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Theorem List for Metamath Proof Explorer - 12801-12900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempcabs 12801 The prime count of an absolute value. (Contributed by Mario Carneiro, 13-Mar-2014.)
 |-  ( ( P  e.  Prime  /\  A  e.  QQ )  ->  ( P  pCnt  ( abs `  A )
 )  =  ( P 
 pCnt  A ) )
 
Theorempcdvdstr 12802 The prime count increases under the divisibility relation. (Contributed by Mario Carneiro, 13-Mar-2014.)
 |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B )
 )  ->  ( P  pCnt  A )  <_  ( P  pCnt  B ) )
 
Theorempcgcd1 12803 The prime count of a GCD is the minimum of the prime counts of the arguments. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( P  pCnt  A )  <_  ( P  pCnt  B ) )  ->  ( P  pCnt  ( A 
 gcd  B ) )  =  ( P  pCnt  A ) )
 
Theorempcgcd 12804 The prime count of a GCD is the minimum of the prime counts of the arguments. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  ( A  gcd  B ) )  =  if ( ( P  pCnt  A )  <_  ( P  pCnt  B ) ,  ( P  pCnt  A ) ,  ( P 
 pCnt  B ) ) )
 
Theorempc2dvds 12805* A characterization of divisibility in terms of prime count. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 3-Oct-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  ||  B 
 <-> 
 A. p  e.  Prime  ( p  pCnt  A )  <_  ( p  pCnt  B ) ) )
 
Theorempc11 12806* The prime count function, viewed as a function from  NN to  ( NN  ^m  Prime ), is one-to-one. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( A  =  B  <->  A. p  e.  Prime  ( p  pCnt  A )  =  ( p  pCnt  B ) ) )
 
Theorempcz 12807* The prime count function can be used as an indicator that a given rational number is an integer. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( A  e.  QQ  ->  ( A  e.  ZZ  <->  A. p  e.  Prime  0  <_  ( p  pCnt  A ) ) )
 
Theorempcprmpw2 12808* Self-referential expression for a prime power. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( E. n  e.  NN0  A  ||  ( P ^ n )  <->  A  =  ( P ^ ( P  pCnt  A ) ) ) )
 
Theorempcprmpw 12809* Self-referential expression for a prime power. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( ( P  e.  Prime  /\  A  e.  NN )  ->  ( E. n  e.  NN0  A  =  ( P ^ n )  <->  A  =  ( P ^ ( P  pCnt  A ) ) ) )
 
Theorempcaddlem 12810 Lemma for pcadd 12811. The original numbers  A and  B have been decomposed using the prime count function as  ( P ^ M )  x.  ( R  /  S ) where  R ,  S are both not divisible by  P and  M  =  ( P  pCnt  A ), and similarly for  B. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  A  =  ( ( P ^ M )  x.  ( R  /  S ) ) )   &    |-  ( ph  ->  B  =  ( ( P ^ N )  x.  ( T  /  U ) ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  ( R  e.  ZZ  /\  -.  P  ||  R ) )   &    |-  ( ph  ->  ( S  e.  NN  /\  -.  P  ||  S ) )   &    |-  ( ph  ->  ( T  e.  ZZ  /\  -.  P  ||  T ) )   &    |-  ( ph  ->  ( U  e.  NN  /\  -.  P  ||  U ) )   =>    |-  ( ph  ->  M 
 <_  ( P  pCnt  ( A  +  B )
 ) )
 
Theorempcadd 12811 An inequality for the prime count of a sum. This is the source of the ultrametric inequality for the p-adic metric. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  A  e.  QQ )   &    |-  ( ph  ->  B  e.  QQ )   &    |-  ( ph  ->  ( P  pCnt  A )  <_  ( P  pCnt  B ) )   =>    |-  ( ph  ->  ( P  pCnt  A )  <_  ( P  pCnt  ( A  +  B ) ) )
 
Theorempcadd2 12812 The inequality of pcadd 12811 becomes an equality when one of the factors has prime count strictly less than the other. (Contributed by Mario Carneiro, 16-Jan-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  A  e.  QQ )   &    |-  ( ph  ->  B  e.  QQ )   &    |-  ( ph  ->  ( P  pCnt  A )  < 
 ( P  pCnt  B ) )   =>    |-  ( ph  ->  ( P  pCnt  A )  =  ( P  pCnt  ( A  +  B )
 ) )
 
Theorempcmptcl 12813 Closure for the prime power map. (Contributed by Mario Carneiro, 12-Mar-2014.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ A ) ,  1 ) )   &    |-  ( ph  ->  A. n  e.  Prime  A  e.  NN0 )   =>    |-  ( ph  ->  ( F : NN --> NN  /\  seq  1 (  x.  ,  F ) : NN --> NN ) )
 
Theorempcmpt 12814* Construct a function with given prime count characteristics. (Contributed by Mario Carneiro, 12-Mar-2014.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ A ) ,  1 ) )   &    |-  ( ph  ->  A. n  e.  Prime  A  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( n  =  P  ->  A  =  B )   =>    |-  ( ph  ->  ( P  pCnt  (  seq  1 (  x.  ,  F ) `
  N ) )  =  if ( P 
 <_  N ,  B , 
 0 ) )
 
Theorempcmpt2 12815* Dividing two prime count maps yields a number with all dividing primes confined to an interval. (Contributed by Mario Carneiro, 14-Mar-2014.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ A ) ,  1 ) )   &    |-  ( ph  ->  A. n  e.  Prime  A  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( n  =  P  ->  A  =  B )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  N )
 )   =>    |-  ( ph  ->  ( P  pCnt  ( (  seq  1 (  x.  ,  F ) `  M )  /  (  seq  1 (  x. 
 ,  F ) `  N ) ) )  =  if ( ( P  <_  M  /\  -.  P  <_  N ) ,  B ,  0 ) )
 
Theorempcmptdvds 12816 The partial products of the prime power map form a divisibility chain. (Contributed by Mario Carneiro, 12-Mar-2014.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ A ) ,  1 ) )   &    |-  ( ph  ->  A. n  e.  Prime  A  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  M  e.  ( ZZ>=
 `  N ) )   =>    |-  ( ph  ->  (  seq  1 (  x.  ,  F ) `  N )  ||  (  seq  1 (  x. 
 ,  F ) `  M ) )
 
Theorempcprod 12817* The product of the primes taken to their respective powers reconstructs the original number. (Contributed by Mario Carneiro, 12-Mar-2014.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ ( n  pCnt  N ) ) ,  1 ) )   =>    |-  ( N  e.  NN  ->  (  seq  1 (  x.  ,  F ) `
  N )  =  N )
 
Theoremsumhash 12818* The sum of 1 over a set is the size of the set. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 20-May-2014.)
 |-  ( ( B  e.  Fin  /\  A  C_  B )  -> 
 sum_ k  e.  B  if ( k  e.  A ,  1 ,  0 )  =  ( # `  A ) )
 
Theoremfldivp1 12819 The difference between the floors of adjacent fractions is either 1 or 0. (Contributed by Mario Carneiro, 8-Mar-2014.)
 |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( |_ `  ( ( M  +  1 )  /  N ) )  -  ( |_ `  ( M  /  N ) ) )  =  if ( N  ||  ( M  +  1
 ) ,  1 ,  0 ) )
 
Theorempcfaclem 12820 Lemma for pcfac 12821. (Contributed by Mario Carneiro, 20-May-2014.)
 |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( |_ `  ( N  /  ( P ^ M ) ) )  =  0 )
 
Theorempcfac 12821* Calculate the prime count of a factorial. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Mario Carneiro, 21-May-2014.)
 |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( P  pCnt  ( ! `  N ) )  =  sum_ k  e.  ( 1 ...
 M ) ( |_ `  ( N  /  ( P ^ k ) ) ) )
 
Theorempcbc 12822* Calculate the prime count of a binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Mario Carneiro, 21-May-2014.)
 |-  ( ( N  e.  NN  /\  K  e.  (
 0 ... N )  /\  P  e.  Prime )  ->  ( P  pCnt  ( N  _C  K ) )  =  sum_ k  e.  (
 1 ... N ) ( ( |_ `  ( N  /  ( P ^
 k ) ) )  -  ( ( |_ `  ( ( N  -  K )  /  ( P ^ k ) ) )  +  ( |_ `  ( K  /  ( P ^ k ) ) ) ) ) )
 
Theoremqexpz 12823 If a power of a rational number is an integer, then the number is an integer. In other words, all n-th roots are irrational unless they are integers (so that the original number is an n-th power). (Contributed by Mario Carneiro, 10-Aug-2015.)
 |-  ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  ->  A  e.  ZZ )
 
Theoremexpnprm 12824 A second or higher power of a rational number is not a prime number. Or by contraposition, the n-th root of a prime number is irrational. Suggested by Norm Megill. (Contributed by Mario Carneiro, 10-Aug-2015.)
 |-  ( ( A  e.  QQ  /\  N  e.  ( ZZ>=
 `  2 ) ) 
 ->  -.  ( A ^ N )  e.  Prime )
 
6.2.6  Pocklington's theorem
 
Theoremprmpwdvds 12825 A relation involving divisibility by a prime power. (Contributed by Mario Carneiro, 2-Mar-2014.)
 |-  ( ( ( K  e.  ZZ  /\  D  e.  ZZ )  /\  ( P  e.  Prime  /\  N  e.  NN )  /\  ( D  ||  ( K  x.  ( P ^ N ) )  /\  -.  D  ||  ( K  x.  ( P ^ ( N  -  1 ) ) ) ) )  ->  ( P ^ N )  ||  D )
 
Theorempockthlem 12826 Lemma for pockthg 12827. (Contributed by Mario Carneiro, 2-Mar-2014.)
 |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  B  <  A )   &    |-  ( ph  ->  N  =  ( ( A  x.  B )  +  1
 ) )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  P 
 ||  N )   &    |-  ( ph  ->  Q  e.  Prime )   &    |-  ( ph  ->  ( Q  pCnt  A )  e.  NN )   &    |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  ( ( C ^ ( N  -  1 ) ) 
 mod  N )  =  1 )   &    |-  ( ph  ->  ( ( ( C ^
 ( ( N  -  1 )  /  Q ) )  -  1 ) 
 gcd  N )  =  1 )   =>    |-  ( ph  ->  ( Q  pCnt  A )  <_  ( Q  pCnt  ( P  -  1 ) ) )
 
Theorempockthg 12827* The generalized Pocklington's theorem. If  N  -  1  =  A  x.  B where  B  <  A, then  N is prime if and only if for every prime factor  p of  A, there is an  x such that  x ^ ( N  -  1 )  =  1 (  mod 
N ) and  gcd  ( x ^ ( ( N  -  1 )  /  p )  -  1 ,  N )  =  1. (Contributed by Mario Carneiro, 2-Mar-2014.)
 |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  B  <  A )   &    |-  ( ph  ->  N  =  ( ( A  x.  B )  +  1
 ) )   &    |-  ( ph  ->  A. p  e.  Prime  ( p  ||  A  ->  E. x  e.  ZZ  ( ( ( x ^ ( N  -  1 ) ) 
 mod  N )  =  1 
 /\  ( ( ( x ^ ( ( N  -  1 ) 
 /  p ) )  -  1 )  gcd  N )  =  1 ) ) )   =>    |-  ( ph  ->  N  e.  Prime )
 
Theorempockthi 12828 Pocklington's theorem, which gives a sufficient criterion for a number  N to be prime. This is the preferred method for verifying large primes, being much more efficient to compute than trial division. This form has been optimized for application to specific large primes; see pockthg 12827 for a more general closed-form version. (Contributed by Mario Carneiro, 2-Mar-2014.)
 |-  P  e.  Prime   &    |-  G  e.  NN   &    |-  M  =  ( G  x.  P )   &    |-  N  =  ( M  +  1 )   &    |-  D  e.  NN   &    |-  E  e.  NN   &    |-  A  e.  NN   &    |-  M  =  ( D  x.  ( P ^ E ) )   &    |-  D  <  ( P ^ E )   &    |-  ( ( A ^ M )  mod  N )  =  ( 1 
 mod  N )   &    |-  ( ( ( A ^ G )  -  1 )  gcd  N )  =  1   =>    |-  N  e.  Prime
 
6.2.7  Infinite primes theorem
 
Theoremunbenlem 12829* Lemma for unben 12830. (Contributed by NM, 5-May-2005.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  G  =  ( rec ( ( x  e. 
 _V  |->  ( x  +  1 ) ) ,  1 )  |`  om )   =>    |-  (
 ( A  C_  NN  /\ 
 A. m  e.  NN  E. n  e.  A  m  <  n )  ->  A  ~~ 
 om )
 
Theoremunben 12830* An unbounded set of natural numbers is infinite. (Contributed by NM, 5-May-2005.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( ( A  C_  NN  /\  A. m  e. 
 NN  E. n  e.  A  m  <  n )  ->  A  ~~  NN )
 
Theoreminfpnlem1 12831* Lemma for infpn 12833. The smallest divisor (greater than 1)  M of  N !  + 
1 is a prime greater than  N. (Contributed by NM, 5-May-2005.)
 |-  K  =  ( ( ! `  N )  +  1 )   =>    |-  ( ( N  e.  NN  /\  M  e.  NN )  ->  (
 ( ( 1  <  M  /\  ( K  /  M )  e.  NN )  /\  A. j  e. 
 NN  ( ( 1  <  j  /\  ( K  /  j )  e. 
 NN )  ->  M  <_  j ) )  ->  ( N  <  M  /\  A. j  e.  NN  (
 ( M  /  j
 )  e.  NN  ->  ( j  =  1  \/  j  =  M ) ) ) ) )
 
Theoreminfpnlem2 12832* Lemma for infpn 12833. For any natural number  N, there exists a prime number  j greater than  N. (Contributed by NM, 5-May-2005.)
 |-  K  =  ( ( ! `  N )  +  1 )   =>    |-  ( N  e.  NN  ->  E. j  e.  NN  ( N  <  j  /\  A. k  e.  NN  (
 ( j  /  k
 )  e.  NN  ->  ( k  =  1  \/  k  =  j ) ) ) )
 
Theoreminfpn 12833* There exist infinitely many prime numbers: for any natural number  N, there exists a prime number  j greater than  N. (See infpn2 12834 for the equinumerosity version.) (Contributed by NM, 1-Jun-2006.)
 |-  ( N  e.  NN  ->  E. j  e.  NN  ( N  <  j  /\  A. k  e.  NN  (
 ( j  /  k
 )  e.  NN  ->  ( k  =  1  \/  k  =  j ) ) ) )
 
Theoreminfpn2 12834* There exist infinitely many prime numbers: the set of all primes  S is unbounded by infpn 12833, so by unben 12830 it is infinite. (Contributed by NM, 5-May-2005.)
 |-  S  =  { n  e.  NN  |  ( 1  <  n  /\  A. m  e.  NN  (
 ( n  /  m )  e.  NN  ->  ( m  =  1  \/  m  =  n ) ) ) }   =>    |-  S  ~~  NN
 
Theoremprmunb 12835* The primes are unbounded. (Contributed by Paul Chapman, 28-Nov-2012.)
 |-  ( N  e.  NN  ->  E. p  e.  Prime  N  <  p )
 
Theoremprminf 12836 There are an infinite number of primes. (Contributed by Paul Chapman, 28-Nov-2012.)
 |- 
 Prime  ~~  NN
 
6.2.8  Sum of prime reciprocals
 
Theoremprmreclem1 12837* Lemma for prmrec 12843. Properties of the "square part" function, which extracts the  m of the decomposition  N  =  r
m ^ 2, with  m maximal and  r squarefree. (Contributed by Mario Carneiro, 5-Aug-2014.)
 |-  Q  =  ( n  e.  NN  |->  sup ( { r  e.  NN  |  ( r ^ 2
 )  ||  n } ,  RR ,  <  )
 )   =>    |-  ( N  e.  NN  ->  ( ( Q `  N )  e.  NN  /\  ( ( Q `  N ) ^ 2
 )  ||  N  /\  ( K  e.  ( ZZ>=
 `  2 )  ->  -.  ( K ^ 2
 )  ||  ( N  /  ( ( Q `  N ) ^ 2
 ) ) ) ) )
 
Theoremprmreclem2 12838* Lemma for prmrec 12843. There are at most  2 ^ K squarefree numbers which divide no primes larger than  K. (We could strengthen this to  2 ^ # ( Prime  i^i  ( 1 ... K ) ) but there's no reason to.) We establish the inequality by showing that the prime counts of the number up to  K completely determine it because all higher prime counts are zero, and they are all at most  1 because no square divides the number, so there are at most  2 ^ K possibilities. (Contributed by Mario Carneiro, 5-Aug-2014.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  (
 1  /  n ) ,  0 ) )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( ph  ->  N  e.  NN )   &    |-  M  =  { n  e.  (
 1 ... N )  | 
 A. p  e.  ( Prime  \  ( 1 ...
 K ) )  -.  p  ||  n }   &    |-  Q  =  ( n  e.  NN  |->  sup ( { r  e. 
 NN  |  ( r ^ 2 )  ||  n } ,  RR ,  <  ) )   =>    |-  ( ph  ->  ( # `
  { x  e.  M  |  ( Q `
  x )  =  1 } )  <_  ( 2 ^ K ) )
 
Theoremprmreclem3 12839* Lemma for prmrec 12843. The main inequality established here is  # M  <_  # { x  e.  M  |  ( Q `  x )  =  1 }  x.  sqr N, where  { x  e.  M  |  ( Q `
 x )  =  1 } is the set of squarefree numbers in  M. This is demonstrated by the map  y  |->  <. y  /  ( Q `  y ) ^ 2 ,  ( Q `  y ) >. where  Q `  y is the largest number whose square divides  y. (Contributed by Mario Carneiro, 5-Aug-2014.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  (
 1  /  n ) ,  0 ) )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( ph  ->  N  e.  NN )   &    |-  M  =  { n  e.  (
 1 ... N )  | 
 A. p  e.  ( Prime  \  ( 1 ...
 K ) )  -.  p  ||  n }   &    |-  Q  =  ( n  e.  NN  |->  sup ( { r  e. 
 NN  |  ( r ^ 2 )  ||  n } ,  RR ,  <  ) )   =>    |-  ( ph  ->  ( # `
  M )  <_  ( ( 2 ^ K )  x.  ( sqr `  N ) ) )
 
Theoremprmreclem4 12840* Lemma for prmrec 12843. Show by induction that the indexed (nondisjoint) union  W `  k is at most the size of the prime reciprocal series. The key counting lemma is hashdvds 12717, to show that the number of numbers in  1 ... N that divide  k is at most  N  /  k. (Contributed by Mario Carneiro, 6-Aug-2014.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  (
 1  /  n ) ,  0 ) )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( ph  ->  N  e.  NN )   &    |-  M  =  { n  e.  (
 1 ... N )  | 
 A. p  e.  ( Prime  \  ( 1 ...
 K ) )  -.  p  ||  n }   &    |-  ( ph  ->  seq  1 (  +  ,  F )  e.  dom  ~~>  )   &    |-  ( ph  ->  sum_
 k  e.  ( ZZ>= `  ( K  +  1
 ) ) if (
 k  e.  Prime ,  (
 1  /  k ) ,  0 )  < 
 ( 1  /  2
 ) )   &    |-  W  =  ( p  e.  NN  |->  { n  e.  ( 1
 ... N )  |  ( p  e.  Prime  /\  p  ||  n ) } )   =>    |-  ( ph  ->  ( N  e.  ( ZZ>= `  K )  ->  ( # ` 
 U_ k  e.  (
 ( K  +  1 ) ... N ) ( W `  k
 ) )  <_  ( N  x.  sum_ k  e.  (
 ( K  +  1 ) ... N ) if ( k  e. 
 Prime ,  ( 1  /  k ) ,  0 ) ) ) )
 
Theoremprmreclem5 12841* Lemma for prmrec 12843. Here we show the inequality  N  / 
2  <  # M by decomposing the set  ( 1 ... N
) into the disjoint union of the set  M of those numbers that are not divisible by any "large" primes (above  K) and the indexed union over  K  <  k of the numbers  W `  k that divide the prime  k. By prmreclem4 12840 the second of these has size less than  N times the prime reciprocal series, which is less than  1  /  2 by assumption, we find that the complementary part  M must be at least  N  /  2 large. (Contributed by Mario Carneiro, 6-Aug-2014.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  (
 1  /  n ) ,  0 ) )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( ph  ->  N  e.  NN )   &    |-  M  =  { n  e.  (
 1 ... N )  | 
 A. p  e.  ( Prime  \  ( 1 ...
 K ) )  -.  p  ||  n }   &    |-  ( ph  ->  seq  1 (  +  ,  F )  e.  dom  ~~>  )   &    |-  ( ph  ->  sum_
 k  e.  ( ZZ>= `  ( K  +  1
 ) ) if (
 k  e.  Prime ,  (
 1  /  k ) ,  0 )  < 
 ( 1  /  2
 ) )   &    |-  W  =  ( p  e.  NN  |->  { n  e.  ( 1
 ... N )  |  ( p  e.  Prime  /\  p  ||  n ) } )   =>    |-  ( ph  ->  ( N  /  2 )  < 
 ( ( 2 ^ K )  x.  ( sqr `  N ) ) )
 
Theoremprmreclem6 12842* Lemma for prmrec 12843. If the series  F was convergent, there would be some  k such that the sum starting from  k  +  1 sums to less than  1  /  2; this is a sufficient hypothesis for prmreclem5 12841 to produce the contradictory bound  N  /  2  < 
( 2 ^ k
) sqr N, which is false for  N  =  2 ^ ( 2 k  +  2 ). (Contributed by Mario Carneiro, 6-Aug-2014.)
 |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  (
 1  /  n ) ,  0 ) )   =>    |-  -. 
 seq  1 (  +  ,  F )  e.  dom  ~~>
 
Theoremprmrec 12843* The sum of the reciprocals of the primes diverges. This is the "second" proof at http://en.wikipedia.org/wiki/Prime_harmonic_series, attributed to Paul Erdős. (Contributed by Mario Carneiro, 6-Aug-2014.)
 |-  F  =  ( n  e.  NN  |->  sum_ k  e.  ( Prime  i^i  ( 1
 ... n ) ) ( 1  /  k
 ) )   =>    |- 
 -.  F  e.  dom  ~~>
 
6.2.9  Fundamental theorem of arithmetic
 
Theorem1arithlem1 12844* Lemma for 1arith 12848. (Contributed by Mario Carneiro, 30-May-2014.)
 |-  M  =  ( n  e.  NN  |->  ( p  e.  Prime  |->  ( p 
 pCnt  n ) ) )   =>    |-  ( N  e.  NN  ->  ( M `  N )  =  ( p  e.  Prime  |->  ( p  pCnt  N ) ) )
 
Theorem1arithlem2 12845* Lemma for 1arith 12848. (Contributed by Mario Carneiro, 30-May-2014.)
 |-  M  =  ( n  e.  NN  |->  ( p  e.  Prime  |->  ( p 
 pCnt  n ) ) )   =>    |-  ( ( N  e.  NN  /\  P  e.  Prime ) 
 ->  ( ( M `  N ) `  P )  =  ( P  pCnt  N ) )
 
Theorem1arithlem3 12846* Lemma for 1arith 12848. (Contributed by Mario Carneiro, 30-May-2014.)
 |-  M  =  ( n  e.  NN  |->  ( p  e.  Prime  |->  ( p 
 pCnt  n ) ) )   =>    |-  ( N  e.  NN  ->  ( M `  N ) : Prime --> NN0 )
 
Theorem1arithlem4 12847* Lemma for 1arith 12848. (Contributed by Mario Carneiro, 30-May-2014.)
 |-  M  =  ( n  e.  NN  |->  ( p  e.  Prime  |->  ( p 
 pCnt  n ) ) )   &    |-  G  =  ( y  e.  NN  |->  if ( y  e. 
 Prime ,  ( y ^
 ( F `  y
 ) ) ,  1 ) )   &    |-  ( ph  ->  F : Prime --> NN0 )   &    |-  ( ph  ->  N  e.  NN )   &    |-  (
 ( ph  /\  ( q  e.  Prime  /\  N  <_  q ) )  ->  ( F `  q )  =  0 )   =>    |-  ( ph  ->  E. x  e.  NN  F  =  ( M `  x ) )
 
Theorem1arith 12848* Fundamental theorem of arithmetic, where a prime factorization is represented as a sequence of prime exponents, for which only finitely many primes have nonzero exponent. The function  M maps the set of positive integers one-to-one onto the set of prime factorizations  R. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 30-May-2014.)
 |-  M  =  ( n  e.  NN  |->  ( p  e.  Prime  |->  ( p 
 pCnt  n ) ) )   &    |-  R  =  { e  e.  ( NN0  ^m  Prime )  |  ( `' e " NN )  e.  Fin }   =>    |-  M : NN -1-1-onto-> R
 
Theorem1arith2 12849* Fundamental theorem of arithmetic, where a prime factorization is represented as a finite monotonic 1-based sequence of primes. Every positive integer has a unique prime factorization. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 30-May-2014.)
 |-  M  =  ( n  e.  NN  |->  ( p  e.  Prime  |->  ( p 
 pCnt  n ) ) )   &    |-  R  =  { e  e.  ( NN0  ^m  Prime )  |  ( `' e " NN )  e.  Fin }   =>    |-  A. z  e.  NN  E! g  e.  R  ( M `  z )  =  g
 
6.2.10  Lagrange's four-square theorem
 
Syntaxcgz 12850 Extend class notation with the set of gaussian integers.
 class  ZZ [ _i ]
 
Definitiondf-gz 12851 Define the set of gaussian integers, which are complex numbers whose real and imaginary parts are integers. (Note that the  [
_i ] is actually part of the symbol token and has no independent meaning.) (Contributed by Mario Carneiro, 14-Jul-2014.)
 |- 
 ZZ [ _i ]  =  { x  e.  CC  |  ( ( Re `  x )  e.  ZZ  /\  ( Im `  x )  e.  ZZ ) }
 
Theoremelgz 12852 Elementhood in the gaussian integers. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  ZZ [ _i ]  <->  ( A  e.  CC  /\  ( Re `  A )  e.  ZZ  /\  ( Im `  A )  e.  ZZ )
 )
 
Theoremgzcn 12853 A gaussian integer is a complex number. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  ZZ [ _i ]  ->  A  e.  CC )
 
Theoremzgz 12854 An integer is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  ZZ  ->  A  e.  ZZ [ _i ] )
 
Theoremigz 12855  _i is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  _i  e.  ZZ [ _i ]
 
Theoremgznegcl 12856 The gaussian integers are closed under negation. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  ZZ [ _i ]  ->  -u A  e.  ZZ [ _i ]
 )
 
Theoremgzcjcl 12857 The gaussian integers are closed under conjugation. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  ZZ [ _i ]  ->  ( * `  A )  e. 
 ZZ [ _i ]
 )
 
Theoremgzaddcl 12858 The gaussian integers are closed under addition. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( ( A  e.  ZZ [ _i ]  /\  B  e.  ZZ [ _i ] )  ->  ( A  +  B )  e. 
 ZZ [ _i ]
 )
 
Theoremgzmulcl 12859 The gaussian integers are closed under multiplication. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( ( A  e.  ZZ [ _i ]  /\  B  e.  ZZ [ _i ] )  ->  ( A  x.  B )  e. 
 ZZ [ _i ]
 )
 
Theoremgzreim 12860 Construct a gaussian integer from real and imaginary parts. (Contributed by Mario Carneiro, 16-Jul-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  +  ( _i  x.  B ) )  e.  ZZ [ _i ] )
 
Theoremgzsubcl 12861 The gaussian integers are closed under subtraction. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( ( A  e.  ZZ [ _i ]  /\  B  e.  ZZ [ _i ] )  ->  ( A  -  B )  e. 
 ZZ [ _i ]
 )
 
Theoremgzabssqcl 12862 The squared norm of a gaussian integer is an integer. (Contributed by Mario Carneiro, 16-Jul-2014.)
 |-  ( A  e.  ZZ [ _i ]  ->  (
 ( abs `  A ) ^ 2 )  e. 
 NN0 )
 
Theorem4sqlem5 12863 Lemma for 4sq 12885. (Contributed by Mario Carneiro, 15-Jul-2014.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  M  e.  NN )   &    |-  B  =  ( ( ( A  +  ( M  / 
 2 ) )  mod  M )  -  ( M 
 /  2 ) )   =>    |-  ( ph  ->  ( B  e.  ZZ  /\  ( ( A  -  B ) 
 /  M )  e. 
 ZZ ) )
 
Theorem4sqlem6 12864 Lemma for 4sq 12885. (Contributed by Mario Carneiro, 15-Jul-2014.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  M  e.  NN )   &    |-  B  =  ( ( ( A  +  ( M  / 
 2 ) )  mod  M )  -  ( M 
 /  2 ) )   =>    |-  ( ph  ->  ( -u ( M  /  2 )  <_  B  /\  B  <  ( M  /  2 ) ) )
 
Theorem4sqlem7 12865 Lemma for 4sq 12885. (Contributed by Mario Carneiro, 15-Jul-2014.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  M  e.  NN )   &    |-  B  =  ( ( ( A  +  ( M  / 
 2 ) )  mod  M )  -  ( M 
 /  2 ) )   =>    |-  ( ph  ->  ( B ^ 2 )  <_  ( ( ( M ^ 2 )  / 
 2 )  /  2
 ) )
 
Theorem4sqlem8 12866 Lemma for 4sq 12885. (Contributed by Mario Carneiro, 15-Jul-2014.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  M  e.  NN )   &    |-  B  =  ( ( ( A  +  ( M  / 
 2 ) )  mod  M )  -  ( M 
 /  2 ) )   =>    |-  ( ph  ->  M  ||  (
 ( A ^ 2
 )  -  ( B ^ 2 ) ) )
 
Theorem4sqlem9 12867 Lemma for 4sq 12885. (Contributed by Mario Carneiro, 15-Jul-2014.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  M  e.  NN )   &    |-  B  =  ( ( ( A  +  ( M  / 
 2 ) )  mod  M )  -  ( M 
 /  2 ) )   &    |-  ( ( ph  /\  ps )  ->  ( B ^
 2 )  =  0 )   =>    |-  ( ( ph  /\  ps )  ->  ( M ^
 2 )  ||  ( A ^ 2 ) )
 
Theorem4sqlem10 12868 Lemma for 4sq 12885. (Contributed by Mario Carneiro, 16-Jul-2014.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  M  e.  NN )   &    |-  B  =  ( ( ( A  +  ( M  / 
 2 ) )  mod  M )  -  ( M 
 /  2 ) )   &    |-  ( ( ph  /\  ps )  ->  ( ( ( ( M ^ 2
 )  /  2 )  /  2 )  -  ( B ^ 2 ) )  =  0 )   =>    |-  ( ( ph  /\  ps )  ->  ( M ^
 2 )  ||  (
 ( A ^ 2
 )  -  ( ( ( M ^ 2
 )  /  2 )  /  2 ) ) )
 
Theorem4sqlem1 12869* Lemma for 4sq 12885. The set  S is the set of all numbers that are expressible as a sum of four squares. Our goal is to show that  S  =  NN0; here we show one subset direction. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  S  =  { n  |  E. x  e.  ZZ  E. y  e.  ZZ  E. z  e.  ZZ  E. w  e.  ZZ  n  =  ( ( ( x ^
 2 )  +  (
 y ^ 2 ) )  +  ( ( z ^ 2 )  +  ( w ^
 2 ) ) ) }   =>    |-  S  C_  NN0
 
Theorem4sqlem2 12870* Lemma for 4sq 12885. Change bound variables in  S. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  S  =  { n  |  E. x  e.  ZZ  E. y  e.  ZZ  E. z  e.  ZZ  E. w  e.  ZZ  n  =  ( ( ( x ^
 2 )  +  (
 y ^ 2 ) )  +  ( ( z ^ 2 )  +  ( w ^
 2 ) ) ) }   =>    |-  ( A  e.  S  <->  E. a  e.  ZZ  E. b  e.  ZZ  E. c  e.  ZZ  E. d  e. 
 ZZ  A  =  ( ( ( a ^
 2 )  +  (
 b ^ 2 ) )  +  ( ( c ^ 2 )  +  ( d ^
 2 ) ) ) )
 
Theorem4sqlem3 12871* Lemma for 4sq 12885. Sufficient condition to be in  S. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  S  =  { n  |  E. x  e.  ZZ  E. y  e.  ZZ  E. z  e.  ZZ  E. w  e.  ZZ  n  =  ( ( ( x ^
 2 )  +  (
 y ^ 2 ) )  +  ( ( z ^ 2 )  +  ( w ^
 2 ) ) ) }   =>    |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  ->  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  +  ( ( C ^ 2 )  +  ( D ^
 2 ) ) )  e.  S )
 
Theorem4sqlem4a 12872* Lemma for 4sqlem4 12873. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  S  =  { n  |  E. x  e.  ZZ  E. y  e.  ZZ  E. z  e.  ZZ  E. w  e.  ZZ  n  =  ( ( ( x ^
 2 )  +  (
 y ^ 2 ) )  +  ( ( z ^ 2 )  +  ( w ^
 2 ) ) ) }   =>    |-  ( ( A  e.  ZZ [ _i ]  /\  B  e.  ZZ [ _i ] )  ->  ( ( ( abs `  A ) ^ 2 )  +  ( ( abs `  B ) ^ 2 ) )  e.  S )
 
Theorem4sqlem4 12873* Lemma for 4sq 12885. We can express the four-square property more compactly in terms of gaussian integers, because the norms of gaussian integers are exactly sums of two squares. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  S  =  { n  |  E. x  e.  ZZ  E. y  e.  ZZ  E. z  e.  ZZ  E. w  e.  ZZ  n  =  ( ( ( x ^
 2 )  +  (
 y ^ 2 ) )  +  ( ( z ^ 2 )  +  ( w ^
 2 ) ) ) }   =>    |-  ( A  e.  S  <->  E. u  e.  ZZ [ _i ]  E. v  e. 
 ZZ [ _i ]  A  =  ( (
 ( abs `  u ) ^ 2 )  +  ( ( abs `  v
 ) ^ 2 ) ) )
 
Theoremmul4sqlem 12874* Lemma for mul4sq 12875: algebraic manipulations. The extra assumptions involving  M are for a part of 4sqlem17 12882 which needs to know not just that the product is a sum of squares, but also that it preserves divisibility by  M. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  S  =  { n  |  E. x  e.  ZZ  E. y  e.  ZZ  E. z  e.  ZZ  E. w  e.  ZZ  n  =  ( ( ( x ^
 2 )  +  (
 y ^ 2 ) )  +  ( ( z ^ 2 )  +  ( w ^
 2 ) ) ) }   &    |-  ( ph  ->  A  e.  ZZ [ _i ] )   &    |-  ( ph  ->  B  e.  ZZ [ _i ] )   &    |-  ( ph  ->  C  e.  ZZ [ _i ] )   &    |-  ( ph  ->  D  e.  ZZ [ _i ] )   &    |-  X  =  ( ( ( abs `  A ) ^ 2 )  +  ( ( abs `  B ) ^ 2 ) )   &    |-  Y  =  ( (
 ( abs `  C ) ^ 2 )  +  ( ( abs `  D ) ^ 2 ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  ( ( A  -  C )  /  M )  e. 
 ZZ [ _i ]
 )   &    |-  ( ph  ->  (
 ( B  -  D )  /  M )  e. 
 ZZ [ _i ]
 )   &    |-  ( ph  ->  ( X  /  M )  e. 
 NN0 )   =>    |-  ( ph  ->  (
 ( X  /  M )  x.  ( Y  /  M ) )  e.  S )
 
Theoremmul4sq 12875* Euler's four-square identity: The product of two sums of four squares is also a sum of four squares. This is usually quoted as an explicit formula involving eight real variables; we save some time by working with complex numbers (gaussian integers) instead, so that we only have to work with four variables, and also hiding the actual formula for the product in the proof of mul4sqlem 12874. (For the curious, the explicit formula that is used is  (  |  a  |  ^ 2  +  |  b  |  ^
2 ) (  |  c  |  ^ 2  +  |  d  |  ^ 2 )  =  |  a *  x.  c  +  b  x.  d *  |  ^ 2  +  | 
a *  x.  d  -  b  x.  c
*  |  ^ 2.) (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  S  =  { n  |  E. x  e.  ZZ  E. y  e.  ZZ  E. z  e.  ZZ  E. w  e.  ZZ  n  =  ( ( ( x ^
 2 )  +  (
 y ^ 2 ) )  +  ( ( z ^ 2 )  +  ( w ^
 2 ) ) ) }   =>    |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  x.  B )  e.  S )
 
Theorem4sqlem11 12876* Lemma for 4sq 12885. Use the pigeonhole principle to show that the sets  { m ^
2  |  m  e.  ( 0 ... N
) } and  { -u 1  -  n ^ 2  |  n  e.  ( 0 ... N ) } have a common element,  mod  P. (Contributed by Mario Carneiro, 15-Jul-2014.)
 |-  S  =  { n  |  E. x  e.  ZZ  E. y  e.  ZZ  E. z  e.  ZZ  E. w  e.  ZZ  n  =  ( ( ( x ^
 2 )  +  (
 y ^ 2 ) )  +  ( ( z ^ 2 )  +  ( w ^
 2 ) ) ) }   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  P  =  ( ( 2  x.  N )  +  1 )
 )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  A  =  { u  |  E. m  e.  ( 0 ... N ) u  =  (
 ( m ^ 2
 )  mod  P ) }   &    |-  F  =  ( v  e.  A  |->  ( ( P  -  1 )  -  v ) )   =>    |-  ( ph  ->  ( A  i^i  ran  F )  =/=  (/) )
 
Theorem4sqlem12 12877* Lemma for 4sq 12885. For any odd prime  P, there is a  k  <  P such that  k P  -  1 is a sum of two squares. (Contributed by Mario Carneiro, 15-Jul-2014.)
 |-  S  =  { n  |  E. x  e.  ZZ  E. y  e.  ZZ  E. z  e.  ZZ  E. w  e.  ZZ  n  =  ( ( ( x ^
 2 )  +  (
 y ^ 2 ) )  +  ( ( z ^ 2 )  +  ( w ^
 2 ) ) ) }   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  P  =  ( ( 2  x.  N )  +  1 )
 )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  A  =  { u  |  E. m  e.  ( 0 ... N ) u  =  (
 ( m ^ 2
 )  mod  P ) }   &    |-  F  =  ( v  e.  A  |->  ( ( P  -  1 )  -  v ) )   =>    |-  ( ph  ->  E. k  e.  ( 1 ... ( P  -  1 ) ) E. u  e.  ZZ [ _i ]  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  (
 k  x.  P ) )
 
Theorem4sqlem13 12878* Lemma for 4sq 12885. (Contributed by Mario Carneiro, 16-Jul-2014.)
 |-  S  =  { n  |  E. x  e.  ZZ  E. y  e.  ZZ  E. z  e.  ZZ  E. w  e.  ZZ  n  =  ( ( ( x ^
 2 )  +  (
 y ^ 2 ) )  +  ( ( z ^ 2 )  +  ( w ^
 2 ) ) ) }   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  P  =  ( ( 2  x.  N )  +  1 )
 )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  ( 0 ... ( 2  x.  N ) ) 
 C_  S )   &    |-  T  =  { i  e.  NN  |  ( i  x.  P )  e.  S }   &    |-  M  =  sup ( T ,  RR ,  `'  <  )   =>    |-  ( ph  ->  ( T  =/=  (/)  /\  M  <  P ) )
 
Theorem4sqlem14 12879* Lemma for 4sq 12885. (Contributed by Mario Carneiro, 16-Jul-2014.)
 |-  S  =  { n  |  E. x  e.  ZZ  E. y  e.  ZZ  E. z  e.  ZZ  E. w  e.  ZZ  n  =  ( ( ( x ^
 2 )  +  (
 y ^ 2 ) )  +  ( ( z ^ 2 )  +  ( w ^
 2 ) ) ) }   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  P  =  ( ( 2  x.  N )  +  1 )
 )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  ( 0 ... ( 2  x.  N ) ) 
 C_  S )   &    |-  T  =  { i  e.  NN  |  ( i  x.  P )  e.  S }   &    |-  M  =  sup ( T ,  RR ,  `'  <  )   &    |-  ( ph  ->  M  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  D  e.  ZZ )   &    |-  E  =  ( ( ( A  +  ( M  /  2
 ) )  mod  M )  -  ( M  / 
 2 ) )   &    |-  F  =  ( ( ( B  +  ( M  / 
 2 ) )  mod  M )  -  ( M 
 /  2 ) )   &    |-  G  =  ( (
 ( C  +  ( M  /  2 ) ) 
 mod  M )  -  ( M  /  2 ) )   &    |-  H  =  ( (
 ( D  +  ( M  /  2 ) ) 
 mod  M )  -  ( M  /  2 ) )   &    |-  R  =  ( (
 ( ( E ^
 2 )  +  ( F ^ 2 ) )  +  ( ( G ^ 2 )  +  ( H ^ 2 ) ) )  /  M )   &    |-  ( ph  ->  ( M  x.  P )  =  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  +  ( ( C ^ 2 )  +  ( D ^
 2 ) ) ) )   =>    |-  ( ph  ->  R  e.  NN0 )
 
Theorem4sqlem15 12880* Lemma for 4sq 12885. (Contributed by Mario Carneiro, 16-Jul-2014.)
 |-  S  =  { n  |  E. x  e.  ZZ  E. y  e.  ZZ  E. z  e.  ZZ  E. w  e.  ZZ  n  =  ( ( ( x ^
 2 )  +  (
 y ^ 2 ) )  +  ( ( z ^ 2 )  +  ( w ^
 2 ) ) ) }   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  P  =  ( ( 2  x.  N )  +  1 )
 )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  ( 0 ... ( 2  x.  N ) ) 
 C_  S )   &    |-  T  =  { i  e.  NN  |  ( i  x.  P )  e.  S }   &    |-  M  =  sup ( T ,  RR ,  `'  <  )   &    |-  ( ph  ->  M  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  D  e.  ZZ )   &    |-  E  =  ( ( ( A  +  ( M  /  2
 ) )  mod  M )  -  ( M  / 
 2 ) )   &    |-  F  =  ( ( ( B  +  ( M  / 
 2 ) )  mod  M )  -  ( M 
 /  2 ) )   &    |-  G  =  ( (
 ( C  +  ( M  /  2 ) ) 
 mod  M )  -  ( M  /  2 ) )   &    |-  H  =  ( (
 ( D  +  ( M  /  2 ) ) 
 mod  M )  -  ( M  /  2 ) )   &    |-  R  =  ( (
 ( ( E ^
 2 )  +  ( F ^ 2 ) )  +  ( ( G ^ 2 )  +  ( H ^ 2 ) ) )  /  M )   &    |-  ( ph  ->  ( M  x.  P )  =  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  +  ( ( C ^ 2 )  +  ( D ^
 2 ) ) ) )   =>    |-  ( ( ph  /\  R  =  M )  ->  (
 ( ( ( ( ( M ^ 2
 )  /  2 )  /  2 )  -  ( E ^ 2 ) )  =  0  /\  ( ( ( ( M ^ 2 ) 
 /  2 )  / 
 2 )  -  ( F ^ 2 ) )  =  0 )  /\  ( ( ( ( ( M ^ 2
 )  /  2 )  /  2 )  -  ( G ^ 2 ) )  =  0  /\  ( ( ( ( M ^ 2 ) 
 /  2 )  / 
 2 )  -  ( H ^ 2 ) )  =  0 ) ) )
 
Theorem4sqlem16 12881* Lemma for 4sq 12885. (Contributed by Mario Carneiro, 16-Jul-2014.)
 |-  S  =  { n  |  E. x  e.  ZZ  E. y  e.  ZZ  E. z  e.  ZZ  E. w  e.  ZZ  n  =  ( ( ( x ^
 2 )  +  (
 y ^ 2 ) )  +  ( ( z ^ 2 )  +  ( w ^
 2 ) ) ) }   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  P  =  ( ( 2  x.  N )  +  1 )
 )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  ( 0 ... ( 2  x.  N ) ) 
 C_  S )   &    |-  T  =  { i  e.  NN  |  ( i  x.  P )  e.  S }   &    |-  M  =  sup ( T ,  RR ,  `'  <  )   &    |-  ( ph  ->  M  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  D  e.  ZZ )   &    |-  E  =  ( ( ( A  +  ( M  /  2
 ) )  mod  M )  -  ( M  / 
 2 ) )   &    |-  F  =  ( ( ( B  +  ( M  / 
 2 ) )  mod  M )  -  ( M 
 /  2 ) )   &    |-  G  =  ( (
 ( C  +  ( M  /  2 ) ) 
 mod  M )  -  ( M  /  2 ) )   &    |-  H  =  ( (
 ( D  +  ( M  /  2 ) ) 
 mod  M )  -  ( M  /  2 ) )   &    |-  R  =  ( (
 ( ( E ^
 2 )  +  ( F ^ 2 ) )  +  ( ( G ^ 2 )  +  ( H ^ 2 ) ) )  /  M )   &    |-  ( ph  ->  ( M  x.  P )  =  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  +  ( ( C ^ 2 )  +  ( D ^
 2 ) ) ) )   =>    |-  ( ph  ->  ( R  <_  M  /\  (
 ( R  =  0  \/  R  =  M )  ->  ( M ^
 2 )  ||  ( M  x.  P ) ) ) )
 
Theorem4sqlem17 12882* Lemma for 4sq 12885. (Contributed by Mario Carneiro, 16-Jul-2014.)
 |-  S  =  { n  |  E. x  e.  ZZ  E. y  e.  ZZ  E. z  e.  ZZ  E. w  e.  ZZ  n  =  ( ( ( x ^
 2 )  +  (
 y ^ 2 ) )  +  ( ( z ^ 2 )  +  ( w ^
 2 ) ) ) }   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  P  =  ( ( 2  x.  N )  +  1 )
 )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  ( 0 ... ( 2  x.  N ) ) 
 C_  S )   &    |-  T  =  { i  e.  NN  |  ( i  x.  P )  e.  S }   &    |-  M  =  sup ( T ,  RR ,  `'  <  )   &    |-  ( ph  ->  M  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  D  e.  ZZ )   &    |-  E  =  ( ( ( A  +  ( M  /  2
 ) )  mod  M )  -  ( M  / 
 2 ) )   &    |-  F  =  ( ( ( B  +  ( M  / 
 2 ) )  mod  M )  -  ( M 
 /  2 ) )   &    |-  G  =  ( (
 ( C  +  ( M  /  2 ) ) 
 mod  M )  -  ( M  /  2 ) )   &    |-  H  =  ( (
 ( D  +  ( M  /  2 ) ) 
 mod  M )  -  ( M  /  2 ) )   &    |-  R  =  ( (
 ( ( E ^
 2 )  +  ( F ^ 2 ) )  +  ( ( G ^ 2 )  +  ( H ^ 2 ) ) )  /  M )   &    |-  ( ph  ->  ( M  x.  P )  =  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  +  ( ( C ^ 2 )  +  ( D ^
 2 ) ) ) )   =>    |- 
 -.  ph
 
Theorem4sqlem18 12883* Lemma for 4sq 12885. Inductive step, odd prime case. (Contributed by Mario Carneiro, 16-Jul-2014.)
 |-  S  =  { n  |  E. x  e.  ZZ  E. y  e.  ZZ  E. z  e.  ZZ  E. w  e.  ZZ  n  =  ( ( ( x ^
 2 )  +  (
 y ^ 2 ) )  +  ( ( z ^ 2 )  +  ( w ^
 2 ) ) ) }   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  P  =  ( ( 2  x.  N )  +  1 )
 )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  ( 0 ... ( 2  x.  N ) ) 
 C_  S )   &    |-  T  =  { i  e.  NN  |  ( i  x.  P )  e.  S }   &    |-  M  =  sup ( T ,  RR ,  `'  <  )   =>    |-  ( ph  ->  P  e.  S )
 
Theorem4sqlem19 12884* Lemma for 4sq 12885. The proof is by strong induction - we show that if all the integers less than  k are in  S, then  k is as well. In this part of the proof we do the induction argument and dispense with all the cases except the odd prime case, which is sent to 4sqlem18 12883. If  k is  0 ,  1 ,  2, we show  k  e.  S directly; otherwise if  k is composite,  k is the product of two numbers less than it (and hence in  S by assumption), so by mul4sq 12875  k  e.  S. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)
 |-  S  =  { n  |  E. x  e.  ZZ  E. y  e.  ZZ  E. z  e.  ZZ  E. w  e.  ZZ  n  =  ( ( ( x ^
 2 )  +  (
 y ^ 2 ) )  +  ( ( z ^ 2 )  +  ( w ^
 2 ) ) ) }   =>    |- 
 NN0  =  S
 
Theorem4sq 12885* Lagrange's four-square theorem, or Bachet's conjecture: every nonnegative integer is expressible as a sum of four squares. (Contributed by Mario Carneiro, 16-Jul-2014.)
 |-  ( A  e.  NN0  <->  E. a  e.  ZZ  E. b  e.  ZZ  E. c  e. 
 ZZ  E. d  e.  ZZ  A  =  ( (
 ( a ^ 2
 )  +  ( b ^ 2 ) )  +  ( ( c ^ 2 )  +  ( d ^ 2
 ) ) ) )
 
6.2.11  Van der Waerden's theorem
 
Syntaxcvdwa 12886 The arithmetic progression function.
 class AP
 
Syntaxcvdwm 12887 The monochromatic arithmetic progression predicate.
 class MonoAP
 
Syntaxcvdwp 12888 The polychromatic arithmetic progression predicate.
 class PolyAP
 
Definitiondf-vdwap 12889* Define the arithmetic progression function, which takes as input a length  k, a start point  a, and a step  d and outputs the set of points in this progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |- AP 
 =  ( k  e. 
 NN0  |->  ( a  e. 
 NN ,  d  e. 
 NN  |->  ran  (  m  e.  ( 0 ... (
 k  -  1 ) )  |->  ( a  +  ( m  x.  d
 ) ) ) ) )
 
Definitiondf-vdwmc 12890* Define the "contains a monochromatic AP" predicate. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |- MonoAP  =  { <. k ,  f >.  |  E. c ( ran  (AP `  k
 )  i^i  ~P ( `' f " { c } ) )  =/=  (/) }
 
Definitiondf-vdwpc 12891* Define the "contains a polychromatic colleciton of APs" predicate. See vdwpc 12901 for more information. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |- PolyAP  =  { <. <. m ,  k >. ,  f >.  |  E. a  e.  NN  E. d  e.  ( NN  ^m  (
 1 ... m ) ) ( A. i  e.  ( 1 ... m ) ( ( a  +  ( d `  i ) ) (AP
 `  k ) ( d `  i ) )  C_  ( `' f " { ( f `
  ( a  +  ( d `  i
 ) ) ) }
 )  /\  ( # `  ran  (  i  e.  (
 1 ... m )  |->  ( f `  ( a  +  ( d `  i ) ) ) ) )  =  m ) }
 
Theoremvdwapfval 12892* Define the arithmetic progression function, which takes as input a length  k, a start point  a, and a step  d and outputs the set of points in this progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |-  ( K  e.  NN0  ->  (AP `  K )  =  ( a  e.  NN ,  d  e.  NN  |->  ran  (  m  e.  (
 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d ) ) ) ) )
 
Theoremvdwapf 12893 The arithmetic progression function is a function. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |-  ( K  e.  NN0  ->  (AP `  K ) : ( NN  X.  NN )
 --> ~P NN )
 
Theoremvdwapval 12894* Value of the arithmetic progression function. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |-  ( ( K  e.  NN0  /\  A  e.  NN  /\  D  e.  NN )  ->  ( X  e.  ( A (AP `  K ) D )  <->  E. m  e.  (
 0 ... ( K  -  1 ) ) X  =  ( A  +  ( m  x.  D ) ) ) )
 
Theoremvdwapun 12895 Remove the first element of an arithmetic progression. (Contributed by Mario Carneiro, 11-Sep-2014.)
 |-  ( ( K  e.  NN0  /\  A  e.  NN  /\  D  e.  NN )  ->  ( A (AP `  ( K  +  1
 ) ) D )  =  ( { A }  u.  ( ( A  +  D ) (AP
 `  K ) D ) ) )
 
Theoremvdwapid1 12896 The first element of an arithmetic progression. (Contributed by Mario Carneiro, 12-Sep-2014.)
 |-  ( ( K  e.  NN  /\  A  e.  NN  /\  D  e.  NN )  ->  A  e.  ( A (AP `  K ) D ) )
 
Theoremvdwap0 12897 Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |-  ( ( A  e.  NN  /\  D  e.  NN )  ->  ( A (AP
 `  0 ) D )  =  (/) )
 
Theoremvdwap1 12898 Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |-  ( ( A  e.  NN  /\  D  e.  NN )  ->  ( A (AP
 `  1 ) D )  =  { A } )
 
Theoremvdwmc 12899* The predicate " The  <. R ,  N >.-coloring  F contains a monochromatic AP of length 
K". (Contributed by Mario Carneiro, 18-Aug-2014.)
 |-  X  e.  _V   &    |-  ( ph  ->  K  e.  NN0 )   &    |-  ( ph  ->  F : X --> R )   =>    |-  ( ph  ->  ( K MonoAP  F  <->  E. c E. a  e.  NN  E. d  e. 
 NN  ( a (AP
 `  K ) d )  C_  ( `' F " { c }
 ) ) )
 
Theoremvdwmc2 12900* Expand out the definition of an arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |-  X  e.  _V   &    |-  ( ph  ->  K  e.  NN0 )   &    |-  ( ph  ->  F : X --> R )   &    |-  ( ph  ->  A  e.  X )   =>    |-  ( ph  ->  ( K MonoAP  F  <->  E. c  e.  R  E. a  e.  NN  E. d  e.  NN  A. m  e.  ( 0 ... ( K  -  1
 ) ) ( a  +  ( m  x.  d ) )  e.  ( `' F " { c } )
 ) )
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