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Theorem vdwpc 13303
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 5928 . . 3  |-  ( ( F : X --> R  /\  X  e.  _V )  ->  F  e.  _V )
63, 4, 5sylancl 644 . 2  |-  ( ph  ->  F  e.  _V )
7 df-br 4173 . . . 4  |-  ( <. M ,  K >. PolyAP  F  <->  <. <. M ,  K >. ,  F >.  e. PolyAP  )
8 df-vdwpc 13293 . . . . 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 2468 . . . 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 6056 . . . . . . . 8  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( 1 ... m
)  =  ( 1 ... M ) )
13 vdwpc.5 . . . . . . . 8  |-  J  =  ( 1 ... M
)
1412, 13syl6eqr 2454 . . . . . . 7  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( 1 ... m
)  =  J )
1514oveq2d 6056 . . . . . 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 5691 . . . . . . . . . 10  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  (AP `  k )  =  (AP `  K
) )
1817oveqd 6057 . . . . . . . . 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 5007 . . . . . . . . . 10  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  `' f  =  `' F )
2119fveq1d 5689 . . . . . . . . . . 11  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( f `  (
a  +  ( d `
 i ) ) )  =  ( F `
 ( a  +  ( d `  i
) ) ) )
2221sneqd 3787 . . . . . . . . . 10  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  { ( f `  ( a  +  ( d `  i ) ) ) }  =  { ( F `  ( a  +  ( d `  i ) ) ) } )
2320, 22imaeq12d 5163 . . . . . . . . 9  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( `' f " { ( f `  ( a  +  ( d `  i ) ) ) } )  =  ( `' F " { ( F `  ( a  +  ( d `  i ) ) ) } ) )
2418, 23sseq12d 3337 . . . . . . . 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 2876 . . . . . . 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 4247 . . . . . . . . . 10  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( i  e.  ( 1 ... m ) 
|->  ( f `  (
a  +  ( d `
 i ) ) ) )  =  ( i  e.  J  |->  ( F `  ( a  +  ( d `  i ) ) ) ) )
2726rneqd 5056 . . . . . . . . 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 5691 . . . . . . . 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 2418 . . . . . . 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 2877 . . . . 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 2687 . . . 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 6119 . . 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 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916    C_ wss 3280   {csn 3774   <.cop 3777   class class class wbr 4172    e. cmpt 4226   `'ccnv 4836   ran crn 4838   "cima 4840   -->wf 5409   ` cfv 5413  (class class class)co 6040   {coprab 6041    ^m cmap 6977   1c1 8947    + caddc 8949   NNcn 9956   NN0cn0 10177   ...cfz 10999   #chash 11573  APcvdwa 13288   PolyAP cvdwp 13290
This theorem is referenced by:  vdwlem6  13309  vdwlem7  13310  vdwlem8  13311  vdwlem11  13314
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-vdwpc 13293
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