| Step | Hyp | Ref
| Expression |
| 1 | | grtriclwlk3.p |
. . . . 5
⊢ (𝜑 → 𝑃:(0..^3)–1-1-onto→𝑇) |
| 2 | | f1ofn 6849 |
. . . . 5
⊢ (𝑃:(0..^3)–1-1-onto→𝑇 → 𝑃 Fn (0..^3)) |
| 3 | | hashfn 14414 |
. . . . 5
⊢ (𝑃 Fn (0..^3) →
(♯‘𝑃) =
(♯‘(0..^3))) |
| 4 | 1, 2, 3 | 3syl 18 |
. . . 4
⊢ (𝜑 → (♯‘𝑃) =
(♯‘(0..^3))) |
| 5 | | 3nn0 12544 |
. . . . 5
⊢ 3 ∈
ℕ0 |
| 6 | | hashfzo0 14469 |
. . . . 5
⊢ (3 ∈
ℕ0 → (♯‘(0..^3)) = 3) |
| 7 | 5, 6 | mp1i 13 |
. . . 4
⊢ (𝜑 → (♯‘(0..^3)) =
3) |
| 8 | 4, 7 | eqtrd 2777 |
. . 3
⊢ (𝜑 → (♯‘𝑃) = 3) |
| 9 | | f1of 6848 |
. . . . . . . . 9
⊢ (𝑃:(0..^3)–1-1-onto→𝑇 → 𝑃:(0..^3)⟶𝑇) |
| 10 | 1, 9 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑃:(0..^3)⟶𝑇) |
| 11 | | grtriclwlk3.t |
. . . . . . . . 9
⊢ (𝜑 → 𝑇 ∈ (GrTriangles‘𝐺)) |
| 12 | | eqid 2737 |
. . . . . . . . . 10
⊢
(Vtx‘𝐺) =
(Vtx‘𝐺) |
| 13 | 12 | grtrissvtx 47911 |
. . . . . . . . 9
⊢ (𝑇 ∈ (GrTriangles‘𝐺) → 𝑇 ⊆ (Vtx‘𝐺)) |
| 14 | 11, 13 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑇 ⊆ (Vtx‘𝐺)) |
| 15 | 10, 14 | jca 511 |
. . . . . . 7
⊢ (𝜑 → (𝑃:(0..^3)⟶𝑇 ∧ 𝑇 ⊆ (Vtx‘𝐺))) |
| 16 | 15 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (𝑃:(0..^3)⟶𝑇 ∧ 𝑇 ⊆ (Vtx‘𝐺))) |
| 17 | | fss 6752 |
. . . . . 6
⊢ ((𝑃:(0..^3)⟶𝑇 ∧ 𝑇 ⊆ (Vtx‘𝐺)) → 𝑃:(0..^3)⟶(Vtx‘𝐺)) |
| 18 | | iswrdi 14556 |
. . . . . 6
⊢ (𝑃:(0..^3)⟶(Vtx‘𝐺) → 𝑃 ∈ Word (Vtx‘𝐺)) |
| 19 | 16, 17, 18 | 3syl 18 |
. . . . 5
⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → 𝑃 ∈ Word (Vtx‘𝐺)) |
| 20 | | oveq1 7438 |
. . . . . . . . . . . 12
⊢
((♯‘𝑃) =
3 → ((♯‘𝑃)
− 1) = (3 − 1)) |
| 21 | | 3m1e2 12394 |
. . . . . . . . . . . 12
⊢ (3
− 1) = 2 |
| 22 | 20, 21 | eqtrdi 2793 |
. . . . . . . . . . 11
⊢
((♯‘𝑃) =
3 → ((♯‘𝑃)
− 1) = 2) |
| 23 | 22 | oveq2d 7447 |
. . . . . . . . . 10
⊢
((♯‘𝑃) =
3 → (0..^((♯‘𝑃) − 1)) = (0..^2)) |
| 24 | | fzo0to2pr 13789 |
. . . . . . . . . 10
⊢ (0..^2) =
{0, 1} |
| 25 | 23, 24 | eqtrdi 2793 |
. . . . . . . . 9
⊢
((♯‘𝑃) =
3 → (0..^((♯‘𝑃) − 1)) = {0, 1}) |
| 26 | 25 | eleq2d 2827 |
. . . . . . . 8
⊢
((♯‘𝑃) =
3 → (𝑖 ∈
(0..^((♯‘𝑃)
− 1)) ↔ 𝑖 ∈
{0, 1})) |
| 27 | 26 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (𝑖 ∈
(0..^((♯‘𝑃)
− 1)) ↔ 𝑖 ∈
{0, 1})) |
| 28 | 11, 1 | jca 511 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑇 ∈ (GrTriangles‘𝐺) ∧ 𝑃:(0..^3)–1-1-onto→𝑇)) |
| 29 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢
(Edg‘𝐺) =
(Edg‘𝐺) |
| 30 | 12, 29 | grtrif1o 47909 |
. . . . . . . . . . . 12
⊢ ((𝑇 ∈ (GrTriangles‘𝐺) ∧ 𝑃:(0..^3)–1-1-onto→𝑇) → ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺))) |
| 31 | | simp1 1137 |
. . . . . . . . . . . 12
⊢ (({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)) → {(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺)) |
| 32 | 28, 30, 31 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → {(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺)) |
| 33 | 32 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺)) |
| 34 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑖 = 0 → (𝑃‘𝑖) = (𝑃‘0)) |
| 35 | | fv0p1e1 12389 |
. . . . . . . . . . . 12
⊢ (𝑖 = 0 → (𝑃‘(𝑖 + 1)) = (𝑃‘1)) |
| 36 | 34, 35 | preq12d 4741 |
. . . . . . . . . . 11
⊢ (𝑖 = 0 → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = {(𝑃‘0), (𝑃‘1)}) |
| 37 | 36 | eleq1d 2826 |
. . . . . . . . . 10
⊢ (𝑖 = 0 → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺))) |
| 38 | 33, 37 | imbitrrid 246 |
. . . . . . . . 9
⊢ (𝑖 = 0 → ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) |
| 39 | | simp3 1139 |
. . . . . . . . . . . 12
⊢ (({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)) → {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)) |
| 40 | 28, 30, 39 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)) |
| 41 | 40 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)) |
| 42 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑖 = 1 → (𝑃‘𝑖) = (𝑃‘1)) |
| 43 | | oveq1 7438 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 1 → (𝑖 + 1) = (1 + 1)) |
| 44 | | 1p1e2 12391 |
. . . . . . . . . . . . . 14
⊢ (1 + 1) =
2 |
| 45 | 43, 44 | eqtrdi 2793 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 1 → (𝑖 + 1) = 2) |
| 46 | 45 | fveq2d 6910 |
. . . . . . . . . . . 12
⊢ (𝑖 = 1 → (𝑃‘(𝑖 + 1)) = (𝑃‘2)) |
| 47 | 42, 46 | preq12d 4741 |
. . . . . . . . . . 11
⊢ (𝑖 = 1 → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = {(𝑃‘1), (𝑃‘2)}) |
| 48 | 47 | eleq1d 2826 |
. . . . . . . . . 10
⊢ (𝑖 = 1 → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺))) |
| 49 | 41, 48 | imbitrrid 246 |
. . . . . . . . 9
⊢ (𝑖 = 1 → ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) |
| 50 | 38, 49 | jaoi 858 |
. . . . . . . 8
⊢ ((𝑖 = 0 ∨ 𝑖 = 1) → ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) |
| 51 | | elpri 4649 |
. . . . . . . 8
⊢ (𝑖 ∈ {0, 1} → (𝑖 = 0 ∨ 𝑖 = 1)) |
| 52 | 50, 51 | syl11 33 |
. . . . . . 7
⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (𝑖 ∈ {0, 1} → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) |
| 53 | 27, 52 | sylbid 240 |
. . . . . 6
⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (𝑖 ∈
(0..^((♯‘𝑃)
− 1)) → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) |
| 54 | 53 | ralrimiv 3145 |
. . . . 5
⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → ∀𝑖 ∈
(0..^((♯‘𝑃)
− 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)) |
| 55 | | ovexd 7466 |
. . . . . . . . . . . 12
⊢ (𝑃:(0..^3)–1-1-onto→𝑇 → (0..^3) ∈
V) |
| 56 | 9, 55 | jca 511 |
. . . . . . . . . . 11
⊢ (𝑃:(0..^3)–1-1-onto→𝑇 → (𝑃:(0..^3)⟶𝑇 ∧ (0..^3) ∈ V)) |
| 57 | | fex 7246 |
. . . . . . . . . . 11
⊢ ((𝑃:(0..^3)⟶𝑇 ∧ (0..^3) ∈ V) →
𝑃 ∈
V) |
| 58 | 1, 56, 57 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑃 ∈ V) |
| 59 | 58 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → 𝑃 ∈ V) |
| 60 | | lsw 14602 |
. . . . . . . . 9
⊢ (𝑃 ∈ V →
(lastS‘𝑃) = (𝑃‘((♯‘𝑃) − 1))) |
| 61 | 59, 60 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (lastS‘𝑃) = (𝑃‘((♯‘𝑃) − 1))) |
| 62 | 22 | fveq2d 6910 |
. . . . . . . . 9
⊢
((♯‘𝑃) =
3 → (𝑃‘((♯‘𝑃) − 1)) = (𝑃‘2)) |
| 63 | 62 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (𝑃‘((♯‘𝑃) − 1)) = (𝑃‘2)) |
| 64 | 61, 63 | eqtrd 2777 |
. . . . . . 7
⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (lastS‘𝑃) = (𝑃‘2)) |
| 65 | 64 | preq1d 4739 |
. . . . . 6
⊢ ((𝜑 ∧ (♯‘𝑃) = 3) →
{(lastS‘𝑃), (𝑃‘0)} = {(𝑃‘2), (𝑃‘0)}) |
| 66 | | prcom 4732 |
. . . . . . . . . . 11
⊢ {(𝑃‘0), (𝑃‘2)} = {(𝑃‘2), (𝑃‘0)} |
| 67 | 66 | eleq1i 2832 |
. . . . . . . . . 10
⊢ ({(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ↔ {(𝑃‘2), (𝑃‘0)} ∈ (Edg‘𝐺)) |
| 68 | 67 | biimpi 216 |
. . . . . . . . 9
⊢ ({(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) → {(𝑃‘2), (𝑃‘0)} ∈ (Edg‘𝐺)) |
| 69 | 68 | 3ad2ant2 1135 |
. . . . . . . 8
⊢ (({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)) → {(𝑃‘2), (𝑃‘0)} ∈ (Edg‘𝐺)) |
| 70 | 28, 30, 69 | 3syl 18 |
. . . . . . 7
⊢ (𝜑 → {(𝑃‘2), (𝑃‘0)} ∈ (Edg‘𝐺)) |
| 71 | 70 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘2), (𝑃‘0)} ∈ (Edg‘𝐺)) |
| 72 | 65, 71 | eqeltrd 2841 |
. . . . 5
⊢ ((𝜑 ∧ (♯‘𝑃) = 3) →
{(lastS‘𝑃), (𝑃‘0)} ∈
(Edg‘𝐺)) |
| 73 | 19, 54, 72 | 3jca 1129 |
. . . 4
⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺))) |
| 74 | | simpr 484 |
. . . 4
⊢ ((𝜑 ∧ (♯‘𝑃) = 3) →
(♯‘𝑃) =
3) |
| 75 | 73, 74 | jca 511 |
. . 3
⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → ((𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈
(0..^((♯‘𝑃)
− 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑃) = 3)) |
| 76 | 8, 75 | mpdan 687 |
. 2
⊢ (𝜑 → ((𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑃) = 3)) |
| 77 | | 3nn 12345 |
. . 3
⊢ 3 ∈
ℕ |
| 78 | 12, 29 | isclwwlknx 30055 |
. . 3
⊢ (3 ∈
ℕ → (𝑃 ∈ (3
ClWWalksN 𝐺) ↔ ((𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈
(0..^((♯‘𝑃)
− 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑃) = 3))) |
| 79 | 77, 78 | mp1i 13 |
. 2
⊢ (𝜑 → (𝑃 ∈ (3 ClWWalksN 𝐺) ↔ ((𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑃) = 3))) |
| 80 | 76, 79 | mpbird 257 |
1
⊢ (𝜑 → 𝑃 ∈ (3 ClWWalksN 𝐺)) |