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Theorem vdwpc 13029
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 5751 . . 3  |-  ( ( F : X --> R  /\  X  e.  _V )  ->  F  e.  _V )
63, 4, 5sylancl 643 . 2  |-  ( ph  ->  F  e.  _V )
7 df-br 4026 . . . 4  |-  ( <. M ,  K >. PolyAP  F  <->  <. <. M ,  K >. ,  F >.  e. PolyAP  )
8 df-vdwpc 13019 . . . . 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 2349 . . . 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 240 . . 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 955 . . . . . . . . 9  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  m  =  M )
1211oveq2d 5876 . . . . . . . 8  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( 1 ... m
)  =  ( 1 ... M ) )
13 vdwpc.5 . . . . . . . 8  |-  J  =  ( 1 ... M
)
1412, 13syl6eqr 2335 . . . . . . 7  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( 1 ... m
)  =  J )
1514oveq2d 5876 . . . . . 6  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( NN  ^m  (
1 ... m ) )  =  ( NN  ^m  J ) )
16 simp2 956 . . . . . . . . . . 11  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  k  =  K )
1716fveq2d 5531 . . . . . . . . . 10  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  (AP `  k )  =  (AP `  K
) )
1817oveqd 5877 . . . . . . . . 9  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( ( a  +  ( d `  i
) ) (AP `  k ) ( d `
 i ) )  =  ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) ) )
19 simp3 957 . . . . . . . . . . . 12  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  f  =  F )
2019cnveqd 4859 . . . . . . . . . . 11  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  `' f  =  `' F )
2120imaeq1d 5013 . . . . . . . . . 10  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( `' f " { ( f `  ( a  +  ( d `  i ) ) ) } )  =  ( `' F " { ( f `  ( a  +  ( d `  i ) ) ) } ) )
2219fveq1d 5529 . . . . . . . . . . . 12  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( f `  (
a  +  ( d `
 i ) ) )  =  ( F `
 ( a  +  ( d `  i
) ) ) )
2322sneqd 3655 . . . . . . . . . . 11  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  { ( f `  ( a  +  ( d `  i ) ) ) }  =  { ( F `  ( a  +  ( d `  i ) ) ) } )
2423imaeq2d 5014 . . . . . . . . . 10  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( `' F " { ( f `  ( a  +  ( d `  i ) ) ) } )  =  ( `' F " { ( F `  ( a  +  ( d `  i ) ) ) } ) )
2521, 24eqtrd 2317 . . . . . . . . 9  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( `' f " { ( f `  ( a  +  ( d `  i ) ) ) } )  =  ( `' F " { ( F `  ( a  +  ( d `  i ) ) ) } ) )
2618, 25sseq12d 3209 . . . . . . . 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
) ) ) } ) ) )
2714, 26raleqbidv 2750 . . . . . . 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 ) ) ) } ) ) )
2814, 22mpteq12dv 4100 . . . . . . . . . 10  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( i  e.  ( 1 ... m ) 
|->  ( f `  (
a  +  ( d `
 i ) ) ) )  =  ( i  e.  J  |->  ( F `  ( a  +  ( d `  i ) ) ) ) )
2928rneqd 4908 . . . . . . . . 9  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ran  ( i  e.  ( 1 ... m
)  |->  ( f `  ( a  +  ( d `  i ) ) ) )  =  ran  ( i  e.  J  |->  ( F `  ( a  +  ( d `  i ) ) ) ) )
3029fveq2d 5531 . . . . . . . 8  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( # `  ran  ( i  e.  ( 1 ... m ) 
|->  ( f `  (
a  +  ( d `
 i ) ) ) ) )  =  ( # `  ran  ( i  e.  J  |->  ( F `  (
a  +  ( d `
 i ) ) ) ) ) )
3130, 11eqeq12d 2299 . . . . . . 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 ) )
3227, 31anbi12d 691 . . . . . 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 ) ) )
3315, 32rexeqbidv 2751 . . . . 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 ) ) )
3433rexbidv 2566 . . . 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 ) ) )
3534eloprabga 5936 . . 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 ) ) )
3610, 35syl5bb 248 . 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 ) ) )
371, 2, 6, 36syl3anc 1182 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 176    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686   A.wral 2545   E.wrex 2546   _Vcvv 2790    C_ wss 3154   {csn 3642   <.cop 3645   class class class wbr 4025    e. cmpt 4079   `'ccnv 4690   ran crn 4692   "cima 4694   -->wf 5253   ` cfv 5257  (class class class)co 5860   {coprab 5861    ^m cmap 6774   1c1 8740    + caddc 8742   NNcn 9748   NN0cn0 9967   ...cfz 10784   #chash 11339  APcvdwa 13014   PolyAP cvdwp 13016
This theorem is referenced by:  vdwlem6  13035  vdwlem7  13036  vdwlem8  13037  vdwlem11  13040
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-vdwpc 13019
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