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Theorem vdwmc 14508
Description: The predicate " The  <. R ,  N >.-coloring  F contains a monochromatic AP of length 
K". (Contributed by Mario Carneiro, 18-Aug-2014.)
Hypotheses
Ref Expression
vdwmc.1  |-  X  e. 
_V
vdwmc.2  |-  ( ph  ->  K  e.  NN0 )
vdwmc.3  |-  ( ph  ->  F : X --> R )
Assertion
Ref Expression
vdwmc  |-  ( ph  ->  ( K MonoAP  F  <->  E. c E. a  e.  NN  E. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } ) ) )
Distinct variable groups:    a, c,
d, F    K, a,
c, d    ph, c
Allowed substitution hints:    ph( a, d)    R( a, c, d)    X( a, c, d)

Proof of Theorem vdwmc
Dummy variables  f 
k  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vdwmc.2 . . 3  |-  ( ph  ->  K  e.  NN0 )
2 vdwmc.3 . . . 4  |-  ( ph  ->  F : X --> R )
3 vdwmc.1 . . . 4  |-  X  e. 
_V
4 fex 6146 . . . 4  |-  ( ( F : X --> R  /\  X  e.  _V )  ->  F  e.  _V )
52, 3, 4sylancl 662 . . 3  |-  ( ph  ->  F  e.  _V )
6 fveq2 5872 . . . . . . . 8  |-  ( k  =  K  ->  (AP `  k )  =  (AP
`  K ) )
76rneqd 5240 . . . . . . 7  |-  ( k  =  K  ->  ran  (AP `  k )  =  ran  (AP `  K
) )
8 cnveq 5186 . . . . . . . . 9  |-  ( f  =  F  ->  `' f  =  `' F
)
98imaeq1d 5346 . . . . . . . 8  |-  ( f  =  F  ->  ( `' f " {
c } )  =  ( `' F " { c } ) )
109pweqd 4020 . . . . . . 7  |-  ( f  =  F  ->  ~P ( `' f " {
c } )  =  ~P ( `' F " { c } ) )
117, 10ineqan12d 3698 . . . . . 6  |-  ( ( k  =  K  /\  f  =  F )  ->  ( ran  (AP `  k )  i^i  ~P ( `' f " {
c } ) )  =  ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) ) )
1211neeq1d 2734 . . . . 5  |-  ( ( k  =  K  /\  f  =  F )  ->  ( ( ran  (AP `  k )  i^i  ~P ( `' f " {
c } ) )  =/=  (/)  <->  ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) )  =/=  (/) ) )
1312exbidv 1715 . . . 4  |-  ( ( k  =  K  /\  f  =  F )  ->  ( E. c ( ran  (AP `  k
)  i^i  ~P ( `' f " {
c } ) )  =/=  (/)  <->  E. c ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) )  =/=  (/) ) )
14 df-vdwmc 14499 . . . 4  |- MonoAP  =  { <. k ,  f >.  |  E. c ( ran  (AP `  k )  i^i  ~P ( `' f " { c } ) )  =/=  (/) }
1513, 14brabga 4770 . . 3  |-  ( ( K  e.  NN0  /\  F  e.  _V )  ->  ( K MonoAP  F  <->  E. c
( ran  (AP `  K
)  i^i  ~P ( `' F " { c } ) )  =/=  (/) ) )
161, 5, 15syl2anc 661 . 2  |-  ( ph  ->  ( K MonoAP  F  <->  E. c
( ran  (AP `  K
)  i^i  ~P ( `' F " { c } ) )  =/=  (/) ) )
17 vdwapf 14502 . . . . 5  |-  ( K  e.  NN0  ->  (AP `  K ) : ( NN  X.  NN ) --> ~P NN )
18 ffn 5737 . . . . 5  |-  ( (AP
`  K ) : ( NN  X.  NN )
--> ~P NN  ->  (AP `  K )  Fn  ( NN  X.  NN ) )
19 selpw 4022 . . . . . . 7  |-  ( z  e.  ~P ( `' F " { c } )  <->  z  C_  ( `' F " { c } ) )
20 sseq1 3520 . . . . . . 7  |-  ( z  =  ( (AP `  K ) `  w
)  ->  ( z  C_  ( `' F " { c } )  <-> 
( (AP `  K
) `  w )  C_  ( `' F " { c } ) ) )
2119, 20syl5bb 257 . . . . . 6  |-  ( z  =  ( (AP `  K ) `  w
)  ->  ( z  e.  ~P ( `' F " { c } )  <-> 
( (AP `  K
) `  w )  C_  ( `' F " { c } ) ) )
2221rexrn 6034 . . . . 5  |-  ( (AP
`  K )  Fn  ( NN  X.  NN )  ->  ( E. z  e.  ran  (AP `  K
) z  e.  ~P ( `' F " { c } )  <->  E. w  e.  ( NN  X.  NN ) ( (AP `  K ) `  w
)  C_  ( `' F " { c } ) ) )
231, 17, 18, 224syl 21 . . . 4  |-  ( ph  ->  ( E. z  e. 
ran  (AP `  K
) z  e.  ~P ( `' F " { c } )  <->  E. w  e.  ( NN  X.  NN ) ( (AP `  K ) `  w
)  C_  ( `' F " { c } ) ) )
24 elin 3683 . . . . . 6  |-  ( z  e.  ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) )  <->  ( z  e.  ran  (AP `  K
)  /\  z  e.  ~P ( `' F " { c } ) ) )
2524exbii 1668 . . . . 5  |-  ( E. z  z  e.  ( ran  (AP `  K
)  i^i  ~P ( `' F " { c } ) )  <->  E. z
( z  e.  ran  (AP `  K )  /\  z  e.  ~P ( `' F " { c } ) ) )
26 n0 3803 . . . . 5  |-  ( ( ran  (AP `  K
)  i^i  ~P ( `' F " { c } ) )  =/=  (/) 
<->  E. z  z  e.  ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) ) )
27 df-rex 2813 . . . . 5  |-  ( E. z  e.  ran  (AP `  K ) z  e. 
~P ( `' F " { c } )  <->  E. z ( z  e. 
ran  (AP `  K
)  /\  z  e.  ~P ( `' F " { c } ) ) )
2825, 26, 273bitr4ri 278 . . . 4  |-  ( E. z  e.  ran  (AP `  K ) z  e. 
~P ( `' F " { c } )  <-> 
( ran  (AP `  K
)  i^i  ~P ( `' F " { c } ) )  =/=  (/) )
29 fveq2 5872 . . . . . . 7  |-  ( w  =  <. a ,  d
>.  ->  ( (AP `  K ) `  w
)  =  ( (AP
`  K ) `  <. a ,  d >.
) )
30 df-ov 6299 . . . . . . 7  |-  ( a (AP `  K ) d )  =  ( (AP `  K ) `
 <. a ,  d
>. )
3129, 30syl6eqr 2516 . . . . . 6  |-  ( w  =  <. a ,  d
>.  ->  ( (AP `  K ) `  w
)  =  ( a (AP `  K ) d ) )
3231sseq1d 3526 . . . . 5  |-  ( w  =  <. a ,  d
>.  ->  ( ( (AP
`  K ) `  w )  C_  ( `' F " { c } )  <->  ( a
(AP `  K )
d )  C_  ( `' F " { c } ) ) )
3332rexxp 5155 . . . 4  |-  ( E. w  e.  ( NN 
X.  NN ) ( (AP `  K ) `
 w )  C_  ( `' F " { c } )  <->  E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) )
3423, 28, 333bitr3g 287 . . 3  |-  ( ph  ->  ( ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) )  =/=  (/) 
<->  E. a  e.  NN  E. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } ) ) )
3534exbidv 1715 . 2  |-  ( ph  ->  ( E. c ( ran  (AP `  K
)  i^i  ~P ( `' F " { c } ) )  =/=  (/) 
<->  E. c E. a  e.  NN  E. d  e.  NN  ( a (AP
`  K ) d )  C_  ( `' F " { c } ) ) )
3616, 35bitrd 253 1  |-  ( ph  ->  ( K MonoAP  F  <->  E. c E. a  e.  NN  E. d  e.  NN  (
a (AP `  K
) d )  C_  ( `' F " { c } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   E.wrex 2808   _Vcvv 3109    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   {csn 4032   <.cop 4038   class class class wbr 4456    X. cxp 5006   `'ccnv 5007   ran crn 5009   "cima 5011    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   NNcn 10556   NN0cn0 10816  APcvdwa 14495   MonoAP cvdwm 14496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-vdwap 14498  df-vdwmc 14499
This theorem is referenced by:  vdwmc2  14509  vdwlem1  14511  vdwlem2  14512  vdwlem9  14519  vdwlem10  14520  vdwlem12  14522  vdwlem13  14523
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