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Theorem List for Metamath Proof Explorer - 12601-12700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmulgcdr 12601 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 12602 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 12603 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 12604 Extend gcdmultiple 12603 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 )
 
Theoremgcdeq 12605  A is equal to its gcd with  B if and only if  A divides  B. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A 
 gcd  B )  =  A  <->  A 
 ||  B ) )
 
Theoremdvdssqim 12606 Unidirectional form of dvdssq 12613. (Contributed by Scott Fenton, 19-Apr-2014.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  ->  ( M ^
 2 )  ||  ( N ^ 2 ) ) )
 
Theoremdvdsmulgcd 12607 A divisibility equivalent for odmulg 14704. (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 12608 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 12609 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 12610 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 12611 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 12612 Lemma for dvdssq 12613. (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 12613 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 ) ) )
 
6.1.8  Algorithms
 
Theoremnn0seqcvgd 12614* 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 )
 
Theoremseq1st 12615 A sequence whose iteration function ignores the second argument is only affected by the first point of the initial value function. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  R  =  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A }
 ) )   =>    |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  R  =  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } )
 )
 
Theoremalgr0 12616 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 }
 ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   =>    |-  ( ph  ->  ( R `  M )  =  A )
 
Theoremalgrf 12617 An algorithm is step a 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 }
 ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  F : S --> S )   =>    |-  ( ph  ->  R : Z --> S )
 
Theoremalgrp1 12618 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 }
 ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  F : S --> S )   =>    |-  ( ( ph  /\  K  e.  Z ) 
 ->  ( R `  ( K  +  1 )
 )  =  ( F `
  ( R `  K ) ) )
 
Theoremalginv 12619* 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 } )
 )   &    |-  F : S --> S   &    |-  I  Fn  S   &    |-  ( x  e.  S  ->  ( I `  ( F `  x ) )  =  ( I `  x ) )   =>    |-  ( ( A  e.  S  /\  K  e.  NN0 )  ->  ( I `  ( R `  K ) )  =  ( I `
  ( R `  0 ) ) )
 
Theoremalgcvg 12620* 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 } ) )   &    |-  C : S --> NN0   &    |-  ( z  e.  S  ->  ( ( C `  ( F `  z ) )  =/=  0  ->  ( C `  ( F `
  z ) )  <  ( C `  z ) ) )   &    |-  N  =  ( C `  A )   =>    |-  ( A  e.  S  ->  ( C `  ( R `  N ) )  =  0 )
 
Theoremalgcvgblem 12621 Lemma for algcvgb 12622. (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 12622 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 non-zero, 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
 ) ) ) )
 
Theoremalgcvga 12623* 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 } ) )   &    |-  C : S --> NN0   &    |-  ( z  e.  S  ->  ( ( C `  ( F `  z ) )  =/=  0  ->  ( C `  ( F `
  z ) )  <  ( C `  z ) ) )   &    |-  N  =  ( C `  A )   =>    |-  ( A  e.  S  ->  ( K  e.  ( ZZ>=
 `  N )  ->  ( C `  ( R `
  K ) )  =  0 ) )
 
Theoremalgfx 12624* 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 } ) )   &    |-  C : S --> NN0   &    |-  ( z  e.  S  ->  ( ( C `  ( F `  z ) )  =/=  0  ->  ( C `  ( F `
  z ) )  <  ( C `  z ) ) )   &    |-  N  =  ( C `  A )   &    |-  ( z  e.  S  ->  ( ( C `  z )  =  0  ->  ( F `  z )  =  z ) )   =>    |-  ( A  e.  S  ->  ( K  e.  ( ZZ>=
 `  N )  ->  ( R `  K )  =  ( R `  N ) ) )
 
6.1.9  Euclid's Algorithm
 
Theoremeucalgval2 12625* 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 12626* 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.

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 12627* 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 12628* 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 12629* 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 )
 ) )
 
Theoremeucalgcvga 12630* Once Euclid's Algorithm halts after 
N steps, the second element of the state remains 0 . (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 29-May-2014.)
 |-  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 } ) )   &    |-  N  =  ( 2nd `  A )   =>    |-  ( A  e.  ( NN0  X.  NN0 )  ->  ( K  e.  ( ZZ>= `  N )  ->  ( 2nd `  ( R `  K ) )  =  0
 ) )
 
Theoremeucalg 12631* Euclid's Algorithm. 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 >.. (Contributed by Paul Chapman, 31-Mar-2011.) (Proof shortened by Mario Carneiro, 29-May-2014.)
 |-  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 } ) )   &    |-  A  =  <. M ,  N >.   =>    |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( 1st `  ( R `  N ) )  =  ( M  gcd  N ) )
 
6.2  Elementary prime number theory
 
6.2.1  Elementary properties
 
Syntaxcprime 12632 Extend the definition of a class to include the set of prime numbers.
 class  Prime
 
Definitiondf-prime 12633* Define the set of prime numbers. (Contributed by Paul Chapman, 22-Jun-2011.)
 |- 
 Prime  =  { p  e.  NN  |  { n  e.  NN  |  n  ||  p }  ~~  2o }
 
Theoremisprm 12634* 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 ) )
 
Theoremprmz 12635 A prime number is an integer. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( P  e.  Prime  ->  P  e.  ZZ )
 
Theoremprmnn 12636 A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( P  e.  Prime  ->  P  e.  NN )
 
Theorem1nprm 12637 1 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.)
 |- 
 -.  1  e.  Prime
 
Theorem1idssfct 12638* The positive divisors of a positive integer include 1 and itself. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( N  e.  NN  ->  { 1 ,  N }  C_  { n  e. 
 NN  |  n  ||  N } )
 
Theoremisprm2lem 12639* Lemma for isprm2 12640. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( P  e.  NN  /\  P  =/=  1
 )  ->  ( { n  e.  NN  |  n  ||  P }  ~~  2o  <->  { n  e.  NN  |  n  ||  P }  =  {
 1 ,  P }
 ) )
 
Theoremisprm2 12640* The predicate "is a prime number". A prime number is an integer greater than or equal to 2 whose only positive divisors are 1 and itself. (Contributed by Paul Chapman, 26-Oct-2012.)
 |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
 
Theoremisprm3 12641* The predicate "is a prime number". A prime number is an integer greater than or equal to 2 with no divisors strictly between 1 and itself. (Contributed by Paul Chapman, 26-Oct-2012.)
 |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  ( 2 ... ( P  -  1 ) )  -.  z  ||  P ) )
 
Theoremisprm4 12642* The predicate "is a prime number". A prime number is an integer greater than or equal to 2 whose only divisor greater than or equal to 2 is itself. (Contributed by Paul Chapman, 26-Oct-2012.)
 |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  ( ZZ>= `  2 )
 ( z  ||  P  ->  z  =  P ) ) )
 
Theoremprmind2 12643* A variation on prmind 12644 assuming complete induction for primes. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( x  =  1 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  z  ->  (
 ph 
 <-> 
 th ) )   &    |-  ( x  =  ( y  x.  z )  ->  ( ph 
 <->  ta ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  et ) )   &    |-  ps   &    |-  (
 ( x  e.  Prime  /\ 
 A. y  e.  (
 1 ... ( x  -  1 ) ) ch )  ->  ph )   &    |-  (
 ( y  e.  ( ZZ>=
 `  2 )  /\  z  e.  ( ZZ>= `  2 ) )  ->  ( ( ch  /\  th )  ->  ta )
 )   =>    |-  ( A  e.  NN  ->  et )
 
Theoremprmind 12644* Perform induction over the multiplicative structure of  NN. If a property  ph ( x ) holds for the primes and  1 and is preserved under multiplication, then it holds for every natural number. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( x  =  1 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  z  ->  (
 ph 
 <-> 
 th ) )   &    |-  ( x  =  ( y  x.  z )  ->  ( ph 
 <->  ta ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  et ) )   &    |-  ps   &    |-  ( x  e.  Prime  ->  ph )   &    |-  (
 ( y  e.  ( ZZ>=
 `  2 )  /\  z  e.  ( ZZ>= `  2 ) )  ->  ( ( ch  /\  th )  ->  ta )
 )   =>    |-  ( A  e.  NN  ->  et )
 
Theoremdvdsprime 12645 If  M divides a prime, then  M is either the prime or one. (Contributed by Scott Fenton, 8-Apr-2014.)
 |-  ( ( P  e.  Prime  /\  M  e.  NN )  ->  ( M  ||  P 
 <->  ( M  =  P  \/  M  =  1 ) ) )
 
Theoremnprm 12646 A product of two integers greater than one is composite. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( ( A  e.  ( ZZ>= `  2 )  /\  B  e.  ( ZZ>= `  2 ) )  ->  -.  ( A  x.  B )  e.  Prime )
 
Theoremnprmi 12647 An inference for compositeness. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)
 |-  A  e.  NN   &    |-  B  e.  NN   &    |-  1  <  A   &    |-  1  <  B   &    |-  ( A  x.  B )  =  N   =>    |-  -.  N  e.  Prime
 
Theorem2prm 12648 2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
 |-  2  e.  Prime
 
Theorem3prm 12649 3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  3  e.  Prime
 
Theoremprmuz2 12650 A prime number is an integer greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)
 |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 ) )
 
Theoremsqnprm 12651 A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( A  e.  ZZ  ->  -.  ( A ^
 2 )  e.  Prime )
 
Theoremdvdsprm 12652 An integer greater than or equal to 2 divides a prime number iff it is equal to it. (Contributed by Paul Chapman, 26-Oct-2012.)
 |-  ( ( N  e.  ( ZZ>= `  2 )  /\  P  e.  Prime )  ->  ( N  ||  P  <->  N  =  P ) )
 
Theoremcoprm 12653 A prime number either divides an integer or is coprime to it, but not both. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  P  ||  N  <->  ( P  gcd  N )  =  1 ) )
 
Theoremprmrp 12654 Unequal prime numbers are relatively prime. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( P  e.  Prime  /\  Q  e.  Prime ) 
 ->  ( ( P  gcd  Q )  =  1  <->  P  =/=  Q ) )
 
Theoremcoprmdvds 12655 If an integer divides the product of two integers and is coprime to one of them, then it divides the other. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  ||  ( M  x.  N )  /\  ( K  gcd  M )  =  1 ) 
 ->  K  ||  N )
 )
 
Theoremcoprmdvds2 12656 If an integer is divisible by two coprime integers, than 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 12657 One half of rpmulgcd2 12658, 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 12658 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 12659 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 12660* 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 ) ) ) )
 
Theoremeuclemma 12661 Euclid's lemma. A prime number divides the product of two integers iff it divides at least one of them. (Contributed by Paul Chapman, 17-Nov-2012.)
 |-  ( ( P  e.  Prime  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( P  ||  ( M  x.  N )  <->  ( P  ||  M  \/  P  ||  N ) ) )
 
Theoremisprm6 12662* A number is prime iff it satisfies Euclid's lemma euclemma 12661. (Contributed by Mario Carneiro, 6-Sep-2015.)
 |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. x  e.  ZZ  A. y  e. 
 ZZ  ( P  ||  ( x  x.  y
 )  ->  ( P  ||  x  \/  P  ||  y ) ) ) )
 
Theoremexprmfct 12663* Every integer greater than or equal to 2 has a prime factor. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 20-Jun-2015.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  ->  E. p  e.  Prime  p 
 ||  N )
 
Theoremnprmdvds1 12664 No prime number divides 1. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 2-Jul-2015.)
 |-  ( P  e.  Prime  ->  -.  P  ||  1 )
 
Theoremisprm5 12665* One need only check prime divisors of  P up to  sqr P in order to ensure primality. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  Prime  ( ( z ^ 2 )  <_  P  ->  -.  z  ||  P ) ) )
 
Theoremmaxprmfct 12666* The set of prime factors of an integer greater than or equal to 2 satisfies the conditions to have a supremum, and that supremum is a member of the set. (Contributed by Paul Chapman, 17-Nov-2012.)
 |-  S  =  { z  e.  Prime  |  z  ||  N }   =>    |-  ( N  e.  ( ZZ>=
 `  2 )  ->  ( ( S  C_  ZZ  /\  S  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  S  y  <_  x )  /\  sup ( S ,  RR ,  <  )  e.  S ) )
 
Theoremprmdvdsexp 12667 A prime divides a positive power of an integer iff it divides the integer. (Contributed by Mario Carneiro, 24-Feb-2014.) (Revised by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  N  e.  NN )  ->  ( P  ||  ( A ^ N )  <->  P  ||  A ) )
 
Theoremprmdvdsexpb 12668 A prime divides a positive power of another iff they are equal. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 24-Feb-2014.)
 |-  ( ( P  e.  Prime  /\  Q  e.  Prime  /\  N  e.  NN )  ->  ( P  ||  ( Q ^ N )  <->  P  =  Q ) )
 
Theoremprmdvdsexpr 12669 If a prime divides a nonnegative power of another then they are equal. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( ( P  e.  Prime  /\  Q  e.  Prime  /\  N  e.  NN0 )  ->  ( P  ||  ( Q ^ N )  ->  P  =  Q )
 )
 
Theoremprmexpb 12670 Two positive prime powers are equal iff the primes and the powers are equal. (Contributed by Paul Chapman, 30-Nov-2012.)
 |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN ) )  ->  ( ( P ^ M )  =  ( Q ^ N )  <->  ( P  =  Q  /\  M  =  N ) ) )
 
Theoremprmfac1 12671 The factorial of a number only contains primes less than the base. (Contributed by Mario Carneiro, 6-Mar-2014.)
 |-  ( ( N  e.  NN0  /\  P  e.  Prime  /\  P  ||  ( ! `  N ) )  ->  P  <_  N )
 
Theoremdivgcdodd 12672 Either  A  /  ( A  gcd  B ) is odd or  B  /  ( A  gcd  B ) is odd. (Contributed by Scott Fenton, 19-Apr-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( -.  2  ||  ( A  /  ( A  gcd  B ) )  \/  -.  2  ||  ( B  /  ( A  gcd  B ) ) ) )
 
Theoremrpexp 12673 If two numbers  A and  B are relatively prime, then they are still relatively prime if raised to a power. (Contributed by Mario Carneiro, 24-Feb-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  ( ( ( A ^ N )  gcd  B )  =  1  <->  ( A  gcd  B )  =  1 ) )
 
Theoremrpexp1i 12674 Relative primality passes to asymmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN0 )  ->  ( ( A  gcd  B )  =  1  ->  ( ( A ^ M )  gcd  B )  =  1 ) )
 
Theoremrpexp12i 12675 Relative primality passes to symmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 )
 )  ->  ( ( A  gcd  B )  =  1  ->  ( ( A ^ M )  gcd  ( B ^ N ) )  =  1 ) )
 
Theoremrpmul 12676 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 12677 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 )
 
6.2.2  Properties of the canonical representation of a rational
 
Syntaxcnumer 12678 Extend class notation to include canonical numerator function.
 class numer
 
Syntaxcdenom 12679 Extend class notation to include canonical denominator function.
 class denom
 
Definitiondf-numer 12680* The canonical numerator of a rational is the numerator of the rational's reduced fraction representation (no common factors, denominator positive). (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- numer  =  ( y  e.  QQ  |->  ( 1st `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  y  =  (
 ( 1st `  x )  /  ( 2nd `  x ) ) ) ) ) )
 
Definitiondf-denom 12681* The canonical denominator of a rational is the denominator of the rational's reduced fraction representation (no common factors, denominator positive). (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- denom  =  ( y  e.  QQ  |->  ( 2nd `  ( iota_ x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  y  =  (
 ( 1st `  x )  /  ( 2nd `  x ) ) ) ) ) )
 
Theoremqnumval 12682* Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (numer `  A )  =  ( 1st `  ( iota_ x  e.  ( ZZ  X. 
 NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) ) ) )
 
Theoremqdenval 12683* Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (denom `  A )  =  ( 2nd `  ( iota_ x  e.  ( ZZ  X. 
 NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) ) ) )
 
Theoremqnumdencl 12684 Lemma for qnumcl 12685 and qdencl 12686. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( (numer `  A )  e.  ZZ  /\  (denom `  A )  e.  NN ) )
 
Theoremqnumcl 12685 The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (numer `  A )  e.  ZZ )
 
Theoremqdencl 12686 The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  (denom `  A )  e.  NN )
 
Theoremfnum 12687 Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- numer : QQ --> ZZ
 
Theoremfden 12688 Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |- denom : QQ --> NN
 
Theoremqnumdenbi 12689 Two numbers are the canonical representation of a rational iff they are coprime and have the right quotient. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( ( ( B 
 gcd  C )  =  1 
 /\  A  =  ( B  /  C ) )  <->  ( (numer `  A )  =  B  /\  (denom `  A )  =  C ) ) )
 
Theoremqnumdencoprm 12690 The canonical representation of a rational is fully reduced. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( (numer `  A )  gcd  (denom `  A ) )  =  1
 )
 
Theoremqeqnumdivden 12691 Recover a rational number from its canonical representation. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  A  =  ( (numer `  A )  /  (denom `  A ) ) )
 
Theoremqmuldeneqnum 12692 Multiplying a rational by its denominator results in an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( A  x.  (denom `  A ) )  =  (numer `  A )
 )
 
Theoremdivnumden 12693 Calculate the reduced form of a quotient using  gcd. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( (numer `  ( A  /  B ) )  =  ( A 
 /  ( A  gcd  B ) )  /\  (denom `  ( A  /  B ) )  =  ( B  /  ( A  gcd  B ) ) ) )
 
Theoremdivdenle 12694 Reducing a quotient never increases the denominator. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  (denom `  ( A  /  B ) )  <_  B )
 
Theoremqnumgt0 12695 A rational is positive iff its canonical numerator is. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( 0  <  A  <->  0  <  (numer `  A ) ) )
 
Theoremqgt0numnn 12696 A rational is positive iff its canonical numerator is a natural number. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( ( A  e.  QQ  /\  0  <  A )  ->  (numer `  A )  e.  NN )
 
Theoremnn0gcdsq 12697 Squaring commutes with GCD, in particular two coprime numbers have coprime squares. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( ( A  gcd  B ) ^ 2 )  =  ( ( A ^ 2 )  gcd  ( B ^ 2 ) ) )
 
Theoremzgcdsq 12698 nn0gcdsq 12697 extended to integers by symmetry. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A 
 gcd  B ) ^ 2
 )  =  ( ( A ^ 2 ) 
 gcd  ( B ^
 2 ) ) )
 
Theoremnumdensq 12699 Squaring a rational squares its canonical components. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  ( (numer `  ( A ^ 2 ) )  =  ( (numer `  A ) ^ 2
 )  /\  (denom `  ( A ^ 2 ) )  =  ( (denom `  A ) ^ 2
 ) ) )
 
Theoremnumsq 12700 Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
 |-  ( A  e.  QQ  ->  (numer `  ( A ^ 2 ) )  =  ( (numer `  A ) ^ 2
 ) )
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