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Theorem vdwpc 14373
 Description: The predicate " The coloring contains a polychromatic -tuple of AP's of length ". A polychromatic -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
vdwmc.2
vdwmc.3
vdwpc.4
vdwpc.5
Assertion
Ref Expression
vdwpc PolyAP AP
Distinct variable groups:   ,,,   ,,,   ,,   ,,,
Allowed substitution hints:   (,,)   (,,)   ()   (,,)

Proof of Theorem vdwpc
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vdwpc.4 . 2
2 vdwmc.2 . 2
3 vdwmc.3 . . 3
4 vdwmc.1 . . 3
5 fex 6144 . . 3
63, 4, 5sylancl 662 . 2
7 df-br 4454 . . . 4 PolyAP PolyAP
8 df-vdwpc 14363 . . . . 5 PolyAP AP
98eleq2i 2545 . . . 4 PolyAP AP
107, 9bitri 249 . . 3 PolyAP AP
11 simp1 996 . . . . . . . . 9
1211oveq2d 6311 . . . . . . . 8
13 vdwpc.5 . . . . . . . 8
1412, 13syl6eqr 2526 . . . . . . 7
1514oveq2d 6311 . . . . . 6
16 simp2 997 . . . . . . . . . . 11
1716fveq2d 5876 . . . . . . . . . 10 AP AP
1817oveqd 6312 . . . . . . . . 9 AP AP
19 simp3 998 . . . . . . . . . . 11
2019cnveqd 5184 . . . . . . . . . 10
2119fveq1d 5874 . . . . . . . . . . 11
2221sneqd 4045 . . . . . . . . . 10
2320, 22imaeq12d 5344 . . . . . . . . 9
2418, 23sseq12d 3538 . . . . . . . 8 AP AP
2514, 24raleqbidv 3077 . . . . . . 7 AP AP
2614, 21mpteq12dv 4531 . . . . . . . . . 10
2726rneqd 5236 . . . . . . . . 9
2827fveq2d 5876 . . . . . . . 8
2928, 11eqeq12d 2489 . . . . . . 7
3025, 29anbi12d 710 . . . . . 6 AP AP
3115, 30rexeqbidv 3078 . . . . 5 AP AP
3231rexbidv 2978 . . . 4 AP AP
3332eloprabga 6384 . . 3 AP AP
3410, 33syl5bb 257 . 2 PolyAP AP
351, 2, 6, 34syl3anc 1228 1 PolyAP AP
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   w3a 973   wceq 1379   wcel 1767  wral 2817  wrex 2818  cvv 3118   wss 3481  csn 4033  cop 4039   class class class wbr 4453   cmpt 4511  ccnv 5004   crn 5006  cima 5008  wf 5590  cfv 5594  (class class class)co 6295  coprab 6296   cmap 7432  c1 9505   caddc 9507  cn 10548  cn0 10807  cfz 11684  chash 12385  APcvdwa 14358   PolyAP cvdwp 14360 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-vdwpc 14363 This theorem is referenced by:  vdwlem6  14379  vdwlem7  14380  vdwlem8  14381  vdwlem11  14384
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