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Theorem vdwpc 13340
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 5961 . . 3  |-  ( ( F : X --> R  /\  X  e.  _V )  ->  F  e.  _V )
63, 4, 5sylancl 644 . 2  |-  ( ph  ->  F  e.  _V )
7 df-br 4205 . . . 4  |-  ( <. M ,  K >. PolyAP  F  <->  <. <. M ,  K >. ,  F >.  e. PolyAP  )
8 df-vdwpc 13330 . . . . 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 2499 . . . 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 6089 . . . . . . . 8  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( 1 ... m
)  =  ( 1 ... M ) )
13 vdwpc.5 . . . . . . . 8  |-  J  =  ( 1 ... M
)
1412, 13syl6eqr 2485 . . . . . . 7  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( 1 ... m
)  =  J )
1514oveq2d 6089 . . . . . 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 5724 . . . . . . . . . 10  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  (AP `  k )  =  (AP `  K
) )
1817oveqd 6090 . . . . . . . . 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 5040 . . . . . . . . . 10  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  `' f  =  `' F )
2119fveq1d 5722 . . . . . . . . . . 11  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( f `  (
a  +  ( d `
 i ) ) )  =  ( F `
 ( a  +  ( d `  i
) ) ) )
2221sneqd 3819 . . . . . . . . . 10  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  { ( f `  ( a  +  ( d `  i ) ) ) }  =  { ( F `  ( a  +  ( d `  i ) ) ) } )
2320, 22imaeq12d 5196 . . . . . . . . 9  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( `' f " { ( f `  ( a  +  ( d `  i ) ) ) } )  =  ( `' F " { ( F `  ( a  +  ( d `  i ) ) ) } ) )
2418, 23sseq12d 3369 . . . . . . . 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 2908 . . . . . . 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 4279 . . . . . . . . . 10  |-  ( ( m  =  M  /\  k  =  K  /\  f  =  F )  ->  ( i  e.  ( 1 ... m ) 
|->  ( f `  (
a  +  ( d `
 i ) ) ) )  =  ( i  e.  J  |->  ( F `  ( a  +  ( d `  i ) ) ) ) )
2726rneqd 5089 . . . . . . . . 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 5724 . . . . . . . 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 2449 . . . . . . 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 2909 . . . . 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 2718 . . . 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 6152 . . 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 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   _Vcvv 2948    C_ wss 3312   {csn 3806   <.cop 3809   class class class wbr 4204    e. cmpt 4258   `'ccnv 4869   ran crn 4871   "cima 4873   -->wf 5442   ` cfv 5446  (class class class)co 6073   {coprab 6074    ^m cmap 7010   1c1 8983    + caddc 8985   NNcn 9992   NN0cn0 10213   ...cfz 11035   #chash 11610  APcvdwa 13325   PolyAP cvdwp 13327
This theorem is referenced by:  vdwlem6  13346  vdwlem7  13347  vdwlem8  13348  vdwlem11  13351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-vdwpc 13330
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