P. 126 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12501-12600   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-bits 12501*	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 12502*	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 12503	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 12504	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 12505	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 12506	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 12507	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 ) )
Theorem	bits0 12508	Value of the zeroth bit. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( N e. ZZ -> ( 0 e. (bits ` N ) <-> -. 2 || N ) )
Theorem	bits0e 12509	The zeroth bit of an even number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( N e. ZZ -> -. 0 e. (bits ` ( 2 x. N ) ) )
Theorem	bits0o 12510	The zeroth bit of an odd number is one. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( N e. ZZ -> 0 e. (bits ` ( ( 2 x. N ) + 1 ) ) )
Theorem	bitsp1 12511	The M + 1-th bit of N is the M-th bit of |_ ( N / 2 ). (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( ( N e. ZZ /\ M e. NN0 ) -> ( ( M + 1 ) e. (bits ` N ) <-> M e. (bits ` ( |_ ` ( N / 2 ) ) ) ) )
Theorem	bitsp1e 12512	The M + 1-th bit of 2 N is the M-th bit of N. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( ( N e. ZZ /\ M e. NN0 ) -> ( ( M + 1 ) e. (bits ` ( 2 x. N ) ) <-> M e. (bits ` N ) ) )
Theorem	bitsp1o 12513	The M + 1-th bit of 2 N + 1 is the M-th bit of N. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( ( N e. ZZ /\ M e. NN0 ) -> ( ( M + 1 ) e. (bits ` ( ( 2 x. N ) + 1 ) ) <-> M e. (bits ` N ) ) )
Theorem	bitsfzolem 12514*	Lemma for bitsfzo 12515. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 1-Oct-2020.)
|- ( ph -> N e. NN0 )   &    |- ( ph -> M e. NN0 )   &    |- ( ph -> (bits ` N ) C_ ( 0..^ M ) )   &    |- S = inf ( { n e. NN0 | N < ( 2 ^ n ) } , RR , < )   =>    |- ( ph -> N e. ( 0..^ ( 2 ^ M ) ) )
Theorem	bitsfzo 12515	The bits of a number are all at positions less than M iff the number is nonnegative and less than 2 ^ M. (Contributed by Mario Carneiro, 5-Sep-2016.) (Proof shortened by AV, 1-Oct-2020.)
|- ( ( N e. ZZ /\ M e. NN0 ) -> ( N e. ( 0..^ ( 2 ^ M ) ) <-> (bits ` N ) C_ ( 0..^ M ) ) )
Theorem	bitsmod 12516	Truncating the bit sequence after some M is equivalent to reducing the argument mod 2 ^ M. (Contributed by Mario Carneiro, 6-Sep-2016.)
|- ( ( N e. ZZ /\ M e. NN0 ) -> (bits ` ( N mod ( 2 ^ M ) ) ) = ( (bits ` N ) i^i ( 0..^ M ) ) )
Theorem	bitsfi 12517	Every number is associated with a finite set of bits. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( N e. NN0 -> (bits ` N ) e. Fin )
Theorem	bitscmp 12518	The bit complement of N is -u N - 1. (Thus, by bitsfi 12517, all negative numbers have cofinite bits representations.) (Contributed by Mario Carneiro, 5-Sep-2016.)
|- ( N e. ZZ -> ( NN0 \ (bits ` N ) ) = (bits ` ( -u N - 1 ) ) )
Theorem	0bits 12519	The bits of zero. (Contributed by Mario Carneiro, 6-Sep-2016.)
|- (bits ` 0 ) = (/)
Theorem	m1bits 12520	The bits of negative one. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- (bits ` -u 1 ) = NN0
Theorem	bitsinv1lem 12521	Lemma for bitsinv1 12522. (Contributed by Mario Carneiro, 22-Sep-2016.)
|- ( ( N e. ZZ /\ M e. NN0 ) -> ( N mod ( 2 ^ ( M + 1 ) ) ) = ( ( N mod ( 2 ^ M ) ) + if ( M e. (bits ` N ) , ( 2 ^ M ) , 0 ) ) )
Theorem	bitsinv1 12522*	There is an explicit inverse to the bits function for nonnegative integers (which can be extended to negative integers using bitscmp 12518), part 1. (Contributed by Mario Carneiro, 7-Sep-2016.)
|- ( N e. NN0 -> sum_ n e. (bits ` N ) ( 2 ^ n ) = N )
5.1.5  The greatest common divisor operator
Syntax	cgcd 12523	Extend the definition of a class to include the greatest common divisor operator.
class gcd
Definition	df-gcd 12524*	Define the gcd operator. For example, ( -u 6 gcd 9 ) = 3 (ex-gcd 16327). (Contributed by Paul Chapman, 21-Mar-2011.)
|- gcd = ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 /\ y = 0 ) , 0 , sup ( { n e. ZZ | ( n || x /\ n || y ) } , RR , < ) ) )
Theorem	gcdmndc 12525	Decidablity lemma used in various proofs related to gcd. (Contributed by Jim Kingdon, 12-Dec-2021.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) )
Theorem	dvdsbnd 12526*	There is an upper bound to the divisors of a nonzero integer. (Contributed by Jim Kingdon, 11-Dec-2021.)
|- ( ( A e. ZZ /\ A =/= 0 ) -> E. n e. NN A. m e. ( ZZ>= ` n ) -. m || A )
Theorem	gcdsupex 12527*	Existence of the supremum used in defining gcd. (Contributed by Jim Kingdon, 12-Dec-2021.)
|- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> E. x e. ZZ ( A. y e. { n e. ZZ | ( n || X /\ n || Y ) } -. x < y /\ A. y e. RR ( y < x -> E. z e. { n e. ZZ | ( n || X /\ n || Y ) } y < z ) ) )
Theorem	gcdsupcl 12528*	Closure of the supremum used in defining gcd. A lemma for gcdval 12529 and gcdn0cl 12532. (Contributed by Jim Kingdon, 11-Dec-2021.)
|- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> sup ( { n e. ZZ | ( n || X /\ n || Y ) } , RR , < ) e. NN )
Theorem	gcdval 12529*	The value of the gcd operator. ( M gcd N ) is the greatest common divisor of M and N. If M and N are both 0, the result is defined conventionally as 0. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by Mario Carneiro, 10-Nov-2013.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = if ( ( M = 0 /\ N = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) ) )
Theorem	gcd0val 12530	The value, by convention, of the gcd operator when both operands are 0. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( 0 gcd 0 ) = 0
Theorem	gcdn0val 12531*	The value of the gcd operator when at least one operand is nonzero. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) )
Theorem	gcdn0cl 12532	Closure of the gcd operator. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) e. NN )
Theorem	gcddvds 12533	The gcd of two integers divides each of them. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
Theorem	dvdslegcd 12534	An integer which divides both operands of the gcd operator is bounded by it. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( ( K || M /\ K || N ) -> K <_ ( M gcd N ) ) )
Theorem	nndvdslegcd 12535	A positive integer which divides both positive operands of the gcd operator is bounded by it. (Contributed by AV, 9-Aug-2020.)
|- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> ( ( K || M /\ K || N ) -> K <_ ( M gcd N ) ) )
Theorem	gcdcl 12536	Closure of the gcd operator. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
Theorem	gcdnncl 12537	Closure of the gcd operator. (Contributed by Thierry Arnoux, 2-Feb-2020.)
|- ( ( M e. NN /\ N e. NN ) -> ( M gcd N ) e. NN )
Theorem	gcdcld 12538	Closure of the gcd operator. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> ( M gcd N ) e. NN0 )
Theorem	gcd2n0cl 12539	Closure of the gcd operator if the second operand is not 0. (Contributed by AV, 10-Jul-2021.)
|- ( ( M e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( M gcd N ) e. NN )
Theorem	zeqzmulgcd 12540*	An integer is the product of an integer and the gcd of it and another integer. (Contributed by AV, 11-Jul-2021.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> E. n e. ZZ A = ( n x. ( A gcd B ) ) )
Theorem	divgcdz 12541	An integer divided by the gcd of it and a nonzero integer is an integer. (Contributed by AV, 11-Jul-2021.)
|- ( ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) -> ( A / ( A gcd B ) ) e. ZZ )
Theorem	gcdf 12542	Domain and codomain of the gcd operator. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 16-Nov-2013.)
|- gcd : ( ZZ X. ZZ ) --> NN0
Theorem	gcdcom 12543	The gcd operator is commutative. Theorem 1.4(a) in [ApostolNT] p. 16. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( N gcd M ) )
Theorem	gcdcomd 12544	The gcd operator is commutative, deduction version. (Contributed by SN, 24-Aug-2024.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> ( M gcd N ) = ( N gcd M ) )
Theorem	divgcdnn 12545	A positive integer divided by the gcd of it and another integer is a positive integer. (Contributed by AV, 10-Jul-2021.)
|- ( ( A e. NN /\ B e. ZZ ) -> ( A / ( A gcd B ) ) e. NN )
Theorem	divgcdnnr 12546	A positive integer divided by the gcd of it and another integer is a positive integer. (Contributed by AV, 10-Jul-2021.)
|- ( ( A e. NN /\ B e. ZZ ) -> ( A / ( B gcd A ) ) e. NN )
Theorem	gcdeq0 12547	The gcd of two integers is zero iff they are both zero. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) = 0 <-> ( M = 0 /\ N = 0 ) ) )
Theorem	gcdn0gt0 12548	The gcd of two integers is positive (nonzero) iff they are not both zero. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 /\ N = 0 ) <-> 0 < ( M gcd N ) ) )
Theorem	gcd0id 12549	The gcd of 0 and an integer is the integer's absolute value. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( N e. ZZ -> ( 0 gcd N ) = ( abs ` N ) )
Theorem	gcdid0 12550	The gcd of an integer and 0 is the integer's absolute value. Theorem 1.4(d)2 in [ApostolNT] p. 16. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( N e. ZZ -> ( N gcd 0 ) = ( abs ` N ) )
Theorem	nn0gcdid0 12551	The gcd of a nonnegative integer with 0 is itself. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( N e. NN0 -> ( N gcd 0 ) = N )
Theorem	gcdneg 12552	Negating one operand of the gcd operator does not alter the result. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd -u N ) = ( M gcd N ) )
Theorem	neggcd 12553	Negating one operand of the gcd operator does not alter the result. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( -u M gcd N ) = ( M gcd N ) )
Theorem	gcdaddm 12554	Adding a multiple of one operand of the gcd operator to the other does not alter the result. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( M gcd ( N + ( K x. M ) ) ) )
Theorem	gcdadd 12555	The GCD of two numbers is the same as the GCD of the left and their sum. (Contributed by Scott Fenton, 20-Apr-2014.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( M gcd ( N + M ) ) )
Theorem	gcdid 12556	The gcd of a number and itself is its absolute value. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( N e. ZZ -> ( N gcd N ) = ( abs ` N ) )
Theorem	gcd1 12557	The gcd of a number with 1 is 1. Theorem 1.4(d)1 in [ApostolNT] p. 16. (Contributed by Mario Carneiro, 19-Feb-2014.)
|- ( M e. ZZ -> ( M gcd 1 ) = 1 )
Theorem	gcdabs 12558	The gcd of two integers is the same as that of their absolute values. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
Theorem	gcdabs1 12559	gcd of the absolute value of the first operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( abs ` N ) gcd M ) = ( N gcd M ) )
Theorem	gcdabs2 12560	gcd of the absolute value of the second operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( N e. ZZ /\ M e. ZZ ) -> ( N gcd ( abs ` M ) ) = ( N gcd M ) )
Theorem	modgcd 12561	The gcd remains unchanged if one operand is replaced with its remainder modulo the other. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( M e. ZZ /\ N e. NN ) -> ( ( M mod N ) gcd N ) = ( M gcd N ) )
Theorem	1gcd 12562	The GCD of one and an integer is one. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( M e. ZZ -> ( 1 gcd M ) = 1 )
Theorem	gcdmultipled 12563	The greatest common divisor of a nonnegative integer M and a multiple of it is M itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- ( ph -> M e. NN0 )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> ( M gcd ( N x. M ) ) = M )
Theorem	dvdsgcdidd 12564	The greatest common divisor of a positive integer and another integer it divides is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- ( ph -> M e. NN )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> M || N )   =>    |- ( ph -> ( M gcd N ) = M )
Theorem	6gcd4e2 12565	The greatest common divisor of six and four is two. To calculate this gcd, a simple form of Euclid's algorithm is used: ( 6 gcd 4 ) = ( ( 4 + 2 ) gcd 4 ) = ( 2 gcd 4 ) and ( 2 gcd 4 ) = ( 2 gcd ( 2 + 2 ) ) = ( 2 gcd 2 ) = 2. (Contributed by AV, 27-Aug-2020.)
|- ( 6 gcd 4 ) = 2
5.1.6  Bézout's identity
Theorem	bezoutlemnewy 12566*	Lemma for Bézout's identity. The is-bezout predicate holds for ( y mod W ). (Contributed by Jim Kingdon, 6-Jan-2022.)
|- ( ph <-> E. s e. ZZ E. t e. ZZ r = ( ( A x. s ) + ( B x. t ) ) )   &    |- ( th -> A e. NN0 )   &    |- ( th -> B e. NN0 )   &    |- ( th -> W e. NN )   &    |- ( th -> [ y / r ] ph )   &    |- ( th -> y e. NN0 )   &    |- ( th -> [. W / r ]. ph )   =>    |- ( th -> [. ( y mod W ) / r ]. ph )
Theorem	bezoutlemstep 12567*	Lemma for Bézout's identity. This is the induction step for the proof by induction. (Contributed by Jim Kingdon, 3-Jan-2022.)
|- ( ph <-> E. s e. ZZ E. t e. ZZ r = ( ( A x. s ) + ( B x. t ) ) )   &    |- ( th -> A e. NN0 )   &    |- ( th -> B e. NN0 )   &    |- ( th -> W e. NN )   &    |- ( th -> [ y / r ] ph )   &    |- ( th -> y e. NN0 )   &    |- ( th -> [. W / r ]. ph )   &    |- ( ps <-> A. z e. NN0 ( z || r -> ( z || x /\ z || y ) ) )   &    |- ( ( th /\ [. ( y mod W ) / r ]. ph ) -> E. r e. NN0 ( [. ( y mod W ) / x ]. [. W / y ]. ps /\ ph ) )   &    |- F/ x th   &    |- F/ r th   =>    |- ( th -> E. r e. NN0 ( [. W / x ]. ps /\ ph ) )
Theorem	bezoutlemmain 12568*	Lemma for Bézout's identity. This is the main result which we prove by induction and which represents the application of the Extended Euclidean algorithm. (Contributed by Jim Kingdon, 30-Dec-2021.)
|- ( ph <-> E. s e. ZZ E. t e. ZZ r = ( ( A x. s ) + ( B x. t ) ) )   &    |- ( ps <-> A. z e. NN0 ( z || r -> ( z || x /\ z || y ) ) )   &    |- ( th -> A e. NN0 )   &    |- ( th -> B e. NN0 )   =>    |- ( th -> A. x e. NN0 ( [ x / r ] ph -> A. y e. NN0 ( [ y / r ] ph -> E. r e. NN0 ( ps /\ ph ) ) ) )
Theorem	bezoutlema 12569*	Lemma for Bézout's identity. The is-bezout condition is satisfied by A. (Contributed by Jim Kingdon, 30-Dec-2021.)
|- ( ph <-> E. s e. ZZ E. t e. ZZ r = ( ( A x. s ) + ( B x. t ) ) )   &    |- ( th -> A e. NN0 )   &    |- ( th -> B e. NN0 )   =>    |- ( th -> [. A / r ]. ph )
Theorem	bezoutlemb 12570*	Lemma for Bézout's identity. The is-bezout condition is satisfied by B. (Contributed by Jim Kingdon, 30-Dec-2021.)
|- ( ph <-> E. s e. ZZ E. t e. ZZ r = ( ( A x. s ) + ( B x. t ) ) )   &    |- ( th -> A e. NN0 )   &    |- ( th -> B e. NN0 )   =>    |- ( th -> [. B / r ]. ph )
Theorem	bezoutlemex 12571*	Lemma for Bézout's identity. Existence of a number which we will later show to be the greater common divisor and its decomposition into cofactors. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jan-2022.)
|- ( ( A e. NN0 /\ B e. NN0 ) -> E. d e. NN0 ( A. z e. NN0 ( z || d -> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) )
Theorem	bezoutlemzz 12572*	Lemma for Bézout's identity. Like bezoutlemex 12571 but where ' z ' is any integer, not just a nonnegative one. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.)
|- ( ( A e. NN0 /\ B e. NN0 ) -> E. d e. NN0 ( A. z e. ZZ ( z || d -> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) )
Theorem	bezoutlemaz 12573*	Lemma for Bézout's identity. Like bezoutlemzz 12572 but where ' A ' can be any integer, not just a nonnegative one. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.)
|- ( ( A e. ZZ /\ B e. NN0 ) -> E. d e. NN0 ( A. z e. ZZ ( z || d -> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) )
Theorem	bezoutlembz 12574*	Lemma for Bézout's identity. Like bezoutlemaz 12573 but where ' B ' can be any integer, not just a nonnegative one. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> E. d e. NN0 ( A. z e. ZZ ( z || d -> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) )
Theorem	bezoutlembi 12575*	Lemma for Bézout's identity. Like bezoutlembz 12574 but the greatest common divisor condition is a biconditional, not just an implication. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> E. d e. NN0 ( A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) /\ E. x e. ZZ E. y e. ZZ d = ( ( A x. x ) + ( B x. y ) ) ) )
Theorem	bezoutlemmo 12576*	Lemma for Bézout's identity. There is at most one nonnegative integer meeting the greatest common divisor condition. (Contributed by Mario Carneiro and Jim Kingdon, 9-Jan-2022.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   &    |- ( ph -> D e. NN0 )   &    |- ( ph -> A. z e. ZZ ( z || D <-> ( z || A /\ z || B ) ) )   &    |- ( ph -> E e. NN0 )   &    |- ( ph -> A. z e. ZZ ( z || E <-> ( z || A /\ z || B ) ) )   =>    |- ( ph -> D = E )
Theorem	bezoutlemeu 12577*	Lemma for Bézout's identity. There is exactly one nonnegative integer meeting the greatest common divisor condition. (Contributed by Mario Carneiro and Jim Kingdon, 9-Jan-2022.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   &    |- ( ph -> D e. NN0 )   &    |- ( ph -> A. z e. ZZ ( z || D <-> ( z || A /\ z || B ) ) )   =>    |- ( ph -> E! d e. NN0 A. z e. ZZ ( z || d <-> ( z || A /\ z || B ) ) )
Theorem	bezoutlemle 12578*	Lemma for Bézout's identity. The number satisfying the greatest common divisor condition is the largest number which divides both A and B. (Contributed by Mario Carneiro and Jim Kingdon, 9-Jan-2022.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   &    |- ( ph -> D e. NN0 )   &    |- ( ph -> A. z e. ZZ ( z || D <-> ( z || A /\ z || B ) ) )   &    |- ( ph -> -. ( A = 0 /\ B = 0 ) )   =>    |- ( ph -> A. z e. ZZ ( ( z || A /\ z || B ) -> z <_ D ) )
Theorem	bezoutlemsup 12579*	Lemma for Bézout's identity. The number satisfying the greatest common divisor condition is the supremum of divisors of both A and B. (Contributed by Mario Carneiro and Jim Kingdon, 9-Jan-2022.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   &    |- ( ph -> D e. NN0 )   &    |- ( ph -> A. z e. ZZ ( z || D <-> ( z || A /\ z || B ) ) )   &    |- ( ph -> -. ( A = 0 /\ B = 0 ) )   =>    |- ( ph -> D = sup ( { z e. ZZ | ( z || A /\ z || B ) } , RR , < ) )
Theorem	dfgcd3 12580*	Alternate definition of the gcd operator. (Contributed by Jim Kingdon, 31-Dec-2021.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( iota_ d e. NN0 A. z e. ZZ ( z || d <-> ( z || M /\ z || N ) ) ) )
Theorem	bezout 12581*	Bézout's identity: For any integers A and B, there are integers x , y such that ( A gcd B ) = A x. x + B x. y. This is Metamath 100 proof #60. The proof is constructive, in the sense that it applies the Extended Euclidian Algorithm to constuct a number which can be shown to be ( A gcd B ) and which satisfies the rest of the theorem. In the presence of excluded middle, it is common to prove Bézout's identity by taking the smallest number which satisfies the Bézout condition, and showing it is the greatest common divisor. But we do not have the ability to show that number exists other than by providing a way to determine it. (Contributed by Mario Carneiro, 22-Feb-2014.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> E. x e. ZZ E. y e. ZZ ( A gcd B ) = ( ( A x. x ) + ( B x. y ) ) )
Theorem	dvdsgcd 12582	An integer which divides each of two others also divides their gcd. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 30-May-2014.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ K || N ) -> K || ( M gcd N ) ) )
Theorem	dvdsgcdb 12583	Biconditional form of dvdsgcd 12582. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || M /\ K || N ) <-> K || ( M gcd N ) ) )
Theorem	dfgcd2 12584*	Alternate definition of the gcd operator, see definition in [ApostolNT] p. 15. (Contributed by AV, 8-Aug-2021.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( D = ( M gcd N ) <-> ( 0 <_ D /\ ( D || M /\ D || N ) /\ A. e e. ZZ ( ( e || M /\ e || N ) -> e || D ) ) ) )
Theorem	gcdass 12585	Associative law for gcd operator. Theorem 1.4(b) in [ApostolNT] p. 16. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N gcd M ) gcd P ) = ( N gcd ( M gcd P ) ) )
Theorem	mulgcd 12586	Distribute multiplication by a nonnegative integer over gcd. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 30-May-2014.)
|- ( ( K e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( K x. ( M gcd N ) ) )
Theorem	absmulgcd 12587	Distribute absolute value of multiplication over gcd. Theorem 1.4(c) in [ApostolNT] p. 16. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) gcd ( K x. N ) ) = ( abs ` ( K x. ( M gcd N ) ) ) )
Theorem	mulgcdr 12588	Reverse distribution law for the gcd operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( A e. ZZ /\ B e. ZZ /\ C e. NN0 ) -> ( ( A x. C ) gcd ( B x. C ) ) = ( ( A gcd B ) x. C ) )
Theorem	gcddiv 12589	Division law for GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. NN ) /\ ( C || A /\ C || B ) ) -> ( ( A gcd B ) / C ) = ( ( A / C ) gcd ( B / C ) ) )
Theorem	gcdmultiple 12590	The GCD of a multiple of a number is the number itself. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( M e. NN /\ N e. NN ) -> ( M gcd ( M x. N ) ) = M )
Theorem	gcdmultiplez 12591	Extend gcdmultiple 12590 so N can be an integer. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. N ) ) = M )
Theorem	gcdzeq 12592	A positive integer A is equal to its gcd with an integer B if and only if A divides B. Generalization of gcdeq 12593. (Contributed by AV, 1-Jul-2020.)
|- ( ( A e. NN /\ B e. ZZ ) -> ( ( A gcd B ) = A <-> A || B ) )
Theorem	gcdeq 12593	A is equal to its gcd with B if and only if A divides B. (Contributed by Mario Carneiro, 23-Feb-2014.) (Proof shortened by AV, 8-Aug-2021.)
|- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) = A <-> A || B ) )
Theorem	dvdssqim 12594	Unidirectional form of dvdssq 12601. (Contributed by Scott Fenton, 19-Apr-2014.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N -> ( M ^ 2 ) || ( N ^ 2 ) ) )
Theorem	dvdsmulgcd 12595	Relationship between the order of an element and that of a multiple. (a divisibility equivalent). (Contributed by Stefan O'Rear, 6-Sep-2015.)
|- ( ( B e. ZZ /\ C e. ZZ ) -> ( A || ( B x. C ) <-> A || ( B x. ( C gcd A ) ) ) )
Theorem	rpmulgcd 12596	If K and M are relatively prime, then the GCD of K and M x. N is the GCD of K and N. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( ( K e. NN /\ M e. NN /\ N e. NN ) /\ ( K gcd M ) = 1 ) -> ( K gcd ( M x. N ) ) = ( K gcd N ) )
Theorem	rplpwr 12597	If A and B are relatively prime, then so are A ^ N and B. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( A e. NN /\ B e. NN /\ N e. NN ) -> ( ( A gcd B ) = 1 -> ( ( A ^ N ) gcd B ) = 1 ) )
Theorem	rppwr 12598	If A and B are relatively prime, then so are A ^ N and B ^ N. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( A e. NN /\ B e. NN /\ N e. NN ) -> ( ( A gcd B ) = 1 -> ( ( A ^ N ) gcd ( B ^ N ) ) = 1 ) )
Theorem	sqgcd 12599	Square distributes over gcd. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( M e. NN /\ N e. NN ) -> ( ( M gcd N ) ^ 2 ) = ( ( M ^ 2 ) gcd ( N ^ 2 ) ) )
Theorem	dvdssqlem 12600	Lemma for dvdssq 12601. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( M e. NN /\ N e. NN ) -> ( M || N <-> ( M ^ 2 ) || ( N ^ 2 ) ) )