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Theorem List for Metamath Proof Explorer - 13001-13100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmul4sq 13001* 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 13000. (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 13002* Lemma for 4sq 13011. 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 13003* Lemma for 4sq 13011. 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 13004* Lemma for 4sq 13011. (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 13005* Lemma for 4sq 13011. (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 13006* Lemma for 4sq 13011. (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 13007* Lemma for 4sq 13011. (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 13008* Lemma for 4sq 13011. (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 13009* Lemma for 4sq 13011. 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 13010* Lemma for 4sq 13011. 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 13009. 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 13001  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 13011* 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 13012 The arithmetic progression function.
 class AP
 
Syntaxcvdwm 13013 The monochromatic arithmetic progression predicate.
 class MonoAP
 
Syntaxcvdwp 13014 The polychromatic arithmetic progression predicate.
 class PolyAP
 
Definitiondf-vdwap 13015* 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 13016* 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 13017* Define the "contains a polychromatic colleciton of APs" predicate. See vdwpc 13027 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 13018* 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 13019 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 13020* 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 13021 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 13022 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 13023 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 13024 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 13025* 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 13026* 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 13027* 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 13028* Lemma for vdw 13041. (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 13029* Lemma for vdw 13041. (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 13030 Lemma for vdw 13041. (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 13031* Lemma for vdw 13041. (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 13032* Lemma for vdw 13041. (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 13033* Lemma for vdw 13041. (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 13034* Lemma for vdw 13041. (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 13035* Lemma for vdw 13041. (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 13036* Lemma for vdw 13041. (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 13037* Lemma for vdw 13041. 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 13038* Lemma for vdw 13041. (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 13039 Lemma for vdw 13041. 
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 13040* Lemma for vdw 13041. 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 13041* 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 13042* Corollary of vdw 13041, and lemma for vdwnn 13045. 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 13043* Lemma for vdwnn 13045. 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 13044* Lemma for vdwnn 13045. (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 13045* 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 13046 Extend class notation with the Ramsey number function.
 class Ramsey
 
Theoremramtlecl 13047* 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 13048* 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 13049* 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 13050* 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 13051* 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 13052* 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 13053* 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 13054* 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 13055* 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 13056* 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 13057* 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 13058* 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 13059* 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 13060* 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 13061* 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 13062* 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 13063 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 13064 The Ramsey number is an extended real number. (This theorem does not imply Ramsey's theorem, unlike ramcl 13076.) (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  e.  RR* )
 
Theoremramubcl 13065 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 13066* 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 13067* 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 13068 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 13069 The Ramsey number when  R  =  (/). (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  ( M  e.  NN0  ->  ( M Ramsey  (/) )  =  M )
 
Theorem0ramcl 13070 Lemma for ramcl 13076: 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 13071 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 13072 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 13073* Lemma for ramub1 13075. (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 } ) ) )
 
Theoremramub1lem2 13074* Lemma for ramub1 13075. (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  ->  E. c  e.  R  E. z  e. 
 ~P  S ( ( F `  c ) 
 <_  ( # `  z
 )  /\  ( z C M )  C_  ( `' K " { c } ) ) )
 
Theoremramub1 13075* Inductive step for Ramsey's theorem, in the form of an explicit upper bound. (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 )   =>    |-  ( ph  ->  ( M Ramsey  F )  <_  (
 ( ( M  -  1 ) Ramsey  G )  +  1 ) )
 
Theoremramcl 13076 Ramsey's theorem: the Ramsey number is an integer for every finite coloring and set of upper bounds. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( ( M  e.  NN0  /\  R  e.  Fin  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  e.  NN0 )
 
Theoremramsey 13077* Ramsey's theorem with the definition Ramsey eliminated. If  M is an integer,  R is a specified finite set of colors, and  F : R --> NN0 is a set of lower bounds for each color, then there is an  n such that for every set  s of size greater than  n and every coloring  f of the set  ( s C M ) of all  M-element subsets of  s, there is a color  c and a subset  x  C_  s such that  x is larger than  F (
c ) and the  M-element subsets of  x are monochromatic with color  c. This is the hypergraph version of Ramsey's theorem; the version for simple graphs is the case  M  =  2. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
 )  =  i }
 )   =>    |-  ( ( M  e.  NN0  /\  R  e.  Fin  /\  F : R --> NN0 )  ->  E. 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 } ) ) ) )
 
6.2.13  Decimal arithmetic (cont.)
 
Theoremdec2dvds 13078 Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  ( B  x.  2
 )  =  C   &    |-  D  =  ( C  +  1 )   =>    |- 
 -.  2  || ; A D
 
Theoremdec5dvds 13079 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  A  e.  NN0   &    |-  B  e.  NN   &    |-  B  <  5   =>    |- 
 -.  5  || ; A B
 
Theoremdec5dvds2 13080 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  A  e.  NN0   &    |-  B  e.  NN   &    |-  B  <  5   &    |-  ( 5  +  B )  =  C   =>    |-  -.  5  || ; A C
 
Theoremdec5nprm 13081 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  A  e.  NN   =>    |-  -. ; A 5  e.  Prime
 
Theoremdec2nprm 13082 Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  A  e.  NN   &    |-  B  e.  NN0   &    |-  ( B  x.  2
 )  =  C   =>    |-  -. ; A C  e.  Prime
 
Theoremmodxai 13083 Add exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.) (Revised by Mario Carneiro, 5-Feb-2015.)
 |-  N  e.  NN   &    |-  A  e.  NN   &    |-  B  e.  NN0   &    |-  D  e.  ZZ   &    |-  K  e.  NN0   &    |-  M  e.  NN0   &    |-  C  e.  NN0   &    |-  L  e.  NN0   &    |-  ( ( A ^ B )  mod  N )  =  ( K  mod  N )   &    |-  ( ( A ^ C )  mod  N )  =  ( L 
 mod  N )   &    |-  ( B  +  C )  =  E   &    |-  (
 ( D  x.  N )  +  M )  =  ( K  x.  L )   =>    |-  ( ( A ^ E )  mod  N )  =  ( M  mod  N )
 
Theoremmod2xi 13084 Double exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)
 |-  N  e.  NN   &    |-  A  e.  NN   &    |-  B  e.  NN0   &    |-  D  e.  ZZ   &    |-  K  e.  NN0   &    |-  M  e.  NN0   &    |-  ( ( A ^ B )  mod  N )  =  ( K  mod  N )   &    |-  ( 2  x.  B )  =  E   &    |-  (
 ( D  x.  N )  +  M )  =  ( K  x.  K )   =>    |-  ( ( A ^ E )  mod  N )  =  ( M  mod  N )
 
Theoremmodxp1i 13085 Add one to an exponent in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)
 |-  N  e.  NN   &    |-  A  e.  NN   &    |-  B  e.  NN0   &    |-  D  e.  ZZ   &    |-  K  e.  NN0   &    |-  M  e.  NN0   &    |-  ( ( A ^ B )  mod  N )  =  ( K  mod  N )   &    |-  ( B  +  1 )  =  E   &    |-  (
 ( D  x.  N )  +  M )  =  ( K  x.  A )   =>    |-  ( ( A ^ E )  mod  N )  =  ( M  mod  N )
 
Theoremmod2xnegi 13086 Version of mod2xi 13084 with a negaive mod value. (Contributed by Mario Carneiro, 21-Feb-2014.)
 |-  A  e.  NN   &    |-  B  e.  NN0   &    |-  D  e.  ZZ   &    |-  K  e.  NN   &    |-  M  e.  NN0   &    |-  L  e.  NN0   &    |-  ( ( A ^ B )  mod  N )  =  ( L  mod  N )   &    |-  ( 2  x.  B )  =  E   &    |-  ( L  +  K )  =  N   &    |-  ( ( D  x.  N )  +  M )  =  ( K  x.  K )   =>    |-  ( ( A ^ E )  mod  N )  =  ( M 
 mod  N )
 
Theoremmodsubi 13087 Subtract from within a mod calculation. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN   &    |-  A  e.  NN   &    |-  B  e.  NN0   &    |-  M  e.  NN0   &    |-  ( A  mod  N )  =  ( K  mod  N )   &    |-  ( M  +  B )  =  K   =>    |-  (
 ( A  -  B )  mod  N )  =  ( M  mod  N )
 
Theoremgcdi 13088 Calculate a GCD via Euclid's algorithm. (Contributed by Mario Carneiro, 19-Feb-2014.)
 |-  K  e.  NN0   &    |-  R  e.  NN0   &    |-  N  e.  NN0   &    |-  ( N  gcd  R )  =  G   &    |-  ( ( K  x.  N )  +  R )  =  M   =>    |-  ( M  gcd  N )  =  G
 
Theoremgcdmodi 13089 Calculate a GCD via Euclid's algorithm. (Contributed by Mario Carneiro, 19-Feb-2014.)
 |-  K  e.  NN0   &    |-  R  e.  NN0   &    |-  N  e.  NN   &    |-  ( K  mod  N )  =  ( R  mod  N )   &    |-  ( N  gcd  R )  =  G   =>    |-  ( K  gcd  N )  =  G
 
Theoremdecexp2 13090 Calculate a power of two. (Contributed by Mario Carneiro, 19-Feb-2014.)
 |-  M  e.  NN0   &    |-  ( M  +  2 )  =  N   =>    |-  ( ( 4  x.  ( 2 ^ M ) )  +  0 )  =  (
 2 ^ N )
 
Theoremnumexp0 13091 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   =>    |-  ( A ^ 0
 )  =  1
 
Theoremnumexp1 13092 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   =>    |-  ( A ^ 1
 )  =  A
 
Theoremnumexpp1 13093 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   &    |-  M  e.  NN0   &    |-  ( M  +  1 )  =  N   &    |-  (
 ( A ^ M )  x.  A )  =  C   =>    |-  ( A ^ N )  =  C
 
Theoremnumexp2x 13094 Double an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   &    |-  M  e.  NN0   &    |-  ( 2  x.  M )  =  N   &    |-  ( A ^ M )  =  D   &    |-  ( D  x.  D )  =  C   =>    |-  ( A ^ N )  =  C
 
Theoremdecsplit0b 13095 Split a decimal number into two parts. Base case:  N  =  0. (Contributed by Mario Carneiro, 16-Jul-2015.)
 |-  A  e.  NN0   =>    |-  ( ( A  x.  ( 10 ^ 0 ) )  +  B )  =  ( A  +  B )
 
Theoremdecsplit0 13096 Split a decimal number into two parts. Base case:  N  =  0. (Contributed by Mario Carneiro, 16-Jul-2015.)
 |-  A  e.  NN0   =>    |-  ( ( A  x.  ( 10 ^ 0 ) )  +  0 )  =  A
 
Theoremdecsplit1 13097 Split a decimal number into two parts. Base case:  N  =  1. (Contributed by Mario Carneiro, 16-Jul-2015.)
 |-  A  e.  NN0   =>    |-  ( ( A  x.  ( 10 ^ 1 ) )  +  B )  = ; A B
 
Theoremdecsplit 13098 Split a decimal number into two parts. Inductive step. (Contributed by Mario Carneiro, 16-Jul-2015.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  D  e.  NN0   &    |-  M  e.  NN0   &    |-  ( M  +  1 )  =  N   &    |-  (
 ( A  x.  ( 10 ^ M ) )  +  B )  =  C   =>    |-  ( ( A  x.  ( 10 ^ N ) )  + ; B D )  = ; C D
 
Theoremkaratsuba 13099 The Karatsuba multiplication algorithm. If  X and 
Y are decomposed into two groups of digits of length  M (only the lower group is known to be this size but the algorithm is most efficient when the partition is chosen near the middle of the digit string), then  X Y can be written in three groups of digits, where each group needs only one multiplication. Thus we can halve both inputs with only three multiplications on the smaller operands, yielding an asymptotic improvement of n^(log2 3) instead of n^2 for the "naive" algorithm decmul1c 10171. (Contributed by Mario Carneiro, 16-Jul-2015.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  D  e.  NN0   &    |-  S  e.  NN0   &    |-  M  e.  NN0   &    |-  ( A  x.  C )  =  R   &    |-  ( B  x.  D )  =  T   &    |-  (
 ( A  +  B )  x.  ( C  +  D ) )  =  ( ( R  +  S )  +  T )   &    |-  ( ( A  x.  ( 10 ^ M ) )  +  B )  =  X   &    |-  ( ( C  x.  ( 10 ^ M ) )  +  D )  =  Y   &    |-  (
 ( R  x.  ( 10 ^ M ) )  +  S )  =  W   &    |-  ( ( W  x.  ( 10 ^ M ) )  +  T )  =  Z   =>    |-  ( X  x.  Y )  =  Z
 
Theorem2exp4 13100 Two to the fourth power is 16. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( 2 ^ 4
 )  = ; 1 6
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