Proof of Theorem clwlkclwwlkf1lem2
Step | Hyp | Ref
| Expression |
1 | | clwlkclwwlkf.c |
. . . . 5
⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤
(♯‘(1st ‘𝑤))} |
2 | | clwlkclwwlkf.a |
. . . . 5
⊢ 𝐴 = (1st ‘𝑈) |
3 | | clwlkclwwlkf.b |
. . . . 5
⊢ 𝐵 = (2nd ‘𝑈) |
4 | 1, 2, 3 | clwlkclwwlkflem 28269 |
. . . 4
⊢ (𝑈 ∈ 𝐶 → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)) |
5 | | clwlkclwwlkf.d |
. . . . 5
⊢ 𝐷 = (1st ‘𝑊) |
6 | | clwlkclwwlkf.e |
. . . . 5
⊢ 𝐸 = (2nd ‘𝑊) |
7 | 1, 5, 6 | clwlkclwwlkflem 28269 |
. . . 4
⊢ (𝑊 ∈ 𝐶 → (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) |
8 | 4, 7 | anim12i 612 |
. . 3
⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ))) |
9 | | eqid 2738 |
. . . . . . 7
⊢
(Vtx‘𝐺) =
(Vtx‘𝐺) |
10 | 9 | wlkpwrd 27887 |
. . . . . 6
⊢ (𝐴(Walks‘𝐺)𝐵 → 𝐵 ∈ Word (Vtx‘𝐺)) |
11 | 10 | 3ad2ant1 1131 |
. . . . 5
⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → 𝐵 ∈ Word (Vtx‘𝐺)) |
12 | 9 | wlkpwrd 27887 |
. . . . . 6
⊢ (𝐷(Walks‘𝐺)𝐸 → 𝐸 ∈ Word (Vtx‘𝐺)) |
13 | 12 | 3ad2ant1 1131 |
. . . . 5
⊢ ((𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ) → 𝐸 ∈ Word (Vtx‘𝐺)) |
14 | 11, 13 | anim12i 612 |
. . . 4
⊢ (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) → (𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝐸 ∈ Word (Vtx‘𝐺))) |
15 | | nnnn0 12170 |
. . . . . 6
⊢
((♯‘𝐴)
∈ ℕ → (♯‘𝐴) ∈
ℕ0) |
16 | 15 | 3ad2ant3 1133 |
. . . . 5
⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) →
(♯‘𝐴) ∈
ℕ0) |
17 | | nnnn0 12170 |
. . . . . 6
⊢
((♯‘𝐷)
∈ ℕ → (♯‘𝐷) ∈
ℕ0) |
18 | 17 | 3ad2ant3 1133 |
. . . . 5
⊢ ((𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ) →
(♯‘𝐷) ∈
ℕ0) |
19 | 16, 18 | anim12i 612 |
. . . 4
⊢ (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) →
((♯‘𝐴) ∈
ℕ0 ∧ (♯‘𝐷) ∈
ℕ0)) |
20 | | wlklenvp1 27888 |
. . . . . . . 8
⊢ (𝐴(Walks‘𝐺)𝐵 → (♯‘𝐵) = ((♯‘𝐴) + 1)) |
21 | | nnre 11910 |
. . . . . . . . . 10
⊢
((♯‘𝐴)
∈ ℕ → (♯‘𝐴) ∈ ℝ) |
22 | 21 | lep1d 11836 |
. . . . . . . . 9
⊢
((♯‘𝐴)
∈ ℕ → (♯‘𝐴) ≤ ((♯‘𝐴) + 1)) |
23 | | breq2 5074 |
. . . . . . . . 9
⊢
((♯‘𝐵) =
((♯‘𝐴) + 1)
→ ((♯‘𝐴)
≤ (♯‘𝐵)
↔ (♯‘𝐴)
≤ ((♯‘𝐴) +
1))) |
24 | 22, 23 | syl5ibr 245 |
. . . . . . . 8
⊢
((♯‘𝐵) =
((♯‘𝐴) + 1)
→ ((♯‘𝐴)
∈ ℕ → (♯‘𝐴) ≤ (♯‘𝐵))) |
25 | 20, 24 | syl 17 |
. . . . . . 7
⊢ (𝐴(Walks‘𝐺)𝐵 → ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ≤ (♯‘𝐵))) |
26 | 25 | a1d 25 |
. . . . . 6
⊢ (𝐴(Walks‘𝐺)𝐵 → ((𝐵‘0) = (𝐵‘(♯‘𝐴)) → ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ≤ (♯‘𝐵)))) |
27 | 26 | 3imp 1109 |
. . . . 5
⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) →
(♯‘𝐴) ≤
(♯‘𝐵)) |
28 | | wlklenvp1 27888 |
. . . . . . . 8
⊢ (𝐷(Walks‘𝐺)𝐸 → (♯‘𝐸) = ((♯‘𝐷) + 1)) |
29 | | nnre 11910 |
. . . . . . . . . 10
⊢
((♯‘𝐷)
∈ ℕ → (♯‘𝐷) ∈ ℝ) |
30 | 29 | lep1d 11836 |
. . . . . . . . 9
⊢
((♯‘𝐷)
∈ ℕ → (♯‘𝐷) ≤ ((♯‘𝐷) + 1)) |
31 | | breq2 5074 |
. . . . . . . . 9
⊢
((♯‘𝐸) =
((♯‘𝐷) + 1)
→ ((♯‘𝐷)
≤ (♯‘𝐸)
↔ (♯‘𝐷)
≤ ((♯‘𝐷) +
1))) |
32 | 30, 31 | syl5ibr 245 |
. . . . . . . 8
⊢
((♯‘𝐸) =
((♯‘𝐷) + 1)
→ ((♯‘𝐷)
∈ ℕ → (♯‘𝐷) ≤ (♯‘𝐸))) |
33 | 28, 32 | syl 17 |
. . . . . . 7
⊢ (𝐷(Walks‘𝐺)𝐸 → ((♯‘𝐷) ∈ ℕ → (♯‘𝐷) ≤ (♯‘𝐸))) |
34 | 33 | a1d 25 |
. . . . . 6
⊢ (𝐷(Walks‘𝐺)𝐸 → ((𝐸‘0) = (𝐸‘(♯‘𝐷)) → ((♯‘𝐷) ∈ ℕ → (♯‘𝐷) ≤ (♯‘𝐸)))) |
35 | 34 | 3imp 1109 |
. . . . 5
⊢ ((𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ) →
(♯‘𝐷) ≤
(♯‘𝐸)) |
36 | 27, 35 | anim12i 612 |
. . . 4
⊢ (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) →
((♯‘𝐴) ≤
(♯‘𝐵) ∧
(♯‘𝐷) ≤
(♯‘𝐸))) |
37 | 14, 19, 36 | 3jca 1126 |
. . 3
⊢ (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) → ((𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝐸 ∈ Word (Vtx‘𝐺)) ∧ ((♯‘𝐴) ∈ ℕ0 ∧
(♯‘𝐷) ∈
ℕ0) ∧ ((♯‘𝐴) ≤ (♯‘𝐵) ∧ (♯‘𝐷) ≤ (♯‘𝐸)))) |
38 | | pfxeq 14337 |
. . 3
⊢ (((𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝐸 ∈ Word (Vtx‘𝐺)) ∧ ((♯‘𝐴) ∈ ℕ0 ∧
(♯‘𝐷) ∈
ℕ0) ∧ ((♯‘𝐴) ≤ (♯‘𝐵) ∧ (♯‘𝐷) ≤ (♯‘𝐸))) → ((𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷)) ↔ ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖)))) |
39 | 8, 37, 38 | 3syl 18 |
. 2
⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → ((𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷)) ↔ ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖)))) |
40 | 39 | biimp3a 1467 |
1
⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖))) |