Step | Hyp | Ref
| Expression |
1 | | wwlksnextbij0.d |
. 2
⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)} |
2 | | 3anass 1035 |
. . . . 5
⊢
(((#‘𝑤) =
(𝑁 + 2) ∧ (𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) ↔ ((#‘𝑤) = (𝑁 + 2) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))) |
3 | 2 | bianass 838 |
. . . 4
⊢ ((𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) ↔ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))) |
4 | | wwlksnextbij0.v |
. . . . . . . . . . 11
⊢ 𝑉 = (Vtx‘𝐺) |
5 | 4 | wwlknbp 41044 |
. . . . . . . . . 10
⊢ (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) |
6 | | simpl 472 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℕ0
∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))) → 𝑁 ∈
ℕ0) |
7 | | simpl 472 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → 𝑤 ∈ Word 𝑉) |
8 | | nn0re 11178 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℝ) |
9 | | 2re 10967 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 2 ∈
ℝ |
10 | 9 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ0
→ 2 ∈ ℝ) |
11 | | nn0ge0 11195 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ0
→ 0 ≤ 𝑁) |
12 | | 2pos 10989 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 0 <
2 |
13 | 12 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ0
→ 0 < 2) |
14 | 8, 10, 11, 13 | addgegt0d 10480 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 ∈ ℕ0
→ 0 < (𝑁 +
2)) |
15 | 14 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈ ℕ0
∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → 0 < (𝑁 + 2)) |
16 | | breq2 4587 |
. . . . . . . . . . . . . . . . . . 19
⊢
((#‘𝑤) =
(𝑁 + 2) → (0 <
(#‘𝑤) ↔ 0 <
(𝑁 + 2))) |
17 | 16 | ad2antll 761 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈ ℕ0
∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → (0 < (#‘𝑤) ↔ 0 < (𝑁 + 2))) |
18 | 15, 17 | mpbird 246 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ ℕ0
∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → 0 < (#‘𝑤)) |
19 | | hashgt0n0 13017 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑤 ∈ Word 𝑉 ∧ 0 < (#‘𝑤)) → 𝑤 ≠ ∅) |
20 | 7, 18, 19 | syl2an2 871 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℕ0
∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → 𝑤 ≠ ∅) |
21 | | lswcl 13208 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑤 ∈ Word 𝑉 ∧ 𝑤 ≠ ∅) → ( lastS ‘𝑤) ∈ 𝑉) |
22 | 7, 20, 21 | syl2an2 871 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℕ0
∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ( lastS ‘𝑤) ∈ 𝑉) |
23 | 22 | adantrr 749 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℕ0
∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))) → ( lastS ‘𝑤) ∈ 𝑉) |
24 | | swrdcl 13271 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑤 ∈ Word 𝑉 → (𝑤 substr 〈0, (𝑁 + 1)〉) ∈ Word 𝑉) |
25 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑊 = (𝑤 substr 〈0, (𝑁 + 1)〉) → (𝑊 ∈ Word 𝑉 ↔ (𝑤 substr 〈0, (𝑁 + 1)〉) ∈ Word 𝑉)) |
26 | 24, 25 | syl5ibr 235 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑊 = (𝑤 substr 〈0, (𝑁 + 1)〉) → (𝑤 ∈ Word 𝑉 → 𝑊 ∈ Word 𝑉)) |
27 | 26 | eqcoms 2618 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 → (𝑤 ∈ Word 𝑉 → 𝑊 ∈ Word 𝑉)) |
28 | 27 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) → (𝑤 ∈ Word 𝑉 → 𝑊 ∈ Word 𝑉)) |
29 | 28 | com12 32 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑤 ∈ Word 𝑉 → (((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) → 𝑊 ∈ Word 𝑉)) |
30 | 29 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → (((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) → 𝑊 ∈ Word 𝑉)) |
31 | 30 | imp 444 |
. . . . . . . . . . . . . . 15
⊢ (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) → 𝑊 ∈ Word 𝑉) |
32 | 31 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℕ0
∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))) → 𝑊 ∈ Word 𝑉) |
33 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑊 = (𝑤 substr 〈0, (𝑁 + 1)〉) → (𝑊 ++ 〈“( lastS ‘𝑤)”〉) = ((𝑤 substr 〈0, (𝑁 + 1)〉) ++ 〈“(
lastS ‘𝑤)”〉)) |
34 | 33 | eqcoms 2618 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 → (𝑊 ++ 〈“( lastS ‘𝑤)”〉) = ((𝑤 substr 〈0, (𝑁 + 1)〉) ++ 〈“(
lastS ‘𝑤)”〉)) |
35 | 34 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) → (𝑊 ++ 〈“( lastS ‘𝑤)”〉) = ((𝑤 substr 〈0, (𝑁 + 1)〉) ++ 〈“(
lastS ‘𝑤)”〉)) |
36 | 35 | ad2antll 761 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℕ0
∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))) → (𝑊 ++ 〈“( lastS ‘𝑤)”〉) = ((𝑤 substr 〈0, (𝑁 + 1)〉) ++ 〈“(
lastS ‘𝑤)”〉)) |
37 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((#‘𝑤) =
(𝑁 + 2) →
((#‘𝑤) − 1) =
((𝑁 + 2) −
1)) |
38 | 37 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → ((#‘𝑤) − 1) = ((𝑁 + 2) − 1)) |
39 | | nn0cn 11179 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℂ) |
40 | | 2cnd 10970 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑁 ∈ ℕ0
→ 2 ∈ ℂ) |
41 | | 1cnd 9935 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑁 ∈ ℕ0
→ 1 ∈ ℂ) |
42 | 39, 40, 41 | addsubassd 10291 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑁 ∈ ℕ0
→ ((𝑁 + 2) − 1)
= (𝑁 + (2 −
1))) |
43 | | 2m1e1 11012 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (2
− 1) = 1 |
44 | 43 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑁 ∈ ℕ0
→ (2 − 1) = 1) |
45 | 44 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + (2 − 1))
= (𝑁 + 1)) |
46 | 42, 45 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈ ℕ0
→ ((𝑁 + 2) − 1)
= (𝑁 + 1)) |
47 | 38, 46 | sylan9eqr 2666 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑁 ∈ ℕ0
∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((#‘𝑤) − 1) = (𝑁 + 1)) |
48 | 47 | opeq2d 4347 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈ ℕ0
∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → 〈0, ((#‘𝑤) − 1)〉 = 〈0,
(𝑁 +
1)〉) |
49 | 48 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈ ℕ0
∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → (𝑤 substr 〈0, ((#‘𝑤) − 1)〉) = (𝑤 substr 〈0, (𝑁 + 1)〉)) |
50 | 49 | oveq1d 6564 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ ℕ0
∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((𝑤 substr 〈0, ((#‘𝑤) − 1)〉) ++ 〈“( lastS
‘𝑤)”〉) =
((𝑤 substr 〈0, (𝑁 + 1)〉) ++ 〈“(
lastS ‘𝑤)”〉)) |
51 | | swrdccatwrd 13320 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑤 ∈ Word 𝑉 ∧ 𝑤 ≠ ∅) → ((𝑤 substr 〈0, ((#‘𝑤) − 1)〉) ++ 〈“( lastS
‘𝑤)”〉) =
𝑤) |
52 | 7, 20, 51 | syl2an2 871 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ ℕ0
∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((𝑤 substr 〈0, ((#‘𝑤) − 1)〉) ++ 〈“( lastS
‘𝑤)”〉) =
𝑤) |
53 | 50, 52 | eqtr3d 2646 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℕ0
∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((𝑤 substr 〈0, (𝑁 + 1)〉) ++ 〈“( lastS
‘𝑤)”〉) =
𝑤) |
54 | 53 | adantrr 749 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℕ0
∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))) → ((𝑤 substr 〈0, (𝑁 + 1)〉) ++ 〈“( lastS
‘𝑤)”〉) =
𝑤) |
55 | 36, 54 | eqtr2d 2645 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℕ0
∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))) → 𝑤 = (𝑊 ++ 〈“( lastS ‘𝑤)”〉)) |
56 | | simprrr 801 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℕ0
∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))) → {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) |
57 | | wwlksnextbij0.e |
. . . . . . . . . . . . . . 15
⊢ 𝐸 = (Edg‘𝐺) |
58 | 4, 57 | wwlksnextbi 41100 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ0
∧ ( lastS ‘𝑤)
∈ 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑤 = (𝑊 ++ 〈“( lastS ‘𝑤)”〉) ∧ {( lastS
‘𝑊), ( lastS
‘𝑤)} ∈ 𝐸)) → (𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ↔ 𝑊 ∈ (𝑁 WWalkSN 𝐺))) |
59 | 6, 23, 32, 55, 56, 58 | syl23anc 1325 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℕ0
∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))) → (𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ↔ 𝑊 ∈ (𝑁 WWalkSN 𝐺))) |
60 | 59 | exbiri 650 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℕ0
→ (((𝑤 ∈ Word
𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) → (𝑊 ∈ (𝑁 WWalkSN 𝐺) → 𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺)))) |
61 | 60 | com23 84 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ0
→ (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) → 𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺)))) |
62 | 61 | 3ad2ant2 1076 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0
∧ 𝑊 ∈ Word 𝑉) → (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) → 𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺)))) |
63 | 5, 62 | mpcom 37 |
. . . . . . . . 9
⊢ (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) → 𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺))) |
64 | 63 | expcomd 453 |
. . . . . . . 8
⊢ (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) → ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → 𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺)))) |
65 | 64 | imp 444 |
. . . . . . 7
⊢ ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) → ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → 𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺))) |
66 | 4, 57 | wwlknp 41045 |
. . . . . . . . . . . 12
⊢ (𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸)) |
67 | 39, 41, 41 | addassd 9941 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ0
→ ((𝑁 + 1) + 1) =
(𝑁 + (1 +
1))) |
68 | | 1p1e2 11011 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (1 + 1) =
2 |
69 | 68 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈ ℕ0
→ (1 + 1) = 2) |
70 | 69 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + (1 + 1)) =
(𝑁 + 2)) |
71 | 67, 70 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 ∈ ℕ0
→ ((𝑁 + 1) + 1) =
(𝑁 + 2)) |
72 | 71 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈ ℕ0
→ ((#‘𝑤) =
((𝑁 + 1) + 1) ↔
(#‘𝑤) = (𝑁 + 2))) |
73 | 72 | biimpd 218 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℕ0
→ ((#‘𝑤) =
((𝑁 + 1) + 1) →
(#‘𝑤) = (𝑁 + 2))) |
74 | 73 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℕ0
∧ 𝑊 ∈ Word 𝑉) → ((#‘𝑤) = ((𝑁 + 1) + 1) → (#‘𝑤) = (𝑁 + 2))) |
75 | 74 | com12 32 |
. . . . . . . . . . . . . . 15
⊢
((#‘𝑤) =
((𝑁 + 1) + 1) →
((𝑁 ∈
ℕ0 ∧ 𝑊
∈ Word 𝑉) →
(#‘𝑤) = (𝑁 + 2))) |
76 | 75 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (#‘𝑤) = (𝑁 + 2))) |
77 | | simpl 472 |
. . . . . . . . . . . . . 14
⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1)) → 𝑤 ∈ Word 𝑉) |
78 | 76, 77 | jctild 564 |
. . . . . . . . . . . . 13
⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)))) |
79 | 78 | 3adant3 1074 |
. . . . . . . . . . . 12
⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)))) |
80 | 66, 79 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)))) |
81 | 80 | com12 32 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ0
∧ 𝑊 ∈ Word 𝑉) → (𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)))) |
82 | 81 | 3adant1 1072 |
. . . . . . . . 9
⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0
∧ 𝑊 ∈ Word 𝑉) → (𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)))) |
83 | 5, 82 | syl 17 |
. . . . . . . 8
⊢ (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)))) |
84 | 83 | adantr 480 |
. . . . . . 7
⊢ ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) → (𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)))) |
85 | 65, 84 | impbid 201 |
. . . . . 6
⊢ ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) → ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ↔ 𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺))) |
86 | 85 | ex 449 |
. . . . 5
⊢ (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) → ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ↔ 𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺)))) |
87 | 86 | pm5.32rd 670 |
. . . 4
⊢ (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) ↔ (𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)))) |
88 | 3, 87 | syl5bb 271 |
. . 3
⊢ (𝑊 ∈ (𝑁 WWalkSN 𝐺) → ((𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) ↔ (𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∧ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)))) |
89 | 88 | rabbidva2 3162 |
. 2
⊢ (𝑊 ∈ (𝑁 WWalkSN 𝐺) → {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)} = {𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}) |
90 | 1, 89 | syl5eq 2656 |
1
⊢ (𝑊 ∈ (𝑁 WWalkSN 𝐺) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr 〈0, (𝑁 + 1)〉) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}) |