Theorem wwlksnwwlksnon 26810

Description: A walk of fixed length is a walk of fixed length between two vertices. (Contributed by Alexander van der Vekens, 21-Feb-2018.) (Revised by AV, 12-May-2021.)

Hypothesis

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

Assertion

Ref	Expression
wwlksnwwlksnon	|- ( ( N e. NN0 /\ G e. U ) -> ( W e. ( N WWalksN G ) <-> E. a e. V E. b e. V W e. ( a ( N WWalksNOn G ) b ) ) )

Distinct variable groups:   G, a, b   N, a, b   U, a, b   V, a, b   W, a, b

Proof of Theorem wwlksnwwlksnon

Step	Hyp	Ref	Expression
1		wwlknbp2 26752	. . . . . 6 |- ( W e. ( N WWalksN G ) -> ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) )
2	1	adantl 482	. . . . 5 |- ( ( ( N e. NN0 /\ G e. U ) /\ W e. ( N WWalksN G ) ) -> ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) )
3		wwlksnwwlksnon.v	. . . . . . . . . . . 12 |- V = (Vtx ` G )
4	3	eqcomi 2631	. . . . . . . . . . 11 |- (Vtx ` G ) = V
5	4	wrdeqi 13328	. . . . . . . . . 10 |- Word (Vtx ` G ) = Word V
6	5	eleq2i 2693	. . . . . . . . 9 |- ( W e. Word (Vtx ` G ) <-> W e. Word V )
7	6	biimpi 206	. . . . . . . 8 |- ( W e. Word (Vtx ` G ) -> W e. Word V )
8	7	adantr 481	. . . . . . 7 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> W e. Word V )
9		nn0p1nn 11332	. . . . . . . . . 10 |- ( N e. NN0 -> ( N + 1 ) e. NN )
10		lbfzo0 12507	. . . . . . . . . 10 |- ( 0 e. ( 0..^ ( N + 1 ) ) <-> ( N + 1 ) e. NN )
11	9, 10	sylibr 224	. . . . . . . . 9 |- ( N e. NN0 -> 0 e. ( 0..^ ( N + 1 ) ) )
12	11	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( N e. NN0 /\ G e. U ) /\ W e. ( N WWalksN G ) ) /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) -> 0 e. ( 0..^ ( N + 1 ) ) )
13		oveq2 6658	. . . . . . . . . 10 |- ( ( # ` W ) = ( N + 1 ) -> ( 0..^ ( # ` W ) ) = ( 0..^ ( N + 1 ) ) )
14	13	eleq2d 2687	. . . . . . . . 9 |- ( ( # ` W ) = ( N + 1 ) -> ( 0 e. ( 0..^ ( # ` W ) ) <-> 0 e. ( 0..^ ( N + 1 ) ) ) )
15	14	ad2antll 765	. . . . . . . 8 |- ( ( ( ( N e. NN0 /\ G e. U ) /\ W e. ( N WWalksN G ) ) /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) -> ( 0 e. ( 0..^ ( # ` W ) ) <-> 0 e. ( 0..^ ( N + 1 ) ) ) )
16	12, 15	mpbird 247	. . . . . . 7 |- ( ( ( ( N e. NN0 /\ G e. U ) /\ W e. ( N WWalksN G ) ) /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) -> 0 e. ( 0..^ ( # ` W ) ) )
17		wrdsymbcl 13318	. . . . . . 7 |- ( ( W e. Word V /\ 0 e. ( 0..^ ( # ` W ) ) ) -> ( W ` 0 ) e. V )
18	8, 16, 17	syl2an2 875	. . . . . 6 |- ( ( ( ( N e. NN0 /\ G e. U ) /\ W e. ( N WWalksN G ) ) /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) -> ( W ` 0 ) e. V )
19		fzonn0p1 12544	. . . . . . . . 9 |- ( N e. NN0 -> N e. ( 0..^ ( N + 1 ) ) )
20	19	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( N e. NN0 /\ G e. U ) /\ W e. ( N WWalksN G ) ) /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) -> N e. ( 0..^ ( N + 1 ) ) )
21	13	eleq2d 2687	. . . . . . . . 9 |- ( ( # ` W ) = ( N + 1 ) -> ( N e. ( 0..^ ( # ` W ) ) <-> N e. ( 0..^ ( N + 1 ) ) ) )
22	21	ad2antll 765	. . . . . . . 8 |- ( ( ( ( N e. NN0 /\ G e. U ) /\ W e. ( N WWalksN G ) ) /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) -> ( N e. ( 0..^ ( # ` W ) ) <-> N e. ( 0..^ ( N + 1 ) ) ) )
23	20, 22	mpbird 247	. . . . . . 7 |- ( ( ( ( N e. NN0 /\ G e. U ) /\ W e. ( N WWalksN G ) ) /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) -> N e. ( 0..^ ( # ` W ) ) )
24		wrdsymbcl 13318	. . . . . . 7 |- ( ( W e. Word V /\ N e. ( 0..^ ( # ` W ) ) ) -> ( W ` N ) e. V )
25	8, 23, 24	syl2an2 875	. . . . . 6 |- ( ( ( ( N e. NN0 /\ G e. U ) /\ W e. ( N WWalksN G ) ) /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) -> ( W ` N ) e. V )
26		simplr 792	. . . . . 6 |- ( ( ( ( N e. NN0 /\ G e. U ) /\ W e. ( N WWalksN G ) ) /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) -> W e. ( N WWalksN G ) )
27		eqidd 2623	. . . . . 6 |- ( ( ( ( N e. NN0 /\ G e. U ) /\ W e. ( N WWalksN G ) ) /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) -> ( W ` 0 ) = ( W ` 0 ) )
28		eqidd 2623	. . . . . 6 |- ( ( ( ( N e. NN0 /\ G e. U ) /\ W e. ( N WWalksN G ) ) /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) -> ( W ` N ) = ( W ` N ) )
29		eqeq2 2633	. . . . . . . 8 |- ( a = ( W ` 0 ) -> ( ( W ` 0 ) = a <-> ( W ` 0 ) = ( W ` 0 ) ) )
30	29	3anbi2d 1404	. . . . . . 7 |- ( a = ( W ` 0 ) -> ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) <-> ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = ( W ` 0 ) /\ ( W ` N ) = b ) ) )
31		eqeq2 2633	. . . . . . . 8 |- ( b = ( W ` N ) -> ( ( W ` N ) = b <-> ( W ` N ) = ( W ` N ) ) )
32	31	3anbi3d 1405	. . . . . . 7 |- ( b = ( W ` N ) -> ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = ( W ` 0 ) /\ ( W ` N ) = b ) <-> ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = ( W ` 0 ) /\ ( W ` N ) = ( W ` N ) ) ) )
33	30, 32	rspc2ev 3324	. . . . . 6 |- ( ( ( W ` 0 ) e. V /\ ( W ` N ) e. V /\ ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = ( W ` 0 ) /\ ( W ` N ) = ( W ` N ) ) ) -> E. a e. V E. b e. V ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) )
34	18, 25, 26, 27, 28, 33	syl113anc 1338	. . . . 5 |- ( ( ( ( N e. NN0 /\ G e. U ) /\ W e. ( N WWalksN G ) ) /\ ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) -> E. a e. V E. b e. V ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) )
35	2, 34	mpdan 702	. . . 4 |- ( ( ( N e. NN0 /\ G e. U ) /\ W e. ( N WWalksN G ) ) -> E. a e. V E. b e. V ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) )
36	35	ex 450	. . 3 |- ( ( N e. NN0 /\ G e. U ) -> ( W e. ( N WWalksN G ) -> E. a e. V E. b e. V ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) ) )
37		simp1 1061	. . . . 5 |- ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) -> W e. ( N WWalksN G ) )
38	37	a1i 11	. . . 4 |- ( ( ( N e. NN0 /\ G e. U ) /\ ( a e. V /\ b e. V ) ) -> ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) -> W e. ( N WWalksN G ) ) )
39	38	rexlimdvva 3038	. . 3 |- ( ( N e. NN0 /\ G e. U ) -> ( E. a e. V E. b e. V ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) -> W e. ( N WWalksN G ) ) )
40	36, 39	impbid 202	. 2 |- ( ( N e. NN0 /\ G e. U ) -> ( W e. ( N WWalksN G ) <-> E. a e. V E. b e. V ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) ) )
41	3	wwlknon 26742	. . . . 5 |- ( ( a e. V /\ b e. V ) -> ( W e. ( a ( N WWalksNOn G ) b ) <-> ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) ) )
42	41	bicomd 213	. . . 4 |- ( ( a e. V /\ b e. V ) -> ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) <-> W e. ( a ( N WWalksNOn G ) b ) ) )
43	42	adantl 482	. . 3 |- ( ( ( N e. NN0 /\ G e. U ) /\ ( a e. V /\ b e. V ) ) -> ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) <-> W e. ( a ( N WWalksNOn G ) b ) ) )
44	43	2rexbidva 3056	. 2 |- ( ( N e. NN0 /\ G e. U ) -> ( E. a e. V E. b e. V ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) <-> E. a e. V E. b e. V W e. ( a ( N WWalksNOn G ) b ) ) )
45	40, 44	bitrd 268	1 |- ( ( N e. NN0 /\ G e. U ) -> ( W e. ( N WWalksN G ) <-> E. a e. V E. b e. V W e. ( a ( N WWalksNOn G ) b ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   NNcn 11020   NN0cn0 11292  ..^cfzo 12465   #chash 13117  Word cword 13291  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-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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-wwlks 26722  df-wwlksn 26723  df-wwlksnon 26724

This theorem is referenced by:  wspthsnwspthsnon  26811  elwwlks2  26861