P. 120 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 11901-12000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	gcdcld 11901	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 11902	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 11903*	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 11904	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 11905	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 11906	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 11907	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 11908	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 11909	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 11910	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 11911	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 11912	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 11913	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 11914	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 11915	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 11916	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 11917	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 11918	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 11919	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 11920	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 11921	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 11922	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 11923	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 11924	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 11925	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 11926	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 11927	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 11928	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
5.1.5  Bézout's identity
Theorem	bezoutlemnewy 11929*	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 11930*	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 11931*	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 11932*	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 11933*	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 11934*	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 11935*	Lemma for Bézout's identity. Like bezoutlemex 11934 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 11936*	Lemma for Bézout's identity. Like bezoutlemzz 11935 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 11937*	Lemma for Bézout's identity. Like bezoutlemaz 11936 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 11938*	Lemma for Bézout's identity. Like bezoutlembz 11937 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 11939*	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 11940*	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 11941*	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 11942*	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 11943*	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 11944*	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 11945	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 11946	Biconditional form of dvdsgcd 11945. (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 11947*	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 11948	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 11949	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 11950	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 11951	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 11952	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 11953	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 11954	Extend gcdmultiple 11953 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 11955	A positive integer A is equal to its gcd with an integer B if and only if A divides B. Generalization of gcdeq 11956. (Contributed by AV, 1-Jul-2020.)
|- ( ( A e. NN /\ B e. ZZ ) -> ( ( A gcd B ) = A <-> A || B ) )
Theorem	gcdeq 11956	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 11957	Unidirectional form of dvdssq 11964. (Contributed by Scott Fenton, 19-Apr-2014.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N -> ( M ^ 2 ) || ( N ^ 2 ) ) )
Theorem	dvdsmulgcd 11958	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 11959	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 11960	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 11961	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 11962	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 11963	Lemma for dvdssq 11964. (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 11964	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 11965	Partial converse to bezout 11944. 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 11966	Converse of bezout 11944 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 ) )
5.1.6  Decidable sets of integers
Theorem	nnmindc 11967*	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 11968*	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 11967. (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 11969*	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 11970*	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 11971*	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 11972*	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 ) ) )
5.1.7  Algorithms
Theorem	nn0seqcvgd 11973*	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 11974	Lemma for ialgr0 11976. Expressing algrflemg 6198 in a form suitable for theorems such as seq3-1 10395 or seqf 10396. (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 11975	Lemma for ialgr0 11976. Closure of a constant function, in a form suitable for theorems such as seq3-1 10395 or seqf 10396. (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 11976	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 )
Theorem	algrf 11977	An algorithm is a step function F : S --> S on a state space S. An algorithm acts on an initial state A e. S by iteratively applying F to give A, ( F ` A ), ( F ` ( F ` A ) ) and so on. An algorithm is said to halt if a fixed point of F is reached after a finite number of iterations. The algorithm iterator R : NN0 --> S "runs" the algorithm F so that ( R ` k ) is the state after k iterations of F on the initial state A. Domain and codomain of the algorithm iterator R. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)
|- 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 : Z --> S )
Theorem	algrp1 11978	The value of the algorithm iterator R at ( K + 1 ). (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 /\ K e. Z ) -> ( R ` ( K + 1 ) ) = ( F ` ( R ` K ) ) )
Theorem	alginv 11979*	If I is an invariant of F, then its value is unchanged after any number of iterations of F. (Contributed by Paul Chapman, 31-Mar-2011.)
|- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) )   &    |- F : S --> S   &    |- ( x e. S -> ( I ` ( F ` x ) ) = ( I ` x ) )   =>    |- ( ( A e. S /\ K e. NN0 ) -> ( I ` ( R ` K ) ) = ( I ` ( R ` 0 ) ) )
Theorem	algcvg 11980*	One way to prove that an algorithm halts is to construct a countdown function C : S --> NN0 whose value is guaranteed to decrease for each iteration of F until it reaches 0. That is, if X e. S is not a fixed point of F, then ( C ` ( F ` X ) ) < ( C ` X ). If C is a countdown function for algorithm F, the sequence ( C ` ( R ` k ) ) reaches 0 after at most N steps, where N is the value of C for the initial state A. (Contributed by Paul Chapman, 22-Jun-2011.)
|- F : S --> S   &    |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) )   &    |- C : S --> NN0   &    |- ( z e. S -> ( ( C ` ( F ` z ) ) =/= 0 -> ( C ` ( F ` z ) ) < ( C ` z ) ) )   &    |- N = ( C ` A )   =>    |- ( A e. S -> ( C ` ( R ` N ) ) = 0 )
Theorem	algcvgblem 11981	Lemma for algcvgb 11982. (Contributed by Paul Chapman, 31-Mar-2011.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( N =/= 0 -> N < M ) <-> ( ( M =/= 0 -> N < M ) /\ ( M = 0 -> N = 0 ) ) ) )
Theorem	algcvgb 11982	Two ways of expressing that C is a countdown function for algorithm F. The first is used in these theorems. The second states the condition more intuitively as a conjunction: if the countdown function's value is currently nonzero, it must decrease at the next step; if it has reached zero, it must remain zero at the next step. (Contributed by Paul Chapman, 31-Mar-2011.)
|- F : S --> S   &    |- C : S --> NN0   =>    |- ( X e. S -> ( ( ( C ` ( F ` X ) ) =/= 0 -> ( C ` ( F ` X ) ) < ( C ` X ) ) <-> ( ( ( C ` X ) =/= 0 -> ( C ` ( F ` X ) ) < ( C ` X ) ) /\ ( ( C ` X ) = 0 -> ( C ` ( F ` X ) ) = 0 ) ) ) )
Theorem	algcvga 11983*	The countdown function C remains 0 after N steps. (Contributed by Paul Chapman, 22-Jun-2011.)
|- F : S --> S   &    |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) )   &    |- C : S --> NN0   &    |- ( z e. S -> ( ( C ` ( F ` z ) ) =/= 0 -> ( C ` ( F ` z ) ) < ( C ` z ) ) )   &    |- N = ( C ` A )   =>    |- ( A e. S -> ( K e. ( ZZ>= ` N ) -> ( C ` ( R ` K ) ) = 0 ) )
Theorem	algfx 11984*	If F reaches a fixed point when the countdown function C reaches 0, F remains fixed after N steps. (Contributed by Paul Chapman, 22-Jun-2011.)
|- F : S --> S   &    |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) )   &    |- C : S --> NN0   &    |- ( z e. S -> ( ( C ` ( F ` z ) ) =/= 0 -> ( C ` ( F ` z ) ) < ( C ` z ) ) )   &    |- N = ( C ` A )   &    |- ( z e. S -> ( ( C ` z ) = 0 -> ( F ` z ) = z ) )   =>    |- ( A e. S -> ( K e. ( ZZ>= ` N ) -> ( R ` K ) = ( R ` N ) ) )
5.1.8  Euclid's Algorithm
Theorem	eucalgval2 11985*	The value of the step function E for Euclid's Algorithm on an ordered pair. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)
|- E = ( x e. NN0 , y e. NN0 |-> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) )   =>    |- ( ( M e. NN0 /\ N e. NN0 ) -> ( M E N ) = if ( N = 0 , <. M , N >. , <. N , ( M mod N ) >. ) )
Theorem	eucalgval 11986*	Euclid's Algorithm eucalg 11991 computes the greatest common divisor of two nonnegative integers by repeatedly replacing the larger of them with its remainder modulo the smaller until the remainder is 0. The value of the step function E for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)
|- E = ( x e. NN0 , y e. NN0 |-> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) )   =>    |- ( X e. ( NN0 X. NN0 ) -> ( E ` X ) = if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) )
Theorem	eucalgf 11987*	Domain and codomain of the step function E for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)
|- E = ( x e. NN0 , y e. NN0 |-> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) )   =>    |- E : ( NN0 X. NN0 ) --> ( NN0 X. NN0 )
Theorem	eucalginv 11988*	The invariant of the step function E for Euclid's Algorithm is the gcd operator applied to the state. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 29-May-2014.)
|- E = ( x e. NN0 , y e. NN0 |-> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) )   =>    |- ( X e. ( NN0 X. NN0 ) -> ( gcd ` ( E ` X ) ) = ( gcd ` X ) )
Theorem	eucalglt 11989*	The second member of the state decreases with each iteration of the step function E for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 29-May-2014.)
|- E = ( x e. NN0 , y e. NN0 |-> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) )   =>    |- ( X e. ( NN0 X. NN0 ) -> ( ( 2nd ` ( E ` X ) ) =/= 0 -> ( 2nd ` ( E ` X ) ) < ( 2nd ` X ) ) )
Theorem	eucalgcvga 11990*	Once Euclid's Algorithm halts after N steps, the second element of the state remains 0 . (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 29-May-2014.)
|- E = ( x e. NN0 , y e. NN0 |-> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) )   &    |- R = seq 0 ( ( E o. 1st ) , ( NN0 X. { A } ) )   &    |- N = ( 2nd ` A )   =>    |- ( A e. ( NN0 X. NN0 ) -> ( K e. ( ZZ>= ` N ) -> ( 2nd ` ( R ` K ) ) = 0 ) )
Theorem	eucalg 11991*	Euclid's Algorithm computes the greatest common divisor of two nonnegative integers by repeatedly replacing the larger of them with its remainder modulo the smaller until the remainder is 0. Theorem 1.15 in [ApostolNT] p. 20. Upon halting, the 1st member of the final state ( R ` N ) is equal to the gcd of the values comprising the input state <. M , N >.. This is Metamath 100 proof #69 (greatest common divisor algorithm). (Contributed by Paul Chapman, 31-Mar-2011.) (Proof shortened by Mario Carneiro, 29-May-2014.)
|- E = ( x e. NN0 , y e. NN0 |-> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) )   &    |- R = seq 0 ( ( E o. 1st ) , ( NN0 X. { A } ) )   &    |- A = <. M , N >.   =>    |- ( ( M e. NN0 /\ N e. NN0 ) -> ( 1st ` ( R ` N ) ) = ( M gcd N ) )
5.1.9  The least common multiple According to Wikipedia ("Least common multiple", 27-Aug-2020, https://en.wikipedia.org/wiki/Least_common_multiple): "In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by lcm(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm(a,0) as 0 for all a, which is the result of taking the lcm to be the least upper bound in the lattice of divisibility." In this section, an operation calculating the least common multiple of two integers (df-lcm 11993). The definition is valid for all integers, including negative integers and 0, obeying the above mentioned convention.
Syntax	clcm 11992	Extend the definition of a class to include the least common multiple operator.
class lcm
Definition	df-lcm 11993*	Define the lcm operator. For example, ( 6 lcm 9 ) = 1 8. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)
|- lcm = ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) ) )
Theorem	lcmmndc 11994	Decidablity lemma used in various proofs related to lcm. (Contributed by Jim Kingdon, 21-Jan-2022.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
Theorem	lcmval 11995*	Value of the lcm operator. ( M lcm N ) is the least common multiple of M and N. If either M or N is 0, the result is defined conventionally as 0. Contrast with df-gcd 11876 and gcdval 11892. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) )
Theorem	lcmcom 11996	The lcm operator is commutative. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = ( N lcm M ) )
Theorem	lcm0val 11997	The value, by convention, of the lcm operator when either operand is 0. (Use lcmcom 11996 for a left-hand 0.) (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
|- ( M e. ZZ -> ( M lcm 0 ) = 0 )
Theorem	lcmn0val 11998*	The value of the lcm operator when both operands are nonzero. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) = inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) )
Theorem	lcmcllem 11999*	Lemma for lcmn0cl 12000 and dvdslcm 12001. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. { n e. NN | ( M || n /\ N || n ) } )
Theorem	lcmn0cl 12000	Closure of the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. NN )