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Theorem vdwmc 15007
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 6155 . . . 4  |-  ( ( F : X --> R  /\  X  e.  _V )  ->  F  e.  _V )
52, 3, 4sylancl 675 . . 3  |-  ( ph  ->  F  e.  _V )
6 fveq2 5879 . . . . . . . 8  |-  ( k  =  K  ->  (AP `  k )  =  (AP
`  K ) )
76rneqd 5068 . . . . . . 7  |-  ( k  =  K  ->  ran  (AP `  k )  =  ran  (AP `  K
) )
8 cnveq 5013 . . . . . . . . 9  |-  ( f  =  F  ->  `' f  =  `' F
)
98imaeq1d 5173 . . . . . . . 8  |-  ( f  =  F  ->  ( `' f " {
c } )  =  ( `' F " { c } ) )
109pweqd 3947 . . . . . . 7  |-  ( f  =  F  ->  ~P ( `' f " {
c } )  =  ~P ( `' F " { c } ) )
117, 10ineqan12d 3627 . . . . . 6  |-  ( ( k  =  K  /\  f  =  F )  ->  ( ran  (AP `  k )  i^i  ~P ( `' f " {
c } ) )  =  ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) ) )
1211neeq1d 2702 . . . . 5  |-  ( ( k  =  K  /\  f  =  F )  ->  ( ( ran  (AP `  k )  i^i  ~P ( `' f " {
c } ) )  =/=  (/)  <->  ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) )  =/=  (/) ) )
1312exbidv 1776 . . . 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 14998 . . . 4  |- MonoAP  =  { <. k ,  f >.  |  E. c ( ran  (AP `  k )  i^i  ~P ( `' f " { c } ) )  =/=  (/) }
1513, 14brabga 4715 . . 3  |-  ( ( K  e.  NN0  /\  F  e.  _V )  ->  ( K MonoAP  F  <->  E. c
( ran  (AP `  K
)  i^i  ~P ( `' F " { c } ) )  =/=  (/) ) )
161, 5, 15syl2anc 673 . 2  |-  ( ph  ->  ( K MonoAP  F  <->  E. c
( ran  (AP `  K
)  i^i  ~P ( `' F " { c } ) )  =/=  (/) ) )
17 vdwapf 15001 . . . . 5  |-  ( K  e.  NN0  ->  (AP `  K ) : ( NN  X.  NN ) --> ~P NN )
18 ffn 5739 . . . . 5  |-  ( (AP
`  K ) : ( NN  X.  NN )
--> ~P NN  ->  (AP `  K )  Fn  ( NN  X.  NN ) )
19 selpw 3949 . . . . . . 7  |-  ( z  e.  ~P ( `' F " { c } )  <->  z  C_  ( `' F " { c } ) )
20 sseq1 3439 . . . . . . 7  |-  ( z  =  ( (AP `  K ) `  w
)  ->  ( z  C_  ( `' F " { c } )  <-> 
( (AP `  K
) `  w )  C_  ( `' F " { c } ) ) )
2119, 20syl5bb 265 . . . . . 6  |-  ( z  =  ( (AP `  K ) `  w
)  ->  ( z  e.  ~P ( `' F " { c } )  <-> 
( (AP `  K
) `  w )  C_  ( `' F " { c } ) ) )
2221rexrn 6039 . . . . 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 19 . . . 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 3608 . . . . . 6  |-  ( z  e.  ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) )  <->  ( z  e.  ran  (AP `  K
)  /\  z  e.  ~P ( `' F " { c } ) ) )
2524exbii 1726 . . . . 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 3732 . . . . 5  |-  ( ( ran  (AP `  K
)  i^i  ~P ( `' F " { c } ) )  =/=  (/) 
<->  E. z  z  e.  ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) ) )
27 df-rex 2762 . . . . 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 286 . . . 4  |-  ( E. z  e.  ran  (AP `  K ) z  e. 
~P ( `' F " { c } )  <-> 
( ran  (AP `  K
)  i^i  ~P ( `' F " { c } ) )  =/=  (/) )
29 fveq2 5879 . . . . . . 7  |-  ( w  =  <. a ,  d
>.  ->  ( (AP `  K ) `  w
)  =  ( (AP
`  K ) `  <. a ,  d >.
) )
30 df-ov 6311 . . . . . . 7  |-  ( a (AP `  K ) d )  =  ( (AP `  K ) `
 <. a ,  d
>. )
3129, 30syl6eqr 2523 . . . . . 6  |-  ( w  =  <. a ,  d
>.  ->  ( (AP `  K ) `  w
)  =  ( a (AP `  K ) d ) )
3231sseq1d 3445 . . . . 5  |-  ( w  =  <. a ,  d
>.  ->  ( ( (AP
`  K ) `  w )  C_  ( `' F " { c } )  <->  ( a
(AP `  K )
d )  C_  ( `' F " { c } ) ) )
3332rexxp 4982 . . . 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 295 . . 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 1776 . 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 261 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 189    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641   E.wrex 2757   _Vcvv 3031    i^i cin 3389    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   {csn 3959   <.cop 3965   class class class wbr 4395    X. cxp 4837   `'ccnv 4838   ran crn 4840   "cima 4842    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   NNcn 10631   NN0cn0 10893  APcvdwa 14994   MonoAP cvdwm 14995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-vdwap 14997  df-vdwmc 14998
This theorem is referenced by:  vdwmc2  15008  vdwlem1  15010  vdwlem2  15011  vdwlem9  15018  vdwlem10  15019  vdwlem12  15021  vdwlem13  15022
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