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Theorem vdwpc 13276
Description: The predicate " The coloring  F contains a polychromatic  M-tuple of AP's of length  K". A polychromatic 
M-tuple of AP's is a set of AP's with the same base point but different step lengths, such that each individual AP is monochromatic, but the AP's all have mutually distinct colors. (The common basepoint is not required to have the same color as any of the AP's.) (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 )
vdwpc.4  |-  ( ph  ->  M  e.  NN )
vdwpc.5  |-  J  =  ( 1 ... M
)
Assertion
Ref Expression
vdwpc  |-  ( ph  ->  ( <. M ,  K >. PolyAP 
F  <->  E. a  e.  NN  E. d  e.  ( NN 
^m  J ) ( A. i  e.  J  ( ( a  +  ( d `  i
) ) (AP `  K ) ( d `
 i ) ) 
C_  ( `' F " { ( F `  ( a  +  ( d `  i ) ) ) } )  /\  ( # `  ran  ( i  e.  J  |->  ( F `  (
a  +  ( d `
 i ) ) ) ) )  =  M ) ) )
Distinct variable groups:    a, d,
i, F    K, a,
d, i    J, d,
i    M, a, d, i
Allowed substitution hints:    ph( i, a, d)    R( i, a, d)    J( a)    X( i, a, d)

Proof of Theorem vdwpc
Dummy variables  f 
k  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vdwpc.4 . 2  |-  ( ph  ->  M  e.  NN )
2 vdwmc.2 . 2  |-  ( ph  ->  K  e.  NN0 )
3 vdwmc.3 . . 3  |-  ( ph  ->  F : X --> R )
4 vdwmc.1 . . 3  |-  X  e. 
_V
5 fex 5909 . . 3  |-  ( ( F : X --> R  /\  X  e.  _V )  ->  F  e.  _V )
63, 4, 5sylancl 644 . 2  |-  ( ph  ->  F  e.  _V )
7 df-br 4155 . . . 4  |-  ( <. M ,  K >. PolyAP  F  <->  <. <. M ,  K >. ,  F >.  e. PolyAP  )
8 df-vdwpc 13266 . . . . 5  |- PolyAP  =  { <. <. m ,  k
>. ,  f >.  |  E. a  e.  NN  E. d  e.  ( NN 
^m  ( 1 ... m ) ) ( A. i  e.  ( 1 ... m ) ( ( a  +  ( d `  i
) ) (AP `  k ) ( d `
 i ) ) 
C_  ( `' f
" { ( f `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... m
)  |->  ( f `  ( a  +  ( d `  i ) ) ) ) )  =  m ) }
98eleq2i 2452 . . . 4  |-  ( <. <. M ,  K >. ,  F >.  e. PolyAP  <->  <. <. M ,  K >. ,  F >.  e. 
{ <. <. m ,  k
>. ,  f >.  |  E. a  e.  NN  E. d  e.  ( NN 
^m  ( 1 ... m ) ) ( A. i  e.  ( 1 ... m ) ( ( a  +  ( d `  i
) ) (AP `  k ) ( d `
 i ) ) 
C_  ( `' f
" { ( f `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... m
)  |->  ( f `  ( a  +  ( d `  i ) ) ) ) )  =  m ) } )
107, 9bitri 241 . . 3  |-  ( <. M ,  K >. PolyAP  F  <->  <. <. M ,  K >. ,  F >.  e.  { <. <.
m ,  k >. ,  f >.  |  E. a  e.  NN  E. d  e.  ( NN  ^m  (
1 ... m ) ) ( A. i  e.  ( 1 ... m
) ( ( a  +  ( d `  i ) ) (AP
`  k ) ( d `  i ) )  C_  ( `' f " { ( f `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... m
)  |->  ( f `  ( a  +  ( d `  i ) ) ) ) )  =  m ) } )
11 simp1 957 . . . . . . . . 9  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  m  =  M )
1211oveq2d 6037 . . . . . . . 8  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( 1 ... m
)  =  ( 1 ... M ) )
13 vdwpc.5 . . . . . . . 8  |-  J  =  ( 1 ... M
)
1412, 13syl6eqr 2438 . . . . . . 7  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( 1 ... m
)  =  J )
1514oveq2d 6037 . . . . . 6  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( NN  ^m  (
1 ... m ) )  =  ( NN  ^m  J ) )
16 simp2 958 . . . . . . . . . . 11  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  k  =  K )
1716fveq2d 5673 . . . . . . . . . 10  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  (AP `  k )  =  (AP `  K
) )
1817oveqd 6038 . . . . . . . . 9  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( ( a  +  ( d `  i
) ) (AP `  k ) ( d `
 i ) )  =  ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) ) )
19 simp3 959 . . . . . . . . . . 11  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  f  =  F )
2019cnveqd 4989 . . . . . . . . . 10  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  `' f  =  `' F )
2119fveq1d 5671 . . . . . . . . . . 11  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( f `  (
a  +  ( d `
 i ) ) )  =  ( F `
 ( a  +  ( d `  i
) ) ) )
2221sneqd 3771 . . . . . . . . . 10  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  { ( f `  ( a  +  ( d `  i ) ) ) }  =  { ( F `  ( a  +  ( d `  i ) ) ) } )
2320, 22imaeq12d 5145 . . . . . . . . 9  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( `' f " { ( f `  ( a  +  ( d `  i ) ) ) } )  =  ( `' F " { ( F `  ( a  +  ( d `  i ) ) ) } ) )
2418, 23sseq12d 3321 . . . . . . . 8  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( ( ( a  +  ( d `  i ) ) (AP
`  k ) ( d `  i ) )  C_  ( `' f " { ( f `
 ( a  +  ( d `  i
) ) ) } )  <->  ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' F " { ( F `
 ( a  +  ( d `  i
) ) ) } ) ) )
2514, 24raleqbidv 2860 . . . . . . 7  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( A. i  e.  ( 1 ... m
) ( ( a  +  ( d `  i ) ) (AP
`  k ) ( d `  i ) )  C_  ( `' f " { ( f `
 ( a  +  ( d `  i
) ) ) } )  <->  A. i  e.  J  ( ( a  +  ( d `  i
) ) (AP `  K ) ( d `
 i ) ) 
C_  ( `' F " { ( F `  ( a  +  ( d `  i ) ) ) } ) ) )
2614, 21mpteq12dv 4229 . . . . . . . . . 10  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( i  e.  ( 1 ... m ) 
|->  ( f `  (
a  +  ( d `
 i ) ) ) )  =  ( i  e.  J  |->  ( F `  ( a  +  ( d `  i ) ) ) ) )
2726rneqd 5038 . . . . . . . . 9  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ran  ( i  e.  ( 1 ... m
)  |->  ( f `  ( a  +  ( d `  i ) ) ) )  =  ran  ( i  e.  J  |->  ( F `  ( a  +  ( d `  i ) ) ) ) )
2827fveq2d 5673 . . . . . . . 8  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( # `  ran  ( i  e.  ( 1 ... m ) 
|->  ( f `  (
a  +  ( d `
 i ) ) ) ) )  =  ( # `  ran  ( i  e.  J  |->  ( F `  (
a  +  ( d `
 i ) ) ) ) ) )
2928, 11eqeq12d 2402 . . . . . . 7  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( ( # `  ran  ( i  e.  ( 1 ... m ) 
|->  ( f `  (
a  +  ( d `
 i ) ) ) ) )  =  m  <->  ( # `  ran  ( i  e.  J  |->  ( F `  (
a  +  ( d `
 i ) ) ) ) )  =  M ) )
3025, 29anbi12d 692 . . . . . 6  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( ( A. i  e.  ( 1 ... m
) ( ( a  +  ( d `  i ) ) (AP
`  k ) ( d `  i ) )  C_  ( `' f " { ( f `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... m
)  |->  ( f `  ( a  +  ( d `  i ) ) ) ) )  =  m )  <->  ( A. i  e.  J  (
( a  +  ( d `  i ) ) (AP `  K
) ( d `  i ) )  C_  ( `' F " { ( F `  ( a  +  ( d `  i ) ) ) } )  /\  ( # `
 ran  ( i  e.  J  |->  ( F `
 ( a  +  ( d `  i
) ) ) ) )  =  M ) ) )
3115, 30rexeqbidv 2861 . . . . 5  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( E. d  e.  ( NN  ^m  (
1 ... m ) ) ( A. i  e.  ( 1 ... m
) ( ( a  +  ( d `  i ) ) (AP
`  k ) ( d `  i ) )  C_  ( `' f " { ( f `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... m
)  |->  ( f `  ( a  +  ( d `  i ) ) ) ) )  =  m )  <->  E. d  e.  ( NN  ^m  J
) ( A. i  e.  J  ( (
a  +  ( d `
 i ) ) (AP `  K ) ( d `  i
) )  C_  ( `' F " { ( F `  ( a  +  ( d `  i ) ) ) } )  /\  ( # `
 ran  ( i  e.  J  |->  ( F `
 ( a  +  ( d `  i
) ) ) ) )  =  M ) ) )
3231rexbidv 2671 . . . 4  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( E. a  e.  NN  E. d  e.  ( NN  ^m  (
1 ... m ) ) ( A. i  e.  ( 1 ... m
) ( ( a  +  ( d `  i ) ) (AP
`  k ) ( d `  i ) )  C_  ( `' f " { ( f `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... m
)  |->  ( f `  ( a  +  ( d `  i ) ) ) ) )  =  m )  <->  E. a  e.  NN  E. d  e.  ( NN  ^m  J
) ( A. i  e.  J  ( (
a  +  ( d `
 i ) ) (AP `  K ) ( d `  i
) )  C_  ( `' F " { ( F `  ( a  +  ( d `  i ) ) ) } )  /\  ( # `
 ran  ( i  e.  J  |->  ( F `
 ( a  +  ( d `  i
) ) ) ) )  =  M ) ) )
3332eloprabga 6100 . . 3  |-  ( ( M  e.  NN  /\  K  e.  NN0  /\  F  e.  _V )  ->  ( <. <. M ,  K >. ,  F >.  e.  { <. <. m ,  k
>. ,  f >.  |  E. a  e.  NN  E. d  e.  ( NN 
^m  ( 1 ... m ) ) ( A. i  e.  ( 1 ... m ) ( ( a  +  ( d `  i
) ) (AP `  k ) ( d `
 i ) ) 
C_  ( `' f
" { ( f `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... m
)  |->  ( f `  ( a  +  ( d `  i ) ) ) ) )  =  m ) }  <->  E. a  e.  NN  E. d  e.  ( NN 
^m  J ) ( A. i  e.  J  ( ( a  +  ( d `  i
) ) (AP `  K ) ( d `
 i ) ) 
C_  ( `' F " { ( F `  ( a  +  ( d `  i ) ) ) } )  /\  ( # `  ran  ( i  e.  J  |->  ( F `  (
a  +  ( d `
 i ) ) ) ) )  =  M ) ) )
3410, 33syl5bb 249 . 2  |-  ( ( M  e.  NN  /\  K  e.  NN0  /\  F  e.  _V )  ->  ( <. M ,  K >. PolyAP  F  <->  E. a  e.  NN  E. d  e.  ( NN  ^m  J ) ( A. i  e.  J  (
( a  +  ( d `  i ) ) (AP `  K
) ( d `  i ) )  C_  ( `' F " { ( F `  ( a  +  ( d `  i ) ) ) } )  /\  ( # `
 ran  ( i  e.  J  |->  ( F `
 ( a  +  ( d `  i
) ) ) ) )  =  M ) ) )
351, 2, 6, 34syl3anc 1184 1  |-  ( ph  ->  ( <. M ,  K >. PolyAP 
F  <->  E. a  e.  NN  E. d  e.  ( NN 
^m  J ) ( A. i  e.  J  ( ( a  +  ( d `  i
) ) (AP `  K ) ( d `
 i ) ) 
C_  ( `' F " { ( F `  ( a  +  ( d `  i ) ) ) } )  /\  ( # `  ran  ( i  e.  J  |->  ( F `  (
a  +  ( d `
 i ) ) ) ) )  =  M ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2650   E.wrex 2651   _Vcvv 2900    C_ wss 3264   {csn 3758   <.cop 3761   class class class wbr 4154    e. cmpt 4208   `'ccnv 4818   ran crn 4820   "cima 4822   -->wf 5391   ` cfv 5395  (class class class)co 6021   {coprab 6022    ^m cmap 6955   1c1 8925    + caddc 8927   NNcn 9933   NN0cn0 10154   ...cfz 10976   #chash 11546  APcvdwa 13261   PolyAP cvdwp 13263
This theorem is referenced by:  vdwlem6  13282  vdwlem7  13283  vdwlem8  13284  vdwlem11  13287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-vdwpc 13266
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