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Theorem List for Intuitionistic Logic Explorer - 10501-10600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnndvdslegcd 10501 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 ) ) )
 
Theoremgcdcl 10502 Closure of the  gcd operator. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e.  NN0 )
 
Theoremgcdnncl 10503 Closure of the  gcd operator. (Contributed by Thierry Arnoux, 2-Feb-2020.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  gcd  N )  e.  NN )
 
Theoremgcdcld 10504 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 )
 
Theoremgcd2n0cl 10505 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 )
 
Theoremzeqzmulgcd 10506* 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 ) ) )
 
Theoremdivgcdz 10507 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 )
 
Theoremgcdf 10508 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
 
Theoremgcdcom 10509 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 ) )
 
Theoremdivgcdnn 10510 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 )
 
Theoremdivgcdnnr 10511 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 )
 
Theoremgcdeq0 10512 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 ) ) )
 
Theoremgcdn0gt0 10513 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 ) ) )
 
Theoremgcd0id 10514 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 ) )
 
Theoremgcdid0 10515 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 ) )
 
Theoremnn0gcdid0 10516 The gcd of a nonnegative integer with 0 is itself. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( N  e.  NN0  ->  ( N  gcd  0 )  =  N )
 
Theoremgcdneg 10517 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 ) )
 
Theoremneggcd 10518 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 ) )
 
Theoremgcdaddm 10519 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 ) ) ) )
 
Theoremgcdadd 10520 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 ) ) )
 
Theoremgcdid 10521 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 ) )
 
Theoremgcd1 10522 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 )
 
Theoremgcdabs 10523 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 ) )
 
Theoremgcdabs1 10524  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 ) )
 
Theoremgcdabs2 10525  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 ) )
 
Theoremmodgcd 10526 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 ) )
 
Theorem1gcd 10527 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 )
 
Theorem6gcd4e2 10528 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
 
4.1.5  Bézout's identity
 
Theorembezoutlemnewy 10529* 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 )
 
Theorembezoutlemstep 10530* 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 ) )
 
Theorembezoutlemmain 10531* 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 ) ) ) )
 
Theorembezoutlema 10532* 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 )
 
Theorembezoutlemb 10533* 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 )
 
Theorembezoutlemex 10534* 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 ) ) ) )
 
Theorembezoutlemzz 10535* Lemma for Bézout's identity. Like bezoutlemex 10534 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 ) ) ) )
 
Theorembezoutlemaz 10536* Lemma for Bézout's identity. Like bezoutlemzz 10535 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 ) ) ) )
 
Theorembezoutlembz 10537* Lemma for Bézout's identity. Like bezoutlemaz 10536 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 ) ) ) )
 
Theorembezoutlembi 10538* Lemma for Bézout's identity. Like bezoutlembz 10537 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 ) ) ) )
 
Theorembezoutlemmo 10539* 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 )
 
Theorembezoutlemeu 10540* 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 ) ) )
 
Theorembezoutlemle 10541* 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 ) )
 
Theorembezoutlemsup 10542* 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 ,  <  ) )
 
Theoremdfgcd3 10543* 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 ) ) ) )
 
Theorembezout 10544* 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 ) ) )
 
Theoremdvdsgcd 10545 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 ) ) )
 
Theoremdvdsgcdb 10546 Biconditional form of dvdsgcd 10545. (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 ) ) )
 
Theoremdfgcd2 10547* 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 ) ) ) )
 
Theoremgcdass 10548 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 ) ) )
 
Theoremmulgcd 10549 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 ) ) )
 
Theoremabsmulgcd 10550 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 ) ) ) )
 
Theoremmulgcdr 10551 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 ) )
 
Theoremgcddiv 10552 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 ) ) )
 
Theoremgcdmultiple 10553 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 )
 
Theoremgcdmultiplez 10554 Extend gcdmultiple 10553 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 )
 
Theoremgcdzeq 10555 A positive integer  A is equal to its gcd with an integer  B if and only if  A divides  B. Generalization of gcdeq 10556. (Contributed by AV, 1-Jul-2020.)
 |-  ( ( A  e.  NN  /\  B  e.  ZZ )  ->  ( ( A 
 gcd  B )  =  A  <->  A 
 ||  B ) )
 
Theoremgcdeq 10556  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 ) )
 
Theoremdvdssqim 10557 Unidirectional form of dvdssq 10564. (Contributed by Scott Fenton, 19-Apr-2014.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  ->  ( M ^
 2 )  ||  ( N ^ 2 ) ) )
 
Theoremdvdsmulgcd 10558 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 ) ) ) )
 
Theoremrpmulgcd 10559 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 ) )
 
Theoremrplpwr 10560 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 ) )
 
Theoremrppwr 10561 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 ) )
 
Theoremsqgcd 10562 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 ) ) )
 
Theoremdvdssqlem 10563 Lemma for dvdssq 10564. (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 ) ) )
 
Theoremdvdssq 10564 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 ) ) )
 
Theorembezoutr 10565 Partial converse to bezout 10544. 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 ) ) )
 
Theorembezoutr1 10566 Converse of bezout 10544 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 )
 )
 
4.1.6  Algorithms
 
Theoremnn0seqcvgd 10567* 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 )
 
Theoremialgrlem1st 10568 Lemma for ialgr0 10570. Expressing algrflemg 5882 in a form suitable for theorems such as iseq1 9533 or iseqfcl 9535. (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 )
 
Theoremialgrlemconst 10569 Lemma for ialgr0 10570. Closure of a constant function, in a form suitable for theorems such as iseq1 9533 or iseqfcl 9535. (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 )
 
Theoremialgr0 10570 The value of the algorithm iterator 
R at  0 is the initial state  A. (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 }
 ) ,  S )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  F : S --> S )   &    |-  ( ph  ->  S  e.  V )   =>    |-  ( ph  ->  ( R `  M )  =  A )
 
Theoremialgrf 10571 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 }
 ) ,  S )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  F : S --> S )   &    |-  ( ph  ->  S  e.  V )   =>    |-  ( ph  ->  R : Z --> S )
 
Theoremialgrp1 10572 The value of the algorithm iterator 
R at  ( K  + 
1 ). (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  R  =  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A }
 ) ,  S )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  F : S --> S )   &    |-  ( ph  ->  S  e.  V )   =>    |-  ( ( ph  /\  K  e.  Z ) 
 ->  ( R `  ( K  +  1 )
 )  =  ( F `
  ( R `  K ) ) )
 
Theoremialginv 10573* If  I is an invariant of  F, 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 } ) ,  S )   &    |-  F : S --> S   &    |-  I  Fn  S   &    |-  ( x  e.  S  ->  ( I `  ( F `
  x ) )  =  ( I `  x ) )   &    |-  S  e.  V   =>    |-  ( ( A  e.  S  /\  K  e.  NN0 )  ->  ( I `  ( R `  K ) )  =  ( I `
  ( R `  0 ) ) )
 
Theoremialgcvg 10574* 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 } ) ,  S )   &    |-  C : S --> NN0   &    |-  ( z  e.  S  ->  ( ( C `  ( F `  z ) )  =/=  0  ->  ( C `  ( F `
  z ) )  <  ( C `  z ) ) )   &    |-  N  =  ( C `  A )   &    |-  S  e.  V   =>    |-  ( A  e.  S  ->  ( C `  ( R `
  N ) )  =  0 )
 
Theoremalgcvgblem 10575 Lemma for algcvgb 10576. (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 ) ) ) )
 
Theoremalgcvgb 10576 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
 ) ) ) )
 
Theoremialgcvga 10577* 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 } ) ,  S )   &    |-  C : S --> NN0   &    |-  ( z  e.  S  ->  ( ( C `  ( F `  z ) )  =/=  0  ->  ( C `  ( F `
  z ) )  <  ( C `  z ) ) )   &    |-  N  =  ( C `  A )   &    |-  S  e.  V   =>    |-  ( A  e.  S  ->  ( K  e.  ( ZZ>= `  N )  ->  ( C `
  ( R `  K ) )  =  0 ) )
 
Theoremialgfx 10578* 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 } ) ,  S )   &    |-  C : S --> NN0   &    |-  ( z  e.  S  ->  ( ( C `  ( F `  z ) )  =/=  0  ->  ( C `  ( F `
  z ) )  <  ( C `  z ) ) )   &    |-  N  =  ( C `  A )   &    |-  S  e.  V   &    |-  (
 z  e.  S  ->  ( ( C `  z
 )  =  0  ->  ( F `  z )  =  z ) )   =>    |-  ( A  e.  S  ->  ( K  e.  ( ZZ>=
 `  N )  ->  ( R `  K )  =  ( R `  N ) ) )
 
4.1.7  Euclid's Algorithm
 
Theoremeucalgval2 10579* 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 ) >. ) )
 
Theoremeucalgval 10580* Euclid's Algorithm eucialg 10585 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 ) >. ) )
 
Theoremeucalgf 10581* 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 )
 
Theoremeucalginv 10582* 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 ) )
 
Theoremeucalglt 10583* 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 )
 ) )
 
Theoremeucialgcvga 10584* Once Euclid's Algorithm halts after 
N steps, the second element of the state remains 0 . (Contributed by Jim Kingdon, 11-Jan-2022.)
 |-  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 } ) ,  ( NN0  X.  NN0 ) )   &    |-  N  =  ( 2nd `  A )   =>    |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( K  e.  ( ZZ>= `  N )  ->  ( 2nd `  ( R `  K ) )  =  0
 ) )
 
Theoremeucialg 10585* 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 Jim Kingdon, 11-Jan-2022.)

 |-  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 } ) ,  ( NN0  X.  NN0 ) )   &    |-  N  =  ( 2nd `  A )   &    |-  A  =  <. M ,  N >.   =>    |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( 1st `  ( R `  N ) )  =  ( M  gcd  N ) )
 
4.1.8  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 10587). The definition is valid for all integers, including negative integers and 0, obeying the above mentioned convention.

 
Syntaxclcm 10586 Extend the definition of a class to include the least common multiple operator.
 class lcm
 
Definitiondf-lcm 10587* 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 ,  <  )
 ) )
 
Theoremlcmmndc 10588 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 ) )
 
Theoremlcmval 10589* 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 10483 and gcdval 10495. (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 ,  <  ) ) )
 
Theoremlcmcom 10590 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 ) )
 
Theoremlcm0val 10591 The value, by convention, of the lcm operator when either operand is 0. (Use lcmcom 10590 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 )
 
Theoremlcmn0val 10592* 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 ,  <  )
 )
 
Theoremlcmcllem 10593* Lemma for lcmn0cl 10594 and dvdslcm 10595. (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 ) } )
 
Theoremlcmn0cl 10594 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 )
 
Theoremdvdslcm 10595 The lcm of two integers is divisible by each of them. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  ( M lcm  N )  /\  N  ||  ( M lcm  N ) ) )
 
Theoremlcmledvds 10596 A positive integer which both operands of the lcm operator divide bounds it. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
 |-  ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 )
 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  <_  K ) )
 
Theoremlcmeq0 10597 The lcm of two integers is zero iff either is zero. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M lcm 
 N )  =  0  <-> 
 ( M  =  0  \/  N  =  0 ) ) )
 
Theoremlcmcl 10598 Closure of the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  NN0 )
 
Theoremgcddvdslcm 10599 The greatest common divisor of two numbers divides their least common multiple. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  ||  ( M lcm  N ) )
 
Theoremlcmneg 10600 Negating one operand of the lcm operator does not alter the result. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  -u N )  =  ( M lcm  N ) )
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