Step | Hyp | Ref
| Expression |
1 | | swrdcl 13995 |
. . . . . 6
⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr 〈𝑋, 𝑌〉) ∈ Word 𝐴) |
2 | 1 | adantr 481 |
. . . . 5
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr 〈𝑋, 𝑌〉) ∈ Word 𝐴) |
3 | | swrdcl 13995 |
. . . . . 6
⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr 〈𝑌, 𝑍〉) ∈ Word 𝐴) |
4 | 3 | adantr 481 |
. . . . 5
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr 〈𝑌, 𝑍〉) ∈ Word 𝐴) |
5 | | ccatcl 13914 |
. . . . 5
⊢ (((𝑆 substr 〈𝑋, 𝑌〉) ∈ Word 𝐴 ∧ (𝑆 substr 〈𝑌, 𝑍〉) ∈ Word 𝐴) → ((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉)) ∈ Word 𝐴) |
6 | 2, 4, 5 | syl2anc 584 |
. . . 4
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉)) ∈ Word 𝐴) |
7 | | wrdfn 13864 |
. . . 4
⊢ (((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉)) ∈ Word 𝐴 → ((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉)) Fn (0..^(♯‘((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉))))) |
8 | 6, 7 | syl 17 |
. . 3
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉)) Fn (0..^(♯‘((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉))))) |
9 | | ccatlen 13915 |
. . . . . . 7
⊢ (((𝑆 substr 〈𝑋, 𝑌〉) ∈ Word 𝐴 ∧ (𝑆 substr 〈𝑌, 𝑍〉) ∈ Word 𝐴) → (♯‘((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉))) = ((♯‘(𝑆 substr 〈𝑋, 𝑌〉)) + (♯‘(𝑆 substr 〈𝑌, 𝑍〉)))) |
10 | 2, 4, 9 | syl2anc 584 |
. . . . . 6
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) →
(♯‘((𝑆 substr
〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉))) = ((♯‘(𝑆 substr 〈𝑋, 𝑌〉)) + (♯‘(𝑆 substr 〈𝑌, 𝑍〉)))) |
11 | | simpl 483 |
. . . . . . . 8
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑆 ∈ Word 𝐴) |
12 | | simpr1 1186 |
. . . . . . . 8
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ (0...𝑌)) |
13 | | simpr2 1187 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ (0...𝑍)) |
14 | | simpr3 1188 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑍 ∈ (0...(♯‘𝑆))) |
15 | | fzass4 12933 |
. . . . . . . . . . 11
⊢ ((𝑌 ∈
(0...(♯‘𝑆))
∧ 𝑍 ∈ (𝑌...(♯‘𝑆))) ↔ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) |
16 | 15 | biimpri 229 |
. . . . . . . . . 10
⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (𝑌 ∈ (0...(♯‘𝑆)) ∧ 𝑍 ∈ (𝑌...(♯‘𝑆)))) |
17 | 16 | simpld 495 |
. . . . . . . . 9
⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → 𝑌 ∈ (0...(♯‘𝑆))) |
18 | 13, 14, 17 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ (0...(♯‘𝑆))) |
19 | | swrdlen 13997 |
. . . . . . . 8
⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr 〈𝑋, 𝑌〉)) = (𝑌 − 𝑋)) |
20 | 11, 12, 18, 19 | syl3anc 1363 |
. . . . . . 7
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) →
(♯‘(𝑆 substr
〈𝑋, 𝑌〉)) = (𝑌 − 𝑋)) |
21 | | swrdlen 13997 |
. . . . . . . 8
⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr 〈𝑌, 𝑍〉)) = (𝑍 − 𝑌)) |
22 | 21 | 3adant3r1 1174 |
. . . . . . 7
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) →
(♯‘(𝑆 substr
〈𝑌, 𝑍〉)) = (𝑍 − 𝑌)) |
23 | 20, 22 | oveq12d 7163 |
. . . . . 6
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) →
((♯‘(𝑆 substr
〈𝑋, 𝑌〉)) + (♯‘(𝑆 substr 〈𝑌, 𝑍〉))) = ((𝑌 − 𝑋) + (𝑍 − 𝑌))) |
24 | | elfzelz 12896 |
. . . . . . . . 9
⊢ (𝑌 ∈ (0...𝑍) → 𝑌 ∈ ℤ) |
25 | 13, 24 | syl 17 |
. . . . . . . 8
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℤ) |
26 | 25 | zcnd 12076 |
. . . . . . 7
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℂ) |
27 | | elfzelz 12896 |
. . . . . . . . 9
⊢ (𝑋 ∈ (0...𝑌) → 𝑋 ∈ ℤ) |
28 | 12, 27 | syl 17 |
. . . . . . . 8
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ ℤ) |
29 | 28 | zcnd 12076 |
. . . . . . 7
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ ℂ) |
30 | | elfzelz 12896 |
. . . . . . . . 9
⊢ (𝑍 ∈
(0...(♯‘𝑆))
→ 𝑍 ∈
ℤ) |
31 | 14, 30 | syl 17 |
. . . . . . . 8
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑍 ∈ ℤ) |
32 | 31 | zcnd 12076 |
. . . . . . 7
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑍 ∈ ℂ) |
33 | 26, 29, 32 | npncan3d 11021 |
. . . . . 6
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑌 − 𝑋) + (𝑍 − 𝑌)) = (𝑍 − 𝑋)) |
34 | 10, 23, 33 | 3eqtrd 2857 |
. . . . 5
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) →
(♯‘((𝑆 substr
〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉))) = (𝑍 − 𝑋)) |
35 | 34 | oveq2d 7161 |
. . . 4
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) →
(0..^(♯‘((𝑆
substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉)))) = (0..^(𝑍 − 𝑋))) |
36 | 35 | fneq2d 6440 |
. . 3
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉)) Fn (0..^(♯‘((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉)))) ↔ ((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉)) Fn (0..^(𝑍 − 𝑋)))) |
37 | 8, 36 | mpbid 233 |
. 2
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉)) Fn (0..^(𝑍 − 𝑋))) |
38 | | swrdcl 13995 |
. . . . 5
⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr 〈𝑋, 𝑍〉) ∈ Word 𝐴) |
39 | 38 | adantr 481 |
. . . 4
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr 〈𝑋, 𝑍〉) ∈ Word 𝐴) |
40 | | wrdfn 13864 |
. . . 4
⊢ ((𝑆 substr 〈𝑋, 𝑍〉) ∈ Word 𝐴 → (𝑆 substr 〈𝑋, 𝑍〉) Fn (0..^(♯‘(𝑆 substr 〈𝑋, 𝑍〉)))) |
41 | 39, 40 | syl 17 |
. . 3
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr 〈𝑋, 𝑍〉) Fn (0..^(♯‘(𝑆 substr 〈𝑋, 𝑍〉)))) |
42 | | fzass4 12933 |
. . . . . . . . 9
⊢ ((𝑋 ∈ (0...𝑍) ∧ 𝑌 ∈ (𝑋...𝑍)) ↔ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍))) |
43 | 42 | biimpri 229 |
. . . . . . . 8
⊢ ((𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)) → (𝑋 ∈ (0...𝑍) ∧ 𝑌 ∈ (𝑋...𝑍))) |
44 | 43 | simpld 495 |
. . . . . . 7
⊢ ((𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)) → 𝑋 ∈ (0...𝑍)) |
45 | 12, 13, 44 | syl2anc 584 |
. . . . . 6
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ (0...𝑍)) |
46 | | swrdlen 13997 |
. . . . . 6
⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr 〈𝑋, 𝑍〉)) = (𝑍 − 𝑋)) |
47 | 11, 45, 14, 46 | syl3anc 1363 |
. . . . 5
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) →
(♯‘(𝑆 substr
〈𝑋, 𝑍〉)) = (𝑍 − 𝑋)) |
48 | 47 | oveq2d 7161 |
. . . 4
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) →
(0..^(♯‘(𝑆
substr 〈𝑋, 𝑍〉))) = (0..^(𝑍 − 𝑋))) |
49 | 48 | fneq2d 6440 |
. . 3
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr 〈𝑋, 𝑍〉) Fn (0..^(♯‘(𝑆 substr 〈𝑋, 𝑍〉))) ↔ (𝑆 substr 〈𝑋, 𝑍〉) Fn (0..^(𝑍 − 𝑋)))) |
50 | 41, 49 | mpbid 233 |
. 2
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr 〈𝑋, 𝑍〉) Fn (0..^(𝑍 − 𝑋))) |
51 | 25, 28 | zsubcld 12080 |
. . . . . 6
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑌 − 𝑋) ∈ ℤ) |
52 | 51 | anim1ci 615 |
. . . . 5
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → (𝑥 ∈ (0..^(𝑍 − 𝑋)) ∧ (𝑌 − 𝑋) ∈ ℤ)) |
53 | | fzospliti 13057 |
. . . . 5
⊢ ((𝑥 ∈ (0..^(𝑍 − 𝑋)) ∧ (𝑌 − 𝑋) ∈ ℤ) → (𝑥 ∈ (0..^(𝑌 − 𝑋)) ∨ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋)))) |
54 | 52, 53 | syl 17 |
. . . 4
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → (𝑥 ∈ (0..^(𝑌 − 𝑋)) ∨ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋)))) |
55 | 1 | ad2antrr 722 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → (𝑆 substr 〈𝑋, 𝑌〉) ∈ Word 𝐴) |
56 | 3 | ad2antrr 722 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → (𝑆 substr 〈𝑌, 𝑍〉) ∈ Word 𝐴) |
57 | 20 | oveq2d 7161 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) →
(0..^(♯‘(𝑆
substr 〈𝑋, 𝑌〉))) = (0..^(𝑌 − 𝑋))) |
58 | 57 | eleq2d 2895 |
. . . . . . . 8
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ (0..^(♯‘(𝑆 substr 〈𝑋, 𝑌〉))) ↔ 𝑥 ∈ (0..^(𝑌 − 𝑋)))) |
59 | 58 | biimpar 478 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑥 ∈ (0..^(♯‘(𝑆 substr 〈𝑋, 𝑌〉)))) |
60 | | ccatval1 13918 |
. . . . . . 7
⊢ (((𝑆 substr 〈𝑋, 𝑌〉) ∈ Word 𝐴 ∧ (𝑆 substr 〈𝑌, 𝑍〉) ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘(𝑆 substr 〈𝑋, 𝑌〉)))) → (((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉))‘𝑥) = ((𝑆 substr 〈𝑋, 𝑌〉)‘𝑥)) |
61 | 55, 56, 59, 60 | syl3anc 1363 |
. . . . . 6
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → (((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉))‘𝑥) = ((𝑆 substr 〈𝑋, 𝑌〉)‘𝑥)) |
62 | | simpll 763 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑆 ∈ Word 𝐴) |
63 | | simplr1 1207 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑋 ∈ (0...𝑌)) |
64 | 18 | adantr 481 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑌 ∈ (0...(♯‘𝑆))) |
65 | | simpr 485 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑥 ∈ (0..^(𝑌 − 𝑋))) |
66 | | swrdfv 13998 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → ((𝑆 substr 〈𝑋, 𝑌〉)‘𝑥) = (𝑆‘(𝑥 + 𝑋))) |
67 | 62, 63, 64, 65, 66 | syl31anc 1365 |
. . . . . 6
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → ((𝑆 substr 〈𝑋, 𝑌〉)‘𝑥) = (𝑆‘(𝑥 + 𝑋))) |
68 | 61, 67 | eqtrd 2853 |
. . . . 5
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → (((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉))‘𝑥) = (𝑆‘(𝑥 + 𝑋))) |
69 | 1 | ad2antrr 722 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑆 substr 〈𝑋, 𝑌〉) ∈ Word 𝐴) |
70 | 3 | ad2antrr 722 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑆 substr 〈𝑌, 𝑍〉) ∈ Word 𝐴) |
71 | 23, 33 | eqtrd 2853 |
. . . . . . . . . 10
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) →
((♯‘(𝑆 substr
〈𝑋, 𝑌〉)) + (♯‘(𝑆 substr 〈𝑌, 𝑍〉))) = (𝑍 − 𝑋)) |
72 | 20, 71 | oveq12d 7163 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) →
((♯‘(𝑆 substr
〈𝑋, 𝑌〉))..^((♯‘(𝑆 substr 〈𝑋, 𝑌〉)) + (♯‘(𝑆 substr 〈𝑌, 𝑍〉)))) = ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) |
73 | 72 | eleq2d 2895 |
. . . . . . . 8
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ ((♯‘(𝑆 substr 〈𝑋, 𝑌〉))..^((♯‘(𝑆 substr 〈𝑋, 𝑌〉)) + (♯‘(𝑆 substr 〈𝑌, 𝑍〉)))) ↔ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋)))) |
74 | 73 | biimpar 478 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑥 ∈ ((♯‘(𝑆 substr 〈𝑋, 𝑌〉))..^((♯‘(𝑆 substr 〈𝑋, 𝑌〉)) + (♯‘(𝑆 substr 〈𝑌, 𝑍〉))))) |
75 | | ccatval2 13920 |
. . . . . . 7
⊢ (((𝑆 substr 〈𝑋, 𝑌〉) ∈ Word 𝐴 ∧ (𝑆 substr 〈𝑌, 𝑍〉) ∈ Word 𝐴 ∧ 𝑥 ∈ ((♯‘(𝑆 substr 〈𝑋, 𝑌〉))..^((♯‘(𝑆 substr 〈𝑋, 𝑌〉)) + (♯‘(𝑆 substr 〈𝑌, 𝑍〉))))) → (((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉))‘𝑥) = ((𝑆 substr 〈𝑌, 𝑍〉)‘(𝑥 − (♯‘(𝑆 substr 〈𝑋, 𝑌〉))))) |
76 | 69, 70, 74, 75 | syl3anc 1363 |
. . . . . 6
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉))‘𝑥) = ((𝑆 substr 〈𝑌, 𝑍〉)‘(𝑥 − (♯‘(𝑆 substr 〈𝑋, 𝑌〉))))) |
77 | | simpll 763 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑆 ∈ Word 𝐴) |
78 | | simplr2 1208 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑌 ∈ (0...𝑍)) |
79 | | simplr3 1209 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑍 ∈ (0...(♯‘𝑆))) |
80 | 20 | oveq2d 7161 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 − (♯‘(𝑆 substr 〈𝑋, 𝑌〉))) = (𝑥 − (𝑌 − 𝑋))) |
81 | 80 | adantr 481 |
. . . . . . . 8
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑥 − (♯‘(𝑆 substr 〈𝑋, 𝑌〉))) = (𝑥 − (𝑌 − 𝑋))) |
82 | 33 | oveq2d 7161 |
. . . . . . . . . . 11
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑌 − 𝑋)..^((𝑌 − 𝑋) + (𝑍 − 𝑌))) = ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) |
83 | 82 | eleq2d 2895 |
. . . . . . . . . 10
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ ((𝑌 − 𝑋)..^((𝑌 − 𝑋) + (𝑍 − 𝑌))) ↔ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋)))) |
84 | 83 | biimpar 478 |
. . . . . . . . 9
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑥 ∈ ((𝑌 − 𝑋)..^((𝑌 − 𝑋) + (𝑍 − 𝑌)))) |
85 | 31, 25 | zsubcld 12080 |
. . . . . . . . . 10
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑍 − 𝑌) ∈ ℤ) |
86 | 85 | adantr 481 |
. . . . . . . . 9
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑍 − 𝑌) ∈ ℤ) |
87 | | fzosubel3 13086 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ((𝑌 − 𝑋)..^((𝑌 − 𝑋) + (𝑍 − 𝑌))) ∧ (𝑍 − 𝑌) ∈ ℤ) → (𝑥 − (𝑌 − 𝑋)) ∈ (0..^(𝑍 − 𝑌))) |
88 | 84, 86, 87 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑥 − (𝑌 − 𝑋)) ∈ (0..^(𝑍 − 𝑌))) |
89 | 81, 88 | eqeltrd 2910 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑥 − (♯‘(𝑆 substr 〈𝑋, 𝑌〉))) ∈ (0..^(𝑍 − 𝑌))) |
90 | | swrdfv 13998 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) ∧ (𝑥 − (♯‘(𝑆 substr 〈𝑋, 𝑌〉))) ∈ (0..^(𝑍 − 𝑌))) → ((𝑆 substr 〈𝑌, 𝑍〉)‘(𝑥 − (♯‘(𝑆 substr 〈𝑋, 𝑌〉)))) = (𝑆‘((𝑥 − (♯‘(𝑆 substr 〈𝑋, 𝑌〉))) + 𝑌))) |
91 | 77, 78, 79, 89, 90 | syl31anc 1365 |
. . . . . 6
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → ((𝑆 substr 〈𝑌, 𝑍〉)‘(𝑥 − (♯‘(𝑆 substr 〈𝑋, 𝑌〉)))) = (𝑆‘((𝑥 − (♯‘(𝑆 substr 〈𝑋, 𝑌〉))) + 𝑌))) |
92 | 80 | oveq1d 7160 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑥 − (♯‘(𝑆 substr 〈𝑋, 𝑌〉))) + 𝑌) = ((𝑥 − (𝑌 − 𝑋)) + 𝑌)) |
93 | 92 | adantr 481 |
. . . . . . . 8
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → ((𝑥 − (♯‘(𝑆 substr 〈𝑋, 𝑌〉))) + 𝑌) = ((𝑥 − (𝑌 − 𝑋)) + 𝑌)) |
94 | | elfzoelz 13026 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋)) → 𝑥 ∈ ℤ) |
95 | 94 | zcnd 12076 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋)) → 𝑥 ∈ ℂ) |
96 | 95 | adantl 482 |
. . . . . . . . 9
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑥 ∈ ℂ) |
97 | 26, 29 | subcld 10985 |
. . . . . . . . . 10
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑌 − 𝑋) ∈ ℂ) |
98 | 97 | adantr 481 |
. . . . . . . . 9
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑌 − 𝑋) ∈ ℂ) |
99 | 26 | adantr 481 |
. . . . . . . . 9
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑌 ∈ ℂ) |
100 | 96, 98, 99 | subadd23d 11007 |
. . . . . . . 8
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → ((𝑥 − (𝑌 − 𝑋)) + 𝑌) = (𝑥 + (𝑌 − (𝑌 − 𝑋)))) |
101 | 26, 29 | nncand 10990 |
. . . . . . . . . 10
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑌 − (𝑌 − 𝑋)) = 𝑋) |
102 | 101 | oveq2d 7161 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 + (𝑌 − (𝑌 − 𝑋))) = (𝑥 + 𝑋)) |
103 | 102 | adantr 481 |
. . . . . . . 8
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑥 + (𝑌 − (𝑌 − 𝑋))) = (𝑥 + 𝑋)) |
104 | 93, 100, 103 | 3eqtrd 2857 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → ((𝑥 − (♯‘(𝑆 substr 〈𝑋, 𝑌〉))) + 𝑌) = (𝑥 + 𝑋)) |
105 | 104 | fveq2d 6667 |
. . . . . 6
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑆‘((𝑥 − (♯‘(𝑆 substr 〈𝑋, 𝑌〉))) + 𝑌)) = (𝑆‘(𝑥 + 𝑋))) |
106 | 76, 91, 105 | 3eqtrd 2857 |
. . . . 5
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉))‘𝑥) = (𝑆‘(𝑥 + 𝑋))) |
107 | 68, 106 | jaodan 951 |
. . . 4
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ (𝑥 ∈ (0..^(𝑌 − 𝑋)) ∨ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋)))) → (((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉))‘𝑥) = (𝑆‘(𝑥 + 𝑋))) |
108 | 54, 107 | syldan 591 |
. . 3
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → (((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉))‘𝑥) = (𝑆‘(𝑥 + 𝑋))) |
109 | | simpll 763 |
. . . 4
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑆 ∈ Word 𝐴) |
110 | 45 | adantr 481 |
. . . 4
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑋 ∈ (0...𝑍)) |
111 | | simplr3 1209 |
. . . 4
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑍 ∈ (0...(♯‘𝑆))) |
112 | | simpr 485 |
. . . 4
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑥 ∈ (0..^(𝑍 − 𝑋))) |
113 | | swrdfv 13998 |
. . . 4
⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → ((𝑆 substr 〈𝑋, 𝑍〉)‘𝑥) = (𝑆‘(𝑥 + 𝑋))) |
114 | 109, 110,
111, 112, 113 | syl31anc 1365 |
. . 3
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → ((𝑆 substr 〈𝑋, 𝑍〉)‘𝑥) = (𝑆‘(𝑥 + 𝑋))) |
115 | 108, 114 | eqtr4d 2856 |
. 2
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → (((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉))‘𝑥) = ((𝑆 substr 〈𝑋, 𝑍〉)‘𝑥)) |
116 | 37, 50, 115 | eqfnfvd 6797 |
1
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉)) = (𝑆 substr 〈𝑋, 𝑍〉)) |