Step | Hyp | Ref
| Expression |
1 | | swrdcl 13847 |
. . . . . 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 13847 |
. . . . . 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 13776 |
. . . . 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 | | wrdf 13716 |
. . . 4
⊢ (((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉)) ∈ Word 𝐴 → ((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉)):(0..^(♯‘((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉))))⟶𝐴) |
8 | | ffn 6389 |
. . . 4
⊢ (((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉)):(0..^(♯‘((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉))))⟶𝐴 → ((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉)) Fn (0..^(♯‘((𝑆 substr 〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉))))) |
9 | 6, 7, 8 | 3syl 18 |
. . 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〉))))) |
10 | | ccatlen 13777 |
. . . . . . 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〉)))) |
11 | 2, 4, 10 | 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〉)))) |
12 | | simpl 483 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑆 ∈ Word 𝐴) |
13 | | simpr1 1187 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ (0...𝑌)) |
14 | | simpr2 1188 |
. . . . . . . . . 10
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ (0...𝑍)) |
15 | | simpr3 1189 |
. . . . . . . . . 10
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑍 ∈ (0...(♯‘𝑆))) |
16 | | fzass4 12799 |
. . . . . . . . . . . 12
⊢ ((𝑌 ∈
(0...(♯‘𝑆))
∧ 𝑍 ∈ (𝑌...(♯‘𝑆))) ↔ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) |
17 | 16 | biimpri 229 |
. . . . . . . . . . 11
⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (𝑌 ∈ (0...(♯‘𝑆)) ∧ 𝑍 ∈ (𝑌...(♯‘𝑆)))) |
18 | 17 | simpld 495 |
. . . . . . . . . 10
⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → 𝑌 ∈ (0...(♯‘𝑆))) |
19 | 14, 15, 18 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ (0...(♯‘𝑆))) |
20 | | swrdlen 13849 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr 〈𝑋, 𝑌〉)) = (𝑌 − 𝑋)) |
21 | 12, 13, 19, 20 | syl3anc 1364 |
. . . . . . . 8
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) →
(♯‘(𝑆 substr
〈𝑋, 𝑌〉)) = (𝑌 − 𝑋)) |
22 | | swrdlen 13849 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr 〈𝑌, 𝑍〉)) = (𝑍 − 𝑌)) |
23 | 12, 14, 15, 22 | syl3anc 1364 |
. . . . . . . 8
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) →
(♯‘(𝑆 substr
〈𝑌, 𝑍〉)) = (𝑍 − 𝑌)) |
24 | 21, 23 | oveq12d 7041 |
. . . . . . 7
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) →
((♯‘(𝑆 substr
〈𝑋, 𝑌〉)) + (♯‘(𝑆 substr 〈𝑌, 𝑍〉))) = ((𝑌 − 𝑋) + (𝑍 − 𝑌))) |
25 | | elfzelz 12762 |
. . . . . . . . . 10
⊢ (𝑌 ∈ (0...𝑍) → 𝑌 ∈ ℤ) |
26 | 14, 25 | syl 17 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℤ) |
27 | 26 | zcnd 11942 |
. . . . . . . 8
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℂ) |
28 | | elfzelz 12762 |
. . . . . . . . . 10
⊢ (𝑋 ∈ (0...𝑌) → 𝑋 ∈ ℤ) |
29 | 13, 28 | syl 17 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ ℤ) |
30 | 29 | zcnd 11942 |
. . . . . . . 8
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ ℂ) |
31 | | elfzelz 12762 |
. . . . . . . . . 10
⊢ (𝑍 ∈
(0...(♯‘𝑆))
→ 𝑍 ∈
ℤ) |
32 | 15, 31 | syl 17 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑍 ∈ ℤ) |
33 | 32 | zcnd 11942 |
. . . . . . . 8
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑍 ∈ ℂ) |
34 | 27, 30, 33 | npncan3d 10887 |
. . . . . . 7
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑌 − 𝑋) + (𝑍 − 𝑌)) = (𝑍 − 𝑋)) |
35 | 24, 34 | eqtrd 2833 |
. . . . . 6
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) →
((♯‘(𝑆 substr
〈𝑋, 𝑌〉)) + (♯‘(𝑆 substr 〈𝑌, 𝑍〉))) = (𝑍 − 𝑋)) |
36 | 11, 35 | eqtrd 2833 |
. . . . 5
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) →
(♯‘((𝑆 substr
〈𝑋, 𝑌〉) ++ (𝑆 substr 〈𝑌, 𝑍〉))) = (𝑍 − 𝑋)) |
37 | 36 | oveq2d 7039 |
. . . 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))) |
38 | 37 | fneq2d 6324 |
. . 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)))) |
39 | 9, 38 | 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))) |
40 | | swrdcl 13847 |
. . . . 5
⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr 〈𝑋, 𝑍〉) ∈ Word 𝐴) |
41 | 40 | adantr 481 |
. . . 4
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr 〈𝑋, 𝑍〉) ∈ Word 𝐴) |
42 | | wrdf 13716 |
. . . 4
⊢ ((𝑆 substr 〈𝑋, 𝑍〉) ∈ Word 𝐴 → (𝑆 substr 〈𝑋, 𝑍〉):(0..^(♯‘(𝑆 substr 〈𝑋, 𝑍〉)))⟶𝐴) |
43 | | ffn 6389 |
. . . 4
⊢ ((𝑆 substr 〈𝑋, 𝑍〉):(0..^(♯‘(𝑆 substr 〈𝑋, 𝑍〉)))⟶𝐴 → (𝑆 substr 〈𝑋, 𝑍〉) Fn (0..^(♯‘(𝑆 substr 〈𝑋, 𝑍〉)))) |
44 | 41, 42, 43 | 3syl 18 |
. . 3
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr 〈𝑋, 𝑍〉) Fn (0..^(♯‘(𝑆 substr 〈𝑋, 𝑍〉)))) |
45 | | fzass4 12799 |
. . . . . . . . 9
⊢ ((𝑋 ∈ (0...𝑍) ∧ 𝑌 ∈ (𝑋...𝑍)) ↔ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍))) |
46 | 45 | biimpri 229 |
. . . . . . . 8
⊢ ((𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)) → (𝑋 ∈ (0...𝑍) ∧ 𝑌 ∈ (𝑋...𝑍))) |
47 | 46 | simpld 495 |
. . . . . . 7
⊢ ((𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)) → 𝑋 ∈ (0...𝑍)) |
48 | 13, 14, 47 | syl2anc 584 |
. . . . . 6
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ (0...𝑍)) |
49 | | swrdlen 13849 |
. . . . . 6
⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr 〈𝑋, 𝑍〉)) = (𝑍 − 𝑋)) |
50 | 12, 48, 15, 49 | syl3anc 1364 |
. . . . 5
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) →
(♯‘(𝑆 substr
〈𝑋, 𝑍〉)) = (𝑍 − 𝑋)) |
51 | 50 | oveq2d 7039 |
. . . 4
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) →
(0..^(♯‘(𝑆
substr 〈𝑋, 𝑍〉))) = (0..^(𝑍 − 𝑋))) |
52 | 51 | fneq2d 6324 |
. . 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)))) |
53 | 44, 52 | mpbid 233 |
. 2
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr 〈𝑋, 𝑍〉) Fn (0..^(𝑍 − 𝑋))) |
54 | | simpr 485 |
. . . . 5
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑥 ∈ (0..^(𝑍 − 𝑋))) |
55 | 26, 29 | zsubcld 11946 |
. . . . . 6
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑌 − 𝑋) ∈ ℤ) |
56 | 55 | adantr 481 |
. . . . 5
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → (𝑌 − 𝑋) ∈ ℤ) |
57 | | fzospliti 12923 |
. . . . 5
⊢ ((𝑥 ∈ (0..^(𝑍 − 𝑋)) ∧ (𝑌 − 𝑋) ∈ ℤ) → (𝑥 ∈ (0..^(𝑌 − 𝑋)) ∨ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋)))) |
58 | 54, 56, 57 | syl2anc 584 |
. . . 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)))) |
59 | 2 | adantr 481 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → (𝑆 substr 〈𝑋, 𝑌〉) ∈ Word 𝐴) |
60 | 4 | adantr 481 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → (𝑆 substr 〈𝑌, 𝑍〉) ∈ Word 𝐴) |
61 | 21 | oveq2d 7039 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) →
(0..^(♯‘(𝑆
substr 〈𝑋, 𝑌〉))) = (0..^(𝑌 − 𝑋))) |
62 | 61 | eleq2d 2870 |
. . . . . . . 8
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ (0..^(♯‘(𝑆 substr 〈𝑋, 𝑌〉))) ↔ 𝑥 ∈ (0..^(𝑌 − 𝑋)))) |
63 | 62 | 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〉)))) |
64 | | ccatval1 13779 |
. . . . . . 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)) |
65 | 59, 60, 63, 64 | syl3anc 1364 |
. . . . . 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)) |
66 | | simpll 763 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑆 ∈ Word 𝐴) |
67 | | simplr1 1208 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑋 ∈ (0...𝑌)) |
68 | 19 | adantr 481 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑌 ∈ (0...(♯‘𝑆))) |
69 | | simpr 485 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑥 ∈ (0..^(𝑌 − 𝑋))) |
70 | | swrdfv 13850 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → ((𝑆 substr 〈𝑋, 𝑌〉)‘𝑥) = (𝑆‘(𝑥 + 𝑋))) |
71 | 66, 67, 68, 69, 70 | syl31anc 1366 |
. . . . . 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))) |
72 | 65, 71 | eqtrd 2833 |
. . . . 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))) |
73 | 2 | adantr 481 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑆 substr 〈𝑋, 𝑌〉) ∈ Word 𝐴) |
74 | 4 | adantr 481 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑆 substr 〈𝑌, 𝑍〉) ∈ Word 𝐴) |
75 | 21, 35 | oveq12d 7041 |
. . . . . . . . 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))) |
76 | 75 | eleq2d 2870 |
. . . . . . . 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)))) |
77 | 76 | 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〉))))) |
78 | | ccatval2 13780 |
. . . . . . 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〉))))) |
79 | 73, 74, 77, 78 | syl3anc 1364 |
. . . . . 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〉))))) |
80 | | simpll 763 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑆 ∈ Word 𝐴) |
81 | | simplr2 1209 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑌 ∈ (0...𝑍)) |
82 | | simplr3 1210 |
. . . . . . 7
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑍 ∈ (0...(♯‘𝑆))) |
83 | 21 | oveq2d 7039 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 − (♯‘(𝑆 substr 〈𝑋, 𝑌〉))) = (𝑥 − (𝑌 − 𝑋))) |
84 | 83 | 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))) |
85 | 34 | oveq2d 7039 |
. . . . . . . . . . 11
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑌 − 𝑋)..^((𝑌 − 𝑋) + (𝑍 − 𝑌))) = ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) |
86 | 85 | eleq2d 2870 |
. . . . . . . . . 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)))) |
87 | 86 | 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)))) |
88 | 32, 26 | zsubcld 11946 |
. . . . . . . . . 10
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑍 − 𝑌) ∈ ℤ) |
89 | 88 | adantr 481 |
. . . . . . . . 9
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑍 − 𝑌) ∈ ℤ) |
90 | | fzosubel3 12952 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ((𝑌 − 𝑋)..^((𝑌 − 𝑋) + (𝑍 − 𝑌))) ∧ (𝑍 − 𝑌) ∈ ℤ) → (𝑥 − (𝑌 − 𝑋)) ∈ (0..^(𝑍 − 𝑌))) |
91 | 87, 89, 90 | 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))) |
92 | 84, 91 | eqeltrd 2885 |
. . . . . . 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))) |
93 | | swrdfv 13850 |
. . . . . . 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))) |
94 | 80, 81, 82, 92, 93 | syl31anc 1366 |
. . . . . 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))) |
95 | 83 | oveq1d 7038 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑥 − (♯‘(𝑆 substr 〈𝑋, 𝑌〉))) + 𝑌) = ((𝑥 − (𝑌 − 𝑋)) + 𝑌)) |
96 | 95 | 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)) |
97 | | elfzoelz 12892 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋)) → 𝑥 ∈ ℤ) |
98 | 97 | zcnd 11942 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋)) → 𝑥 ∈ ℂ) |
99 | 98 | adantl 482 |
. . . . . . . . 9
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑥 ∈ ℂ) |
100 | 27, 30 | subcld 10851 |
. . . . . . . . . 10
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑌 − 𝑋) ∈ ℂ) |
101 | 100 | adantr 481 |
. . . . . . . . 9
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑌 − 𝑋) ∈ ℂ) |
102 | 27 | adantr 481 |
. . . . . . . . 9
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑌 ∈ ℂ) |
103 | 99, 101, 102 | subadd23d 10873 |
. . . . . . . 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)))) |
104 | 27, 30 | nncand 10856 |
. . . . . . . . . 10
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑌 − (𝑌 − 𝑋)) = 𝑋) |
105 | 104 | oveq2d 7039 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 + (𝑌 − (𝑌 − 𝑋))) = (𝑥 + 𝑋)) |
106 | 105 | 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)) |
107 | 96, 103, 106 | 3eqtrd 2837 |
. . . . . . 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)) |
108 | 107 | fveq2d 6549 |
. . . . . 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))) |
109 | 79, 94, 108 | 3eqtrd 2837 |
. . . . 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))) |
110 | 72, 109 | jaodan 952 |
. . . 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))) |
111 | 58, 110 | 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))) |
112 | | simpll 763 |
. . . 4
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑆 ∈ Word 𝐴) |
113 | 48 | adantr 481 |
. . . 4
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑋 ∈ (0...𝑍)) |
114 | | simplr3 1210 |
. . . 4
⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑍 ∈ (0...(♯‘𝑆))) |
115 | | swrdfv 13850 |
. . . 4
⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → ((𝑆 substr 〈𝑋, 𝑍〉)‘𝑥) = (𝑆‘(𝑥 + 𝑋))) |
116 | 112, 113,
114, 54, 115 | syl31anc 1366 |
. . 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))) |
117 | 111, 116 | eqtr4d 2836 |
. 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)) |
118 | 39, 53, 117 | eqfnfvd 6677 |
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〉)) |