Theorem wlkeq 26529

Description: Conditions for two walks (within the same graph) being the same. (Contributed by AV, 1-Jul-2018.) (Revised by AV, 16-May-2019.) (Revised by AV, 14-Apr-2021.)

Assertion

Ref	Expression
wlkeq	|- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( A = B <-> ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )

Distinct variable groups:   x, A   x, B   x, N

Allowed substitution hint:   G( x)

Proof of Theorem wlkeq

Step	Hyp	Ref	Expression
1		wlkop 26523	. . . . 5 |- ( A e. (Walks ` G ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
2		1st2ndb 7206	. . . . 5 |- ( A e. ( _V X. _V ) <-> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
3	1, 2	sylibr 224	. . . 4 |- ( A e. (Walks ` G ) -> A e. ( _V X. _V ) )
4		wlkop 26523	. . . . 5 |- ( B e. (Walks ` G ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
5		1st2ndb 7206	. . . . 5 |- ( B e. ( _V X. _V ) <-> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
6	4, 5	sylibr 224	. . . 4 |- ( B e. (Walks ` G ) -> B e. ( _V X. _V ) )
7		xpopth 7207	. . . . 5 |- ( ( A e. ( _V X. _V ) /\ B e. ( _V X. _V ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> A = B ) )
8	7	bicomd 213	. . . 4 |- ( ( A e. ( _V X. _V ) /\ B e. ( _V X. _V ) ) -> ( A = B <-> ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) )
9	3, 6, 8	syl2an 494	. . 3 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) -> ( A = B <-> ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) )
10	9	3adant3 1081	. 2 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( A = B <-> ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) )
11		eqid 2622	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
12		eqid 2622	. . . . . . 7 |- (iEdg ` G ) = (iEdg ` G )
13		eqid 2622	. . . . . . 7 |- ( 1st ` A ) = ( 1st ` A )
14		eqid 2622	. . . . . . 7 |- ( 2nd ` A ) = ( 2nd ` A )
15	11, 12, 13, 14	wlkelwrd 26528	. . . . . 6 |- ( A e. (Walks ` G ) -> ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) )
16		eqid 2622	. . . . . . 7 |- ( 1st ` B ) = ( 1st ` B )
17		eqid 2622	. . . . . . 7 |- ( 2nd ` B ) = ( 2nd ` B )
18	11, 12, 16, 17	wlkelwrd 26528	. . . . . 6 |- ( B e. (Walks ` G ) -> ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) )
19	15, 18	anim12i 590	. . . . 5 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) -> ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) /\ ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) ) )
20		eleq1 2689	. . . . . . . 8 |- ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. -> ( A e. (Walks ` G ) <-> <. ( 1st ` A ) , ( 2nd ` A ) >. e. (Walks ` G ) ) )
21		df-br 4654	. . . . . . . . 9 |- ( ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) <-> <. ( 1st ` A ) , ( 2nd ` A ) >. e. (Walks ` G ) )
22		wlklenvm1 26517	. . . . . . . . 9 |- ( ( 1st ` A ) (Walks ` G ) ( 2nd ` A ) -> ( # ` ( 1st ` A ) ) = ( ( # ` ( 2nd ` A ) ) - 1 ) )
23	21, 22	sylbir 225	. . . . . . . 8 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. (Walks ` G ) -> ( # ` ( 1st ` A ) ) = ( ( # ` ( 2nd ` A ) ) - 1 ) )
24	20, 23	syl6bi 243	. . . . . . 7 |- ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. -> ( A e. (Walks ` G ) -> ( # ` ( 1st ` A ) ) = ( ( # ` ( 2nd ` A ) ) - 1 ) ) )
25	1, 24	mpcom 38	. . . . . 6 |- ( A e. (Walks ` G ) -> ( # ` ( 1st ` A ) ) = ( ( # ` ( 2nd ` A ) ) - 1 ) )
26		eleq1 2689	. . . . . . . 8 |- ( B = <. ( 1st ` B ) , ( 2nd ` B ) >. -> ( B e. (Walks ` G ) <-> <. ( 1st ` B ) , ( 2nd ` B ) >. e. (Walks ` G ) ) )
27		df-br 4654	. . . . . . . . 9 |- ( ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) <-> <. ( 1st ` B ) , ( 2nd ` B ) >. e. (Walks ` G ) )
28		wlklenvm1 26517	. . . . . . . . 9 |- ( ( 1st ` B ) (Walks ` G ) ( 2nd ` B ) -> ( # ` ( 1st ` B ) ) = ( ( # ` ( 2nd ` B ) ) - 1 ) )
29	27, 28	sylbir 225	. . . . . . . 8 |- ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. (Walks ` G ) -> ( # ` ( 1st ` B ) ) = ( ( # ` ( 2nd ` B ) ) - 1 ) )
30	26, 29	syl6bi 243	. . . . . . 7 |- ( B = <. ( 1st ` B ) , ( 2nd ` B ) >. -> ( B e. (Walks ` G ) -> ( # ` ( 1st ` B ) ) = ( ( # ` ( 2nd ` B ) ) - 1 ) ) )
31	4, 30	mpcom 38	. . . . . 6 |- ( B e. (Walks ` G ) -> ( # ` ( 1st ` B ) ) = ( ( # ` ( 2nd ` B ) ) - 1 ) )
32	25, 31	anim12i 590	. . . . 5 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) -> ( ( # ` ( 1st ` A ) ) = ( ( # ` ( 2nd ` A ) ) - 1 ) /\ ( # ` ( 1st ` B ) ) = ( ( # ` ( 2nd ` B ) ) - 1 ) ) )
33		eqwrd 13346	. . . . . . . 8 |- ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 1st ` B ) e. Word dom (iEdg ` G ) ) -> ( ( 1st ` A ) = ( 1st ` B ) <-> ( ( # ` ( 1st ` A ) ) = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) ) )
34	33	ad2ant2r 783	. . . . . . 7 |- ( ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) /\ ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) ) -> ( ( 1st ` A ) = ( 1st ` B ) <-> ( ( # ` ( 1st ` A ) ) = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) ) )
35	34	adantr 481	. . . . . 6 |- ( ( ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) /\ ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) ) /\ ( ( # ` ( 1st ` A ) ) = ( ( # ` ( 2nd ` A ) ) - 1 ) /\ ( # ` ( 1st ` B ) ) = ( ( # ` ( 2nd ` B ) ) - 1 ) ) ) -> ( ( 1st ` A ) = ( 1st ` B ) <-> ( ( # ` ( 1st ` A ) ) = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) ) )
36		lencl 13324	. . . . . . . . . . 11 |- ( ( 1st ` A ) e. Word dom (iEdg ` G ) -> ( # ` ( 1st ` A ) ) e. NN0 )
37	36	adantr 481	. . . . . . . . . 10 |- ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) -> ( # ` ( 1st ` A ) ) e. NN0 )
38	37	adantr 481	. . . . . . . . 9 |- ( ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) /\ ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) ) -> ( # ` ( 1st ` A ) ) e. NN0 )
39		simplr 792	. . . . . . . . 9 |- ( ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) /\ ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) ) -> ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) )
40		simprr 796	. . . . . . . . 9 |- ( ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) /\ ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) ) -> ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) )
41	38, 39, 40	3jca 1242	. . . . . . . 8 |- ( ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) /\ ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) ) -> ( ( # ` ( 1st ` A ) ) e. NN0 /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) )
42	41	adantr 481	. . . . . . 7 |- ( ( ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) /\ ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) ) /\ ( ( # ` ( 1st ` A ) ) = ( ( # ` ( 2nd ` A ) ) - 1 ) /\ ( # ` ( 1st ` B ) ) = ( ( # ` ( 2nd ` B ) ) - 1 ) ) ) -> ( ( # ` ( 1st ` A ) ) e. NN0 /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) )
43		2ffzeq 12460	. . . . . . 7 |- ( ( ( # ` ( 1st ` A ) ) e. NN0 /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) -> ( ( 2nd ` A ) = ( 2nd ` B ) <-> ( ( # ` ( 1st ` A ) ) = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0 ... ( # ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
44	42, 43	syl 17	. . . . . 6 |- ( ( ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) /\ ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) ) /\ ( ( # ` ( 1st ` A ) ) = ( ( # ` ( 2nd ` A ) ) - 1 ) /\ ( # ` ( 1st ` B ) ) = ( ( # ` ( 2nd ` B ) ) - 1 ) ) ) -> ( ( 2nd ` A ) = ( 2nd ` B ) <-> ( ( # ` ( 1st ` A ) ) = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0 ... ( # ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
45	35, 44	anbi12d 747	. . . . 5 |- ( ( ( ( ( 1st ` A ) e. Word dom (iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> (Vtx ` G ) ) /\ ( ( 1st ` B ) e. Word dom (iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> (Vtx ` G ) ) ) /\ ( ( # ` ( 1st ` A ) ) = ( ( # ` ( 2nd ` A ) ) - 1 ) /\ ( # ` ( 1st ` B ) ) = ( ( # ` ( 2nd ` B ) ) - 1 ) ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( ( # ` ( 1st ` A ) ) = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( ( # ` ( 1st ` A ) ) = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0 ... ( # ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) )
46	19, 32, 45	syl2anc 693	. . . 4 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( ( # ` ( 1st ` A ) ) = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( ( # ` ( 1st ` A ) ) = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0 ... ( # ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) )
47	46	3adant3 1081	. . 3 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( ( # ` ( 1st ` A ) ) = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( ( # ` ( 1st ` A ) ) = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0 ... ( # ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) )
48		eqeq1 2626	. . . . . . 7 |- ( N = ( # ` ( 1st ` A ) ) -> ( N = ( # ` ( 1st ` B ) ) <-> ( # ` ( 1st ` A ) ) = ( # ` ( 1st ` B ) ) ) )
49		oveq2 6658	. . . . . . . 8 |- ( N = ( # ` ( 1st ` A ) ) -> ( 0..^ N ) = ( 0..^ ( # ` ( 1st ` A ) ) ) )
50	49	raleqdv 3144	. . . . . . 7 |- ( N = ( # ` ( 1st ` A ) ) -> ( A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) <-> A. x e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) )
51	48, 50	anbi12d 747	. . . . . 6 |- ( N = ( # ` ( 1st ` A ) ) -> ( ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) <-> ( ( # ` ( 1st ` A ) ) = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) ) )
52		oveq2 6658	. . . . . . . 8 |- ( N = ( # ` ( 1st ` A ) ) -> ( 0 ... N ) = ( 0 ... ( # ` ( 1st ` A ) ) ) )
53	52	raleqdv 3144	. . . . . . 7 |- ( N = ( # ` ( 1st ` A ) ) -> ( A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) <-> A. x e. ( 0 ... ( # ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) )
54	48, 53	anbi12d 747	. . . . . 6 |- ( N = ( # ` ( 1st ` A ) ) -> ( ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) <-> ( ( # ` ( 1st ` A ) ) = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0 ... ( # ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
55	51, 54	anbi12d 747	. . . . 5 |- ( N = ( # ` ( 1st ` A ) ) -> ( ( ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) <-> ( ( ( # ` ( 1st ` A ) ) = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( ( # ` ( 1st ` A ) ) = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0 ... ( # ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) )
56	55	bibi2d 332	. . . 4 |- ( N = ( # ` ( 1st ` A ) ) -> ( ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) <-> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( ( # ` ( 1st ` A ) ) = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( ( # ` ( 1st ` A ) ) = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0 ... ( # ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) ) )
57	56	3ad2ant3 1084	. . 3 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) <-> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( ( # ` ( 1st ` A ) ) = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ ( # ` ( 1st ` A ) ) ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( ( # ` ( 1st ` A ) ) = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0 ... ( # ` ( 1st ` A ) ) ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) ) )
58	47, 57	mpbird 247	. 2 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> ( ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) ) )
59		3anass 1042	. . . 4 |- ( ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) <-> ( N = ( # ` ( 1st ` B ) ) /\ ( A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
60		anandi 871	. . . 4 |- ( ( N = ( # ` ( 1st ` B ) ) /\ ( A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) <-> ( ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
61	59, 60	bitr2i 265	. . 3 |- ( ( ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) <-> ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) )
62	61	a1i 11	. 2 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( ( ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) ) /\ ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) <-> ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
63	10, 58, 62	3bitrd 294	1 |- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( A = B <-> ( N = ( # ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   <.cop 4183   class class class wbr 4653   X. cxp 5112   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   0cc0 9936   1c1 9937   - cmin 10266   NN0cn0 11292   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875  Walkscwlks 26492

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-wlks 26495

This theorem is referenced by:  uspgr2wlkeq  26542