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Theorem List for Metamath Proof Explorer - 12901-13000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem1arith 12901* 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 12902* 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 12903 Extend class notation with the set of gaussian integers.
 class  ZZ [ _i ]
 
Definitiondf-gz 12904 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 12905 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 12906 A gaussian integer is a complex number. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  ZZ [ _i ]  ->  A  e.  CC )
 
Theoremzgz 12907 An integer is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  ZZ  ->  A  e.  ZZ [ _i ] )
 
Theoremigz 12908  _i is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  _i  e.  ZZ [ _i ]
 
Theoremgznegcl 12909 The gaussian integers are closed under negation. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  ZZ [ _i ]  ->  -u A  e.  ZZ [ _i ]
 )
 
Theoremgzcjcl 12910 The gaussian integers are closed under conjugation. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  ZZ [ _i ]  ->  ( * `  A )  e. 
 ZZ [ _i ]
 )
 
Theoremgzaddcl 12911 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 12912 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 12913 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 12914 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 12915 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 12916 Lemma for 4sq 12938. (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 12917 Lemma for 4sq 12938. (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 12918 Lemma for 4sq 12938. (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 12919 Lemma for 4sq 12938. (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 12920 Lemma for 4sq 12938. (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 12921 Lemma for 4sq 12938. (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 12922* Lemma for 4sq 12938. 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 12923* Lemma for 4sq 12938. 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 12924* Lemma for 4sq 12938. 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 12925* Lemma for 4sqlem4 12926. (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 12926* Lemma for 4sq 12938. 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 12927* Lemma for mul4sq 12928: algebraic manipulations. The extra assumptions involving  M are for a part of 4sqlem17 12935 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 12928* 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 12927. (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 12929* Lemma for 4sq 12938. 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 12930* Lemma for 4sq 12938. 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 12931* Lemma for 4sq 12938. (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 12932* Lemma for 4sq 12938. (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 12933* Lemma for 4sq 12938. (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 12934* Lemma for 4sq 12938. (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 12935* Lemma for 4sq 12938. (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 12936* Lemma for 4sq 12938. 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 12937* Lemma for 4sq 12938. 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 12936. 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 12928  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 12938* 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 12939 The arithmetic progression function.
 class AP
 
Syntaxcvdwm 12940 The monochromatic arithmetic progression predicate.
 class MonoAP
 
Syntaxcvdwp 12941 The polychromatic arithmetic progression predicate.
 class PolyAP
 
Definitiondf-vdwap 12942* 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 12943* 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 12944* Define the "contains a polychromatic colleciton of APs" predicate. See vdwpc 12954 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 12945* 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 12946 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 12947* 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 12948 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 12949 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 12950 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 12951 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 12952* 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 12953* 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 } )
 ) )
 
Theoremvdwpc 12954* The predicate " The coloring 
F contains a polychromatic  M-tuple of AP's of length  K". A polychromatic 
M-tuple of AP's is a set of AP's with the same base point but different step lengths, such that each individual AP is monochromatic, but the AP's all have mutually distinct colors. (The common basepoint is not required to have the same color as any of the AP's.) (Contributed by Mario Carneiro, 18-Aug-2014.)
 |-  X  e.  _V   &    |-  ( ph  ->  K  e.  NN0 )   &    |-  ( ph  ->  F : X --> R )   &    |-  ( ph  ->  M  e.  NN )   &    |-  J  =  ( 1
 ... M )   =>    |-  ( ph  ->  (
 <. M ,  K >. PolyAP  F  <->  E. a  e.  NN  E. d  e.  ( NN  ^m  J ) ( A. i  e.  J  (
 ( a  +  (
 d `  i )
 ) (AP `  K ) ( d `  i ) )  C_  ( `' F " { ( F `  ( a  +  ( d `  i
 ) ) ) }
 )  /\  ( # `  ran  (  i  e.  J  |->  ( F `  ( a  +  ( d `  i ) ) ) ) )  =  M ) ) )
 
Theoremvdwlem1 12955* Lemma for vdw 12968. (Contributed by Mario Carneiro, 12-Sep-2014.)
 |-  ( ph  ->  R  e.  Fin )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( ph  ->  W  e.  NN )   &    |-  ( ph  ->  F : ( 1 ...
 W ) --> R )   &    |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  D : ( 1 ... M ) --> NN )   &    |-  ( ph  ->  A. i  e.  ( 1
 ... M ) ( ( A  +  ( D `  i ) ) (AP `  K )
 ( D `  i
 ) )  C_  ( `' F " { ( F `  ( A  +  ( D `  i ) ) ) } )
 )   &    |-  ( ph  ->  I  e.  ( 1 ... M ) )   &    |-  ( ph  ->  ( F `  A )  =  ( F `  ( A  +  ( D `  I ) ) ) )   =>    |-  ( ph  ->  ( K  +  1 ) MonoAP  F )
 
Theoremvdwlem2 12956* Lemma for vdw 12968. (Contributed by Mario Carneiro, 12-Sep-2014.)
 |-  ( ph  ->  R  e.  Fin )   &    |-  ( ph  ->  K  e.  NN0 )   &    |-  ( ph  ->  W  e.  NN )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  F : ( 1 ...
 M ) --> R )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  ( W  +  N ) ) )   &    |-  G  =  ( x  e.  ( 1 ... W )  |->  ( F `  ( x  +  N ) ) )   =>    |-  ( ph  ->  ( K MonoAP  G  ->  K MonoAP  F ) )
 
Theoremvdwlem3 12957 Lemma for vdw 12968. (Contributed by Mario Carneiro, 13-Sep-2014.)
 |-  ( ph  ->  V  e.  NN )   &    |-  ( ph  ->  W  e.  NN )   &    |-  ( ph  ->  A  e.  (
 1 ... V ) )   &    |-  ( ph  ->  B  e.  ( 1 ... W ) )   =>    |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  -  1 )  +  V ) ) )  e.  ( 1 ... ( W  x.  ( 2  x.  V ) ) ) )
 
Theoremvdwlem4 12958* Lemma for vdw 12968. (Contributed by Mario Carneiro, 12-Sep-2014.)
 |-  ( ph  ->  V  e.  NN )   &    |-  ( ph  ->  W  e.  NN )   &    |-  ( ph  ->  R  e.  Fin )   &    |-  ( ph  ->  H : ( 1 ... ( W  x.  (
 2  x.  V ) ) ) --> R )   &    |-  F  =  ( x  e.  ( 1 ... V )  |->  ( y  e.  ( 1 ... W )  |->  ( H `  ( y  +  ( W  x.  ( ( x  -  1 )  +  V ) ) ) ) ) )   =>    |-  ( ph  ->  F : ( 1 ...
 V ) --> ( R 
 ^m  ( 1 ...
 W ) ) )
 
Theoremvdwlem5 12959* Lemma for vdw 12968. (Contributed by Mario Carneiro, 12-Sep-2014.)
 |-  ( ph  ->  V  e.  NN )   &    |-  ( ph  ->  W  e.  NN )   &    |-  ( ph  ->  R  e.  Fin )   &    |-  ( ph  ->  H : ( 1 ... ( W  x.  (
 2  x.  V ) ) ) --> R )   &    |-  F  =  ( x  e.  ( 1 ... V )  |->  ( y  e.  ( 1 ... W )  |->  ( H `  ( y  +  ( W  x.  ( ( x  -  1 )  +  V ) ) ) ) ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  G : ( 1 ...
 W ) --> R )   &    |-  ( ph  ->  K  e.  ( ZZ>= `  2 )
 )   &    |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  D  e.  NN )   &    |-  ( ph  ->  ( A (AP
 `  K ) D )  C_  ( `' F " { G }
 ) )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  E : ( 1 ... M ) --> NN )   &    |-  ( ph  ->  A. i  e.  ( 1
 ... M ) ( ( B  +  ( E `  i ) ) (AP `  K )
 ( E `  i
 ) )  C_  ( `' G " { ( G `  ( B  +  ( E `  i ) ) ) } )
 )   &    |-  J  =  ( i  e.  ( 1 ...
 M )  |->  ( G `
  ( B  +  ( E `  i ) ) ) )   &    |-  ( ph  ->  ( # `  ran  J )  =  M )   &    |-  T  =  ( B  +  ( W  x.  (
 ( A  +  ( V  -  D ) )  -  1 ) ) )   &    |-  P  =  ( j  e.  ( 1
 ... ( M  +  1 ) )  |->  ( if ( j  =  ( M  +  1 ) ,  0 ,  ( E `  j
 ) )  +  ( W  x.  D ) ) )   =>    |-  ( ph  ->  T  e.  NN )
 
Theoremvdwlem6 12960* Lemma for vdw 12968. (Contributed by Mario Carneiro, 13-Sep-2014.)
 |-  ( ph  ->  V  e.  NN )   &    |-  ( ph  ->  W  e.  NN )   &    |-  ( ph  ->  R  e.  Fin )   &    |-  ( ph  ->  H : ( 1 ... ( W  x.  (
 2  x.  V ) ) ) --> R )   &    |-  F  =  ( x  e.  ( 1 ... V )  |->  ( y  e.  ( 1 ... W )  |->  ( H `  ( y  +  ( W  x.  ( ( x  -  1 )  +  V ) ) ) ) ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  G : ( 1 ...
 W ) --> R )   &    |-  ( ph  ->  K  e.  ( ZZ>= `  2 )
 )   &    |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  D  e.  NN )   &    |-  ( ph  ->  ( A (AP
 `  K ) D )  C_  ( `' F " { G }
 ) )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  E : ( 1 ... M ) --> NN )   &    |-  ( ph  ->  A. i  e.  ( 1
 ... M ) ( ( B  +  ( E `  i ) ) (AP `  K )
 ( E `  i
 ) )  C_  ( `' G " { ( G `  ( B  +  ( E `  i ) ) ) } )
 )   &    |-  J  =  ( i  e.  ( 1 ...
 M )  |->  ( G `
  ( B  +  ( E `  i ) ) ) )   &    |-  ( ph  ->  ( # `  ran  J )  =  M )   &    |-  T  =  ( B  +  ( W  x.  (
 ( A  +  ( V  -  D ) )  -  1 ) ) )   &    |-  P  =  ( j  e.  ( 1
 ... ( M  +  1 ) )  |->  ( if ( j  =  ( M  +  1 ) ,  0 ,  ( E `  j
 ) )  +  ( W  x.  D ) ) )   =>    |-  ( ph  ->  ( <. ( M  +  1 ) ,  K >. PolyAP  H  \/  ( K  +  1 ) MonoAP  G ) )
 
Theoremvdwlem7 12961* Lemma for vdw 12968. (Contributed by Mario Carneiro, 12-Sep-2014.)
 |-  ( ph  ->  V  e.  NN )   &    |-  ( ph  ->  W  e.  NN )   &    |-  ( ph  ->  R  e.  Fin )   &    |-  ( ph  ->  H : ( 1 ... ( W  x.  (
 2  x.  V ) ) ) --> R )   &    |-  F  =  ( x  e.  ( 1 ... V )  |->  ( y  e.  ( 1 ... W )  |->  ( H `  ( y  +  ( W  x.  ( ( x  -  1 )  +  V ) ) ) ) ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  G : ( 1 ...
 W ) --> R )   &    |-  ( ph  ->  K  e.  ( ZZ>= `  2 )
 )   &    |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  D  e.  NN )   &    |-  ( ph  ->  ( A (AP
 `  K ) D )  C_  ( `' F " { G }
 ) )   =>    |-  ( ph  ->  ( <. M ,  K >. PolyAP  G  ->  ( <. ( M  +  1 ) ,  K >. PolyAP 
 H  \/  ( K  +  1 ) MonoAP  G ) ) )
 
Theoremvdwlem8 12962* Lemma for vdw 12968. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |-  ( ph  ->  R  e.  Fin )   &    |-  ( ph  ->  K  e.  ( ZZ>= `  2
 ) )   &    |-  ( ph  ->  W  e.  NN )   &    |-  ( ph  ->  F : ( 1 ... ( 2  x.  W ) ) --> R )   &    |-  C  e.  _V   &    |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  D  e.  NN )   &    |-  ( ph  ->  ( A (AP `  K ) D )  C_  ( `' G " { C } ) )   &    |-  G  =  ( x  e.  (
 1 ... W )  |->  ( F `  ( x  +  W ) ) )   =>    |-  ( ph  ->  <. 1 ,  K >. PolyAP  F )
 
Theoremvdwlem9 12963* Lemma for vdw 12968. (Contributed by Mario Carneiro, 12-Sep-2014.)
 |-  ( ph  ->  R  e.  Fin )   &    |-  ( ph  ->  K  e.  ( ZZ>= `  2
 ) )   &    |-  ( ph  ->  A. s  e.  Fin  E. n  e.  NN  A. f  e.  ( s  ^m  (
 1 ... n ) ) K MonoAP  f )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  W  e.  NN )   &    |-  ( ph  ->  A. g  e.  ( R 
 ^m  ( 1 ...
 W ) ) (
 <. M ,  K >. PolyAP  g  \/  ( K  +  1 ) MonoAP  g ) )   &    |-  ( ph  ->  V  e.  NN )   &    |-  ( ph  ->  A. f  e.  ( ( R  ^m  ( 1
 ... W ) ) 
 ^m  ( 1 ...
 V ) ) K MonoAP 
 f )   &    |-  ( ph  ->  H : ( 1 ... ( W  x.  (
 2  x.  V ) ) ) --> R )   &    |-  F  =  ( x  e.  ( 1 ... V )  |->  ( y  e.  ( 1 ... W )  |->  ( H `  ( y  +  ( W  x.  ( ( x  -  1 )  +  V ) ) ) ) ) )   =>    |-  ( ph  ->  (
 <. ( M  +  1 ) ,  K >. PolyAP  H  \/  ( K  +  1 ) MonoAP  H ) )
 
Theoremvdwlem10 12964* Lemma for vdw 12968. Set up secondary induction on  M. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |-  ( ph  ->  R  e.  Fin )   &    |-  ( ph  ->  K  e.  ( ZZ>= `  2
 ) )   &    |-  ( ph  ->  A. s  e.  Fin  E. n  e.  NN  A. f  e.  ( s  ^m  (
 1 ... n ) ) K MonoAP  f )   &    |-  ( ph  ->  M  e.  NN )   =>    |-  ( ph  ->  E. n  e.  NN  A. f  e.  ( R  ^m  (
 1 ... n ) ) ( <. M ,  K >. PolyAP 
 f  \/  ( K  +  1 ) MonoAP  f
 ) )
 
Theoremvdwlem11 12965* Lemma for vdw 12968. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |-  ( ph  ->  R  e.  Fin )   &    |-  ( ph  ->  K  e.  ( ZZ>= `  2
 ) )   &    |-  ( ph  ->  A. s  e.  Fin  E. n  e.  NN  A. f  e.  ( s  ^m  (
 1 ... n ) ) K MonoAP  f )   =>    |-  ( ph  ->  E. n  e.  NN  A. f  e.  ( R  ^m  ( 1 ... n ) ) ( K  +  1 ) MonoAP  f
 )
 
Theoremvdwlem12 12966 Lemma for vdw 12968. 
K  =  2 base case of induction. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |-  ( ph  ->  R  e.  Fin )   &    |-  ( ph  ->  F : ( 1 ... ( ( # `  R )  +  1 )
 ) --> R )   &    |-  ( ph  ->  -.  2 MonoAP  F )   =>    |-  -.  ph
 
Theoremvdwlem13 12967* Lemma for vdw 12968. Main induction on  K;  K  =  0,  K  =  1 base cases. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |-  ( ph  ->  R  e.  Fin )   &    |-  ( ph  ->  K  e.  NN0 )   =>    |-  ( ph  ->  E. n  e.  NN  A. f  e.  ( R  ^m  (
 1 ... n ) ) K MonoAP  f )
 
Theoremvdw 12968* Van der Waerden's theorem. For any finite coloring  R and integer  K, there is an  N such that every coloring function from  1 ... N to  R contains a monochromatic arithmetic progression (which written out in full means that there is a color  c and base, increment values  a ,  d such that all the numbers  a ,  a  +  d ,  ... ,  a  +  ( k  -  1 ) d lie in the preimage of  {
c }, i.e. they are all in  1 ... N and  f evaluated at each one yields  c). (Contributed by Mario Carneiro, 13-Sep-2014.)
 |-  ( ( R  e.  Fin  /\  K  e.  NN0 )  ->  E. n  e.  NN  A. f  e.  ( R 
 ^m  ( 1 ... n ) ) 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 } )
 )
 
Theoremvdwnnlem1 12969* Corollary of vdw 12968, and lemma for vdwnn 12972. If  F is a coloring of the integers, then there are arbitrarily long monochromatic APs in  F. (Contributed by Mario Carneiro, 13-Sep-2014.)
 |-  ( ( R  e.  Fin  /\  F : NN --> R  /\  K  e.  NN0 )  ->  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 } )
 )
 
Theoremvdwnnlem2 12970* Lemma for vdwnn 12972. The set of all "bad"  k for the theorem is upwards-closed, because a long AP implies a short AP. (Contributed by Mario Carneiro, 13-Sep-2014.)
 |-  ( ph  ->  R  e.  Fin )   &    |-  ( ph  ->  F : NN --> R )   &    |-  S  =  { k  e.  NN  |  -.  E. a  e.  NN  E. d  e.  NN  A. m  e.  ( 0 ... (
 k  -  1 ) ) ( a  +  ( m  x.  d
 ) )  e.  ( `' F " { c } ) }   =>    |-  ( ( ph  /\  B  e.  ( ZZ>= `  A ) )  ->  ( A  e.  S  ->  B  e.  S ) )
 
Theoremvdwnnlem3 12971* Lemma for vdwnn 12972. (Contributed by Mario Carneiro, 13-Sep-2014.)
 |-  ( ph  ->  R  e.  Fin )   &    |-  ( ph  ->  F : NN --> R )   &    |-  S  =  { k  e.  NN  |  -.  E. a  e.  NN  E. d  e.  NN  A. m  e.  ( 0 ... (
 k  -  1 ) ) ( a  +  ( m  x.  d
 ) )  e.  ( `' F " { c } ) }   &    |-  ( ph  ->  A. c  e.  R  S  =/=  (/) )   =>    |- 
 -.  ph
 
Theoremvdwnn 12972* Van der Waerden's theorem, infinitary version. For any finite coloring  F of the natural numbers, there is a color  c that contains arbitrarily long arithmetic progressions. (Contributed by Mario Carneiro, 13-Sep-2014.)
 |-  ( ( R  e.  Fin  /\  F : NN --> R ) 
 ->  E. c  e.  R  A. k  e.  NN  E. a  e.  NN  E. d  e.  NN  A. m  e.  ( 0 ... (
 k  -  1 ) ) ( a  +  ( m  x.  d
 ) )  e.  ( `' F " { c } ) )
 
6.2.12  Ramsey's theorem
 
Syntaxcram 12973 Extend class notation with the Ramsey number function.
 class Ramsey
 
Theoremramtlecl 12974* The set  T of numbers with the Ramsey number property is upward-closed. (Contributed by Mario Carneiro, 21-Apr-2015.)
 |-  T  =  { n  e.  NN0  |  A. s
 ( n  <_  ( # `
  s )  ->  ph ) }   =>    |-  ( M  e.  T  ->  ( ZZ>= `  M )  C_  T )
 
Definitiondf-ram 12975* Define the Ramsey number function. The input is a number  m for the size of the edges of the hypergraph, and a tuple  r from the finite color set to lower bounds for each color. The Ramsey number  ( M Ramsey  R
) is the smallest number such that for any set  S with  ( M Ramsey  R
)  <_  # S and any coloring  F of the set of  M-element subsets of  S (with color set  dom  R), there is a color  c  e.  dom  R and a subset  x  C_  S such that  R ( c )  <_  # x and all the hyperedges of  x (that is, subsets of  x of size  M) have color  c. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |- Ramsey  =  ( m  e.  NN0 ,  r  e.  _V  |->  sup ( { n  e. 
 NN0  |  A. s ( n  <_  ( # `  s
 )  ->  A. f  e.  ( dom  r  ^m  { y  e.  ~P s  |  ( # `  y
 )  =  m }
 ) E. c  e. 
 dom  r E. x  e.  ~P  s ( ( r `  c ) 
 <_  ( # `  x )  /\  A. y  e. 
 ~P  x ( ( # `  y )  =  m  ->  ( f `  y )  =  c ) ) ) } ,  RR* ,  `'  <  ) )
 
Theoremhashbcval 12976* Value of the "binomial set", the set of all  N-element subsets of  A. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   =>    |-  ( ( A  e.  V  /\  N  e.  NN0 )  ->  ( A C N )  =  { x  e.  ~P A  |  ( # `  x )  =  N }
 )
 
Theoremhashbccl 12977* The binomial set is a finite set. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   =>    |-  ( ( A  e.  Fin  /\  N  e.  NN0 )  ->  ( A C N )  e.  Fin )
 
Theoremhashbcss 12978* Subset relation for the binomial set. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   =>    |-  ( ( A  e.  V  /\  B  C_  A  /\  N  e.  NN0 )  ->  ( B C N )  C_  ( A C N ) )
 
Theoremhashbc0 12979* The set of subsets of size zero is the singleton of the empty set. (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   =>    |-  ( A  e.  V  ->  ( A C 0 )  =  { (/) } )
 
Theoremhashbc2 12980* The size of the binomial set is the binomial coefficient. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   =>    |-  ( ( A  e.  Fin  /\  N  e.  NN0 )  ->  ( # `  ( A C N ) )  =  ( ( # `  A )  _C  N ) )
 
Theorem0hashbc 12981* There are no subsets of the empty set with size greater than zero. (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   =>    |-  ( N  e.  NN  ->  ( (/) C N )  =  (/) )
 
Theoremramval 12982* The value of the Ramsey number function. (Contributed by Mario Carneiro, 21-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   &    |-  T  =  { n  e.  NN0  |  A. s
 ( n  <_  ( # `
  s )  ->  A. f  e.  ( R  ^m  ( s C M ) ) E. c  e.  R  E. x  e.  ~P  s ( ( F `  c ) 
 <_  ( # `  x )  /\  ( x C M )  C_  ( `' f " { c } ) ) ) }   =>    |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  =  sup ( T ,  RR*
 ,  `'  <  )
 )
 
Theoremramcl2lem 12983* Lemma for extended real closure of the Ramsey number function. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   &    |-  T  =  { n  e.  NN0  |  A. s
 ( n  <_  ( # `
  s )  ->  A. f  e.  ( R  ^m  ( s C M ) ) E. c  e.  R  E. x  e.  ~P  s ( ( F `  c ) 
 <_  ( # `  x )  /\  ( x C M )  C_  ( `' f " { c } ) ) ) }   =>    |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  =  if ( T  =  (/)
 ,  +oo ,  sup ( T ,  RR ,  `'  <  ) ) )
 
Theoremramtcl 12984* The Ramsey number has the Ramsey number property if any number does. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   &    |-  T  =  { n  e.  NN0  |  A. s
 ( n  <_  ( # `
  s )  ->  A. f  e.  ( R  ^m  ( s C M ) ) E. c  e.  R  E. x  e.  ~P  s ( ( F `  c ) 
 <_  ( # `  x )  /\  ( x C M )  C_  ( `' f " { c } ) ) ) }   =>    |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( ( M Ramsey  F )  e.  T  <->  T  =/=  (/) ) )
 
Theoremramtcl2 12985* The Ramsey number is an integer iff there is a number with the Ramsey number property. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   &    |-  T  =  { n  e.  NN0  |  A. s
 ( n  <_  ( # `
  s )  ->  A. f  e.  ( R  ^m  ( s C M ) ) E. c  e.  R  E. x  e.  ~P  s ( ( F `  c ) 
 <_  ( # `  x )  /\  ( x C M )  C_  ( `' f " { c } ) ) ) }   =>    |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( ( M Ramsey  F )  e.  NN0  <->  T  =/=  (/) ) )
 
Theoremramtub 12986* The Ramsey number is a lower bound on the set of all numbers with the Ramsey number property. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   &    |-  T  =  { n  e.  NN0  |  A. s
 ( n  <_  ( # `
  s )  ->  A. f  e.  ( R  ^m  ( s C M ) ) E. c  e.  R  E. x  e.  ~P  s ( ( F `  c ) 
 <_  ( # `  x )  /\  ( x C M )  C_  ( `' f " { c } ) ) ) }   =>    |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  A  e.  T )  ->  ( M Ramsey  F )  <_  A )
 
Theoremramub 12987* The Ramsey number is a lower bound on the set of all numbers with the Ramsey number property. (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  F : R --> NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ( ph  /\  ( N  <_  ( # `
  s )  /\  f : ( s C M ) --> R ) )  ->  E. c  e.  R  E. x  e. 
 ~P  s ( ( F `  c ) 
 <_  ( # `  x )  /\  ( x C M )  C_  ( `' f " { c } ) ) )   =>    |-  ( ph  ->  ( M Ramsey  F )  <_  N )
 
Theoremramub2 12988* It is sufficient to check the Ramsey property on finite sets of size equal to the upper bound. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  F : R --> NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ( ph  /\  ( ( # `  s
 )  =  N  /\  f : ( s C M ) --> R ) )  ->  E. c  e.  R  E. x  e. 
 ~P  s ( ( F `  c ) 
 <_  ( # `  x )  /\  ( x C M )  C_  ( `' f " { c } ) ) )   =>    |-  ( ph  ->  ( M Ramsey  F )  <_  N )
 
Theoremrami 12989* The defining property of a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  F : R --> NN0 )   &    |-  ( ph  ->  ( M Ramsey  F )  e.  NN0 )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  ( M Ramsey  F )  <_  ( # `
  S ) )   &    |-  ( ph  ->  G :
 ( S C M )
 --> R )   =>    |-  ( ph  ->  E. c  e.  R  E. x  e. 
 ~P  S ( ( F `  c ) 
 <_  ( # `  x )  /\  ( x C M )  C_  ( `' G " { c } ) ) )
 
Theoremramcl2 12990 The Ramsey number is either a nonnegative integer or plus infinity. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  e.  ( NN0  u.  {  +oo } ) )
 
Theoremramxrcl 12991 The Ramsey number is an extended real number. (This theorem does not imply Ramsey's theorem, unlike ramcl 13003.) (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  e.  RR* )
 
Theoremramubcl 12992 If the Ramsey number is upper bounded, then it is an integer. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( A  e.  NN0  /\  ( M Ramsey  F )  <_  A ) )  ->  ( M Ramsey  F )  e.  NN0 )
 
Theoremramlb 12993* Establish a lower bound on a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  F : R --> NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  G : ( ( 1
 ... N ) C M ) --> R )   &    |-  ( ( ph  /\  (
 c  e.  R  /\  x  C_  ( 1 ...
 N ) ) ) 
 ->  ( ( x C M )  C_  ( `' G " { c } )  ->  ( # `  x )  <  ( F `  c ) ) )   =>    |-  ( ph  ->  N  <  ( M Ramsey  F )
 )
 
Theorem0ram 12994* The Ramsey number when  M  =  0. (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  ( ( ( R  e.  V  /\  R  =/= 
 (/)  /\  F : R --> NN0 )  /\  E. x  e.  ZZ  A. y  e. 
 ran  F  y  <_  x )  ->  ( 0 Ramsey  F )  =  sup ( ran  F ,  RR ,  <  ) )
 
Theorem0ram2 12995 The Ramsey number when  M  =  0. (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R --> NN0 )  ->  (
 0 Ramsey  F )  =  sup ( ran  F ,  RR ,  <  ) )
 
Theoremram0 12996 The Ramsey number when  R  =  (/). (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  ( M  e.  NN0  ->  ( M Ramsey  (/) )  =  M )
 
Theorem0ramcl 12997 Lemma for ramcl 13003: Existence of the Ramsey number when  M  =  0. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( ( R  e.  Fin  /\  F : R --> NN0 )  ->  ( 0 Ramsey  F )  e.  NN0 )
 
Theoremramz2 12998 The Ramsey number when  F has value zero for some color  C. (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R
 --> NN0 )  /\  ( C  e.  R  /\  ( F `  C )  =  0 ) ) 
 ->  ( M Ramsey  F )  =  0 )
 
Theoremramz 12999 The Ramsey number when  F is the zero function. (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  ( ( M  e.  NN0  /\  R  e.  V  /\  R  =/=  (/) )  ->  ( M Ramsey  ( R  X.  {
 0 } ) )  =  0 )
 
Theoremramub1lem1 13000* Lemma for ramub1 13002. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  R  e.  Fin )   &    |-  ( ph  ->  F : R --> NN )   &    |-  G  =  ( x  e.  R  |->  ( M Ramsey  ( y  e.  R  |->  if ( y  =  x ,  ( ( F `  x )  -  1 ) ,  ( F `  y
 ) ) ) ) )   &    |-  ( ph  ->  G : R --> NN0 )   &    |-  ( ph  ->  ( ( M  -  1 ) Ramsey  G )  e.  NN0 )   &    |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b )  =  i } )   &    |-  ( ph  ->  S  e.  Fin )   &    |-  ( ph  ->  ( # `
  S )  =  ( ( ( M  -  1 ) Ramsey  G )  +  1 )
 )   &    |-  ( ph  ->  K : ( S C M ) --> R )   &    |-  ( ph  ->  X  e.  S )   &    |-  H  =  ( u  e.  ( ( S  \  { X } ) C ( M  -  1 ) )  |->  ( K `  ( u  u.  { X } ) ) )   &    |-  ( ph  ->  D  e.  R )   &    |-  ( ph  ->  W 
 C_  ( S  \  { X } ) )   &    |-  ( ph  ->  ( G `  D )  <_  ( # `
  W ) )   &    |-  ( ph  ->  ( W C ( M  -  1 ) )  C_  ( `' H " { D } ) )   &    |-  ( ph  ->  E  e.  R )   &    |-  ( ph  ->  V  C_  W )   &    |-  ( ph  ->  if ( E  =  D ,  ( ( F `  D )  -  1
 ) ,  ( F `
  E ) ) 
 <_  ( # `  V ) )   &    |-  ( ph  ->  ( V C M ) 
 C_  ( `' K " { E } )
 )   =>    |-  ( ph  ->  E. z  e.  ~P  S ( ( F `  E ) 
 <_  ( # `  z
 )  /\  ( z C M )  C_  ( `' K " { E } ) ) )
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