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Theorem List for Metamath Proof Explorer - 14501-14600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremvdwapfval 14501* 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 14502 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 14503* 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 14504 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 14505 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 14506 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 14507 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 14508* 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 14509* 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 14510* 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 14511* Lemma for vdw 14524. (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 14512* Lemma for vdw 14524. (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 14513 Lemma for vdw 14524. (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 14514* Lemma for vdw 14524. (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 14515* Lemma for vdw 14524. (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 14516* Lemma for vdw 14524. (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 14517* Lemma for vdw 14524. (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 14518* Lemma for vdw 14524. (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 14519* Lemma for vdw 14524. (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 14520* Lemma for vdw 14524. 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 14521* Lemma for vdw 14524. (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 14522 Lemma for vdw 14524. 
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 14523* Lemma for vdw 14524. 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 14524* 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 14525* Corollary of vdw 14524, and lemma for vdwnn 14528. 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 14526* Lemma for vdwnn 14528. 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 14527* Lemma for vdwnn 14528. (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 14528* Van der Waerden's theorem, infinitary version. For any finite coloring  F of the positive integers, 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 } ) )
 
6.2.13  Ramsey's theorem
 
Syntaxcram 14529 Extend class notation with the Ramsey number function.
 class Ramsey
 
Theoremramtlecl 14530* 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 14531* 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 14532* 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 14533* 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 14534* 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 14535* 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 14536* 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 14537* 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 14538* 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 14539* 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 14540* 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 14541* 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 14542* 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 14543* 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 14544* 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 14545* 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 14546 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 14547 The Ramsey number is an extended real number. (This theorem does not imply Ramsey's theorem, unlike ramcl 14559.) (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  e.  RR* )
 
Theoremramubcl 14548 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 14549* 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 14550* 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 14551 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 14552 The Ramsey number when  R  =  (/). (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  ( M  e.  NN0  ->  ( M Ramsey  (/) )  =  M )
 
Theorem0ramcl 14553 Lemma for ramcl 14559: 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 14554 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 14555 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 14556* Lemma for ramub1 14558. (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 14557* Lemma for ramub1 14558. (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 14558* 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 14559 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 14560* 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. This is Metamath 100 proof #31. (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 } ) ) ) )
 
6.2.14  Decimal arithmetic (cont.)
 
Theoremdec2dvds 14561 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 14562 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 14563 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 14564 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  A  e.  NN   =>    |-  -. ; A 5  e.  Prime
 
Theoremdec2nprm 14565 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 14566 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 14567 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 14568 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 14569 Version of mod2xi 14567 with a negative 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 14570 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 14571 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 14572 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 14573 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 14574 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   =>    |-  ( A ^ 0
 )  =  1
 
Theoremnumexp1 14575 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   =>    |-  ( A ^ 1
 )  =  A
 
Theoremnumexpp1 14576 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 14577 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 14578 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 14579 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 14580 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 14581 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 14582 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 11047. (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 14583 Two to the fourth power is 16. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( 2 ^ 4
 )  = ; 1 6
 
Theorem2exp6 14584 Two to the sixth power is 64. (Contributed by Mario Carneiro, 20-Apr-2015.) (Proof shortened by OpenAI, 25-Mar-2020.)
 |-  ( 2 ^ 6
 )  = ; 6 4
 
Theorem2exp6OLD 14585 Two to the sixth power is 64. (Contributed by Mario Carneiro, 20-Apr-2015.) Obsolete version of 2exp6 14584 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( 2 ^ 6
 )  = ; 6 4
 
Theorem2exp8 14586 Two to the eighth power is 256. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( 2 ^ 8
 )  = ;; 2 5 6
 
Theorem2exp16 14587 Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( 2 ^; 1 6 )  = ;;;; 6 5 5 3 6
 
Theorem3exp3 14588 Three to the third power is 27. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( 3 ^ 3
 )  = ; 2 7
 
Theorem2expltfac 14589 The factorial grows faster than two to the power  N. (Contributed by Mario Carneiro, 15-Sep-2016.)
 |-  ( N  e.  ( ZZ>=
 `  4 )  ->  ( 2 ^ N )  <  ( ! `  N ) )
 
6.2.15  Cyclical shifts of words (cont.)
 
Theoremcshwsidrepsw 14590 If cyclically shifting a word of length being a prime number by a number of positions which is not divisible by the prime number results in the word itself, the word is a "repeated symbol word". (Contributed by AV, 18-May-2018.) (Revised by AV, 10-Nov-2018.)
 |-  ( ( W  e. Word  V 
 /\  ( # `  W )  e.  Prime )  ->  ( ( L  e.  ZZ  /\  ( L  mod  ( # `  W ) )  =/=  0  /\  ( W cyclShift  L )  =  W )  ->  W  =  ( ( W `  0 ) repeatS  ( # `  W ) ) ) )
 
Theoremcshwsidrepswmod0 14591 If cyclically shifting a word of length being a prime number results in the word itself, the shift must be either by 0 (modulo the length of the word) or the word must be a "repeated symbol word". (Contributed by AV, 18-May-2018.) (Revised by AV, 10-Nov-2018.)
 |-  ( ( W  e. Word  V 
 /\  ( # `  W )  e.  Prime  /\  L  e.  ZZ )  ->  (
 ( W cyclShift  L )  =  W  ->  ( ( L  mod  ( # `  W ) )  =  0  \/  W  =  ( ( W `  0 ) repeatS  ( # `  W ) ) ) ) )
 
Theoremcshwshashlem1 14592* If cyclically shifting a word of length being a prime number not consisting of identical symbols by at least one position (and not by as many positions as the length of the word), the result will not be the word itself. (Contributed by AV, 19-May-2018.) (Revised by AV, 8-Jun-2018.) (Revised by AV, 10-Nov-2018.)
 |-  ( ph  ->  ( W  e. Word  V  /\  ( # `
  W )  e. 
 Prime ) )   =>    |-  ( ( ph  /\  E. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =/=  ( W `  0 )  /\  L  e.  ( 1..^ ( # `  W ) ) )  ->  ( W cyclShift  L )  =/=  W )
 
Theoremcshwshashlem2 14593* If cyclically shifting a word of length being a prime number and not of identical symbols by different numbers of positions, the resulting words are different. (Contributed by Alexander van der Vekens, 19-May-2018.) (Revised by Alexander van der Vekens, 8-Jun-2018.)
 |-  ( ph  ->  ( W  e. Word  V  /\  ( # `
  W )  e. 
 Prime ) )   =>    |-  ( ( ph  /\  E. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =/=  ( W `  0 ) ) 
 ->  ( ( L  e.  ( 0..^ ( # `  W ) )  /\  K  e.  ( 0..^ ( # `  W ) )  /\  K  <  L )  ->  ( W cyclShift  L )  =/=  ( W cyclShift  K ) ) )
 
Theoremcshwshashlem3 14594* If cyclically shifting a word of length being a prime number and not of identical symbols by different numbers of positions, the resulting words are different. (Contributed by Alexander van der Vekens, 19-May-2018.) (Revised by Alexander van der Vekens, 8-Jun-2018.)
 |-  ( ph  ->  ( W  e. Word  V  /\  ( # `
  W )  e. 
 Prime ) )   =>    |-  ( ( ph  /\  E. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =/=  ( W `  0 ) ) 
 ->  ( ( L  e.  ( 0..^ ( # `  W ) )  /\  K  e.  ( 0..^ ( # `  W ) )  /\  K  =/=  L )  ->  ( W cyclShift  L )  =/=  ( W cyclShift  K ) ) )
 
Theoremcshwsdisj 14595* The singletons resulting by cyclically shifting a given word of length being a prime number and not consisting of identical symbols is a disjoint collection. (Contributed by Alexander van der Vekens, 19-May-2018.) (Revised by Alexander van der Vekens, 8-Jun-2018.)
 |-  ( ph  ->  ( W  e. Word  V  /\  ( # `
  W )  e. 
 Prime ) )   =>    |-  ( ( ph  /\  E. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =/=  ( W `  0 ) ) 
 -> Disj 
 n  e.  ( 0..^ ( # `  W ) ) { ( W cyclShift  n ) } )
 
Theoremcshwsiun 14596* The set of (different!) words resulting by cyclically shifting a given word is an indexed union. (Contributed by AV, 19-May-2018.) (Revised by AV, 8-Jun-2018.) (Proof shortened by AV, 8-Nov-2018.)
 |-  M  =  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }   =>    |-  ( W  e. Word  V  ->  M  =  U_ n  e.  ( 0..^ ( # `  W ) ) {
 ( W cyclShift  n ) }
 )
 
Theoremcshwsex 14597* The class of (different!) words resulting by cyclically shifting a given word is a set. (Contributed by AV, 8-Jun-2018.) (Revised by AV, 8-Nov-2018.)
 |-  M  =  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }   =>    |-  ( W  e. Word  V  ->  M  e.  _V )
 
Theoremcshws0 14598* The size of the set of (different!) words resulting by cyclically shifting an empty word is 0. (Contributed by AV, 8-Nov-2018.)
 |-  M  =  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }   =>    |-  ( W  =  (/)  ->  ( # `  M )  =  0 )
 
Theoremcshwrepswhash1 14599* The size of the set of (different!) words resulting by cyclically shifting a nonempty "repeated symbol word" is 1. (Contributed by AV, 18-May-2018.) (Revised by AV, 8-Nov-2018.)
 |-  M  =  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }   =>    |-  ( ( A  e.  V  /\  N  e.  NN  /\  W  =  ( A repeatS  N ) )  ->  ( # `  M )  =  1 )
 
Theoremcshwshashnsame 14600* If a word (not consisting of identical symbols) has a length being a prime number, the size of the set of (different!) words resulting by cyclically shifting the original word equals the length of the original word. (Contributed by AV, 19-May-2018.) (Revised by AV, 10-Nov-2018.)
 |-  M  =  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }   =>    |-  ( ( W  e. Word  V 
 /\  ( # `  W )  e.  Prime )  ->  ( E. i  e.  (
 0..^ ( # `  W ) ) ( W `
  i )  =/=  ( W `  0
 )  ->  ( # `  M )  =  ( # `  W ) ) )
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