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Theorem vdw 13290
Description: 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.)
Assertion
Ref Expression
vdw  |-  ( ( 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 } ) )
Distinct variable groups:    a, c,
d, f, m, n, K    R, a, c, d, f, n
Allowed substitution hint:    R( m)

Proof of Theorem vdw
StepHypRef Expression
1 simpl 444 . . 3  |-  ( ( R  e.  Fin  /\  K  e.  NN0 )  ->  R  e.  Fin )
2 simpr 448 . . 3  |-  ( ( R  e.  Fin  /\  K  e.  NN0 )  ->  K  e.  NN0 )
31, 2vdwlem13 13289 . 2  |-  ( ( R  e.  Fin  /\  K  e.  NN0 )  ->  E. n  e.  NN  A. f  e.  ( R  ^m  ( 1 ... n ) ) K MonoAP 
f )
4 ovex 6046 . . . . 5  |-  ( 1 ... n )  e. 
_V
5 simpllr 736 . . . . 5  |-  ( ( ( ( R  e. 
Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  /\  f  e.  ( R  ^m  (
1 ... n ) ) )  ->  K  e.  NN0 )
6 simpll 731 . . . . . . 7  |-  ( ( ( R  e.  Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  ->  R  e.  Fin )
7 elmapg 6968 . . . . . . 7  |-  ( ( R  e.  Fin  /\  ( 1 ... n
)  e.  _V )  ->  ( f  e.  ( R  ^m  ( 1 ... n ) )  <-> 
f : ( 1 ... n ) --> R ) )
86, 4, 7sylancl 644 . . . . . 6  |-  ( ( ( R  e.  Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  ->  ( f  e.  ( R  ^m  (
1 ... n ) )  <-> 
f : ( 1 ... n ) --> R ) )
98biimpa 471 . . . . 5  |-  ( ( ( ( R  e. 
Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  /\  f  e.  ( R  ^m  (
1 ... n ) ) )  ->  f :
( 1 ... n
) --> R )
10 simplr 732 . . . . . . 7  |-  ( ( ( ( R  e. 
Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  /\  f  e.  ( R  ^m  (
1 ... n ) ) )  ->  n  e.  NN )
11 nnuz 10454 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
1210, 11syl6eleq 2478 . . . . . 6  |-  ( ( ( ( R  e. 
Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  /\  f  e.  ( R  ^m  (
1 ... n ) ) )  ->  n  e.  ( ZZ>= `  1 )
)
13 eluzfz1 10997 . . . . . 6  |-  ( n  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... n
) )
1412, 13syl 16 . . . . 5  |-  ( ( ( ( R  e. 
Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  /\  f  e.  ( R  ^m  (
1 ... n ) ) )  ->  1  e.  ( 1 ... n
) )
154, 5, 9, 14vdwmc2 13275 . . . 4  |-  ( ( ( ( R  e. 
Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  /\  f  e.  ( R  ^m  (
1 ... n ) ) )  ->  ( 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 } ) ) )
1615ralbidva 2666 . . 3  |-  ( ( ( R  e.  Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  ->  ( A. f  e.  ( R  ^m  (
1 ... n ) ) K MonoAP  f  <->  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 } ) ) )
1716rexbidva 2667 . 2  |-  ( ( R  e.  Fin  /\  K  e.  NN0 )  -> 
( E. n  e.  NN  A. f  e.  ( R  ^m  (
1 ... n ) ) K MonoAP  f  <->  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 } ) ) )
183, 17mpbid 202 1  |-  ( ( 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 } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1717   A.wral 2650   E.wrex 2651   _Vcvv 2900   {csn 3758   class class class wbr 4154   `'ccnv 4818   "cima 4822   -->wf 5391   ` cfv 5395  (class class class)co 6021    ^m cmap 6955   Fincfn 7046   0cc0 8924   1c1 8925    + caddc 8927    x. cmul 8929    - cmin 9224   NNcn 9933   NN0cn0 10154   ZZ>=cuz 10421   ...cfz 10976   MonoAP cvdwm 13262
This theorem is referenced by:  vdwnnlem1  13291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-fz 10977  df-hash 11547  df-vdwap 13264  df-vdwmc 13265  df-vdwpc 13266
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