Step | Hyp | Ref
| Expression |
1 | | ccatcl 13359 |
. . . . 5
⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉) |
2 | 1 | adantr 481 |
. . . 4
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐴 ++ 𝐵) ∈ Word 𝑉) |
3 | | elfz0fzfz0 12444 |
. . . . 5
⊢ ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → 𝑀 ∈ (0...𝑁)) |
4 | 3 | adantl 482 |
. . . 4
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝑀 ∈ (0...𝑁)) |
5 | | elfzuz2 12346 |
. . . . . . . . 9
⊢ (𝑀 ∈ (0...𝐿) → 𝐿 ∈
(ℤ≥‘0)) |
6 | 5 | adantl 482 |
. . . . . . . 8
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → 𝐿 ∈
(ℤ≥‘0)) |
7 | | fzss1 12380 |
. . . . . . . 8
⊢ (𝐿 ∈
(ℤ≥‘0) → (𝐿...(𝐿 + (#‘𝐵))) ⊆ (0...(𝐿 + (#‘𝐵)))) |
8 | 6, 7 | syl 17 |
. . . . . . 7
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (𝐿...(𝐿 + (#‘𝐵))) ⊆ (0...(𝐿 + (#‘𝐵)))) |
9 | | ccatlen 13360 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵))) |
10 | | swrdccatin12.l |
. . . . . . . . . . . 12
⊢ 𝐿 = (#‘𝐴) |
11 | 10 | eqcomi 2631 |
. . . . . . . . . . 11
⊢
(#‘𝐴) = 𝐿 |
12 | 11 | oveq1i 6660 |
. . . . . . . . . 10
⊢
((#‘𝐴) +
(#‘𝐵)) = (𝐿 + (#‘𝐵)) |
13 | 9, 12 | syl6eq 2672 |
. . . . . . . . 9
⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (#‘(𝐴 ++ 𝐵)) = (𝐿 + (#‘𝐵))) |
14 | 13 | adantr 481 |
. . . . . . . 8
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (#‘(𝐴 ++ 𝐵)) = (𝐿 + (#‘𝐵))) |
15 | 14 | oveq2d 6666 |
. . . . . . 7
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (0...(#‘(𝐴 ++ 𝐵))) = (0...(𝐿 + (#‘𝐵)))) |
16 | 8, 15 | sseqtr4d 3642 |
. . . . . 6
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (𝐿...(𝐿 + (#‘𝐵))) ⊆ (0...(#‘(𝐴 ++ 𝐵)))) |
17 | 16 | sseld 3602 |
. . . . 5
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵))))) |
18 | 17 | impr 649 |
. . . 4
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))) |
19 | | swrdvalfn 13426 |
. . . 4
⊢ (((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) Fn (0..^(𝑁 − 𝑀))) |
20 | 2, 4, 18, 19 | syl3anc 1326 |
. . 3
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) Fn (0..^(𝑁 − 𝑀))) |
21 | | swrdcl 13419 |
. . . . . . 7
⊢ (𝐴 ∈ Word 𝑉 → (𝐴 substr 〈𝑀, 𝐿〉) ∈ Word 𝑉) |
22 | | swrdcl 13419 |
. . . . . . 7
⊢ (𝐵 ∈ Word 𝑉 → (𝐵 substr 〈0, (𝑁 − 𝐿)〉) ∈ Word 𝑉) |
23 | 21, 22 | anim12i 590 |
. . . . . 6
⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝐴 substr 〈𝑀, 𝐿〉) ∈ Word 𝑉 ∧ (𝐵 substr 〈0, (𝑁 − 𝐿)〉) ∈ Word 𝑉)) |
24 | 23 | adantr 481 |
. . . . 5
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 substr 〈𝑀, 𝐿〉) ∈ Word 𝑉 ∧ (𝐵 substr 〈0, (𝑁 − 𝐿)〉) ∈ Word 𝑉)) |
25 | | ccatvalfn 13365 |
. . . . 5
⊢ (((𝐴 substr 〈𝑀, 𝐿〉) ∈ Word 𝑉 ∧ (𝐵 substr 〈0, (𝑁 − 𝐿)〉) ∈ Word 𝑉) → ((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 substr 〈0, (𝑁 − 𝐿)〉)) Fn (0..^((#‘(𝐴 substr 〈𝑀, 𝐿〉)) + (#‘(𝐵 substr 〈0, (𝑁 − 𝐿)〉))))) |
26 | 24, 25 | syl 17 |
. . . 4
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 substr 〈0, (𝑁 − 𝐿)〉)) Fn (0..^((#‘(𝐴 substr 〈𝑀, 𝐿〉)) + (#‘(𝐵 substr 〈0, (𝑁 − 𝐿)〉))))) |
27 | | simpll 790 |
. . . . . . . . 9
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐴 ∈ Word 𝑉) |
28 | | simprl 794 |
. . . . . . . . 9
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝑀 ∈ (0...𝐿)) |
29 | | lencl 13324 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈
ℕ0) |
30 | | elnn0uz 11725 |
. . . . . . . . . . . . . 14
⊢
((#‘𝐴) ∈
ℕ0 ↔ (#‘𝐴) ∈
(ℤ≥‘0)) |
31 | | eluzfz2 12349 |
. . . . . . . . . . . . . 14
⊢
((#‘𝐴) ∈
(ℤ≥‘0) → (#‘𝐴) ∈ (0...(#‘𝐴))) |
32 | 30, 31 | sylbi 207 |
. . . . . . . . . . . . 13
⊢
((#‘𝐴) ∈
ℕ0 → (#‘𝐴) ∈ (0...(#‘𝐴))) |
33 | 10, 32 | syl5eqel 2705 |
. . . . . . . . . . . 12
⊢
((#‘𝐴) ∈
ℕ0 → 𝐿 ∈ (0...(#‘𝐴))) |
34 | 29, 33 | syl 17 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ Word 𝑉 → 𝐿 ∈ (0...(#‘𝐴))) |
35 | 34 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝐿 ∈ (0...(#‘𝐴))) |
36 | 35 | adantr 481 |
. . . . . . . . 9
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐿 ∈ (0...(#‘𝐴))) |
37 | | swrdlen 13423 |
. . . . . . . . 9
⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝐴))) → (#‘(𝐴 substr 〈𝑀, 𝐿〉)) = (𝐿 − 𝑀)) |
38 | 27, 28, 36, 37 | syl3anc 1326 |
. . . . . . . 8
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (#‘(𝐴 substr 〈𝑀, 𝐿〉)) = (𝐿 − 𝑀)) |
39 | | simpr 477 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝐵 ∈ Word 𝑉) |
40 | 39 | adantr 481 |
. . . . . . . . 9
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐵 ∈ Word 𝑉) |
41 | | lencl 13324 |
. . . . . . . . . . . . 13
⊢ (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈
ℕ0) |
42 | 41 | nn0zd 11480 |
. . . . . . . . . . . 12
⊢ (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℤ) |
43 | 42 | adantl 482 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (#‘𝐵) ∈ ℤ) |
44 | | simpr 477 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) |
45 | 43, 44 | anim12i 590 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((#‘𝐵) ∈ ℤ ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) |
46 | | elfzmlbp 12450 |
. . . . . . . . . 10
⊢
(((#‘𝐵) ∈
ℤ ∧ 𝑁 ∈
(𝐿...(𝐿 + (#‘𝐵)))) → (𝑁 − 𝐿) ∈ (0...(#‘𝐵))) |
47 | 45, 46 | syl 17 |
. . . . . . . . 9
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝑁 − 𝐿) ∈ (0...(#‘𝐵))) |
48 | | swrd0len 13422 |
. . . . . . . . 9
⊢ ((𝐵 ∈ Word 𝑉 ∧ (𝑁 − 𝐿) ∈ (0...(#‘𝐵))) → (#‘(𝐵 substr 〈0, (𝑁 − 𝐿)〉)) = (𝑁 − 𝐿)) |
49 | 40, 47, 48 | syl2anc 693 |
. . . . . . . 8
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (#‘(𝐵 substr 〈0, (𝑁 − 𝐿)〉)) = (𝑁 − 𝐿)) |
50 | 38, 49 | oveq12d 6668 |
. . . . . . 7
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((#‘(𝐴 substr 〈𝑀, 𝐿〉)) + (#‘(𝐵 substr 〈0, (𝑁 − 𝐿)〉))) = ((𝐿 − 𝑀) + (𝑁 − 𝐿))) |
51 | | elfz2nn0 12431 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ (0...𝐿) ↔ (𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0
∧ 𝑀 ≤ 𝐿)) |
52 | | nn0cn 11302 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐿 ∈ ℕ0
→ 𝐿 ∈
ℂ) |
53 | 52 | adantl 482 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑀 ∈ ℕ0
∧ 𝐿 ∈
ℕ0) → 𝐿 ∈ ℂ) |
54 | 53 | adantl 482 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0
∧ 𝐿 ∈
ℕ0)) → 𝐿 ∈ ℂ) |
55 | | nn0cn 11302 |
. . . . . . . . . . . . . . . 16
⊢ (𝑀 ∈ ℕ0
→ 𝑀 ∈
ℂ) |
56 | 55 | ad2antrl 764 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0
∧ 𝐿 ∈
ℕ0)) → 𝑀 ∈ ℂ) |
57 | | zcn 11382 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈ ℤ → 𝑁 ∈
ℂ) |
58 | 57 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0
∧ 𝐿 ∈
ℕ0)) → 𝑁 ∈ ℂ) |
59 | 54, 56, 58 | 3jca 1242 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0
∧ 𝐿 ∈
ℕ0)) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)) |
60 | 59 | ex 450 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℤ → ((𝑀 ∈ ℕ0
∧ 𝐿 ∈
ℕ0) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))) |
61 | | elfzelz 12342 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → 𝑁 ∈ ℤ) |
62 | 60, 61 | syl11 33 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℕ0
∧ 𝐿 ∈
ℕ0) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))) |
63 | 62 | 3adant3 1081 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℕ0
∧ 𝐿 ∈
ℕ0 ∧ 𝑀
≤ 𝐿) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))) |
64 | 51, 63 | sylbi 207 |
. . . . . . . . . 10
⊢ (𝑀 ∈ (0...𝐿) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))) |
65 | 64 | imp 445 |
. . . . . . . . 9
⊢ ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)) |
66 | 65 | adantl 482 |
. . . . . . . 8
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)) |
67 | | npncan3 10319 |
. . . . . . . 8
⊢ ((𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝐿 − 𝑀) + (𝑁 − 𝐿)) = (𝑁 − 𝑀)) |
68 | 66, 67 | syl 17 |
. . . . . . 7
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐿 − 𝑀) + (𝑁 − 𝐿)) = (𝑁 − 𝑀)) |
69 | 50, 68 | eqtr2d 2657 |
. . . . . 6
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝑁 − 𝑀) = ((#‘(𝐴 substr 〈𝑀, 𝐿〉)) + (#‘(𝐵 substr 〈0, (𝑁 − 𝐿)〉)))) |
70 | 69 | oveq2d 6666 |
. . . . 5
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (0..^(𝑁 − 𝑀)) = (0..^((#‘(𝐴 substr 〈𝑀, 𝐿〉)) + (#‘(𝐵 substr 〈0, (𝑁 − 𝐿)〉))))) |
71 | 70 | fneq2d 5982 |
. . . 4
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 substr 〈0, (𝑁 − 𝐿)〉)) Fn (0..^(𝑁 − 𝑀)) ↔ ((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 substr 〈0, (𝑁 − 𝐿)〉)) Fn (0..^((#‘(𝐴 substr 〈𝑀, 𝐿〉)) + (#‘(𝐵 substr 〈0, (𝑁 − 𝐿)〉)))))) |
72 | 26, 71 | mpbird 247 |
. . 3
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 substr 〈0, (𝑁 − 𝐿)〉)) Fn (0..^(𝑁 − 𝑀))) |
73 | | simprl 794 |
. . . . . 6
⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))))) |
74 | | simpr 477 |
. . . . . . . 8
⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → 𝑘 ∈ (0..^(𝑁 − 𝑀))) |
75 | 74 | anim2i 593 |
. . . . . . 7
⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) |
76 | 75 | ancomd 467 |
. . . . . 6
⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (𝑘 ∈ (0..^(𝑁 − 𝑀)) ∧ 𝑘 ∈ (0..^(𝐿 − 𝑀)))) |
77 | 10 | swrdccatin12lem3 13490 |
. . . . . 6
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝑘 ∈ (0..^(𝑁 − 𝑀)) ∧ 𝑘 ∈ (0..^(𝐿 − 𝑀))) → (((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉)‘𝑘) = ((𝐴 substr 〈𝑀, 𝐿〉)‘𝑘))) |
78 | 73, 76, 77 | sylc 65 |
. . . . 5
⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉)‘𝑘) = ((𝐴 substr 〈𝑀, 𝐿〉)‘𝑘)) |
79 | 24 | ad2antrl 764 |
. . . . . . 7
⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → ((𝐴 substr 〈𝑀, 𝐿〉) ∈ Word 𝑉 ∧ (𝐵 substr 〈0, (𝑁 − 𝐿)〉) ∈ Word 𝑉)) |
80 | | simpl 473 |
. . . . . . . 8
⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → 𝑘 ∈ (0..^(𝐿 − 𝑀))) |
81 | | nn0fz0 12437 |
. . . . . . . . . . . . . . . 16
⊢
((#‘𝐴) ∈
ℕ0 ↔ (#‘𝐴) ∈ (0...(#‘𝐴))) |
82 | 29, 81 | sylib 208 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ (0...(#‘𝐴))) |
83 | 10, 82 | syl5eqel 2705 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ Word 𝑉 → 𝐿 ∈ (0...(#‘𝐴))) |
84 | 83 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝐿 ∈ (0...(#‘𝐴))) |
85 | 84 | adantr 481 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐿 ∈ (0...(#‘𝐴))) |
86 | 27, 28, 85 | 3jca 1242 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝐴)))) |
87 | 86 | ad2antrl 764 |
. . . . . . . . . 10
⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝐴)))) |
88 | 87, 37 | syl 17 |
. . . . . . . . 9
⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (#‘(𝐴 substr 〈𝑀, 𝐿〉)) = (𝐿 − 𝑀)) |
89 | 88 | oveq2d 6666 |
. . . . . . . 8
⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (0..^(#‘(𝐴 substr 〈𝑀, 𝐿〉))) = (0..^(𝐿 − 𝑀))) |
90 | 80, 89 | eleqtrrd 2704 |
. . . . . . 7
⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → 𝑘 ∈ (0..^(#‘(𝐴 substr 〈𝑀, 𝐿〉)))) |
91 | | df-3an 1039 |
. . . . . . 7
⊢ (((𝐴 substr 〈𝑀, 𝐿〉) ∈ Word 𝑉 ∧ (𝐵 substr 〈0, (𝑁 − 𝐿)〉) ∈ Word 𝑉 ∧ 𝑘 ∈ (0..^(#‘(𝐴 substr 〈𝑀, 𝐿〉)))) ↔ (((𝐴 substr 〈𝑀, 𝐿〉) ∈ Word 𝑉 ∧ (𝐵 substr 〈0, (𝑁 − 𝐿)〉) ∈ Word 𝑉) ∧ 𝑘 ∈ (0..^(#‘(𝐴 substr 〈𝑀, 𝐿〉))))) |
92 | 79, 90, 91 | sylanbrc 698 |
. . . . . 6
⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → ((𝐴 substr 〈𝑀, 𝐿〉) ∈ Word 𝑉 ∧ (𝐵 substr 〈0, (𝑁 − 𝐿)〉) ∈ Word 𝑉 ∧ 𝑘 ∈ (0..^(#‘(𝐴 substr 〈𝑀, 𝐿〉))))) |
93 | | ccatval1 13361 |
. . . . . 6
⊢ (((𝐴 substr 〈𝑀, 𝐿〉) ∈ Word 𝑉 ∧ (𝐵 substr 〈0, (𝑁 − 𝐿)〉) ∈ Word 𝑉 ∧ 𝑘 ∈ (0..^(#‘(𝐴 substr 〈𝑀, 𝐿〉)))) → (((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 substr 〈0, (𝑁 − 𝐿)〉))‘𝑘) = ((𝐴 substr 〈𝑀, 𝐿〉)‘𝑘)) |
94 | 92, 93 | syl 17 |
. . . . 5
⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 substr 〈0, (𝑁 − 𝐿)〉))‘𝑘) = ((𝐴 substr 〈𝑀, 𝐿〉)‘𝑘)) |
95 | 78, 94 | eqtr4d 2659 |
. . . 4
⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉)‘𝑘) = (((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 substr 〈0, (𝑁 − 𝐿)〉))‘𝑘)) |
96 | | simprl 794 |
. . . . . 6
⊢ ((¬
𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))))) |
97 | 74 | anim2i 593 |
. . . . . . 7
⊢ ((¬
𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (¬ 𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) |
98 | 97 | ancomd 467 |
. . . . . 6
⊢ ((¬
𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (𝑘 ∈ (0..^(𝑁 − 𝑀)) ∧ ¬ 𝑘 ∈ (0..^(𝐿 − 𝑀)))) |
99 | 10 | swrdccatin12lem2 13489 |
. . . . . 6
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝑘 ∈ (0..^(𝑁 − 𝑀)) ∧ ¬ 𝑘 ∈ (0..^(𝐿 − 𝑀))) → (((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉)‘𝑘) = ((𝐵 substr 〈0, (𝑁 − 𝐿)〉)‘(𝑘 − (#‘(𝐴 substr 〈𝑀, 𝐿〉)))))) |
100 | 96, 98, 99 | sylc 65 |
. . . . 5
⊢ ((¬
𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉)‘𝑘) = ((𝐵 substr 〈0, (𝑁 − 𝐿)〉)‘(𝑘 − (#‘(𝐴 substr 〈𝑀, 𝐿〉))))) |
101 | 24 | ad2antrl 764 |
. . . . . . 7
⊢ ((¬
𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → ((𝐴 substr 〈𝑀, 𝐿〉) ∈ Word 𝑉 ∧ (𝐵 substr 〈0, (𝑁 − 𝐿)〉) ∈ Word 𝑉)) |
102 | | elfzuz 12338 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → 𝑁 ∈ (ℤ≥‘𝐿)) |
103 | | eluzelz 11697 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈
(ℤ≥‘𝐿) → 𝑁 ∈ ℤ) |
104 | | simpll 790 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐿 ∈ ℕ0
∧ 𝑀 ∈
ℕ0) ∧ 𝑁 ∈ ℤ) → 𝐿 ∈
ℕ0) |
105 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐿 ∈ ℕ0
∧ 𝑀 ∈
ℕ0) → 𝑀 ∈
ℕ0) |
106 | 105 | adantr 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐿 ∈ ℕ0
∧ 𝑀 ∈
ℕ0) ∧ 𝑁 ∈ ℤ) → 𝑀 ∈
ℕ0) |
107 | | simpr 477 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐿 ∈ ℕ0
∧ 𝑀 ∈
ℕ0) ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ) |
108 | 104, 106,
107 | 3jca 1242 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐿 ∈ ℕ0
∧ 𝑀 ∈
ℕ0) ∧ 𝑁 ∈ ℤ) → (𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℤ)) |
109 | 108 | ex 450 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐿 ∈ ℕ0
∧ 𝑀 ∈
ℕ0) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℤ))) |
110 | 109 | ancoms 469 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑀 ∈ ℕ0
∧ 𝐿 ∈
ℕ0) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℤ))) |
111 | 110 | 3adant3 1081 |
. . . . . . . . . . . . . . 15
⊢ ((𝑀 ∈ ℕ0
∧ 𝐿 ∈
ℕ0 ∧ 𝑀
≤ 𝐿) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0
∧ 𝑀 ∈
ℕ0 ∧ 𝑁
∈ ℤ))) |
112 | 51, 111 | sylbi 207 |
. . . . . . . . . . . . . 14
⊢ (𝑀 ∈ (0...𝐿) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℤ))) |
113 | 103, 112 | syl5com 31 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈
(ℤ≥‘𝐿) → (𝑀 ∈ (0...𝐿) → (𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℤ))) |
114 | 102, 113 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝑀 ∈ (0...𝐿) → (𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℤ))) |
115 | 114 | impcom 446 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → (𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℤ)) |
116 | 115 | adantl 482 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℤ)) |
117 | 116 | ad2antrl 764 |
. . . . . . . . 9
⊢ ((¬
𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℤ)) |
118 | | swrdccatin12lem1 13484 |
. . . . . . . . 9
⊢ ((𝐿 ∈ ℕ0
∧ 𝑀 ∈
ℕ0 ∧ 𝑁
∈ ℤ) → ((𝑘
∈ (0..^(𝑁 −
𝑀)) ∧ ¬ 𝑘 ∈ (0..^(𝐿 − 𝑀))) → 𝑘 ∈ ((𝐿 − 𝑀)..^((𝐿 − 𝑀) + (𝑁 − 𝐿))))) |
119 | 117, 98, 118 | sylc 65 |
. . . . . . . 8
⊢ ((¬
𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → 𝑘 ∈ ((𝐿 − 𝑀)..^((𝐿 − 𝑀) + (𝑁 − 𝐿)))) |
120 | 27, 28, 85, 37 | syl3anc 1326 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (#‘(𝐴 substr 〈𝑀, 𝐿〉)) = (𝐿 − 𝑀)) |
121 | 39 | adantl 482 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → 𝐵 ∈ Word 𝑉) |
122 | 43 | adantl 482 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → (#‘𝐵) ∈ ℤ) |
123 | | simpl 473 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) |
124 | 122, 123,
46 | syl2anc 693 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → (𝑁 − 𝐿) ∈ (0...(#‘𝐵))) |
125 | 121, 124 | jca 554 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → (𝐵 ∈ Word 𝑉 ∧ (𝑁 − 𝐿) ∈ (0...(#‘𝐵)))) |
126 | 125 | ex 450 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐵 ∈ Word 𝑉 ∧ (𝑁 − 𝐿) ∈ (0...(#‘𝐵))))) |
127 | 126 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐵 ∈ Word 𝑉 ∧ (𝑁 − 𝐿) ∈ (0...(#‘𝐵))))) |
128 | 127 | impcom 446 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐵 ∈ Word 𝑉 ∧ (𝑁 − 𝐿) ∈ (0...(#‘𝐵)))) |
129 | 128, 48 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (#‘(𝐵 substr 〈0, (𝑁 − 𝐿)〉)) = (𝑁 − 𝐿)) |
130 | 120, 129 | oveq12d 6668 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((#‘(𝐴 substr 〈𝑀, 𝐿〉)) + (#‘(𝐵 substr 〈0, (𝑁 − 𝐿)〉))) = ((𝐿 − 𝑀) + (𝑁 − 𝐿))) |
131 | 120, 130 | oveq12d 6668 |
. . . . . . . . 9
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((#‘(𝐴 substr 〈𝑀, 𝐿〉))..^((#‘(𝐴 substr 〈𝑀, 𝐿〉)) + (#‘(𝐵 substr 〈0, (𝑁 − 𝐿)〉)))) = ((𝐿 − 𝑀)..^((𝐿 − 𝑀) + (𝑁 − 𝐿)))) |
132 | 131 | ad2antrl 764 |
. . . . . . . 8
⊢ ((¬
𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → ((#‘(𝐴 substr 〈𝑀, 𝐿〉))..^((#‘(𝐴 substr 〈𝑀, 𝐿〉)) + (#‘(𝐵 substr 〈0, (𝑁 − 𝐿)〉)))) = ((𝐿 − 𝑀)..^((𝐿 − 𝑀) + (𝑁 − 𝐿)))) |
133 | 119, 132 | eleqtrrd 2704 |
. . . . . . 7
⊢ ((¬
𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → 𝑘 ∈ ((#‘(𝐴 substr 〈𝑀, 𝐿〉))..^((#‘(𝐴 substr 〈𝑀, 𝐿〉)) + (#‘(𝐵 substr 〈0, (𝑁 − 𝐿)〉))))) |
134 | | df-3an 1039 |
. . . . . . 7
⊢ (((𝐴 substr 〈𝑀, 𝐿〉) ∈ Word 𝑉 ∧ (𝐵 substr 〈0, (𝑁 − 𝐿)〉) ∈ Word 𝑉 ∧ 𝑘 ∈ ((#‘(𝐴 substr 〈𝑀, 𝐿〉))..^((#‘(𝐴 substr 〈𝑀, 𝐿〉)) + (#‘(𝐵 substr 〈0, (𝑁 − 𝐿)〉))))) ↔ (((𝐴 substr 〈𝑀, 𝐿〉) ∈ Word 𝑉 ∧ (𝐵 substr 〈0, (𝑁 − 𝐿)〉) ∈ Word 𝑉) ∧ 𝑘 ∈ ((#‘(𝐴 substr 〈𝑀, 𝐿〉))..^((#‘(𝐴 substr 〈𝑀, 𝐿〉)) + (#‘(𝐵 substr 〈0, (𝑁 − 𝐿)〉)))))) |
135 | 101, 133,
134 | sylanbrc 698 |
. . . . . 6
⊢ ((¬
𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → ((𝐴 substr 〈𝑀, 𝐿〉) ∈ Word 𝑉 ∧ (𝐵 substr 〈0, (𝑁 − 𝐿)〉) ∈ Word 𝑉 ∧ 𝑘 ∈ ((#‘(𝐴 substr 〈𝑀, 𝐿〉))..^((#‘(𝐴 substr 〈𝑀, 𝐿〉)) + (#‘(𝐵 substr 〈0, (𝑁 − 𝐿)〉)))))) |
136 | | ccatval2 13362 |
. . . . . 6
⊢ (((𝐴 substr 〈𝑀, 𝐿〉) ∈ Word 𝑉 ∧ (𝐵 substr 〈0, (𝑁 − 𝐿)〉) ∈ Word 𝑉 ∧ 𝑘 ∈ ((#‘(𝐴 substr 〈𝑀, 𝐿〉))..^((#‘(𝐴 substr 〈𝑀, 𝐿〉)) + (#‘(𝐵 substr 〈0, (𝑁 − 𝐿)〉))))) → (((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 substr 〈0, (𝑁 − 𝐿)〉))‘𝑘) = ((𝐵 substr 〈0, (𝑁 − 𝐿)〉)‘(𝑘 − (#‘(𝐴 substr 〈𝑀, 𝐿〉))))) |
137 | 135, 136 | syl 17 |
. . . . 5
⊢ ((¬
𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 substr 〈0, (𝑁 − 𝐿)〉))‘𝑘) = ((𝐵 substr 〈0, (𝑁 − 𝐿)〉)‘(𝑘 − (#‘(𝐴 substr 〈𝑀, 𝐿〉))))) |
138 | 100, 137 | eqtr4d 2659 |
. . . 4
⊢ ((¬
𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉)‘𝑘) = (((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 substr 〈0, (𝑁 − 𝐿)〉))‘𝑘)) |
139 | 95, 138 | pm2.61ian 831 |
. . 3
⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉)‘𝑘) = (((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 substr 〈0, (𝑁 − 𝐿)〉))‘𝑘)) |
140 | 20, 72, 139 | eqfnfvd 6314 |
. 2
⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = ((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 substr 〈0, (𝑁 − 𝐿)〉))) |
141 | 140 | ex 450 |
1
⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr 〈𝑀, 𝑁〉) = ((𝐴 substr 〈𝑀, 𝐿〉) ++ (𝐵 substr 〈0, (𝑁 − 𝐿)〉)))) |