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Theorem List for Intuitionistic Logic Explorer - 11601-11700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremn2dvds3 11601 2 does not divide 3, i.e. 3 is an odd number. (Contributed by AV, 28-Feb-2021.)
 |- 
 -.  2  ||  3
 
Theoremz4even 11602 4 is an even number. (Contributed by AV, 23-Jul-2020.) (Revised by AV, 4-Jul-2021.)
 |-  2  ||  4
 
Theorem4dvdseven 11603 An integer which is divisible by 4 is an even integer. (Contributed by AV, 4-Jul-2021.)
 |-  ( 4  ||  N  ->  2  ||  N )
 
5.1.3  The division algorithm
 
Theoremdivalglemnn 11604* Lemma for divalg 11610. Existence for a positive denominator. (Contributed by Jim Kingdon, 30-Nov-2021.)
 |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  E. r  e.  ZZ  E. q  e.  ZZ  (
 0  <_  r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  r )
 ) )
 
Theoremdivalglemqt 11605 Lemma for divalg 11610. The  Q  =  T case involved in showing uniqueness. (Contributed by Jim Kingdon, 5-Dec-2021.)
 |-  ( ph  ->  D  e.  ZZ )   &    |-  ( ph  ->  R  e.  ZZ )   &    |-  ( ph  ->  S  e.  ZZ )   &    |-  ( ph  ->  Q  e.  ZZ )   &    |-  ( ph  ->  T  e.  ZZ )   &    |-  ( ph  ->  Q  =  T )   &    |-  ( ph  ->  (
 ( Q  x.  D )  +  R )  =  ( ( T  x.  D )  +  S ) )   =>    |-  ( ph  ->  R  =  S )
 
Theoremdivalglemnqt 11606 Lemma for divalg 11610. The  Q  <  T case involved in showing uniqueness. (Contributed by Jim Kingdon, 4-Dec-2021.)
 |-  ( ph  ->  D  e.  NN )   &    |-  ( ph  ->  R  e.  ZZ )   &    |-  ( ph  ->  S  e.  ZZ )   &    |-  ( ph  ->  Q  e.  ZZ )   &    |-  ( ph  ->  T  e.  ZZ )   &    |-  ( ph  ->  0  <_  S )   &    |-  ( ph  ->  R  <  D )   &    |-  ( ph  ->  ( ( Q  x.  D )  +  R )  =  ( ( T  x.  D )  +  S ) )   =>    |-  ( ph  ->  -.  Q  <  T )
 
Theoremdivalglemeunn 11607* Lemma for divalg 11610. Uniqueness for a positive denominator. (Contributed by Jim Kingdon, 4-Dec-2021.)
 |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  E! r  e. 
 ZZ  E. q  e.  ZZ  ( 0  <_  r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  r
 ) ) )
 
Theoremdivalglemex 11608* Lemma for divalg 11610. The quotient and remainder exist. (Contributed by Jim Kingdon, 30-Nov-2021.)
 |-  ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 ) 
 ->  E. r  e.  ZZ  E. q  e.  ZZ  (
 0  <_  r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  r )
 ) )
 
Theoremdivalglemeuneg 11609* Lemma for divalg 11610. Uniqueness for a negative denominator. (Contributed by Jim Kingdon, 4-Dec-2021.)
 |-  ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  <  0 ) 
 ->  E! r  e.  ZZ  E. q  e.  ZZ  (
 0  <_  r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  r )
 ) )
 
Theoremdivalg 11610* The division algorithm (theorem). Dividing an integer  N by a nonzero integer  D produces a (unique) quotient  q and a unique remainder  0  <_  r  <  ( abs `  D
). Theorem 1.14 in [ApostolNT] p. 19. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 ) 
 ->  E! r  e.  ZZ  E. q  e.  ZZ  (
 0  <_  r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  r )
 ) )
 
Theoremdivalgb 11611* Express the division algorithm as stated in divalg 11610 in terms of  ||. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 ) 
 ->  ( E! r  e. 
 ZZ  E. q  e.  ZZ  ( 0  <_  r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  r
 ) )  <->  E! r  e.  NN0  ( r  <  ( abs `  D )  /\  D  ||  ( N  -  r
 ) ) ) )
 
Theoremdivalg2 11612* The division algorithm (theorem) for a positive divisor. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  E! r  e. 
 NN0  ( r  <  D  /\  D  ||  ( N  -  r ) ) )
 
Theoremdivalgmod 11613 The result of the  mod operator satisfies the requirements for the remainder  R in the division algorithm for a positive divisor (compare divalg2 11612 and divalgb 11611). This demonstration theorem justifies the use of  mod to yield an explicit remainder from this point forward. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by AV, 21-Aug-2021.)
 |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( R  =  ( N  mod  D )  <-> 
 ( R  e.  NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R ) ) ) ) )
 
Theoremdivalgmodcl 11614 The result of the  mod operator satisfies the requirements for the remainder  R in the division algorithm for a positive divisor. Variant of divalgmod 11613. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof shortened by AV, 21-Aug-2021.)
 |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  R  e.  NN0 )  ->  ( R  =  ( N  mod  D )  <-> 
 ( R  <  D  /\  D  ||  ( N  -  R ) ) ) )
 
Theoremmodremain 11615* The result of the modulo operation is the remainder of the division algorithm. (Contributed by AV, 19-Aug-2021.)
 |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  ( R  e.  NN0  /\  R  <  D ) )  ->  ( ( N  mod  D )  =  R  <->  E. z  e.  ZZ  ( ( z  x.  D )  +  R )  =  N )
 )
 
Theoremndvdssub 11616 Corollary of the division algorithm. If an integer  D greater than  1 divides  N, then it does not divide any of  N  -  1,  N  -  2...  N  -  ( D  -  1 ). (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  ( K  e.  NN  /\  K  <  D ) )  ->  ( D  ||  N  ->  -.  D  ||  ( N  -  K ) ) )
 
Theoremndvdsadd 11617 Corollary of the division algorithm. If an integer  D greater than  1 divides  N, then it does not divide any of  N  +  1,  N  +  2...  N  +  ( D  -  1 ). (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  ( K  e.  NN  /\  K  <  D ) )  ->  ( D  ||  N  ->  -.  D  ||  ( N  +  K ) ) )
 
Theoremndvdsp1 11618 Special case of ndvdsadd 11617. If an integer  D greater than  1 divides  N, it does not divide  N  +  1. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  1  <  D ) 
 ->  ( D  ||  N  ->  -.  D  ||  ( N  +  1 )
 ) )
 
Theoremndvdsi 11619 A quick test for non-divisibility. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  A  e.  NN   &    |-  Q  e.  NN0   &    |-  R  e.  NN   &    |-  (
 ( A  x.  Q )  +  R )  =  B   &    |-  R  <  A   =>    |-  -.  A  ||  B
 
Theoremflodddiv4 11620 The floor of an odd integer divided by 4. (Contributed by AV, 17-Jun-2021.)
 |-  ( ( M  e.  ZZ  /\  N  =  ( ( 2  x.  M )  +  1 )
 )  ->  ( |_ `  ( N  /  4
 ) )  =  if ( 2  ||  M ,  ( M  /  2
 ) ,  ( ( M  -  1 ) 
 /  2 ) ) )
 
Theoremfldivndvdslt 11621 The floor of an integer divided by a nonzero integer not dividing the first integer is less than the integer divided by the positive integer. (Contributed by AV, 4-Jul-2021.)
 |-  ( ( K  e.  ZZ  /\  ( L  e.  ZZ  /\  L  =/=  0
 )  /\  -.  L  ||  K )  ->  ( |_ `  ( K  /  L ) )  <  ( K 
 /  L ) )
 
Theoremflodddiv4lt 11622 The floor of an odd number divided by 4 is less than the odd number divided by 4. (Contributed by AV, 4-Jul-2021.)
 |-  ( ( N  e.  ZZ  /\  -.  2  ||  N )  ->  ( |_ `  ( N  /  4
 ) )  <  ( N  /  4 ) )
 
Theoremflodddiv4t2lthalf 11623 The floor of an odd number divided by 4, multiplied by 2 is less than the half of the odd number. (Contributed by AV, 4-Jul-2021.)
 |-  ( ( N  e.  ZZ  /\  -.  2  ||  N )  ->  ( ( |_ `  ( N 
 /  4 ) )  x.  2 )  < 
 ( N  /  2
 ) )
 
5.1.4  The greatest common divisor operator
 
Syntaxcgcd 11624 Extend the definition of a class to include the greatest common divisor operator.
 class  gcd
 
Definitiondf-gcd 11625* Define the  gcd operator. For example,  ( -u 6  gcd  9 )  =  3 (ex-gcd 12932). (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 ,  <  ) ) )
 
Theoremgcdmndc 11626 Decidablity lemma used in various proofs related to  gcd. (Contributed by Jim Kingdon, 12-Dec-2021.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID 
 ( M  =  0 
 /\  N  =  0 ) )
 
Theoremzsupcllemstep 11627* Lemma for zsupcl 11629. Induction step. (Contributed by Jim Kingdon, 7-Dec-2021.)
 |-  ( ( ph  /\  n  e.  ( ZZ>= `  M )
 )  -> DECID  ps )   =>    |-  ( K  e.  ( ZZ>=
 `  M )  ->  ( ( ( ph  /\ 
 A. n  e.  ( ZZ>=
 `  K )  -.  ps )  ->  E. x  e.  ZZ  ( A. y  e.  { n  e.  ZZ  |  ps }  -.  x  <  y  /\  A. y  e.  RR  ( y  < 
 x  ->  E. z  e.  { n  e.  ZZ  |  ps } y  < 
 z ) ) ) 
 ->  ( ( ph  /\  A. n  e.  ( ZZ>= `  ( K  +  1
 ) )  -.  ps )  ->  E. x  e.  ZZ  ( A. y  e.  { n  e.  ZZ  |  ps }  -.  x  <  y  /\  A. y  e.  RR  ( y  <  x  ->  E. z  e.  { n  e.  ZZ  |  ps }
 y  <  z )
 ) ) ) )
 
Theoremzsupcllemex 11628* Lemma for zsupcl 11629. Existence of the supremum. (Contributed by Jim Kingdon, 7-Dec-2021.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( n  =  M  ->  ( ps  <->  ch ) )   &    |-  ( ph  ->  ch )   &    |-  ( ( ph  /\  n  e.  ( ZZ>= `  M ) )  -> DECID  ps )   &    |-  ( ph  ->  E. j  e.  ( ZZ>= `  M ) A. n  e.  ( ZZ>=
 `  j )  -.  ps )   =>    |-  ( ph  ->  E. x  e.  ZZ  ( A. y  e.  { n  e.  ZZ  |  ps }  -.  x  <  y  /\  A. y  e.  RR  ( y  < 
 x  ->  E. z  e.  { n  e.  ZZ  |  ps } y  < 
 z ) ) )
 
Theoremzsupcl 11629* Closure of supremum for decidable integer properties. The property which defines the set we are taking the supremum of must (a) be true at  M (which corresponds to the nonempty condition of classical supremum theorems), (b) decidable at each value after  M, and (c) be false after  j (which corresponds to the upper bound condition found in classical supremum theorems). (Contributed by Jim Kingdon, 7-Dec-2021.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( n  =  M  ->  ( ps  <->  ch ) )   &    |-  ( ph  ->  ch )   &    |-  ( ( ph  /\  n  e.  ( ZZ>= `  M ) )  -> DECID  ps )   &    |-  ( ph  ->  E. j  e.  ( ZZ>= `  M ) A. n  e.  ( ZZ>=
 `  j )  -.  ps )   =>    |-  ( ph  ->  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  )  e.  ( ZZ>=
 `  M ) )
 
Theoremzssinfcl 11630* The infimum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 16-Jan-2022.)
 |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  B  -.  y  < 
 x  /\  A. y  e. 
 RR  ( x  < 
 y  ->  E. z  e.  B  z  <  y
 ) ) )   &    |-  ( ph  ->  B  C_  ZZ )   &    |-  ( ph  -> inf ( B ,  RR ,  <  )  e.  ZZ )   =>    |-  ( ph  -> inf ( B ,  RR ,  <  )  e.  B )
 
Theoreminfssuzex 11631* Existence of the infimum of a subset of an upper set of integers. (Contributed by Jim Kingdon, 13-Jan-2022.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  S  =  { n  e.  ( ZZ>= `  M )  |  ps }   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  n  e.  ( M
 ... A ) ) 
 -> DECID  ps )   =>    |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  S  -.  y  < 
 x  /\  A. y  e. 
 RR  ( x  < 
 y  ->  E. z  e.  S  z  <  y
 ) ) )
 
Theoreminfssuzledc 11632* The infimum of a subset of an upper set of integers is less than or equal to all members of the subset. (Contributed by Jim Kingdon, 13-Jan-2022.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  S  =  { n  e.  ( ZZ>= `  M )  |  ps }   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  n  e.  ( M
 ... A ) ) 
 -> DECID  ps )   =>    |-  ( ph  -> inf ( S ,  RR ,  <  ) 
 <_  A )
 
Theoreminfssuzcldc 11633* The infimum of a subset of an upper set of integers belongs to the subset. (Contributed by Jim Kingdon, 20-Jan-2022.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  S  =  { n  e.  ( ZZ>= `  M )  |  ps }   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  n  e.  ( M
 ... A ) ) 
 -> DECID  ps )   =>    |-  ( ph  -> inf ( S ,  RR ,  <  )  e.  S )
 
Theoremdvdsbnd 11634* There is an upper bound to the divisors of a nonzero integer. (Contributed by Jim Kingdon, 11-Dec-2021.)
 |-  ( ( A  e.  ZZ  /\  A  =/=  0
 )  ->  E. n  e.  NN  A. m  e.  ( ZZ>= `  n )  -.  m  ||  A )
 
Theoremgcdsupex 11635* Existence of the supremum used in defining  gcd. (Contributed by Jim Kingdon, 12-Dec-2021.)
 |-  ( ( ( X  e.  ZZ  /\  Y  e.  ZZ )  /\  -.  ( X  =  0  /\  Y  =  0 ) )  ->  E. x  e.  ZZ  ( A. y  e.  { n  e.  ZZ  |  ( n  ||  X  /\  n  ||  Y ) }  -.  x  < 
 y  /\  A. y  e. 
 RR  ( y  < 
 x  ->  E. z  e.  { n  e.  ZZ  |  ( n  ||  X  /\  n  ||  Y ) } y  <  z
 ) ) )
 
Theoremgcdsupcl 11636* Closure of the supremum used in defining  gcd. A lemma for gcdval 11637 and gcdn0cl 11640. (Contributed by Jim Kingdon, 11-Dec-2021.)
 |-  ( ( ( X  e.  ZZ  /\  Y  e.  ZZ )  /\  -.  ( X  =  0  /\  Y  =  0 ) )  ->  sup ( { n  e.  ZZ  |  ( n  ||  X  /\  n  ||  Y ) } ,  RR ,  <  )  e.  NN )
 
Theoremgcdval 11637* 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 11638 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 11639* 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 ,  <  ) )
 
Theoremgcdn0cl 11640 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 11641 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 11642 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 11643 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 11644 Closure of the  gcd operator. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e.  NN0 )
 
Theoremgcdnncl 11645 Closure of the  gcd operator. (Contributed by Thierry Arnoux, 2-Feb-2020.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  gcd  N )  e.  NN )
 
Theoremgcdcld 11646 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 11647 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 11648* 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 11649 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 11650 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 11651 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 11652 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 11653 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 11654 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 11655 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 11656 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 11657 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 11658 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 11659 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 11660 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 11661 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 11662 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 11663 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 11664 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 11665 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 11666  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 11667  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 11668 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 11669 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 )
 
Theoremgcdmultipled 11670 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 )
 
Theoremdvdsgcdidd 11671 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 )
 
Theorem6gcd4e2 11672 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
 
Theorembezoutlemnewy 11673* 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 11674* 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 11675* 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 11676* 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 11677* 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 11678* 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 11679* Lemma for Bézout's identity. Like bezoutlemex 11678 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 11680* Lemma for Bézout's identity. Like bezoutlemzz 11679 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 11681* Lemma for Bézout's identity. Like bezoutlemaz 11680 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 11682* Lemma for Bézout's identity. Like bezoutlembz 11681 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 11683* 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 11684* 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 11685* 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 11686* 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 11687* 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 11688* 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 11689 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 11690 Biconditional form of dvdsgcd 11689. (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 11691* 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 11692 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 11693 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 11694 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 11695 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 11696 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 11697 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 11698 Extend gcdmultiple 11697 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 11699 A positive integer  A is equal to its gcd with an integer  B if and only if  A divides  B. Generalization of gcdeq 11700. (Contributed by AV, 1-Jul-2020.)
 |-  ( ( A  e.  NN  /\  B  e.  ZZ )  ->  ( ( A 
 gcd  B )  =  A  <->  A 
 ||  B ) )
 
Theoremgcdeq 11700  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 ) )
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