Proof of Theorem 2cshwcshw
Step | Hyp | Ref
| Expression |
1 | | difelfznle 12322 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐾 ∈ (0...𝑁) ∧ 𝑚 ∈ (0...𝑁) ∧ ¬ 𝐾 ≤ 𝑚) → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁)) |
2 | 1 | 3exp 1256 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (¬ 𝐾 ≤ 𝑚 → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁)))) |
3 | 2 | ad2antrr 758 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑚 ∈ (0...𝑁) → (¬ 𝐾 ≤ 𝑚 → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁)))) |
4 | 3 | imp 444 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) → (¬ 𝐾 ≤ 𝑚 → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁))) |
5 | 4 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐾 ∈
(0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → (¬ 𝐾 ≤ 𝑚 → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁))) |
6 | 5 | com12 32 |
. . . . . . . . . . . . . 14
⊢ (¬
𝐾 ≤ 𝑚 → (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁))) |
7 | 6 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((¬
𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚) → (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁))) |
8 | 7 | imp 444 |
. . . . . . . . . . . 12
⊢ (((¬
𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚) ∧ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚))) → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁)) |
9 | | simprl 790 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → 𝑌 ∈ Word 𝑉) |
10 | 9 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → 𝑌 ∈ Word 𝑉) |
11 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℤ) |
12 | 11 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → 𝐾 ∈ ℤ) |
13 | 12 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → 𝐾 ∈ ℤ) |
14 | | elfz2 12204 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝐾 ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (0 ≤
𝐾 ∧ 𝐾 ≤ 𝑁))) |
15 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℤ) |
16 | | zaddcl 11294 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑚 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑚 + 𝑁) ∈ ℤ) |
17 | 16 | adantrr 749 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑚 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝑚 + 𝑁) ∈ ℤ) |
18 | | simprr 792 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑚 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → 𝐾 ∈
ℤ) |
19 | 17, 18 | zsubcld 11363 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑚 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ) |
20 | 19 | ex 449 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑚 ∈ ℤ → ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ)) |
21 | 15, 20 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑚 ∈ (0...𝑁) → ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ)) |
22 | 21 | com12 32 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑚 ∈ (0...𝑁) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ)) |
23 | 22 | 3adant1 1072 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((0
∈ ℤ ∧ 𝑁
∈ ℤ ∧ 𝐾
∈ ℤ) → (𝑚
∈ (0...𝑁) →
((𝑚 + 𝑁) − 𝐾) ∈ ℤ)) |
24 | 23 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((0
∈ ℤ ∧ 𝑁
∈ ℤ ∧ 𝐾
∈ ℤ) ∧ (0 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → (𝑚 ∈ (0...𝑁) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ)) |
25 | 14, 24 | sylbi 206 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ)) |
26 | 25 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) → (𝑚 ∈ (0...𝑁) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ)) |
27 | 26 | imp 444 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ) |
28 | | 2cshw 13410 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑌 ∈ Word 𝑉 ∧ 𝐾 ∈ ℤ ∧ ((𝑚 + 𝑁) − 𝐾) ∈ ℤ) → ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾)) = (𝑌 cyclShift (𝐾 + ((𝑚 + 𝑁) − 𝐾)))) |
29 | 10, 13, 27, 28 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾)) = (𝑌 cyclShift (𝐾 + ((𝑚 + 𝑁) − 𝐾)))) |
30 | 18, 19 | zaddcld 11362 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑚 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ) |
31 | 30 | ex 449 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑚 ∈ ℤ → ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ)) |
32 | 15, 31 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑚 ∈ (0...𝑁) → ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ)) |
33 | 32 | com12 32 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑚 ∈ (0...𝑁) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ)) |
34 | 33 | 3adant1 1072 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((0
∈ ℤ ∧ 𝑁
∈ ℤ ∧ 𝐾
∈ ℤ) → (𝑚
∈ (0...𝑁) →
(𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ)) |
35 | 34 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((0
∈ ℤ ∧ 𝑁
∈ ℤ ∧ 𝐾
∈ ℤ) ∧ (0 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → (𝑚 ∈ (0...𝑁) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ)) |
36 | 14, 35 | sylbi 206 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ)) |
37 | 36 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) → (𝑚 ∈ (0...𝑁) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ)) |
38 | 37 | imp 444 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ) |
39 | | cshwsublen 13393 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑌 ∈ Word 𝑉 ∧ (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ) → (𝑌 cyclShift (𝐾 + ((𝑚 + 𝑁) − 𝐾))) = (𝑌 cyclShift ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (#‘𝑌)))) |
40 | 10, 38, 39 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → (𝑌 cyclShift (𝐾 + ((𝑚 + 𝑁) − 𝐾))) = (𝑌 cyclShift ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (#‘𝑌)))) |
41 | 29, 40 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾)) = (𝑌 cyclShift ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (#‘𝑌)))) |
42 | | elfz2nn0 12300 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0
∧ 𝐾 ≤ 𝑁)) |
43 | | elfznn0 12302 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℕ0) |
44 | | nn0cn 11179 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑚 ∈ ℕ0
→ 𝑚 ∈
ℂ) |
45 | | nn0cn 11179 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝐾 ∈ ℕ0
→ 𝐾 ∈
ℂ) |
46 | | nn0cn 11179 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℂ) |
47 | 45, 46 | anim12i 588 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝐾 ∈ ℕ0
∧ 𝑁 ∈
ℕ0) → (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) |
48 | | simprl 790 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → 𝐾 ∈
ℂ) |
49 | | addcl 9897 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑚 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑚 + 𝑁) ∈ ℂ) |
50 | 49 | adantrl 748 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → (𝑚 + 𝑁) ∈ ℂ) |
51 | 48, 50 | pncan3d 10274 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) = (𝑚 + 𝑁)) |
52 | 51 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = ((𝑚 + 𝑁) − 𝑁)) |
53 | | pncan 10166 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((𝑚 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑚 + 𝑁) − 𝑁) = 𝑚) |
54 | 53 | adantrl 748 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → ((𝑚 + 𝑁) − 𝑁) = 𝑚) |
55 | 52, 54 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚) |
56 | 44, 47, 55 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑚 ∈ ℕ0
∧ (𝐾 ∈
ℕ0 ∧ 𝑁
∈ ℕ0)) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚) |
57 | 56 | ex 449 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑚 ∈ ℕ0
→ ((𝐾 ∈
ℕ0 ∧ 𝑁
∈ ℕ0) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)) |
58 | 43, 57 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑚 ∈ (0...𝑁) → ((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)
→ ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)) |
59 | 58 | com12 32 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝐾 ∈ ℕ0
∧ 𝑁 ∈
ℕ0) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)) |
60 | 59 | 3adant3 1074 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝐾 ∈ ℕ0
∧ 𝑁 ∈
ℕ0 ∧ 𝐾
≤ 𝑁) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)) |
61 | 42, 60 | sylbi 206 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)) |
62 | 61 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)) |
63 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
((#‘𝑌) = 𝑁 → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (#‘𝑌)) = ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁)) |
64 | 63 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
((#‘𝑌) = 𝑁 → (((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (#‘𝑌)) = 𝑚 ↔ ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)) |
65 | 64 | imbi2d 329 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
((#‘𝑌) = 𝑁 → ((𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (#‘𝑌)) = 𝑚) ↔ (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚))) |
66 | 65 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → ((𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (#‘𝑌)) = 𝑚) ↔ (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚))) |
67 | 66 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → ((𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (#‘𝑌)) = 𝑚) ↔ (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚))) |
68 | 62, 67 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (#‘𝑌)) = 𝑚)) |
69 | 68 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (#‘𝑌)) = 𝑚)) |
70 | 69 | imp 444 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (#‘𝑌)) = 𝑚) |
71 | 70 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → (𝑌 cyclShift ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (#‘𝑌))) = (𝑌 cyclShift 𝑚)) |
72 | 41, 71 | eqtr2d 2645 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾))) |
73 | 72 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝐾 ∈
(0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾))) |
74 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑋 = (𝑌 cyclShift 𝐾) → (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)) = ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾))) |
75 | 74 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝐾 ∈
(0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)) = ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾))) |
76 | 73, 75 | eqtr4d 2647 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝐾 ∈
(0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑌 cyclShift 𝑚) = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾))) |
77 | 76 | exp41 636 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → ((¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚) → (𝑚 ∈ (0...𝑁) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑌 cyclShift 𝑚) = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)))))) |
78 | 77 | com24 93 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑚 ∈ (0...𝑁) → ((¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚) → (𝑌 cyclShift 𝑚) = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)))))) |
79 | 78 | imp41 617 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝐾 ∈
(0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) → (𝑌 cyclShift 𝑚) = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾))) |
80 | 79 | eqeq2d 2620 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐾 ∈
(0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) → (𝑍 = (𝑌 cyclShift 𝑚) ↔ 𝑍 = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)))) |
81 | 80 | biimpd 218 |
. . . . . . . . . . . . . 14
⊢
(((((𝐾 ∈
(0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) → (𝑍 = (𝑌 cyclShift 𝑚) → 𝑍 = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)))) |
82 | 81 | impancom 455 |
. . . . . . . . . . . . 13
⊢
(((((𝐾 ∈
(0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → ((¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚) → 𝑍 = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)))) |
83 | 82 | impcom 445 |
. . . . . . . . . . . 12
⊢ (((¬
𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚) ∧ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚))) → 𝑍 = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾))) |
84 | | oveq2 6557 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = ((𝑚 + 𝑁) − 𝐾) → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾))) |
85 | 84 | eqeq2d 2620 |
. . . . . . . . . . . . 13
⊢ (𝑛 = ((𝑚 + 𝑁) − 𝐾) → (𝑍 = (𝑋 cyclShift 𝑛) ↔ 𝑍 = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)))) |
86 | 85 | rspcev 3282 |
. . . . . . . . . . . 12
⊢ ((((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁) ∧ 𝑍 = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾))) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)) |
87 | 8, 83, 86 | syl2anc 691 |
. . . . . . . . . . 11
⊢ (((¬
𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚) ∧ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚))) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)) |
88 | 87 | exp31 628 |
. . . . . . . . . 10
⊢ (¬
𝑚 = 0 → (¬ 𝐾 ≤ 𝑚 → (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))) |
89 | | oveq2 6557 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = 0 → (𝑌 cyclShift 𝑚) = (𝑌 cyclShift 0)) |
90 | 89 | eqeq2d 2620 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = 0 → (𝑍 = (𝑌 cyclShift 𝑚) ↔ 𝑍 = (𝑌 cyclShift 0))) |
91 | | cshw0 13391 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑌 ∈ Word 𝑉 → (𝑌 cyclShift 0) = 𝑌) |
92 | 91 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → (𝑌 cyclShift 0) = 𝑌) |
93 | 92 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → (𝑍 = (𝑌 cyclShift 0) ↔ 𝑍 = 𝑌)) |
94 | | fznn0sub2 12315 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ (0...𝑁)) |
95 | 94 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) ∈ (0...𝑁)) |
96 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
((#‘𝑌) = 𝑁 → ((#‘𝑌) − 𝐾) = (𝑁 − 𝐾)) |
97 | 96 | eleq1d 2672 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
((#‘𝑌) = 𝑁 → (((#‘𝑌) − 𝐾) ∈ (0...𝑁) ↔ (𝑁 − 𝐾) ∈ (0...𝑁))) |
98 | 97 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) → (((#‘𝑌) − 𝐾) ∈ (0...𝑁) ↔ (𝑁 − 𝐾) ∈ (0...𝑁))) |
99 | 95, 98 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) → ((#‘𝑌) − 𝐾) ∈ (0...𝑁)) |
100 | 99 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → ((#‘𝑌) − 𝐾) ∈ (0...𝑁)) |
101 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑋 = (𝑌 cyclShift 𝐾) → (𝑋 cyclShift ((#‘𝑌) − 𝐾)) = ((𝑌 cyclShift 𝐾) cyclShift ((#‘𝑌) − 𝐾))) |
102 | | simpl 472 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → 𝑌 ∈ Word 𝑉) |
103 | | 2cshwid 13411 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑌 ∈ Word 𝑉 ∧ 𝐾 ∈ ℤ) → ((𝑌 cyclShift 𝐾) cyclShift ((#‘𝑌) − 𝐾)) = 𝑌) |
104 | 102, 11, 103 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) → ((𝑌 cyclShift 𝐾) cyclShift ((#‘𝑌) − 𝐾)) = 𝑌) |
105 | 101, 104 | sylan9eqr 2666 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑋 cyclShift ((#‘𝑌) − 𝐾)) = 𝑌) |
106 | 105 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → 𝑌 = (𝑋 cyclShift ((#‘𝑌) − 𝐾))) |
107 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑛 = ((#‘𝑌) − 𝐾) → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift ((#‘𝑌) − 𝐾))) |
108 | 107 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑛 = ((#‘𝑌) − 𝐾) → (𝑌 = (𝑋 cyclShift 𝑛) ↔ 𝑌 = (𝑋 cyclShift ((#‘𝑌) − 𝐾)))) |
109 | 108 | rspcev 3282 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((((#‘𝑌)
− 𝐾) ∈
(0...𝑁) ∧ 𝑌 = (𝑋 cyclShift ((#‘𝑌) − 𝐾))) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) |
110 | 100, 106,
109 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) |
111 | 110 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝑌 ∈ Word
𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑍 = 𝑌) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) |
112 | | eqeq1 2614 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑍 = 𝑌 → (𝑍 = (𝑋 cyclShift 𝑛) ↔ 𝑌 = (𝑋 cyclShift 𝑛))) |
113 | 112 | rexbidv 3034 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑍 = 𝑌 → (∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))) |
114 | 113 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝑌 ∈ Word
𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑍 = 𝑌) → (∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))) |
115 | 111, 114 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑌 ∈ Word
𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑍 = 𝑌) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)) |
116 | 115 | exp41 636 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → (𝐾 ∈ (0...𝑁) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑍 = 𝑌 → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))))) |
117 | 116 | com24 93 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → (𝑍 = 𝑌 → (𝑋 = (𝑌 cyclShift 𝐾) → (𝐾 ∈ (0...𝑁) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))))) |
118 | 93, 117 | sylbid 229 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → (𝑍 = (𝑌 cyclShift 0) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝐾 ∈ (0...𝑁) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))))) |
119 | 118 | com24 93 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → (𝐾 ∈ (0...𝑁) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑍 = (𝑌 cyclShift 0) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))))) |
120 | 119 | impcom 445 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑍 = (𝑌 cyclShift 0) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))) |
121 | 120 | com13 86 |
. . . . . . . . . . . . . . . 16
⊢ (𝑍 = (𝑌 cyclShift 0) → (𝑋 = (𝑌 cyclShift 𝐾) → ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))) |
122 | 121 | a1d 25 |
. . . . . . . . . . . . . . 15
⊢ (𝑍 = (𝑌 cyclShift 0) → (𝑚 ∈ (0...𝑁) → (𝑋 = (𝑌 cyclShift 𝐾) → ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))))) |
123 | 90, 122 | syl6bi 242 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = 0 → (𝑍 = (𝑌 cyclShift 𝑚) → (𝑚 ∈ (0...𝑁) → (𝑋 = (𝑌 cyclShift 𝐾) → ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))))) |
124 | 123 | com24 93 |
. . . . . . . . . . . . 13
⊢ (𝑚 = 0 → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑚 ∈ (0...𝑁) → (𝑍 = (𝑌 cyclShift 𝑚) → ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))))) |
125 | 124 | com15 99 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑚 ∈ (0...𝑁) → (𝑍 = (𝑌 cyclShift 𝑚) → (𝑚 = 0 → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))))) |
126 | 125 | imp41 617 |
. . . . . . . . . . 11
⊢
(((((𝐾 ∈
(0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → (𝑚 = 0 → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))) |
127 | 126 | com12 32 |
. . . . . . . . . 10
⊢ (𝑚 = 0 → (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))) |
128 | | difelfzle 12321 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐾 ∈ (0...𝑁) ∧ 𝑚 ∈ (0...𝑁) ∧ 𝐾 ≤ 𝑚) → (𝑚 − 𝐾) ∈ (0...𝑁)) |
129 | 128 | 3exp 1256 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (𝐾 ≤ 𝑚 → (𝑚 − 𝐾) ∈ (0...𝑁)))) |
130 | 129 | ad2antrr 758 |
. . . . . . . . . . . . . . 15
⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑚 ∈ (0...𝑁) → (𝐾 ≤ 𝑚 → (𝑚 − 𝐾) ∈ (0...𝑁)))) |
131 | 130 | imp 444 |
. . . . . . . . . . . . . 14
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) → (𝐾 ≤ 𝑚 → (𝑚 − 𝐾) ∈ (0...𝑁))) |
132 | 131 | adantr 480 |
. . . . . . . . . . . . 13
⊢
(((((𝐾 ∈
(0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → (𝐾 ≤ 𝑚 → (𝑚 − 𝐾) ∈ (0...𝑁))) |
133 | 132 | impcom 445 |
. . . . . . . . . . . 12
⊢ ((𝐾 ≤ 𝑚 ∧ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚))) → (𝑚 − 𝐾) ∈ (0...𝑁)) |
134 | 9 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → 𝑌 ∈ Word 𝑉) |
135 | 12 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → 𝐾 ∈ ℤ) |
136 | | zsubcl 11296 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑚 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑚 − 𝐾) ∈ ℤ) |
137 | 136 | ex 449 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑚 ∈ ℤ → (𝐾 ∈ ℤ → (𝑚 − 𝐾) ∈ ℤ)) |
138 | 15, 11, 137 | syl2imc 40 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (𝑚 − 𝐾) ∈ ℤ)) |
139 | 138 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) → (𝑚 ∈ (0...𝑁) → (𝑚 − 𝐾) ∈ ℤ)) |
140 | 139 | imp 444 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → (𝑚 − 𝐾) ∈ ℤ) |
141 | | 2cshw 13410 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑌 ∈ Word 𝑉 ∧ 𝐾 ∈ ℤ ∧ (𝑚 − 𝐾) ∈ ℤ) → ((𝑌 cyclShift 𝐾) cyclShift (𝑚 − 𝐾)) = (𝑌 cyclShift (𝐾 + (𝑚 − 𝐾)))) |
142 | 134, 135,
140, 141 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑌 cyclShift 𝐾) cyclShift (𝑚 − 𝐾)) = (𝑌 cyclShift (𝐾 + (𝑚 − 𝐾)))) |
143 | | zcn 11259 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝐾 ∈ ℤ → 𝐾 ∈
ℂ) |
144 | 15 | zcnd 11359 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℂ) |
145 | | pncan3 10168 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝐾 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝐾 + (𝑚 − 𝐾)) = 𝑚) |
146 | 143, 144,
145 | syl2anr 494 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑚 ∈ (0...𝑁) ∧ 𝐾 ∈ ℤ) → (𝐾 + (𝑚 − 𝐾)) = 𝑚) |
147 | 146 | ex 449 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑚 ∈ (0...𝑁) → (𝐾 ∈ ℤ → (𝐾 + (𝑚 − 𝐾)) = 𝑚)) |
148 | 11, 147 | syl5com 31 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (𝐾 + (𝑚 − 𝐾)) = 𝑚)) |
149 | 148 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) → (𝑚 ∈ (0...𝑁) → (𝐾 + (𝑚 − 𝐾)) = 𝑚)) |
150 | 149 | imp 444 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → (𝐾 + (𝑚 − 𝐾)) = 𝑚) |
151 | 150 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → (𝑌 cyclShift (𝐾 + (𝑚 − 𝐾))) = (𝑌 cyclShift 𝑚)) |
152 | 142, 151 | eqtr2d 2645 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift (𝑚 − 𝐾))) |
153 | 152 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝐾 ∈
(0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift (𝑚 − 𝐾))) |
154 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑋 = (𝑌 cyclShift 𝐾) → (𝑋 cyclShift (𝑚 − 𝐾)) = ((𝑌 cyclShift 𝐾) cyclShift (𝑚 − 𝐾))) |
155 | 154 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑋 = (𝑌 cyclShift 𝐾) → ((𝑌 cyclShift 𝑚) = (𝑋 cyclShift (𝑚 − 𝐾)) ↔ (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift (𝑚 − 𝐾)))) |
156 | 155 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝐾 ∈
(0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → ((𝑌 cyclShift 𝑚) = (𝑋 cyclShift (𝑚 − 𝐾)) ↔ (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift (𝑚 − 𝐾)))) |
157 | 153, 156 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝐾 ∈
(0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑌 cyclShift 𝑚) = (𝑋 cyclShift (𝑚 − 𝐾))) |
158 | 157 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝐾 ∈
(0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑍 = (𝑌 cyclShift 𝑚) ↔ 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾)))) |
159 | 158 | biimpd 218 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝐾 ∈
(0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑍 = (𝑌 cyclShift 𝑚) → 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾)))) |
160 | 159 | exp41 636 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → (𝐾 ≤ 𝑚 → (𝑚 ∈ (0...𝑁) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑍 = (𝑌 cyclShift 𝑚) → 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾))))))) |
161 | 160 | com24 93 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑚 ∈ (0...𝑁) → (𝐾 ≤ 𝑚 → (𝑍 = (𝑌 cyclShift 𝑚) → 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾))))))) |
162 | 161 | imp31 447 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) → (𝐾 ≤ 𝑚 → (𝑍 = (𝑌 cyclShift 𝑚) → 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾))))) |
163 | 162 | com23 84 |
. . . . . . . . . . . . . 14
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) → (𝑍 = (𝑌 cyclShift 𝑚) → (𝐾 ≤ 𝑚 → 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾))))) |
164 | 163 | imp 444 |
. . . . . . . . . . . . 13
⊢
(((((𝐾 ∈
(0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → (𝐾 ≤ 𝑚 → 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾)))) |
165 | 164 | impcom 445 |
. . . . . . . . . . . 12
⊢ ((𝐾 ≤ 𝑚 ∧ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚))) → 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾))) |
166 | | oveq2 6557 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = (𝑚 − 𝐾) → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift (𝑚 − 𝐾))) |
167 | 166 | eqeq2d 2620 |
. . . . . . . . . . . . 13
⊢ (𝑛 = (𝑚 − 𝐾) → (𝑍 = (𝑋 cyclShift 𝑛) ↔ 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾)))) |
168 | 167 | rspcev 3282 |
. . . . . . . . . . . 12
⊢ (((𝑚 − 𝐾) ∈ (0...𝑁) ∧ 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾))) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)) |
169 | 133, 165,
168 | syl2anc 691 |
. . . . . . . . . . 11
⊢ ((𝐾 ≤ 𝑚 ∧ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚))) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)) |
170 | 169 | ex 449 |
. . . . . . . . . 10
⊢ (𝐾 ≤ 𝑚 → (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))) |
171 | 88, 127, 170 | pm2.61ii 176 |
. . . . . . . . 9
⊢
(((((𝐾 ∈
(0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)) |
172 | 171 | ex 449 |
. . . . . . . 8
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) → (𝑍 = (𝑌 cyclShift 𝑚) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))) |
173 | 172 | rexlimdva 3013 |
. . . . . . 7
⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))) |
174 | 173 | ex 449 |
. . . . . 6
⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → (𝑋 = (𝑌 cyclShift 𝐾) → (∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))) |
175 | 174 | com23 84 |
. . . . 5
⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → (∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚) → (𝑋 = (𝑌 cyclShift 𝐾) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))) |
176 | 175 | ex 449 |
. . . 4
⊢ (𝐾 ∈ (0...𝑁) → ((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → (∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚) → (𝑋 = (𝑌 cyclShift 𝐾) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))))) |
177 | 176 | com24 93 |
. . 3
⊢ (𝐾 ∈ (0...𝑁) → (𝑋 = (𝑌 cyclShift 𝐾) → (∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚) → ((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))))) |
178 | 177 | 3imp 1249 |
. 2
⊢ ((𝐾 ∈ (0...𝑁) ∧ 𝑋 = (𝑌 cyclShift 𝐾) ∧ ∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚)) → ((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))) |
179 | 178 | com12 32 |
1
⊢ ((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → ((𝐾 ∈ (0...𝑁) ∧ 𝑋 = (𝑌 cyclShift 𝐾) ∧ ∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))) |