P. 128 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12701-12800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bitsmod 12701	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 12702	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 12703	The bit complement of N is -u N - 1. (Thus, by bitsfi 12702, 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 12704	The bits of zero. (Contributed by Mario Carneiro, 6-Sep-2016.)
|- (bits ` 0 ) = (/)
Theorem	m1bits 12705	The bits of negative one. (Contributed by Mario Carneiro, 5-Sep-2016.)
|- (bits ` -u 1 ) = NN0
Theorem	bitsinv1lem 12706	Lemma for bitsinv1 12707. (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 12707*	There is an explicit inverse to the bits function for nonnegative integers (which can be extended to negative integers using bitscmp 12703), 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 12708	Extend the definition of a class to include the greatest common divisor operator.
class gcd
Definition	df-gcd 12709*	Define the gcd operator. For example, ( -u 6 gcd 9 ) = 3 (ex-gcd 16659). (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 12710	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 12711*	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 12712*	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 12713*	Closure of the supremum used in defining gcd. A lemma for gcdval 12714 and gcdn0cl 12717. (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 12714*	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 12715	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 12716*	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 12717	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 12718	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 12719	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 12720	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 12721	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 12722	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 12723	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 12724	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 12725*	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 12726	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 12727	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 12728	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 12729	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 12730	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 12731	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 12732	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 12733	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 12734	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 12735	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 12736	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 12737	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 12738	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 12739	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 12740	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 12741	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 12742	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 12743	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 12744	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 12745	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 12746	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 12747	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 12748	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 12749	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 12750	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 12751*	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 12752*	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 12753*	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 12754*	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 12755*	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 12756*	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 12757*	Lemma for Bézout's identity. Like bezoutlemex 12756 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 12758*	Lemma for Bézout's identity. Like bezoutlemzz 12757 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 12759*	Lemma for Bézout's identity. Like bezoutlemaz 12758 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 12760*	Lemma for Bézout's identity. Like bezoutlembz 12759 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 12761*	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 12762*	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 12763*	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 12764*	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 12765*	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 12766*	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 12767	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 12768	Biconditional form of dvdsgcd 12767. (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 12769*	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 12770	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 12771	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 12772	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 12773	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 12774	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 12775	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 12776	Extend gcdmultiple 12775 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 12777	A positive integer A is equal to its gcd with an integer B if and only if A divides B. Generalization of gcdeq 12778. (Contributed by AV, 1-Jul-2020.)
|- ( ( A e. NN /\ B e. ZZ ) -> ( ( A gcd B ) = A <-> A || B ) )
Theorem	gcdeq 12778	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 12779	Unidirectional form of dvdssq 12786. (Contributed by Scott Fenton, 19-Apr-2014.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N -> ( M ^ 2 ) || ( N ^ 2 ) ) )
Theorem	dvdsmulgcd 12780	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 12781	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 12782	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 12783	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 12784	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 12785	Lemma for dvdssq 12786. (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 ) ) )
Theorem	dvdssq 12786	Two numbers are divisible iff their squares are. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> ( M ^ 2 ) || ( N ^ 2 ) ) )
Theorem	bezoutr 12787	Partial converse to bezout 12766. Existence of a linear combination does not set the GCD, but it does upper bound it. (Contributed by Stefan O'Rear, 23-Sep-2014.)
|- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) -> ( A gcd B ) || ( ( A x. X ) + ( B x. Y ) ) )
Theorem	bezoutr1 12788	Converse of bezout 12766 for when the greater common divisor is one (sufficient condition for relative primality). (Contributed by Stefan O'Rear, 23-Sep-2014.)
|- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( X e. ZZ /\ Y e. ZZ ) ) -> ( ( ( A x. X ) + ( B x. Y ) ) = 1 -> ( A gcd B ) = 1 ) )
5.1.7  Decidable sets of integers
Theorem	nnmindc 12789*	An inhabited decidable subset of the natural numbers has a minimum. (Contributed by Jim Kingdon, 23-Sep-2024.)
|- ( ( A C_ NN /\ A. x e. NN DECID x e. A /\ E. y y e. A ) -> inf ( A , RR , < ) e. A )
Theorem	nnminle 12790*	The infimum of a decidable subset of the natural numbers is less than an element of the set. The infimum is also a minimum as shown at nnmindc 12789. (Contributed by Jim Kingdon, 26-Sep-2024.)
|- ( ( A C_ NN /\ A. x e. NN DECID x e. A /\ B e. A ) -> inf ( A , RR , < ) <_ B )
Theorem	nnwodc 12791*	Well-ordering principle: any inhabited decidable set of positive integers has a least element. Theorem I.37 (well-ordering principle) of [Apostol] p. 34. (Contributed by NM, 17-Aug-2001.) (Revised by Jim Kingdon, 23-Oct-2024.)
|- ( ( A C_ NN /\ E. w w e. A /\ A. j e. NN DECID j e. A ) -> E. x e. A A. y e. A x <_ y )
Theorem	uzwodc 12792*	Well-ordering principle: any inhabited decidable subset of an upper set of integers has a least element. (Contributed by NM, 8-Oct-2005.) (Revised by Jim Kingdon, 22-Oct-2024.)
|- ( ( S C_ ( ZZ>= ` M ) /\ E. x x e. S /\ A. x e. ( ZZ>= ` M )DECID x e. S ) -> E. j e. S A. k e. S j <_ k )
Theorem	nnwofdc 12793*	Well-ordering principle: any inhabited decidable set of positive integers has a least element. This version allows x and y to be present in A as long as they are effectively not free. (Contributed by NM, 17-Aug-2001.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x A   &    |- F/_ y A   =>    |- ( ( A C_ NN /\ E. z z e. A /\ A. j e. NN DECID j e. A ) -> E. x e. A A. y e. A x <_ y )
Theorem	nnwosdc 12794*	Well-ordering principle: any inhabited decidable set of positive integers has a least element (schema form). (Contributed by NM, 17-Aug-2001.) (Revised by Jim Kingdon, 25-Oct-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( ( E. x e. NN ph /\ A. x e. NN DECID ph ) -> E. x e. NN ( ph /\ A. y e. NN ( ps -> x <_ y ) ) )
Theorem	nninfctlemfo 12795*	Lemma for nninfct 12796. (Contributed by Jim Kingdon, 10-Jul-2025.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- F = ( n e. om |-> ( i e. om |-> if ( i e. n , 1o , (/) ) ) )   &    |- I = ( ( F o. `' G ) u. { <. +oo , ( om X. { 1o } ) >. } )   =>    |- ( om e. Omni -> I :NN0* -onto->ℕ∞ )
Theorem	nninfct 12796	The limited principle of omniscience (LPO) implies that ℕ∞ is countable. (Contributed by Jim Kingdon, 8-Jul-2025.)
|- ( om e. Omni -> E. f f : om -onto-> (ℕ∞ ⊔ 1o ) )
5.1.8  Algorithms
Theorem	nn0seqcvgd 12797*	A strictly-decreasing nonnegative integer sequence with initial term N reaches zero by the N th term. Deduction version. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ph -> F : NN0 --> NN0 )   &    |- ( ph -> N = ( F ` 0 ) )   &    |- ( ( ph /\ k e. NN0 ) -> ( ( F ` ( k + 1 ) ) =/= 0 -> ( F ` ( k + 1 ) ) < ( F ` k ) ) )   =>    |- ( ph -> ( F ` N ) = 0 )
Theorem	ialgrlem1st 12798	Lemma for ialgr0 12800. Expressing algrflemg 6456 in a form suitable for theorems such as seq3-1 10877 or seqf 10879. (Contributed by Jim Kingdon, 22-Jul-2021.)
|- ( ph -> F : S --> S )   =>    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x ( F o. 1st ) y ) e. S )
Theorem	ialgrlemconst 12799	Lemma for ialgr0 12800. Closure of a constant function, in a form suitable for theorems such as seq3-1 10877 or seqf 10879. (Contributed by Jim Kingdon, 22-Jul-2021.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> A e. S )   =>    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( Z X. { A } ) ` x ) e. S )
Theorem	ialgr0 12800	The value of the algorithm iterator R at 0 is the initial state A. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Jim Kingdon, 12-Mar-2023.)
|- Z = ( ZZ>= ` M )   &    |- R = seq M ( ( F o. 1st ) , ( Z X. { A } ) )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> F : S --> S )   =>    |- ( ph -> ( R ` M ) = A )