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Theorem List for Metamath Proof Explorer - 14501-14600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsmup0 14501* The initial element of the partial sum sequence. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  P  =  seq 0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m )  e.  B ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )   =>    |-  ( ph  ->  ( P `  0 )  =  (/) )
 
Theoremsmupp1 14502* The initial element of the partial sum sequence. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  P  =  seq 0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m )  e.  B ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( P `  ( N  +  1 ) )  =  ( ( P `  N ) sadd  { n  e. 
 NN0  |  ( N  e.  A  /\  ( n  -  N )  e.  B ) } )
 )
 
Theoremsmuval 14503* Define the addition of two bit sequences, using df-had 1507 and df-cad 1520 bit operations. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  P  =  seq 0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m )  e.  B ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( N  e.  ( A smul  B )  <->  N  e.  ( P `  ( N  +  1 ) ) ) )
 
Theoremsmuval2 14504* The partial sum sequence stabilizes at  N after the  N  +  1-th element of the sequence; this stable value is the value of the sequence multiplication. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  P  =  seq 0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m )  e.  B ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  ( N  +  1 )
 ) )   =>    |-  ( ph  ->  ( N  e.  ( A smul  B )  <->  N  e.  ( P `  M ) ) )
 
Theoremsmupvallem 14505* If  A only has elements less than  N, then all elements of the partial sum sequence past  N already equal the final value. (Contributed by Mario Carneiro, 20-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  P  =  seq 0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m )  e.  B ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  A 
 C_  ( 0..^ N ) )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  N ) )   =>    |-  ( ph  ->  ( P `  M )  =  ( A smul  B ) )
 
Theoremsmucl 14506 The product of two sequences is a sequence. (Contributed by Mario Carneiro, 19-Sep-2016.)
 |-  ( ( A  C_  NN0  /\  B  C_  NN0 )  ->  ( A smul  B )  C_  NN0 )
 
Theoremsmu01lem 14507* Lemma for smu01 14508 and smu02 14509. (Contributed by Mario Carneiro, 19-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  ( ( ph  /\  ( k  e.  NN0  /\  n  e.  NN0 )
 )  ->  -.  (
 k  e.  A  /\  ( n  -  k
 )  e.  B ) )   =>    |-  ( ph  ->  ( A smul  B )  =  (/) )
 
Theoremsmu01 14508 Multiplication of a sequence by  0 on the right. (Contributed by Mario Carneiro, 19-Sep-2016.)
 |-  ( A  C_  NN0  ->  ( A smul  (/) )  =  (/) )
 
Theoremsmu02 14509 Multiplication of a sequence by  0 on the left. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( A  C_  NN0  ->  ( (/) smul  A )  =  (/) )
 
Theoremsmupval 14510* Rewrite the elements of the partial sum sequence in terms of sequence multiplication. (Contributed by Mario Carneiro, 20-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  P  =  seq 0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m )  e.  B ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( P `  N )  =  ( ( A  i^i  ( 0..^ N ) ) smul 
 B ) )
 
Theoremsmup1 14511* Rewrite smupp1 14502 using only smul instead of the internal recursive function  P. (Contributed by Mario Carneiro, 20-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( A  i^i  (
 0..^ ( N  +  1 ) ) ) smul 
 B )  =  ( ( ( A  i^i  ( 0..^ N ) ) smul 
 B ) sadd  { n  e.  NN0  |  ( N  e.  A  /\  ( n  -  N )  e.  B ) } )
 )
 
Theoremsmueqlem 14512* Any element of a sequence multiplication only depends on the values of the argument sequences up to and including that point. (Contributed by Mario Carneiro, 20-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  P  =  seq 0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m )  e.  B ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )   &    |-  Q  =  seq 0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  A  /\  ( n  -  m )  e.  ( B  i^i  ( 0..^ N ) ) ) } )
 ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   =>    |-  ( ph  ->  (
 ( A smul  B )  i^i  ( 0..^ N ) )  =  ( ( ( A  i^i  (
 0..^ N ) ) smul 
 ( B  i^i  (
 0..^ N ) ) )  i^i  ( 0..^ N ) ) )
 
Theoremsmueq 14513 Any element of a sequence multiplication only depends on the values of the argument sequences up to and including that point. (Contributed by Mario Carneiro, 20-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( A smul  B )  i^i  ( 0..^ N ) )  =  ( ( ( A  i^i  (
 0..^ N ) ) smul 
 ( B  i^i  (
 0..^ N ) ) )  i^i  ( 0..^ N ) ) )
 
Theoremsmumullem 14514 Lemma for smumul 14515. (Contributed by Mario Carneiro, 22-Sep-2016.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( (bits `  A )  i^i  ( 0..^ N ) ) smul  (bits `  B ) )  =  (bits `  ( ( A  mod  ( 2 ^ N ) )  x.  B ) ) )
 
Theoremsmumul 14515 For sequences that correspond to valid integers, the sequence multiplication function produces the sequence for the product. This is effectively a proof of the correctness of the multiplication process, implemented in terms of logic gates for df-sad 14473, whose correctness is verified in sadadd 14489.

Outside this range, the sequences cannot be representing integers, but the smul function still "works". This extended function is best interpreted in terms of the ring structure of the 2-adic integers. (Contributed by Mario Carneiro, 22-Sep-2016.)

 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( (bits `  A ) smul  (bits `  B ) )  =  (bits `  ( A  x.  B ) ) )
 
6.1.6  The greatest common divisor operator
 
Syntaxcgcd 14516 Extend the definition of a class to include the greatest common divisor operator.
 class  gcd
 
Definitiondf-gcd 14517* Define the  gcd operator. (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 ,  <  ) ) )
 
Theoremgcdval 14518* 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 ,  <  ) ) )
 
Theoremgcd0val 14519 The value, by convention, of the 
gcd operator when both operands are 0. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( 0  gcd  0
 )  =  0
 
Theoremgcdn0val 14520* 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 ,  <  ) )
 
Theoremgcdcllem1 14521* Lemma for gcdn0cl 14524, gcddvds 14525 and dvdslegcd 14526. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  S  =  { z  e.  ZZ  |  A. n  e.  A  z  ||  n }   =>    |-  ( ( A  C_  ZZ  /\  E. n  e.  A  n  =/=  0
 )  ->  ( S  =/= 
 (/)  /\  E. x  e. 
 ZZ  A. y  e.  S  y  <_  x ) )
 
Theoremgcdcllem2 14522* Lemma for gcdn0cl 14524, gcddvds 14525 and dvdslegcd 14526. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  S  =  { z  e.  ZZ  |  A. n  e.  { M ,  N } z  ||  n }   &    |-  R  =  { z  e.  ZZ  |  ( z  ||  M  /\  z  ||  N ) }   =>    |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  R  =  S )
 
Theoremgcdcllem3 14523* Lemma for gcdn0cl 14524, gcddvds 14525 and dvdslegcd 14526. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  S  =  { z  e.  ZZ  |  A. n  e.  { M ,  N } z  ||  n }   &    |-  R  =  { z  e.  ZZ  |  ( z  ||  M  /\  z  ||  N ) }   =>    |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( sup ( R ,  RR ,  <  )  e.  NN  /\  ( sup ( R ,  RR ,  <  )  ||  M  /\  sup ( R ,  RR ,  <  ) 
 ||  N )  /\  ( ( K  e.  ZZ  /\  K  ||  M  /\  K  ||  N )  ->  K  <_  sup ( R ,  RR ,  <  ) ) ) )
 
Theoremgcdn0cl 14524 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 )
 
Theoremgcddvds 14525 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 ) )
 
Theoremdvdslegcd 14526 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 ) ) )
 
Theoremnndvdslegcd 14527 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 14528 Closure of the  gcd operator. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e.  NN0 )
 
Theoremgcdnncl 14529 Closure of the  gcd operator. (Contributed by Thierry Arnoux, 2-Feb-2020.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  gcd  N )  e.  NN )
 
Theoremgcdcld 14530 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 )
 
Theoremgcdf 14531 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 14532 The  gcd operator is commutative. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  =  ( N 
 gcd  M ) )
 
Theoremgcdeq0 14533 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 14534 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 14535 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 14536 The gcd of an integer and 0 is the integer's absolute value. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( N  e.  ZZ  ->  ( N  gcd  0
 )  =  ( abs `  N ) )
 
Theoremnn0gcdid0 14537 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 14538 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 14539 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 ) )
 
Theoremgcdaddmlem 14540 Lemma for gcdaddm 14541. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  K  e.  ZZ   &    |-  M  e.  ZZ   &    |-  N  e.  ZZ   =>    |-  ( M  gcd  N )  =  ( M  gcd  (
 ( K  x.  M )  +  N )
 )
 
Theoremgcdaddm 14541 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 14542 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 14543 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 14544 The gcd of a number with 1 is 1. (Contributed by Mario Carneiro, 19-Feb-2014.)
 |-  ( M  e.  ZZ  ->  ( M  gcd  1
 )  =  1 )
 
Theoremgcdabs 14545 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 14546  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 14547  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 14548 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 14549 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 14550 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
 
6.1.7  Bézout's identity
 
Theorembezoutlem1 14551* Lemma for bezout 14558. (Contributed by Mario Carneiro, 15-Mar-2014.)
 |-  M  =  { z  e.  NN  |  E. x  e.  ZZ  E. y  e. 
 ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y ) ) }   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   =>    |-  ( ph  ->  ( A  =/=  0  ->  ( abs `  A )  e.  M ) )
 
Theorembezoutlem2OLD 14552* Lemma for bezout 14558. (Contributed by Mario Carneiro, 15-Mar-2014.) Obsolete version of bezoutlem2 14555 as of 30-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  M  =  { z  e.  NN  |  E. x  e.  ZZ  E. y  e. 
 ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y ) ) }   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  G  =  sup ( M ,  RR ,  `'  <  )   &    |-  ( ph  ->  -.  ( A  =  0 
 /\  B  =  0 ) )   =>    |-  ( ph  ->  G  e.  M )
 
Theorembezoutlem3OLD 14553* Lemma for bezout 14558. (Contributed by Mario Carneiro, 22-Feb-2014.) Obsolete version of bezoutlem3 14556 as of 30-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  M  =  { z  e.  NN  |  E. x  e.  ZZ  E. y  e. 
 ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y ) ) }   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  G  =  sup ( M ,  RR ,  `'  <  )   &    |-  ( ph  ->  -.  ( A  =  0 
 /\  B  =  0 ) )   =>    |-  ( ph  ->  ( C  e.  M  ->  G 
 ||  C ) )
 
Theorembezoutlem4OLD 14554* Lemma for bezout 14558. (Contributed by Mario Carneiro, 22-Feb-2014.) Obsolete version of bezoutlem4 14557 as of 30-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  M  =  { z  e.  NN  |  E. x  e.  ZZ  E. y  e. 
 ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y ) ) }   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  G  =  sup ( M ,  RR ,  `'  <  )   &    |-  ( ph  ->  -.  ( A  =  0 
 /\  B  =  0 ) )   =>    |-  ( ph  ->  ( A  gcd  B )  e.  M )
 
Theorembezoutlem2 14555* Lemma for bezout 14558. (Contributed by Mario Carneiro, 15-Mar-2014.) ( Revised by AV, 30-Sep-2020.)
 |-  M  =  { z  e.  NN  |  E. x  e.  ZZ  E. y  e. 
 ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y ) ) }   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  G  = inf ( M ,  RR ,  <  )   &    |-  ( ph  ->  -.  ( A  =  0  /\  B  =  0 )
 )   =>    |-  ( ph  ->  G  e.  M )
 
Theorembezoutlem3 14556* Lemma for bezout 14558. (Contributed by Mario Carneiro, 22-Feb-2014.) ( Revised by AV, 30-Sep-2020.)
 |-  M  =  { z  e.  NN  |  E. x  e.  ZZ  E. y  e. 
 ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y ) ) }   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  G  = inf ( M ,  RR ,  <  )   &    |-  ( ph  ->  -.  ( A  =  0  /\  B  =  0 )
 )   =>    |-  ( ph  ->  ( C  e.  M  ->  G 
 ||  C ) )
 
Theorembezoutlem4 14557* Lemma for bezout 14558. (Contributed by Mario Carneiro, 22-Feb-2014.)
 |-  M  =  { z  e.  NN  |  E. x  e.  ZZ  E. y  e. 
 ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y ) ) }   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  G  = inf ( M ,  RR ,  <  )   &    |-  ( ph  ->  -.  ( A  =  0  /\  B  =  0 )
 )   =>    |-  ( ph  ->  ( A  gcd  B )  e.  M )
 
Theorembezout 14558* 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. (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 14559 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 14560 Biconditional form of dvdsgcd 14559. (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 ) ) )
 
Theoremgcdass 14561 Associative law for  gcd operator. (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 14562 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 14563 Distribute absolute value of multiplication over gcd. (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 14564 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 14565 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 14566 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 14567 Extend gcdmultiple 14566 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 )
 
Theoremgcdeq 14568  A is equal to its gcd with  B if and only if  A divides  B. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A 
 gcd  B )  =  A  <->  A 
 ||  B ) )
 
Theoremdvdssqim 14569 Unidirectional form of dvdssq 14576. (Contributed by Scott Fenton, 19-Apr-2014.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  ->  ( M ^
 2 )  ||  ( N ^ 2 ) ) )
 
Theoremdvdsmulgcd 14570 A divisibility equivalent for odmulg 17255. (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 14571 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 14572 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 14573 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 14574 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 14575 Lemma for dvdssq 14576. (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 14576 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 ) ) )
 
6.1.8  Algorithms
 
Theoremnn0seqcvgd 14577* 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 )
 
Theoremseq1st 14578 A sequence whose iteration function ignores the second argument is only affected by the first point of the initial value function. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  R  =  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A }
 ) )   =>    |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  R  =  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } )
 )
 
Theoremalgr0 14579 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 }
 ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   =>    |-  ( ph  ->  ( R `  M )  =  A )
 
Theoremalgrf 14580 An algorithm is step a 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 )
 
Theoremalgrp1 14581 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 }
 ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  F : S --> S )   =>    |-  ( ( ph  /\  K  e.  Z ) 
 ->  ( R `  ( K  +  1 )
 )  =  ( F `
  ( R `  K ) ) )
 
Theoremalginv 14582* 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 } )
 )   &    |-  F : S --> S   &    |-  I  Fn  S   &    |-  ( x  e.  S  ->  ( I `  ( F `  x ) )  =  ( I `  x ) )   =>    |-  ( ( A  e.  S  /\  K  e.  NN0 )  ->  ( I `  ( R `  K ) )  =  ( I `
  ( R `  0 ) ) )
 
Theoremalgcvg 14583* 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 )
 
Theoremalgcvgblem 14584 Lemma for algcvgb 14585. (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 14585 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 non-zero, 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
 ) ) ) )
 
Theoremalgcvga 14586* 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 ) )
 
Theoremalgfx 14587* 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 ) ) )
 
6.1.9  Euclid's Algorithm
 
Theoremeucalgval2 14588* 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 14589* Euclid's Algorithm eucalg 14594 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 14590* 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 14591* 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 14592* 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 )
 ) )
 
Theoremeucalgcvga 14593* 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
 ) )
 
Theoremeucalg 14594* 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.

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 ) )
 
6.1.10  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. ... The lcm of more than two integers is also well-defined: it is the smallest positive integer that is divisible by each of them."

In this section, an operation calculating the least common multiple of two integers (df-lcm 14599) as well as a function mapping a set of integers to their least common multiple (df-lcmf 14601) are provided. Both definitions are valid for all integers, including negative integers and 0, obeying the above mentioned convention. It is shown by lcmfpr 14648 that the two definitions are compatible.

It seems a little confusing at the first glance that the least common multiple is defined by a supremum. Actually, it is an infimum, because the inverse of "less than"  `'  < is used to turn supremum into infimum - currently we don't have infimum defined separately...

 
Syntaxclcm 14595 Extend the definition of a class to include the least common multiple operator.
 class lcm
 
Syntaxclcmf 14596 Extend the definition of a class to include the least common multiple function.
 class lcm
 
Syntaxclcmold 14597 Extend the definition of a class to include the least common multiple operator (old version.
 class lcm
 
Syntaxclcmfold 14598 Extend the definition of a class to include the least common multiple function (old version).
 class lcm
 
Definitiondf-lcm 14599* Define the lcm operator. (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 ,  <  )
 ) )
 
Definitiondf-lcmOLD 14600* Define the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) Obsolete version of df-lcm 14599 as of 16-Sep-2020. (New usage is discouraged.)
 |- lcm 
 =  ( x  e. 
 ZZ ,  y  e. 
 ZZ  |->  if ( ( x  =  0  \/  y  =  0 ) ,  0 ,  sup ( { n  e.  NN  |  ( x  ||  n  /\  y  ||  n ) } ,  RR ,  `'  <  ) ) )
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