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 30023 | . . . 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 30023 | . . . 4
⊢ (𝑊 ∈ 𝐶 → (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) | 
| 8 | 4, 7 | anim12i 613 | . . 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 2737 | . . . . . . 7
⊢
(Vtx‘𝐺) =
(Vtx‘𝐺) | 
| 10 | 9 | wlkpwrd 29635 | . . . . . 6
⊢ (𝐴(Walks‘𝐺)𝐵 → 𝐵 ∈ Word (Vtx‘𝐺)) | 
| 11 | 10 | 3ad2ant1 1134 | . . . . 5
⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → 𝐵 ∈ Word (Vtx‘𝐺)) | 
| 12 | 9 | wlkpwrd 29635 | . . . . . 6
⊢ (𝐷(Walks‘𝐺)𝐸 → 𝐸 ∈ Word (Vtx‘𝐺)) | 
| 13 | 12 | 3ad2ant1 1134 | . . . . 5
⊢ ((𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ) → 𝐸 ∈ Word (Vtx‘𝐺)) | 
| 14 | 11, 13 | anim12i 613 | . . . 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 12533 | . . . . . 6
⊢
((♯‘𝐴)
∈ ℕ → (♯‘𝐴) ∈
ℕ0) | 
| 16 | 15 | 3ad2ant3 1136 | . . . . 5
⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) →
(♯‘𝐴) ∈
ℕ0) | 
| 17 |  | nnnn0 12533 | . . . . . 6
⊢
((♯‘𝐷)
∈ ℕ → (♯‘𝐷) ∈
ℕ0) | 
| 18 | 17 | 3ad2ant3 1136 | . . . . 5
⊢ ((𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ) →
(♯‘𝐷) ∈
ℕ0) | 
| 19 | 16, 18 | anim12i 613 | . . . 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 29636 | . . . . . . . 8
⊢ (𝐴(Walks‘𝐺)𝐵 → (♯‘𝐵) = ((♯‘𝐴) + 1)) | 
| 21 |  | nnre 12273 | . . . . . . . . . 10
⊢
((♯‘𝐴)
∈ ℕ → (♯‘𝐴) ∈ ℝ) | 
| 22 | 21 | lep1d 12199 | . . . . . . . . 9
⊢
((♯‘𝐴)
∈ ℕ → (♯‘𝐴) ≤ ((♯‘𝐴) + 1)) | 
| 23 |  | breq2 5147 | . . . . . . . . 9
⊢
((♯‘𝐵) =
((♯‘𝐴) + 1)
→ ((♯‘𝐴)
≤ (♯‘𝐵)
↔ (♯‘𝐴)
≤ ((♯‘𝐴) +
1))) | 
| 24 | 22, 23 | imbitrrid 246 | . . . . . . . 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 1111 | . . . . 5
⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) →
(♯‘𝐴) ≤
(♯‘𝐵)) | 
| 28 |  | wlklenvp1 29636 | . . . . . . . 8
⊢ (𝐷(Walks‘𝐺)𝐸 → (♯‘𝐸) = ((♯‘𝐷) + 1)) | 
| 29 |  | nnre 12273 | . . . . . . . . . 10
⊢
((♯‘𝐷)
∈ ℕ → (♯‘𝐷) ∈ ℝ) | 
| 30 | 29 | lep1d 12199 | . . . . . . . . 9
⊢
((♯‘𝐷)
∈ ℕ → (♯‘𝐷) ≤ ((♯‘𝐷) + 1)) | 
| 31 |  | breq2 5147 | . . . . . . . . 9
⊢
((♯‘𝐸) =
((♯‘𝐷) + 1)
→ ((♯‘𝐷)
≤ (♯‘𝐸)
↔ (♯‘𝐷)
≤ ((♯‘𝐷) +
1))) | 
| 32 | 30, 31 | imbitrrid 246 | . . . . . . . 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 1111 | . . . . 5
⊢ ((𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ) →
(♯‘𝐷) ≤
(♯‘𝐸)) | 
| 36 | 27, 35 | anim12i 613 | . . . 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 1129 | . . 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 14734 | . . 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 1471 | 1
⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖))) |