Theorem wlknewwlksn 26773

Description: If a walk in a pseudograph has length N, then the sequence of the vertices of the walk is a word representing the walk as word of length N. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 11-Apr-2021.)

Assertion

Ref	Expression
wlknewwlksn	|- ( ( ( G e. UPGraph /\ W e. (Walks ` G ) ) /\ ( N e. NN0 /\ ( # ` ( 1st ` W ) ) = N ) ) -> ( 2nd ` W ) e. ( N WWalksN G ) )

Proof of Theorem wlknewwlksn

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wlkcpr 26524	. . . . . 6 |- ( W e. (Walks ` G ) <-> ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) )
2		wlkn0 26516	. . . . . 6 |- ( ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) -> ( 2nd ` W ) =/= (/) )
3	1, 2	sylbi 207	. . . . 5 |- ( W e. (Walks ` G ) -> ( 2nd ` W ) =/= (/) )
4	3	adantl 482	. . . 4 |- ( ( G e. UPGraph /\ W e. (Walks ` G ) ) -> ( 2nd ` W ) =/= (/) )
5		eqid 2622	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
6		eqid 2622	. . . . . . 7 |- (iEdg ` G ) = (iEdg ` G )
7		eqid 2622	. . . . . . 7 |- ( 1st ` W ) = ( 1st ` W )
8		eqid 2622	. . . . . . 7 |- ( 2nd ` W ) = ( 2nd ` W )
9	5, 6, 7, 8	wlkelwrd 26528	. . . . . 6 |- ( W e. (Walks ` G ) -> ( ( 1st ` W ) e. Word dom (iEdg ` G ) /\ ( 2nd ` W ) : ( 0 ... ( # ` ( 1st ` W ) ) ) --> (Vtx ` G ) ) )
10		ffz0iswrd 13332	. . . . . . 7 |- ( ( 2nd ` W ) : ( 0 ... ( # ` ( 1st ` W ) ) ) --> (Vtx ` G ) -> ( 2nd ` W ) e. Word (Vtx ` G ) )
11	10	adantl 482	. . . . . 6 |- ( ( ( 1st ` W ) e. Word dom (iEdg ` G ) /\ ( 2nd ` W ) : ( 0 ... ( # ` ( 1st ` W ) ) ) --> (Vtx ` G ) ) -> ( 2nd ` W ) e. Word (Vtx ` G ) )
12	9, 11	syl 17	. . . . 5 |- ( W e. (Walks ` G ) -> ( 2nd ` W ) e. Word (Vtx ` G ) )
13	12	adantl 482	. . . 4 |- ( ( G e. UPGraph /\ W e. (Walks ` G ) ) -> ( 2nd ` W ) e. Word (Vtx ` G ) )
14		eqid 2622	. . . . . . 7 |- (Edg ` G ) = (Edg ` G )
15	14	upgrwlkvtxedg 26541	. . . . . 6 |- ( ( G e. UPGraph /\ ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) ) -> A. i e. ( 0..^ ( # ` ( 1st ` W ) ) ) { ( ( 2nd ` W ) ` i ) , ( ( 2nd ` W ) ` ( i + 1 ) ) } e. (Edg ` G ) )
16		wlklenvm1 26517	. . . . . . . . 9 |- ( ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) -> ( # ` ( 1st ` W ) ) = ( ( # ` ( 2nd ` W ) ) - 1 ) )
17	16	adantl 482	. . . . . . . 8 |- ( ( G e. UPGraph /\ ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) ) -> ( # ` ( 1st ` W ) ) = ( ( # ` ( 2nd ` W ) ) - 1 ) )
18	17	oveq2d 6666	. . . . . . 7 |- ( ( G e. UPGraph /\ ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) ) -> ( 0..^ ( # ` ( 1st ` W ) ) ) = ( 0..^ ( ( # ` ( 2nd ` W ) ) - 1 ) ) )
19	18	raleqdv 3144	. . . . . 6 |- ( ( G e. UPGraph /\ ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) ) -> ( A. i e. ( 0..^ ( # ` ( 1st ` W ) ) ) { ( ( 2nd ` W ) ` i ) , ( ( 2nd ` W ) ` ( i + 1 ) ) } e. (Edg ` G ) <-> A. i e. ( 0..^ ( ( # ` ( 2nd ` W ) ) - 1 ) ) { ( ( 2nd ` W ) ` i ) , ( ( 2nd ` W ) ` ( i + 1 ) ) } e. (Edg ` G ) ) )
20	15, 19	mpbid 222	. . . . 5 |- ( ( G e. UPGraph /\ ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) ) -> A. i e. ( 0..^ ( ( # ` ( 2nd ` W ) ) - 1 ) ) { ( ( 2nd ` W ) ` i ) , ( ( 2nd ` W ) ` ( i + 1 ) ) } e. (Edg ` G ) )
21	1, 20	sylan2b 492	. . . 4 |- ( ( G e. UPGraph /\ W e. (Walks ` G ) ) -> A. i e. ( 0..^ ( ( # ` ( 2nd ` W ) ) - 1 ) ) { ( ( 2nd ` W ) ` i ) , ( ( 2nd ` W ) ` ( i + 1 ) ) } e. (Edg ` G ) )
22	4, 13, 21	3jca 1242	. . 3 |- ( ( G e. UPGraph /\ W e. (Walks ` G ) ) -> ( ( 2nd ` W ) =/= (/) /\ ( 2nd ` W ) e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` ( 2nd ` W ) ) - 1 ) ) { ( ( 2nd ` W ) ` i ) , ( ( 2nd ` W ) ` ( i + 1 ) ) } e. (Edg ` G ) ) )
23	22	adantr 481	. 2 |- ( ( ( G e. UPGraph /\ W e. (Walks ` G ) ) /\ ( N e. NN0 /\ ( # ` ( 1st ` W ) ) = N ) ) -> ( ( 2nd ` W ) =/= (/) /\ ( 2nd ` W ) e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` ( 2nd ` W ) ) - 1 ) ) { ( ( 2nd ` W ) ` i ) , ( ( 2nd ` W ) ` ( i + 1 ) ) } e. (Edg ` G ) ) )
24		simpl 473	. . . . . . 7 |- ( ( N e. NN0 /\ ( # ` ( 1st ` W ) ) = N ) -> N e. NN0 )
25		oveq2 6658	. . . . . . . . . . . . 13 |- ( ( # ` ( 1st ` W ) ) = N -> ( 0 ... ( # ` ( 1st ` W ) ) ) = ( 0 ... N ) )
26	25	adantl 482	. . . . . . . . . . . 12 |- ( ( ( 1st ` W ) e. Word dom (iEdg ` G ) /\ ( # ` ( 1st ` W ) ) = N ) -> ( 0 ... ( # ` ( 1st ` W ) ) ) = ( 0 ... N ) )
27	26	feq2d 6031	. . . . . . . . . . 11 |- ( ( ( 1st ` W ) e. Word dom (iEdg ` G ) /\ ( # ` ( 1st ` W ) ) = N ) -> ( ( 2nd ` W ) : ( 0 ... ( # ` ( 1st ` W ) ) ) --> (Vtx ` G ) <-> ( 2nd ` W ) : ( 0 ... N ) --> (Vtx ` G ) ) )
28	27	biimpd 219	. . . . . . . . . 10 |- ( ( ( 1st ` W ) e. Word dom (iEdg ` G ) /\ ( # ` ( 1st ` W ) ) = N ) -> ( ( 2nd ` W ) : ( 0 ... ( # ` ( 1st ` W ) ) ) --> (Vtx ` G ) -> ( 2nd ` W ) : ( 0 ... N ) --> (Vtx ` G ) ) )
29	28	impancom 456	. . . . . . . . 9 |- ( ( ( 1st ` W ) e. Word dom (iEdg ` G ) /\ ( 2nd ` W ) : ( 0 ... ( # ` ( 1st ` W ) ) ) --> (Vtx ` G ) ) -> ( ( # ` ( 1st ` W ) ) = N -> ( 2nd ` W ) : ( 0 ... N ) --> (Vtx ` G ) ) )
30	29	adantld 483	. . . . . . . 8 |- ( ( ( 1st ` W ) e. Word dom (iEdg ` G ) /\ ( 2nd ` W ) : ( 0 ... ( # ` ( 1st ` W ) ) ) --> (Vtx ` G ) ) -> ( ( N e. NN0 /\ ( # ` ( 1st ` W ) ) = N ) -> ( 2nd ` W ) : ( 0 ... N ) --> (Vtx ` G ) ) )
31	30	imp 445	. . . . . . 7 |- ( ( ( ( 1st ` W ) e. Word dom (iEdg ` G ) /\ ( 2nd ` W ) : ( 0 ... ( # ` ( 1st ` W ) ) ) --> (Vtx ` G ) ) /\ ( N e. NN0 /\ ( # ` ( 1st ` W ) ) = N ) ) -> ( 2nd ` W ) : ( 0 ... N ) --> (Vtx ` G ) )
32		ffz0hash 13231	. . . . . . 7 |- ( ( N e. NN0 /\ ( 2nd ` W ) : ( 0 ... N ) --> (Vtx ` G ) ) -> ( # ` ( 2nd ` W ) ) = ( N + 1 ) )
33	24, 31, 32	syl2an2 875	. . . . . 6 |- ( ( ( ( 1st ` W ) e. Word dom (iEdg ` G ) /\ ( 2nd ` W ) : ( 0 ... ( # ` ( 1st ` W ) ) ) --> (Vtx ` G ) ) /\ ( N e. NN0 /\ ( # ` ( 1st ` W ) ) = N ) ) -> ( # ` ( 2nd ` W ) ) = ( N + 1 ) )
34	33	ex 450	. . . . 5 |- ( ( ( 1st ` W ) e. Word dom (iEdg ` G ) /\ ( 2nd ` W ) : ( 0 ... ( # ` ( 1st ` W ) ) ) --> (Vtx ` G ) ) -> ( ( N e. NN0 /\ ( # ` ( 1st ` W ) ) = N ) -> ( # ` ( 2nd ` W ) ) = ( N + 1 ) ) )
35	9, 34	syl 17	. . . 4 |- ( W e. (Walks ` G ) -> ( ( N e. NN0 /\ ( # ` ( 1st ` W ) ) = N ) -> ( # ` ( 2nd ` W ) ) = ( N + 1 ) ) )
36	35	adantl 482	. . 3 |- ( ( G e. UPGraph /\ W e. (Walks ` G ) ) -> ( ( N e. NN0 /\ ( # ` ( 1st ` W ) ) = N ) -> ( # ` ( 2nd ` W ) ) = ( N + 1 ) ) )
37	36	imp 445	. 2 |- ( ( ( G e. UPGraph /\ W e. (Walks ` G ) ) /\ ( N e. NN0 /\ ( # ` ( 1st ` W ) ) = N ) ) -> ( # ` ( 2nd ` W ) ) = ( N + 1 ) )
38	24	adantl 482	. . 3 |- ( ( ( G e. UPGraph /\ W e. (Walks ` G ) ) /\ ( N e. NN0 /\ ( # ` ( 1st ` W ) ) = N ) ) -> N e. NN0 )
39		iswwlksn 26730	. . . 4 |- ( N e. NN0 -> ( ( 2nd ` W ) e. ( N WWalksN G ) <-> ( ( 2nd ` W ) e. (WWalks ` G ) /\ ( # ` ( 2nd ` W ) ) = ( N + 1 ) ) ) )
40	5, 14	iswwlks 26728	. . . . . 6 |- ( ( 2nd ` W ) e. (WWalks ` G ) <-> ( ( 2nd ` W ) =/= (/) /\ ( 2nd ` W ) e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` ( 2nd ` W ) ) - 1 ) ) { ( ( 2nd ` W ) ` i ) , ( ( 2nd ` W ) ` ( i + 1 ) ) } e. (Edg ` G ) ) )
41	40	a1i 11	. . . . 5 |- ( N e. NN0 -> ( ( 2nd ` W ) e. (WWalks ` G ) <-> ( ( 2nd ` W ) =/= (/) /\ ( 2nd ` W ) e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` ( 2nd ` W ) ) - 1 ) ) { ( ( 2nd ` W ) ` i ) , ( ( 2nd ` W ) ` ( i + 1 ) ) } e. (Edg ` G ) ) ) )
42	41	anbi1d 741	. . . 4 |- ( N e. NN0 -> ( ( ( 2nd ` W ) e. (WWalks ` G ) /\ ( # ` ( 2nd ` W ) ) = ( N + 1 ) ) <-> ( ( ( 2nd ` W ) =/= (/) /\ ( 2nd ` W ) e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` ( 2nd ` W ) ) - 1 ) ) { ( ( 2nd ` W ) ` i ) , ( ( 2nd ` W ) ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` ( 2nd ` W ) ) = ( N + 1 ) ) ) )
43	39, 42	bitrd 268	. . 3 |- ( N e. NN0 -> ( ( 2nd ` W ) e. ( N WWalksN G ) <-> ( ( ( 2nd ` W ) =/= (/) /\ ( 2nd ` W ) e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` ( 2nd ` W ) ) - 1 ) ) { ( ( 2nd ` W ) ` i ) , ( ( 2nd ` W ) ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` ( 2nd ` W ) ) = ( N + 1 ) ) ) )
44	38, 43	syl 17	. 2 |- ( ( ( G e. UPGraph /\ W e. (Walks ` G ) ) /\ ( N e. NN0 /\ ( # ` ( 1st ` W ) ) = N ) ) -> ( ( 2nd ` W ) e. ( N WWalksN G ) <-> ( ( ( 2nd ` W ) =/= (/) /\ ( 2nd ` W ) e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` ( 2nd ` W ) ) - 1 ) ) { ( ( 2nd ` W ) ` i ) , ( ( 2nd ` W ) ` ( i + 1 ) ) } e. (Edg ` G ) ) /\ ( # ` ( 2nd ` W ) ) = ( N + 1 ) ) ) )
45	23, 37, 44	mpbir2and 957	1 |- ( ( ( G e. UPGraph /\ W e. (Walks ` G ) ) /\ ( N e. NN0 /\ ( # ` ( 1st ` W ) ) = N ) ) -> ( 2nd ` W ) e. ( N WWalksN G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   (/)c0 3915   {cpr 4179   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NN0cn0 11292   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UPGraph cupgr 25975  Walkscwlks 26492  WWalkscwwlks 26717   WWalksN cwwlksn 26718

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-ifp 1013  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-rmo 2920  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-2o 7561  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-cda 8990  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-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-wlks 26495  df-wwlks 26722  df-wwlksn 26723

This theorem is referenced by:  wlknwwlksnfun  26774  wlkwwlkfun  26781