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Theorem vdwmc 14044
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 5955 . . . 4  |-  ( ( F : X --> R  /\  X  e.  _V )  ->  F  e.  _V )
52, 3, 4sylancl 662 . . 3  |-  ( ph  ->  F  e.  _V )
6 fveq2 5696 . . . . . . . 8  |-  ( k  =  K  ->  (AP `  k )  =  (AP
`  K ) )
76rneqd 5072 . . . . . . 7  |-  ( k  =  K  ->  ran  (AP `  k )  =  ran  (AP `  K
) )
8 cnveq 5018 . . . . . . . . 9  |-  ( f  =  F  ->  `' f  =  `' F
)
98imaeq1d 5173 . . . . . . . 8  |-  ( f  =  F  ->  ( `' f " {
c } )  =  ( `' F " { c } ) )
109pweqd 3870 . . . . . . 7  |-  ( f  =  F  ->  ~P ( `' f " {
c } )  =  ~P ( `' F " { c } ) )
117, 10ineqan12d 3559 . . . . . 6  |-  ( ( k  =  K  /\  f  =  F )  ->  ( ran  (AP `  k )  i^i  ~P ( `' f " {
c } ) )  =  ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) ) )
1211neeq1d 2626 . . . . 5  |-  ( ( k  =  K  /\  f  =  F )  ->  ( ( ran  (AP `  k )  i^i  ~P ( `' f " {
c } ) )  =/=  (/)  <->  ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) )  =/=  (/) ) )
1312exbidv 1680 . . . 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 14035 . . . 4  |- MonoAP  =  { <. k ,  f >.  |  E. c ( ran  (AP `  k )  i^i  ~P ( `' f " { c } ) )  =/=  (/) }
1513, 14brabga 4608 . . 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 14038 . . . . 5  |-  ( K  e.  NN0  ->  (AP `  K ) : ( NN  X.  NN ) --> ~P NN )
18 ffn 5564 . . . . 5  |-  ( (AP
`  K ) : ( NN  X.  NN )
--> ~P NN  ->  (AP `  K )  Fn  ( NN  X.  NN ) )
19 selpw 3872 . . . . . . 7  |-  ( z  e.  ~P ( `' F " { c } )  <->  z  C_  ( `' F " { c } ) )
20 sseq1 3382 . . . . . . 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 5850 . . . . 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 3544 . . . . . 6  |-  ( z  e.  ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) )  <->  ( z  e.  ran  (AP `  K
)  /\  z  e.  ~P ( `' F " { c } ) ) )
2524exbii 1634 . . . . 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 3651 . . . . 5  |-  ( ( ran  (AP `  K
)  i^i  ~P ( `' F " { c } ) )  =/=  (/) 
<->  E. z  z  e.  ( ran  (AP `  K )  i^i  ~P ( `' F " { c } ) ) )
27 df-rex 2726 . . . . 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 5696 . . . . . . 7  |-  ( w  =  <. a ,  d
>.  ->  ( (AP `  K ) `  w
)  =  ( (AP
`  K ) `  <. a ,  d >.
) )
30 df-ov 6099 . . . . . . 7  |-  ( a (AP `  K ) d )  =  ( (AP `  K ) `
 <. a ,  d
>. )
3129, 30syl6eqr 2493 . . . . . 6  |-  ( w  =  <. a ,  d
>.  ->  ( (AP `  K ) `  w
)  =  ( a (AP `  K ) d ) )
3231sseq1d 3388 . . . . 5  |-  ( w  =  <. a ,  d
>.  ->  ( ( (AP
`  K ) `  w )  C_  ( `' F " { c } )  <->  ( a
(AP `  K )
d )  C_  ( `' F " { c } ) ) )
3332rexxp 4987 . . . 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 1680 . 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 1369   E.wex 1586    e. wcel 1756    =/= wne 2611   E.wrex 2721   _Vcvv 2977    i^i cin 3332    C_ wss 3333   (/)c0 3642   ~Pcpw 3865   {csn 3882   <.cop 3888   class class class wbr 4297    X. cxp 4843   `'ccnv 4844   ran crn 4846   "cima 4848    Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096   NNcn 10327   NN0cn0 10584  APcvdwa 14031   MonoAP cvdwm 14032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-vdwap 14034  df-vdwmc 14035
This theorem is referenced by:  vdwmc2  14045  vdwlem1  14047  vdwlem2  14048  vdwlem9  14055  vdwlem10  14056  vdwlem12  14058  vdwlem13  14059
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