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Theorem List for Intuitionistic Logic Explorer - 10801-10900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdivalg2 10801* 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 10802 The result of the  mod operator satisfies the requirements for the remainder  R in the division algorithm for a positive divisor (compare divalg2 10801 and divalgb 10800). 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 10803 The result of the  mod operator satisfies the requirements for the remainder  R in the division algorithm for a positive divisor. Variant of divalgmod 10802. (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 10804* 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 10805 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 10806 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 10807 Special case of ndvdsadd 10806. 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 10808 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 10809 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 10810 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 10811 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 10812 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
 ) )
 
4.1.4  The greatest common divisor operator
 
Syntaxcgcd 10813 Extend the definition of a class to include the greatest common divisor operator.
 class  gcd
 
Definitiondf-gcd 10814* Define the  gcd operator. For example,  ( -u 6  gcd  9 )  =  3 (ex-gcd 11096). (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 10815 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 10816* Lemma for zsupcl 10818. 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 10817* Lemma for zsupcl 10818. 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 10818* 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 10819* 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 10820* 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 10821* 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 10822* 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 10823* 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 10824* 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 10825* Closure of the supremum used in defining  gcd. A lemma for gcdval 10826 and gcdn0cl 10829. (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 10826* 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 10827 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 10828* 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 10829 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 10830 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 10831 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 10832 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 10833 Closure of the  gcd operator. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e.  NN0 )
 
Theoremgcdnncl 10834 Closure of the  gcd operator. (Contributed by Thierry Arnoux, 2-Feb-2020.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  gcd  N )  e.  NN )
 
Theoremgcdcld 10835 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 10836 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 10837* 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 10838 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 10839 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 10840 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 10841 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 10842 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 10843 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 10844 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 10845 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 10846 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 10847 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 10848 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 10849 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 10850 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 10851 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 10852 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 10853 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 10854 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 10855  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 10856  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 10857 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 10858 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 10859 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 10860* 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 10861* 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 10862* 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 10863* 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 10864* 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 10865* 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 10866* Lemma for Bézout's identity. Like bezoutlemex 10865 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 10867* Lemma for Bézout's identity. Like bezoutlemzz 10866 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 10868* Lemma for Bézout's identity. Like bezoutlemaz 10867 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 10869* Lemma for Bézout's identity. Like bezoutlembz 10868 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 10870* 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 10871* 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 10872* 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 10873* 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 10874* 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 10875* 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 10876 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 10877 Biconditional form of dvdsgcd 10876. (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 10878* 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 10879 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 10880 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 10881 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 10882 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 10883 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 10884 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 10885 Extend gcdmultiple 10884 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 10886 A positive integer  A is equal to its gcd with an integer  B if and only if  A divides  B. Generalization of gcdeq 10887. (Contributed by AV, 1-Jul-2020.)
 |-  ( ( A  e.  NN  /\  B  e.  ZZ )  ->  ( ( A 
 gcd  B )  =  A  <->  A 
 ||  B ) )
 
Theoremgcdeq 10887  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 10888 Unidirectional form of dvdssq 10895. (Contributed by Scott Fenton, 19-Apr-2014.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  ->  ( M ^
 2 )  ||  ( N ^ 2 ) ) )
 
Theoremdvdsmulgcd 10889 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 10890 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 10891 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 10892 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 10893 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 10894 Lemma for dvdssq 10895. (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 10895 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 10896 Partial converse to bezout 10875. 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 10897 Converse of bezout 10875 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 10898* 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 10899 Lemma for ialgr0 10901. Expressing algrflemg 5952 in a form suitable for theorems such as iseq1 9791 or iseqfcl 9793. (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 10900 Lemma for ialgr0 10901. Closure of a constant function, in a form suitable for theorems such as iseq1 9791 or iseqfcl 9793. (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 )
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