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Theorem vdwmc 14367
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 6143 . . . 4  |-  ( ( F : X --> R  /\  X  e.  _V )  ->  F  e.  _V )
52, 3, 4sylancl 662 . . 3  |-  ( ph  ->  F  e.  _V )
6 fveq2 5871 . . . . . . . 8  |-  ( k  =  K  ->  (AP `  k )  =  (AP
`  K ) )
76rneqd 5235 . . . . . . 7  |-  ( k  =  K  ->  ran  (AP `  k )  =  ran  (AP `  K
) )
8 cnveq 5181 . . . . . . . . 9  |-  ( f  =  F  ->  `' f  =  `' F
)
98imaeq1d 5341 . . . . . . . 8  |-  ( f  =  F  ->  ( `' f " {
c } )  =  ( `' F " { c } ) )
109pweqd 4020 . . . . . . 7  |-  ( f  =  F  ->  ~P ( `' f " {
c } )  =  ~P ( `' F " { c } ) )
117, 10ineqan12d 3707 . . . . . 6  |-  ( ( k  =  K  /\  f  =  F )  ->  ( ran  (AP `  k )  i^i  ~P ( `' f " {
c } ) )  =  ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) ) )
1211neeq1d 2744 . . . . 5  |-  ( ( k  =  K  /\  f  =  F )  ->  ( ( ran  (AP `  k )  i^i  ~P ( `' f " {
c } ) )  =/=  (/)  <->  ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) )  =/=  (/) ) )
1312exbidv 1690 . . . 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 14358 . . . 4  |- MonoAP  =  { <. k ,  f >.  |  E. c ( ran  (AP `  k )  i^i  ~P ( `' f " { c } ) )  =/=  (/) }
1513, 14brabga 4766 . . 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 14361 . . . . 5  |-  ( K  e.  NN0  ->  (AP `  K ) : ( NN  X.  NN ) --> ~P NN )
18 ffn 5736 . . . . 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 3530 . . . . . . 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 6033 . . . . 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 3692 . . . . . 6  |-  ( z  e.  ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) )  <->  ( z  e.  ran  (AP `  K
)  /\  z  e.  ~P ( `' F " { c } ) ) )
2524exbii 1644 . . . . 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 3799 . . . . 5  |-  ( ( ran  (AP `  K
)  i^i  ~P ( `' F " { c } ) )  =/=  (/) 
<->  E. z  z  e.  ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) ) )
27 df-rex 2823 . . . . 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 5871 . . . . . . 7  |-  ( w  =  <. a ,  d
>.  ->  ( (AP `  K ) `  w
)  =  ( (AP
`  K ) `  <. a ,  d >.
) )
30 df-ov 6297 . . . . . . 7  |-  ( a (AP `  K ) d )  =  ( (AP `  K ) `
 <. a ,  d
>. )
3129, 30syl6eqr 2526 . . . . . 6  |-  ( w  =  <. a ,  d
>.  ->  ( (AP `  K ) `  w
)  =  ( a (AP `  K ) d ) )
3231sseq1d 3536 . . . . 5  |-  ( w  =  <. a ,  d
>.  ->  ( ( (AP
`  K ) `  w )  C_  ( `' F " { c } )  <->  ( a
(AP `  K )
d )  C_  ( `' F " { c } ) ) )
3332rexxp 5150 . . . 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 1690 . 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 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   E.wrex 2818   _Vcvv 3118    i^i cin 3480    C_ wss 3481   (/)c0 3790   ~Pcpw 4015   {csn 4032   <.cop 4038   class class class wbr 4452    X. cxp 5002   `'ccnv 5003   ran crn 5005   "cima 5007    Fn wfn 5588   -->wf 5589   ` cfv 5593  (class class class)co 6294   NNcn 10546   NN0cn0 10805  APcvdwa 14354   MonoAP cvdwm 14355
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 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-n0 10806  df-z 10875  df-uz 11093  df-fz 11683  df-vdwap 14357  df-vdwmc 14358
This theorem is referenced by:  vdwmc2  14368  vdwlem1  14370  vdwlem2  14371  vdwlem9  14378  vdwlem10  14379  vdwlem12  14381  vdwlem13  14382
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