Theorem wwlks2onv 26847

Description: If a length 3 string represents a walk of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.)

Hypothesis

Ref	Expression
wwlks2onv.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
wwlks2onv	|- ( ( B e. U /\ <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) ) -> ( A e. V /\ B e. V /\ C e. V ) )

Proof of Theorem wwlks2onv

Dummy variables a c w b g n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2nn0 11309	. . . . . . . 8 |- 2 e. NN0
2		wwlks2onv.v	. . . . . . . . 9 |- V = (Vtx ` G )
3	2	wwlksnon 26738	. . . . . . . 8 |- ( ( 2 e. NN0 /\ G e. _V ) -> ( 2 WWalksNOn G ) = ( a e. V , c e. V |-> { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` 2 ) = c ) } ) )
4	1, 3	mpan 706	. . . . . . 7 |- ( G e. _V -> ( 2 WWalksNOn G ) = ( a e. V , c e. V |-> { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` 2 ) = c ) } ) )
5	4	oveqd 6667	. . . . . 6 |- ( G e. _V -> ( A ( 2 WWalksNOn G ) C ) = ( A ( a e. V , c e. V |-> { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` 2 ) = c ) } ) C ) )
6	5	eleq2d 2687	. . . . 5 |- ( G e. _V -> ( <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) <-> <" A B C "> e. ( A ( a e. V , c e. V |-> { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` 2 ) = c ) } ) C ) ) )
7		eqid 2622	. . . . . . 7 |- ( a e. V , c e. V |-> { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` 2 ) = c ) } ) = ( a e. V , c e. V |-> { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` 2 ) = c ) } )
8	7	elmpt2cl 6876	. . . . . 6 |- ( <" A B C "> e. ( A ( a e. V , c e. V |-> { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` 2 ) = c ) } ) C ) -> ( A e. V /\ C e. V ) )
9		simprl 794	. . . . . . . 8 |- ( ( ( <" A B C "> e. ( A ( a e. V , c e. V |-> { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` 2 ) = c ) } ) C ) /\ B e. U ) /\ ( A e. V /\ C e. V ) ) -> A e. V )
10		eqeq2 2633	. . . . . . . . . . . . . . 15 |- ( a = A -> ( ( w ` 0 ) = a <-> ( w ` 0 ) = A ) )
11	10	anbi1d 741	. . . . . . . . . . . . . 14 |- ( a = A -> ( ( ( w ` 0 ) = a /\ ( w ` 2 ) = c ) <-> ( ( w ` 0 ) = A /\ ( w ` 2 ) = c ) ) )
12	11	rabbidv 3189	. . . . . . . . . . . . 13 |- ( a = A -> { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` 2 ) = c ) } = { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` 2 ) = c ) } )
13		eqeq2 2633	. . . . . . . . . . . . . . 15 |- ( c = C -> ( ( w ` 2 ) = c <-> ( w ` 2 ) = C ) )
14	13	anbi2d 740	. . . . . . . . . . . . . 14 |- ( c = C -> ( ( ( w ` 0 ) = A /\ ( w ` 2 ) = c ) <-> ( ( w ` 0 ) = A /\ ( w ` 2 ) = C ) ) )
15	14	rabbidv 3189	. . . . . . . . . . . . 13 |- ( c = C -> { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` 2 ) = c ) } = { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` 2 ) = C ) } )
16		ovex 6678	. . . . . . . . . . . . . 14 |- ( 2 WWalksN G ) e. _V
17	16	rabex 4813	. . . . . . . . . . . . 13 |- { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` 2 ) = C ) } e. _V
18	12, 15, 7, 17	ovmpt2 6796	. . . . . . . . . . . 12 |- ( ( A e. V /\ C e. V ) -> ( A ( a e. V , c e. V |-> { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` 2 ) = c ) } ) C ) = { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` 2 ) = C ) } )
19	18	eleq2d 2687	. . . . . . . . . . 11 |- ( ( A e. V /\ C e. V ) -> ( <" A B C "> e. ( A ( a e. V , c e. V |-> { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` 2 ) = c ) } ) C ) <-> <" A B C "> e. { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` 2 ) = C ) } ) )
20		fveq1 6190	. . . . . . . . . . . . . . 15 |- ( w = <" A B C "> -> ( w ` 0 ) = ( <" A B C "> ` 0 ) )
21	20	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( w = <" A B C "> -> ( ( w ` 0 ) = A <-> ( <" A B C "> ` 0 ) = A ) )
22		fveq1 6190	. . . . . . . . . . . . . . 15 |- ( w = <" A B C "> -> ( w ` 2 ) = ( <" A B C "> ` 2 ) )
23	22	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( w = <" A B C "> -> ( ( w ` 2 ) = C <-> ( <" A B C "> ` 2 ) = C ) )
24	21, 23	anbi12d 747	. . . . . . . . . . . . 13 |- ( w = <" A B C "> -> ( ( ( w ` 0 ) = A /\ ( w ` 2 ) = C ) <-> ( ( <" A B C "> ` 0 ) = A /\ ( <" A B C "> ` 2 ) = C ) ) )
25	24	elrab 3363	. . . . . . . . . . . 12 |- ( <" A B C "> e. { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` 2 ) = C ) } <-> ( <" A B C "> e. ( 2 WWalksN G ) /\ ( ( <" A B C "> ` 0 ) = A /\ ( <" A B C "> ` 2 ) = C ) ) )
26		wwlknbp2 26752	. . . . . . . . . . . . . . 15 |- ( <" A B C "> e. ( 2 WWalksN G ) -> ( <" A B C "> e. Word (Vtx ` G ) /\ ( # ` <" A B C "> ) = ( 2 + 1 ) ) )
27		s3fv1 13637	. . . . . . . . . . . . . . . . . . 19 |- ( B e. U -> ( <" A B C "> ` 1 ) = B )
28	27	eqcomd 2628	. . . . . . . . . . . . . . . . . 18 |- ( B e. U -> B = ( <" A B C "> ` 1 ) )
29	28	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( ( <" A B C "> e. Word (Vtx ` G ) /\ ( # ` <" A B C "> ) = ( 2 + 1 ) ) /\ B e. U ) -> B = ( <" A B C "> ` 1 ) )
30		1ex 10035	. . . . . . . . . . . . . . . . . . . . 21 |- 1 e. _V
31	30	tpid2 4304	. . . . . . . . . . . . . . . . . . . 20 |- 1 e. { 0 , 1 , 2 }
32		id 22	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( # ` <" A B C "> ) = ( 2 + 1 ) -> ( # ` <" A B C "> ) = ( 2 + 1 ) )
33		2p1e3 11151	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( 2 + 1 ) = 3
34	32, 33	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( # ` <" A B C "> ) = ( 2 + 1 ) -> ( # ` <" A B C "> ) = 3 )
35	34	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` <" A B C "> ) = ( 2 + 1 ) -> ( 0..^ ( # ` <" A B C "> ) ) = ( 0..^ 3 ) )
36		fzo0to3tp 12554	. . . . . . . . . . . . . . . . . . . . 21 |- ( 0..^ 3 ) = { 0 , 1 , 2 }
37	35, 36	syl6eq 2672	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` <" A B C "> ) = ( 2 + 1 ) -> ( 0..^ ( # ` <" A B C "> ) ) = { 0 , 1 , 2 } )
38	31, 37	syl5eleqr 2708	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` <" A B C "> ) = ( 2 + 1 ) -> 1 e. ( 0..^ ( # ` <" A B C "> ) ) )
39		wrdsymbcl 13318	. . . . . . . . . . . . . . . . . . . 20 |- ( ( <" A B C "> e. Word (Vtx ` G ) /\ 1 e. ( 0..^ ( # ` <" A B C "> ) ) ) -> ( <" A B C "> ` 1 ) e. (Vtx ` G ) )
40	39, 2	syl6eleqr 2712	. . . . . . . . . . . . . . . . . . 19 |- ( ( <" A B C "> e. Word (Vtx ` G ) /\ 1 e. ( 0..^ ( # ` <" A B C "> ) ) ) -> ( <" A B C "> ` 1 ) e. V )
41	38, 40	sylan2 491	. . . . . . . . . . . . . . . . . 18 |- ( ( <" A B C "> e. Word (Vtx ` G ) /\ ( # ` <" A B C "> ) = ( 2 + 1 ) ) -> ( <" A B C "> ` 1 ) e. V )
42	41	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ( <" A B C "> e. Word (Vtx ` G ) /\ ( # ` <" A B C "> ) = ( 2 + 1 ) ) /\ B e. U ) -> ( <" A B C "> ` 1 ) e. V )
43	29, 42	eqeltrd 2701	. . . . . . . . . . . . . . . 16 |- ( ( ( <" A B C "> e. Word (Vtx ` G ) /\ ( # ` <" A B C "> ) = ( 2 + 1 ) ) /\ B e. U ) -> B e. V )
44	43	ex 450	. . . . . . . . . . . . . . 15 |- ( ( <" A B C "> e. Word (Vtx ` G ) /\ ( # ` <" A B C "> ) = ( 2 + 1 ) ) -> ( B e. U -> B e. V ) )
45	26, 44	syl 17	. . . . . . . . . . . . . 14 |- ( <" A B C "> e. ( 2 WWalksN G ) -> ( B e. U -> B e. V ) )
46	45	adantr 481	. . . . . . . . . . . . 13 |- ( ( <" A B C "> e. ( 2 WWalksN G ) /\ ( ( <" A B C "> ` 0 ) = A /\ ( <" A B C "> ` 2 ) = C ) ) -> ( B e. U -> B e. V ) )
47	46	a1i 11	. . . . . . . . . . . 12 |- ( ( A e. V /\ C e. V ) -> ( ( <" A B C "> e. ( 2 WWalksN G ) /\ ( ( <" A B C "> ` 0 ) = A /\ ( <" A B C "> ` 2 ) = C ) ) -> ( B e. U -> B e. V ) ) )
48	25, 47	syl5bi 232	. . . . . . . . . . 11 |- ( ( A e. V /\ C e. V ) -> ( <" A B C "> e. { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` 2 ) = C ) } -> ( B e. U -> B e. V ) ) )
49	19, 48	sylbid 230	. . . . . . . . . 10 |- ( ( A e. V /\ C e. V ) -> ( <" A B C "> e. ( A ( a e. V , c e. V |-> { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` 2 ) = c ) } ) C ) -> ( B e. U -> B e. V ) ) )
50	49	impd 447	. . . . . . . . 9 |- ( ( A e. V /\ C e. V ) -> ( ( <" A B C "> e. ( A ( a e. V , c e. V |-> { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` 2 ) = c ) } ) C ) /\ B e. U ) -> B e. V ) )
51	50	impcom 446	. . . . . . . 8 |- ( ( ( <" A B C "> e. ( A ( a e. V , c e. V |-> { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` 2 ) = c ) } ) C ) /\ B e. U ) /\ ( A e. V /\ C e. V ) ) -> B e. V )
52		simprr 796	. . . . . . . 8 |- ( ( ( <" A B C "> e. ( A ( a e. V , c e. V |-> { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` 2 ) = c ) } ) C ) /\ B e. U ) /\ ( A e. V /\ C e. V ) ) -> C e. V )
53	9, 51, 52	3jca 1242	. . . . . . 7 |- ( ( ( <" A B C "> e. ( A ( a e. V , c e. V |-> { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` 2 ) = c ) } ) C ) /\ B e. U ) /\ ( A e. V /\ C e. V ) ) -> ( A e. V /\ B e. V /\ C e. V ) )
54	53	exp31 630	. . . . . 6 |- ( <" A B C "> e. ( A ( a e. V , c e. V |-> { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` 2 ) = c ) } ) C ) -> ( B e. U -> ( ( A e. V /\ C e. V ) -> ( A e. V /\ B e. V /\ C e. V ) ) ) )
55	8, 54	mpid 44	. . . . 5 |- ( <" A B C "> e. ( A ( a e. V , c e. V |-> { w e. ( 2 WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` 2 ) = c ) } ) C ) -> ( B e. U -> ( A e. V /\ B e. V /\ C e. V ) ) )
56	6, 55	syl6bi 243	. . . 4 |- ( G e. _V -> ( <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) -> ( B e. U -> ( A e. V /\ B e. V /\ C e. V ) ) ) )
57	56	com23 86	. . 3 |- ( G e. _V -> ( B e. U -> ( <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) -> ( A e. V /\ B e. V /\ C e. V ) ) ) )
58	57	impd 447	. 2 |- ( G e. _V -> ( ( B e. U /\ <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) ) -> ( A e. V /\ B e. V /\ C e. V ) ) )
59		df-wwlksnon 26724	. . . . . . . . . 10 |- WWalksNOn = ( n e. NN0 , g e. _V |-> ( a e. (Vtx ` g ) , b e. (Vtx ` g ) |-> { w e. ( n WWalksN g ) | ( ( w ` 0 ) = a /\ ( w ` n ) = b ) } ) )
60	59	reldmmpt2 6771	. . . . . . . . 9 |- Rel dom WWalksNOn
61	60	ovprc2 6685	. . . . . . . 8 |- ( -. G e. _V -> ( 2 WWalksNOn G ) = (/) )
62	61	oveqd 6667	. . . . . . 7 |- ( -. G e. _V -> ( A ( 2 WWalksNOn G ) C ) = ( A (/) C ) )
63		0ov 6682	. . . . . . 7 |- ( A (/) C ) = (/)
64	62, 63	syl6eq 2672	. . . . . 6 |- ( -. G e. _V -> ( A ( 2 WWalksNOn G ) C ) = (/) )
65	64	eleq2d 2687	. . . . 5 |- ( -. G e. _V -> ( <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) <-> <" A B C "> e. (/) ) )
66		noel 3919	. . . . . 6 |- -. <" A B C "> e. (/)
67	66	pm2.21i 116	. . . . 5 |- ( <" A B C "> e. (/) -> ( B e. U -> ( A e. V /\ B e. V /\ C e. V ) ) )
68	65, 67	syl6bi 243	. . . 4 |- ( -. G e. _V -> ( <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) -> ( B e. U -> ( A e. V /\ B e. V /\ C e. V ) ) ) )
69	68	com23 86	. . 3 |- ( -. G e. _V -> ( B e. U -> ( <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) -> ( A e. V /\ B e. V /\ C e. V ) ) ) )
70	69	impd 447	. 2 |- ( -. G e. _V -> ( ( B e. U /\ <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) ) -> ( A e. V /\ B e. V /\ C e. V ) ) )
71	58, 70	pm2.61i 176	1 |- ( ( B e. U /\ <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) ) -> ( A e. V /\ B e. V /\ C e. V ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   (/)c0 3915   {ctp 4181   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   + caddc 9939   2c2 11070   3c3 11071   NN0cn0 11292  ..^cfzo 12465   #chash 13117  Word cword 13291   <"cs3 13587  Vtxcvtx 25874   WWalksN cwwlksn 26718   WWalksNOn cwwlksnon 26719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-wwlks 26722  df-wwlksn 26723  df-wwlksnon 26724

This theorem is referenced by:  frgr2wwlkeqm  27195