| Step | Hyp | Ref
| Expression |
| 1 | | wwlksnext.v |
. . . . 5
⊢ 𝑉 = (Vtx‘𝐺) |
| 2 | | wwlksnext.e |
. . . . 5
⊢ 𝐸 = (Edg‘𝐺) |
| 3 | 1, 2 | wwlknp 29863 |
. . . 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 29912 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ0
→ (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺))) |
| 5 | 4 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺))) |
| 6 | | fveqeq2 6915 |
. . . . . . . . . . . . . . . 16
⊢ (𝑊 = (𝑇 ++ 〈“𝑆”〉) → ((♯‘𝑊) = ((𝑁 + 1) + 1) ↔ (♯‘(𝑇 ++ 〈“𝑆”〉)) = ((𝑁 + 1) + 1))) |
| 7 | 6 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . 15
⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → ((♯‘𝑊) = ((𝑁 + 1) + 1) ↔ (♯‘(𝑇 ++ 〈“𝑆”〉)) = ((𝑁 + 1) + 1))) |
| 8 | 7 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ((♯‘𝑊) = ((𝑁 + 1) + 1) ↔ (♯‘(𝑇 ++ 〈“𝑆”〉)) = ((𝑁 + 1) + 1))) |
| 9 | | s1cl 14640 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑆 ∈ 𝑉 → 〈“𝑆”〉 ∈ Word 𝑉) |
| 10 | 9 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) → 〈“𝑆”〉 ∈ Word 𝑉) |
| 11 | 10 | anim1ci 616 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (𝑇 ∈ Word 𝑉 ∧ 〈“𝑆”〉 ∈ Word 𝑉)) |
| 12 | | ccatlen 14613 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑇 ∈ Word 𝑉 ∧ 〈“𝑆”〉 ∈ Word 𝑉) → (♯‘(𝑇 ++ 〈“𝑆”〉)) = ((♯‘𝑇) +
(♯‘〈“𝑆”〉))) |
| 13 | 11, 12 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (♯‘(𝑇 ++ 〈“𝑆”〉)) = ((♯‘𝑇) +
(♯‘〈“𝑆”〉))) |
| 14 | 13 | eqeq1d 2739 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘(𝑇 ++ 〈“𝑆”〉)) = ((𝑁 + 1) + 1) ↔ ((♯‘𝑇) +
(♯‘〈“𝑆”〉)) = ((𝑁 + 1) + 1))) |
| 15 | | s1len 14644 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(♯‘〈“𝑆”〉) = 1 |
| 16 | 15 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (♯‘〈“𝑆”〉) =
1) |
| 17 | 16 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘𝑇) + (♯‘〈“𝑆”〉)) =
((♯‘𝑇) +
1)) |
| 18 | 17 | eqeq1d 2739 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (((♯‘𝑇) + (♯‘〈“𝑆”〉)) = ((𝑁 + 1) + 1) ↔
((♯‘𝑇) + 1) =
((𝑁 + 1) +
1))) |
| 19 | | lencl 14571 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑇 ∈ Word 𝑉 → (♯‘𝑇) ∈
ℕ0) |
| 20 | 19 | nn0cnd 12589 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑇 ∈ Word 𝑉 → (♯‘𝑇) ∈ ℂ) |
| 21 | 20 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (♯‘𝑇) ∈ ℂ) |
| 22 | | peano2nn0 12566 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
ℕ0) |
| 23 | 22 | nn0cnd 12589 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
ℂ) |
| 24 | 23 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (𝑁 + 1) ∈ ℂ) |
| 25 | | 1cnd 11256 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → 1 ∈ ℂ) |
| 26 | 21, 24, 25 | addcan2d 11465 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (((♯‘𝑇) + 1) = ((𝑁 + 1) + 1) ↔ (♯‘𝑇) = (𝑁 + 1))) |
| 27 | 14, 18, 26 | 3bitrd 305 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘(𝑇 ++ 〈“𝑆”〉)) = ((𝑁 + 1) + 1) ↔ (♯‘𝑇) = (𝑁 + 1))) |
| 28 | | oveq2 7439 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 + 1) = (♯‘𝑇) → ((𝑇 ++ 〈“𝑆”〉) prefix (𝑁 + 1)) = ((𝑇 ++ 〈“𝑆”〉) prefix (♯‘𝑇))) |
| 29 | 28 | eqcoms 2745 |
. . . . . . . . . . . . . . . . . 18
⊢
((♯‘𝑇) =
(𝑁 + 1) → ((𝑇 ++ 〈“𝑆”〉) prefix (𝑁 + 1)) = ((𝑇 ++ 〈“𝑆”〉) prefix (♯‘𝑇))) |
| 30 | | pfxccat1 14740 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑇 ∈ Word 𝑉 ∧ 〈“𝑆”〉 ∈ Word 𝑉) → ((𝑇 ++ 〈“𝑆”〉) prefix (♯‘𝑇)) = 𝑇) |
| 31 | 11, 30 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((𝑇 ++ 〈“𝑆”〉) prefix (♯‘𝑇)) = 𝑇) |
| 32 | 29, 31 | sylan9eqr 2799 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) ∧ (♯‘𝑇) = (𝑁 + 1)) → ((𝑇 ++ 〈“𝑆”〉) prefix (𝑁 + 1)) = 𝑇) |
| 33 | 32 | ex 412 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘𝑇) = (𝑁 + 1) → ((𝑇 ++ 〈“𝑆”〉) prefix (𝑁 + 1)) = 𝑇)) |
| 34 | 27, 33 | sylbid 240 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘(𝑇 ++ 〈“𝑆”〉)) = ((𝑁 + 1) + 1) → ((𝑇 ++ 〈“𝑆”〉) prefix (𝑁 + 1)) = 𝑇)) |
| 35 | 34 | 3ad2antr1 1189 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ((♯‘(𝑇 ++ 〈“𝑆”〉)) = ((𝑁 + 1) + 1) → ((𝑇 ++ 〈“𝑆”〉) prefix (𝑁 + 1)) = 𝑇)) |
| 36 | 8, 35 | sylbid 240 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ((♯‘𝑊) = ((𝑁 + 1) + 1) → ((𝑇 ++ 〈“𝑆”〉) prefix (𝑁 + 1)) = 𝑇)) |
| 37 | 36 | imp 406 |
. . . . . . . . . . . 12
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑇 ++ 〈“𝑆”〉) prefix (𝑁 + 1)) = 𝑇) |
| 38 | | oveq1 7438 |
. . . . . . . . . . . . . . 15
⊢ (𝑊 = (𝑇 ++ 〈“𝑆”〉) → (𝑊 prefix (𝑁 + 1)) = ((𝑇 ++ 〈“𝑆”〉) prefix (𝑁 + 1))) |
| 39 | 38 | eqeq1d 2739 |
. . . . . . . . . . . . . 14
⊢ (𝑊 = (𝑇 ++ 〈“𝑆”〉) → ((𝑊 prefix (𝑁 + 1)) = 𝑇 ↔ ((𝑇 ++ 〈“𝑆”〉) prefix (𝑁 + 1)) = 𝑇)) |
| 40 | 39 | 3ad2ant2 1135 |
. . . . . . . . . . . . 13
⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → ((𝑊 prefix (𝑁 + 1)) = 𝑇 ↔ ((𝑇 ++ 〈“𝑆”〉) prefix (𝑁 + 1)) = 𝑇)) |
| 41 | 40 | ad2antlr 727 |
. . . . . . . . . . . 12
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 prefix (𝑁 + 1)) = 𝑇 ↔ ((𝑇 ++ 〈“𝑆”〉) prefix (𝑁 + 1)) = 𝑇)) |
| 42 | 37, 41 | mpbird 257 |
. . . . . . . . . . 11
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → (𝑊 prefix (𝑁 + 1)) = 𝑇) |
| 43 | 42 | eleq1d 2826 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺) ↔ 𝑇 ∈ (𝑁 WWalksN 𝐺))) |
| 44 | 43 | biimpd 229 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺) → 𝑇 ∈ (𝑁 WWalksN 𝐺))) |
| 45 | 44 | ex 412 |
. . . . . . . 8
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ((♯‘𝑊) = ((𝑁 + 1) + 1) → ((𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺) → 𝑇 ∈ (𝑁 WWalksN 𝐺)))) |
| 46 | 45 | com23 86 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ((𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺) → ((♯‘𝑊) = ((𝑁 + 1) + 1) → 𝑇 ∈ (𝑁 WWalksN 𝐺)))) |
| 47 | 5, 46 | 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)))) |
| 48 | 47 | 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)))) |
| 49 | 48 | 3ad2ant2 1135 |
. . . 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)))) |
| 50 | 3, 49 | 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))) |
| 51 | 50 | 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))) |
| 52 | 1, 2 | wwlksnext 29913 |
. . . . . . . . . . 11
⊢ ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ 〈“𝑆”〉) ∈ ((𝑁 + 1) WWalksN 𝐺)) |
| 53 | | eleq1 2829 |
. . . . . . . . . . 11
⊢ (𝑊 = (𝑇 ++ 〈“𝑆”〉) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ (𝑇 ++ 〈“𝑆”〉) ∈ ((𝑁 + 1) WWalksN 𝐺))) |
| 54 | 52, 53 | syl5ibrcom 247 |
. . . . . . . . . 10
⊢ ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑊 = (𝑇 ++ 〈“𝑆”〉) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) |
| 55 | 54 | 3exp 1120 |
. . . . . . . . 9
⊢ (𝑇 ∈ (𝑁 WWalksN 𝐺) → (𝑆 ∈ 𝑉 → ({(lastS‘𝑇), 𝑆} ∈ 𝐸 → (𝑊 = (𝑇 ++ 〈“𝑆”〉) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))))) |
| 56 | 55 | com23 86 |
. . . . . . . 8
⊢ (𝑇 ∈ (𝑁 WWalksN 𝐺) → ({(lastS‘𝑇), 𝑆} ∈ 𝐸 → (𝑆 ∈ 𝑉 → (𝑊 = (𝑇 ++ 〈“𝑆”〉) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))))) |
| 57 | 56 | com14 96 |
. . . . . . 7
⊢ (𝑊 = (𝑇 ++ 〈“𝑆”〉) → ({(lastS‘𝑇), 𝑆} ∈ 𝐸 → (𝑆 ∈ 𝑉 → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))))) |
| 58 | 57 | imp 406 |
. . . . . 6
⊢ ((𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑆 ∈ 𝑉 → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))) |
| 59 | 58 | 3adant1 1131 |
. . . . 5
⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑆 ∈ 𝑉 → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))) |
| 60 | 59 | com12 32 |
. . . 4
⊢ (𝑆 ∈ 𝑉 → ((𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))) |
| 61 | 60 | adantl 481 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) → ((𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))) |
| 62 | 61 | imp 406 |
. 2
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) |
| 63 | 51, 62 | impbid 212 |
1
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ 〈“𝑆”〉) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ 𝑇 ∈ (𝑁 WWalksN 𝐺))) |