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Theorem vdw 13003
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 445 . . 3  |-  ( ( R  e.  Fin  /\  K  e.  NN0 )  ->  R  e.  Fin )
2 simpr 449 . . 3  |-  ( ( R  e.  Fin  /\  K  e.  NN0 )  ->  K  e.  NN0 )
31, 2vdwlem13 13002 . 2  |-  ( ( R  e.  Fin  /\  K  e.  NN0 )  ->  E. n  e.  NN  A. f  e.  ( R  ^m  ( 1 ... n ) ) K MonoAP 
f )
4 ovex 5817 . . . . 5  |-  ( 1 ... n )  e. 
_V
5 simpllr 738 . . . . 5  |-  ( ( ( ( R  e. 
Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  /\  f  e.  ( R  ^m  (
1 ... n ) ) )  ->  K  e.  NN0 )
6 simpll 733 . . . . . . 7  |-  ( ( ( R  e.  Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  ->  R  e.  Fin )
7 elmapg 6753 . . . . . . 7  |-  ( ( R  e.  Fin  /\  ( 1 ... n
)  e.  _V )  ->  ( f  e.  ( R  ^m  ( 1 ... n ) )  <-> 
f : ( 1 ... n ) --> R ) )
86, 4, 7sylancl 646 . . . . . 6  |-  ( ( ( R  e.  Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  ->  ( f  e.  ( R  ^m  (
1 ... n ) )  <-> 
f : ( 1 ... n ) --> R ) )
98biimpa 472 . . . . 5  |-  ( ( ( ( R  e. 
Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  /\  f  e.  ( R  ^m  (
1 ... n ) ) )  ->  f :
( 1 ... n
) --> R )
10 simplr 734 . . . . . . 7  |-  ( ( ( ( R  e. 
Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  /\  f  e.  ( R  ^m  (
1 ... n ) ) )  ->  n  e.  NN )
11 nnuz 10230 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
1210, 11syl6eleq 2348 . . . . . 6  |-  ( ( ( ( R  e. 
Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  /\  f  e.  ( R  ^m  (
1 ... n ) ) )  ->  n  e.  ( ZZ>= `  1 )
)
13 eluzfz1 10769 . . . . . 6  |-  ( n  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... n
) )
1412, 13syl 17 . . . . 5  |-  ( ( ( ( R  e. 
Fin  /\  K  e.  NN0 )  /\  n  e.  NN )  /\  f  e.  ( R  ^m  (
1 ... n ) ) )  ->  1  e.  ( 1 ... n
) )
154, 5, 9, 14vdwmc2 12988 . . . 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 2534 . . 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 2535 . 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 203 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 6    <-> wb 178    /\ wa 360    e. wcel 1621   A.wral 2518   E.wrex 2519   _Vcvv 2763   {csn 3614   class class class wbr 3997   `'ccnv 4660   "cima 4664   -->wf 4669   ` cfv 4673  (class class class)co 5792    ^m cmap 6740   Fincfn 6831   0cc0 8705   1c1 8706    + caddc 8708    x. cmul 8710    - cmin 9005   NNcn 9714   NN0cn0 9932   ZZ>=cuz 10197   ...cfz 10748   MonoAP cvdwm 12975
This theorem is referenced by:  vdwnnlem1  13004
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-er 6628  df-map 6742  df-pm 6743  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-card 7540  df-cda 7762  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-2 9772  df-n0 9933  df-z 9992  df-uz 10198  df-rp 10322  df-fz 10749  df-hash 11304  df-vdwap 12977  df-vdwmc 12978  df-vdwpc 12979
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