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