Proof of Theorem 2wlkeq
Step | Hyp | Ref
| Expression |
1 | | wlkop 26056 |
. . . . 5
⊢ (𝐴 ∈ (𝑉 Walks 𝐸) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
2 | | 1st2ndb 7097 |
. . . . 5
⊢ (𝐴 ∈ (V × V) ↔
𝐴 = 〈(1st
‘𝐴), (2nd
‘𝐴)〉) |
3 | 1, 2 | sylibr 223 |
. . . 4
⊢ (𝐴 ∈ (𝑉 Walks 𝐸) → 𝐴 ∈ (V × V)) |
4 | | wlkop 26056 |
. . . . 5
⊢ (𝐵 ∈ (𝑉 Walks 𝐸) → 𝐵 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉) |
5 | | 1st2ndb 7097 |
. . . . 5
⊢ (𝐵 ∈ (V × V) ↔
𝐵 = 〈(1st
‘𝐵), (2nd
‘𝐵)〉) |
6 | 4, 5 | sylibr 223 |
. . . 4
⊢ (𝐵 ∈ (𝑉 Walks 𝐸) → 𝐵 ∈ (V × V)) |
7 | | xpopth 7098 |
. . . . 5
⊢ ((𝐴 ∈ (V × V) ∧
𝐵 ∈ (V × V))
→ (((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵)) ↔ 𝐴 = 𝐵)) |
8 | 7 | bicomd 212 |
. . . 4
⊢ ((𝐴 ∈ (V × V) ∧
𝐵 ∈ (V × V))
→ (𝐴 = 𝐵 ↔ ((1st
‘𝐴) = (1st
‘𝐵) ∧
(2nd ‘𝐴) =
(2nd ‘𝐵)))) |
9 | 3, 6, 8 | syl2an 493 |
. . 3
⊢ ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) → (𝐴 = 𝐵 ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd
‘𝐴) = (2nd
‘𝐵)))) |
10 | 9 | 3adant3 1074 |
. 2
⊢ ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸) ∧ 𝑁 = (#‘(1st ‘𝐴))) → (𝐴 = 𝐵 ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd
‘𝐴) = (2nd
‘𝐵)))) |
11 | | wlkelwrd 26058 |
. . . . . 6
⊢ (𝐴 ∈ (𝑉 Walks 𝐸) → ((1st ‘𝐴) ∈ Word dom 𝐸 ∧ (2nd
‘𝐴):(0...(#‘(1st ‘𝐴)))⟶𝑉)) |
12 | | wlkelwrd 26058 |
. . . . . 6
⊢ (𝐵 ∈ (𝑉 Walks 𝐸) → ((1st ‘𝐵) ∈ Word dom 𝐸 ∧ (2nd
‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉)) |
13 | 11, 12 | anim12i 588 |
. . . . 5
⊢ ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) → (((1st ‘𝐴) ∈ Word dom 𝐸 ∧ (2nd
‘𝐴):(0...(#‘(1st ‘𝐴)))⟶𝑉) ∧ ((1st ‘𝐵) ∈ Word dom 𝐸 ∧ (2nd
‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉))) |
14 | | eleq1 2676 |
. . . . . . . 8
⊢ (𝐴 = 〈(1st
‘𝐴), (2nd
‘𝐴)〉 →
(𝐴 ∈ (𝑉 Walks 𝐸) ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ (𝑉 Walks 𝐸))) |
15 | | df-br 4584 |
. . . . . . . . 9
⊢
((1st ‘𝐴)(𝑉 Walks 𝐸)(2nd ‘𝐴) ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ (𝑉 Walks 𝐸)) |
16 | | wlklenvm1 26060 |
. . . . . . . . 9
⊢
((1st ‘𝐴)(𝑉 Walks 𝐸)(2nd ‘𝐴) → (#‘(1st
‘𝐴)) =
((#‘(2nd ‘𝐴)) − 1)) |
17 | 15, 16 | sylbir 224 |
. . . . . . . 8
⊢
(〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ (𝑉 Walks 𝐸) → (#‘(1st
‘𝐴)) =
((#‘(2nd ‘𝐴)) − 1)) |
18 | 14, 17 | syl6bi 242 |
. . . . . . 7
⊢ (𝐴 = 〈(1st
‘𝐴), (2nd
‘𝐴)〉 →
(𝐴 ∈ (𝑉 Walks 𝐸) → (#‘(1st
‘𝐴)) =
((#‘(2nd ‘𝐴)) − 1))) |
19 | 1, 18 | mpcom 37 |
. . . . . 6
⊢ (𝐴 ∈ (𝑉 Walks 𝐸) → (#‘(1st
‘𝐴)) =
((#‘(2nd ‘𝐴)) − 1)) |
20 | | eleq1 2676 |
. . . . . . . 8
⊢ (𝐵 = 〈(1st
‘𝐵), (2nd
‘𝐵)〉 →
(𝐵 ∈ (𝑉 Walks 𝐸) ↔ 〈(1st ‘𝐵), (2nd ‘𝐵)〉 ∈ (𝑉 Walks 𝐸))) |
21 | | df-br 4584 |
. . . . . . . . 9
⊢
((1st ‘𝐵)(𝑉 Walks 𝐸)(2nd ‘𝐵) ↔ 〈(1st ‘𝐵), (2nd ‘𝐵)〉 ∈ (𝑉 Walks 𝐸)) |
22 | | wlklenvm1 26060 |
. . . . . . . . 9
⊢
((1st ‘𝐵)(𝑉 Walks 𝐸)(2nd ‘𝐵) → (#‘(1st
‘𝐵)) =
((#‘(2nd ‘𝐵)) − 1)) |
23 | 21, 22 | sylbir 224 |
. . . . . . . 8
⊢
(〈(1st ‘𝐵), (2nd ‘𝐵)〉 ∈ (𝑉 Walks 𝐸) → (#‘(1st
‘𝐵)) =
((#‘(2nd ‘𝐵)) − 1)) |
24 | 20, 23 | syl6bi 242 |
. . . . . . 7
⊢ (𝐵 = 〈(1st
‘𝐵), (2nd
‘𝐵)〉 →
(𝐵 ∈ (𝑉 Walks 𝐸) → (#‘(1st
‘𝐵)) =
((#‘(2nd ‘𝐵)) − 1))) |
25 | 4, 24 | mpcom 37 |
. . . . . 6
⊢ (𝐵 ∈ (𝑉 Walks 𝐸) → (#‘(1st
‘𝐵)) =
((#‘(2nd ‘𝐵)) − 1)) |
26 | 19, 25 | anim12i 588 |
. . . . 5
⊢ ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) → ((#‘(1st
‘𝐴)) =
((#‘(2nd ‘𝐴)) − 1) ∧ (#‘(1st
‘𝐵)) =
((#‘(2nd ‘𝐵)) − 1))) |
27 | | eqwrd 13201 |
. . . . . . . 8
⊢
(((1st ‘𝐴) ∈ Word dom 𝐸 ∧ (1st ‘𝐵) ∈ Word dom 𝐸) → ((1st
‘𝐴) = (1st
‘𝐵) ↔
((#‘(1st ‘𝐴)) = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈
(0..^(#‘(1st ‘𝐴)))((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥)))) |
28 | 27 | ad2ant2r 779 |
. . . . . . 7
⊢
((((1st ‘𝐴) ∈ Word dom 𝐸 ∧ (2nd ‘𝐴):(0...(#‘(1st
‘𝐴)))⟶𝑉) ∧ ((1st
‘𝐵) ∈ Word dom
𝐸 ∧ (2nd
‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉)) → ((1st ‘𝐴) = (1st ‘𝐵) ↔
((#‘(1st ‘𝐴)) = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈
(0..^(#‘(1st ‘𝐴)))((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥)))) |
29 | 28 | adantr 480 |
. . . . . 6
⊢
(((((1st ‘𝐴) ∈ Word dom 𝐸 ∧ (2nd ‘𝐴):(0...(#‘(1st
‘𝐴)))⟶𝑉) ∧ ((1st
‘𝐵) ∈ Word dom
𝐸 ∧ (2nd
‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉)) ∧ ((#‘(1st
‘𝐴)) =
((#‘(2nd ‘𝐴)) − 1) ∧ (#‘(1st
‘𝐵)) =
((#‘(2nd ‘𝐵)) − 1))) → ((1st
‘𝐴) = (1st
‘𝐵) ↔
((#‘(1st ‘𝐴)) = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈
(0..^(#‘(1st ‘𝐴)))((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥)))) |
30 | | lencl 13179 |
. . . . . . . . . . 11
⊢
((1st ‘𝐴) ∈ Word dom 𝐸 → (#‘(1st
‘𝐴)) ∈
ℕ0) |
31 | 30 | adantr 480 |
. . . . . . . . . 10
⊢
(((1st ‘𝐴) ∈ Word dom 𝐸 ∧ (2nd ‘𝐴):(0...(#‘(1st
‘𝐴)))⟶𝑉) →
(#‘(1st ‘𝐴)) ∈
ℕ0) |
32 | 31 | adantr 480 |
. . . . . . . . 9
⊢
((((1st ‘𝐴) ∈ Word dom 𝐸 ∧ (2nd ‘𝐴):(0...(#‘(1st
‘𝐴)))⟶𝑉) ∧ ((1st
‘𝐵) ∈ Word dom
𝐸 ∧ (2nd
‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉)) → (#‘(1st
‘𝐴)) ∈
ℕ0) |
33 | | simplr 788 |
. . . . . . . . 9
⊢
((((1st ‘𝐴) ∈ Word dom 𝐸 ∧ (2nd ‘𝐴):(0...(#‘(1st
‘𝐴)))⟶𝑉) ∧ ((1st
‘𝐵) ∈ Word dom
𝐸 ∧ (2nd
‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉)) → (2nd ‘𝐴):(0...(#‘(1st
‘𝐴)))⟶𝑉) |
34 | | simprr 792 |
. . . . . . . . 9
⊢
((((1st ‘𝐴) ∈ Word dom 𝐸 ∧ (2nd ‘𝐴):(0...(#‘(1st
‘𝐴)))⟶𝑉) ∧ ((1st
‘𝐵) ∈ Word dom
𝐸 ∧ (2nd
‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉)) → (2nd ‘𝐵):(0...(#‘(1st
‘𝐵)))⟶𝑉) |
35 | 32, 33, 34 | 3jca 1235 |
. . . . . . . 8
⊢
((((1st ‘𝐴) ∈ Word dom 𝐸 ∧ (2nd ‘𝐴):(0...(#‘(1st
‘𝐴)))⟶𝑉) ∧ ((1st
‘𝐵) ∈ Word dom
𝐸 ∧ (2nd
‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉)) → ((#‘(1st
‘𝐴)) ∈
ℕ0 ∧ (2nd ‘𝐴):(0...(#‘(1st ‘𝐴)))⟶𝑉 ∧ (2nd ‘𝐵):(0...(#‘(1st
‘𝐵)))⟶𝑉)) |
36 | 35 | adantr 480 |
. . . . . . 7
⊢
(((((1st ‘𝐴) ∈ Word dom 𝐸 ∧ (2nd ‘𝐴):(0...(#‘(1st
‘𝐴)))⟶𝑉) ∧ ((1st
‘𝐵) ∈ Word dom
𝐸 ∧ (2nd
‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉)) ∧ ((#‘(1st
‘𝐴)) =
((#‘(2nd ‘𝐴)) − 1) ∧ (#‘(1st
‘𝐵)) =
((#‘(2nd ‘𝐵)) − 1))) →
((#‘(1st ‘𝐴)) ∈ ℕ0 ∧
(2nd ‘𝐴):(0...(#‘(1st ‘𝐴)))⟶𝑉 ∧ (2nd ‘𝐵):(0...(#‘(1st
‘𝐵)))⟶𝑉)) |
37 | | 2ffzeq 12329 |
. . . . . . 7
⊢
(((#‘(1st ‘𝐴)) ∈ ℕ0 ∧
(2nd ‘𝐴):(0...(#‘(1st ‘𝐴)))⟶𝑉 ∧ (2nd ‘𝐵):(0...(#‘(1st
‘𝐵)))⟶𝑉) → ((2nd
‘𝐴) = (2nd
‘𝐵) ↔
((#‘(1st ‘𝐴)) = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈
(0...(#‘(1st ‘𝐴)))((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥)))) |
38 | 36, 37 | syl 17 |
. . . . . 6
⊢
(((((1st ‘𝐴) ∈ Word dom 𝐸 ∧ (2nd ‘𝐴):(0...(#‘(1st
‘𝐴)))⟶𝑉) ∧ ((1st
‘𝐵) ∈ Word dom
𝐸 ∧ (2nd
‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉)) ∧ ((#‘(1st
‘𝐴)) =
((#‘(2nd ‘𝐴)) − 1) ∧ (#‘(1st
‘𝐵)) =
((#‘(2nd ‘𝐵)) − 1))) → ((2nd
‘𝐴) = (2nd
‘𝐵) ↔
((#‘(1st ‘𝐴)) = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈
(0...(#‘(1st ‘𝐴)))((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥)))) |
39 | 29, 38 | anbi12d 743 |
. . . . 5
⊢
(((((1st ‘𝐴) ∈ Word dom 𝐸 ∧ (2nd ‘𝐴):(0...(#‘(1st
‘𝐴)))⟶𝑉) ∧ ((1st
‘𝐵) ∈ Word dom
𝐸 ∧ (2nd
‘𝐵):(0...(#‘(1st ‘𝐵)))⟶𝑉)) ∧ ((#‘(1st
‘𝐴)) =
((#‘(2nd ‘𝐴)) − 1) ∧ (#‘(1st
‘𝐵)) =
((#‘(2nd ‘𝐵)) − 1))) → (((1st
‘𝐴) = (1st
‘𝐵) ∧
(2nd ‘𝐴) =
(2nd ‘𝐵))
↔ (((#‘(1st ‘𝐴)) = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈
(0..^(#‘(1st ‘𝐴)))((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥)) ∧ ((#‘(1st
‘𝐴)) =
(#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1st
‘𝐴)))((2nd
‘𝐴)‘𝑥) = ((2nd
‘𝐵)‘𝑥))))) |
40 | 13, 26, 39 | syl2anc 691 |
. . . 4
⊢ ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) → (((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd
‘𝐴) = (2nd
‘𝐵)) ↔
(((#‘(1st ‘𝐴)) = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈
(0..^(#‘(1st ‘𝐴)))((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥)) ∧ ((#‘(1st
‘𝐴)) =
(#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1st
‘𝐴)))((2nd
‘𝐴)‘𝑥) = ((2nd
‘𝐵)‘𝑥))))) |
41 | 40 | 3adant3 1074 |
. . 3
⊢ ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸) ∧ 𝑁 = (#‘(1st ‘𝐴))) → (((1st
‘𝐴) = (1st
‘𝐵) ∧
(2nd ‘𝐴) =
(2nd ‘𝐵))
↔ (((#‘(1st ‘𝐴)) = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈
(0..^(#‘(1st ‘𝐴)))((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥)) ∧ ((#‘(1st
‘𝐴)) =
(#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1st
‘𝐴)))((2nd
‘𝐴)‘𝑥) = ((2nd
‘𝐵)‘𝑥))))) |
42 | | eqeq1 2614 |
. . . . . . 7
⊢ (𝑁 = (#‘(1st
‘𝐴)) → (𝑁 = (#‘(1st
‘𝐵)) ↔
(#‘(1st ‘𝐴)) = (#‘(1st ‘𝐵)))) |
43 | | oveq2 6557 |
. . . . . . . 8
⊢ (𝑁 = (#‘(1st
‘𝐴)) →
(0..^𝑁) =
(0..^(#‘(1st ‘𝐴)))) |
44 | 43 | raleqdv 3121 |
. . . . . . 7
⊢ (𝑁 = (#‘(1st
‘𝐴)) →
(∀𝑥 ∈
(0..^𝑁)((1st
‘𝐴)‘𝑥) = ((1st
‘𝐵)‘𝑥) ↔ ∀𝑥 ∈
(0..^(#‘(1st ‘𝐴)))((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥))) |
45 | 42, 44 | anbi12d 743 |
. . . . . 6
⊢ (𝑁 = (#‘(1st
‘𝐴)) → ((𝑁 = (#‘(1st
‘𝐵)) ∧
∀𝑥 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥)) ↔ ((#‘(1st
‘𝐴)) =
(#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1st
‘𝐴)))((1st
‘𝐴)‘𝑥) = ((1st
‘𝐵)‘𝑥)))) |
46 | | oveq2 6557 |
. . . . . . . 8
⊢ (𝑁 = (#‘(1st
‘𝐴)) →
(0...𝑁) =
(0...(#‘(1st ‘𝐴)))) |
47 | 46 | raleqdv 3121 |
. . . . . . 7
⊢ (𝑁 = (#‘(1st
‘𝐴)) →
(∀𝑥 ∈
(0...𝑁)((2nd
‘𝐴)‘𝑥) = ((2nd
‘𝐵)‘𝑥) ↔ ∀𝑥 ∈
(0...(#‘(1st ‘𝐴)))((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥))) |
48 | 42, 47 | anbi12d 743 |
. . . . . 6
⊢ (𝑁 = (#‘(1st
‘𝐴)) → ((𝑁 = (#‘(1st
‘𝐵)) ∧
∀𝑥 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥)) ↔ ((#‘(1st
‘𝐴)) =
(#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1st
‘𝐴)))((2nd
‘𝐴)‘𝑥) = ((2nd
‘𝐵)‘𝑥)))) |
49 | 45, 48 | anbi12d 743 |
. . . . 5
⊢ (𝑁 = (#‘(1st
‘𝐴)) → (((𝑁 = (#‘(1st
‘𝐵)) ∧
∀𝑥 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥))) ↔ (((#‘(1st
‘𝐴)) =
(#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1st
‘𝐴)))((1st
‘𝐴)‘𝑥) = ((1st
‘𝐵)‘𝑥)) ∧
((#‘(1st ‘𝐴)) = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈
(0...(#‘(1st ‘𝐴)))((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥))))) |
50 | 49 | bibi2d 331 |
. . . 4
⊢ (𝑁 = (#‘(1st
‘𝐴)) →
((((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵)) ↔ ((𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥)))) ↔ (((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd
‘𝐴) = (2nd
‘𝐵)) ↔
(((#‘(1st ‘𝐴)) = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈
(0..^(#‘(1st ‘𝐴)))((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥)) ∧ ((#‘(1st
‘𝐴)) =
(#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1st
‘𝐴)))((2nd
‘𝐴)‘𝑥) = ((2nd
‘𝐵)‘𝑥)))))) |
51 | 50 | 3ad2ant3 1077 |
. . 3
⊢ ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸) ∧ 𝑁 = (#‘(1st ‘𝐴))) → ((((1st
‘𝐴) = (1st
‘𝐵) ∧
(2nd ‘𝐴) =
(2nd ‘𝐵))
↔ ((𝑁 =
(#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥)))) ↔ (((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd
‘𝐴) = (2nd
‘𝐵)) ↔
(((#‘(1st ‘𝐴)) = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈
(0..^(#‘(1st ‘𝐴)))((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥)) ∧ ((#‘(1st
‘𝐴)) =
(#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1st
‘𝐴)))((2nd
‘𝐴)‘𝑥) = ((2nd
‘𝐵)‘𝑥)))))) |
52 | 41, 51 | mpbird 246 |
. 2
⊢ ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸) ∧ 𝑁 = (#‘(1st ‘𝐴))) → (((1st
‘𝐴) = (1st
‘𝐵) ∧
(2nd ‘𝐴) =
(2nd ‘𝐵))
↔ ((𝑁 =
(#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥))))) |
53 | | 3anass 1035 |
. . . 4
⊢ ((𝑁 = (#‘(1st
‘𝐵)) ∧
∀𝑥 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥)) ↔ (𝑁 = (#‘(1st ‘𝐵)) ∧ (∀𝑥 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥)))) |
54 | | anandi 867 |
. . . 4
⊢ ((𝑁 = (#‘(1st
‘𝐵)) ∧
(∀𝑥 ∈
(0..^𝑁)((1st
‘𝐴)‘𝑥) = ((1st
‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥))) ↔ ((𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥)))) |
55 | 53, 54 | bitr2i 264 |
. . 3
⊢ (((𝑁 = (#‘(1st
‘𝐵)) ∧
∀𝑥 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥))) ↔ (𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥))) |
56 | 55 | a1i 11 |
. 2
⊢ ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸) ∧ 𝑁 = (#‘(1st ‘𝐴))) → (((𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥))) ↔ (𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥)))) |
57 | 10, 52, 56 | 3bitrd 293 |
1
⊢ ((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸) ∧ 𝑁 = (#‘(1st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥)))) |