Step | Hyp | Ref
| Expression |
1 | | 3anan32 1050 |
. . 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
⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 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 | | wlkeq 26529 |
. . . 4
⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 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 1265 |
. . 3
⊢ (((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 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 1079 |
. 2
⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 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 12565 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (0..^𝑁) → (𝑥 + 1) ∈ (0...𝑁)) |
7 | 6 | adantl 482 |
. . . . . . . . . . 11
⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (0...𝑁)) |
8 | | fveq2 6191 |
. . . . . . . . . . . . 13
⊢ (𝑦 = (𝑥 + 1) → ((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐴)‘(𝑥 + 1))) |
9 | | fveq2 6191 |
. . . . . . . . . . . . 13
⊢ (𝑦 = (𝑥 + 1) → ((2nd ‘𝐵)‘𝑦) = ((2nd ‘𝐵)‘(𝑥 + 1))) |
10 | 8, 9 | eqeq12d 2637 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝑥 + 1) → (((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) ↔ ((2nd ‘𝐴)‘(𝑥 + 1)) = ((2nd ‘𝐵)‘(𝑥 + 1)))) |
11 | 10 | adantl 482 |
. . . . . . . . . . 11
⊢
(((((𝐺 ∈
USPGraph ∧ (𝐴 ∈
(Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 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 3312 |
. . . . . . . . . 10
⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 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 456 |
. . . . . . . . 9
⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 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 2965 |
. . . . . . . 8
⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 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 6657 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑥 → (𝑦 + 1) = (𝑥 + 1)) |
16 | 15 | fveq2d 6195 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑥 → ((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐴)‘(𝑥 + 1))) |
17 | 15 | fveq2d 6195 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑥 → ((2nd ‘𝐵)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑥 + 1))) |
18 | 16, 17 | eqeq12d 2637 |
. . . . . . . . 9
⊢ (𝑦 = 𝑥 → (((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1)) ↔ ((2nd ‘𝐴)‘(𝑥 + 1)) = ((2nd ‘𝐵)‘(𝑥 + 1)))) |
19 | 18 | cbvralv 3171 |
. . . . . . . 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 224 |
. . . . . . 7
⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 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 12488 |
. . . . . . . . . 10
⊢
(0..^𝑁) ⊆
(0...𝑁) |
22 | | ssralv 3666 |
. . . . . . . . . 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
⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 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 3064 |
. . . . . . . . . . 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 4270 |
. . . . . . . . . . . . 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
⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 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 2963 |
. . . . . . . . . . 11
⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 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 233 |
. . . . . . . . . 10
⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 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 452 |
. . . . . . . . 9
⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 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 47 |
. . . . . . . 8
⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 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 445 |
. . . . . . 7
⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 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
⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 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 450 |
. . . . 5
⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 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 | | uspgrupgr 26071 |
. . . . . . . 8
⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph
) |
35 | | eqid 2622 |
. . . . . . . . . 10
⊢
(Vtx‘𝐺) =
(Vtx‘𝐺) |
36 | | eqid 2622 |
. . . . . . . . . 10
⊢
(iEdg‘𝐺) =
(iEdg‘𝐺) |
37 | | eqid 2622 |
. . . . . . . . . 10
⊢
(1st ‘𝐴) = (1st ‘𝐴) |
38 | | eqid 2622 |
. . . . . . . . . 10
⊢
(2nd ‘𝐴) = (2nd ‘𝐴) |
39 | 35, 36, 37, 38 | upgrwlkcompim 26539 |
. . . . . . . . 9
⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ (Walks‘𝐺)) → ((1st
‘𝐴) ∈ Word dom
(iEdg‘𝐺) ∧
(2nd ‘𝐴):(0...(#‘(1st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈
(0..^(#‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))})) |
40 | 39 | ex 450 |
. . . . . . . 8
⊢ (𝐺 ∈ UPGraph → (𝐴 ∈ (Walks‘𝐺) → ((1st
‘𝐴) ∈ Word dom
(iEdg‘𝐺) ∧
(2nd ‘𝐴):(0...(#‘(1st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈
(0..^(#‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}))) |
41 | 34, 40 | syl 17 |
. . . . . . 7
⊢ (𝐺 ∈ USPGraph → (𝐴 ∈ (Walks‘𝐺) → ((1st
‘𝐴) ∈ Word dom
(iEdg‘𝐺) ∧
(2nd ‘𝐴):(0...(#‘(1st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈
(0..^(#‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}))) |
42 | | eqid 2622 |
. . . . . . . . . 10
⊢
(1st ‘𝐵) = (1st ‘𝐵) |
43 | | eqid 2622 |
. . . . . . . . . 10
⊢
(2nd ‘𝐵) = (2nd ‘𝐵) |
44 | 35, 36, 42, 43 | upgrwlkcompim 26539 |
. . . . . . . . 9
⊢ ((𝐺 ∈ UPGraph ∧ 𝐵 ∈ (Walks‘𝐺)) → ((1st
‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈
(0..^(#‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))})) |
45 | 44 | ex 450 |
. . . . . . . 8
⊢ (𝐺 ∈ UPGraph → (𝐵 ∈ (Walks‘𝐺) → ((1st
‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈
(0..^(#‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}))) |
46 | 34, 45 | syl 17 |
. . . . . . 7
⊢ (𝐺 ∈ USPGraph → (𝐵 ∈ (Walks‘𝐺) → ((1st
‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈
(0..^(#‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}))) |
47 | | oveq2 6658 |
. . . . . . . . . . . . . . . . . . 19
⊢
((#‘(1st ‘𝐵)) = 𝑁 → (0..^(#‘(1st
‘𝐵))) = (0..^𝑁)) |
48 | 47 | eqcoms 2630 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 = (#‘(1st
‘𝐵)) →
(0..^(#‘(1st ‘𝐵))) = (0..^𝑁)) |
49 | 48 | raleqdv 3144 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 = (#‘(1st
‘𝐵)) →
(∀𝑦 ∈
(0..^(#‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ↔ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))})) |
50 | | oveq2 6658 |
. . . . . . . . . . . . . . . . . . 19
⊢
((#‘(1st ‘𝐴)) = 𝑁 → (0..^(#‘(1st
‘𝐴))) = (0..^𝑁)) |
51 | 50 | eqcoms 2630 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 = (#‘(1st
‘𝐴)) →
(0..^(#‘(1st ‘𝐴))) = (0..^𝑁)) |
52 | 51 | raleqdv 3144 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 = (#‘(1st
‘𝐴)) →
(∀𝑦 ∈
(0..^(#‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} ↔ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))})) |
53 | 49, 52 | bi2anan9r 918 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 = (#‘(1st
‘𝐴)) ∧ 𝑁 = (#‘(1st
‘𝐵))) →
((∀𝑦 ∈
(0..^(#‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(#‘(1st
‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) ↔ (∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}))) |
54 | | r19.26 3064 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑦 ∈
(0..^𝑁)(((iEdg‘𝐺)‘((1st
‘𝐵)‘𝑦)) = {((2nd
‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) ↔ (∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))})) |
55 | | eqeq2 2633 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
({((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1st
‘𝐴)‘𝑦)) = {((2nd
‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} ↔ ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))})) |
56 | | eqeq2 2633 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
({((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) → (((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ↔ ((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)))) |
57 | 56 | eqcoms 2630 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1st
‘𝐵)‘𝑦)) = {((2nd
‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ↔ ((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)))) |
58 | 57 | biimpd 219 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1st
‘𝐵)‘𝑦)) = {((2nd
‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)))) |
59 | 55, 58 | syl6bi 243 |
. . . . . . . . . . . . . . . . . . . 20
⊢
({((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1st
‘𝐴)‘𝑦)) = {((2nd
‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1st
‘𝐵)‘𝑦)) = {((2nd
‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦))))) |
60 | 59 | com13 88 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1st
‘𝐴)‘𝑦)) = {((2nd
‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} → ({((2nd
‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦))))) |
61 | 60 | imp 445 |
. . . . . . . . . . . . . . . . . 18
⊢
((((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) → ({((2nd
‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)))) |
62 | 61 | ral2imi 2947 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑦 ∈
(0..^𝑁)(((iEdg‘𝐺)‘((1st
‘𝐵)‘𝑦)) = {((2nd
‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ((iEdg‘𝐺)‘((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)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)))) |
63 | 54, 62 | sylbir 225 |
. . . . . . . . . . . . . . . 16
⊢
((∀𝑦 ∈
(0..^𝑁)((iEdg‘𝐺)‘((1st
‘𝐵)‘𝑦)) = {((2nd
‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((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)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)))) |
64 | 53, 63 | syl6bi 243 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 = (#‘(1st
‘𝐴)) ∧ 𝑁 = (#‘(1st
‘𝐵))) →
((∀𝑦 ∈
(0..^(#‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(#‘(1st
‘𝐴)))((iEdg‘𝐺)‘((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)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦))))) |
65 | 64 | com12 32 |
. . . . . . . . . . . . . 14
⊢
((∀𝑦 ∈
(0..^(#‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(#‘(1st
‘𝐴)))((iEdg‘𝐺)‘((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)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦))))) |
66 | 65 | ex 450 |
. . . . . . . . . . . . 13
⊢
(∀𝑦 ∈
(0..^(#‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → (∀𝑦 ∈ (0..^(#‘(1st
‘𝐴)))((iEdg‘𝐺)‘((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)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)))))) |
67 | 66 | 3ad2ant3 1084 |
. . . . . . . . . . . 12
⊢
(((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd
‘𝐵):(0...(#‘(1st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈
(0..^(#‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^(#‘(1st
‘𝐴)))((iEdg‘𝐺)‘((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)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)))))) |
68 | 67 | com12 32 |
. . . . . . . . . . 11
⊢
(∀𝑦 ∈
(0..^(#‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} → (((1st ‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈
(0..^(#‘(1st ‘𝐵)))((iEdg‘𝐺)‘((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)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)))))) |
69 | 68 | 3ad2ant3 1084 |
. . . . . . . . . 10
⊢
(((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd
‘𝐴):(0...(#‘(1st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈
(0..^(#‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) → (((1st
‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈
(0..^(#‘(1st ‘𝐵)))((iEdg‘𝐺)‘((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)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)))))) |
70 | 69 | imp 445 |
. . . . . . . . 9
⊢
((((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd
‘𝐴):(0...(#‘(1st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈
(0..^(#‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) ∧ ((1st ‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈
(0..^(#‘(1st ‘𝐵)))((iEdg‘𝐺)‘((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)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦))))) |
71 | 70 | expd 452 |
. . . . . . . 8
⊢
((((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd
‘𝐴):(0...(#‘(1st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈
(0..^(#‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) ∧ ((1st ‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈
(0..^(#‘(1st ‘𝐵)))((iEdg‘𝐺)‘((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)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)))))) |
72 | 71 | a1i 11 |
. . . . . . 7
⊢ (𝐺 ∈ USPGraph →
((((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd
‘𝐴):(0...(#‘(1st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈
(0..^(#‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) ∧ ((1st ‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈
(0..^(#‘(1st ‘𝐵)))((iEdg‘𝐺)‘((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)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦))))))) |
73 | 41, 46, 72 | syl2and 500 |
. . . . . 6
⊢ (𝐺 ∈ USPGraph → ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (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)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦))))))) |
74 | 73 | 3imp1 1280 |
. . . . 5
⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 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)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)))) |
75 | | eqcom 2629 |
. . . . . . 7
⊢
(((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) ↔ ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦))) |
76 | 36 | uspgrf1oedg 26068 |
. . . . . . . . . . . 12
⊢ (𝐺 ∈ USPGraph →
(iEdg‘𝐺):dom
(iEdg‘𝐺)–1-1-onto→(Edg‘𝐺)) |
77 | | f1of1 6136 |
. . . . . . . . . . . 12
⊢
((iEdg‘𝐺):dom
(iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺)) |
78 | 76, 77 | syl 17 |
. . . . . . . . . . 11
⊢ (𝐺 ∈ USPGraph →
(iEdg‘𝐺):dom
(iEdg‘𝐺)–1-1→(Edg‘𝐺)) |
79 | | eqidd 2623 |
. . . . . . . . . . . 12
⊢ (𝐺 ∈ USPGraph →
(iEdg‘𝐺) =
(iEdg‘𝐺)) |
80 | | eqidd 2623 |
. . . . . . . . . . . 12
⊢ (𝐺 ∈ USPGraph → dom
(iEdg‘𝐺) = dom
(iEdg‘𝐺)) |
81 | | edgval 25941 |
. . . . . . . . . . . . . 14
⊢
(Edg‘𝐺) = ran
(iEdg‘𝐺) |
82 | 81 | eqcomi 2631 |
. . . . . . . . . . . . 13
⊢ ran
(iEdg‘𝐺) =
(Edg‘𝐺) |
83 | 82 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝐺 ∈ USPGraph → ran
(iEdg‘𝐺) =
(Edg‘𝐺)) |
84 | 79, 80, 83 | f1eq123d 6131 |
. . . . . . . . . . 11
⊢ (𝐺 ∈ USPGraph →
((iEdg‘𝐺):dom
(iEdg‘𝐺)–1-1→ran (iEdg‘𝐺) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺))) |
85 | 78, 84 | mpbird 247 |
. . . . . . . . . 10
⊢ (𝐺 ∈ USPGraph →
(iEdg‘𝐺):dom
(iEdg‘𝐺)–1-1→ran (iEdg‘𝐺)) |
86 | 85 | 3ad2ant1 1082 |
. . . . . . . . 9
⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (#‘(1st ‘𝐴))) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺)) |
87 | 86 | adantr 481 |
. . . . . . . 8
⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺)) |
88 | 35, 36, 37, 38 | wlkelwrd 26528 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ∈ (Walks‘𝐺) → ((1st
‘𝐴) ∈ Word dom
(iEdg‘𝐺) ∧
(2nd ‘𝐴):(0...(#‘(1st ‘𝐴)))⟶(Vtx‘𝐺))) |
89 | 35, 36, 42, 43 | wlkelwrd 26528 |
. . . . . . . . . . . . . . 15
⊢ (𝐵 ∈ (Walks‘𝐺) → ((1st
‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶(Vtx‘𝐺))) |
90 | | oveq2 6658 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑁 = (#‘(1st
‘𝐴)) →
(0..^𝑁) =
(0..^(#‘(1st ‘𝐴)))) |
91 | 90 | eleq2d 2687 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑁 = (#‘(1st
‘𝐴)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(#‘(1st
‘𝐴))))) |
92 | | wrdsymbcl 13318 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(#‘(1st
‘𝐴)))) →
((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)) |
93 | 92 | expcom 451 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 ∈
(0..^(#‘(1st ‘𝐴))) → ((1st ‘𝐴) ∈ Word dom
(iEdg‘𝐺) →
((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))) |
94 | 91, 93 | syl6bi 243 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑁 = (#‘(1st
‘𝐴)) → (𝑦 ∈ (0..^𝑁) → ((1st ‘𝐴) ∈ Word dom
(iEdg‘𝐺) →
((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))) |
95 | 94 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑁 = (#‘(1st
‘𝐴)) ∧ 𝑁 = (#‘(1st
‘𝐵))) → (𝑦 ∈ (0..^𝑁) → ((1st ‘𝐴) ∈ Word dom
(iEdg‘𝐺) →
((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))) |
96 | 95 | imp 445 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑁 = (#‘(1st
‘𝐴)) ∧ 𝑁 = (#‘(1st
‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st ‘𝐴) ∈ Word dom
(iEdg‘𝐺) →
((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))) |
97 | 96 | com12 32 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → (((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))) |
98 | 97 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (1st
‘𝐴) ∈ Word dom
(iEdg‘𝐺)) →
(((𝑁 =
(#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))) |
99 | | oveq2 6658 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑁 = (#‘(1st
‘𝐵)) →
(0..^𝑁) =
(0..^(#‘(1st ‘𝐵)))) |
100 | 99 | eleq2d 2687 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑁 = (#‘(1st
‘𝐵)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(#‘(1st
‘𝐵))))) |
101 | | wrdsymbcl 13318 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(#‘(1st
‘𝐵)))) →
((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)) |
102 | 101 | expcom 451 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 ∈
(0..^(#‘(1st ‘𝐵))) → ((1st ‘𝐵) ∈ Word dom
(iEdg‘𝐺) →
((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))) |
103 | 100, 102 | syl6bi 243 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑁 = (#‘(1st
‘𝐵)) → (𝑦 ∈ (0..^𝑁) → ((1st ‘𝐵) ∈ Word dom
(iEdg‘𝐺) →
((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))) |
104 | 103 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑁 = (#‘(1st
‘𝐴)) ∧ 𝑁 = (#‘(1st
‘𝐵))) → (𝑦 ∈ (0..^𝑁) → ((1st ‘𝐵) ∈ Word dom
(iEdg‘𝐺) →
((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))) |
105 | 104 | imp 445 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑁 = (#‘(1st
‘𝐴)) ∧ 𝑁 = (#‘(1st
‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st ‘𝐵) ∈ Word dom
(iEdg‘𝐺) →
((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))) |
106 | 105 | com12 32 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → (((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))) |
107 | 106 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (1st
‘𝐴) ∈ Word dom
(iEdg‘𝐺)) →
(((𝑁 =
(#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))) |
108 | 98, 107 | jcad 555 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (1st
‘𝐴) ∈ Word dom
(iEdg‘𝐺)) →
(((𝑁 =
(#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))) |
109 | 108 | ex 450 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → ((1st
‘𝐴) ∈ Word dom
(iEdg‘𝐺) →
(((𝑁 =
(#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))) |
110 | 109 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢
(((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd
‘𝐵):(0...(#‘(1st ‘𝐵)))⟶(Vtx‘𝐺)) → ((1st
‘𝐴) ∈ Word dom
(iEdg‘𝐺) →
(((𝑁 =
(#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))) |
111 | 110 | com12 32 |
. . . . . . . . . . . . . . . . 17
⊢
((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → (((1st
‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶(Vtx‘𝐺)) → (((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))) |
112 | 111 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢
(((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd
‘𝐴):(0...(#‘(1st ‘𝐴)))⟶(Vtx‘𝐺)) → (((1st
‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶(Vtx‘𝐺)) → (((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))) |
113 | 112 | imp 445 |
. . . . . . . . . . . . . . 15
⊢
((((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd
‘𝐴):(0...(#‘(1st ‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1st
‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2nd ‘𝐵):(0...(#‘(1st ‘𝐵)))⟶(Vtx‘𝐺))) → (((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))) |
114 | 88, 89, 113 | syl2an 494 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))) |
115 | 114 | expd 452 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → ((𝑁 = (#‘(1st ‘𝐴)) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (𝑦 ∈ (0..^𝑁) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))) |
116 | 115 | expd 452 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (𝑁 = (#‘(1st ‘𝐴)) → (𝑁 = (#‘(1st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))) |
117 | 116 | imp 445 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (#‘(1st ‘𝐴))) → (𝑁 = (#‘(1st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))) |
118 | 117 | 3adant1 1079 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (#‘(1st ‘𝐴))) → (𝑁 = (#‘(1st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))) |
119 | 118 | imp 445 |
. . . . . . . . 9
⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (𝑦 ∈ (0..^𝑁) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))) |
120 | 119 | imp 445 |
. . . . . . . 8
⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))) |
121 | | f1veqaeq 6514 |
. . . . . . . 8
⊢
(((iEdg‘𝐺):dom
(iEdg‘𝐺)–1-1→ran (iEdg‘𝐺) ∧ (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))) → (((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) → ((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦))) |
122 | 87, 120, 121 | syl2an2r 876 |
. . . . . . 7
⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) → ((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦))) |
123 | 75, 122 | syl5bi 232 |
. . . . . 6
⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) → ((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦))) |
124 | 123 | ralimdva 2962 |
. . . . 5
⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦))) |
125 | 33, 74, 124 | 3syld 60 |
. . . 4
⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (#‘(1st ‘𝐴))) ∧ 𝑁 = (#‘(1st ‘𝐵))) → (∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦))) |
126 | 125 | expimpd 629 |
. . 3
⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (#‘(1st ‘𝐴))) → ((𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦))) |
127 | 126 | pm4.71d 666 |
. 2
⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 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)))) |
128 | 2, 5, 127 | 3bitr4d 300 |
1
⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (#‘(1st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)))) |