| Step | Hyp | Ref
| Expression |
| 1 | | 1nn 12277 |
. . 3
⊢ 1 ∈
ℕ |
| 2 | | eqid 2737 |
. . . 4
⊢
(Vtx‘𝐺) =
(Vtx‘𝐺) |
| 3 | | eqid 2737 |
. . . 4
⊢
(Edg‘𝐺) =
(Edg‘𝐺) |
| 4 | 2, 3 | isclwwlknx 30055 |
. . 3
⊢ (1 ∈
ℕ → (𝑊 ∈ (1
ClWWalksN 𝐺) ↔ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑊) = 1))) |
| 5 | 1, 4 | ax-mp 5 |
. 2
⊢ (𝑊 ∈ (1 ClWWalksN 𝐺) ↔ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑊) = 1)) |
| 6 | | 3anass 1095 |
. . . 4
⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺)) ↔ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺)))) |
| 7 | | ral0 4513 |
. . . . . . . 8
⊢
∀𝑖 ∈
∅ {(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) |
| 8 | | oveq1 7438 |
. . . . . . . . . . . . 13
⊢
((♯‘𝑊) =
1 → ((♯‘𝑊)
− 1) = (1 − 1)) |
| 9 | | 1m1e0 12338 |
. . . . . . . . . . . . 13
⊢ (1
− 1) = 0 |
| 10 | 8, 9 | eqtrdi 2793 |
. . . . . . . . . . . 12
⊢
((♯‘𝑊) =
1 → ((♯‘𝑊)
− 1) = 0) |
| 11 | 10 | oveq2d 7447 |
. . . . . . . . . . 11
⊢
((♯‘𝑊) =
1 → (0..^((♯‘𝑊) − 1)) = (0..^0)) |
| 12 | | fzo0 13723 |
. . . . . . . . . . 11
⊢ (0..^0) =
∅ |
| 13 | 11, 12 | eqtrdi 2793 |
. . . . . . . . . 10
⊢
((♯‘𝑊) =
1 → (0..^((♯‘𝑊) − 1)) = ∅) |
| 14 | 13 | raleqdv 3326 |
. . . . . . . . 9
⊢
((♯‘𝑊) =
1 → (∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ ∅ {(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺))) |
| 15 | 14 | adantr 480 |
. . . . . . . 8
⊢
(((♯‘𝑊)
= 1 ∧ 𝑊 ∈ Word
(Vtx‘𝐺)) →
(∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ ∅ {(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺))) |
| 16 | 7, 15 | mpbiri 258 |
. . . . . . 7
⊢
(((♯‘𝑊)
= 1 ∧ 𝑊 ∈ Word
(Vtx‘𝐺)) →
∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) |
| 17 | 16 | biantrurd 532 |
. . . . . 6
⊢
(((♯‘𝑊)
= 1 ∧ 𝑊 ∈ Word
(Vtx‘𝐺)) →
({(lastS‘𝑊), (𝑊‘0)} ∈
(Edg‘𝐺) ↔
(∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺)))) |
| 18 | | lsw1 14605 |
. . . . . . . . . 10
⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 1) → (lastS‘𝑊) = (𝑊‘0)) |
| 19 | 18 | ancoms 458 |
. . . . . . . . 9
⊢
(((♯‘𝑊)
= 1 ∧ 𝑊 ∈ Word
(Vtx‘𝐺)) →
(lastS‘𝑊) = (𝑊‘0)) |
| 20 | 19 | preq1d 4739 |
. . . . . . . 8
⊢
(((♯‘𝑊)
= 1 ∧ 𝑊 ∈ Word
(Vtx‘𝐺)) →
{(lastS‘𝑊), (𝑊‘0)} = {(𝑊‘0), (𝑊‘0)}) |
| 21 | | dfsn2 4639 |
. . . . . . . 8
⊢ {(𝑊‘0)} = {(𝑊‘0), (𝑊‘0)} |
| 22 | 20, 21 | eqtr4di 2795 |
. . . . . . 7
⊢
(((♯‘𝑊)
= 1 ∧ 𝑊 ∈ Word
(Vtx‘𝐺)) →
{(lastS‘𝑊), (𝑊‘0)} = {(𝑊‘0)}) |
| 23 | 22 | eleq1d 2826 |
. . . . . 6
⊢
(((♯‘𝑊)
= 1 ∧ 𝑊 ∈ Word
(Vtx‘𝐺)) →
({(lastS‘𝑊), (𝑊‘0)} ∈
(Edg‘𝐺) ↔
{(𝑊‘0)} ∈
(Edg‘𝐺))) |
| 24 | 17, 23 | bitr3d 281 |
. . . . 5
⊢
(((♯‘𝑊)
= 1 ∧ 𝑊 ∈ Word
(Vtx‘𝐺)) →
((∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺)) ↔ {(𝑊‘0)} ∈ (Edg‘𝐺))) |
| 25 | 24 | pm5.32da 579 |
. . . 4
⊢
((♯‘𝑊) =
1 → ((𝑊 ∈ Word
(Vtx‘𝐺) ∧
(∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺))) ↔ (𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)))) |
| 26 | 6, 25 | bitrid 283 |
. . 3
⊢
((♯‘𝑊) =
1 → ((𝑊 ∈ Word
(Vtx‘𝐺) ∧
∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺)) ↔ (𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)))) |
| 27 | 26 | pm5.32ri 575 |
. 2
⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑊) = 1) ↔ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑊) = 1)) |
| 28 | | 3anass 1095 |
. . 3
⊢
(((♯‘𝑊)
= 1 ∧ 𝑊 ∈ Word
(Vtx‘𝐺) ∧ {(𝑊‘0)} ∈
(Edg‘𝐺)) ↔
((♯‘𝑊) = 1
∧ (𝑊 ∈ Word
(Vtx‘𝐺) ∧ {(𝑊‘0)} ∈
(Edg‘𝐺)))) |
| 29 | | ancom 460 |
. . 3
⊢
(((♯‘𝑊)
= 1 ∧ (𝑊 ∈ Word
(Vtx‘𝐺) ∧ {(𝑊‘0)} ∈
(Edg‘𝐺))) ↔
((𝑊 ∈ Word
(Vtx‘𝐺) ∧ {(𝑊‘0)} ∈
(Edg‘𝐺)) ∧
(♯‘𝑊) =
1)) |
| 30 | 28, 29 | bitr2i 276 |
. 2
⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑊) = 1) ↔
((♯‘𝑊) = 1
∧ 𝑊 ∈ Word
(Vtx‘𝐺) ∧ {(𝑊‘0)} ∈
(Edg‘𝐺))) |
| 31 | 5, 27, 30 | 3bitri 297 |
1
⊢ (𝑊 ∈ (1 ClWWalksN 𝐺) ↔ ((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺))) |