Theorem wlkiswwlks1 26753

Description: The sequence of vertices in a walk is a walk as word in a pseudograph. (Contributed by Alexander van der Vekens, 20-Jul-2018.) (Revised by AV, 9-Apr-2021.)

Assertion

Ref	Expression
wlkiswwlks1	|- ( G e. UPGraph -> ( F (Walks ` G ) P -> P e. (WWalks ` G ) ) )

Proof of Theorem wlkiswwlks1

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wlkn0 26516	. 2 |- ( F (Walks ` G ) P -> P =/= (/) )
2		eqid 2622	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
3		eqid 2622	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
4	2, 3	upgriswlk 26537	. . 3 |- ( G e. UPGraph -> ( F (Walks ` G ) P <-> ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) )
5		simpr 477	. . . . . 6 |- ( ( ( G e. UPGraph /\ ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) /\ P =/= (/) ) -> P =/= (/) )
6		ffz0iswrd 13332	. . . . . . . 8 |- ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) -> P e. Word (Vtx ` G ) )
7	6	3ad2ant2 1083	. . . . . . 7 |- ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> P e. Word (Vtx ` G ) )
8	7	ad2antlr 763	. . . . . 6 |- ( ( ( G e. UPGraph /\ ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) /\ P =/= (/) ) -> P e. Word (Vtx ` G ) )
9		upgruhgr 25997	. . . . . . . . . . . . . . . . . 18 |- ( G e. UPGraph -> G e. UHGraph )
10	3	uhgrfun 25961	. . . . . . . . . . . . . . . . . 18 |- ( G e. UHGraph -> Fun (iEdg ` G ) )
11		funfn 5918	. . . . . . . . . . . . . . . . . . 19 |- ( Fun (iEdg ` G ) <-> (iEdg ` G ) Fn dom (iEdg ` G ) )
12	11	biimpi 206	. . . . . . . . . . . . . . . . . 18 |- ( Fun (iEdg ` G ) -> (iEdg ` G ) Fn dom (iEdg ` G ) )
13	9, 10, 12	3syl 18	. . . . . . . . . . . . . . . . 17 |- ( G e. UPGraph -> (iEdg ` G ) Fn dom (iEdg ` G ) )
14	13	ad2antlr 763	. . . . . . . . . . . . . . . 16 |- ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ G e. UPGraph ) /\ i e. ( 0..^ ( # ` F ) ) ) -> (iEdg ` G ) Fn dom (iEdg ` G ) )
15		wrdsymbcl 13318	. . . . . . . . . . . . . . . . 17 |- ( ( F e. Word dom (iEdg ` G ) /\ i e. ( 0..^ ( # ` F ) ) ) -> ( F ` i ) e. dom (iEdg ` G ) )
16	15	ad4ant14 1293	. . . . . . . . . . . . . . . 16 |- ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ G e. UPGraph ) /\ i e. ( 0..^ ( # ` F ) ) ) -> ( F ` i ) e. dom (iEdg ` G ) )
17		fnfvelrn 6356	. . . . . . . . . . . . . . . 16 |- ( ( (iEdg ` G ) Fn dom (iEdg ` G ) /\ ( F ` i ) e. dom (iEdg ` G ) ) -> ( (iEdg ` G ) ` ( F ` i ) ) e. ran (iEdg ` G ) )
18	14, 16, 17	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ G e. UPGraph ) /\ i e. ( 0..^ ( # ` F ) ) ) -> ( (iEdg ` G ) ` ( F ` i ) ) e. ran (iEdg ` G ) )
19		edgval 25941	. . . . . . . . . . . . . . 15 |- (Edg ` G ) = ran (iEdg ` G )
20	18, 19	syl6eleqr 2712	. . . . . . . . . . . . . 14 |- ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ G e. UPGraph ) /\ i e. ( 0..^ ( # ` F ) ) ) -> ( (iEdg ` G ) ` ( F ` i ) ) e. (Edg ` G ) )
21		eleq1 2689	. . . . . . . . . . . . . . 15 |- ( { ( P ` i ) , ( P ` ( i + 1 ) ) } = ( (iEdg ` G ) ` ( F ` i ) ) -> ( { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) <-> ( (iEdg ` G ) ` ( F ` i ) ) e. (Edg ` G ) ) )
22	21	eqcoms 2630	. . . . . . . . . . . . . 14 |- ( ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } -> ( { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) <-> ( (iEdg ` G ) ` ( F ` i ) ) e. (Edg ` G ) ) )
23	20, 22	syl5ibrcom 237	. . . . . . . . . . . . 13 |- ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ G e. UPGraph ) /\ i e. ( 0..^ ( # ` F ) ) ) -> ( ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } -> { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) )
24	23	ralimdva 2962	. . . . . . . . . . . 12 |- ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ G e. UPGraph ) -> ( A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } -> A. i e. ( 0..^ ( # ` F ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) )
25	24	ex 450	. . . . . . . . . . 11 |- ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) -> ( G e. UPGraph -> ( A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } -> A. i e. ( 0..^ ( # ` F ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) )
26	25	com23 86	. . . . . . . . . 10 |- ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) -> ( A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } -> ( G e. UPGraph -> A. i e. ( 0..^ ( # ` F ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) ) )
27	26	3impia 1261	. . . . . . . . 9 |- ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( G e. UPGraph -> A. i e. ( 0..^ ( # ` F ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) )
28	27	impcom 446	. . . . . . . 8 |- ( ( G e. UPGraph /\ ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> A. i e. ( 0..^ ( # ` F ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) )
29		lencl 13324	. . . . . . . . . . . . . 14 |- ( F e. Word dom (iEdg ` G ) -> ( # ` F ) e. NN0 )
30		ffz0hash 13231	. . . . . . . . . . . . . . . 16 |- ( ( ( # ` F ) e. NN0 /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) -> ( # ` P ) = ( ( # ` F ) + 1 ) )
31	30	ex 450	. . . . . . . . . . . . . . 15 |- ( ( # ` F ) e. NN0 -> ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) -> ( # ` P ) = ( ( # ` F ) + 1 ) ) )
32		oveq1 6657	. . . . . . . . . . . . . . . . 17 |- ( ( # ` P ) = ( ( # ` F ) + 1 ) -> ( ( # ` P ) - 1 ) = ( ( ( # ` F ) + 1 ) - 1 ) )
33		nn0cn 11302	. . . . . . . . . . . . . . . . . 18 |- ( ( # ` F ) e. NN0 -> ( # ` F ) e. CC )
34		pncan1 10454	. . . . . . . . . . . . . . . . . 18 |- ( ( # ` F ) e. CC -> ( ( ( # ` F ) + 1 ) - 1 ) = ( # ` F ) )
35	33, 34	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( # ` F ) e. NN0 -> ( ( ( # ` F ) + 1 ) - 1 ) = ( # ` F ) )
36	32, 35	sylan9eqr 2678	. . . . . . . . . . . . . . . 16 |- ( ( ( # ` F ) e. NN0 /\ ( # ` P ) = ( ( # ` F ) + 1 ) ) -> ( ( # ` P ) - 1 ) = ( # ` F ) )
37	36	ex 450	. . . . . . . . . . . . . . 15 |- ( ( # ` F ) e. NN0 -> ( ( # ` P ) = ( ( # ` F ) + 1 ) -> ( ( # ` P ) - 1 ) = ( # ` F ) ) )
38	31, 37	syld 47	. . . . . . . . . . . . . 14 |- ( ( # ` F ) e. NN0 -> ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) -> ( ( # ` P ) - 1 ) = ( # ` F ) ) )
39	29, 38	syl 17	. . . . . . . . . . . . 13 |- ( F e. Word dom (iEdg ` G ) -> ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) -> ( ( # ` P ) - 1 ) = ( # ` F ) ) )
40	39	imp 445	. . . . . . . . . . . 12 |- ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) -> ( ( # ` P ) - 1 ) = ( # ` F ) )
41	40	oveq2d 6666	. . . . . . . . . . 11 |- ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) -> ( 0..^ ( ( # ` P ) - 1 ) ) = ( 0..^ ( # ` F ) ) )
42	41	raleqdv 3144	. . . . . . . . . 10 |- ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) <-> A. i e. ( 0..^ ( # ` F ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) )
43	42	3adant3 1081	. . . . . . . . 9 |- ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) <-> A. i e. ( 0..^ ( # ` F ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) )
44	43	adantl 482	. . . . . . . 8 |- ( ( G e. UPGraph /\ ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) <-> A. i e. ( 0..^ ( # ` F ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) )
45	28, 44	mpbird 247	. . . . . . 7 |- ( ( G e. UPGraph /\ ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) )
46	45	adantr 481	. . . . . 6 |- ( ( ( G e. UPGraph /\ ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) /\ P =/= (/) ) -> A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) )
47		eqid 2622	. . . . . . 7 |- (Edg ` G ) = (Edg ` G )
48	2, 47	iswwlks 26728	. . . . . 6 |- ( P e. (WWalks ` G ) <-> ( P =/= (/) /\ P e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. (Edg ` G ) ) )
49	5, 8, 46, 48	syl3anbrc 1246	. . . . 5 |- ( ( ( G e. UPGraph /\ ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) /\ P =/= (/) ) -> P e. (WWalks ` G ) )
50	49	ex 450	. . . 4 |- ( ( G e. UPGraph /\ ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> ( P =/= (/) -> P e. (WWalks ` G ) ) )
51	50	ex 450	. . 3 |- ( G e. UPGraph -> ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. i e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( P =/= (/) -> P e. (WWalks ` G ) ) ) )
52	4, 51	sylbid 230	. 2 |- ( G e. UPGraph -> ( F (Walks ` G ) P -> ( P =/= (/) -> P e. (WWalks ` G ) ) ) )
53	1, 52	mpdi 45	1 |- ( G e. UPGraph -> ( F (Walks ` G ) P -> P e. (WWalks ` 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   ran crn 5115   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   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   UHGraph cuhgr 25951   UPGraph cupgr 25975  Walkscwlks 26492  WWalkscwwlks 26717

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

This theorem is referenced by:  wlklnwwlkln1  26754  wlkiswwlks  26762  wlkiswwlkupgr  26764  elwspths2spth  26862