Step | Hyp | Ref
| Expression |
1 | | wlkop 26758 |
. . . . 5
⊢ (𝐴 ∈ (Walks‘𝐺) → 𝐴 = ⟨(1^{st} ‘𝐴), (2^{nd} ‘𝐴)⟩) |
2 | | 1st2ndb 7355 |
. . . . 5
⊢ (𝐴 ∈ (V × V) ↔
𝐴 = ⟨(1^{st}
‘𝐴), (2^{nd}
‘𝐴)⟩) |
3 | 1, 2 | sylibr 224 |
. . . 4
⊢ (𝐴 ∈ (Walks‘𝐺) → 𝐴 ∈ (V × V)) |
4 | | wlkop 26758 |
. . . . 5
⊢ (𝐵 ∈ (Walks‘𝐺) → 𝐵 = ⟨(1^{st} ‘𝐵), (2^{nd} ‘𝐵)⟩) |
5 | | 1st2ndb 7355 |
. . . . 5
⊢ (𝐵 ∈ (V × V) ↔
𝐵 = ⟨(1^{st}
‘𝐵), (2^{nd}
‘𝐵)⟩) |
6 | 4, 5 | sylibr 224 |
. . . 4
⊢ (𝐵 ∈ (Walks‘𝐺) → 𝐵 ∈ (V × V)) |
7 | | xpopth 7356 |
. . . . 5
⊢ ((𝐴 ∈ (V × V) ∧
𝐵 ∈ (V × V))
→ (((1^{st} ‘𝐴) = (1^{st} ‘𝐵) ∧ (2^{nd} ‘𝐴) = (2^{nd} ‘𝐵)) ↔ 𝐴 = 𝐵)) |
8 | 7 | bicomd 213 |
. . . 4
⊢ ((𝐴 ∈ (V × V) ∧
𝐵 ∈ (V × V))
→ (𝐴 = 𝐵 ↔ ((1^{st}
‘𝐴) = (1^{st}
‘𝐵) ∧
(2^{nd} ‘𝐴) =
(2^{nd} ‘𝐵)))) |
9 | 3, 6, 8 | syl2an 583 |
. . 3
⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (𝐴 = 𝐵 ↔ ((1^{st} ‘𝐴) = (1^{st} ‘𝐵) ∧ (2^{nd}
‘𝐴) = (2^{nd}
‘𝐵)))) |
10 | 9 | 3adant3 1126 |
. 2
⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (♯‘(1^{st}
‘𝐴))) → (𝐴 = 𝐵 ↔ ((1^{st} ‘𝐴) = (1^{st} ‘𝐵) ∧ (2^{nd}
‘𝐴) = (2^{nd}
‘𝐵)))) |
11 | | eqid 2771 |
. . . . . . 7
⊢
(Vtx‘𝐺) =
(Vtx‘𝐺) |
12 | | eqid 2771 |
. . . . . . 7
⊢
(iEdg‘𝐺) =
(iEdg‘𝐺) |
13 | | eqid 2771 |
. . . . . . 7
⊢
(1^{st} ‘𝐴) = (1^{st} ‘𝐴) |
14 | | eqid 2771 |
. . . . . . 7
⊢
(2^{nd} ‘𝐴) = (2^{nd} ‘𝐴) |
15 | 11, 12, 13, 14 | wlkelwrd 26763 |
. . . . . 6
⊢ (𝐴 ∈ (Walks‘𝐺) → ((1^{st}
‘𝐴) ∈ Word dom
(iEdg‘𝐺) ∧
(2^{nd} ‘𝐴):(0...(♯‘(1^{st}
‘𝐴)))⟶(Vtx‘𝐺))) |
16 | | eqid 2771 |
. . . . . . 7
⊢
(1^{st} ‘𝐵) = (1^{st} ‘𝐵) |
17 | | eqid 2771 |
. . . . . . 7
⊢
(2^{nd} ‘𝐵) = (2^{nd} ‘𝐵) |
18 | 11, 12, 16, 17 | wlkelwrd 26763 |
. . . . . 6
⊢ (𝐵 ∈ (Walks‘𝐺) → ((1^{st}
‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2^{nd} ‘𝐵):(0...(♯‘(1^{st}
‘𝐵)))⟶(Vtx‘𝐺))) |
19 | 15, 18 | anim12i 600 |
. . . . 5
⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (((1^{st} ‘𝐴) ∈ Word dom
(iEdg‘𝐺) ∧
(2^{nd} ‘𝐴):(0...(♯‘(1^{st}
‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^{st} ‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2^{nd} ‘𝐵):(0...(♯‘(1^{st}
‘𝐵)))⟶(Vtx‘𝐺)))) |
20 | | eleq1 2838 |
. . . . . . . 8
⊢ (𝐴 = ⟨(1^{st}
‘𝐴), (2^{nd}
‘𝐴)⟩ →
(𝐴 ∈
(Walks‘𝐺) ↔
⟨(1^{st} ‘𝐴), (2^{nd} ‘𝐴)⟩ ∈ (Walks‘𝐺))) |
21 | | df-br 4787 |
. . . . . . . . 9
⊢
((1^{st} ‘𝐴)(Walks‘𝐺)(2^{nd} ‘𝐴) ↔ ⟨(1^{st} ‘𝐴), (2^{nd} ‘𝐴)⟩ ∈
(Walks‘𝐺)) |
22 | | wlklenvm1 26752 |
. . . . . . . . 9
⊢
((1^{st} ‘𝐴)(Walks‘𝐺)(2^{nd} ‘𝐴) → (♯‘(1^{st}
‘𝐴)) =
((♯‘(2^{nd} ‘𝐴)) − 1)) |
23 | 21, 22 | sylbir 225 |
. . . . . . . 8
⊢
(⟨(1^{st} ‘𝐴), (2^{nd} ‘𝐴)⟩ ∈ (Walks‘𝐺) →
(♯‘(1^{st} ‘𝐴)) = ((♯‘(2^{nd}
‘𝐴)) −
1)) |
24 | 20, 23 | syl6bi 243 |
. . . . . . 7
⊢ (𝐴 = ⟨(1^{st}
‘𝐴), (2^{nd}
‘𝐴)⟩ →
(𝐴 ∈
(Walks‘𝐺) →
(♯‘(1^{st} ‘𝐴)) = ((♯‘(2^{nd}
‘𝐴)) −
1))) |
25 | 1, 24 | mpcom 38 |
. . . . . 6
⊢ (𝐴 ∈ (Walks‘𝐺) →
(♯‘(1^{st} ‘𝐴)) = ((♯‘(2^{nd}
‘𝐴)) −
1)) |
26 | | eleq1 2838 |
. . . . . . . 8
⊢ (𝐵 = ⟨(1^{st}
‘𝐵), (2^{nd}
‘𝐵)⟩ →
(𝐵 ∈
(Walks‘𝐺) ↔
⟨(1^{st} ‘𝐵), (2^{nd} ‘𝐵)⟩ ∈ (Walks‘𝐺))) |
27 | | df-br 4787 |
. . . . . . . . 9
⊢
((1^{st} ‘𝐵)(Walks‘𝐺)(2^{nd} ‘𝐵) ↔ ⟨(1^{st} ‘𝐵), (2^{nd} ‘𝐵)⟩ ∈
(Walks‘𝐺)) |
28 | | wlklenvm1 26752 |
. . . . . . . . 9
⊢
((1^{st} ‘𝐵)(Walks‘𝐺)(2^{nd} ‘𝐵) → (♯‘(1^{st}
‘𝐵)) =
((♯‘(2^{nd} ‘𝐵)) − 1)) |
29 | 27, 28 | sylbir 225 |
. . . . . . . 8
⊢
(⟨(1^{st} ‘𝐵), (2^{nd} ‘𝐵)⟩ ∈ (Walks‘𝐺) →
(♯‘(1^{st} ‘𝐵)) = ((♯‘(2^{nd}
‘𝐵)) −
1)) |
30 | 26, 29 | syl6bi 243 |
. . . . . . 7
⊢ (𝐵 = ⟨(1^{st}
‘𝐵), (2^{nd}
‘𝐵)⟩ →
(𝐵 ∈
(Walks‘𝐺) →
(♯‘(1^{st} ‘𝐵)) = ((♯‘(2^{nd}
‘𝐵)) −
1))) |
31 | 4, 30 | mpcom 38 |
. . . . . 6
⊢ (𝐵 ∈ (Walks‘𝐺) →
(♯‘(1^{st} ‘𝐵)) = ((♯‘(2^{nd}
‘𝐵)) −
1)) |
32 | 25, 31 | anim12i 600 |
. . . . 5
⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → ((♯‘(1^{st}
‘𝐴)) =
((♯‘(2^{nd} ‘𝐴)) − 1) ∧
(♯‘(1^{st} ‘𝐵)) = ((♯‘(2^{nd}
‘𝐵)) −
1))) |
33 | | eqwrd 13543 |
. . . . . . . 8
⊢
(((1^{st} ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (1^{st}
‘𝐵) ∈ Word dom
(iEdg‘𝐺)) →
((1^{st} ‘𝐴)
= (1^{st} ‘𝐵)
↔ ((♯‘(1^{st} ‘𝐴)) = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈
(0..^(♯‘(1^{st} ‘𝐴)))((1^{st} ‘𝐴)‘𝑥) = ((1^{st} ‘𝐵)‘𝑥)))) |
34 | 33 | ad2ant2r 741 |
. . . . . . 7
⊢
((((1^{st} ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^{nd}
‘𝐴):(0...(♯‘(1^{st}
‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^{st} ‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2^{nd} ‘𝐵):(0...(♯‘(1^{st}
‘𝐵)))⟶(Vtx‘𝐺))) → ((1^{st} ‘𝐴) = (1^{st} ‘𝐵) ↔
((♯‘(1^{st} ‘𝐴)) = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈
(0..^(♯‘(1^{st} ‘𝐴)))((1^{st} ‘𝐴)‘𝑥) = ((1^{st} ‘𝐵)‘𝑥)))) |
35 | 34 | adantr 466 |
. . . . . 6
⊢
(((((1^{st} ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^{nd}
‘𝐴):(0...(♯‘(1^{st}
‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^{st} ‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2^{nd} ‘𝐵):(0...(♯‘(1^{st}
‘𝐵)))⟶(Vtx‘𝐺))) ∧ ((♯‘(1^{st}
‘𝐴)) =
((♯‘(2^{nd} ‘𝐴)) − 1) ∧
(♯‘(1^{st} ‘𝐵)) = ((♯‘(2^{nd}
‘𝐵)) − 1)))
→ ((1^{st} ‘𝐴) = (1^{st} ‘𝐵) ↔ ((♯‘(1^{st}
‘𝐴)) =
(♯‘(1^{st} ‘𝐵)) ∧ ∀𝑥 ∈ (0..^(♯‘(1^{st}
‘𝐴)))((1^{st}
‘𝐴)‘𝑥) = ((1^{st}
‘𝐵)‘𝑥)))) |
36 | | lencl 13520 |
. . . . . . . . . . 11
⊢
((1^{st} ‘𝐴) ∈ Word dom (iEdg‘𝐺) →
(♯‘(1^{st} ‘𝐴)) ∈
ℕ_{0}) |
37 | 36 | adantr 466 |
. . . . . . . . . 10
⊢
(((1^{st} ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^{nd}
‘𝐴):(0...(♯‘(1^{st}
‘𝐴)))⟶(Vtx‘𝐺)) → (♯‘(1^{st}
‘𝐴)) ∈
ℕ_{0}) |
38 | 37 | adantr 466 |
. . . . . . . . 9
⊢
((((1^{st} ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^{nd}
‘𝐴):(0...(♯‘(1^{st}
‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^{st} ‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2^{nd} ‘𝐵):(0...(♯‘(1^{st}
‘𝐵)))⟶(Vtx‘𝐺))) → (♯‘(1^{st}
‘𝐴)) ∈
ℕ_{0}) |
39 | | simplr 752 |
. . . . . . . . 9
⊢
((((1^{st} ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^{nd}
‘𝐴):(0...(♯‘(1^{st}
‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^{st} ‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2^{nd} ‘𝐵):(0...(♯‘(1^{st}
‘𝐵)))⟶(Vtx‘𝐺))) → (2^{nd} ‘𝐴):(0...(♯‘(1^{st}
‘𝐴)))⟶(Vtx‘𝐺)) |
40 | | simprr 756 |
. . . . . . . . 9
⊢
((((1^{st} ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^{nd}
‘𝐴):(0...(♯‘(1^{st}
‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^{st} ‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2^{nd} ‘𝐵):(0...(♯‘(1^{st}
‘𝐵)))⟶(Vtx‘𝐺))) → (2^{nd} ‘𝐵):(0...(♯‘(1^{st}
‘𝐵)))⟶(Vtx‘𝐺)) |
41 | 38, 39, 40 | 3jca 1122 |
. . . . . . . 8
⊢
((((1^{st} ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^{nd}
‘𝐴):(0...(♯‘(1^{st}
‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^{st} ‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2^{nd} ‘𝐵):(0...(♯‘(1^{st}
‘𝐵)))⟶(Vtx‘𝐺))) → ((♯‘(1^{st}
‘𝐴)) ∈
ℕ_{0} ∧ (2^{nd} ‘𝐴):(0...(♯‘(1^{st}
‘𝐴)))⟶(Vtx‘𝐺) ∧ (2^{nd} ‘𝐵):(0...(♯‘(1^{st}
‘𝐵)))⟶(Vtx‘𝐺))) |
42 | 41 | adantr 466 |
. . . . . . 7
⊢
(((((1^{st} ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^{nd}
‘𝐴):(0...(♯‘(1^{st}
‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^{st} ‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2^{nd} ‘𝐵):(0...(♯‘(1^{st}
‘𝐵)))⟶(Vtx‘𝐺))) ∧ ((♯‘(1^{st}
‘𝐴)) =
((♯‘(2^{nd} ‘𝐴)) − 1) ∧
(♯‘(1^{st} ‘𝐵)) = ((♯‘(2^{nd}
‘𝐵)) − 1)))
→ ((♯‘(1^{st} ‘𝐴)) ∈ ℕ_{0} ∧
(2^{nd} ‘𝐴):(0...(♯‘(1^{st}
‘𝐴)))⟶(Vtx‘𝐺) ∧ (2^{nd} ‘𝐵):(0...(♯‘(1^{st}
‘𝐵)))⟶(Vtx‘𝐺))) |
43 | | 2ffzeq 12668 |
. . . . . . 7
⊢
(((♯‘(1^{st} ‘𝐴)) ∈ ℕ_{0} ∧
(2^{nd} ‘𝐴):(0...(♯‘(1^{st}
‘𝐴)))⟶(Vtx‘𝐺) ∧ (2^{nd} ‘𝐵):(0...(♯‘(1^{st}
‘𝐵)))⟶(Vtx‘𝐺)) → ((2^{nd} ‘𝐴) = (2^{nd} ‘𝐵) ↔
((♯‘(1^{st} ‘𝐴)) = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈
(0...(♯‘(1^{st} ‘𝐴)))((2^{nd} ‘𝐴)‘𝑥) = ((2^{nd} ‘𝐵)‘𝑥)))) |
44 | 42, 43 | syl 17 |
. . . . . 6
⊢
(((((1^{st} ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^{nd}
‘𝐴):(0...(♯‘(1^{st}
‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^{st} ‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2^{nd} ‘𝐵):(0...(♯‘(1^{st}
‘𝐵)))⟶(Vtx‘𝐺))) ∧ ((♯‘(1^{st}
‘𝐴)) =
((♯‘(2^{nd} ‘𝐴)) − 1) ∧
(♯‘(1^{st} ‘𝐵)) = ((♯‘(2^{nd}
‘𝐵)) − 1)))
→ ((2^{nd} ‘𝐴) = (2^{nd} ‘𝐵) ↔ ((♯‘(1^{st}
‘𝐴)) =
(♯‘(1^{st} ‘𝐵)) ∧ ∀𝑥 ∈ (0...(♯‘(1^{st}
‘𝐴)))((2^{nd}
‘𝐴)‘𝑥) = ((2^{nd}
‘𝐵)‘𝑥)))) |
45 | 35, 44 | anbi12d 616 |
. . . . 5
⊢
(((((1^{st} ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^{nd}
‘𝐴):(0...(♯‘(1^{st}
‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^{st} ‘𝐵) ∈ Word dom
(iEdg‘𝐺) ∧
(2^{nd} ‘𝐵):(0...(♯‘(1^{st}
‘𝐵)))⟶(Vtx‘𝐺))) ∧ ((♯‘(1^{st}
‘𝐴)) =
((♯‘(2^{nd} ‘𝐴)) − 1) ∧
(♯‘(1^{st} ‘𝐵)) = ((♯‘(2^{nd}
‘𝐵)) − 1)))
→ (((1^{st} ‘𝐴) = (1^{st} ‘𝐵) ∧ (2^{nd} ‘𝐴) = (2^{nd} ‘𝐵)) ↔
(((♯‘(1^{st} ‘𝐴)) = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈
(0..^(♯‘(1^{st} ‘𝐴)))((1^{st} ‘𝐴)‘𝑥) = ((1^{st} ‘𝐵)‘𝑥)) ∧ ((♯‘(1^{st}
‘𝐴)) =
(♯‘(1^{st} ‘𝐵)) ∧ ∀𝑥 ∈ (0...(♯‘(1^{st}
‘𝐴)))((2^{nd}
‘𝐴)‘𝑥) = ((2^{nd}
‘𝐵)‘𝑥))))) |
46 | 19, 32, 45 | syl2anc 573 |
. . . 4
⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (((1^{st} ‘𝐴) = (1^{st} ‘𝐵) ∧ (2^{nd}
‘𝐴) = (2^{nd}
‘𝐵)) ↔
(((♯‘(1^{st} ‘𝐴)) = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈
(0..^(♯‘(1^{st} ‘𝐴)))((1^{st} ‘𝐴)‘𝑥) = ((1^{st} ‘𝐵)‘𝑥)) ∧ ((♯‘(1^{st}
‘𝐴)) =
(♯‘(1^{st} ‘𝐵)) ∧ ∀𝑥 ∈ (0...(♯‘(1^{st}
‘𝐴)))((2^{nd}
‘𝐴)‘𝑥) = ((2^{nd}
‘𝐵)‘𝑥))))) |
47 | 46 | 3adant3 1126 |
. . 3
⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (♯‘(1^{st}
‘𝐴))) →
(((1^{st} ‘𝐴)
= (1^{st} ‘𝐵)
∧ (2^{nd} ‘𝐴) = (2^{nd} ‘𝐵)) ↔ (((♯‘(1^{st}
‘𝐴)) =
(♯‘(1^{st} ‘𝐵)) ∧ ∀𝑥 ∈ (0..^(♯‘(1^{st}
‘𝐴)))((1^{st}
‘𝐴)‘𝑥) = ((1^{st}
‘𝐵)‘𝑥)) ∧
((♯‘(1^{st} ‘𝐴)) = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈
(0...(♯‘(1^{st} ‘𝐴)))((2^{nd} ‘𝐴)‘𝑥) = ((2^{nd} ‘𝐵)‘𝑥))))) |
48 | | eqeq1 2775 |
. . . . . . 7
⊢ (𝑁 =
(♯‘(1^{st} ‘𝐴)) → (𝑁 = (♯‘(1^{st}
‘𝐵)) ↔
(♯‘(1^{st} ‘𝐴)) = (♯‘(1^{st}
‘𝐵)))) |
49 | | oveq2 6801 |
. . . . . . . 8
⊢ (𝑁 =
(♯‘(1^{st} ‘𝐴)) → (0..^𝑁) = (0..^(♯‘(1^{st}
‘𝐴)))) |
50 | 49 | raleqdv 3293 |
. . . . . . 7
⊢ (𝑁 =
(♯‘(1^{st} ‘𝐴)) → (∀𝑥 ∈ (0..^𝑁)((1^{st} ‘𝐴)‘𝑥) = ((1^{st} ‘𝐵)‘𝑥) ↔ ∀𝑥 ∈ (0..^(♯‘(1^{st}
‘𝐴)))((1^{st}
‘𝐴)‘𝑥) = ((1^{st}
‘𝐵)‘𝑥))) |
51 | 48, 50 | anbi12d 616 |
. . . . . 6
⊢ (𝑁 =
(♯‘(1^{st} ‘𝐴)) → ((𝑁 = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈ (0..^𝑁)((1^{st} ‘𝐴)‘𝑥) = ((1^{st} ‘𝐵)‘𝑥)) ↔ ((♯‘(1^{st}
‘𝐴)) =
(♯‘(1^{st} ‘𝐵)) ∧ ∀𝑥 ∈ (0..^(♯‘(1^{st}
‘𝐴)))((1^{st}
‘𝐴)‘𝑥) = ((1^{st}
‘𝐵)‘𝑥)))) |
52 | | oveq2 6801 |
. . . . . . . 8
⊢ (𝑁 =
(♯‘(1^{st} ‘𝐴)) → (0...𝑁) = (0...(♯‘(1^{st}
‘𝐴)))) |
53 | 52 | raleqdv 3293 |
. . . . . . 7
⊢ (𝑁 =
(♯‘(1^{st} ‘𝐴)) → (∀𝑥 ∈ (0...𝑁)((2^{nd} ‘𝐴)‘𝑥) = ((2^{nd} ‘𝐵)‘𝑥) ↔ ∀𝑥 ∈ (0...(♯‘(1^{st}
‘𝐴)))((2^{nd}
‘𝐴)‘𝑥) = ((2^{nd}
‘𝐵)‘𝑥))) |
54 | 48, 53 | anbi12d 616 |
. . . . . 6
⊢ (𝑁 =
(♯‘(1^{st} ‘𝐴)) → ((𝑁 = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈ (0...𝑁)((2^{nd} ‘𝐴)‘𝑥) = ((2^{nd} ‘𝐵)‘𝑥)) ↔ ((♯‘(1^{st}
‘𝐴)) =
(♯‘(1^{st} ‘𝐵)) ∧ ∀𝑥 ∈ (0...(♯‘(1^{st}
‘𝐴)))((2^{nd}
‘𝐴)‘𝑥) = ((2^{nd}
‘𝐵)‘𝑥)))) |
55 | 51, 54 | anbi12d 616 |
. . . . 5
⊢ (𝑁 =
(♯‘(1^{st} ‘𝐴)) → (((𝑁 = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈ (0..^𝑁)((1^{st} ‘𝐴)‘𝑥) = ((1^{st} ‘𝐵)‘𝑥)) ∧ (𝑁 = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈ (0...𝑁)((2^{nd} ‘𝐴)‘𝑥) = ((2^{nd} ‘𝐵)‘𝑥))) ↔ (((♯‘(1^{st}
‘𝐴)) =
(♯‘(1^{st} ‘𝐵)) ∧ ∀𝑥 ∈ (0..^(♯‘(1^{st}
‘𝐴)))((1^{st}
‘𝐴)‘𝑥) = ((1^{st}
‘𝐵)‘𝑥)) ∧
((♯‘(1^{st} ‘𝐴)) = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈
(0...(♯‘(1^{st} ‘𝐴)))((2^{nd} ‘𝐴)‘𝑥) = ((2^{nd} ‘𝐵)‘𝑥))))) |
56 | 55 | bibi2d 331 |
. . . 4
⊢ (𝑁 =
(♯‘(1^{st} ‘𝐴)) → ((((1^{st} ‘𝐴) = (1^{st} ‘𝐵) ∧ (2^{nd}
‘𝐴) = (2^{nd}
‘𝐵)) ↔ ((𝑁 =
(♯‘(1^{st} ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1^{st} ‘𝐴)‘𝑥) = ((1^{st} ‘𝐵)‘𝑥)) ∧ (𝑁 = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈ (0...𝑁)((2^{nd} ‘𝐴)‘𝑥) = ((2^{nd} ‘𝐵)‘𝑥)))) ↔ (((1^{st} ‘𝐴) = (1^{st} ‘𝐵) ∧ (2^{nd}
‘𝐴) = (2^{nd}
‘𝐵)) ↔
(((♯‘(1^{st} ‘𝐴)) = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈
(0..^(♯‘(1^{st} ‘𝐴)))((1^{st} ‘𝐴)‘𝑥) = ((1^{st} ‘𝐵)‘𝑥)) ∧ ((♯‘(1^{st}
‘𝐴)) =
(♯‘(1^{st} ‘𝐵)) ∧ ∀𝑥 ∈ (0...(♯‘(1^{st}
‘𝐴)))((2^{nd}
‘𝐴)‘𝑥) = ((2^{nd}
‘𝐵)‘𝑥)))))) |
57 | 56 | 3ad2ant3 1129 |
. . 3
⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (♯‘(1^{st}
‘𝐴))) →
((((1^{st} ‘𝐴) = (1^{st} ‘𝐵) ∧ (2^{nd} ‘𝐴) = (2^{nd} ‘𝐵)) ↔ ((𝑁 = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈ (0..^𝑁)((1^{st} ‘𝐴)‘𝑥) = ((1^{st} ‘𝐵)‘𝑥)) ∧ (𝑁 = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈ (0...𝑁)((2^{nd} ‘𝐴)‘𝑥) = ((2^{nd} ‘𝐵)‘𝑥)))) ↔ (((1^{st} ‘𝐴) = (1^{st} ‘𝐵) ∧ (2^{nd}
‘𝐴) = (2^{nd}
‘𝐵)) ↔
(((♯‘(1^{st} ‘𝐴)) = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈
(0..^(♯‘(1^{st} ‘𝐴)))((1^{st} ‘𝐴)‘𝑥) = ((1^{st} ‘𝐵)‘𝑥)) ∧ ((♯‘(1^{st}
‘𝐴)) =
(♯‘(1^{st} ‘𝐵)) ∧ ∀𝑥 ∈ (0...(♯‘(1^{st}
‘𝐴)))((2^{nd}
‘𝐴)‘𝑥) = ((2^{nd}
‘𝐵)‘𝑥)))))) |
58 | 47, 57 | mpbird 247 |
. 2
⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (♯‘(1^{st}
‘𝐴))) →
(((1^{st} ‘𝐴)
= (1^{st} ‘𝐵)
∧ (2^{nd} ‘𝐴) = (2^{nd} ‘𝐵)) ↔ ((𝑁 = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈ (0..^𝑁)((1^{st} ‘𝐴)‘𝑥) = ((1^{st} ‘𝐵)‘𝑥)) ∧ (𝑁 = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈ (0...𝑁)((2^{nd} ‘𝐴)‘𝑥) = ((2^{nd} ‘𝐵)‘𝑥))))) |
59 | | 3anass 1080 |
. . . 4
⊢ ((𝑁 =
(♯‘(1^{st} ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1^{st} ‘𝐴)‘𝑥) = ((1^{st} ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2^{nd} ‘𝐴)‘𝑥) = ((2^{nd} ‘𝐵)‘𝑥)) ↔ (𝑁 = (♯‘(1^{st}
‘𝐵)) ∧
(∀𝑥 ∈
(0..^𝑁)((1^{st}
‘𝐴)‘𝑥) = ((1^{st}
‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2^{nd} ‘𝐴)‘𝑥) = ((2^{nd} ‘𝐵)‘𝑥)))) |
60 | | anandi 655 |
. . . 4
⊢ ((𝑁 =
(♯‘(1^{st} ‘𝐵)) ∧ (∀𝑥 ∈ (0..^𝑁)((1^{st} ‘𝐴)‘𝑥) = ((1^{st} ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2^{nd} ‘𝐴)‘𝑥) = ((2^{nd} ‘𝐵)‘𝑥))) ↔ ((𝑁 = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈ (0..^𝑁)((1^{st} ‘𝐴)‘𝑥) = ((1^{st} ‘𝐵)‘𝑥)) ∧ (𝑁 = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈ (0...𝑁)((2^{nd} ‘𝐴)‘𝑥) = ((2^{nd} ‘𝐵)‘𝑥)))) |
61 | 59, 60 | bitr2i 265 |
. . 3
⊢ (((𝑁 =
(♯‘(1^{st} ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1^{st} ‘𝐴)‘𝑥) = ((1^{st} ‘𝐵)‘𝑥)) ∧ (𝑁 = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈ (0...𝑁)((2^{nd} ‘𝐴)‘𝑥) = ((2^{nd} ‘𝐵)‘𝑥))) ↔ (𝑁 = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈ (0..^𝑁)((1^{st} ‘𝐴)‘𝑥) = ((1^{st} ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2^{nd} ‘𝐴)‘𝑥) = ((2^{nd} ‘𝐵)‘𝑥))) |
62 | 61 | a1i 11 |
. 2
⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (♯‘(1^{st}
‘𝐴))) → (((𝑁 =
(♯‘(1^{st} ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1^{st} ‘𝐴)‘𝑥) = ((1^{st} ‘𝐵)‘𝑥)) ∧ (𝑁 = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈ (0...𝑁)((2^{nd} ‘𝐴)‘𝑥) = ((2^{nd} ‘𝐵)‘𝑥))) ↔ (𝑁 = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈ (0..^𝑁)((1^{st} ‘𝐴)‘𝑥) = ((1^{st} ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2^{nd} ‘𝐴)‘𝑥) = ((2^{nd} ‘𝐵)‘𝑥)))) |
63 | 10, 58, 62 | 3bitrd 294 |
1
⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (♯‘(1^{st}
‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1^{st}
‘𝐵)) ∧
∀𝑥 ∈ (0..^𝑁)((1^{st} ‘𝐴)‘𝑥) = ((1^{st} ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2^{nd} ‘𝐴)‘𝑥) = ((2^{nd} ‘𝐵)‘𝑥)))) |