Theorem clwlkclwwlk 26903

Description: A closed walk as word of length at least 2 corresponds to a closed walk in a simple pseudograph. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 24-Apr-2021.)

Hypotheses

Ref	Expression
clwlkclwwlk.v	|- V = (Vtx ` G )
clwlkclwwlk.e	|- E = (iEdg ` G )

Assertion

Ref	Expression
clwlkclwwlk	|- ( ( G e. USPGraph /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( E. f f (ClWalks ` G ) P <-> ( ( lastS ` P ) = ( P ` 0 ) /\ ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. (ClWWalks ` G ) ) ) )

Distinct variable groups:   f, E   P, f   f, V   f, G

Proof of Theorem clwlkclwwlk

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwlkclwwlk.e	. . . . . 6 |- E = (iEdg ` G )
2	1	uspgrf1oedg 26068	. . . . 5 |- ( G e. USPGraph -> E : dom E -1-1-onto-> (Edg ` G ) )
3		f1of1 6136	. . . . 5 |- ( E : dom E -1-1-onto-> (Edg ` G ) -> E : dom E -1-1-> (Edg ` G ) )
4	2, 3	syl 17	. . . 4 |- ( G e. USPGraph -> E : dom E -1-1-> (Edg ` G ) )
5		clwlkclwwlklem3 26902	. . . 4 |- ( ( E : dom E -1-1-> (Edg ` G ) /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( E. f ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) ) )
6	4, 5	syl3an1 1359	. . 3 |- ( ( G e. USPGraph /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( E. f ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) ) )
7		lencl 13324	. . . . . . . . . . . . . 14 |- ( P e. Word V -> ( # ` P ) e. NN0 )
8		ige2m1fz 12430	. . . . . . . . . . . . . 14 |- ( ( ( # ` P ) e. NN0 /\ 2 <_ ( # ` P ) ) -> ( ( # ` P ) - 1 ) e. ( 0 ... ( # ` P ) ) )
9	7, 8	sylan 488	. . . . . . . . . . . . 13 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( # ` P ) - 1 ) e. ( 0 ... ( # ` P ) ) )
10		swrd0len 13422	. . . . . . . . . . . . 13 |- ( ( P e. Word V /\ ( ( # ` P ) - 1 ) e. ( 0 ... ( # ` P ) ) ) -> ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) = ( ( # ` P ) - 1 ) )
11	9, 10	syldan 487	. . . . . . . . . . . 12 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) = ( ( # ` P ) - 1 ) )
12	7	nn0cnd 11353	. . . . . . . . . . . . . . . 16 |- ( P e. Word V -> ( # ` P ) e. CC )
13		1cnd 10056	. . . . . . . . . . . . . . . 16 |- ( P e. Word V -> 1 e. CC )
14	12, 13	subcld 10392	. . . . . . . . . . . . . . 15 |- ( P e. Word V -> ( ( # ` P ) - 1 ) e. CC )
15	14	subid1d 10381	. . . . . . . . . . . . . 14 |- ( P e. Word V -> ( ( ( # ` P ) - 1 ) - 0 ) = ( ( # ` P ) - 1 ) )
16	15	eqcomd 2628	. . . . . . . . . . . . 13 |- ( P e. Word V -> ( ( # ` P ) - 1 ) = ( ( ( # ` P ) - 1 ) - 0 ) )
17	16	adantr 481	. . . . . . . . . . . 12 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( # ` P ) - 1 ) = ( ( ( # ` P ) - 1 ) - 0 ) )
18	11, 17	eqtrd 2656	. . . . . . . . . . 11 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) = ( ( ( # ` P ) - 1 ) - 0 ) )
19	18	oveq1d 6665	. . . . . . . . . 10 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) = ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) )
20	19	oveq2d 6666	. . . . . . . . 9 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) = ( 0..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) )
21	11	oveq1d 6665	. . . . . . . . . . . . 13 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) = ( ( ( # ` P ) - 1 ) - 1 ) )
22	21	oveq2d 6666	. . . . . . . . . . . 12 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) = ( 0..^ ( ( ( # ` P ) - 1 ) - 1 ) ) )
23	22	eleq2d 2687	. . . . . . . . . . 11 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) <-> i e. ( 0..^ ( ( ( # ` P ) - 1 ) - 1 ) ) ) )
24		simpll 790	. . . . . . . . . . . . . . 15 |- ( ( ( P e. Word V /\ 2 <_ ( # ` P ) ) /\ i e. ( 0..^ ( ( ( # ` P ) - 1 ) - 1 ) ) ) -> P e. Word V )
25		wrdlenge2n0 13341	. . . . . . . . . . . . . . . 16 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> P =/= (/) )
26	25	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( P e. Word V /\ 2 <_ ( # ` P ) ) /\ i e. ( 0..^ ( ( ( # ` P ) - 1 ) - 1 ) ) ) -> P =/= (/) )
27		nn0z 11400	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` P ) e. NN0 -> ( # ` P ) e. ZZ )
28		peano2zm 11420	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` P ) e. ZZ -> ( ( # ` P ) - 1 ) e. ZZ )
29	27, 28	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( # ` P ) e. NN0 -> ( ( # ` P ) - 1 ) e. ZZ )
30	7, 29	syl 17	. . . . . . . . . . . . . . . . 17 |- ( P e. Word V -> ( ( # ` P ) - 1 ) e. ZZ )
31	30	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( # ` P ) - 1 ) e. ZZ )
32		elfzom1elfzo 12535	. . . . . . . . . . . . . . . 16 |- ( ( ( ( # ` P ) - 1 ) e. ZZ /\ i e. ( 0..^ ( ( ( # ` P ) - 1 ) - 1 ) ) ) -> i e. ( 0..^ ( ( # ` P ) - 1 ) ) )
33	31, 32	sylan 488	. . . . . . . . . . . . . . 15 |- ( ( ( P e. Word V /\ 2 <_ ( # ` P ) ) /\ i e. ( 0..^ ( ( ( # ` P ) - 1 ) - 1 ) ) ) -> i e. ( 0..^ ( ( # ` P ) - 1 ) ) )
34		swrdtrcfv 13441	. . . . . . . . . . . . . . 15 |- ( ( P e. Word V /\ P =/= (/) /\ i e. ( 0..^ ( ( # ` P ) - 1 ) ) ) -> ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) = ( P ` i ) )
35	24, 26, 33, 34	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ( ( P e. Word V /\ 2 <_ ( # ` P ) ) /\ i e. ( 0..^ ( ( ( # ` P ) - 1 ) - 1 ) ) ) -> ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) = ( P ` i ) )
36	7	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( # ` P ) e. NN0 )
37		elfzom1elp1fzo 12534	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( # ` P ) - 1 ) e. ZZ /\ i e. ( 0..^ ( ( ( # ` P ) - 1 ) - 1 ) ) ) -> ( i + 1 ) e. ( 0..^ ( ( # ` P ) - 1 ) ) )
38	29, 37	sylan 488	. . . . . . . . . . . . . . . 16 |- ( ( ( # ` P ) e. NN0 /\ i e. ( 0..^ ( ( ( # ` P ) - 1 ) - 1 ) ) ) -> ( i + 1 ) e. ( 0..^ ( ( # ` P ) - 1 ) ) )
39	36, 38	sylan 488	. . . . . . . . . . . . . . 15 |- ( ( ( P e. Word V /\ 2 <_ ( # ` P ) ) /\ i e. ( 0..^ ( ( ( # ` P ) - 1 ) - 1 ) ) ) -> ( i + 1 ) e. ( 0..^ ( ( # ` P ) - 1 ) ) )
40		swrdtrcfv 13441	. . . . . . . . . . . . . . 15 |- ( ( P e. Word V /\ P =/= (/) /\ ( i + 1 ) e. ( 0..^ ( ( # ` P ) - 1 ) ) ) -> ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) = ( P ` ( i + 1 ) ) )
41	24, 26, 39, 40	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ( ( P e. Word V /\ 2 <_ ( # ` P ) ) /\ i e. ( 0..^ ( ( ( # ` P ) - 1 ) - 1 ) ) ) -> ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) = ( P ` ( i + 1 ) ) )
42	35, 41	preq12d 4276	. . . . . . . . . . . . 13 |- ( ( ( P e. Word V /\ 2 <_ ( # ` P ) ) /\ i e. ( 0..^ ( ( ( # ` P ) - 1 ) - 1 ) ) ) -> { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } = { ( P ` i ) , ( P ` ( i + 1 ) ) } )
43	42	eleq1d 2686	. . . . . . . . . . . 12 |- ( ( ( P e. Word V /\ 2 <_ ( # ` P ) ) /\ i e. ( 0..^ ( ( ( # ` P ) - 1 ) - 1 ) ) ) -> ( { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. ran E <-> { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) )
44	43	ex 450	. . . . . . . . . . 11 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( i e. ( 0..^ ( ( ( # ` P ) - 1 ) - 1 ) ) -> ( { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. ran E <-> { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) ) )
45	23, 44	sylbid 230	. . . . . . . . . 10 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) -> ( { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. ran E <-> { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) ) )
46	45	imp 445	. . . . . . . . 9 |- ( ( ( P e. Word V /\ 2 <_ ( # ` P ) ) /\ i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) ) -> ( { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. ran E <-> { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) )
47	20, 46	raleqbidva 3154	. . . . . . . 8 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( A. i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. ran E <-> A. i e. ( 0..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) )
48		swrdtrcfvl 13450	. . . . . . . . . 10 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) = ( P ` ( ( # ` P ) - 2 ) ) )
49		swrdtrcfv0 13442	. . . . . . . . . 10 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) = ( P ` 0 ) )
50	48, 49	preq12d 4276	. . . . . . . . 9 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> { ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) } = { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } )
51	50	eleq1d 2686	. . . . . . . 8 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( { ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) } e. ran E <-> { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) )
52	47, 51	anbi12d 747	. . . . . . 7 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( A. i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) } e. ran E ) <-> ( A. i e. ( 0..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) )
53	52	bicomd 213	. . . . . 6 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( A. i e. ( 0..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) <-> ( A. i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) } e. ran E ) ) )
54	53	3adant1 1079	. . . . 5 |- ( ( G e. USPGraph /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( A. i e. ( 0..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) <-> ( A. i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) } e. ran E ) ) )
55		swrdcl 13419	. . . . . . 7 |- ( P e. Word V -> ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. Word V )
56	55	3ad2ant2 1083	. . . . . 6 |- ( ( G e. USPGraph /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. Word V )
57	56	3biant1d 1441	. . . . 5 |- ( ( G e. USPGraph /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( A. i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) } e. ran E ) <-> ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. Word V /\ A. i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) } e. ran E ) ) )
58	54, 57	bitrd 268	. . . 4 |- ( ( G e. USPGraph /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( A. i e. ( 0..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) <-> ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. Word V /\ A. i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) } e. ran E ) ) )
59	58	anbi2d 740	. . 3 |- ( ( G e. USPGraph /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( ( lastS ` P ) = ( P ` 0 ) /\ ( A. i e. ( 0..^ ( ( ( ( # ` P ) - 1 ) - 0 ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( P ` ( ( # ` P ) - 2 ) ) , ( P ` 0 ) } e. ran E ) ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. Word V /\ A. i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) } e. ran E ) ) ) )
60	6, 59	bitrd 268	. 2 |- ( ( G e. USPGraph /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( E. f ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. Word V /\ A. i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) } e. ran E ) ) ) )
61		uspgrupgr 26071	. . . . . 6 |- ( G e. USPGraph -> G e. UPGraph )
62		clwlkclwwlk.v	. . . . . . . 8 |- V = (Vtx ` G )
63	62, 1	isclwlkupgr 26674	. . . . . . 7 |- ( G e. UPGraph -> ( f (ClWalks ` G ) P <-> ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V ) /\ ( A. i e. ( 0..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) ) ) )
64		3an4anass 1291	. . . . . . 7 |- ( ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) <-> ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V ) /\ ( A. i e. ( 0..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) ) )
65	63, 64	syl6bbr 278	. . . . . 6 |- ( G e. UPGraph -> ( f (ClWalks ` G ) P <-> ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) ) )
66	61, 65	syl 17	. . . . 5 |- ( G e. USPGraph -> ( f (ClWalks ` G ) P <-> ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) ) )
67	66	adantr 481	. . . 4 |- ( ( G e. USPGraph /\ P e. Word V ) -> ( f (ClWalks ` G ) P <-> ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) ) )
68	67	exbidv 1850	. . 3 |- ( ( G e. USPGraph /\ P e. Word V ) -> ( E. f f (ClWalks ` G ) P <-> E. f ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) ) )
69	68	3adant3 1081	. 2 |- ( ( G e. USPGraph /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( E. f f (ClWalks ` G ) P <-> E. f ( ( f e. Word dom E /\ P : ( 0 ... ( # ` f ) ) --> V /\ A. i e. ( 0..^ ( # ` f ) ) ( E ` ( f ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ ( P ` 0 ) = ( P ` ( # ` f ) ) ) ) )
70		eqid 2622	. . . . . 6 |- (Edg ` G ) = (Edg ` G )
71	62, 70	isclwwlks 26880	. . . . 5 |- ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. (ClWWalks ` G ) <-> ( ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. Word V /\ ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) =/= (/) ) /\ A. i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) } e. (Edg ` G ) ) )
72		simpl 473	. . . . . . . . . . 11 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> P e. Word V )
73		nn0ge2m1nn 11360	. . . . . . . . . . . 12 |- ( ( ( # ` P ) e. NN0 /\ 2 <_ ( # ` P ) ) -> ( ( # ` P ) - 1 ) e. NN )
74	7, 73	sylan 488	. . . . . . . . . . 11 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( # ` P ) - 1 ) e. NN )
75		nn0re 11301	. . . . . . . . . . . . . . 15 |- ( ( # ` P ) e. NN0 -> ( # ` P ) e. RR )
76	75	lem1d 10957	. . . . . . . . . . . . . 14 |- ( ( # ` P ) e. NN0 -> ( ( # ` P ) - 1 ) <_ ( # ` P ) )
77	76	a1d 25	. . . . . . . . . . . . 13 |- ( ( # ` P ) e. NN0 -> ( 2 <_ ( # ` P ) -> ( ( # ` P ) - 1 ) <_ ( # ` P ) ) )
78	7, 77	syl 17	. . . . . . . . . . . 12 |- ( P e. Word V -> ( 2 <_ ( # ` P ) -> ( ( # ` P ) - 1 ) <_ ( # ` P ) ) )
79	78	imp 445	. . . . . . . . . . 11 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( # ` P ) - 1 ) <_ ( # ` P ) )
80	72, 74, 79	3jca 1242	. . . . . . . . . 10 |- ( ( P e. Word V /\ 2 <_ ( # ` P ) ) -> ( P e. Word V /\ ( ( # ` P ) - 1 ) e. NN /\ ( ( # ` P ) - 1 ) <_ ( # ` P ) ) )
81	80	3adant1 1079	. . . . . . . . 9 |- ( ( G e. USPGraph /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( P e. Word V /\ ( ( # ` P ) - 1 ) e. NN /\ ( ( # ` P ) - 1 ) <_ ( # ` P ) ) )
82		swrdn0 13430	. . . . . . . . 9 |- ( ( P e. Word V /\ ( ( # ` P ) - 1 ) e. NN /\ ( ( # ` P ) - 1 ) <_ ( # ` P ) ) -> ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) =/= (/) )
83	81, 82	syl 17	. . . . . . . 8 |- ( ( G e. USPGraph /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) =/= (/) )
84	83	biantrud 528	. . . . . . 7 |- ( ( G e. USPGraph /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. Word V <-> ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. Word V /\ ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) =/= (/) ) ) )
85	84	bicomd 213	. . . . . 6 |- ( ( G e. USPGraph /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. Word V /\ ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) =/= (/) ) <-> ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. Word V ) )
86	85	3anbi1d 1403	. . . . 5 |- ( ( G e. USPGraph /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. Word V /\ ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) =/= (/) ) /\ A. i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) } e. (Edg ` G ) ) <-> ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. Word V /\ A. i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) } e. (Edg ` G ) ) ) )
87	71, 86	syl5bb 272	. . . 4 |- ( ( G e. USPGraph /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. (ClWWalks ` G ) <-> ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. Word V /\ A. i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) } e. (Edg ` G ) ) ) )
88		biid 251	. . . . 5 |- ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. Word V <-> ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. Word V )
89		edgval 25941	. . . . . . . 8 |- (Edg ` G ) = ran (iEdg ` G )
90	1	eqcomi 2631	. . . . . . . . 9 |- (iEdg ` G ) = E
91	90	rneqi 5352	. . . . . . . 8 |- ran (iEdg ` G ) = ran E
92	89, 91	eqtri 2644	. . . . . . 7 |- (Edg ` G ) = ran E
93	92	eleq2i 2693	. . . . . 6 |- ( { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. (Edg ` G ) <-> { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. ran E )
94	93	ralbii 2980	. . . . 5 |- ( A. i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. (Edg ` G ) <-> A. i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. ran E )
95	92	eleq2i 2693	. . . . 5 |- ( { ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) } e. (Edg ` G ) <-> { ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) } e. ran E )
96	88, 94, 95	3anbi123i 1251	. . . 4 |- ( ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. Word V /\ A. i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) } e. (Edg ` G ) ) <-> ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. Word V /\ A. i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) } e. ran E ) )
97	87, 96	syl6bb 276	. . 3 |- ( ( G e. USPGraph /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. (ClWWalks ` G ) <-> ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. Word V /\ A. i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) } e. ran E ) ) )
98	97	anbi2d 740	. 2 |- ( ( G e. USPGraph /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( ( ( lastS ` P ) = ( P ` 0 ) /\ ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. (ClWWalks ` G ) ) <-> ( ( lastS ` P ) = ( P ` 0 ) /\ ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. Word V /\ A. i e. ( 0..^ ( ( # ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) - 1 ) ) { ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` i ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ) , ( ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) ` 0 ) } e. ran E ) ) ) )
99	60, 69, 98	3bitr4d 300	1 |- ( ( G e. USPGraph /\ P e. Word V /\ 2 <_ ( # ` P ) ) -> ( E. f f (ClWalks ` G ) P <-> ( ( lastS ` P ) = ( P ` 0 ) /\ ( P substr <. 0 , ( ( # ` P ) - 1 ) >. ) e. (ClWWalks ` G ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   (/)c0 3915   {cpr 4179   <.cop 4183   class class class wbr 4653   dom cdm 5114   ran crn 5115   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   <_ cle 10075   - cmin 10266   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   lastS clsw 13292   substr csubstr 13295  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UPGraph cupgr 25975   USPGraph cuspgr 26043  ClWalkscclwlks 26666  ClWWalkscclwwlks 26875

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-lsw 13300  df-substr 13303  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-uspgr 26045  df-wlks 26495  df-clwlks 26667  df-clwwlks 26877

This theorem is referenced by:  clwlkclwwlk2  26904