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Theorem List for Metamath Proof Explorer - 12901-13000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempczcl 12901 Closure of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0
 ) )  ->  ( P  pCnt  N )  e. 
 NN0 )
 
Theorempccl 12902 Closure of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( P  e.  Prime  /\  N  e.  NN )  ->  ( P  pCnt  N )  e.  NN0 )
 
Theorempccld 12903 Closure of the prime power function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  ( P  pCnt  N )  e.  NN0 )
 
Theorempcmul 12904 Multiplication property of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0
 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  ->  ( P  pCnt  ( A  x.  B ) )  =  ( ( P 
 pCnt  A )  +  ( P  pCnt  B ) ) )
 
Theorempcdiv 12905 Division property of the prime power function. (Contributed by Mario Carneiro, 1-Mar-2014.)
 |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0
 )  /\  B  e.  NN )  ->  ( P 
 pCnt  ( A  /  B ) )  =  (
 ( P  pCnt  A )  -  ( P  pCnt  B ) ) )
 
Theorempcqmul 12906 Multiplication property of the prime power function. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0
 )  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( P  pCnt  ( A  x.  B ) )  =  ( ( P 
 pCnt  A )  +  ( P  pCnt  B ) ) )
 
Theorempc0 12907 The value of the prime power function at zero. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( P  e.  Prime  ->  ( P  pCnt  0 )  =  +oo )
 
Theorempc1 12908 Value of the prime count function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( P  e.  Prime  ->  ( P  pCnt  1 )  =  0 )
 
Theorempcqcl 12909 Closure of the general prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0
 ) )  ->  ( P  pCnt  N )  e. 
 ZZ )
 
Theorempcqdiv 12910 Division property of the prime power function. (Contributed by Mario Carneiro, 10-Aug-2015.)
 |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0
 )  /\  ( B  e.  QQ  /\  B  =/=  0 ) )  ->  ( P  pCnt  ( A 
 /  B ) )  =  ( ( P 
 pCnt  A )  -  ( P  pCnt  B ) ) )
 
Theorempcrec 12911 Prime power of a reciprocal. (Contributed by Mario Carneiro, 10-Aug-2015.)
 |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0
 ) )  ->  ( P  pCnt  ( 1  /  A ) )  =  -u ( P  pCnt  A ) )
 
Theorempcexp 12912 Prime power of an exponential. (Contributed by Mario Carneiro, 10-Aug-2015.)
 |-  ( ( P  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0
 )  /\  N  e.  ZZ )  ->  ( P 
 pCnt  ( A ^ N ) )  =  ( N  x.  ( P  pCnt  A ) ) )
 
Theorempcxcl 12913 Extended real closure of the general prime count function. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( ( P  e.  Prime  /\  N  e.  QQ )  ->  ( P  pCnt  N )  e.  RR* )
 
Theorempcge0 12914 The prime count of an integer is greater or equal to zero. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  0  <_  ( P  pCnt  N ) )
 
Theorempczdvds 12915 Defining property of the prime count function. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0
 ) )  ->  ( P ^ ( P  pCnt  N ) )  ||  N )
 
Theorempcdvds 12916 Defining property of the prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( P  e.  Prime  /\  N  e.  NN )  ->  ( P ^
 ( P  pCnt  N ) )  ||  N )
 
Theorempczndvds 12917 Defining property of the prime count function. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0
 ) )  ->  -.  ( P ^ ( ( P 
 pCnt  N )  +  1 ) )  ||  N )
 
Theorempcndvds 12918 Defining property of the prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( P  e.  Prime  /\  N  e.  NN )  ->  -.  ( P ^ ( ( P 
 pCnt  N )  +  1 ) )  ||  N )
 
Theorempczndvds2 12919 The remainder after dividing out all factors of  P is not divisible by  P. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0
 ) )  ->  -.  P  ||  ( N  /  ( P ^ ( P  pCnt  N ) ) ) )
 
Theorempcndvds2 12920 The remainder after dividing out all factors of  P is not divisible by  P. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( P  e.  Prime  /\  N  e.  NN )  ->  -.  P  ||  ( N  /  ( P ^
 ( P  pCnt  N ) ) ) )
 
Theorempcdvdsb 12921  P ^ A divides  N if and only if  A is at most the count of  P. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  ->  ( A  <_  ( P  pCnt  N )  <->  ( P ^ A )  ||  N ) )
 
Theorempcelnn 12922 There are a positive number of powers of a prime  P in  N iff  P divides  N. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( P  e.  Prime  /\  N  e.  NN )  ->  ( ( P 
 pCnt  N )  e.  NN  <->  P  ||  N ) )
 
Theorempceq0 12923 There are zero powers of a prime  P in  N iff  P does not divide  N. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( P  e.  Prime  /\  N  e.  NN )  ->  ( ( P 
 pCnt  N )  =  0  <->  -.  P  ||  N )
 )
 
Theorempcidlem 12924 The prime count of a prime power. (Contributed by Mario Carneiro, 12-Mar-2014.)
 |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
 
Theorempcid 12925 The prime count of a prime power. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
 
Theorempcneg 12926 The prime count of a negative number. (Contributed by Mario Carneiro, 13-Mar-2014.)
 |-  ( ( P  e.  Prime  /\  A  e.  QQ )  ->  ( P  pCnt  -u A )  =  ( P  pCnt  A )
 )
 
Theorempcabs 12927 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 12928 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 12929 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 12930 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 12931* 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 12932* 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 12933* 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 12934* 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 12935* 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 12936 Lemma for pcadd 12937. 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 12937 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 12938 The inequality of pcadd 12937 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 12939 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 12940* 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 12941* 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 12942 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 12943* 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 12944* 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 12945 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 12946 Lemma for pcfac 12947. (Contributed by Mario Carneiro, 20-May-2014.)
 |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( |_ `  ( N  /  ( P ^ M ) ) )  =  0 )
 
Theorempcfac 12947* 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 12948* 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 12949 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 12950 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 12951 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 12952 Lemma for pockthg 12953. (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 12953* 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 12954 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 12953 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 12955* Lemma for unben 12956. (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 12956* 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 12957* Lemma for infpn 12959. 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 12958* Lemma for infpn 12959. 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 12959* There exist infinitely many prime numbers: for any natural number  N, there exists a prime number  j greater than  N. (See infpn2 12960 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 12960* There exist infinitely many prime numbers: the set of all primes  S is unbounded by infpn 12959, so by unben 12956 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 12961* The primes are unbounded. (Contributed by Paul Chapman, 28-Nov-2012.)
 |-  ( N  e.  NN  ->  E. p  e.  Prime  N  <  p )
 
Theoremprminf 12962 There are an infinite number of primes. (Contributed by Paul Chapman, 28-Nov-2012.)
 |- 
 Prime  ~~  NN
 
6.2.8  Sum of prime reciprocals
 
Theoremprmreclem1 12963* Lemma for prmrec 12969. 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 12964* Lemma for prmrec 12969. 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 12965* Lemma for prmrec 12969. 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 12966* Lemma for prmrec 12969. 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 12843, 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 12967* Lemma for prmrec 12969. 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 12966 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 12968* Lemma for prmrec 12969. 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 12967 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 12969* 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 12970* Lemma for 1arith 12974. (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 12971* Lemma for 1arith 12974. (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 12972* Lemma for 1arith 12974. (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 12973* Lemma for 1arith 12974. (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 12974* 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 12975* 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 12976 Extend class notation with the set of gaussian integers.
 class  ZZ [ _i ]
 
Definitiondf-gz 12977 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 12978 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 12979 A gaussian integer is a complex number. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  ZZ [ _i ]  ->  A  e.  CC )
 
Theoremzgz 12980 An integer is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  ZZ  ->  A  e.  ZZ [ _i ] )
 
Theoremigz 12981  _i is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  _i  e.  ZZ [ _i ]
 
Theoremgznegcl 12982 The gaussian integers are closed under negation. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  ZZ [ _i ]  ->  -u A  e.  ZZ [ _i ]
 )
 
Theoremgzcjcl 12983 The gaussian integers are closed under conjugation. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  ZZ [ _i ]  ->  ( * `  A )  e. 
 ZZ [ _i ]
 )
 
Theoremgzaddcl 12984 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 12985 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 12986 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 12987 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 12988 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 12989 Lemma for 4sq 13011. (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 12990 Lemma for 4sq 13011. (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 12991 Lemma for 4sq 13011. (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 12992 Lemma for 4sq 13011. (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 12993 Lemma for 4sq 13011. (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 12994 Lemma for 4sq 13011. (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 12995* Lemma for 4sq 13011. 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 12996* Lemma for 4sq 13011. 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 12997* Lemma for 4sq 13011. 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 12998* Lemma for 4sqlem4 12999. (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 12999* Lemma for 4sq 13011. 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 13000* Lemma for mul4sq 13001: algebraic manipulations. The extra assumptions involving  M are for a part of 4sqlem17 13008 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 )
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