Theorem clwlksf1clwwlklem 26968

Description: Lemma for clwlksf1clwwlk 26969. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-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
clwlksf1clwwlklem	|- ( ( N e. NN /\ U e. C /\ W e. C ) -> ( ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) -> A. y e. ( 0 ... N ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) ) )

Distinct variable groups:   G, c   N, c   W, c   C, c   F, c   y, G   y, N   U, c, y   y, W

Allowed substitution hints:   A( y, c)   B( y, c)   C( y)   F( y)

Proof of Theorem clwlksf1clwwlklem

Step	Hyp	Ref	Expression
1		clwlksfclwwlk.1	. . . . . . . . . . . 12 |- A = ( 1st ` c )
2		clwlksfclwwlk.2	. . . . . . . . . . . 12 |- B = ( 2nd ` c )
3		clwlksfclwwlk.c	. . . . . . . . . . . 12 |- C = { c e. (ClWalks ` G ) | ( # ` A ) = N }
4		clwlksfclwwlk.f	. . . . . . . . . . . 12 |- F = ( c e. C |-> ( B substr <. 0 , ( # ` A ) >. ) )
5	1, 2, 3, 4	clwlksf1clwwlklem3 26967	. . . . . . . . . . 11 |- ( W e. C -> ( 2nd ` W ) e. Word (Vtx ` G ) )
6	1, 2, 3, 4	clwlksf1clwwlklem3 26967	. . . . . . . . . . 11 |- ( U e. C -> ( 2nd ` U ) e. Word (Vtx ` G ) )
7	5, 6	anim12ci 591	. . . . . . . . . 10 |- ( ( W e. C /\ U e. C ) -> ( ( 2nd ` U ) e. Word (Vtx ` G ) /\ ( 2nd ` W ) e. Word (Vtx ` G ) ) )
8	7	adantr 481	. . . . . . . . 9 |- ( ( ( W e. C /\ U e. C ) /\ N e. NN ) -> ( ( 2nd ` U ) e. Word (Vtx ` G ) /\ ( 2nd ` W ) e. Word (Vtx ` G ) ) )
9		nnnn0 11299	. . . . . . . . . . 11 |- ( N e. NN -> N e. NN0 )
10	9	adantl 482	. . . . . . . . . 10 |- ( ( ( W e. C /\ U e. C ) /\ N e. NN ) -> N e. NN0 )
11	1, 2, 3, 4	clwlksf1clwwlklem1 26965	. . . . . . . . . . . 12 |- ( U e. C -> N <_ ( # ` ( 2nd ` U ) ) )
12	11	adantl 482	. . . . . . . . . . 11 |- ( ( W e. C /\ U e. C ) -> N <_ ( # ` ( 2nd ` U ) ) )
13	12	adantr 481	. . . . . . . . . 10 |- ( ( ( W e. C /\ U e. C ) /\ N e. NN ) -> N <_ ( # ` ( 2nd ` U ) ) )
14	1, 2, 3, 4	clwlksf1clwwlklem1 26965	. . . . . . . . . . . 12 |- ( W e. C -> N <_ ( # ` ( 2nd ` W ) ) )
15	14	adantr 481	. . . . . . . . . . 11 |- ( ( W e. C /\ U e. C ) -> N <_ ( # ` ( 2nd ` W ) ) )
16	15	adantr 481	. . . . . . . . . 10 |- ( ( ( W e. C /\ U e. C ) /\ N e. NN ) -> N <_ ( # ` ( 2nd ` W ) ) )
17	10, 13, 16	3jca 1242	. . . . . . . . 9 |- ( ( ( W e. C /\ U e. C ) /\ N e. NN ) -> ( N e. NN0 /\ N <_ ( # ` ( 2nd ` U ) ) /\ N <_ ( # ` ( 2nd ` W ) ) ) )
18	8, 17	jca 554	. . . . . . . 8 |- ( ( ( W e. C /\ U e. C ) /\ N e. NN ) -> ( ( ( 2nd ` U ) e. Word (Vtx ` G ) /\ ( 2nd ` W ) e. Word (Vtx ` G ) ) /\ ( N e. NN0 /\ N <_ ( # ` ( 2nd ` U ) ) /\ N <_ ( # ` ( 2nd ` W ) ) ) ) )
19	18	exp31 630	. . . . . . 7 |- ( W e. C -> ( U e. C -> ( N e. NN -> ( ( ( 2nd ` U ) e. Word (Vtx ` G ) /\ ( 2nd ` W ) e. Word (Vtx ` G ) ) /\ ( N e. NN0 /\ N <_ ( # ` ( 2nd ` U ) ) /\ N <_ ( # ` ( 2nd ` W ) ) ) ) ) ) )
20	19	3imp31 1257	. . . . . 6 |- ( ( N e. NN /\ U e. C /\ W e. C ) -> ( ( ( 2nd ` U ) e. Word (Vtx ` G ) /\ ( 2nd ` W ) e. Word (Vtx ` G ) ) /\ ( N e. NN0 /\ N <_ ( # ` ( 2nd ` U ) ) /\ N <_ ( # ` ( 2nd ` W ) ) ) ) )
21	20	adantr 481	. . . . 5 |- ( ( ( N e. NN /\ U e. C /\ W e. C ) /\ ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) ) -> ( ( ( 2nd ` U ) e. Word (Vtx ` G ) /\ ( 2nd ` W ) e. Word (Vtx ` G ) ) /\ ( N e. NN0 /\ N <_ ( # ` ( 2nd ` U ) ) /\ N <_ ( # ` ( 2nd ` W ) ) ) ) )
22	1, 2, 3, 4	clwlksfclwwlk1hashn 26959	. . . . . . . . . 10 |- ( U e. C -> ( # ` ( 1st ` U ) ) = N )
23	22	3ad2ant2 1083	. . . . . . . . 9 |- ( ( N e. NN /\ U e. C /\ W e. C ) -> ( # ` ( 1st ` U ) ) = N )
24	23	opeq2d 4409	. . . . . . . 8 |- ( ( N e. NN /\ U e. C /\ W e. C ) -> <. 0 , ( # ` ( 1st ` U ) ) >. = <. 0 , N >. )
25	24	oveq2d 6666	. . . . . . 7 |- ( ( N e. NN /\ U e. C /\ W e. C ) -> ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` U ) substr <. 0 , N >. ) )
26	1, 2, 3, 4	clwlksfclwwlk1hashn 26959	. . . . . . . . . 10 |- ( W e. C -> ( # ` ( 1st ` W ) ) = N )
27	26	3ad2ant3 1084	. . . . . . . . 9 |- ( ( N e. NN /\ U e. C /\ W e. C ) -> ( # ` ( 1st ` W ) ) = N )
28	27	opeq2d 4409	. . . . . . . 8 |- ( ( N e. NN /\ U e. C /\ W e. C ) -> <. 0 , ( # ` ( 1st ` W ) ) >. = <. 0 , N >. )
29	28	oveq2d 6666	. . . . . . 7 |- ( ( N e. NN /\ U e. C /\ W e. C ) -> ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , N >. ) )
30	25, 29	eqeq12d 2637	. . . . . 6 |- ( ( N e. NN /\ U e. C /\ W e. C ) -> ( ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) <-> ( ( 2nd ` U ) substr <. 0 , N >. ) = ( ( 2nd ` W ) substr <. 0 , N >. ) ) )
31	30	biimpa 501	. . . . 5 |- ( ( ( N e. NN /\ U e. C /\ W e. C ) /\ ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) ) -> ( ( 2nd ` U ) substr <. 0 , N >. ) = ( ( 2nd ` W ) substr <. 0 , N >. ) )
32		simpl 473	. . . . . . 7 |- ( ( ( ( 2nd ` U ) e. Word (Vtx ` G ) /\ ( 2nd ` W ) e. Word (Vtx ` G ) ) /\ ( N e. NN0 /\ N <_ ( # ` ( 2nd ` U ) ) /\ N <_ ( # ` ( 2nd ` W ) ) ) ) -> ( ( 2nd ` U ) e. Word (Vtx ` G ) /\ ( 2nd ` W ) e. Word (Vtx ` G ) ) )
33		id 22	. . . . . . . . . 10 |- ( N e. NN0 -> N e. NN0 )
34	33, 33	jca 554	. . . . . . . . 9 |- ( N e. NN0 -> ( N e. NN0 /\ N e. NN0 ) )
35	34	3ad2ant1 1082	. . . . . . . 8 |- ( ( N e. NN0 /\ N <_ ( # ` ( 2nd ` U ) ) /\ N <_ ( # ` ( 2nd ` W ) ) ) -> ( N e. NN0 /\ N e. NN0 ) )
36	35	adantl 482	. . . . . . 7 |- ( ( ( ( 2nd ` U ) e. Word (Vtx ` G ) /\ ( 2nd ` W ) e. Word (Vtx ` G ) ) /\ ( N e. NN0 /\ N <_ ( # ` ( 2nd ` U ) ) /\ N <_ ( # ` ( 2nd ` W ) ) ) ) -> ( N e. NN0 /\ N e. NN0 ) )
37		3simpc 1060	. . . . . . . 8 |- ( ( N e. NN0 /\ N <_ ( # ` ( 2nd ` U ) ) /\ N <_ ( # ` ( 2nd ` W ) ) ) -> ( N <_ ( # ` ( 2nd ` U ) ) /\ N <_ ( # ` ( 2nd ` W ) ) ) )
38	37	adantl 482	. . . . . . 7 |- ( ( ( ( 2nd ` U ) e. Word (Vtx ` G ) /\ ( 2nd ` W ) e. Word (Vtx ` G ) ) /\ ( N e. NN0 /\ N <_ ( # ` ( 2nd ` U ) ) /\ N <_ ( # ` ( 2nd ` W ) ) ) ) -> ( N <_ ( # ` ( 2nd ` U ) ) /\ N <_ ( # ` ( 2nd ` W ) ) ) )
39		swrdeq 13444	. . . . . . 7 |- ( ( ( ( 2nd ` U ) e. Word (Vtx ` G ) /\ ( 2nd ` W ) e. Word (Vtx ` G ) ) /\ ( N e. NN0 /\ N e. NN0 ) /\ ( N <_ ( # ` ( 2nd ` U ) ) /\ N <_ ( # ` ( 2nd ` W ) ) ) ) -> ( ( ( 2nd ` U ) substr <. 0 , N >. ) = ( ( 2nd ` W ) substr <. 0 , N >. ) <-> ( N = N /\ A. y e. ( 0..^ N ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) ) ) )
40	32, 36, 38, 39	syl3anc 1326	. . . . . 6 |- ( ( ( ( 2nd ` U ) e. Word (Vtx ` G ) /\ ( 2nd ` W ) e. Word (Vtx ` G ) ) /\ ( N e. NN0 /\ N <_ ( # ` ( 2nd ` U ) ) /\ N <_ ( # ` ( 2nd ` W ) ) ) ) -> ( ( ( 2nd ` U ) substr <. 0 , N >. ) = ( ( 2nd ` W ) substr <. 0 , N >. ) <-> ( N = N /\ A. y e. ( 0..^ N ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) ) ) )
41		simpr 477	. . . . . 6 |- ( ( N = N /\ A. y e. ( 0..^ N ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) ) -> A. y e. ( 0..^ N ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) )
42	40, 41	syl6bi 243	. . . . 5 |- ( ( ( ( 2nd ` U ) e. Word (Vtx ` G ) /\ ( 2nd ` W ) e. Word (Vtx ` G ) ) /\ ( N e. NN0 /\ N <_ ( # ` ( 2nd ` U ) ) /\ N <_ ( # ` ( 2nd ` W ) ) ) ) -> ( ( ( 2nd ` U ) substr <. 0 , N >. ) = ( ( 2nd ` W ) substr <. 0 , N >. ) -> A. y e. ( 0..^ N ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) ) )
43	21, 31, 42	sylc 65	. . . 4 |- ( ( ( N e. NN /\ U e. C /\ W e. C ) /\ ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) ) -> A. y e. ( 0..^ N ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) )
44		lbfzo0 12507	. . . . . . . . 9 |- ( 0 e. ( 0..^ N ) <-> N e. NN )
45	44	biimpri 218	. . . . . . . 8 |- ( N e. NN -> 0 e. ( 0..^ N ) )
46	45	3ad2ant1 1082	. . . . . . 7 |- ( ( N e. NN /\ U e. C /\ W e. C ) -> 0 e. ( 0..^ N ) )
47	46	adantr 481	. . . . . 6 |- ( ( ( N e. NN /\ U e. C /\ W e. C ) /\ ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) ) -> 0 e. ( 0..^ N ) )
48		fveq2 6191	. . . . . . . 8 |- ( y = 0 -> ( ( 2nd ` U ) ` y ) = ( ( 2nd ` U ) ` 0 ) )
49		fveq2 6191	. . . . . . . 8 |- ( y = 0 -> ( ( 2nd ` W ) ` y ) = ( ( 2nd ` W ) ` 0 ) )
50	48, 49	eqeq12d 2637	. . . . . . 7 |- ( y = 0 -> ( ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) <-> ( ( 2nd ` U ) ` 0 ) = ( ( 2nd ` W ) ` 0 ) ) )
51	50	rspcv 3305	. . . . . 6 |- ( 0 e. ( 0..^ N ) -> ( A. y e. ( 0..^ N ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) -> ( ( 2nd ` U ) ` 0 ) = ( ( 2nd ` W ) ` 0 ) ) )
52	47, 51	syl 17	. . . . 5 |- ( ( ( N e. NN /\ U e. C /\ W e. C ) /\ ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) ) -> ( A. y e. ( 0..^ N ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) -> ( ( 2nd ` U ) ` 0 ) = ( ( 2nd ` W ) ` 0 ) ) )
53	1, 2, 3, 4	clwlksf1clwwlklem2 26966	. . . . . . . 8 |- ( U e. C -> ( ( 2nd ` U ) ` 0 ) = ( ( 2nd ` U ) ` N ) )
54	53	3ad2ant2 1083	. . . . . . 7 |- ( ( N e. NN /\ U e. C /\ W e. C ) -> ( ( 2nd ` U ) ` 0 ) = ( ( 2nd ` U ) ` N ) )
55	54	adantr 481	. . . . . 6 |- ( ( ( N e. NN /\ U e. C /\ W e. C ) /\ ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) ) -> ( ( 2nd ` U ) ` 0 ) = ( ( 2nd ` U ) ` N ) )
56	1, 2, 3, 4	clwlksf1clwwlklem2 26966	. . . . . . . 8 |- ( W e. C -> ( ( 2nd ` W ) ` 0 ) = ( ( 2nd ` W ) ` N ) )
57	56	3ad2ant3 1084	. . . . . . 7 |- ( ( N e. NN /\ U e. C /\ W e. C ) -> ( ( 2nd ` W ) ` 0 ) = ( ( 2nd ` W ) ` N ) )
58	57	adantr 481	. . . . . 6 |- ( ( ( N e. NN /\ U e. C /\ W e. C ) /\ ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) ) -> ( ( 2nd ` W ) ` 0 ) = ( ( 2nd ` W ) ` N ) )
59	55, 58	eqeq12d 2637	. . . . 5 |- ( ( ( N e. NN /\ U e. C /\ W e. C ) /\ ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) ) -> ( ( ( 2nd ` U ) ` 0 ) = ( ( 2nd ` W ) ` 0 ) <-> ( ( 2nd ` U ) ` N ) = ( ( 2nd ` W ) ` N ) ) )
60	52, 59	sylibd 229	. . . 4 |- ( ( ( N e. NN /\ U e. C /\ W e. C ) /\ ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) ) -> ( A. y e. ( 0..^ N ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) -> ( ( 2nd ` U ) ` N ) = ( ( 2nd ` W ) ` N ) ) )
61	43, 60	jcai 559	. . 3 |- ( ( ( N e. NN /\ U e. C /\ W e. C ) /\ ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) ) -> ( A. y e. ( 0..^ N ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) /\ ( ( 2nd ` U ) ` N ) = ( ( 2nd ` W ) ` N ) ) )
62		elnn0uz 11725	. . . . . . . . 9 |- ( N e. NN0 <-> N e. ( ZZ>= ` 0 ) )
63	9, 62	sylib 208	. . . . . . . 8 |- ( N e. NN -> N e. ( ZZ>= ` 0 ) )
64	63	3ad2ant1 1082	. . . . . . 7 |- ( ( N e. NN /\ U e. C /\ W e. C ) -> N e. ( ZZ>= ` 0 ) )
65	64	adantr 481	. . . . . 6 |- ( ( ( N e. NN /\ U e. C /\ W e. C ) /\ ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) ) -> N e. ( ZZ>= ` 0 ) )
66		fzisfzounsn 12580	. . . . . 6 |- ( N e. ( ZZ>= ` 0 ) -> ( 0 ... N ) = ( ( 0..^ N ) u. { N } ) )
67	65, 66	syl 17	. . . . 5 |- ( ( ( N e. NN /\ U e. C /\ W e. C ) /\ ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) ) -> ( 0 ... N ) = ( ( 0..^ N ) u. { N } ) )
68	67	raleqdv 3144	. . . 4 |- ( ( ( N e. NN /\ U e. C /\ W e. C ) /\ ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) ) -> ( A. y e. ( 0 ... N ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) <-> A. y e. ( ( 0..^ N ) u. { N } ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) ) )
69		simpl1 1064	. . . . 5 |- ( ( ( N e. NN /\ U e. C /\ W e. C ) /\ ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) ) -> N e. NN )
70		fveq2 6191	. . . . . . 7 |- ( y = N -> ( ( 2nd ` U ) ` y ) = ( ( 2nd ` U ) ` N ) )
71		fveq2 6191	. . . . . . 7 |- ( y = N -> ( ( 2nd ` W ) ` y ) = ( ( 2nd ` W ) ` N ) )
72	70, 71	eqeq12d 2637	. . . . . 6 |- ( y = N -> ( ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) <-> ( ( 2nd ` U ) ` N ) = ( ( 2nd ` W ) ` N ) ) )
73	72	ralunsn 4422	. . . . 5 |- ( N e. NN -> ( A. y e. ( ( 0..^ N ) u. { N } ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) <-> ( A. y e. ( 0..^ N ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) /\ ( ( 2nd ` U ) ` N ) = ( ( 2nd ` W ) ` N ) ) ) )
74	69, 73	syl 17	. . . 4 |- ( ( ( N e. NN /\ U e. C /\ W e. C ) /\ ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) ) -> ( A. y e. ( ( 0..^ N ) u. { N } ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) <-> ( A. y e. ( 0..^ N ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) /\ ( ( 2nd ` U ) ` N ) = ( ( 2nd ` W ) ` N ) ) ) )
75	68, 74	bitrd 268	. . 3 |- ( ( ( N e. NN /\ U e. C /\ W e. C ) /\ ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) ) -> ( A. y e. ( 0 ... N ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) <-> ( A. y e. ( 0..^ N ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) /\ ( ( 2nd ` U ) ` N ) = ( ( 2nd ` W ) ` N ) ) ) )
76	61, 75	mpbird 247	. 2 |- ( ( ( N e. NN /\ U e. C /\ W e. C ) /\ ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) ) -> A. y e. ( 0 ... N ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) )
77	76	ex 450	1 |- ( ( N e. NN /\ U e. C /\ W e. C ) -> ( ( ( 2nd ` U ) substr <. 0 , ( # ` ( 1st ` U ) ) >. ) = ( ( 2nd ` W ) substr <. 0 , ( # ` ( 1st ` W ) ) >. ) -> A. y e. ( 0 ... N ) ( ( 2nd ` U ) ` y ) = ( ( 2nd ` W ) ` y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   u. cun 3572   {csn 4177   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   0cc0 9936   <_ cle 10075   NNcn 11020   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   substr csubstr 13295  Vtxcvtx 25874  ClWalkscclwlks 26666

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-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-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-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-substr 13303  df-wlks 26495  df-clwlks 26667

This theorem is referenced by:  clwlksf1clwwlk  26969