Theorem usg2wlkeq 26236

Description: Conditions for two walks within the same undirected simple graph being the same. It is sufficient that the vertices (in the same order) are identical. (Contributed by AV, 3-Jul-2018.)

Assertion

Ref	Expression
usg2wlkeq	⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦))))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝑁   𝑦,𝐸   𝑦,𝑉

Proof of Theorem usg2wlkeq

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		3anan32 1043	. . 3 ⊢ ((𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) ↔ ((𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦)))
2	1	a1i 11	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) → ((𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) ↔ ((𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦))))
3		2wlkeq 26235	. . . 4 ⊢ ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸) ∧ 𝑁 = (#‘(1st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦))))
4	3	3expa 1257	. . 3 ⊢ (((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦))))
5	4	3adant1 1072	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦))))
6		fzofzp1 12431	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^𝑁) → (𝑥 + 1) ∈ (0...𝑁))
7	6	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (0...𝑁))
8		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 + 1) → ((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐴)‘(𝑥 + 1)))
9		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 + 1) → ((2nd ‘𝐵)‘𝑦) = ((2nd ‘𝐵)‘(𝑥 + 1)))
10	8, 9	eqeq12d 2625	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 + 1) → (((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) ↔ ((2nd ‘𝐴)‘(𝑥 + 1)) = ((2nd ‘𝐵)‘(𝑥 + 1))))
11	10	adantl 481	. . . . . . . . . . 11 ⊢ (((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) ∧ 𝑦 = (𝑥 + 1)) → (((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) ↔ ((2nd ‘𝐴)‘(𝑥 + 1)) = ((2nd ‘𝐵)‘(𝑥 + 1))))
12	7, 11	rspcdv 3285	. . . . . . . . . 10 ⊢ ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) → (∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) → ((2nd ‘𝐴)‘(𝑥 + 1)) = ((2nd ‘𝐵)‘(𝑥 + 1))))
13	12	impancom 455	. . . . . . . . 9 ⊢ ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) → (𝑥 ∈ (0..^𝑁) → ((2nd ‘𝐴)‘(𝑥 + 1)) = ((2nd ‘𝐵)‘(𝑥 + 1))))
14	13	ralrimiv 2948	. . . . . . . 8 ⊢ ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) → ∀𝑥 ∈ (0..^𝑁)((2nd ‘𝐴)‘(𝑥 + 1)) = ((2nd ‘𝐵)‘(𝑥 + 1)))
15		oveq1 6556	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦 + 1) = (𝑥 + 1))
16	15	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐴)‘(𝑥 + 1)))
17	15	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((2nd ‘𝐵)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑥 + 1)))
18	16, 17	eqeq12d 2625	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1)) ↔ ((2nd ‘𝐴)‘(𝑥 + 1)) = ((2nd ‘𝐵)‘(𝑥 + 1))))
19	18	cbvralv 3147	. . . . . . . 8 ⊢ (∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1)) ↔ ∀𝑥 ∈ (0..^𝑁)((2nd ‘𝐴)‘(𝑥 + 1)) = ((2nd ‘𝐵)‘(𝑥 + 1)))
20	14, 19	sylibr 223	. . . . . . 7 ⊢ ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1)))
21		fzossfz 12357	. . . . . . . . . 10 ⊢ (0..^𝑁) ⊆ (0...𝑁)
22		ssralv 3629	. . . . . . . . . 10 ⊢ ((0..^𝑁) ⊆ (0...𝑁) → (∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)))
23	21, 22	mp1i 13	. . . . . . . . 9 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)))
24		r19.26 3046	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ (0..^𝑁)(((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) ∧ ((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1))) ↔ (∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1))))
25		preq12 4214	. . . . . . . . . . . . 13 ⊢ ((((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) ∧ ((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1))) → {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))})
26	25	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) → ((((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) ∧ ((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1))) → {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}))
27	26	ralimdv 2946	. . . . . . . . . . 11 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁)(((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) ∧ ((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1))) → ∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}))
28	24, 27	syl5bir 232	. . . . . . . . . 10 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) → ((∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1))) → ∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}))
29	28	expd 451	. . . . . . . . 9 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) → (∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1)) → ∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))})))
30	23, 29	syld 46	. . . . . . . 8 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) → (∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1)) → ∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))})))
31	30	imp 444	. . . . . . 7 ⊢ ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) → (∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1)) → ∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}))
32	20, 31	mpd 15	. . . . . 6 ⊢ ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))})
33	32	ex 449	. . . . 5 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}))
34		eqid 2610	. . . . . . . . . 10 ⊢ (1st ‘𝐴) = (1st ‘𝐴)
35		eqid 2610	. . . . . . . . . 10 ⊢ (2nd ‘𝐴) = (2nd ‘𝐴)
36	34, 35	wlkcompim 26054	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑉 Walks 𝐸) → ((1st ‘𝐴) ∈ Word dom 𝐸 ∧ (2nd ‘𝐴):(0...(#‘(1st ‘𝐴)))⟶𝑉 ∧ ∀𝑦 ∈ (0..^(#‘(1st ‘𝐴)))(𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}))
37		eqid 2610	. . . . . . . . . 10 ⊢ (1st ‘𝐵) = (1st ‘𝐵)
38		eqid 2610	. . . . . . . . . 10 ⊢ (2nd ‘𝐵) = (2nd ‘𝐵)
39	37, 38	wlkcompim 26054	. . . . . . . . 9 ⊢ (𝐵 ∈ (𝑉 Walks 𝐸) → ((1st ‘𝐵) ∈ Word dom 𝐸 ∧ (2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉 ∧ ∀𝑦 ∈ (0..^(#‘(1st ‘𝐵)))(𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}))
40		oveq2 6557	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘(1st ‘𝐵)) = 𝑁 → (0..^(#‘(1st ‘𝐵))) = (0..^𝑁))
41	40	eqcoms 2618	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (#‘(1st ‘𝐵)) → (0..^(#‘(1st ‘𝐵))) = (0..^𝑁))
42	41	raleqdv 3121	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (#‘(1st ‘𝐵)) → (∀𝑦 ∈ (0..^(#‘(1st ‘𝐵)))(𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ↔ ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}))
43		oveq2 6557	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘(1st ‘𝐴)) = 𝑁 → (0..^(#‘(1st ‘𝐴))) = (0..^𝑁))
44	43	eqcoms 2618	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (#‘(1st ‘𝐴)) → (0..^(#‘(1st ‘𝐴))) = (0..^𝑁))
45	44	raleqdv 3121	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (#‘(1st ‘𝐴)) → (∀𝑦 ∈ (0..^(#‘(1st ‘𝐴)))(𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} ↔ ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}))
46	42, 45	bi2anan9r 914	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) → ((∀𝑦 ∈ (0..^(#‘(1st ‘𝐵)))(𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(#‘(1st ‘𝐴)))(𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) ↔ (∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))})))
47		r19.26 3046	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 ∈ (0..^𝑁)((𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ (𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) ↔ (∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}))
48		eqeq2 2621	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ((𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} ↔ (𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}))
49		eqeq2 2621	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ({((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} = (𝐸‘((1st ‘𝐴)‘𝑦)) → ((𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ↔ (𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦))))
50	49	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ((𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ↔ (𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦))))
51	50	biimpd 218	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ((𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → (𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦))))
52	48, 51	syl6bi 242	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ((𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} → ((𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → (𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦)))))
53	52	com13 86	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ((𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} → ({((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → (𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦)))))
54	53	imp 444	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ (𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) → ({((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → (𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦))))
55	54	ral2imi 2931	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 ∈ (0..^𝑁)((𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ (𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦))))
56	47, 55	sylbir 224	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦))))
57	46, 56	syl6bi 242	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) → ((∀𝑦 ∈ (0..^(#‘(1st ‘𝐵)))(𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(#‘(1st ‘𝐴)))(𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦)))))
58	57	com12 32	. . . . . . . . . . . . . 14 ⊢ ((∀𝑦 ∈ (0..^(#‘(1st ‘𝐵)))(𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(#‘(1st ‘𝐴)))(𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) → ((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦)))))
59	58	ex 449	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ (0..^(#‘(1st ‘𝐵)))(𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → (∀𝑦 ∈ (0..^(#‘(1st ‘𝐴)))(𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} → ((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦))))))
60	59	3ad2ant3 1077	. . . . . . . . . . . 12 ⊢ (((1st ‘𝐵) ∈ Word dom 𝐸 ∧ (2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉 ∧ ∀𝑦 ∈ (0..^(#‘(1st ‘𝐵)))(𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^(#‘(1st ‘𝐴)))(𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} → ((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦))))))
61	60	com12 32	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ (0..^(#‘(1st ‘𝐴)))(𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} → (((1st ‘𝐵) ∈ Word dom 𝐸 ∧ (2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉 ∧ ∀𝑦 ∈ (0..^(#‘(1st ‘𝐵)))(𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}) → ((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦))))))
62	61	3ad2ant3 1077	. . . . . . . . . 10 ⊢ (((1st ‘𝐴) ∈ Word dom 𝐸 ∧ (2nd ‘𝐴):(0...(#‘(1st ‘𝐴)))⟶𝑉 ∧ ∀𝑦 ∈ (0..^(#‘(1st ‘𝐴)))(𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) → (((1st ‘𝐵) ∈ Word dom 𝐸 ∧ (2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉 ∧ ∀𝑦 ∈ (0..^(#‘(1st ‘𝐵)))(𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}) → ((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦))))))
63	62	imp 444	. . . . . . . . 9 ⊢ ((((1st ‘𝐴) ∈ Word dom 𝐸 ∧ (2nd ‘𝐴):(0...(#‘(1st ‘𝐴)))⟶𝑉 ∧ ∀𝑦 ∈ (0..^(#‘(1st ‘𝐴)))(𝐸‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) ∧ ((1st ‘𝐵) ∈ Word dom 𝐸 ∧ (2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉 ∧ ∀𝑦 ∈ (0..^(#‘(1st ‘𝐵)))(𝐸‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))})) → ((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦)))))
64	36, 39, 63	syl2an 493	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) → ((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦)))))
65	64	expd 451	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) → (𝑁 = (#‘(1st ‘𝐴)) → (𝑁 = (#‘(1st ‘𝐵)) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦))))))
66	65	a1i 11	. . . . . 6 ⊢ (𝑉 USGrph 𝐸 → ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) → (𝑁 = (#‘(1st ‘𝐴)) → (𝑁 = (#‘(1st ‘𝐵)) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦)))))))
67	66	3imp1 1272	. . . . 5 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦))))
68		eqcom 2617	. . . . . . 7 ⊢ ((𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦)) ↔ (𝐸‘((1st ‘𝐴)‘𝑦)) = (𝐸‘((1st ‘𝐵)‘𝑦)))
69		usgraf1 25889	. . . . . . . . . . 11 ⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸–1-1→ran 𝐸)
70	69	3ad2ant1 1075	. . . . . . . . . 10 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) → 𝐸:dom 𝐸–1-1→ran 𝐸)
71	70	adantr 480	. . . . . . . . 9 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) → 𝐸:dom 𝐸–1-1→ran 𝐸)
72	71	adantr 480	. . . . . . . 8 ⊢ ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → 𝐸:dom 𝐸–1-1→ran 𝐸)
73		wlkelwrd 26058	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (𝑉 Walks 𝐸) → ((1st ‘𝐴) ∈ Word dom 𝐸 ∧ (2nd ‘𝐴):(0...(#‘(1st ‘𝐴)))⟶𝑉))
74		wlkelwrd 26058	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ (𝑉 Walks 𝐸) → ((1st ‘𝐵) ∈ Word dom 𝐸 ∧ (2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉))
75		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 = (#‘(1st ‘𝐴)) → (0..^𝑁) = (0..^(#‘(1st ‘𝐴))))
76	75	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 = (#‘(1st ‘𝐴)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(#‘(1st ‘𝐴)))))
77		wrdsymbcl 13173	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1st ‘𝐴) ∈ Word dom 𝐸 ∧ 𝑦 ∈ (0..^(#‘(1st ‘𝐴)))) → ((1st ‘𝐴)‘𝑦) ∈ dom 𝐸)
78	77	expcom 450	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^(#‘(1st ‘𝐴))) → ((1st ‘𝐴) ∈ Word dom 𝐸 → ((1st ‘𝐴)‘𝑦) ∈ dom 𝐸))
79	76, 78	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 = (#‘(1st ‘𝐴)) → (𝑦 ∈ (0..^𝑁) → ((1st ‘𝐴) ∈ Word dom 𝐸 → ((1st ‘𝐴)‘𝑦) ∈ dom 𝐸)))
80	79	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (𝑦 ∈ (0..^𝑁) → ((1st ‘𝐴) ∈ Word dom 𝐸 → ((1st ‘𝐴)‘𝑦) ∈ dom 𝐸)))
81	80	imp 444	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st ‘𝐴) ∈ Word dom 𝐸 → ((1st ‘𝐴)‘𝑦) ∈ dom 𝐸))
82	81	com12 32	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1st ‘𝐴) ∈ Word dom 𝐸 → (((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st ‘𝐴)‘𝑦) ∈ dom 𝐸))
83	82	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1st ‘𝐵) ∈ Word dom 𝐸 ∧ (1st ‘𝐴) ∈ Word dom 𝐸) → (((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st ‘𝐴)‘𝑦) ∈ dom 𝐸))
84		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 = (#‘(1st ‘𝐵)) → (0..^𝑁) = (0..^(#‘(1st ‘𝐵))))
85	84	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 = (#‘(1st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(#‘(1st ‘𝐵)))))
86		wrdsymbcl 13173	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1st ‘𝐵) ∈ Word dom 𝐸 ∧ 𝑦 ∈ (0..^(#‘(1st ‘𝐵)))) → ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸)
87	86	expcom 450	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^(#‘(1st ‘𝐵))) → ((1st ‘𝐵) ∈ Word dom 𝐸 → ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸))
88	85, 87	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 = (#‘(1st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) → ((1st ‘𝐵) ∈ Word dom 𝐸 → ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸)))
89	88	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (𝑦 ∈ (0..^𝑁) → ((1st ‘𝐵) ∈ Word dom 𝐸 → ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸)))
90	89	imp 444	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st ‘𝐵) ∈ Word dom 𝐸 → ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸))
91	90	com12 32	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1st ‘𝐵) ∈ Word dom 𝐸 → (((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸))
92	91	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1st ‘𝐵) ∈ Word dom 𝐸 ∧ (1st ‘𝐴) ∈ Word dom 𝐸) → (((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸))
93	83, 92	jcad 554	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1st ‘𝐵) ∈ Word dom 𝐸 ∧ (1st ‘𝐴) ∈ Word dom 𝐸) → (((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸)))
94	93	ex 449	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘𝐵) ∈ Word dom 𝐸 → ((1st ‘𝐴) ∈ Word dom 𝐸 → (((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸))))
95	94	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((1st ‘𝐵) ∈ Word dom 𝐸 ∧ (2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉) → ((1st ‘𝐴) ∈ Word dom 𝐸 → (((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸))))
96	95	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝐴) ∈ Word dom 𝐸 → (((1st ‘𝐵) ∈ Word dom 𝐸 ∧ (2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉) → (((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸))))
97	96	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘𝐴) ∈ Word dom 𝐸 ∧ (2nd ‘𝐴):(0...(#‘(1st ‘𝐴)))⟶𝑉) → (((1st ‘𝐵) ∈ Word dom 𝐸 ∧ (2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉) → (((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸))))
98	97	imp 444	. . . . . . . . . . . . . . 15 ⊢ ((((1st ‘𝐴) ∈ Word dom 𝐸 ∧ (2nd ‘𝐴):(0...(#‘(1st ‘𝐴)))⟶𝑉) ∧ ((1st ‘𝐵) ∈ Word dom 𝐸 ∧ (2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉)) → (((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸)))
99	73, 74, 98	syl2an 493	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) → (((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸)))
100	99	expd 451	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) → ((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (𝑦 ∈ (0..^𝑁) → (((1st ‘𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸))))
101	100	expd 451	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) → (𝑁 = (#‘(1st ‘𝐴)) → (𝑁 = (#‘(1st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1st ‘𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸)))))
102	101	imp 444	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) → (𝑁 = (#‘(1st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1st ‘𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸))))
103	102	3adant1 1072	. . . . . . . . . 10 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) → (𝑁 = (#‘(1st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1st ‘𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸))))
104	103	imp 444	. . . . . . . . 9 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (𝑦 ∈ (0..^𝑁) → (((1st ‘𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸)))
105	104	imp 444	. . . . . . . 8 ⊢ ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸))
106		f1veqaeq 6418	. . . . . . . 8 ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ (((1st ‘𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st ‘𝐵)‘𝑦) ∈ dom 𝐸)) → ((𝐸‘((1st ‘𝐴)‘𝑦)) = (𝐸‘((1st ‘𝐵)‘𝑦)) → ((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦)))
107	72, 105, 106	syl2anc 691	. . . . . . 7 ⊢ ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝐸‘((1st ‘𝐴)‘𝑦)) = (𝐸‘((1st ‘𝐵)‘𝑦)) → ((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦)))
108	68, 107	syl5bi 231	. . . . . 6 ⊢ ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦)) → ((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦)))
109	108	ralimdva 2945	. . . . 5 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st ‘𝐵)‘𝑦)) = (𝐸‘((1st ‘𝐴)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦)))
110	33, 67, 109	3syld 58	. . . 4 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦)))
111	110	expimpd 627	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) → ((𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦)))
112	111	pm4.71d 664	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) → ((𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) ↔ ((𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦))))
113	2, 5, 112	3bitr4d 299	1 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  {cpr 4127   class class class wbr 4583  dom cdm 5038  ran crn 5039  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058  0cc0 9815  1c1 9816   + caddc 9818  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   USGrph cusg 25859   Walks cwalk 26026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036

This theorem is referenced by:  usg2wlkeq2  26237  clwlkf1clwwlk  26377