Theorem clwlksfoclwwlk 26963

Description: There is an onto function between the set of closed walks (defined as words) of length n and the set of closed walks of length n. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 2-May-2021.)

Hypotheses

Ref	Expression
clwlksfclwwlk.1	|- A = ( 1st ` c )
clwlksfclwwlk.2	|- B = ( 2nd ` c )
clwlksfclwwlk.c	|- C = { c e. (ClWalks ` G ) | ( # ` A ) = N }
clwlksfclwwlk.f	|- F = ( c e. C |-> ( B substr <. 0 , ( # ` A ) >. ) )

Assertion

Ref	Expression
clwlksfoclwwlk	|- ( ( G e. FinUSGraph /\ N e. Prime ) -> F : C -onto-> ( N ClWWalksN G ) )

Distinct variable groups:   G, c   N, c   C, c   F, c

Allowed substitution hints:   A( c)   B( c)

Proof of Theorem clwlksfoclwwlk

Dummy variables f w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwlksfclwwlk.1	. . 3 |- A = ( 1st ` c )
2		clwlksfclwwlk.2	. . 3 |- B = ( 2nd ` c )
3		clwlksfclwwlk.c	. . 3 |- C = { c e. (ClWalks ` G ) | ( # ` A ) = N }
4		clwlksfclwwlk.f	. . 3 |- F = ( c e. C |-> ( B substr <. 0 , ( # ` A ) >. ) )
5	1, 2, 3, 4	clwlksfclwwlk 26962	. 2 |- ( ( G e. FinUSGraph /\ N e. Prime ) -> F : C --> ( N ClWWalksN G ) )
6		eqid 2622	. . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
7	6	clwwlknbp 26885	. . . . 5 |- ( w e. ( N ClWWalksN G ) -> ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) )
8	7	adantl 482	. . . 4 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ w e. ( N ClWWalksN G ) ) -> ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) )
9		prmnn 15388	. . . . . . . . 9 |- ( N e. Prime -> N e. NN )
10	9	ad2antlr 763	. . . . . . . 8 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> N e. NN )
11		isclwwlksn 26882	. . . . . . . 8 |- ( N e. NN -> ( w e. ( N ClWWalksN G ) <-> ( w e. (ClWWalks ` G ) /\ ( # ` w ) = N ) ) )
12	10, 11	syl 17	. . . . . . 7 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> ( w e. ( N ClWWalksN G ) <-> ( w e. (ClWWalks ` G ) /\ ( # ` w ) = N ) ) )
13		fusgrusgr 26214	. . . . . . . . . . . . 13 |- ( G e. FinUSGraph -> G e. USGraph )
14		usgruspgr 26073	. . . . . . . . . . . . 13 |- ( G e. USGraph -> G e. USPGraph )
15	13, 14	syl 17	. . . . . . . . . . . 12 |- ( G e. FinUSGraph -> G e. USPGraph )
16	15	adantr 481	. . . . . . . . . . 11 |- ( ( G e. FinUSGraph /\ N e. Prime ) -> G e. USPGraph )
17	16	adantr 481	. . . . . . . . . 10 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> G e. USPGraph )
18		simprl 794	. . . . . . . . . 10 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> w e. Word (Vtx ` G ) )
19		eleq1 2689	. . . . . . . . . . . . . . 15 |- ( ( # ` w ) = N -> ( ( # ` w ) e. Prime <-> N e. Prime ) )
20		prmnn 15388	. . . . . . . . . . . . . . . 16 |- ( ( # ` w ) e. Prime -> ( # ` w ) e. NN )
21	20	nnge1d 11063	. . . . . . . . . . . . . . 15 |- ( ( # ` w ) e. Prime -> 1 <_ ( # ` w ) )
22	19, 21	syl6bir 244	. . . . . . . . . . . . . 14 |- ( ( # ` w ) = N -> ( N e. Prime -> 1 <_ ( # ` w ) ) )
23	22	adantl 482	. . . . . . . . . . . . 13 |- ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) -> ( N e. Prime -> 1 <_ ( # ` w ) ) )
24	23	com12 32	. . . . . . . . . . . 12 |- ( N e. Prime -> ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) -> 1 <_ ( # ` w ) ) )
25	24	adantl 482	. . . . . . . . . . 11 |- ( ( G e. FinUSGraph /\ N e. Prime ) -> ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) -> 1 <_ ( # ` w ) ) )
26	25	imp 445	. . . . . . . . . 10 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> 1 <_ ( # ` w ) )
27		eqid 2622	. . . . . . . . . . 11 |- (iEdg ` G ) = (iEdg ` G )
28	6, 27	clwlkclwwlk2 26904	. . . . . . . . . 10 |- ( ( G e. USPGraph /\ w e. Word (Vtx ` G ) /\ 1 <_ ( # ` w ) ) -> ( E. f f (ClWalks ` G ) ( w ++ <" ( w ` 0 ) "> ) <-> w e. (ClWWalks ` G ) ) )
29	17, 18, 26, 28	syl3anc 1326	. . . . . . . . 9 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> ( E. f f (ClWalks ` G ) ( w ++ <" ( w ` 0 ) "> ) <-> w e. (ClWWalks ` G ) ) )
30	29	bicomd 213	. . . . . . . 8 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> ( w e. (ClWWalks ` G ) <-> E. f f (ClWalks ` G ) ( w ++ <" ( w ` 0 ) "> ) ) )
31	30	anbi1d 741	. . . . . . 7 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> ( ( w e. (ClWWalks ` G ) /\ ( # ` w ) = N ) <-> ( E. f f (ClWalks ` G ) ( w ++ <" ( w ` 0 ) "> ) /\ ( # ` w ) = N ) ) )
32	12, 31	bitrd 268	. . . . . 6 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> ( w e. ( N ClWWalksN G ) <-> ( E. f f (ClWalks ` G ) ( w ++ <" ( w ` 0 ) "> ) /\ ( # ` w ) = N ) ) )
33		df-br 4654	. . . . . . . . 9 |- ( f (ClWalks ` G ) ( w ++ <" ( w ` 0 ) "> ) <-> <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) )
34		simpl 473	. . . . . . . . . . . . 13 |- ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) -> <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) )
35	9	nnge1d 11063	. . . . . . . . . . . . . . . . . . 19 |- ( N e. Prime -> 1 <_ N )
36	35	ad2antlr 763	. . . . . . . . . . . . . . . . . 18 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> 1 <_ N )
37		breq2 4657	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` w ) = N -> ( 1 <_ ( # ` w ) <-> 1 <_ N ) )
38	37	ad2antll 765	. . . . . . . . . . . . . . . . . 18 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> ( 1 <_ ( # ` w ) <-> 1 <_ N ) )
39	36, 38	mpbird 247	. . . . . . . . . . . . . . . . 17 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> 1 <_ ( # ` w ) )
40	18, 39	jca 554	. . . . . . . . . . . . . . . 16 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> ( w e. Word (Vtx ` G ) /\ 1 <_ ( # ` w ) ) )
41		clwlkwlk 26671	. . . . . . . . . . . . . . . 16 |- ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (Walks ` G ) )
42		wlklenvclwlk 26551	. . . . . . . . . . . . . . . 16 |- ( ( w e. Word (Vtx ` G ) /\ 1 <_ ( # ` w ) ) -> ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (Walks ` G ) -> ( # ` f ) = ( # ` w ) ) )
43	40, 41, 42	syl2im 40	. . . . . . . . . . . . . . 15 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> ( # ` f ) = ( # ` w ) ) )
44	43	impcom 446	. . . . . . . . . . . . . 14 |- ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) -> ( # ` f ) = ( # ` w ) )
45		vex 3203	. . . . . . . . . . . . . . . . . 18 |- f e. _V
46		ovex 6678	. . . . . . . . . . . . . . . . . 18 |- ( w ++ <" ( w ` 0 ) "> ) e. _V
47	45, 46	op1st 7176	. . . . . . . . . . . . . . . . 17 |- ( 1st ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) = f
48	47	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> ( 1st ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) = f )
49	48	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> ( # ` ( 1st ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) ) = ( # ` f ) )
50	49	adantl 482	. . . . . . . . . . . . . 14 |- ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) -> ( # ` ( 1st ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) ) = ( # ` f ) )
51		eqcom 2629	. . . . . . . . . . . . . . . . 17 |- ( ( # ` w ) = N <-> N = ( # ` w ) )
52	51	biimpi 206	. . . . . . . . . . . . . . . 16 |- ( ( # ` w ) = N -> N = ( # ` w ) )
53	52	ad2antll 765	. . . . . . . . . . . . . . 15 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> N = ( # ` w ) )
54	53	adantl 482	. . . . . . . . . . . . . 14 |- ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) -> N = ( # ` w ) )
55	44, 50, 54	3eqtr4d 2666	. . . . . . . . . . . . 13 |- ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) -> ( # ` ( 1st ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) ) = N )
56	1	fveq2i 6194	. . . . . . . . . . . . . . . 16 |- ( # ` A ) = ( # ` ( 1st ` c ) )
57	56	eqeq1i 2627	. . . . . . . . . . . . . . 15 |- ( ( # ` A ) = N <-> ( # ` ( 1st ` c ) ) = N )
58		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( c = <. f , ( w ++ <" ( w ` 0 ) "> ) >. -> ( 1st ` c ) = ( 1st ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) )
59	58	fveq2d 6195	. . . . . . . . . . . . . . . 16 |- ( c = <. f , ( w ++ <" ( w ` 0 ) "> ) >. -> ( # ` ( 1st ` c ) ) = ( # ` ( 1st ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) ) )
60	59	eqeq1d 2624	. . . . . . . . . . . . . . 15 |- ( c = <. f , ( w ++ <" ( w ` 0 ) "> ) >. -> ( ( # ` ( 1st ` c ) ) = N <-> ( # ` ( 1st ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) ) = N ) )
61	57, 60	syl5bb 272	. . . . . . . . . . . . . 14 |- ( c = <. f , ( w ++ <" ( w ` 0 ) "> ) >. -> ( ( # ` A ) = N <-> ( # ` ( 1st ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) ) = N ) )
62	61, 3	elrab2 3366	. . . . . . . . . . . . 13 |- ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. C <-> ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( # ` ( 1st ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) ) = N ) )
63	34, 55, 62	sylanbrc 698	. . . . . . . . . . . 12 |- ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) -> <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. C )
64	44	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) /\ <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) ) -> ( # ` f ) = ( # ` w ) )
65	64	opeq2d 4409	. . . . . . . . . . . . . . . . 17 |- ( ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) /\ <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) ) -> <. 0 , ( # ` f ) >. = <. 0 , ( # ` w ) >. )
66	65	oveq2d 6666	. . . . . . . . . . . . . . . 16 |- ( ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) /\ <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) ) -> ( ( w ++ <" ( w ` 0 ) "> ) substr <. 0 , ( # ` f ) >. ) = ( ( w ++ <" ( w ` 0 ) "> ) substr <. 0 , ( # ` w ) >. ) )
67		simpr 477	. . . . . . . . . . . . . . . . . 18 |- ( ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) /\ <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) ) -> <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) )
68	43	adantl 482	. . . . . . . . . . . . . . . . . . . 20 |- ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) -> ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> ( # ` f ) = ( # ` w ) ) )
69		eqeq2 2633	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( N = ( # ` w ) -> ( ( # ` f ) = N <-> ( # ` f ) = ( # ` w ) ) )
70	69	eqcoms 2630	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( # ` w ) = N -> ( ( # ` f ) = N <-> ( # ` f ) = ( # ` w ) ) )
71	70	imbi2d 330	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( # ` w ) = N -> ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> ( # ` f ) = N ) <-> ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> ( # ` f ) = ( # ` w ) ) ) )
72	71	ad2antll 765	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> ( # ` f ) = N ) <-> ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> ( # ` f ) = ( # ` w ) ) ) )
73	72	adantl 482	. . . . . . . . . . . . . . . . . . . 20 |- ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) -> ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> ( # ` f ) = N ) <-> ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> ( # ` f ) = ( # ` w ) ) ) )
74	68, 73	mpbird 247	. . . . . . . . . . . . . . . . . . 19 |- ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) -> ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> ( # ` f ) = N ) )
75	74	imp 445	. . . . . . . . . . . . . . . . . 18 |- ( ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) /\ <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) ) -> ( # ` f ) = N )
76	47	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( c = <. f , ( w ++ <" ( w ` 0 ) "> ) >. -> ( 1st ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) = f )
77	76	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . 22 |- ( c = <. f , ( w ++ <" ( w ` 0 ) "> ) >. -> ( # ` ( 1st ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) ) = ( # ` f ) )
78	59, 77	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . 21 |- ( c = <. f , ( w ++ <" ( w ` 0 ) "> ) >. -> ( # ` ( 1st ` c ) ) = ( # ` f ) )
79	78	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . 20 |- ( c = <. f , ( w ++ <" ( w ` 0 ) "> ) >. -> ( ( # ` ( 1st ` c ) ) = N <-> ( # ` f ) = N ) )
80	57, 79	syl5bb 272	. . . . . . . . . . . . . . . . . . 19 |- ( c = <. f , ( w ++ <" ( w ` 0 ) "> ) >. -> ( ( # ` A ) = N <-> ( # ` f ) = N ) )
81	80, 3	elrab2 3366	. . . . . . . . . . . . . . . . . 18 |- ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. C <-> ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( # ` f ) = N ) )
82	67, 75, 81	sylanbrc 698	. . . . . . . . . . . . . . . . 17 |- ( ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) /\ <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) ) -> <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. C )
83		ovex 6678	. . . . . . . . . . . . . . . . 17 |- ( ( w ++ <" ( w ` 0 ) "> ) substr <. 0 , ( # ` f ) >. ) e. _V
84	56	opeq2i 4406	. . . . . . . . . . . . . . . . . . . 20 |- <. 0 , ( # ` A ) >. = <. 0 , ( # ` ( 1st ` c ) ) >.
85	2, 84	oveq12i 6662	. . . . . . . . . . . . . . . . . . 19 |- ( B substr <. 0 , ( # ` A ) >. ) = ( ( 2nd ` c ) substr <. 0 , ( # ` ( 1st ` c ) ) >. )
86		fveq2 6191	. . . . . . . . . . . . . . . . . . . . 21 |- ( c = <. f , ( w ++ <" ( w ` 0 ) "> ) >. -> ( 2nd ` c ) = ( 2nd ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) )
87	59	opeq2d 4409	. . . . . . . . . . . . . . . . . . . . 21 |- ( c = <. f , ( w ++ <" ( w ` 0 ) "> ) >. -> <. 0 , ( # ` ( 1st ` c ) ) >. = <. 0 , ( # ` ( 1st ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) ) >. )
88	86, 87	oveq12d 6668	. . . . . . . . . . . . . . . . . . . 20 |- ( c = <. f , ( w ++ <" ( w ` 0 ) "> ) >. -> ( ( 2nd ` c ) substr <. 0 , ( # ` ( 1st ` c ) ) >. ) = ( ( 2nd ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) substr <. 0 , ( # ` ( 1st ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) ) >. ) )
89	45, 46	op2nd 7177	. . . . . . . . . . . . . . . . . . . . 21 |- ( 2nd ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) = ( w ++ <" ( w ` 0 ) "> )
90	47	fveq2i 6194	. . . . . . . . . . . . . . . . . . . . . 22 |- ( # ` ( 1st ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) ) = ( # ` f )
91	90	opeq2i 4406	. . . . . . . . . . . . . . . . . . . . 21 |- <. 0 , ( # ` ( 1st ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) ) >. = <. 0 , ( # ` f ) >.
92	89, 91	oveq12i 6662	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 2nd ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) substr <. 0 , ( # ` ( 1st ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) ) >. ) = ( ( w ++ <" ( w ` 0 ) "> ) substr <. 0 , ( # ` f ) >. )
93	88, 92	syl6eq 2672	. . . . . . . . . . . . . . . . . . 19 |- ( c = <. f , ( w ++ <" ( w ` 0 ) "> ) >. -> ( ( 2nd ` c ) substr <. 0 , ( # ` ( 1st ` c ) ) >. ) = ( ( w ++ <" ( w ` 0 ) "> ) substr <. 0 , ( # ` f ) >. ) )
94	85, 93	syl5eq 2668	. . . . . . . . . . . . . . . . . 18 |- ( c = <. f , ( w ++ <" ( w ` 0 ) "> ) >. -> ( B substr <. 0 , ( # ` A ) >. ) = ( ( w ++ <" ( w ` 0 ) "> ) substr <. 0 , ( # ` f ) >. ) )
95	94, 4	fvmptg 6280	. . . . . . . . . . . . . . . . 17 |- ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. C /\ ( ( w ++ <" ( w ` 0 ) "> ) substr <. 0 , ( # ` f ) >. ) e. _V ) -> ( F ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) = ( ( w ++ <" ( w ` 0 ) "> ) substr <. 0 , ( # ` f ) >. ) )
96	82, 83, 95	sylancl 694	. . . . . . . . . . . . . . . 16 |- ( ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) /\ <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) ) -> ( F ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) = ( ( w ++ <" ( w ` 0 ) "> ) substr <. 0 , ( # ` f ) >. ) )
97	40	ad2antlr 763	. . . . . . . . . . . . . . . . 17 |- ( ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) /\ <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) ) -> ( w e. Word (Vtx ` G ) /\ 1 <_ ( # ` w ) ) )
98		simpl 473	. . . . . . . . . . . . . . . . . . 19 |- ( ( w e. Word (Vtx ` G ) /\ 1 <_ ( # ` w ) ) -> w e. Word (Vtx ` G ) )
99		wrdsymb1 13342	. . . . . . . . . . . . . . . . . . . 20 |- ( ( w e. Word (Vtx ` G ) /\ 1 <_ ( # ` w ) ) -> ( w ` 0 ) e. (Vtx ` G ) )
100	99	s1cld 13383	. . . . . . . . . . . . . . . . . . 19 |- ( ( w e. Word (Vtx ` G ) /\ 1 <_ ( # ` w ) ) -> <" ( w ` 0 ) "> e. Word (Vtx ` G ) )
101		eqidd 2623	. . . . . . . . . . . . . . . . . . 19 |- ( ( w e. Word (Vtx ` G ) /\ 1 <_ ( # ` w ) ) -> ( # ` w ) = ( # ` w ) )
102		swrdccatid 13497	. . . . . . . . . . . . . . . . . . 19 |- ( ( w e. Word (Vtx ` G ) /\ <" ( w ` 0 ) "> e. Word (Vtx ` G ) /\ ( # ` w ) = ( # ` w ) ) -> ( ( w ++ <" ( w ` 0 ) "> ) substr <. 0 , ( # ` w ) >. ) = w )
103	98, 100, 101, 102	syl3anc 1326	. . . . . . . . . . . . . . . . . 18 |- ( ( w e. Word (Vtx ` G ) /\ 1 <_ ( # ` w ) ) -> ( ( w ++ <" ( w ` 0 ) "> ) substr <. 0 , ( # ` w ) >. ) = w )
104	103	eqcomd 2628	. . . . . . . . . . . . . . . . 17 |- ( ( w e. Word (Vtx ` G ) /\ 1 <_ ( # ` w ) ) -> w = ( ( w ++ <" ( w ` 0 ) "> ) substr <. 0 , ( # ` w ) >. ) )
105	97, 104	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) /\ <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) ) -> w = ( ( w ++ <" ( w ` 0 ) "> ) substr <. 0 , ( # ` w ) >. ) )
106	66, 96, 105	3eqtr4rd 2667	. . . . . . . . . . . . . . 15 |- ( ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) /\ <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) ) -> w = ( F ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) )
107	106	ex 450	. . . . . . . . . . . . . 14 |- ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) -> ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> w = ( F ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) ) )
108	107	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) /\ c = <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) -> ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> w = ( F ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) ) )
109		fveq2 6191	. . . . . . . . . . . . . . . 16 |- ( c = <. f , ( w ++ <" ( w ` 0 ) "> ) >. -> ( F ` c ) = ( F ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) )
110	109	eqeq2d 2632	. . . . . . . . . . . . . . 15 |- ( c = <. f , ( w ++ <" ( w ` 0 ) "> ) >. -> ( w = ( F ` c ) <-> w = ( F ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) ) )
111	110	imbi2d 330	. . . . . . . . . . . . . 14 |- ( c = <. f , ( w ++ <" ( w ` 0 ) "> ) >. -> ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> w = ( F ` c ) ) <-> ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> w = ( F ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) ) ) )
112	111	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) /\ c = <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) -> ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> w = ( F ` c ) ) <-> ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> w = ( F ` <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) ) ) )
113	108, 112	mpbird 247	. . . . . . . . . . . 12 |- ( ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) /\ c = <. f , ( w ++ <" ( w ` 0 ) "> ) >. ) -> ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> w = ( F ` c ) ) )
114	63, 113	rspcimedv 3311	. . . . . . . . . . 11 |- ( ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) /\ ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) ) -> ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> E. c e. C w = ( F ` c ) ) )
115	114	ex 450	. . . . . . . . . 10 |- ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> E. c e. C w = ( F ` c ) ) ) )
116	115	pm2.43b 55	. . . . . . . . 9 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> ( <. f , ( w ++ <" ( w ` 0 ) "> ) >. e. (ClWalks ` G ) -> E. c e. C w = ( F ` c ) ) )
117	33, 116	syl5bi 232	. . . . . . . 8 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> ( f (ClWalks ` G ) ( w ++ <" ( w ` 0 ) "> ) -> E. c e. C w = ( F ` c ) ) )
118	117	exlimdv 1861	. . . . . . 7 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> ( E. f f (ClWalks ` G ) ( w ++ <" ( w ` 0 ) "> ) -> E. c e. C w = ( F ` c ) ) )
119	118	adantrd 484	. . . . . 6 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> ( ( E. f f (ClWalks ` G ) ( w ++ <" ( w ` 0 ) "> ) /\ ( # ` w ) = N ) -> E. c e. C w = ( F ` c ) ) )
120	32, 119	sylbid 230	. . . . 5 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) ) -> ( w e. ( N ClWWalksN G ) -> E. c e. C w = ( F ` c ) ) )
121	120	impancom 456	. . . 4 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ w e. ( N ClWWalksN G ) ) -> ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = N ) -> E. c e. C w = ( F ` c ) ) )
122	8, 121	mpd 15	. . 3 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ w e. ( N ClWWalksN G ) ) -> E. c e. C w = ( F ` c ) )
123	122	ralrimiva 2966	. 2 |- ( ( G e. FinUSGraph /\ N e. Prime ) -> A. w e. ( N ClWWalksN G ) E. c e. C w = ( F ` c ) )
124		dffo3 6374	. 2 |- ( F : C -onto-> ( N ClWWalksN G ) <-> ( F : C --> ( N ClWWalksN G ) /\ A. w e. ( N ClWWalksN G ) E. c e. C w = ( F ` c ) ) )
125	5, 123, 124	sylanbrc 698	1 |- ( ( G e. FinUSGraph /\ N e. Prime ) -> F : C -onto-> ( N ClWWalksN G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   0cc0 9936   1c1 9937   <_ cle 10075   NNcn 11020   #chash 13117  Word cword 13291   ++ cconcat 13293   <"cs1 13294   substr csubstr 13295   Primecprime 15385  Vtxcvtx 25874  iEdgciedg 25875   USPGraph cuspgr 26043   USGraph cusgr 26044   FinUSGraph cfusgr 26208  Walkscwlks 26492  ClWalkscclwlks 26666  ClWWalkscclwwlks 26875   ClWWalksN cclwwlksn 26876

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  ax-pre-sup 10014

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-sup 8348  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-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-prm 15386  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-uspgr 26045  df-usgr 26046  df-fusgr 26209  df-wlks 26495  df-clwlks 26667  df-clwwlks 26877  df-clwwlksn 26878

This theorem is referenced by:  clwlksf1oclwwlk  26970