Theorem vdwpc 14498

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

Distinct variable groups:   ,,,   ,,,   J,,   M,,,

Proof of Theorem vdwpc

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vdwpc.4	. 2
2		vdwmc.2	. 2
3		vdwmc.3	. . 3
4		vdwmc.1	. . 3
5		fex 6145	. . 3
6	3, 4, 5	sylancl 662	. 2
7		df-br 4453	. . . 4
8		df-vdwpc 14488	. . . . 5
9	8	eleq2i 2535	. . . 4
10	7, 9	bitri 249	. . 3
11		simp1 996	. . . . . . . . 9
12	11	oveq2d 6312	. . . . . . . 8
13		vdwpc.5	. . . . . . . 8
14	12, 13	syl6eqr 2516	. . . . . . 7
15	14	oveq2d 6312	. . . . . 6
16		simp2 997	. . . . . . . . . . 11
17	16	fveq2d 5875	. . . . . . . . . 10
18	17	oveqd 6313	. . . . . . . . 9
19		simp3 998	. . . . . . . . . . 11
20	19	cnveqd 5183	. . . . . . . . . 10
21	19	fveq1d 5873	. . . . . . . . . . 11
22	21	sneqd 4041	. . . . . . . . . 10
23	20, 22	imaeq12d 5343	. . . . . . . . 9
24	18, 23	sseq12d 3532	. . . . . . . 8
25	14, 24	raleqbidv 3068	. . . . . . 7
26	14, 21	mpteq12dv 4530	. . . . . . . . . 10
27	26	rneqd 5235	. . . . . . . . 9
28	27	fveq2d 5875	. . . . . . . 8
29	28, 11	eqeq12d 2479	. . . . . . 7
30	25, 29	anbi12d 710	. . . . . 6
31	15, 30	rexeqbidv 3069	. . . . 5
32	31	rexbidv 2968	. . . 4
33	32	eloprabga 6389	. . 3
34	10, 33	syl5bb 257	. 2
35	1, 2, 6, 34	syl3anc 1228	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  A.wral 2807  E.wrex 2808  cvv 3109  C_wss 3475  {csn 4029  <.cop 4035   class class class wbr 4452  e.cmpt 4510  `'ccnv 5003  rancrn 5005  "cima 5007  -->wf 5589  `cfv 5593  (class class class)co 6296  {coprab 6297  cmap 7439  1c1 9514  caddc 9516  cn 10561  cn0 10820  cfz 11701  chash 12405  cvdwa 14483  cvdwp 14485

This theorem is referenced by:  vdwlem6  14504  vdwlem7  14505  vdwlem8  14506  vdwlem11  14509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6299  df-oprab 6300  df-vdwpc 14488