HomeHome Intuitionistic Logic Explorer
Theorem List (p. 107 of 111)
< Previous  Next >
Browser slow? Try the
Unicode version.

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

Theorem List for Intuitionistic Logic Explorer - 10601-10700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembezoutlembi 10601* Lemma for Bézout's identity. Like bezoutlembz 10600 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 10602* 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 10603* 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 10604* 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 10605* 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 10606* 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 10607* 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 10608 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 10609 Biconditional form of dvdsgcd 10608. (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 10610* 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 10611 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 10612 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 10613 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 10614 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 10615 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 10616 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 10617 Extend gcdmultiple 10616 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 10618 A positive integer  A is equal to its gcd with an integer  B if and only if  A divides  B. Generalization of gcdeq 10619. (Contributed by AV, 1-Jul-2020.)
 |-  ( ( A  e.  NN  /\  B  e.  ZZ )  ->  ( ( A 
 gcd  B )  =  A  <->  A 
 ||  B ) )
 
Theoremgcdeq 10619  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 10620 Unidirectional form of dvdssq 10627. (Contributed by Scott Fenton, 19-Apr-2014.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  ->  ( M ^
 2 )  ||  ( N ^ 2 ) ) )
 
Theoremdvdsmulgcd 10621 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 10622 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 10623 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 10624 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 10625 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 10626 Lemma for dvdssq 10627. (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 10627 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 10628 Partial converse to bezout 10607. 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 10629 Converse of bezout 10607 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 10630* 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 10631 Lemma for ialgr0 10633. Expressing algrflemg 5902 in a form suitable for theorems such as iseq1 9585 or iseqfcl 9587. (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 10632 Lemma for ialgr0 10633. Closure of a constant function, in a form suitable for theorems such as iseq1 9585 or iseqfcl 9587. (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 )
 
Theoremialgr0 10633 The value of the algorithm iterator 
R at  0 is the initial state  A. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  R  =  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A }
 ) ,  S )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  F : S --> S )   &    |-  ( ph  ->  S  e.  V )   =>    |-  ( ph  ->  ( R `  M )  =  A )
 
Theoremialgrf 10634 An algorithm is a step function  F : S --> S on a state space  S. An algorithm acts on an initial state  A  e.  S by iteratively applying  F to give  A,  ( F `  A ),  ( F `  ( F `  A )
) and so on. An algorithm is said to halt if a fixed point of  F is reached after a finite number of iterations.

The algorithm iterator  R : NN0 --> S "runs" the algorithm  F so that  ( R `  k ) is the state after  k iterations of  F on the initial state  A.

Domain and codomain of the algorithm iterator  R. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)

 |-  Z  =  ( ZZ>= `  M )   &    |-  R  =  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A }
 ) ,  S )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  F : S --> S )   &    |-  ( ph  ->  S  e.  V )   =>    |-  ( ph  ->  R : Z --> S )
 
Theoremialgrp1 10635 The value of the algorithm iterator 
R at  ( K  + 
1 ). (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  R  =  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A }
 ) ,  S )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  F : S --> S )   &    |-  ( ph  ->  S  e.  V )   =>    |-  ( ( ph  /\  K  e.  Z ) 
 ->  ( R `  ( K  +  1 )
 )  =  ( F `
  ( R `  K ) ) )
 
Theoremialginv 10636* If  I is an invariant of  F, its value is unchanged after any number of iterations of  F. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  R  =  seq 0
 ( ( F  o.  1st ) ,  ( NN0  X. 
 { A } ) ,  S )   &    |-  F : S --> S   &    |-  I  Fn  S   &    |-  ( x  e.  S  ->  ( I `  ( F `
  x ) )  =  ( I `  x ) )   &    |-  S  e.  V   =>    |-  ( ( A  e.  S  /\  K  e.  NN0 )  ->  ( I `  ( R `  K ) )  =  ( I `
  ( R `  0 ) ) )
 
Theoremialgcvg 10637* One way to prove that an algorithm halts is to construct a countdown function  C : S --> NN0 whose value is guaranteed to decrease for each iteration of  F until it reaches  0. That is, if  X  e.  S is not a fixed point of  F, then  ( C `  ( F `  X ) )  <  ( C `
 X ).

If  C is a countdown function for algorithm  F, the sequence  ( C `  ( R `  k ) ) reaches  0 after at most  N steps, where  N is the value of  C for the initial state  A. (Contributed by Paul Chapman, 22-Jun-2011.)

 |-  F : S --> S   &    |-  R  =  seq 0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) ,  S )   &    |-  C : S --> NN0   &    |-  ( z  e.  S  ->  ( ( C `  ( F `  z ) )  =/=  0  ->  ( C `  ( F `
  z ) )  <  ( C `  z ) ) )   &    |-  N  =  ( C `  A )   &    |-  S  e.  V   =>    |-  ( A  e.  S  ->  ( C `  ( R `
  N ) )  =  0 )
 
Theoremalgcvgblem 10638 Lemma for algcvgb 10639. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( ( N  =/=  0  ->  N  <  M ) 
 <->  ( ( M  =/=  0  ->  N  <  M )  /\  ( M  =  0  ->  N  =  0 ) ) ) )
 
Theoremalgcvgb 10639 Two ways of expressing that  C is a countdown function for algorithm  F. The first is used in these theorems. The second states the condition more intuitively as a conjunction: if the countdown function's value is currently nonzero, it must decrease at the next step; if it has reached zero, it must remain zero at the next step. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  F : S --> S   &    |-  C : S --> NN0   =>    |-  ( X  e.  S  ->  ( ( ( C `
  ( F `  X ) )  =/=  0  ->  ( C `  ( F `  X ) )  <  ( C `
  X ) )  <-> 
 ( ( ( C `
  X )  =/=  0  ->  ( C `  ( F `  X ) )  <  ( C `
  X ) ) 
 /\  ( ( C `
  X )  =  0  ->  ( C `  ( F `  X ) )  =  0
 ) ) ) )
 
Theoremialgcvga 10640* The countdown function  C remains  0 after  N steps. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  F : S --> S   &    |-  R  =  seq 0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) ,  S )   &    |-  C : S --> NN0   &    |-  ( z  e.  S  ->  ( ( C `  ( F `  z ) )  =/=  0  ->  ( C `  ( F `
  z ) )  <  ( C `  z ) ) )   &    |-  N  =  ( C `  A )   &    |-  S  e.  V   =>    |-  ( A  e.  S  ->  ( K  e.  ( ZZ>= `  N )  ->  ( C `
  ( R `  K ) )  =  0 ) )
 
Theoremialgfx 10641* If  F reaches a fixed point when the countdown function 
C reaches  0,  F remains fixed after  N steps. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  F : S --> S   &    |-  R  =  seq 0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) ,  S )   &    |-  C : S --> NN0   &    |-  ( z  e.  S  ->  ( ( C `  ( F `  z ) )  =/=  0  ->  ( C `  ( F `
  z ) )  <  ( C `  z ) ) )   &    |-  N  =  ( C `  A )   &    |-  S  e.  V   &    |-  (
 z  e.  S  ->  ( ( C `  z
 )  =  0  ->  ( F `  z )  =  z ) )   =>    |-  ( A  e.  S  ->  ( K  e.  ( ZZ>=
 `  N )  ->  ( R `  K )  =  ( R `  N ) ) )
 
4.1.7  Euclid's Algorithm
 
Theoremeucalgval2 10642* The value of the step function  E for Euclid's Algorithm on an ordered pair. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)
 |-  E  =  ( x  e.  NN0 ,  y  e. 
 NN0  |->  if ( y  =  0 ,  <. x ,  y >. ,  <. y ,  ( x  mod  y
 ) >. ) )   =>    |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M E N )  =  if ( N  =  0 ,  <. M ,  N >. ,  <. N ,  ( M  mod  N ) >. ) )
 
Theoremeucalgval 10643* Euclid's Algorithm eucialg 10648 computes the greatest common divisor of two nonnegative integers by repeatedly replacing the larger of them with its remainder modulo the smaller until the remainder is 0.

The value of the step function  E for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)

 |-  E  =  ( x  e.  NN0 ,  y  e. 
 NN0  |->  if ( y  =  0 ,  <. x ,  y >. ,  <. y ,  ( x  mod  y
 ) >. ) )   =>    |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( E `  X )  =  if (
 ( 2nd `  X )  =  0 ,  X ,  <. ( 2nd `  X ) ,  (  mod  `  X ) >. ) )
 
Theoremeucalgf 10644* Domain and codomain of the step function  E for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)
 |-  E  =  ( x  e.  NN0 ,  y  e. 
 NN0  |->  if ( y  =  0 ,  <. x ,  y >. ,  <. y ,  ( x  mod  y
 ) >. ) )   =>    |-  E : (
 NN0  X.  NN0 ) --> ( NN0  X. 
 NN0 )
 
Theoremeucalginv 10645* The invariant of the step function 
E for Euclid's Algorithm is the  gcd operator applied to the state. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 29-May-2014.)
 |-  E  =  ( x  e.  NN0 ,  y  e. 
 NN0  |->  if ( y  =  0 ,  <. x ,  y >. ,  <. y ,  ( x  mod  y
 ) >. ) )   =>    |-  ( X  e.  ( NN0  X.  NN0 )  ->  (  gcd  `  ( E `  X ) )  =  (  gcd  `  X ) )
 
Theoremeucalglt 10646* The second member of the state decreases with each iteration of the step function  E for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 29-May-2014.)
 |-  E  =  ( x  e.  NN0 ,  y  e. 
 NN0  |->  if ( y  =  0 ,  <. x ,  y >. ,  <. y ,  ( x  mod  y
 ) >. ) )   =>    |-  ( X  e.  ( NN0  X.  NN0 )  ->  ( ( 2nd `  ( E `  X ) )  =/=  0  ->  ( 2nd `  ( E `  X ) )  < 
 ( 2nd `  X )
 ) )
 
Theoremeucialgcvga 10647* Once Euclid's Algorithm halts after 
N steps, the second element of the state remains 0 . (Contributed by Jim Kingdon, 11-Jan-2022.)
 |-  E  =  ( x  e.  NN0 ,  y  e. 
 NN0  |->  if ( y  =  0 ,  <. x ,  y >. ,  <. y ,  ( x  mod  y
 ) >. ) )   &    |-  R  =  seq 0 ( ( E  o.  1st ) ,  ( NN0  X.  { A } ) ,  ( NN0  X.  NN0 ) )   &    |-  N  =  ( 2nd `  A )   =>    |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( K  e.  ( ZZ>= `  N )  ->  ( 2nd `  ( R `  K ) )  =  0
 ) )
 
Theoremeucialg 10648* Euclid's Algorithm computes the greatest common divisor of two nonnegative integers by repeatedly replacing the larger of them with its remainder modulo the smaller until the remainder is 0. Theorem 1.15 in [ApostolNT] p. 20.

Upon halting, the 1st member of the final state  ( R `  N ) is equal to the gcd of the values comprising the input state  <. M ,  N >.. This is Metamath 100 proof #69 (greatest common divisor algorithm). (Contributed by Jim Kingdon, 11-Jan-2022.)

 |-  E  =  ( x  e.  NN0 ,  y  e. 
 NN0  |->  if ( y  =  0 ,  <. x ,  y >. ,  <. y ,  ( x  mod  y
 ) >. ) )   &    |-  R  =  seq 0 ( ( E  o.  1st ) ,  ( NN0  X.  { A } ) ,  ( NN0  X.  NN0 ) )   &    |-  N  =  ( 2nd `  A )   &    |-  A  =  <. M ,  N >.   =>    |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( 1st `  ( R `  N ) )  =  ( M  gcd  N ) )
 
4.1.8  The least common multiple

According to Wikipedia ("Least common multiple", 27-Aug-2020, https://en.wikipedia.org/wiki/Least_common_multiple): "In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by lcm(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm(a,0) as 0 for all a, which is the result of taking the lcm to be the least upper bound in the lattice of divisibility."

In this section, an operation calculating the least common multiple of two integers (df-lcm 10650). The definition is valid for all integers, including negative integers and 0, obeying the above mentioned convention.

 
Syntaxclcm 10649 Extend the definition of a class to include the least common multiple operator.
 class lcm
 
Definitiondf-lcm 10650* Define the lcm operator. For example,  ( 6 lcm  9 )  =  1 8. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)
 |- lcm 
 =  ( x  e. 
 ZZ ,  y  e. 
 ZZ  |->  if ( ( x  =  0  \/  y  =  0 ) ,  0 , inf ( { n  e.  NN  |  ( x  ||  n  /\  y  ||  n ) } ,  RR ,  <  )
 ) )
 
Theoremlcmmndc 10651 Decidablity lemma used in various proofs related to lcm. (Contributed by Jim Kingdon, 21-Jan-2022.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID 
 ( M  =  0  \/  N  =  0 ) )
 
Theoremlcmval 10652* Value of the lcm operator.  ( M lcm  N ) is the least common multiple of  M and  N. If either  M or  N is  0, the result is defined conventionally as  0. Contrast with df-gcd 10546 and gcdval 10558. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  =  if (
 ( M  =  0  \/  N  =  0 ) ,  0 , inf ( { n  e. 
 NN  |  ( M 
 ||  n  /\  N  ||  n ) } ,  RR ,  <  ) ) )
 
Theoremlcmcom 10653 The lcm operator is commutative. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  =  ( N lcm  M ) )
 
Theoremlcm0val 10654 The value, by convention, of the lcm operator when either operand is 0. (Use lcmcom 10653 for a left-hand 0.) (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
 |-  ( M  e.  ZZ  ->  ( M lcm  0 )  =  0 )
 
Theoremlcmn0val 10655* The value of the lcm operator when both operands are nonzero. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)
 |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M lcm  N )  = inf ( { n  e.  NN  |  ( M  ||  n  /\  N  ||  n ) } ,  RR ,  <  )
 )
 
Theoremlcmcllem 10656* Lemma for lcmn0cl 10657 and dvdslcm 10658. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
 |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M lcm  N )  e.  { n  e.  NN  |  ( M 
 ||  n  /\  N  ||  n ) } )
 
Theoremlcmn0cl 10657 Closure of the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M lcm  N )  e.  NN )
 
Theoremdvdslcm 10658 The lcm of two integers is divisible by each of them. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  ( M lcm  N )  /\  N  ||  ( M lcm  N ) ) )
 
Theoremlcmledvds 10659 A positive integer which both operands of the lcm operator divide bounds it. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
 |-  ( ( ( K  e.  NN  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 )
 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  <_  K ) )
 
Theoremlcmeq0 10660 The lcm of two integers is zero iff either is zero. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M lcm 
 N )  =  0  <-> 
 ( M  =  0  \/  N  =  0 ) ) )
 
Theoremlcmcl 10661 Closure of the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  NN0 )
 
Theoremgcddvdslcm 10662 The greatest common divisor of two numbers divides their least common multiple. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N )  ||  ( M lcm  N ) )
 
Theoremlcmneg 10663 Negating one operand of the lcm operator does not alter the result. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  -u N )  =  ( M lcm  N ) )
 
Theoremneglcm 10664 Negating one operand of the lcm operator does not alter the result. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u M lcm  N )  =  ( M lcm 
 N ) )
 
Theoremlcmabs 10665 The lcm of two integers is the same as that of their absolute values. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M ) lcm  ( abs `  N ) )  =  ( M lcm  N ) )
 
Theoremlcmgcdlem 10666 Lemma for lcmgcd 10667 and lcmdvds 10668. Prove them for positive  M,  N, and  K. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( ( ( M lcm  N )  x.  ( M  gcd  N ) )  =  ( abs `  ( M  x.  N ) )  /\  ( ( K  e.  NN  /\  ( M  ||  K  /\  N  ||  K ) )  ->  ( M lcm 
 N )  ||  K ) ) )
 
Theoremlcmgcd 10667 The product of two numbers' least common multiple and greatest common divisor is the absolute value of the product of the two numbers. In particular, that absolute value is the least common multiple of two coprime numbers, for which  ( M  gcd  N
)  =  1.

Multiple methods exist for proving this, and it is often proven either as a consequence of the fundamental theorem of arithmetic or of Bézout's identity bezout 10607; see e.g. https://proofwiki.org/wiki/Product_of_GCD_and_LCM and https://math.stackexchange.com/a/470827. This proof uses the latter to first confirm it for positive integers  M and 
N (the "Second Proof" in the above Stack Exchange page), then shows that implies it for all nonzero integer inputs, then finally uses lcm0val 10654 to show it applies when either or both inputs are zero. (Contributed by Steve Rodriguez, 20-Jan-2020.)

 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M lcm 
 N )  x.  ( M  gcd  N ) )  =  ( abs `  ( M  x.  N ) ) )
 
Theoremlcmdvds 10668 The lcm of two integers divides any integer the two divide. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K ) )
 
Theoremlcmid 10669 The lcm of an integer and itself is its absolute value. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( M  e.  ZZ  ->  ( M lcm  M )  =  ( abs `  M ) )
 
Theoremlcm1 10670 The lcm of an integer and 1 is the absolute value of the integer. (Contributed by AV, 23-Aug-2020.)
 |-  ( M  e.  ZZ  ->  ( M lcm  1 )  =  ( abs `  M ) )
 
Theoremlcmgcdnn 10671 The product of two positive integers' least common multiple and greatest common divisor is the product of the two integers. (Contributed by AV, 27-Aug-2020.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( ( M lcm 
 N )  x.  ( M  gcd  N ) )  =  ( M  x.  N ) )
 
Theoremlcmgcdeq 10672 Two integers' absolute values are equal iff their least common multiple and greatest common divisor are equal. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M lcm 
 N )  =  ( M  gcd  N )  <-> 
 ( abs `  M )  =  ( abs `  N ) ) )
 
Theoremlcmdvdsb 10673 Biconditional form of lcmdvds 10668. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  ||  K  /\  N  ||  K ) 
 <->  ( M lcm  N ) 
 ||  K ) )
 
Theoremlcmass 10674 Associative law for lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
 |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( N lcm  M ) lcm  P )  =  ( N lcm  ( M lcm  P ) ) )
 
Theorem3lcm2e6woprm 10675 The least common multiple of three and two is six. This proof does not use the property of 2 and 3 being prime. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 27-Aug-2020.)
 |-  ( 3 lcm  2 )  =  6
 
Theorem6lcm4e12 10676 The least common multiple of six and four is twelve. (Contributed by AV, 27-Aug-2020.)
 |-  ( 6 lcm  4 )  = ; 1 2
 
4.1.9  Coprimality and Euclid's lemma

According to Wikipedia "Coprime integers", see https://en.wikipedia.org/wiki/Coprime_integers (16-Aug-2020) "[...] two integers a and b are said to be relatively prime, mutually prime, or coprime [...] if the only positive integer (factor) that divides both of them is 1. Consequently, any prime number that divides one does not divide the other. This is equivalent to their greatest common divisor (gcd) being 1.". In the following, we use this equivalent characterization to say that  A  e.  ZZ and  B  e.  ZZ are coprime (or relatively prime) if  ( A  gcd  B )  =  1. The equivalence of the definitions is shown by coprmgcdb 10677. The negation, i.e. two integers are not coprime, can be expressed either by  ( A  gcd  B )  =/=  1, see ncoprmgcdne1b 10678, or equivalently by  1  <  ( A  gcd  B ), see ncoprmgcdgt1b 10679.

A proof of Euclid's lemma based on coprimality is provided in coprmdvds 10681 (as opposed to Euclid's lemma for primes).

 
Theoremcoprmgcdb 10677* Two positive integers are coprime, i.e. the only positive integer that divides both of them is 1, iff their greatest common divisor is 1. (Contributed by AV, 9-Aug-2020.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A. i  e.  NN  ( ( i 
 ||  A  /\  i  ||  B )  ->  i  =  1 )  <->  ( A  gcd  B )  =  1 ) )
 
Theoremncoprmgcdne1b 10678* Two positive integers are not coprime, i.e. there is an integer greater than 1 which divides both integers, iff their greatest common divisor is not 1. (Contributed by AV, 9-Aug-2020.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( E. i  e.  ( ZZ>= `  2 )
 ( i  ||  A  /\  i  ||  B )  <-> 
 ( A  gcd  B )  =/=  1 ) )
 
Theoremncoprmgcdgt1b 10679* Two positive integers are not coprime, i.e. there is an integer greater than 1 which divides both integers, iff their greatest common divisor is greater than 1. (Contributed by AV, 9-Aug-2020.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( E. i  e.  ( ZZ>= `  2 )
 ( i  ||  A  /\  i  ||  B )  <-> 
 1  <  ( A  gcd  B ) ) )
 
Theoremcoprmdvds1 10680 If two positive integers are coprime, i.e. their greatest common divisor is 1, the only positive integer that divides both of them is 1. (Contributed by AV, 4-Aug-2021.)
 |-  ( ( F  e.  NN  /\  G  e.  NN  /\  ( F  gcd  G )  =  1 )  ->  ( ( I  e. 
 NN  /\  I  ||  F  /\  I  ||  G ) 
 ->  I  =  1
 ) )
 
Theoremcoprmdvds 10681 Euclid's Lemma (see ProofWiki "Euclid's Lemma", 10-Jul-2021, https://proofwiki.org/wiki/Euclid's_Lemma): If an integer divides the product of two integers and is coprime to one of them, then it divides the other. See also theorem 1.5 in [ApostolNT] p. 16. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by AV, 10-Jul-2021.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  ||  ( M  x.  N )  /\  ( K  gcd  M )  =  1 ) 
 ->  K  ||  N )
 )
 
Theoremcoprmdvds2 10682 If an integer is divisible by two coprime integers, then it is divisible by their product. (Contributed by Mario Carneiro, 24-Feb-2014.)
 |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M 
 gcd  N )  =  1 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M  x.  N )  ||  K ) )
 
Theoremmulgcddvds 10683 One half of rpmulgcd2 10684, which does not need the coprimality assumption. (Contributed by Mario Carneiro, 2-Jul-2015.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  gcd  ( M  x.  N ) ) 
 ||  ( ( K 
 gcd  M )  x.  ( K  gcd  N ) ) )
 
Theoremrpmulgcd2 10684 If  M is relatively prime to  N, then the GCD of  K with  M  x.  N is the product of the GCDs with  M and  N respectively. (Contributed by Mario Carneiro, 2-Jul-2015.)
 |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M 
 gcd  N )  =  1 )  ->  ( K  gcd  ( M  x.  N ) )  =  (
 ( K  gcd  M )  x.  ( K  gcd  N ) ) )
 
Theoremqredeq 10685 Two equal reduced fractions have the same numerator and denominator. (Contributed by Jeff Hankins, 29-Sep-2013.)
 |-  ( ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M 
 gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P 
 gcd  Q )  =  1 )  /\  ( M 
 /  N )  =  ( P  /  Q ) )  ->  ( M  =  P  /\  N  =  Q ) )
 
Theoremqredeu 10686* Every rational number has a unique reduced form. (Contributed by Jeff Hankins, 29-Sep-2013.)
 |-  ( A  e.  QQ  ->  E! x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) )
 
Theoremrpmul 10687 If  K is relatively prime to  M and to  N, it is also relatively prime to their product. (Contributed by Mario Carneiro, 24-Feb-2014.) (Proof shortened by Mario Carneiro, 2-Jul-2015.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( K 
 gcd  M )  =  1 
 /\  ( K  gcd  N )  =  1 ) 
 ->  ( K  gcd  ( M  x.  N ) )  =  1 ) )
 
Theoremrpdvds 10688 If  K is relatively prime to  N then it is also relatively prime to any divisor  M of  N. (Contributed by Mario Carneiro, 19-Jun-2015.)
 |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( K  gcd  N )  =  1  /\  M  ||  N ) )  ->  ( K  gcd  M )  =  1 )
 
4.1.10  Cancellability of congruences
 
Theoremcongr 10689* Definition of congruence by integer multiple (see ProofWiki "Congruence (Number Theory)", 11-Jul-2021, https://proofwiki.org/wiki/Definition:Congruence_(Number_Theory)): An integer  A is congruent to an integer  B modulo  M if their difference is a multiple of 
M. See also the definition in [ApostolNT] p. 104: "...  a is congruent to  b modulo  m, and we write  a  ==  b (mod  m) if  m divides the difference  a  -  b", or Wikipedia "Modular arithmetic - Congruence", https://en.wikipedia.org/wiki/Modular_arithmetic#Congruence, 11-Jul-2021,: "Given an integer n > 1, called a modulus, two integers are said to be congruent modulo n, if n is a divisor of their difference (i.e., if there is an integer k such that a-b = kn)". (Contributed by AV, 11-Jul-2021.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN )  ->  ( ( A  mod  M )  =  ( B 
 mod  M )  <->  E. n  e.  ZZ  ( n  x.  M )  =  ( A  -  B ) ) )
 
Theoremdivgcdcoprm0 10690 Integers divided by gcd are coprime. (Contributed by AV, 12-Jul-2021.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  B  =/=  0 ) 
 ->  ( ( A  /  ( A  gcd  B ) )  gcd  ( B 
 /  ( A  gcd  B ) ) )  =  1 )
 
Theoremdivgcdcoprmex 10691* Integers divided by gcd are coprime (see ProofWiki "Integers Divided by GCD are Coprime", 11-Jul-2021, https://proofwiki.org/wiki/Integers_Divided_by_GCD_are_Coprime): Any pair of integers, not both zero, can be reduced to a pair of coprime ones by dividing them by their gcd. (Contributed by AV, 12-Jul-2021.)
 |-  ( ( A  e.  ZZ  /\  ( B  e.  ZZ  /\  B  =/=  0
 )  /\  M  =  ( A  gcd  B ) )  ->  E. a  e.  ZZ  E. b  e. 
 ZZ  ( A  =  ( M  x.  a
 )  /\  B  =  ( M  x.  b
 )  /\  ( a  gcd  b )  =  1 ) )
 
Theoremcncongr1 10692 One direction of the bicondition in cncongr 10694. Theorem 5.4 in [ApostolNT] p. 109. (Contributed by AV, 13-Jul-2021.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( N  e.  NN  /\  M  =  ( N  /  ( C  gcd  N ) ) ) )  ->  (
 ( ( A  x.  C )  mod  N )  =  ( ( B  x.  C )  mod  N )  ->  ( A  mod  M )  =  ( B  mod  M ) ) )
 
Theoremcncongr2 10693 The other direction of the bicondition in cncongr 10694. (Contributed by AV, 11-Jul-2021.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( N  e.  NN  /\  M  =  ( N  /  ( C  gcd  N ) ) ) )  ->  (
 ( A  mod  M )  =  ( B  mod  M )  ->  (
 ( A  x.  C )  mod  N )  =  ( ( B  x.  C )  mod  N ) ) )
 
Theoremcncongr 10694 Cancellability of Congruences (see ProofWiki "Cancellability of Congruences, https://proofwiki.org/wiki/Cancellability_of_Congruences, 10-Jul-2021): Two products with a common factor are congruent modulo a positive integer iff the other factors are congruent modulo the integer divided by the greates common divisor of the integer and the common factor. See also Theorem 5.4 "Cancellation law" in [ApostolNT] p. 109. (Contributed by AV, 13-Jul-2021.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( N  e.  NN  /\  M  =  ( N  /  ( C  gcd  N ) ) ) )  ->  (
 ( ( A  x.  C )  mod  N )  =  ( ( B  x.  C )  mod  N )  <->  ( A  mod  M )  =  ( B 
 mod  M ) ) )
 
Theoremcncongrcoprm 10695 Corollary 1 of Cancellability of Congruences: Two products with a common factor are congruent modulo an integer being coprime to the common factor iff the other factors are congruent modulo the integer. (Contributed by AV, 13-Jul-2021.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( N  e.  NN  /\  ( C  gcd  N )  =  1 ) )  ->  ( ( ( A  x.  C )  mod  N )  =  ( ( B  x.  C ) 
 mod  N )  <->  ( A  mod  N )  =  ( B 
 mod  N ) ) )
 
4.2  Elementary prime number theory
 
4.2.1  Elementary properties

Remark: to represent odd prime numbers, i.e., all prime numbers except  2, the idiom  P  e.  ( Prime  \  { 2 } ) is used. It is a little bit shorter than  ( P  e. 
Prime  /\  P  =/=  2
). Both representations can be converted into each other by eldifsn 3535.

 
Syntaxcprime 10696 Extend the definition of a class to include the set of prime numbers.
 class  Prime
 
Definitiondf-prm 10697* Define the set of prime numbers. (Contributed by Paul Chapman, 22-Jun-2011.)
 |- 
 Prime  =  { p  e.  NN  |  { n  e.  NN  |  n  ||  p }  ~~  2o }
 
Theoremisprm 10698* The predicate "is a prime number". A prime number is a positive integer with exactly two positive divisors. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( P  e.  Prime  <->  ( P  e.  NN  /\  { n  e.  NN  |  n  ||  P }  ~~  2o ) )
 
Theoremprmnn 10699 A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( P  e.  Prime  ->  P  e.  NN )
 
Theoremprmz 10700 A prime number is an integer. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Jonathan Yan, 16-Jul-2017.)
 |-  ( P  e.  Prime  ->  P  e.  ZZ )
    < 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 11001-11062
  Copyright terms: Public domain < Previous  Next >