Step | Hyp | Ref
| Expression |
1 | | upgr2wlk.v |
. . . 4
⊢ 𝑉 = (Vtx‘𝐺) |
2 | | upgr2wlk.i |
. . . 4
⊢ 𝐼 = (iEdg‘𝐺) |
3 | 1, 2 | upgriswlk 27988 |
. . 3
⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) |
4 | 3 | anbi1d 629 |
. 2
⊢ (𝐺 ∈ UPGraph → ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) ↔ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ∧ (♯‘𝐹) = 2))) |
5 | | iswrdb 14204 |
. . . . . . . . 9
⊢ (𝐹 ∈ Word dom 𝐼 ↔ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) |
6 | | oveq2 7276 |
. . . . . . . . . 10
⊢
((♯‘𝐹) =
2 → (0..^(♯‘𝐹)) = (0..^2)) |
7 | 6 | feq2d 6582 |
. . . . . . . . 9
⊢
((♯‘𝐹) =
2 → (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 ↔ 𝐹:(0..^2)⟶dom 𝐼)) |
8 | 5, 7 | syl5bb 282 |
. . . . . . . 8
⊢
((♯‘𝐹) =
2 → (𝐹 ∈ Word dom
𝐼 ↔ 𝐹:(0..^2)⟶dom 𝐼)) |
9 | | oveq2 7276 |
. . . . . . . . 9
⊢
((♯‘𝐹) =
2 → (0...(♯‘𝐹)) = (0...2)) |
10 | 9 | feq2d 6582 |
. . . . . . . 8
⊢
((♯‘𝐹) =
2 → (𝑃:(0...(♯‘𝐹))⟶𝑉 ↔ 𝑃:(0...2)⟶𝑉)) |
11 | | fzo0to2pr 13453 |
. . . . . . . . . . 11
⊢ (0..^2) =
{0, 1} |
12 | 6, 11 | eqtrdi 2795 |
. . . . . . . . . 10
⊢
((♯‘𝐹) =
2 → (0..^(♯‘𝐹)) = {0, 1}) |
13 | 12 | raleqdv 3346 |
. . . . . . . . 9
⊢
((♯‘𝐹) =
2 → (∀𝑘 ∈
(0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ {0, 1} (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) |
14 | | 2wlklem 28015 |
. . . . . . . . 9
⊢
(∀𝑘 ∈
{0, 1} (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) |
15 | 13, 14 | bitrdi 286 |
. . . . . . . 8
⊢
((♯‘𝐹) =
2 → (∀𝑘 ∈
(0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) |
16 | 8, 10, 15 | 3anbi123d 1434 |
. . . . . . 7
⊢
((♯‘𝐹) =
2 → ((𝐹 ∈ Word
dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))) |
17 | 16 | adantl 481 |
. . . . . 6
⊢ ((𝐺 ∈ UPGraph ∧
(♯‘𝐹) = 2)
→ ((𝐹 ∈ Word dom
𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))) |
18 | | 3anass 1093 |
. . . . . 6
⊢ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))) |
19 | 17, 18 | bitrdi 286 |
. . . . 5
⊢ ((𝐺 ∈ UPGraph ∧
(♯‘𝐹) = 2)
→ ((𝐹 ∈ Word dom
𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))) |
20 | 19 | ex 412 |
. . . 4
⊢ (𝐺 ∈ UPGraph →
((♯‘𝐹) = 2
→ ((𝐹 ∈ Word dom
𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))))) |
21 | 20 | pm5.32rd 577 |
. . 3
⊢ (𝐺 ∈ UPGraph → (((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ∧ (♯‘𝐹) = 2) ↔ ((𝐹:(0..^2)⟶dom 𝐼 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) ∧ (♯‘𝐹) = 2))) |
22 | | 3anass 1093 |
. . . 4
⊢ (((𝐹:(0..^2)⟶dom 𝐼 ∧ (♯‘𝐹) = 2) ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ↔ ((𝐹:(0..^2)⟶dom 𝐼 ∧ (♯‘𝐹) = 2) ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))) |
23 | | an32 642 |
. . . 4
⊢ (((𝐹:(0..^2)⟶dom 𝐼 ∧ (♯‘𝐹) = 2) ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) ↔ ((𝐹:(0..^2)⟶dom 𝐼 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) ∧ (♯‘𝐹) = 2)) |
24 | 22, 23 | bitri 274 |
. . 3
⊢ (((𝐹:(0..^2)⟶dom 𝐼 ∧ (♯‘𝐹) = 2) ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ↔ ((𝐹:(0..^2)⟶dom 𝐼 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) ∧ (♯‘𝐹) = 2)) |
25 | 21, 24 | bitr4di 288 |
. 2
⊢ (𝐺 ∈ UPGraph → (((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ∧ (♯‘𝐹) = 2) ↔ ((𝐹:(0..^2)⟶dom 𝐼 ∧ (♯‘𝐹) = 2) ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))) |
26 | | 2nn0 12233 |
. . . . . . 7
⊢ 2 ∈
ℕ0 |
27 | | fnfzo0hash 14143 |
. . . . . . 7
⊢ ((2
∈ ℕ0 ∧ 𝐹:(0..^2)⟶dom 𝐼) → (♯‘𝐹) = 2) |
28 | 26, 27 | mpan 686 |
. . . . . 6
⊢ (𝐹:(0..^2)⟶dom 𝐼 → (♯‘𝐹) = 2) |
29 | 28 | pm4.71i 559 |
. . . . 5
⊢ (𝐹:(0..^2)⟶dom 𝐼 ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ (♯‘𝐹) = 2)) |
30 | 29 | bicomi 223 |
. . . 4
⊢ ((𝐹:(0..^2)⟶dom 𝐼 ∧ (♯‘𝐹) = 2) ↔ 𝐹:(0..^2)⟶dom 𝐼) |
31 | 30 | a1i 11 |
. . 3
⊢ (𝐺 ∈ UPGraph → ((𝐹:(0..^2)⟶dom 𝐼 ∧ (♯‘𝐹) = 2) ↔ 𝐹:(0..^2)⟶dom 𝐼)) |
32 | 31 | 3anbi1d 1438 |
. 2
⊢ (𝐺 ∈ UPGraph → (((𝐹:(0..^2)⟶dom 𝐼 ∧ (♯‘𝐹) = 2) ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))) |
33 | 4, 25, 32 | 3bitrd 304 |
1
⊢ (𝐺 ∈ UPGraph → ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))) |