 12012	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.)
|- ( ( M || N /\ N || M ) -> ( abs ` M ) = ( abs ` N ) )
Theorem	dvdseq 12013	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.)
|- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( M || N /\ N || M ) ) -> M = N )
Theorem	divconjdvds 12014	If a nonzero integer M divides another integer N, the other integer N divided by the nonzero integer M (i.e. the divisor conjugate of N to M) divides the other integer N. Theorem 1.1(k) in [ApostolNT] p. 14. (Contributed by AV, 7-Aug-2021.)
|- ( ( M || N /\ M =/= 0 ) -> ( N / M ) || N )
Theorem	dvdsdivcl 12015*	The complement of a divisor of N is also a divisor of N. (Contributed by Mario Carneiro, 2-Jul-2015.) (Proof shortened by AV, 9-Aug-2021.)
|- ( ( N e. NN /\ A e. { x e. NN | x || N } ) -> ( N / A ) e. { x e. NN | x || N } )
Theorem	dvdsflip 12016*	An involution of the divisors of a number. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 13-May-2016.)
|- A = { x e. NN | x || N }   &    |- F = ( y e. A |-> ( N / y ) )   =>    |- ( N e. NN -> F : A -1-1-onto-> A )
Theorem	dvdsssfz1 12017*	The set of divisors of a number is a subset of a finite set. (Contributed by Mario Carneiro, 22-Sep-2014.)
|- ( A e. NN -> { p e. NN | p || A } C_ ( 1 ... A ) )
Theorem	dvds1 12018	The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.)
|- ( M e. NN0 -> ( M || 1 <-> M = 1 ) )
Theorem	alzdvds 12019*	Only 0 is divisible by all integers. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( N e. ZZ -> ( A. x e. ZZ x || N <-> N = 0 ) )
Theorem	dvdsext 12020*	Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.)
|- ( ( A e. NN0 /\ B e. NN0 ) -> ( A = B <-> A. x e. NN0 ( A || x <-> B || x ) ) )
Theorem	fzm1ndvds 12021	No number between 1 and M - 1 divides M. (Contributed by Mario Carneiro, 24-Jan-2015.)
|- ( ( M e. NN /\ N e. ( 1 ... ( M - 1 ) ) ) -> -. M || N )
Theorem	fzo0dvdseq 12022	Zero is the only one of the first A nonnegative integers that is divisible by A. (Contributed by Stefan O'Rear, 6-Sep-2015.)
|- ( B e. ( 0..^ A ) -> ( A || B <-> B = 0 ) )
Theorem	fzocongeq 12023	Two different elements of a half-open range are not congruent mod its length. (Contributed by Stefan O'Rear, 6-Sep-2015.)
|- ( ( A e. ( C..^ D ) /\ B e. ( C..^ D ) ) -> ( ( D - C ) || ( A - B ) <-> A = B ) )
Theorem	addmodlteqALT 12024	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 10490 based on the "divides" relation. (Contributed by AV, 14-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( ( I e. ( 0..^ N ) /\ J e. ( 0..^ N ) /\ S e. ZZ ) -> ( ( ( I + S ) mod N ) = ( ( J + S ) mod N ) <-> I = J ) )
Theorem	dvdsfac 12025	A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.)
|- ( ( K e. NN /\ N e. ( ZZ>= ` K ) ) -> K || ( ! ` N ) )
Theorem	dvdsexp 12026	A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.)
|- ( ( A e. ZZ /\ M e. NN0 /\ N e. ( ZZ>= ` M ) ) -> ( A ^ M ) || ( A ^ N ) )
Theorem	dvdsmod 12027	Any number K whose mod base N is divisible by a divisor P of the base is also divisible by P. This means that primes will also be relatively prime to the base when reduced mod N for any base. (Contributed by Mario Carneiro, 13-Mar-2014.)
|- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( P || ( K mod N ) <-> P || K ) )
Theorem	mulmoddvds 12028	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.)
|- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> ( N || A -> ( ( A x. B ) mod N ) = 0 ) )
Theorem	3dvds 12029*	A rule for divisibility by 3 of a number written in base 10. This is Metamath 100 proof #85. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 17-Jan-2015.) (Revised by AV, 8-Sep-2021.)
|- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> ( 3 || sum_ k e. ( 0 ... N ) ( ( F ` k ) x. (; 1 0 ^ k ) ) <-> 3 || sum_ k e. ( 0 ... N ) ( F ` k ) ) )
Theorem	3dvdsdec 12030	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 A and B actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers A and B, especially if A is itself a decimal number, e.g., A = ; C D. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 8-Sep-2021.)
|- A e. NN0   &    |- B e. NN0   =>    |- ( 3 || ; A B <-> 3 || ( A + B ) )
Theorem	3dvds2dec 12031	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 A, B and C actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers A, B and C. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
|- A e. NN0   &    |- B e. NN0   &    |- C e. NN0   =>    |- ( 3 || ;; A B C <-> 3 || ( ( A + B ) + C ) )
5.1.2  Even and odd numbers The set ZZ of integers can be partitioned into the set of even numbers and the set of odd numbers, see zeo4 12035. Instead of defining new class variables Even and Odd to represent these sets, we use the idiom 2 || N to say that " N is even" (which implies N e. ZZ, see evenelz 12032) and -. 2 || N to say that " N is odd" (under the assumption that N e. ZZ). The previously proven theorems about even and odd numbers, like zneo 9427, zeo 9431, zeo2 9432, etc. use different representations, which are equivalent with the representations using the divides relation, see evend2 12054 and oddp1d2 12055. The corresponding theorems are zeneo 12036, zeo3 12033 and zeo4 12035.
Theorem	evenelz 12032	An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 11957. (Contributed by AV, 22-Jun-2021.)
|- ( 2 || N -> N e. ZZ )
Theorem	zeo3 12033	An integer is even or odd. (Contributed by AV, 17-Jun-2021.)
|- ( N e. ZZ -> ( 2 || N \/ -. 2 || N ) )
Theorem	zeoxor 12034	An integer is even or odd but not both. (Contributed by Jim Kingdon, 10-Nov-2021.)
|- ( N e. ZZ -> ( 2 || N \/_ -. 2 || N ) )
Theorem	zeo4 12035	An integer is even or odd but not both. (Contributed by AV, 17-Jun-2021.)
|- ( N e. ZZ -> ( 2 || N <-> -. -. 2 || N ) )
Theorem	zeneo 12036	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 9427 follows immediately from the fact that a contradiction implies anything, see pm2.21i 647. (Contributed by AV, 22-Jun-2021.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( 2 || A /\ -. 2 || B ) -> A =/= B ) )
Theorem	odd2np1lem 12037*	Lemma for odd2np1 12038. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( N e. NN0 -> ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N \/ E. k e. ZZ ( k x. 2 ) = N ) )
Theorem	odd2np1 12038*	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.)
|- ( N e. ZZ -> ( -. 2 || N <-> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
Theorem	even2n 12039*	An integer is even iff it is twice another integer. (Contributed by AV, 25-Jun-2020.)
|- ( 2 || N <-> E. n e. ZZ ( 2 x. n ) = N )
Theorem	oddm1even 12040	An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( N e. ZZ -> ( -. 2 || N <-> 2 || ( N - 1 ) ) )
Theorem	oddp1even 12041	An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( N e. ZZ -> ( -. 2 || N <-> 2 || ( N + 1 ) ) )
Theorem	oexpneg 12042	The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.)
|- ( ( A e. CC /\ N e. NN /\ -. 2 || N ) -> ( -u A ^ N ) = -u ( A ^ N ) )
Theorem	mod2eq0even 12043	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.)
|- ( N e. ZZ -> ( ( N mod 2 ) = 0 <-> 2 || N ) )
Theorem	mod2eq1n2dvds 12044	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.)
|- ( N e. ZZ -> ( ( N mod 2 ) = 1 <-> -. 2 || N ) )
Theorem	oddnn02np1 12045*	A nonnegative integer is odd iff it is one plus twice another nonnegative integer. (Contributed by AV, 19-Jun-2021.)
|- ( N e. NN0 -> ( -. 2 || N <-> E. n e. NN0 ( ( 2 x. n ) + 1 ) = N ) )
Theorem	oddge22np1 12046*	An integer greater than one is odd iff it is one plus twice a positive integer. (Contributed by AV, 16-Aug-2021.)
|- ( N e. ( ZZ>= ` 2 ) -> ( -. 2 || N <-> E. n e. NN ( ( 2 x. n ) + 1 ) = N ) )
Theorem	evennn02n 12047*	A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.)
|- ( N e. NN0 -> ( 2 || N <-> E. n e. NN0 ( 2 x. n ) = N ) )
Theorem	evennn2n 12048*	A positive integer is even iff it is twice another positive integer. (Contributed by AV, 12-Aug-2021.)
|- ( N e. NN -> ( 2 || N <-> E. n e. NN ( 2 x. n ) = N ) )
Theorem	2tp1odd 12049	A number which is twice an integer increased by 1 is odd. (Contributed by AV, 16-Jul-2021.)
|- ( ( A e. ZZ /\ B = ( ( 2 x. A ) + 1 ) ) -> -. 2 || B )
Theorem	mulsucdiv2z 12050	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.)
|- ( N e. ZZ -> ( ( N x. ( N + 1 ) ) / 2 ) e. ZZ )
Theorem	sqoddm1div8z 12051	A squared odd number minus 1 divided by 8 is an integer. (Contributed by AV, 19-Jul-2021.)
|- ( ( N e. ZZ /\ -. 2 || N ) -> ( ( ( N ^ 2 ) - 1 ) / 8 ) e. ZZ )
Theorem	2teven 12052	A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021.)
|- ( ( A e. ZZ /\ B = ( 2 x. A ) ) -> 2 || B )
Theorem	zeo5 12053	An integer is either even or odd, version of zeo3 12033 avoiding the negation of the representation of an odd number. (Proposed by BJ, 21-Jun-2021.) (Contributed by AV, 26-Jun-2020.)
|- ( N e. ZZ -> ( 2 || N \/ 2 || ( N + 1 ) ) )
Theorem	evend2 12054	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 9431 and zeo2 9432. (Contributed by AV, 22-Jun-2021.)
|- ( N e. ZZ -> ( 2 || N <-> ( N / 2 ) e. ZZ ) )
Theorem	oddp1d2 12055	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 9431 and zeo2 9432. (Contributed by AV, 22-Jun-2021.)
|- ( N e. ZZ -> ( -. 2 || N <-> ( ( N + 1 ) / 2 ) e. ZZ ) )
Theorem	zob 12056	Alternate characterizations of an odd number. (Contributed by AV, 7-Jun-2020.)
|- ( N e. ZZ -> ( ( ( N + 1 ) / 2 ) e. ZZ <-> ( ( N - 1 ) / 2 ) e. ZZ ) )
Theorem	oddm1d2 12057	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.)
|- ( N e. ZZ -> ( -. 2 || N <-> ( ( N - 1 ) / 2 ) e. ZZ ) )
Theorem	ltoddhalfle 12058	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.)
|- ( ( N e. ZZ /\ -. 2 || N /\ M e. ZZ ) -> ( M < ( N / 2 ) <-> M <_ ( ( N - 1 ) / 2 ) ) )
Theorem	halfleoddlt 12059	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.)
|- ( ( N e. ZZ /\ -. 2 || N /\ M e. ZZ ) -> ( ( N / 2 ) <_ M <-> ( N / 2 ) < M ) )
Theorem	opoe 12060	The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ -. 2 || B ) ) -> 2 || ( A + B ) )
Theorem	omoe 12061	The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ -. 2 || B ) ) -> 2 || ( A - B ) )
Theorem	opeo 12062	The sum of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ 2 || B ) ) -> -. 2 || ( A + B ) )
Theorem	omeo 12063	The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ 2 || B ) ) -> -. 2 || ( A - B ) )
Theorem	m1expe 12064	Exponentiation of -1 by an even power. Variant of m1expeven 10678. (Contributed by AV, 25-Jun-2021.)
|- ( 2 || N -> ( -u 1 ^ N ) = 1 )
Theorem	m1expo 12065	Exponentiation of -1 by an odd power. (Contributed by AV, 26-Jun-2021.)
|- ( ( N e. ZZ /\ -. 2 || N ) -> ( -u 1 ^ N ) = -u 1 )
Theorem	m1exp1 12066	Exponentiation of negative one is one iff the exponent is even. (Contributed by AV, 20-Jun-2021.)
|- ( N e. ZZ -> ( ( -u 1 ^ N ) = 1 <-> 2 || N ) )
Theorem	nn0enne 12067	A positive integer is an even nonnegative integer iff it is an even positive integer. (Contributed by AV, 30-May-2020.)
|- ( N e. NN -> ( ( N / 2 ) e. NN0 <-> ( N / 2 ) e. NN ) )
Theorem	nn0ehalf 12068	The half of an even nonnegative integer is a nonnegative integer. (Contributed by AV, 22-Jun-2020.) (Revised by AV, 28-Jun-2021.)
|- ( ( N e. NN0 /\ 2 || N ) -> ( N / 2 ) e. NN0 )
Theorem	nnehalf 12069	The half of an even positive integer is a positive integer. (Contributed by AV, 28-Jun-2021.)
|- ( ( N e. NN /\ 2 || N ) -> ( N / 2 ) e. NN )
Theorem	nn0o1gt2 12070	An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.)
|- ( ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) -> ( N = 1 \/ 2 < N ) )
Theorem	nno 12071	An alternate characterization of an odd integer greater than 1. (Contributed by AV, 2-Jun-2020.)
|- ( ( N e. ( ZZ>= ` 2 ) /\ ( ( N + 1 ) / 2 ) e. NN0 ) -> ( ( N - 1 ) / 2 ) e. NN )
Theorem	nn0o 12072	An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Proof shortened by AV, 2-Jun-2020.)
|- ( ( N e. NN0 /\ ( ( N + 1 ) / 2 ) e. NN0 ) -> ( ( N - 1 ) / 2 ) e. NN0 )
Theorem	nn0ob 12073	Alternate characterizations of an odd nonnegative integer. (Contributed by AV, 4-Jun-2020.)
|- ( N e. NN0 -> ( ( ( N + 1 ) / 2 ) e. NN0 <-> ( ( N - 1 ) / 2 ) e. NN0 ) )
Theorem	nn0oddm1d2 12074	A positive integer is odd iff its predecessor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.)
|- ( N e. NN0 -> ( -. 2 || N <-> ( ( N - 1 ) / 2 ) e. NN0 ) )
Theorem	nnoddm1d2 12075	A positive integer is odd iff its successor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.)
|- ( N e. NN -> ( -. 2 || N <-> ( ( N + 1 ) / 2 ) e. NN ) )
Theorem	z0even 12076	0 is even. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 23-Jun-2021.)
|- 2 || 0
Theorem	n2dvds1 12077	2 does not divide 1 (common case). That means 1 is odd. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -. 2 || 1
Theorem	n2dvdsm1 12078	2 does not divide -1. That means -1 is odd. (Contributed by AV, 15-Aug-2021.)
|- -. 2 || -u 1
Theorem	z2even 12079	2 is even. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 23-Jun-2021.)
|- 2 || 2
Theorem	n2dvds3 12080	2 does not divide 3, i.e. 3 is an odd number. (Contributed by AV, 28-Feb-2021.)
|- -. 2 || 3
Theorem	z4even 12081	4 is an even number. (Contributed by AV, 23-Jul-2020.) (Revised by AV, 4-Jul-2021.)
|- 2 || 4
Theorem	4dvdseven 12082	An integer which is divisible by 4 is an even integer. (Contributed by AV, 4-Jul-2021.)
|- ( 4 || N -> 2 || N )
5.1.3  The division algorithm
Theorem	divalglemnn 12083*	Lemma for divalg 12089. Existence for a positive denominator. (Contributed by Jim Kingdon, 30-Nov-2021.)
|- ( ( N e. ZZ /\ D e. NN ) -> E. r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) )
Theorem	divalglemqt 12084	Lemma for divalg 12089. The Q = T case involved in showing uniqueness. (Contributed by Jim Kingdon, 5-Dec-2021.)
|- ( ph -> D e. ZZ )   &    |- ( ph -> R e. ZZ )   &    |- ( ph -> S e. ZZ )   &    |- ( ph -> Q e. ZZ )   &    |- ( ph -> T e. ZZ )   &    |- ( ph -> Q = T )   &    |- ( ph -> ( ( Q x. D ) + R ) = ( ( T x. D ) + S ) )   =>    |- ( ph -> R = S )
Theorem	divalglemnqt 12085	Lemma for divalg 12089. The Q < T case involved in showing uniqueness. (Contributed by Jim Kingdon, 4-Dec-2021.)
|- ( ph -> D e. NN )   &    |- ( ph -> R e. ZZ )   &    |- ( ph -> S e. ZZ )   &    |- ( ph -> Q e. ZZ )   &    |- ( ph -> T e. ZZ )   &    |- ( ph -> 0 <_ S )   &    |- ( ph -> R < D )   &    |- ( ph -> ( ( Q x. D ) + R ) = ( ( T x. D ) + S ) )   =>    |- ( ph -> -. Q < T )
Theorem	divalglemeunn 12086*	Lemma for divalg 12089. Uniqueness for a positive denominator. (Contributed by Jim Kingdon, 4-Dec-2021.)
|- ( ( N e. ZZ /\ D e. NN ) -> E! r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) )
Theorem	divalglemex 12087*	Lemma for divalg 12089. The quotient and remainder exist. (Contributed by Jim Kingdon, 30-Nov-2021.)
|- ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) -> E. r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) )
Theorem	divalglemeuneg 12088*	Lemma for divalg 12089. Uniqueness for a negative denominator. (Contributed by Jim Kingdon, 4-Dec-2021.)
|- ( ( N e. ZZ /\ D e. ZZ /\ D < 0 ) -> E! r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) )
Theorem	divalg 12089*	The division algorithm (theorem). Dividing an integer N by a nonzero integer D produces a (unique) quotient q and a unique remainder 0 <_ r < ( abs ` D ). Theorem 1.14 in [ApostolNT] p. 19. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) -> E! r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) )
Theorem	divalgb 12090*	Express the division algorithm as stated in divalg 12089 in terms of ||. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) -> ( E! r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) <-> E! r e. NN0 ( r < ( abs ` D ) /\ D || ( N - r ) ) ) )
Theorem	divalg2 12091*	The division algorithm (theorem) for a positive divisor. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( N e. ZZ /\ D e. NN ) -> E! r e. NN0 ( r < D /\ D || ( N - r ) ) )
Theorem	divalgmod 12092	The result of the mod operator satisfies the requirements for the remainder R in the division algorithm for a positive divisor (compare divalg2 12091 and divalgb 12090). This demonstration theorem justifies the use of mod to yield an explicit remainder from this point forward. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by AV, 21-Aug-2021.)
|- ( ( N e. ZZ /\ D e. NN ) -> ( R = ( N mod D ) <-> ( R e. NN0 /\ ( R < D /\ D || ( N - R ) ) ) ) )
Theorem	divalgmodcl 12093	The result of the mod operator satisfies the requirements for the remainder R in the division algorithm for a positive divisor. Variant of divalgmod 12092. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof shortened by AV, 21-Aug-2021.)
|- ( ( N e. ZZ /\ D e. NN /\ R e. NN0 ) -> ( R = ( N mod D ) <-> ( R < D /\ D || ( N - R ) ) ) )
Theorem	modremain 12094*	The result of the modulo operation is the remainder of the division algorithm. (Contributed by AV, 19-Aug-2021.)
|- ( ( N e. ZZ /\ D e. NN /\ ( R e. NN0 /\ R < D ) ) -> ( ( N mod D ) = R <-> E. z e. ZZ ( ( z x. D ) + R ) = N ) )
Theorem	ndvdssub 12095	Corollary of the division algorithm. If an integer D greater than 1 divides N, then it does not divide any of N - 1, N - 2... N - ( D - 1 ). (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( N e. ZZ /\ D e. NN /\ ( K e. NN /\ K < D ) ) -> ( D || N -> -. D || ( N - K ) ) )
Theorem	ndvdsadd 12096	Corollary of the division algorithm. If an integer D greater than 1 divides N, then it does not divide any of N + 1, N + 2... N + ( D - 1 ). (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( N e. ZZ /\ D e. NN /\ ( K e. NN /\ K < D ) ) -> ( D || N -> -. D || ( N + K ) ) )
Theorem	ndvdsp1 12097	Special case of ndvdsadd 12096. If an integer D greater than 1 divides N, it does not divide N + 1. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( N e. ZZ /\ D e. NN /\ 1 < D ) -> ( D || N -> -. D || ( N + 1 ) ) )
Theorem	ndvdsi 12098	A quick test for non-divisibility. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- A e. NN   &    |- Q e. NN0   &    |- R e. NN   &    |- ( ( A x. Q ) + R ) = B   &    |- R < A   =>    |- -. A || B
Theorem	5ndvds3 12099	5 does not divide 3. (Contributed by AV, 8-Sep-2025.)
|- -. 5 || 3
Theorem	5ndvds6 12100	5 does not divide 6. (Contributed by AV, 8-Sep-2025.)
|- -. 5 || 6