P. 122 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12101-12200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dvdseq 12101	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 12102	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 12103*	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 12104*	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 12105*	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 12106	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 12107*	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 12108*	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 12109	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 12110	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 12111	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 12112	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 10541 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 12113	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 12114	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 12115	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 12116	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 12117*	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 12118	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 12119	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 12123. 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 12120) 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 9473, zeo 9477, zeo2 9478, etc. use different representations, which are equivalent with the representations using the divides relation, see evend2 12142 and oddp1d2 12143. The corresponding theorems are zeneo 12124, zeo3 12121 and zeo4 12123.
Theorem	evenelz 12120	An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 12045. (Contributed by AV, 22-Jun-2021.)
|- ( 2 || N -> N e. ZZ )
Theorem	zeo3 12121	An integer is even or odd. (Contributed by AV, 17-Jun-2021.)
|- ( N e. ZZ -> ( 2 || N \/ -. 2 || N ) )
Theorem	zeoxor 12122	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 12123	An integer is even or odd but not both. (Contributed by AV, 17-Jun-2021.)
|- ( N e. ZZ -> ( 2 || N <-> -. -. 2 || N ) )
Theorem	zeneo 12124	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 9473 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 12125*	Lemma for odd2np1 12126. (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 12126*	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 12127*	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 12128	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 12129	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 12130	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 12131	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 12132	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 12133*	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 12134*	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 12135*	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 12136*	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 12137	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 12138	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 12139	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 12140	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 12141	An integer is either even or odd, version of zeo3 12121 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 12142	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 9477 and zeo2 9478. (Contributed by AV, 22-Jun-2021.)
|- ( N e. ZZ -> ( 2 || N <-> ( N / 2 ) e. ZZ ) )
Theorem	oddp1d2 12143	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 9477 and zeo2 9478. (Contributed by AV, 22-Jun-2021.)
|- ( N e. ZZ -> ( -. 2 || N <-> ( ( N + 1 ) / 2 ) e. ZZ ) )
Theorem	zob 12144	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 12145	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 12146	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 12147	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 12148	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 12149	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 12150	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 12151	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 12152	Exponentiation of -1 by an even power. Variant of m1expeven 10729. (Contributed by AV, 25-Jun-2021.)
|- ( 2 || N -> ( -u 1 ^ N ) = 1 )
Theorem	m1expo 12153	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 12154	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 12155	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 12156	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 12157	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 12158	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 12159	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 12160	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 12161	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 12162	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 12163	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 12164	0 is even. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 23-Jun-2021.)
|- 2 || 0
Theorem	n2dvds1 12165	2 does not divide 1 (common case). That means 1 is odd. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -. 2 || 1
Theorem	n2dvdsm1 12166	2 does not divide -1. That means -1 is odd. (Contributed by AV, 15-Aug-2021.)
|- -. 2 || -u 1
Theorem	z2even 12167	2 is even. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 23-Jun-2021.)
|- 2 || 2
Theorem	n2dvds3 12168	2 does not divide 3, i.e. 3 is an odd number. (Contributed by AV, 28-Feb-2021.)
|- -. 2 || 3
Theorem	z4even 12169	4 is an even number. (Contributed by AV, 23-Jul-2020.) (Revised by AV, 4-Jul-2021.)
|- 2 || 4
Theorem	4dvdseven 12170	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 12171*	Lemma for divalg 12177. 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 12172	Lemma for divalg 12177. 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 12173	Lemma for divalg 12177. 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 12174*	Lemma for divalg 12177. 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 12175*	Lemma for divalg 12177. 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 12176*	Lemma for divalg 12177. 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 12177*	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 12178*	Express the division algorithm as stated in divalg 12177 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 12179*	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 12180	The result of the mod operator satisfies the requirements for the remainder R in the division algorithm for a positive divisor (compare divalg2 12179 and divalgb 12178). 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 12181	The result of the mod operator satisfies the requirements for the remainder R in the division algorithm for a positive divisor. Variant of divalgmod 12180. (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 12182*	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 12183	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 12184	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 12185	Special case of ndvdsadd 12184. 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 12186	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 12187	5 does not divide 3. (Contributed by AV, 8-Sep-2025.)
|- -. 5 || 3
Theorem	5ndvds6 12188	5 does not divide 6. (Contributed by AV, 8-Sep-2025.)
|- -. 5 || 6
Theorem	flodddiv4 12189	The floor of an odd integer divided by 4. (Contributed by AV, 17-Jun-2021.)
|- ( ( M e. ZZ /\ N = ( ( 2 x. M ) + 1 ) ) -> ( |_ ` ( N / 4 ) ) = if ( 2 || M , ( M / 2 ) , ( ( M - 1 ) / 2 ) ) )
Theorem	fldivndvdslt 12190	The floor of an integer divided by a nonzero integer not dividing the first integer is less than the integer divided by the positive integer. (Contributed by AV, 4-Jul-2021.)
|- ( ( K e. ZZ /\ ( L e. ZZ /\ L =/= 0 ) /\ -. L || K ) -> ( |_ ` ( K / L ) ) < ( K / L ) )
Theorem	flodddiv4lt 12191	The floor of an odd number divided by 4 is less than the odd number divided by 4. (Contributed by AV, 4-Jul-2021.)
|- ( ( N e. ZZ /\ -. 2 || N ) -> ( |_ ` ( N / 4 ) ) < ( N / 4 ) )
Theorem	flodddiv4t2lthalf 12192	The floor of an odd number divided by 4, multiplied by 2 is less than the half of the odd number. (Contributed by AV, 4-Jul-2021.)
|- ( ( N e. ZZ /\ -. 2 || N ) -> ( ( |_ ` ( N / 4 ) ) x. 2 ) < ( N / 2 ) )
5.1.4  Bit sequences
Syntax	cbits 12193	Define the binary bits of an integer.
class bits
Definition	df-bits 12194*	Define the binary bits of an integer. The expression M e. (bits ` N ) means that the M-th bit of N is 1 (and its negation means the bit is 0). (Contributed by Mario Carneiro, 4-Sep-2016.)
|- bits = ( n e. ZZ |-> { m e. NN0 | -. 2 || ( |_ ` ( n / ( 2 ^ m ) ) ) } )
Theorem	bitsfval 12195*	Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( N e. ZZ -> (bits ` N ) = { m e. NN0 | -. 2 || ( |_ ` ( N / ( 2 ^ m ) ) ) } )
Theorem	bitsval 12196	Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( M e. (bits ` N ) <-> ( N e. ZZ /\ M e. NN0 /\ -. 2 || ( |_ ` ( N / ( 2 ^ M ) ) ) ) )
Theorem	bitsval2 12197	Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( ( N e. ZZ /\ M e. NN0 ) -> ( M e. (bits ` N ) <-> -. 2 || ( |_ ` ( N / ( 2 ^ M ) ) ) ) )
Theorem	bitsss 12198	The set of bits of an integer is a subset of NN0. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- (bits ` N ) C_ NN0
Theorem	bitsf 12199	The bits function is a function from integers to subsets of nonnegative integers. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- bits : ZZ --> ~P NN0
Theorem	bitsdc 12200	Whether a bit is set is decidable. (Contributed by Jim Kingdon, 31-Oct-2025.)
|- ( ( N e. ZZ /\ M e. NN0 ) -> DECID M e. (bits ` N ) )