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Theorem vdw 14367
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 14366 . 2  |-  ( ( R  e.  Fin  /\  K  e.  NN0 )  ->  E. n  e.  NN  A. f  e.  ( R  ^m  ( 1 ... n ) ) K MonoAP 
f )
4 ovex 6307 . . . . 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 7430 . . . . . . 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 11113 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
1210, 11syl6eleq 2565 . . . . . 6  |-  ( ( ( ( R  e. 
Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  /\  f  e.  ( R  ^m  (
1 ... n ) ) )  ->  n  e.  ( ZZ>= `  1 )
)
13 eluzfz1 11689 . . . . . 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 14352 . . . 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 2900 . . 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 2970 . 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 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113   {csn 4027   class class class wbr 4447   `'ccnv 4998   "cima 5002   -->wf 5582   ` cfv 5586  (class class class)co 6282    ^m cmap 7417   Fincfn 7513   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    - cmin 9801   NNcn 10532   NN0cn0 10791   ZZ>=cuz 11078   ...cfz 11668   MonoAP cvdwm 14339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-hash 12370  df-vdwap 14341  df-vdwmc 14342  df-vdwpc 14343
This theorem is referenced by:  vdwnnlem1  14368
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