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Theorem List for Metamath Proof Explorer - 14401-14500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembits0e 14401 The zeroth bit of an even number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  ZZ  ->  -.  0  e.  (bits `  ( 2  x.  N ) ) )
 
Theorembits0o 14402 The zeroth bit of an odd number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  ZZ  ->  0  e.  (bits `  ( ( 2  x.  N )  +  1 ) ) )
 
Theorembitsp1 14403 The  M  +  1-th bit of  N is the  M-th bit of  |_ ( N  / 
2 ). (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  ( ( M  +  1 )  e.  (bits `  N )  <->  M  e.  (bits `  ( |_ `  ( N  / 
 2 ) ) ) ) )
 
Theorembitsp1e 14404 The  M  +  1-th bit of  2 N is the  M-th bit of  N. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  ( ( M  +  1 )  e.  (bits `  ( 2  x.  N ) )  <->  M  e.  (bits `  N ) ) )
 
Theorembitsp1o 14405 The  M  +  1-th bit of  2 N  +  1 is the  M-th bit of  N. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  ( ( M  +  1 )  e.  (bits `  ( (
 2  x.  N )  +  1 ) )  <->  M  e.  (bits `  N ) ) )
 
Theorembitsfzolem 14406* Lemma for bitsfzo 14408. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 1-Oct-2020.)
 |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  (bits `  N )  C_  (
 0..^ M ) )   &    |-  S  = inf ( { n  e.  NN0  |  N  <  ( 2 ^ n ) } ,  RR ,  <  )   =>    |-  ( ph  ->  N  e.  ( 0..^ ( 2 ^ M ) ) )
 
TheorembitsfzolemOLD 14407* Lemma for bitsfzo 14408. (Contributed by Mario Carneiro, 5-Sep-2016.) Obsolete version of bitsfzolem 14406 as of 1-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  (bits `  N )  C_  (
 0..^ M ) )   &    |-  S  =  sup ( { n  e.  NN0  |  N  <  ( 2 ^ n ) } ,  RR ,  `'  <  )   =>    |-  ( ph  ->  N  e.  ( 0..^ ( 2 ^ M ) ) )
 
Theorembitsfzo 14408 The bits of a number are all less than  M iff the number is nonnegative and less than  2 ^ M. (Contributed by Mario Carneiro, 5-Sep-2016.) (Proof shortened by AV, 1-Oct-2020.)
 |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  ( N  e.  ( 0..^ ( 2 ^ M ) )  <->  (bits `  N )  C_  ( 0..^ M ) ) )
 
Theorembitsmod 14409 Truncating the bit sequence after some  M is equivalent to reducing the argument  mod  2 ^ M. (Contributed by Mario Carneiro, 6-Sep-2016.)
 |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  (bits `  ( N  mod  ( 2 ^ M ) ) )  =  ( (bits `  N )  i^i  ( 0..^ M ) ) )
 
Theorembitsfi 14410 Every number is associated with a finite set of bits. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  NN0  ->  (bits `  N )  e. 
 Fin )
 
Theorembitscmp 14411 The bit complement of  N is  -u N  - 
1. (Thus, by bitsfi 14410, all negative numbers have cofinite bits representations.) (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  ZZ  ->  ( NN0  \  (bits `  N ) )  =  (bits `  ( -u N  -  1 ) ) )
 
Theorem0bits 14412 The bits of zero. (Contributed by Mario Carneiro, 6-Sep-2016.)
 |-  (bits `  0 )  =  (/)
 
Theoremm1bits 14413 The bits of negative one. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  (bits `  -u 1 )  =  NN0
 
Theorembitsinv1lem 14414 Lemma for bitsinv1 14415. (Contributed by Mario Carneiro, 22-Sep-2016.)
 |-  ( ( N  e.  ZZ  /\  M  e.  NN0 )  ->  ( N  mod  ( 2 ^ ( M  +  1 )
 ) )  =  ( ( N  mod  (
 2 ^ M ) )  +  if ( M  e.  (bits `  N ) ,  ( 2 ^ M ) ,  0 ) ) )
 
Theorembitsinv1 14415* There is an explicit inverse to the bits function for nonnegative integers (which can be extended to negative integers using bitscmp 14411), part 1. (Contributed by Mario Carneiro, 7-Sep-2016.)
 |-  ( N  e.  NN0  ->  sum_ n  e.  (bits `  N ) ( 2 ^ n )  =  N )
 
Theorembitsinv2 14416* There is an explicit inverse to the bits function for nonnegative integers, part 2. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  ( A  e.  ( ~P NN0  i^i  Fin )  ->  (bits `  sum_ n  e.  A  ( 2 ^ n ) )  =  A )
 
Theorembitsf1ocnv 14417* The bits function restricted to nonnegative integers is a bijection from the integers to the finite sets of integers. It is in fact the inverse of the Ackermann bijection ackbijnn 13885. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  ( (bits  |`  NN0 ) : NN0
 -1-1-onto-> ( ~P NN0  i^i  Fin )  /\  `' (bits  |`  NN0 )  =  ( x  e.  ( ~P
 NN0  i^i  Fin )  |->  sum_ n  e.  x  ( 2 ^ n ) ) )
 
Theorembitsf1o 14418 The bits function restricted to nonnegative integers is a bijection from the integers to the finite sets of integers. It is in fact the inverse of the Ackermann bijection ackbijnn 13885. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  (bits  |`  NN0 ) : NN0 -1-1-onto-> ( ~P NN0  i^i  Fin )
 
Theorembitsf1 14419 The bits function is an injection from  ZZ to  ~P NN0. It is obviously not a bijection (by Cantor's theorem canth2 7734), and in fact its range is the set of finite and cofinite subsets of  NN0. (Contributed by Mario Carneiro, 22-Sep-2016.)
 |- bits : ZZ -1-1-> ~P NN0
 
Theorem2ebits 14420 The bits of a power of two. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  NN0  ->  (bits `  ( 2 ^ N ) )  =  { N } )
 
Theorembitsinv 14421* The inverse of the bits function. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  K  =  `' (bits  |` 
 NN0 )   =>    |-  ( A  e.  ( ~P NN0  i^i  Fin )  ->  ( K `  A )  =  sum_ k  e.  A  ( 2 ^ k
 ) )
 
Theorembitsinvp1 14422 Recursive definition of the inverse of the bits function. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  K  =  `' (bits  |` 
 NN0 )   =>    |-  ( ( A  C_  NN0  /\  N  e.  NN0 )  ->  ( K `  ( A  i^i  ( 0..^ ( N  +  1 ) ) ) )  =  ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  if ( N  e.  A ,  (
 2 ^ N ) ,  0 ) ) )
 
Theoremsadadd2lem2 14423 The core of the proof of sadadd2 14433. The intuitive justification for this is that cadd is true if at least two arguments are true, and hadd is true if an odd number of arguments are true, so altogether the result is  n  x.  A where  n is the number of true arguments, which is equivalently obtained by adding together one  A for each true argument, on the right side. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  ( A  e.  CC  ->  ( if (hadd ( ph ,  ps ,  ch ) ,  A , 
 0 )  +  if (cadd ( ph ,  ps ,  ch ) ,  (
 2  x.  A ) ,  0 ) )  =  ( ( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) )  +  if ( ch ,  A ,  0 ) ) )
 
Definitiondf-sad 14424* Define the addition of two bit sequences, using df-had 1490 and df-cad 1503 bit operations. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |- sadd  =  ( x  e.  ~P NN0
 ,  y  e.  ~P NN0  |->  { k  e.  NN0  | hadd ( k  e.  x ,  k  e.  y ,  (/)  e.  (  seq 0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  x ,  m  e.  y ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) ) `  k ) ) } )
 
Theoremsadfval 14425* Define the addition of two bit sequences, using df-had 1490 and df-cad 1503 bit operations. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq 0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   =>    |-  ( ph  ->  ( A sadd  B )  =  {
 k  e.  NN0  | hadd ( k  e.  A ,  k  e.  B ,  (/)  e.  ( C `
  k ) ) } )
 
Theoremsadcf 14426* The carry sequence is a sequence of elements of  2o encoding a "sequence of wffs". (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq 0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   =>    |-  ( ph  ->  C : NN0 --> 2o )
 
Theoremsadc0 14427* The initial element of the carry sequence is F.. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq 0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   =>    |-  ( ph  ->  -.  (/)  e.  ( C `  0 ) )
 
Theoremsadcp1 14428* The carry sequence (which is a sequence of wffs, encoded as  1o and  (/)) is defined recursively as the carry operation applied to the previous carry and the two current inputs. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq 0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( (/) 
 e.  ( C `  ( N  +  1
 ) )  <-> cadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `
  N ) ) ) )
 
Theoremsadval 14429* The full adder sequence is the half adder function applied to the inputs and the carry sequence. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq 0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( N  e.  ( A sadd  B )  <-> hadd ( N  e.  A ,  N  e.  B ,  (/)  e.  ( C `
  N ) ) ) )
 
Theoremsadcaddlem 14430* Lemma for sadcadd 14431. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq 0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  K  =  `' (bits  |`  NN0 )   &    |-  ( ph  ->  ( (/)  e.  ( C `  N )  <->  ( 2 ^ N )  <_  ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  ( K `  ( B  i^i  ( 0..^ N ) ) ) ) ) )   =>    |-  ( ph  ->  ( (/) 
 e.  ( C `  ( N  +  1
 ) )  <->  ( 2 ^
 ( N  +  1 ) )  <_  (
 ( K `  ( A  i^i  ( 0..^ ( N  +  1 ) ) ) )  +  ( K `  ( B  i^i  ( 0..^ ( N  +  1 ) ) ) ) ) ) )
 
Theoremsadcadd 14431* Non-recursive definition of the carry sequence. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq 0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  K  =  `' (bits  |`  NN0 )   =>    |-  ( ph  ->  ( (/) 
 e.  ( C `  N )  <->  ( 2 ^ N )  <_  ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  ( K `  ( B  i^i  ( 0..^ N ) ) ) ) ) )
 
Theoremsadadd2lem 14432* Lemma for sadadd2 14433. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq 0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  K  =  `' (bits  |`  NN0 )   &    |-  ( ph  ->  ( ( K `  (
 ( A sadd  B )  i^i  ( 0..^ N ) ) )  +  if ( (/)  e.  ( C `
  N ) ,  ( 2 ^ N ) ,  0 )
 )  =  ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  ( K `  ( B  i^i  ( 0..^ N ) ) ) ) )   =>    |-  ( ph  ->  (
 ( K `  (
 ( A sadd  B )  i^i  ( 0..^ ( N  +  1 ) ) ) )  +  if ( (/)  e.  ( C `
  ( N  +  1 ) ) ,  ( 2 ^ ( N  +  1 )
 ) ,  0 ) )  =  ( ( K `  ( A  i^i  ( 0..^ ( N  +  1 ) ) ) )  +  ( K `  ( B  i^i  ( 0..^ ( N  +  1 ) ) ) ) ) )
 
Theoremsadadd2 14433* Sum of initial segments of the sadd sequence. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq 0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  K  =  `' (bits  |`  NN0 )   =>    |-  ( ph  ->  (
 ( K `  (
 ( A sadd  B )  i^i  ( 0..^ N ) ) )  +  if ( (/)  e.  ( C `
  N ) ,  ( 2 ^ N ) ,  0 )
 )  =  ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  ( K `  ( B  i^i  ( 0..^ N ) ) ) ) )
 
Theoremsadadd3 14434* Sum of initial segments of the sadd sequence. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  C  =  seq 0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  K  =  `' (bits  |`  NN0 )   =>    |-  ( ph  ->  (
 ( K `  (
 ( A sadd  B )  i^i  ( 0..^ N ) ) )  mod  (
 2 ^ N ) )  =  ( ( ( K `  ( A  i^i  ( 0..^ N ) ) )  +  ( K `  ( B  i^i  ( 0..^ N ) ) ) ) 
 mod  ( 2 ^ N ) ) )
 
Theoremsadcl 14435 The sum of two sequences is a sequence. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( A  C_  NN0  /\  B  C_  NN0 )  ->  ( A sadd  B )  C_  NN0 )
 
Theoremsadcom 14436 The adder sequence function is commutative. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( A  C_  NN0  /\  B  C_  NN0 )  ->  ( A sadd  B )  =  ( B sadd  A ) )
 
Theoremsaddisjlem 14437* Lemma for sadadd 14440. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  C  =  seq 0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c
 ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/)
 ,  ( n  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( N  e.  ( A sadd  B )  <->  N  e.  ( A  u.  B ) ) )
 
Theoremsaddisj 14438 The sum of disjoint sequences is the union of the sequences. (In this case, there are no carried bits.) (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   =>    |-  ( ph  ->  ( A sadd  B )  =  ( A  u.  B ) )
 
Theoremsadaddlem 14439* Lemma for sadadd 14440. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  C  =  seq 0
 ( ( c  e. 
 2o ,  m  e. 
 NN0  |->  if (cadd ( m  e.  (bits `  A ) ,  m  e.  (bits `  B ) ,  (/)  e.  c ) ,  1o ,  (/) ) ) ,  ( n  e. 
 NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )   &    |-  K  =  `' (bits  |`  NN0 )   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( (bits `  A ) sadd  (bits `  B )
 )  i^i  ( 0..^ N ) )  =  (bits `  ( ( A  +  B )  mod  ( 2 ^ N ) ) ) )
 
Theoremsadadd 14440 For sequences that correspond to valid integers, the adder sequence function produces the sequence for the sum. This is effectively a proof of the correctness of the ripple carry adder, implemented with logic gates corresponding to df-had 1490 and df-cad 1503.

It is interesting to consider in what sense the sadd function can be said to be "adding" things outside the range of the bits function, that is, when adding sequences that are not eventually constant and so do not denote any integer. The correct interpretation is that the sequences are representations of 2-adic integers, which have a natural ring structure. (Contributed by Mario Carneiro, 9-Sep-2016.)

 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( (bits `  A ) sadd  (bits `  B ) )  =  (bits `  ( A  +  B ) ) )
 
Theoremsadid1 14441 The adder sequence function has a left identity, the empty set, which is the representation of the integer zero. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( A  C_  NN0  ->  ( A sadd  (/) )  =  A )
 
Theoremsadid2 14442 The adder sequence function has a right identity, the empty set, which is the representation of the integer zero. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( A  C_  NN0  ->  ( (/) sadd  A )  =  A )
 
Theoremsadasslem 14443 Lemma for sadass 14444. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  ( ph  ->  C 
 C_  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( ( A sadd  B ) sadd  C )  i^i  (
 0..^ N ) )  =  ( ( A sadd 
 ( B sadd  C )
 )  i^i  ( 0..^ N ) ) )
 
Theoremsadass 14444 Sequence addition is associative. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ( A  C_  NN0  /\  B  C_  NN0  /\  C  C_ 
 NN0 )  ->  (
 ( A sadd  B ) sadd  C )  =  ( A sadd 
 ( B sadd  C )
 ) )
 
Theoremsadeq 14445 Any element of a sequence sum only depends on the values of the argument sequences up to and including that point. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  C_ 
 NN0 )   &    |-  ( ph  ->  B 
 C_  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( A sadd  B )  i^i  ( 0..^ N ) )  =  ( ( ( A  i^i  (
 0..^ N ) ) sadd 
 ( B  i^i  (
 0..^ N ) ) )  i^i  ( 0..^ N ) ) )
 
Theorembitsres 14446 Restrict the bits of a number to an upper integer set. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  ( (bits `  A )  i^i  ( ZZ>= `  N ) )  =  (bits `  ( ( |_ `  ( A  /  ( 2 ^ N ) ) )  x.  ( 2 ^ N ) ) ) )
 
Theorembitsuz 14447 The bits of a number are all at least  N iff the number is divisible by  2 ^ N. (Contributed by Mario Carneiro, 21-Sep-2016.)
 |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  ( ( 2 ^ N )  ||  A 
 <->  (bits `  A )  C_  ( ZZ>= `  N )
 ) )
 
Theorembitsshft 14448* Shifting a bit sequence to the left (toward the more significant bits) causes the number to be multiplied by a power of two. (Contributed by Mario Carneiro, 22-Sep-2016.)
 |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  { n  e. 
 NN0  |  ( n  -  N )  e.  (bits `  A ) }  =  (bits `  ( A  x.  ( 2 ^ N ) ) ) )
 
Definitiondf-smu 14449* Define the multiplication of two bit sequences, using repeated sequence addition. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |- smul  =  ( x  e.  ~P NN0
 ,  y  e.  ~P NN0  |->  { k  e.  NN0  |  k  e.  (  seq 0 ( ( p  e.  ~P NN0 ,  m  e.  NN0  |->  ( p sadd  { n  e.  NN0  |  ( m  e.  x  /\  ( n  -  m )  e.  y ) } ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) ) `  (
 k  +  1 ) ) } )
 
Theoremsmufval 14450* The multiplication of two bit sequences as repeated sequence addition. (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  ->  ( A smul  B )  =  {
 k  e.  NN0  |  k  e.  ( P `  ( k  +  1 ) ) } )
 
Theoremsmupf 14451* The sequence of partial sums 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  ->  P : NN0 --> ~P NN0 )
 
Theoremsmup0 14452* 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 14453* 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 14454* Define the addition of two bit sequences, using df-had 1490 and df-cad 1503 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 14455* 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 14456* 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 14457 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 14458* Lemma for smu01 14459 and smu02 14460. (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 14459 Multiplication of a sequence by  0 on the right. (Contributed by Mario Carneiro, 19-Sep-2016.)
 |-  ( A  C_  NN0  ->  ( A smul  (/) )  =  (/) )
 
Theoremsmu02 14460 Multiplication of a sequence by  0 on the left. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( A  C_  NN0  ->  ( (/) smul  A )  =  (/) )
 
Theoremsmupval 14461* 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 14462* Rewrite smupp1 14453 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 14463* 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 14464 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 14465 Lemma for smumul 14466. (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 14466 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 14424, whose correctness is verified in sadadd 14440.

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 14467 Extend the definition of a class to include the greatest common divisor operator.
 class  gcd
 
Definitiondf-gcd 14468* 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 14469* 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 14470 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 14471* 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 14472* Lemma for gcdn0cl 14475, gcddvds 14476 and dvdslegcd 14477. (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 14473* Lemma for gcdn0cl 14475, gcddvds 14476 and dvdslegcd 14477. (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 14474* Lemma for gcdn0cl 14475, gcddvds 14476 and dvdslegcd 14477. (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 14475 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 14476 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 14477 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 14478 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 14479 Closure of the  gcd operator. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e.  NN0 )
 
Theoremgcdnncl 14480 Closure of the  gcd operator. (Contributed by Thierry Arnoux, 2-Feb-2020.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  gcd  N )  e.  NN )
 
Theoremgcdcld 14481 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 14482 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 14483 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 14484 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 14485 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 14486 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 14487 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 14488 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 14489 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 14490 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 14491 Lemma for gcdaddm 14492. (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 14492 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 14493 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 14494 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 14495 The gcd of a number with 1 is 1. (Contributed by Mario Carneiro, 19-Feb-2014.)
 |-  ( M  e.  ZZ  ->  ( M  gcd  1
 )  =  1 )
 
Theoremgcdabs 14496 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 14497  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 14498  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 14499 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 14500 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 )
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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