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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 = (#‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘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.
StepHypRef Expression
1 3anan32 1043 . . 3 ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) ↔ ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦)))
21a1i 11 . 2 ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) → ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) ↔ ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦))))
3 2wlkeq 26235 . . . 4 ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸) ∧ 𝑁 = (#‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦))))
433expa 1257 . . 3 (((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦))))
543adant1 1072 . 2 ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦))))
6 fzofzp1 12431 . . . . . . . . . . . 12 (𝑥 ∈ (0..^𝑁) → (𝑥 + 1) ∈ (0...𝑁))
76adantl 481 . . . . . . . . . . 11 ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (0...𝑁))
8 fveq2 6103 . . . . . . . . . . . . 13 (𝑦 = (𝑥 + 1) → ((2nd𝐴)‘𝑦) = ((2nd𝐴)‘(𝑥 + 1)))
9 fveq2 6103 . . . . . . . . . . . . 13 (𝑦 = (𝑥 + 1) → ((2nd𝐵)‘𝑦) = ((2nd𝐵)‘(𝑥 + 1)))
108, 9eqeq12d 2625 . . . . . . . . . . . 12 (𝑦 = (𝑥 + 1) → (((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ↔ ((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1))))
1110adantl 481 . . . . . . . . . . 11 (((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) ∧ 𝑦 = (𝑥 + 1)) → (((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ↔ ((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1))))
127, 11rspcdv 3285 . . . . . . . . . 10 ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) → (∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) → ((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1))))
1312impancom 455 . . . . . . . . 9 ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) → (𝑥 ∈ (0..^𝑁) → ((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1))))
1413ralrimiv 2948 . . . . . . . 8 ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) → ∀𝑥 ∈ (0..^𝑁)((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1)))
15 oveq1 6556 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑦 + 1) = (𝑥 + 1))
1615fveq2d 6107 . . . . . . . . . 10 (𝑦 = 𝑥 → ((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐴)‘(𝑥 + 1)))
1715fveq2d 6107 . . . . . . . . . 10 (𝑦 = 𝑥 → ((2nd𝐵)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑥 + 1)))
1816, 17eqeq12d 2625 . . . . . . . . 9 (𝑦 = 𝑥 → (((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1)) ↔ ((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1))))
1918cbvralv 3147 . . . . . . . 8 (∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1)) ↔ ∀𝑥 ∈ (0..^𝑁)((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1)))
2014, 19sylibr 223 . . . . . . 7 ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1)))
21 fzossfz 12357 . . . . . . . . . 10 (0..^𝑁) ⊆ (0...𝑁)
22 ssralv 3629 . . . . . . . . . 10 ((0..^𝑁) ⊆ (0...𝑁) → (∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)))
2321, 22mp1i 13 . . . . . . . . 9 (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)))
24 r19.26 3046 . . . . . . . . . . 11 (∀𝑦 ∈ (0..^𝑁)(((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ∧ ((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1))) ↔ (∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1))))
25 preq12 4214 . . . . . . . . . . . . 13 ((((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ∧ ((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1))) → {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))})
2625a1i 11 . . . . . . . . . . . 12 (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → ((((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ∧ ((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1))) → {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}))
2726ralimdv 2946 . . . . . . . . . . 11 (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁)(((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ∧ ((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1))) → ∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}))
2824, 27syl5bir 232 . . . . . . . . . 10 (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → ((∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1))) → ∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}))
2928expd 451 . . . . . . . . 9 (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) → (∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1)) → ∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))})))
3023, 29syld 46 . . . . . . . 8 (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) → (∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1)) → ∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))})))
3130imp 444 . . . . . . 7 ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) → (∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1)) → ∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}))
3220, 31mpd 15 . . . . . 6 ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))})
3332ex 449 . . . . 5 (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}))
34 eqid 2610 . . . . . . . . . 10 (1st𝐴) = (1st𝐴)
35 eqid 2610 . . . . . . . . . 10 (2nd𝐴) = (2nd𝐴)
3634, 35wlkcompim 26054 . . . . . . . . 9 (𝐴 ∈ (𝑉 Walks 𝐸) → ((1st𝐴) ∈ Word dom 𝐸 ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶𝑉 ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐴)))(𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}))
37 eqid 2610 . . . . . . . . . 10 (1st𝐵) = (1st𝐵)
38 eqid 2610 . . . . . . . . . 10 (2nd𝐵) = (2nd𝐵)
3937, 38wlkcompim 26054 . . . . . . . . 9 (𝐵 ∈ (𝑉 Walks 𝐸) → ((1st𝐵) ∈ Word dom 𝐸 ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶𝑉 ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐵)))(𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}))
40 oveq2 6557 . . . . . . . . . . . . . . . . . . 19 ((#‘(1st𝐵)) = 𝑁 → (0..^(#‘(1st𝐵))) = (0..^𝑁))
4140eqcoms 2618 . . . . . . . . . . . . . . . . . 18 (𝑁 = (#‘(1st𝐵)) → (0..^(#‘(1st𝐵))) = (0..^𝑁))
4241raleqdv 3121 . . . . . . . . . . . . . . . . 17 (𝑁 = (#‘(1st𝐵)) → (∀𝑦 ∈ (0..^(#‘(1st𝐵)))(𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ↔ ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}))
43 oveq2 6557 . . . . . . . . . . . . . . . . . . 19 ((#‘(1st𝐴)) = 𝑁 → (0..^(#‘(1st𝐴))) = (0..^𝑁))
4443eqcoms 2618 . . . . . . . . . . . . . . . . . 18 (𝑁 = (#‘(1st𝐴)) → (0..^(#‘(1st𝐴))) = (0..^𝑁))
4544raleqdv 3121 . . . . . . . . . . . . . . . . 17 (𝑁 = (#‘(1st𝐴)) → (∀𝑦 ∈ (0..^(#‘(1st𝐴)))(𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} ↔ ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}))
4642, 45bi2anan9r 914 . . . . . . . . . . . . . . . 16 ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → ((∀𝑦 ∈ (0..^(#‘(1st𝐵)))(𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐴)))(𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}) ↔ (∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))})))
47 r19.26 3046 . . . . . . . . . . . . . . . . 17 (∀𝑦 ∈ (0..^𝑁)((𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ∧ (𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}) ↔ (∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}))
48 eqeq2 2621 . . . . . . . . . . . . . . . . . . . . 21 ({((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ((𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} ↔ (𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}))
49 eqeq2 2621 . . . . . . . . . . . . . . . . . . . . . . 23 ({((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} = (𝐸‘((1st𝐴)‘𝑦)) → ((𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ↔ (𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦))))
5049eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ((𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ↔ (𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦))))
5150biimpd 218 . . . . . . . . . . . . . . . . . . . . 21 ((𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ((𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → (𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦))))
5248, 51syl6bi 242 . . . . . . . . . . . . . . . . . . . 20 ({((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ((𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} → ((𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → (𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦)))))
5352com13 86 . . . . . . . . . . . . . . . . . . 19 ((𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ((𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} → ({((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → (𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦)))))
5453imp 444 . . . . . . . . . . . . . . . . . 18 (((𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ∧ (𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}) → ({((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → (𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦))))
5554ral2imi 2931 . . . . . . . . . . . . . . . . 17 (∀𝑦 ∈ (0..^𝑁)((𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ∧ (𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦))))
5647, 55sylbir 224 . . . . . . . . . . . . . . . 16 ((∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦))))
5746, 56syl6bi 242 . . . . . . . . . . . . . . 15 ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → ((∀𝑦 ∈ (0..^(#‘(1st𝐵)))(𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐴)))(𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦)))))
5857com12 32 . . . . . . . . . . . . . 14 ((∀𝑦 ∈ (0..^(#‘(1st𝐵)))(𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐴)))(𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}) → ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦)))))
5958ex 449 . . . . . . . . . . . . 13 (∀𝑦 ∈ (0..^(#‘(1st𝐵)))(𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → (∀𝑦 ∈ (0..^(#‘(1st𝐴)))(𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} → ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦))))))
60593ad2ant3 1077 . . . . . . . . . . . 12 (((1st𝐵) ∈ Word dom 𝐸 ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶𝑉 ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐵)))(𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^(#‘(1st𝐴)))(𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} → ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦))))))
6160com12 32 . . . . . . . . . . 11 (∀𝑦 ∈ (0..^(#‘(1st𝐴)))(𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} → (((1st𝐵) ∈ Word dom 𝐸 ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶𝑉 ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐵)))(𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}) → ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦))))))
62613ad2ant3 1077 . . . . . . . . . 10 (((1st𝐴) ∈ Word dom 𝐸 ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶𝑉 ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐴)))(𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}) → (((1st𝐵) ∈ Word dom 𝐸 ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶𝑉 ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐵)))(𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}) → ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦))))))
6362imp 444 . . . . . . . . 9 ((((1st𝐴) ∈ Word dom 𝐸 ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶𝑉 ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐴)))(𝐸‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}) ∧ ((1st𝐵) ∈ Word dom 𝐸 ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶𝑉 ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐵)))(𝐸‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))})) → ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦)))))
6436, 39, 63syl2an 493 . . . . . . . 8 ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) → ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦)))))
6564expd 451 . . . . . . 7 ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) → (𝑁 = (#‘(1st𝐴)) → (𝑁 = (#‘(1st𝐵)) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦))))))
6665a1i 11 . . . . . 6 (𝑉 USGrph 𝐸 → ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) → (𝑁 = (#‘(1st𝐴)) → (𝑁 = (#‘(1st𝐵)) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦)))))))
67663imp1 1272 . . . . 5 (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦))))
68 eqcom 2617 . . . . . . 7 ((𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦)) ↔ (𝐸‘((1st𝐴)‘𝑦)) = (𝐸‘((1st𝐵)‘𝑦)))
69 usgraf1 25889 . . . . . . . . . . 11 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→ran 𝐸)
70693ad2ant1 1075 . . . . . . . . . 10 ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) → 𝐸:dom 𝐸1-1→ran 𝐸)
7170adantr 480 . . . . . . . . 9 (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → 𝐸:dom 𝐸1-1→ran 𝐸)
7271adantr 480 . . . . . . . 8 ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → 𝐸:dom 𝐸1-1→ran 𝐸)
73 wlkelwrd 26058 . . . . . . . . . . . . . . 15 (𝐴 ∈ (𝑉 Walks 𝐸) → ((1st𝐴) ∈ Word dom 𝐸 ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶𝑉))
74 wlkelwrd 26058 . . . . . . . . . . . . . . 15 (𝐵 ∈ (𝑉 Walks 𝐸) → ((1st𝐵) ∈ Word dom 𝐸 ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶𝑉))
75 oveq2 6557 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 = (#‘(1st𝐴)) → (0..^𝑁) = (0..^(#‘(1st𝐴))))
7675eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 = (#‘(1st𝐴)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(#‘(1st𝐴)))))
77 wrdsymbcl 13173 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝐴) ∈ Word dom 𝐸𝑦 ∈ (0..^(#‘(1st𝐴)))) → ((1st𝐴)‘𝑦) ∈ dom 𝐸)
7877expcom 450 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (0..^(#‘(1st𝐴))) → ((1st𝐴) ∈ Word dom 𝐸 → ((1st𝐴)‘𝑦) ∈ dom 𝐸))
7976, 78syl6bi 242 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 = (#‘(1st𝐴)) → (𝑦 ∈ (0..^𝑁) → ((1st𝐴) ∈ Word dom 𝐸 → ((1st𝐴)‘𝑦) ∈ dom 𝐸)))
8079adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → (𝑦 ∈ (0..^𝑁) → ((1st𝐴) ∈ Word dom 𝐸 → ((1st𝐴)‘𝑦) ∈ dom 𝐸)))
8180imp 444 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st𝐴) ∈ Word dom 𝐸 → ((1st𝐴)‘𝑦) ∈ dom 𝐸))
8281com12 32 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝐴) ∈ Word dom 𝐸 → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st𝐴)‘𝑦) ∈ dom 𝐸))
8382adantl 481 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝐵) ∈ Word dom 𝐸 ∧ (1st𝐴) ∈ Word dom 𝐸) → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st𝐴)‘𝑦) ∈ dom 𝐸))
84 oveq2 6557 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 = (#‘(1st𝐵)) → (0..^𝑁) = (0..^(#‘(1st𝐵))))
8584eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 = (#‘(1st𝐵)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(#‘(1st𝐵)))))
86 wrdsymbcl 13173 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝐵) ∈ Word dom 𝐸𝑦 ∈ (0..^(#‘(1st𝐵)))) → ((1st𝐵)‘𝑦) ∈ dom 𝐸)
8786expcom 450 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (0..^(#‘(1st𝐵))) → ((1st𝐵) ∈ Word dom 𝐸 → ((1st𝐵)‘𝑦) ∈ dom 𝐸))
8885, 87syl6bi 242 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 = (#‘(1st𝐵)) → (𝑦 ∈ (0..^𝑁) → ((1st𝐵) ∈ Word dom 𝐸 → ((1st𝐵)‘𝑦) ∈ dom 𝐸)))
8988adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → (𝑦 ∈ (0..^𝑁) → ((1st𝐵) ∈ Word dom 𝐸 → ((1st𝐵)‘𝑦) ∈ dom 𝐸)))
9089imp 444 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st𝐵) ∈ Word dom 𝐸 → ((1st𝐵)‘𝑦) ∈ dom 𝐸))
9190com12 32 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝐵) ∈ Word dom 𝐸 → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st𝐵)‘𝑦) ∈ dom 𝐸))
9291adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝐵) ∈ Word dom 𝐸 ∧ (1st𝐴) ∈ Word dom 𝐸) → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st𝐵)‘𝑦) ∈ dom 𝐸))
9383, 92jcad 554 . . . . . . . . . . . . . . . . . . . 20 (((1st𝐵) ∈ Word dom 𝐸 ∧ (1st𝐴) ∈ Word dom 𝐸) → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st𝐵)‘𝑦) ∈ dom 𝐸)))
9493ex 449 . . . . . . . . . . . . . . . . . . 19 ((1st𝐵) ∈ Word dom 𝐸 → ((1st𝐴) ∈ Word dom 𝐸 → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st𝐵)‘𝑦) ∈ dom 𝐸))))
9594adantr 480 . . . . . . . . . . . . . . . . . 18 (((1st𝐵) ∈ Word dom 𝐸 ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶𝑉) → ((1st𝐴) ∈ Word dom 𝐸 → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st𝐵)‘𝑦) ∈ dom 𝐸))))
9695com12 32 . . . . . . . . . . . . . . . . 17 ((1st𝐴) ∈ Word dom 𝐸 → (((1st𝐵) ∈ Word dom 𝐸 ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶𝑉) → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st𝐵)‘𝑦) ∈ dom 𝐸))))
9796adantr 480 . . . . . . . . . . . . . . . 16 (((1st𝐴) ∈ Word dom 𝐸 ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶𝑉) → (((1st𝐵) ∈ Word dom 𝐸 ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶𝑉) → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st𝐵)‘𝑦) ∈ dom 𝐸))))
9897imp 444 . . . . . . . . . . . . . . 15 ((((1st𝐴) ∈ Word dom 𝐸 ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶𝑉) ∧ ((1st𝐵) ∈ Word dom 𝐸 ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶𝑉)) → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st𝐵)‘𝑦) ∈ dom 𝐸)))
9973, 74, 98syl2an 493 . . . . . . . . . . . . . 14 ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st𝐵)‘𝑦) ∈ dom 𝐸)))
10099expd 451 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) → ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → (𝑦 ∈ (0..^𝑁) → (((1st𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st𝐵)‘𝑦) ∈ dom 𝐸))))
101100expd 451 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) → (𝑁 = (#‘(1st𝐴)) → (𝑁 = (#‘(1st𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1st𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st𝐵)‘𝑦) ∈ dom 𝐸)))))
102101imp 444 . . . . . . . . . . 11 (((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) → (𝑁 = (#‘(1st𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1st𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st𝐵)‘𝑦) ∈ dom 𝐸))))
1031023adant1 1072 . . . . . . . . . 10 ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) → (𝑁 = (#‘(1st𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1st𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st𝐵)‘𝑦) ∈ dom 𝐸))))
104103imp 444 . . . . . . . . 9 (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → (𝑦 ∈ (0..^𝑁) → (((1st𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st𝐵)‘𝑦) ∈ dom 𝐸)))
105104imp 444 . . . . . . . 8 ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st𝐵)‘𝑦) ∈ dom 𝐸))
106 f1veqaeq 6418 . . . . . . . 8 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ (((1st𝐴)‘𝑦) ∈ dom 𝐸 ∧ ((1st𝐵)‘𝑦) ∈ dom 𝐸)) → ((𝐸‘((1st𝐴)‘𝑦)) = (𝐸‘((1st𝐵)‘𝑦)) → ((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦)))
10772, 105, 106syl2anc 691 . . . . . . 7 ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝐸‘((1st𝐴)‘𝑦)) = (𝐸‘((1st𝐵)‘𝑦)) → ((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦)))
10868, 107syl5bi 231 . . . . . 6 ((((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦)) → ((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦)))
109108ralimdva 2945 . . . . 5 (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁)(𝐸‘((1st𝐵)‘𝑦)) = (𝐸‘((1st𝐴)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦)))
11033, 67, 1093syld 58 . . . 4 (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦)))
111110expimpd 627 . . 3 ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) → ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦)))
112111pm4.71d 664 . 2 ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) → ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) ↔ ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦))))
1132, 5, 1123bitr4d 299 1 ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦))))
 Colors of variables: wff setvar class 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  1st c1st 7057  2nd 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
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