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Theorem vdw 14060
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 457 . . 3  |-  ( ( R  e.  Fin  /\  K  e.  NN0 )  ->  R  e.  Fin )
2 simpr 461 . . 3  |-  ( ( R  e.  Fin  /\  K  e.  NN0 )  ->  K  e.  NN0 )
31, 2vdwlem13 14059 . 2  |-  ( ( R  e.  Fin  /\  K  e.  NN0 )  ->  E. n  e.  NN  A. f  e.  ( R  ^m  ( 1 ... n ) ) K MonoAP 
f )
4 ovex 6121 . . . . 5  |-  ( 1 ... n )  e. 
_V
5 simpllr 758 . . . . 5  |-  ( ( ( ( R  e. 
Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  /\  f  e.  ( R  ^m  (
1 ... n ) ) )  ->  K  e.  NN0 )
6 simpll 753 . . . . . . 7  |-  ( ( ( R  e.  Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  ->  R  e.  Fin )
7 elmapg 7232 . . . . . . 7  |-  ( ( R  e.  Fin  /\  ( 1 ... n
)  e.  _V )  ->  ( f  e.  ( R  ^m  ( 1 ... n ) )  <-> 
f : ( 1 ... n ) --> R ) )
86, 4, 7sylancl 662 . . . . . 6  |-  ( ( ( R  e.  Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  ->  ( f  e.  ( R  ^m  (
1 ... n ) )  <-> 
f : ( 1 ... n ) --> R ) )
98biimpa 484 . . . . 5  |-  ( ( ( ( R  e. 
Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  /\  f  e.  ( R  ^m  (
1 ... n ) ) )  ->  f :
( 1 ... n
) --> R )
10 simplr 754 . . . . . . 7  |-  ( ( ( ( R  e. 
Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  /\  f  e.  ( R  ^m  (
1 ... n ) ) )  ->  n  e.  NN )
11 nnuz 10901 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
1210, 11syl6eleq 2533 . . . . . 6  |-  ( ( ( ( R  e. 
Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  /\  f  e.  ( R  ^m  (
1 ... n ) ) )  ->  n  e.  ( ZZ>= `  1 )
)
13 eluzfz1 11463 . . . . . 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 14045 . . . 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 2736 . . 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 2737 . 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 210 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1756   A.wral 2720   E.wrex 2721   _Vcvv 2977   {csn 3882   class class class wbr 4297   `'ccnv 4844   "cima 4848   -->wf 5419   ` cfv 5423  (class class class)co 6096    ^m cmap 7219   Fincfn 7315   0cc0 9287   1c1 9288    + caddc 9290    x. cmul 9292    - cmin 9600   NNcn 10327   NN0cn0 10584   ZZ>=cuz 10866   ...cfz 11442   MonoAP cvdwm 14032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fz 11443  df-hash 12109  df-vdwap 14034  df-vdwmc 14035  df-vdwpc 14036
This theorem is referenced by:  vdwnnlem1  14061
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