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Theorem List for Metamath Proof Explorer - 13201-13300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem4sqlem9 13201 Lemma for 4sq 13219. (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 13202 Lemma for 4sq 13219. (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 13203* Lemma for 4sq 13219. 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 13204* Lemma for 4sq 13219. 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 13205* Lemma for 4sq 13219. 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 13206* Lemma for 4sqlem4 13207. (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 13207* Lemma for 4sq 13219. 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 13208* Lemma for mul4sq 13209: algebraic manipulations. The extra assumptions involving  M are for a part of 4sqlem17 13216 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 13209* 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 13208. (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 13210* Lemma for 4sq 13219. 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 13211* Lemma for 4sq 13219. 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 13212* Lemma for 4sq 13219. (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 13213* Lemma for 4sq 13219. (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 13214* Lemma for 4sq 13219. (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 13215* Lemma for 4sq 13219. (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 13216* Lemma for 4sq 13219. (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 13217* Lemma for 4sq 13219. 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 13218* Lemma for 4sq 13219. 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 13217. 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 13209  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 13219* 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 13220 The arithmetic progression function.
 class AP
 
Syntaxcvdwm 13221 The monochromatic arithmetic progression predicate.
 class MonoAP
 
Syntaxcvdwp 13222 The polychromatic arithmetic progression predicate.
 class PolyAP
 
Definitiondf-vdwap 13223* 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 13224* 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 13225* Define the "contains a polychromatic colleciton of APs" predicate. See vdwpc 13235 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 13226* 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 13227 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 13228* 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 13229 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 13230 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 13231 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 13232 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 13233* 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 13234* 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 13235* 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 13236* Lemma for vdw 13249. (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 13237* Lemma for vdw 13249. (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 13238 Lemma for vdw 13249. (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 13239* Lemma for vdw 13249. (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 13240* Lemma for vdw 13249. (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 13241* Lemma for vdw 13249. (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 13242* Lemma for vdw 13249. (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 13243* Lemma for vdw 13249. (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 13244* Lemma for vdw 13249. (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 13245* Lemma for vdw 13249. 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 13246* Lemma for vdw 13249. (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 13247 Lemma for vdw 13249. 
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 13248* Lemma for vdw 13249. 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 13249* 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 13250* Corollary of vdw 13249, and lemma for vdwnn 13253. 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 13251* Lemma for vdwnn 13253. 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 13252* Lemma for vdwnn 13253. (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 13253* 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 13254 Extend class notation with the Ramsey number function.
 class Ramsey
 
Theoremramtlecl 13255* 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 13256* 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 13257* 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 13258* 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 13259* 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 13260* 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 13261* 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 13262* 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 13263* 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 13264* 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 13265* 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 13266* 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 13267* 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 13268* 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 13269* 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 13270* 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 13271 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 13272 The Ramsey number is an extended real number. (This theorem does not imply Ramsey's theorem, unlike ramcl 13284.) (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  e.  RR* )
 
Theoremramubcl 13273 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 13274* 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 13275* 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 13276 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 13277 The Ramsey number when  R  =  (/). (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  ( M  e.  NN0  ->  ( M Ramsey  (/) )  =  M )
 
Theorem0ramcl 13278 Lemma for ramcl 13284: 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 13279 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 13280 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 13281* Lemma for ramub1 13283. (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 13282* Lemma for ramub1 13283. (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 13283* 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 13284 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 13285* 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 13286 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 13287 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 13288 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 13289 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  A  e.  NN   =>    |-  -. ; A 5  e.  Prime
 
Theoremdec2nprm 13290 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 13291 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 13292 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 13293 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 13294 Version of mod2xi 13292 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 13295 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 13296 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 13297 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 13298 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 13299 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   =>    |-  ( A ^ 0
 )  =  1
 
Theoremnumexp1 13300 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   =>    |-  ( A ^ 1
 )  =  A
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