HomeHome Metamath Proof Explorer
Theorem List (p. 126 of 310)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21328)
  Hilbert Space Explorer  Hilbert Space Explorer
(21329-22851)
  Users' Mathboxes  Users' Mathboxes
(22852-30955)
 

Theorem List for Metamath Proof Explorer - 12501-12600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembitsmod 12501 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 12502 Every number is associated to a finite set of bits. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  NN0  ->  (bits `  N )  e. 
 Fin )
 
Theorembitscmp 12503 The bit complement of  N is  -u N  - 
1. (Thus, by bitsfi 12502, 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 12504 The bits of zero. (Contributed by Mario Carneiro, 6-Sep-2016.)
 |-  (bits `  0 )  =  (/)
 
Theoremm1bits 12505 The bits of negative one. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  (bits `  -u 1 )  =  NN0
 
Theorembitsinv1lem 12506 Lemma for bitsinv1 12507. (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 12507* There is an explicit inverse to the bits function for nonnegative integers (which can be extended to negative integers using bitscmp 12503), part 1. (Contributed by Mario Carneiro, 7-Sep-2016.)
 |-  ( N  e.  NN0  ->  sum_ n  e.  (bits `  N ) ( 2 ^ n )  =  N )
 
Theorembitsinv2 12508* 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 12509* 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 12163. (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 12510 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 12163. (Contributed by Mario Carneiro, 8-Sep-2016.)
 |-  (bits  |`  NN0 ) : NN0 -1-1-onto-> ( ~P NN0  i^i  Fin )
 
Theorembitsf1 12511 The bits function is an injection from  ZZ to  ~P NN0. It is obviously not a bijection (by Cantor's theorem canth2 6899), 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 12512 The bits of a power of two. (Contributed by Mario Carneiro, 5-Sep-2016.)
 |-  ( N  e.  NN0  ->  (bits `  ( 2 ^ N ) )  =  { N } )
 
Theorembitsinv 12513* 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 12514 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 12515 The core of the proof of sadadd2 12525. 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 12516* Define the addition of two bit sequences, using df-had 1376 and df-cad 1377 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 12517* Define the addition of two bit sequences, using df-had 1376 and df-cad 1377 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 12518* 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 12519* 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 12520* 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 12521* 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 12522* Lemma for sadcadd 12523. (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 12523* 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 12524* Lemma for sadadd2 12525. (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 12525* 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 12526* 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 12527 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 12528 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 12529* Lemma for sadadd 12532. (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 12530 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 12531* Lemma for sadadd 12532. (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 12532 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 1376 and df-cad 1377.

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 12533 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 12534 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 12535 Lemma for sadass 12536. (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 12536 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 12537 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 12538 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 12539 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 12540* 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 12541* 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 12542* Define the addition of two bit sequences, using df-had 1376 and df-cad 1377 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  ->  ( A smul  B )  =  {
 k  e.  NN0  |  k  e.  ( P `  ( k  +  1 ) ) } )
 
Theoremsmupf 12543* 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 12544* 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 12545* 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 12546* Define the addition of two bit sequences, using df-had 1376 and df-cad 1377 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 12547* 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 12548* 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 12549 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 12550* Lemma for smu01 12551 and smu02 12552. (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 12551 Multiplication of a sequence by  0 on the right. (Contributed by Mario Carneiro, 19-Sep-2016.)
 |-  ( A  C_  NN0  ->  ( A smul  (/) )  =  (/) )
 
Theoremsmu02 12552 Multiplication of a sequence by  0 on the left. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( A  C_  NN0  ->  ( (/) smul  A )  =  (/) )
 
Theoremsmupval 12553* 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 12554* Rewrite smupp1 12545 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 12555* 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 12556 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 12557 Lemma for smumul 12558. (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 12558 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 12516, whose correctness is verified in sadadd 12532.

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 12559 Extend the definition of a class to include the greatest common divisor operator.
 class  gcd
 
Definitiondf-gcd 12560* 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 12561* 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 12562 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 12563* 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 12564* Lemma for gcdn0cl 12567, gcddvds 12568 and dvdslegcd 12569. (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 12565* Lemma for gcdn0cl 12567, gcddvds 12568 and dvdslegcd 12569. (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 12566* Lemma for gcdn0cl 12567, gcddvds 12568 and dvdslegcd 12569. (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 12567 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 12568 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 12569 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 ) ) )
 
Theoremgcdcl 12570 Closure of the  gcd operator. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  e.  NN0 )
 
Theoremgcdcld 12571 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 12572 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 12573 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 12574 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 12575 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 12576 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 12577 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 12578 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 12579 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 12580 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 12581 Lemma for gcdaddm 12582. (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 12582 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 12583 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 12584 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 12585 The gcd of a number with 1 is 1. (Contributed by Mario Carneiro, 19-Feb-2014.)
 |-  ( M  e.  ZZ  ->  ( M  gcd  1
 )  =  1 )
 
Theoremgcdabs 12586 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 12587  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 12588  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 12589 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 12590 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 )
 
6.1.7  Bézout's identity
 
Theorembezoutlem1 12591* Lemma for bezout 12595. (Contributed by Mario Carneiro, 15-Mar-2014.)
 |-  M  =  { z  e.  NN  |  E. x  e.  ZZ  E. y  e. 
 ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y ) ) }   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   =>    |-  ( ph  ->  ( A  =/=  0  ->  ( abs `  A )  e.  M ) )
 
Theorembezoutlem2 12592* Lemma for bezout 12595. (Contributed by Mario Carneiro, 15-Mar-2014.)
 |-  M  =  { z  e.  NN  |  E. x  e.  ZZ  E. y  e. 
 ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y ) ) }   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  G  =  sup ( M ,  RR ,  `'  <  )   &    |-  ( ph  ->  -.  ( A  =  0 
 /\  B  =  0 ) )   =>    |-  ( ph  ->  G  e.  M )
 
Theorembezoutlem3 12593* Lemma for bezout 12595. (Contributed by Mario Carneiro, 22-Feb-2014.)
 |-  M  =  { z  e.  NN  |  E. x  e.  ZZ  E. y  e. 
 ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y ) ) }   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  G  =  sup ( M ,  RR ,  `'  <  )   &    |-  ( ph  ->  -.  ( A  =  0 
 /\  B  =  0 ) )   =>    |-  ( ph  ->  ( C  e.  M  ->  G 
 ||  C ) )
 
Theorembezoutlem4 12594* Lemma for bezout 12595. (Contributed by Mario Carneiro, 22-Feb-2014.)
 |-  M  =  { z  e.  NN  |  E. x  e.  ZZ  E. y  e. 
 ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y ) ) }   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  G  =  sup ( M ,  RR ,  `'  <  )   &    |-  ( ph  ->  -.  ( A  =  0 
 /\  B  =  0 ) )   =>    |-  ( ph  ->  ( A  gcd  B )  e.  M )
 
Theorembezout 12595* 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. (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 12596 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 12597 Biconditional form of dvdsgcd 12596. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  ||  M  /\  K  ||  N ) 
 <->  K  ||  ( M  gcd  N ) ) )
 
Theoremgcdass 12598 Associative law for  gcd operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( N  gcd  M )  gcd  P )  =  ( N  gcd  ( M  gcd  P ) ) )
 
Theoremmulgcd 12599 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 12600 Distribute absolute value of multiplication over gcd. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  gcd  ( K  x.  N ) )  =  ( abs `  ( K  x.  ( M  gcd  N ) ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 30901-30955
  Copyright terms: Public domain < Previous  Next >