Step | Hyp | Ref
| Expression |
1 | | wwlksnext.v |
. . . . 5
⊢ 𝑉 = (Vtx‘𝐺) |
2 | | wwlksnext.e |
. . . . 5
⊢ 𝐸 = (Edg‘𝐺) |
3 | 1, 2 | wwlknp 26867 |
. . . 4
⊢ (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) |
4 | | wwlksnred 26931 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ0
→ (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 substr 〈0, (𝑁 + 1)〉) ∈ (𝑁 WWalksN 𝐺))) |
5 | 4 | ad2antrr 764 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 substr 〈0, (𝑁 + 1)〉) ∈ (𝑁 WWalksN 𝐺))) |
6 | | fveq2 6304 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑊 = (𝑇 ++ 〈“𝑆”〉) → (♯‘𝑊) = (♯‘(𝑇 ++ 〈“𝑆”〉))) |
7 | 6 | eqeq1d 2726 |
. . . . . . . . . . . . . . . 16
⊢ (𝑊 = (𝑇 ++ 〈“𝑆”〉) → ((♯‘𝑊) = ((𝑁 + 1) + 1) ↔ (♯‘(𝑇 ++ 〈“𝑆”〉)) = ((𝑁 + 1) + 1))) |
8 | 7 | 3ad2ant2 1126 |
. . . . . . . . . . . . . . 15
⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → ((♯‘𝑊) = ((𝑁 + 1) + 1) ↔ (♯‘(𝑇 ++ 〈“𝑆”〉)) = ((𝑁 + 1) + 1))) |
9 | 8 | adantl 473 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → ((♯‘𝑊) = ((𝑁 + 1) + 1) ↔ (♯‘(𝑇 ++ 〈“𝑆”〉)) = ((𝑁 + 1) + 1))) |
10 | | s1cl 13493 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑆 ∈ 𝑉 → 〈“𝑆”〉 ∈ Word 𝑉) |
11 | 10 | adantl 473 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) → 〈“𝑆”〉 ∈ Word 𝑉) |
12 | 11 | anim2i 594 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑇 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉)) → (𝑇 ∈ Word 𝑉 ∧ 〈“𝑆”〉 ∈ Word 𝑉)) |
13 | 12 | ancoms 468 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (𝑇 ∈ Word 𝑉 ∧ 〈“𝑆”〉 ∈ Word 𝑉)) |
14 | | ccatlen 13468 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑇 ∈ Word 𝑉 ∧ 〈“𝑆”〉 ∈ Word 𝑉) → (♯‘(𝑇 ++ 〈“𝑆”〉)) = ((♯‘𝑇) +
(♯‘〈“𝑆”〉))) |
15 | 13, 14 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (♯‘(𝑇 ++ 〈“𝑆”〉)) = ((♯‘𝑇) +
(♯‘〈“𝑆”〉))) |
16 | 15 | eqeq1d 2726 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘(𝑇 ++ 〈“𝑆”〉)) = ((𝑁 + 1) + 1) ↔ ((♯‘𝑇) +
(♯‘〈“𝑆”〉)) = ((𝑁 + 1) + 1))) |
17 | | s1len 13497 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(♯‘〈“𝑆”〉) = 1 |
18 | 17 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (♯‘〈“𝑆”〉) =
1) |
19 | 18 | oveq2d 6781 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘𝑇) + (♯‘〈“𝑆”〉)) =
((♯‘𝑇) +
1)) |
20 | 19 | eqeq1d 2726 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (((♯‘𝑇) + (♯‘〈“𝑆”〉)) = ((𝑁 + 1) + 1) ↔
((♯‘𝑇) + 1) =
((𝑁 + 1) +
1))) |
21 | | lencl 13431 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑇 ∈ Word 𝑉 → (♯‘𝑇) ∈
ℕ0) |
22 | 21 | nn0cnd 11466 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑇 ∈ Word 𝑉 → (♯‘𝑇) ∈ ℂ) |
23 | 22 | adantl 473 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (♯‘𝑇) ∈ ℂ) |
24 | | peano2nn0 11446 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
ℕ0) |
25 | 24 | nn0cnd 11466 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
ℂ) |
26 | 25 | ad2antrr 764 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (𝑁 + 1) ∈ ℂ) |
27 | | 1cnd 10169 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → 1 ∈ ℂ) |
28 | 23, 26, 27 | addcan2d 10353 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (((♯‘𝑇) + 1) = ((𝑁 + 1) + 1) ↔ (♯‘𝑇) = (𝑁 + 1))) |
29 | 16, 20, 28 | 3bitrd 294 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘(𝑇 ++ 〈“𝑆”〉)) = ((𝑁 + 1) + 1) ↔ (♯‘𝑇) = (𝑁 + 1))) |
30 | | opeq2 4510 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑁 + 1) = (♯‘𝑇) → 〈0, (𝑁 + 1)〉 = 〈0,
(♯‘𝑇)〉) |
31 | 30 | eqcoms 2732 |
. . . . . . . . . . . . . . . . . . 19
⊢
((♯‘𝑇) =
(𝑁 + 1) → 〈0,
(𝑁 + 1)〉 = 〈0,
(♯‘𝑇)〉) |
32 | 31 | oveq2d 6781 |
. . . . . . . . . . . . . . . . . 18
⊢
((♯‘𝑇) =
(𝑁 + 1) → ((𝑇 ++ 〈“𝑆”〉) substr 〈0,
(𝑁 + 1)〉) = ((𝑇 ++ 〈“𝑆”〉) substr 〈0,
(♯‘𝑇)〉)) |
33 | | swrdccat1 13578 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑇 ∈ Word 𝑉 ∧ 〈“𝑆”〉 ∈ Word 𝑉) → ((𝑇 ++ 〈“𝑆”〉) substr 〈0,
(♯‘𝑇)〉) =
𝑇) |
34 | 13, 33 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((𝑇 ++ 〈“𝑆”〉) substr 〈0,
(♯‘𝑇)〉) =
𝑇) |
35 | 32, 34 | sylan9eqr 2780 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) ∧ (♯‘𝑇) = (𝑁 + 1)) → ((𝑇 ++ 〈“𝑆”〉) substr 〈0, (𝑁 + 1)〉) = 𝑇) |
36 | 35 | ex 449 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘𝑇) = (𝑁 + 1) → ((𝑇 ++ 〈“𝑆”〉) substr 〈0, (𝑁 + 1)〉) = 𝑇)) |
37 | 29, 36 | sylbid 230 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘(𝑇 ++ 〈“𝑆”〉)) = ((𝑁 + 1) + 1) → ((𝑇 ++ 〈“𝑆”〉) substr 〈0, (𝑁 + 1)〉) = 𝑇)) |
38 | 37 | 3ad2antr1 1180 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → ((♯‘(𝑇 ++ 〈“𝑆”〉)) = ((𝑁 + 1) + 1) → ((𝑇 ++ 〈“𝑆”〉) substr 〈0, (𝑁 + 1)〉) = 𝑇)) |
39 | 9, 38 | sylbid 230 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → ((♯‘𝑊) = ((𝑁 + 1) + 1) → ((𝑇 ++ 〈“𝑆”〉) substr 〈0, (𝑁 + 1)〉) = 𝑇)) |
40 | 39 | imp 444 |
. . . . . . . . . . . 12
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑇 ++ 〈“𝑆”〉) substr 〈0, (𝑁 + 1)〉) = 𝑇) |
41 | | oveq1 6772 |
. . . . . . . . . . . . . . 15
⊢ (𝑊 = (𝑇 ++ 〈“𝑆”〉) → (𝑊 substr 〈0, (𝑁 + 1)〉) = ((𝑇 ++ 〈“𝑆”〉) substr 〈0, (𝑁 + 1)〉)) |
42 | 41 | eqeq1d 2726 |
. . . . . . . . . . . . . 14
⊢ (𝑊 = (𝑇 ++ 〈“𝑆”〉) → ((𝑊 substr 〈0, (𝑁 + 1)〉) = 𝑇 ↔ ((𝑇 ++ 〈“𝑆”〉) substr 〈0, (𝑁 + 1)〉) = 𝑇)) |
43 | 42 | 3ad2ant2 1126 |
. . . . . . . . . . . . 13
⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → ((𝑊 substr 〈0, (𝑁 + 1)〉) = 𝑇 ↔ ((𝑇 ++ 〈“𝑆”〉) substr 〈0, (𝑁 + 1)〉) = 𝑇)) |
44 | 43 | ad2antlr 765 |
. . . . . . . . . . . 12
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 substr 〈0, (𝑁 + 1)〉) = 𝑇 ↔ ((𝑇 ++ 〈“𝑆”〉) substr 〈0, (𝑁 + 1)〉) = 𝑇)) |
45 | 40, 44 | mpbird 247 |
. . . . . . . . . . 11
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → (𝑊 substr 〈0, (𝑁 + 1)〉) = 𝑇) |
46 | 45 | eleq1d 2788 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 substr 〈0, (𝑁 + 1)〉) ∈ (𝑁 WWalksN 𝐺) ↔ 𝑇 ∈ (𝑁 WWalksN 𝐺))) |
47 | 46 | biimpd 219 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 substr 〈0, (𝑁 + 1)〉) ∈ (𝑁 WWalksN 𝐺) → 𝑇 ∈ (𝑁 WWalksN 𝐺))) |
48 | 47 | ex 449 |
. . . . . . . 8
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → ((♯‘𝑊) = ((𝑁 + 1) + 1) → ((𝑊 substr 〈0, (𝑁 + 1)〉) ∈ (𝑁 WWalksN 𝐺) → 𝑇 ∈ (𝑁 WWalksN 𝐺)))) |
49 | 48 | com23 86 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → ((𝑊 substr 〈0, (𝑁 + 1)〉) ∈ (𝑁 WWalksN 𝐺) → ((♯‘𝑊) = ((𝑁 + 1) + 1) → 𝑇 ∈ (𝑁 WWalksN 𝐺)))) |
50 | 5, 49 | syld 47 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → ((♯‘𝑊) = ((𝑁 + 1) + 1) → 𝑇 ∈ (𝑁 WWalksN 𝐺)))) |
51 | 50 | com13 88 |
. . . . 5
⊢
((♯‘𝑊) =
((𝑁 + 1) + 1) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → 𝑇 ∈ (𝑁 WWalksN 𝐺)))) |
52 | 51 | 3ad2ant2 1126 |
. . . 4
⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → 𝑇 ∈ (𝑁 WWalksN 𝐺)))) |
53 | 3, 52 | mpcom 38 |
. . 3
⊢ (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → 𝑇 ∈ (𝑁 WWalksN 𝐺))) |
54 | 53 | com12 32 |
. 2
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → 𝑇 ∈ (𝑁 WWalksN 𝐺))) |
55 | 1, 2 | wwlksnext 26932 |
. . . . . . . . . . 11
⊢ ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆 ∈ 𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ 〈“𝑆”〉) ∈ ((𝑁 + 1) WWalksN 𝐺)) |
56 | | eleq1 2791 |
. . . . . . . . . . 11
⊢ (𝑊 = (𝑇 ++ 〈“𝑆”〉) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ (𝑇 ++ 〈“𝑆”〉) ∈ ((𝑁 + 1) WWalksN 𝐺))) |
57 | 55, 56 | syl5ibrcom 237 |
. . . . . . . . . 10
⊢ ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆 ∈ 𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑊 = (𝑇 ++ 〈“𝑆”〉) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) |
58 | 57 | 3exp 1112 |
. . . . . . . . 9
⊢ (𝑇 ∈ (𝑁 WWalksN 𝐺) → (𝑆 ∈ 𝑉 → ({( lastS ‘𝑇), 𝑆} ∈ 𝐸 → (𝑊 = (𝑇 ++ 〈“𝑆”〉) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))))) |
59 | 58 | com23 86 |
. . . . . . . 8
⊢ (𝑇 ∈ (𝑁 WWalksN 𝐺) → ({( lastS ‘𝑇), 𝑆} ∈ 𝐸 → (𝑆 ∈ 𝑉 → (𝑊 = (𝑇 ++ 〈“𝑆”〉) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))))) |
60 | 59 | com14 96 |
. . . . . . 7
⊢ (𝑊 = (𝑇 ++ 〈“𝑆”〉) → ({( lastS ‘𝑇), 𝑆} ∈ 𝐸 → (𝑆 ∈ 𝑉 → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))))) |
61 | 60 | imp 444 |
. . . . . 6
⊢ ((𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑆 ∈ 𝑉 → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))) |
62 | 61 | 3adant1 1122 |
. . . . 5
⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑆 ∈ 𝑉 → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))) |
63 | 62 | com12 32 |
. . . 4
⊢ (𝑆 ∈ 𝑉 → ((𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))) |
64 | 63 | adantl 473 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) → ((𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))) |
65 | 64 | imp 444 |
. 2
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) |
66 | 54, 65 | impbid 202 |
1
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ 𝑇 ∈ (𝑁 WWalksN 𝐺))) |