Step | Hyp | Ref
| Expression |
1 | | wwlknimp 26215 |
. . . 4
⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)) |
2 | | fzonn0p1 12411 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈ (0..^(𝑁 + 1))) |
3 | 2 | adantl 481 |
. . . . . . . . . 10
⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ (0..^(𝑁 + 1))) |
4 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑁 → (𝑊‘𝑖) = (𝑊‘𝑁)) |
5 | | oveq1 6556 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 𝑁 → (𝑖 + 1) = (𝑁 + 1)) |
6 | 5 | fveq2d 6107 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑁 → (𝑊‘(𝑖 + 1)) = (𝑊‘(𝑁 + 1))) |
7 | 4, 6 | preq12d 4220 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑁 → {(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} = {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))}) |
8 | 7 | eleq1d 2672 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝑁 → ({(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ ran 𝐸)) |
9 | 8 | rspcv 3278 |
. . . . . . . . . 10
⊢ (𝑁 ∈ (0..^(𝑁 + 1)) → (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 → {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ ran 𝐸)) |
10 | 3, 9 | syl 17 |
. . . . . . . . 9
⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) →
(∀𝑖 ∈
(0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 → {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ ran 𝐸)) |
11 | 10 | imp 444 |
. . . . . . . 8
⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) ∧
∀𝑖 ∈
(0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ ran 𝐸) |
12 | | simpll 786 |
. . . . . . . . . . . . . 14
⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) → 𝑊 ∈ Word 𝑉) |
13 | | 1zzd 11285 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) → 1 ∈
ℤ) |
14 | | lencl 13179 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈
ℕ0) |
15 | 14 | nn0zd 11356 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℤ) |
16 | 15 | ad2antrr 758 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) →
(#‘𝑊) ∈
ℤ) |
17 | | peano2nn0 11210 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
ℕ0) |
18 | 17 | nn0zd 11356 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
ℤ) |
19 | 18 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ∈
ℤ) |
20 | 13, 16, 19 | 3jca 1235 |
. . . . . . . . . . . . . . 15
⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) → (1
∈ ℤ ∧ (#‘𝑊) ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ)) |
21 | | nn0ge0 11195 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈ ℕ0
→ 0 ≤ 𝑁) |
22 | | 1red 9934 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 ∈ ℕ0
→ 1 ∈ ℝ) |
23 | | nn0re 11178 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℝ) |
24 | 22, 23 | addge02d 10495 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈ ℕ0
→ (0 ≤ 𝑁 ↔ 1
≤ (𝑁 +
1))) |
25 | 21, 24 | mpbid 221 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℕ0
→ 1 ≤ (𝑁 +
1)) |
26 | 25 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) → 1 ≤
(𝑁 + 1)) |
27 | 17 | nn0red 11229 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
ℝ) |
28 | 27 | lep1d 10834 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ≤ ((𝑁 + 1) + 1)) |
29 | | breq2 4587 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((#‘𝑊) =
((𝑁 + 1) + 1) →
((𝑁 + 1) ≤
(#‘𝑊) ↔ (𝑁 + 1) ≤ ((𝑁 + 1) + 1))) |
30 | 28, 29 | syl5ibrcom 236 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ0
→ ((#‘𝑊) =
((𝑁 + 1) + 1) → (𝑁 + 1) ≤ (#‘𝑊))) |
31 | 30 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢
((#‘𝑊) ∈
ℕ0 → (𝑁 ∈ ℕ0 →
((#‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 + 1) ≤ (#‘𝑊)))) |
32 | 31 | com23 84 |
. . . . . . . . . . . . . . . . . 18
⊢
((#‘𝑊) ∈
ℕ0 → ((#‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 ∈ ℕ0 → (𝑁 + 1) ≤ (#‘𝑊)))) |
33 | 14, 32 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑊 ∈ Word 𝑉 → ((#‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 ∈ ℕ0 → (𝑁 + 1) ≤ (#‘𝑊)))) |
34 | 33 | imp31 447 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ≤ (#‘𝑊)) |
35 | 26, 34 | jca 553 |
. . . . . . . . . . . . . . 15
⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) → (1 ≤
(𝑁 + 1) ∧ (𝑁 + 1) ≤ (#‘𝑊))) |
36 | | elfz2 12204 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 + 1) ∈ (1...(#‘𝑊)) ↔ ((1 ∈ ℤ
∧ (#‘𝑊) ∈
ℤ ∧ (𝑁 + 1)
∈ ℤ) ∧ (1 ≤ (𝑁 + 1) ∧ (𝑁 + 1) ≤ (#‘𝑊)))) |
37 | 20, 35, 36 | sylanbrc 695 |
. . . . . . . . . . . . . 14
⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ∈ (1...(#‘𝑊))) |
38 | 12, 37 | jca 553 |
. . . . . . . . . . . . 13
⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (1...(#‘𝑊)))) |
39 | | swrd0fvlsw 13295 |
. . . . . . . . . . . . 13
⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (1...(#‘𝑊))) → ( lastS ‘(𝑊 substr 〈0, (𝑁 + 1)〉)) = (𝑊‘((𝑁 + 1) − 1))) |
40 | 38, 39 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) → ( lastS
‘(𝑊 substr 〈0,
(𝑁 + 1)〉)) = (𝑊‘((𝑁 + 1) − 1))) |
41 | | nn0cn 11179 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℂ) |
42 | | 1cnd 9935 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℕ0
→ 1 ∈ ℂ) |
43 | 41, 42 | pncand 10272 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℕ0
→ ((𝑁 + 1) − 1)
= 𝑁) |
44 | 43 | fveq2d 6107 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℕ0
→ (𝑊‘((𝑁 + 1) − 1)) = (𝑊‘𝑁)) |
45 | 44 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) → (𝑊‘((𝑁 + 1) − 1)) = (𝑊‘𝑁)) |
46 | 40, 45 | eqtrd 2644 |
. . . . . . . . . . 11
⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) → ( lastS
‘(𝑊 substr 〈0,
(𝑁 + 1)〉)) = (𝑊‘𝑁)) |
47 | | lsw 13204 |
. . . . . . . . . . . . 13
⊢ (𝑊 ∈ Word 𝑉 → ( lastS ‘𝑊) = (𝑊‘((#‘𝑊) − 1))) |
48 | 47 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) → ( lastS
‘𝑊) = (𝑊‘((#‘𝑊) − 1))) |
49 | | oveq1 6556 |
. . . . . . . . . . . . . . 15
⊢
((#‘𝑊) =
((𝑁 + 1) + 1) →
((#‘𝑊) − 1) =
(((𝑁 + 1) + 1) −
1)) |
50 | 49 | fveq2d 6107 |
. . . . . . . . . . . . . 14
⊢
((#‘𝑊) =
((𝑁 + 1) + 1) → (𝑊‘((#‘𝑊) − 1)) = (𝑊‘(((𝑁 + 1) + 1) − 1))) |
51 | 50 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑊‘((#‘𝑊) − 1)) = (𝑊‘(((𝑁 + 1) + 1) − 1))) |
52 | 17 | nn0cnd 11230 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
ℂ) |
53 | 52, 42 | pncand 10272 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℕ0
→ (((𝑁 + 1) + 1)
− 1) = (𝑁 +
1)) |
54 | 53 | fveq2d 6107 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℕ0
→ (𝑊‘(((𝑁 + 1) + 1) − 1)) = (𝑊‘(𝑁 + 1))) |
55 | 51, 54 | sylan9eq 2664 |
. . . . . . . . . . . 12
⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) → (𝑊‘((#‘𝑊) − 1)) = (𝑊‘(𝑁 + 1))) |
56 | 48, 55 | eqtrd 2644 |
. . . . . . . . . . 11
⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) → ( lastS
‘𝑊) = (𝑊‘(𝑁 + 1))) |
57 | 46, 56 | preq12d 4220 |
. . . . . . . . . 10
⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) → {( lastS
‘(𝑊 substr 〈0,
(𝑁 + 1)〉)), ( lastS
‘𝑊)} = {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))}) |
58 | 57 | eleq1d 2672 |
. . . . . . . . 9
⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) → ({(
lastS ‘(𝑊 substr
〈0, (𝑁 + 1)〉)), (
lastS ‘𝑊)} ∈ ran
𝐸 ↔ {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ ran 𝐸)) |
59 | 58 | adantr 480 |
. . . . . . . 8
⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) ∧
∀𝑖 ∈
(0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → ({( lastS ‘(𝑊 substr 〈0, (𝑁 + 1)〉)), ( lastS ‘𝑊)} ∈ ran 𝐸 ↔ {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ ran 𝐸)) |
60 | 11, 59 | mpbird 246 |
. . . . . . 7
⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ0) ∧
∀𝑖 ∈
(0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → {( lastS ‘(𝑊 substr 〈0, (𝑁 + 1)〉)), ( lastS ‘𝑊)} ∈ ran 𝐸) |
61 | 60 | exp31 628 |
. . . . . 6
⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ0 →
(∀𝑖 ∈
(0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 → {( lastS ‘(𝑊 substr 〈0, (𝑁 + 1)〉)), ( lastS ‘𝑊)} ∈ ran 𝐸))) |
62 | 61 | com23 84 |
. . . . 5
⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 → (𝑁 ∈ ℕ0 → {( lastS
‘(𝑊 substr 〈0,
(𝑁 + 1)〉)), ( lastS
‘𝑊)} ∈ ran 𝐸))) |
63 | 62 | 3impia 1253 |
. . . 4
⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → (𝑁 ∈ ℕ0 → {( lastS
‘(𝑊 substr 〈0,
(𝑁 + 1)〉)), ( lastS
‘𝑊)} ∈ ran 𝐸)) |
64 | 1, 63 | syl 17 |
. . 3
⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑁 ∈ ℕ0 → {( lastS
‘(𝑊 substr 〈0,
(𝑁 + 1)〉)), ( lastS
‘𝑊)} ∈ ran 𝐸)) |
65 | | wwlkextprop.x |
. . 3
⊢ 𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) |
66 | 64, 65 | eleq2s 2706 |
. 2
⊢ (𝑊 ∈ 𝑋 → (𝑁 ∈ ℕ0 → {( lastS
‘(𝑊 substr 〈0,
(𝑁 + 1)〉)), ( lastS
‘𝑊)} ∈ ran 𝐸)) |
67 | 66 | imp 444 |
1
⊢ ((𝑊 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → {( lastS
‘(𝑊 substr 〈0,
(𝑁 + 1)〉)), ( lastS
‘𝑊)} ∈ ran 𝐸) |