Proof of Theorem isclwwlknx
Step | Hyp | Ref
| Expression |
1 | | eleq1 2828 |
. . . . . . . . . 10
⊢
((♯‘𝑊) =
𝑁 →
((♯‘𝑊) ∈
ℕ ↔ 𝑁 ∈
ℕ)) |
2 | | len0nnbi 14252 |
. . . . . . . . . . 11
⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ≠ ∅ ↔ (♯‘𝑊) ∈
ℕ)) |
3 | 2 | biimprcd 249 |
. . . . . . . . . 10
⊢
((♯‘𝑊)
∈ ℕ → (𝑊
∈ Word 𝑉 → 𝑊 ≠ ∅)) |
4 | 1, 3 | syl6bir 253 |
. . . . . . . . 9
⊢
((♯‘𝑊) =
𝑁 → (𝑁 ∈ ℕ → (𝑊 ∈ Word 𝑉 → 𝑊 ≠ ∅))) |
5 | 4 | impcom 408 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧
(♯‘𝑊) = 𝑁) → (𝑊 ∈ Word 𝑉 → 𝑊 ≠ ∅)) |
6 | 5 | imp 407 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ ∧
(♯‘𝑊) = 𝑁) ∧ 𝑊 ∈ Word 𝑉) → 𝑊 ≠ ∅) |
7 | 6 | biantrurd 533 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ ∧
(♯‘𝑊) = 𝑁) ∧ 𝑊 ∈ Word 𝑉) → ((∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸) ↔ (𝑊 ≠ ∅ ∧ (∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸)))) |
8 | 7 | bicomd 222 |
. . . . 5
⊢ (((𝑁 ∈ ℕ ∧
(♯‘𝑊) = 𝑁) ∧ 𝑊 ∈ Word 𝑉) → ((𝑊 ≠ ∅ ∧ (∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸)) ↔ (∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸))) |
9 | 8 | pm5.32da 579 |
. . . 4
⊢ ((𝑁 ∈ ℕ ∧
(♯‘𝑊) = 𝑁) → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 ≠ ∅ ∧ (∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸))) ↔ (𝑊 ∈ Word 𝑉 ∧ (∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸)))) |
10 | 9 | ex 413 |
. . 3
⊢ (𝑁 ∈ ℕ →
((♯‘𝑊) = 𝑁 → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 ≠ ∅ ∧ (∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸))) ↔ (𝑊 ∈ Word 𝑉 ∧ (∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸))))) |
11 | 10 | pm5.32rd 578 |
. 2
⊢ (𝑁 ∈ ℕ → (((𝑊 ∈ Word 𝑉 ∧ (𝑊 ≠ ∅ ∧ (∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸))) ∧ (♯‘𝑊) = 𝑁) ↔ ((𝑊 ∈ Word 𝑉 ∧ (∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸)) ∧ (♯‘𝑊) = 𝑁))) |
12 | | isclwwlkn 28387 |
. . 3
⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 𝑁)) |
13 | | isclwwlknx.v |
. . . . . 6
⊢ 𝑉 = (Vtx‘𝐺) |
14 | | isclwwlknx.e |
. . . . . 6
⊢ 𝐸 = (Edg‘𝐺) |
15 | 13, 14 | isclwwlk 28344 |
. . . . 5
⊢ (𝑊 ∈ (ClWWalks‘𝐺) ↔ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) ∧ ∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸)) |
16 | | 3anass 1094 |
. . . . . 6
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) ∧ ∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸) ↔ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) ∧ (∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸))) |
17 | | anass 469 |
. . . . . 6
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) ∧ (∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸)) ↔ (𝑊 ∈ Word 𝑉 ∧ (𝑊 ≠ ∅ ∧ (∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸)))) |
18 | 16, 17 | bitri 274 |
. . . . 5
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) ∧ ∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸) ↔ (𝑊 ∈ Word 𝑉 ∧ (𝑊 ≠ ∅ ∧ (∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸)))) |
19 | 15, 18 | bitri 274 |
. . . 4
⊢ (𝑊 ∈ (ClWWalks‘𝐺) ↔ (𝑊 ∈ Word 𝑉 ∧ (𝑊 ≠ ∅ ∧ (∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸)))) |
20 | 19 | anbi1i 624 |
. . 3
⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 𝑁) ↔ ((𝑊 ∈ Word 𝑉 ∧ (𝑊 ≠ ∅ ∧ (∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸))) ∧ (♯‘𝑊) = 𝑁)) |
21 | 12, 20 | bitri 274 |
. 2
⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ ((𝑊 ∈ Word 𝑉 ∧ (𝑊 ≠ ∅ ∧ (∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸))) ∧ (♯‘𝑊) = 𝑁)) |
22 | | 3anass 1094 |
. . 3
⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸) ↔ (𝑊 ∈ Word 𝑉 ∧ (∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸))) |
23 | 22 | anbi1i 624 |
. 2
⊢ (((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (♯‘𝑊) = 𝑁) ↔ ((𝑊 ∈ Word 𝑉 ∧ (∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸)) ∧ (♯‘𝑊) = 𝑁)) |
24 | 11, 21, 23 | 3bitr4g 314 |
1
⊢ (𝑁 ∈ ℕ → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (♯‘𝑊) = 𝑁))) |