Proof of Theorem 2cshwcshw
Step | Hyp | Ref
| Expression |
1 | | difelfznle 12453 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐾 ∈ (0...𝑁) ∧ 𝑚 ∈ (0...𝑁) ∧ ¬ 𝐾 ≤ 𝑚) → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁)) |
2 | 1 | 3exp 1264 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (¬ 𝐾 ≤ 𝑚 → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁)))) |
3 | 2 | ad2antrr 762 |
. . . . . . . . . . . . . . . . 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 445 |
. . . . . . . . . . . . . . . 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 481 |
. . . . . . . . . . . . . . 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 482 |
. . . . . . . . . . . . 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 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))) → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁)) |
9 | | simprl 794 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → 𝑌 ∈ Word 𝑉) |
10 | 9 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → 𝑌 ∈ Word 𝑉) |
11 | | elfzelz 12342 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℤ) |
12 | 11 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → 𝐾 ∈ ℤ) |
13 | 12 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → 𝐾 ∈ ℤ) |
14 | | elfz2 12333 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝐾 ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (0 ≤
𝐾 ∧ 𝐾 ≤ 𝑁))) |
15 | | zaddcl 11417 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑚 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑚 + 𝑁) ∈ ℤ) |
16 | 15 | adantrr 753 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑚 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝑚 + 𝑁) ∈ ℤ) |
17 | | simprr 796 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑚 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → 𝐾 ∈
ℤ) |
18 | 16, 17 | zsubcld 11487 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑚 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ) |
19 | 18 | ex 450 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑚 ∈ ℤ → ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ)) |
20 | | elfzelz 12342 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℤ) |
21 | 19, 20 | syl11 33 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑚 ∈ (0...𝑁) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ)) |
22 | 21 | 3adant1 1079 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((0
∈ ℤ ∧ 𝑁
∈ ℤ ∧ 𝐾
∈ ℤ) → (𝑚
∈ (0...𝑁) →
((𝑚 + 𝑁) − 𝐾) ∈ ℤ)) |
23 | 22 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((0
∈ ℤ ∧ 𝑁
∈ ℤ ∧ 𝐾
∈ ℤ) ∧ (0 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → (𝑚 ∈ (0...𝑁) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ)) |
24 | 14, 23 | sylbi 207 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ)) |
25 | 24 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) → (𝑚 ∈ (0...𝑁) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ)) |
26 | 25 | imp 445 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ) |
27 | | 2cshw 13559 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑌 ∈ Word 𝑉 ∧ 𝐾 ∈ ℤ ∧ ((𝑚 + 𝑁) − 𝐾) ∈ ℤ) → ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾)) = (𝑌 cyclShift (𝐾 + ((𝑚 + 𝑁) − 𝐾)))) |
28 | 10, 13, 26, 27 | syl3anc 1326 |
. . . . . . . . . . . . . . . . . . . . . . 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)))) |
29 | 17, 18 | zaddcld 11486 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑚 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ) |
30 | 29 | ex 450 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑚 ∈ ℤ → ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ)) |
31 | 30, 20 | syl11 33 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑚 ∈ (0...𝑁) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ)) |
32 | 31 | 3adant1 1079 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((0
∈ ℤ ∧ 𝑁
∈ ℤ ∧ 𝐾
∈ ℤ) → (𝑚
∈ (0...𝑁) →
(𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ)) |
33 | 32 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((0
∈ ℤ ∧ 𝑁
∈ ℤ ∧ 𝐾
∈ ℤ) ∧ (0 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → (𝑚 ∈ (0...𝑁) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ)) |
34 | 14, 33 | sylbi 207 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ)) |
35 | 34 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) → (𝑚 ∈ (0...𝑁) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ)) |
36 | 35 | imp 445 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ) |
37 | | cshwsublen 13542 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑌 ∈ Word 𝑉 ∧ (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ) → (𝑌 cyclShift (𝐾 + ((𝑚 + 𝑁) − 𝐾))) = (𝑌 cyclShift ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (#‘𝑌)))) |
38 | 10, 36, 37 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . . . . . 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)))) |
39 | 28, 38 | eqtrd 2656 |
. . . . . . . . . . . . . . . . . . . . . 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)))) |
40 | | elfz2nn0 12431 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0
∧ 𝐾 ≤ 𝑁)) |
41 | | nn0cn 11302 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑚 ∈ ℕ0
→ 𝑚 ∈
ℂ) |
42 | | nn0cn 11302 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝐾 ∈ ℕ0
→ 𝐾 ∈
ℂ) |
43 | | nn0cn 11302 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℂ) |
44 | 42, 43 | anim12i 590 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝐾 ∈ ℕ0
∧ 𝑁 ∈
ℕ0) → (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) |
45 | | simprl 794 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → 𝐾 ∈
ℂ) |
46 | | addcl 10018 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝑚 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑚 + 𝑁) ∈ ℂ) |
47 | 46 | adantrl 752 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → (𝑚 + 𝑁) ∈ ℂ) |
48 | 45, 47 | pncan3d 10395 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) = (𝑚 + 𝑁)) |
49 | 48 | oveq1d 6665 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = ((𝑚 + 𝑁) − 𝑁)) |
50 | | pncan 10287 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑚 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑚 + 𝑁) − 𝑁) = 𝑚) |
51 | 50 | adantrl 752 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → ((𝑚 + 𝑁) − 𝑁) = 𝑚) |
52 | 49, 51 | eqtrd 2656 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚) |
53 | 41, 44, 52 | syl2an 494 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑚 ∈ ℕ0
∧ (𝐾 ∈
ℕ0 ∧ 𝑁
∈ ℕ0)) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚) |
54 | 53 | ex 450 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑚 ∈ ℕ0
→ ((𝐾 ∈
ℕ0 ∧ 𝑁
∈ ℕ0) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)) |
55 | | elfznn0 12433 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℕ0) |
56 | 54, 55 | syl11 33 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝐾 ∈ ℕ0
∧ 𝑁 ∈
ℕ0) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)) |
57 | 56 | 3adant3 1081 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝐾 ∈ ℕ0
∧ 𝑁 ∈
ℕ0 ∧ 𝐾
≤ 𝑁) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)) |
58 | 40, 57 | sylbi 207 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)) |
59 | 58 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)) |
60 | | oveq2 6658 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
((#‘𝑌) = 𝑁 → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (#‘𝑌)) = ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁)) |
61 | 60 | eqeq1d 2624 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
((#‘𝑌) = 𝑁 → (((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (#‘𝑌)) = 𝑚 ↔ ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)) |
62 | 61 | imbi2d 330 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
((#‘𝑌) = 𝑁 → ((𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (#‘𝑌)) = 𝑚) ↔ (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚))) |
63 | 62 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → ((𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (#‘𝑌)) = 𝑚) ↔ (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚))) |
64 | 63 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . 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))) |
65 | 59, 64 | mpbird 247 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (#‘𝑌)) = 𝑚)) |
66 | 65 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (#‘𝑌)) = 𝑚)) |
67 | 66 | imp 445 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (#‘𝑌)) = 𝑚) |
68 | 67 | oveq2d 6666 |
. . . . . . . . . . . . . . . . . . . . . 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)) |
69 | 39, 68 | eqtr2d 2657 |
. . . . . . . . . . . . . . . . . . . . 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))) |
70 | 69 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 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))) |
71 | | oveq1 6657 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑋 = (𝑌 cyclShift 𝐾) → (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)) = ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾))) |
72 | 71 | adantl 482 |
. . . . . . . . . . . . . . . . . . . 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))) |
73 | 70, 72 | eqtr4d 2659 |
. . . . . . . . . . . . . . . . . . 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))) |
74 | 73 | exp41 638 |
. . . . . . . . . . . . . . . . . 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)))))) |
75 | 74 | com24 95 |
. . . . . . . . . . . . . . . . 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)))))) |
76 | 75 | imp41 619 |
. . . . . . . . . . . . . . . 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))) |
77 | 76 | eqeq2d 2632 |
. . . . . . . . . . . . . . 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)))) |
78 | 77 | biimpd 219 |
. . . . . . . . . . . . . 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)))) |
79 | 78 | impancom 456 |
. . . . . . . . . . . . 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)))) |
80 | 79 | impcom 446 |
. . . . . . . . . . . 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))) |
81 | | oveq2 6658 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = ((𝑚 + 𝑁) − 𝐾) → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾))) |
82 | 81 | eqeq2d 2632 |
. . . . . . . . . . . . 13
⊢ (𝑛 = ((𝑚 + 𝑁) − 𝐾) → (𝑍 = (𝑋 cyclShift 𝑛) ↔ 𝑍 = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)))) |
83 | 82 | rspcev 3309 |
. . . . . . . . . . . 12
⊢ ((((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁) ∧ 𝑍 = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾))) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)) |
84 | 8, 80, 83 | syl2anc 693 |
. . . . . . . . . . 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)) |
85 | 84 | exp31 630 |
. . . . . . . . . 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)))) |
86 | | oveq2 6658 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = 0 → (𝑌 cyclShift 𝑚) = (𝑌 cyclShift 0)) |
87 | 86 | eqeq2d 2632 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = 0 → (𝑍 = (𝑌 cyclShift 𝑚) ↔ 𝑍 = (𝑌 cyclShift 0))) |
88 | | cshw0 13540 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑌 ∈ Word 𝑉 → (𝑌 cyclShift 0) = 𝑌) |
89 | 88 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → (𝑌 cyclShift 0) = 𝑌) |
90 | 89 | eqeq2d 2632 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → (𝑍 = (𝑌 cyclShift 0) ↔ 𝑍 = 𝑌)) |
91 | | fznn0sub2 12446 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ (0...𝑁)) |
92 | 91 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) ∈ (0...𝑁)) |
93 | | oveq1 6657 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
((#‘𝑌) = 𝑁 → ((#‘𝑌) − 𝐾) = (𝑁 − 𝐾)) |
94 | 93 | eleq1d 2686 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
((#‘𝑌) = 𝑁 → (((#‘𝑌) − 𝐾) ∈ (0...𝑁) ↔ (𝑁 − 𝐾) ∈ (0...𝑁))) |
95 | 94 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) → (((#‘𝑌) − 𝐾) ∈ (0...𝑁) ↔ (𝑁 − 𝐾) ∈ (0...𝑁))) |
96 | 92, 95 | mpbird 247 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) → ((#‘𝑌) − 𝐾) ∈ (0...𝑁)) |
97 | 96 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → ((#‘𝑌) − 𝐾) ∈ (0...𝑁)) |
98 | | oveq1 6657 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑋 = (𝑌 cyclShift 𝐾) → (𝑋 cyclShift ((#‘𝑌) − 𝐾)) = ((𝑌 cyclShift 𝐾) cyclShift ((#‘𝑌) − 𝐾))) |
99 | | simpl 473 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → 𝑌 ∈ Word 𝑉) |
100 | | 2cshwid 13560 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑌 ∈ Word 𝑉 ∧ 𝐾 ∈ ℤ) → ((𝑌 cyclShift 𝐾) cyclShift ((#‘𝑌) − 𝐾)) = 𝑌) |
101 | 99, 11, 100 | syl2an 494 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) → ((𝑌 cyclShift 𝐾) cyclShift ((#‘𝑌) − 𝐾)) = 𝑌) |
102 | 98, 101 | sylan9eqr 2678 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑋 cyclShift ((#‘𝑌) − 𝐾)) = 𝑌) |
103 | 102 | eqcomd 2628 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → 𝑌 = (𝑋 cyclShift ((#‘𝑌) − 𝐾))) |
104 | | oveq2 6658 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑛 = ((#‘𝑌) − 𝐾) → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift ((#‘𝑌) − 𝐾))) |
105 | 104 | eqeq2d 2632 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑛 = ((#‘𝑌) − 𝐾) → (𝑌 = (𝑋 cyclShift 𝑛) ↔ 𝑌 = (𝑋 cyclShift ((#‘𝑌) − 𝐾)))) |
106 | 105 | rspcev 3309 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((((#‘𝑌)
− 𝐾) ∈
(0...𝑁) ∧ 𝑌 = (𝑋 cyclShift ((#‘𝑌) − 𝐾))) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) |
107 | 97, 103, 106 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) |
108 | 107 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝑌 ∈ Word
𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑍 = 𝑌) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) |
109 | | eqeq1 2626 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑍 = 𝑌 → (𝑍 = (𝑋 cyclShift 𝑛) ↔ 𝑌 = (𝑋 cyclShift 𝑛))) |
110 | 109 | rexbidv 3052 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑍 = 𝑌 → (∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))) |
111 | 110 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . 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))) |
112 | 108, 111 | mpbird 247 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑌 ∈ Word
𝑉 ∧ (#‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑍 = 𝑌) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)) |
113 | 112 | exp41 638 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → (𝐾 ∈ (0...𝑁) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑍 = 𝑌 → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))))) |
114 | 113 | com24 95 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → (𝑍 = 𝑌 → (𝑋 = (𝑌 cyclShift 𝐾) → (𝐾 ∈ (0...𝑁) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))))) |
115 | 90, 114 | sylbid 230 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → (𝑍 = (𝑌 cyclShift 0) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝐾 ∈ (0...𝑁) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))))) |
116 | 115 | com24 95 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → (𝐾 ∈ (0...𝑁) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑍 = (𝑌 cyclShift 0) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))))) |
117 | 116 | impcom 446 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑍 = (𝑌 cyclShift 0) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))) |
118 | 117 | com13 88 |
. . . . . . . . . . . . . . . 16
⊢ (𝑍 = (𝑌 cyclShift 0) → (𝑋 = (𝑌 cyclShift 𝐾) → ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))) |
119 | 118 | 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))))) |
120 | 87, 119 | syl6bi 243 |
. . . . . . . . . . . . . 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)))))) |
121 | 120 | com24 95 |
. . . . . . . . . . . . 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)))))) |
122 | 121 | com15 101 |
. . . . . . . . . . . 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)))))) |
123 | 122 | imp41 619 |
. . . . . . . . . . 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))) |
124 | 123 | 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))) |
125 | | difelfzle 12452 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐾 ∈ (0...𝑁) ∧ 𝑚 ∈ (0...𝑁) ∧ 𝐾 ≤ 𝑚) → (𝑚 − 𝐾) ∈ (0...𝑁)) |
126 | 125 | 3exp 1264 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (𝐾 ≤ 𝑚 → (𝑚 − 𝐾) ∈ (0...𝑁)))) |
127 | 126 | ad2antrr 762 |
. . . . . . . . . . . . . . 15
⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑚 ∈ (0...𝑁) → (𝐾 ≤ 𝑚 → (𝑚 − 𝐾) ∈ (0...𝑁)))) |
128 | 127 | imp 445 |
. . . . . . . . . . . . . 14
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) → (𝐾 ≤ 𝑚 → (𝑚 − 𝐾) ∈ (0...𝑁))) |
129 | 128 | adantr 481 |
. . . . . . . . . . . . 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))) |
130 | 129 | impcom 446 |
. . . . . . . . . . . 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)) |
131 | 9 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → 𝑌 ∈ Word 𝑉) |
132 | 12 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → 𝐾 ∈ ℤ) |
133 | | zsubcl 11419 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑚 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑚 − 𝐾) ∈ ℤ) |
134 | 133 | ex 450 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑚 ∈ ℤ → (𝐾 ∈ ℤ → (𝑚 − 𝐾) ∈ ℤ)) |
135 | 20, 11, 134 | syl2imc 41 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (𝑚 − 𝐾) ∈ ℤ)) |
136 | 135 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) → (𝑚 ∈ (0...𝑁) → (𝑚 − 𝐾) ∈ ℤ)) |
137 | 136 | imp 445 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → (𝑚 − 𝐾) ∈ ℤ) |
138 | | 2cshw 13559 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑌 ∈ Word 𝑉 ∧ 𝐾 ∈ ℤ ∧ (𝑚 − 𝐾) ∈ ℤ) → ((𝑌 cyclShift 𝐾) cyclShift (𝑚 − 𝐾)) = (𝑌 cyclShift (𝐾 + (𝑚 − 𝐾)))) |
139 | 131, 132,
137, 138 | syl3anc 1326 |
. . . . . . . . . . . . . . . . . . . . . . 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)))) |
140 | | zcn 11382 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝐾 ∈ ℤ → 𝐾 ∈
ℂ) |
141 | 20 | zcnd 11483 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℂ) |
142 | | pncan3 10289 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝐾 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝐾 + (𝑚 − 𝐾)) = 𝑚) |
143 | 140, 141,
142 | syl2anr 495 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑚 ∈ (0...𝑁) ∧ 𝐾 ∈ ℤ) → (𝐾 + (𝑚 − 𝐾)) = 𝑚) |
144 | 143 | ex 450 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑚 ∈ (0...𝑁) → (𝐾 ∈ ℤ → (𝐾 + (𝑚 − 𝐾)) = 𝑚)) |
145 | 11, 144 | syl5com 31 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (𝐾 + (𝑚 − 𝐾)) = 𝑚)) |
146 | 145 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) → (𝑚 ∈ (0...𝑁) → (𝐾 + (𝑚 − 𝐾)) = 𝑚)) |
147 | 146 | imp 445 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → (𝐾 + (𝑚 − 𝐾)) = 𝑚) |
148 | 147 | oveq2d 6666 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → (𝑌 cyclShift (𝐾 + (𝑚 − 𝐾))) = (𝑌 cyclShift 𝑚)) |
149 | 139, 148 | eqtr2d 2657 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift (𝑚 − 𝐾))) |
150 | 149 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . 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))) |
151 | | oveq1 6657 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑋 = (𝑌 cyclShift 𝐾) → (𝑋 cyclShift (𝑚 − 𝐾)) = ((𝑌 cyclShift 𝐾) cyclShift (𝑚 − 𝐾))) |
152 | 151 | eqeq2d 2632 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑋 = (𝑌 cyclShift 𝐾) → ((𝑌 cyclShift 𝑚) = (𝑋 cyclShift (𝑚 − 𝐾)) ↔ (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift (𝑚 − 𝐾)))) |
153 | 152 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . 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)))) |
154 | 150, 153 | mpbird 247 |
. . . . . . . . . . . . . . . . . . . 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))) |
155 | 154 | eqeq2d 2632 |
. . . . . . . . . . . . . . . . . . 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)))) |
156 | 155 | biimpd 219 |
. . . . . . . . . . . . . . . . . 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)))) |
157 | 156 | exp41 638 |
. . . . . . . . . . . . . . . . 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))))))) |
158 | 157 | com24 95 |
. . . . . . . . . . . . . . . 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))))))) |
159 | 158 | imp31 448 |
. . . . . . . . . . . . . . 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))))) |
160 | 159 | com23 86 |
. . . . . . . . . . . . . 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))))) |
161 | 160 | imp 445 |
. . . . . . . . . . . . 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)))) |
162 | 161 | impcom 446 |
. . . . . . . . . . . 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))) |
163 | | oveq2 6658 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = (𝑚 − 𝐾) → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift (𝑚 − 𝐾))) |
164 | 163 | eqeq2d 2632 |
. . . . . . . . . . . . 13
⊢ (𝑛 = (𝑚 − 𝐾) → (𝑍 = (𝑋 cyclShift 𝑛) ↔ 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾)))) |
165 | 164 | rspcev 3309 |
. . . . . . . . . . . 12
⊢ (((𝑚 − 𝐾) ∈ (0...𝑁) ∧ 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾))) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)) |
166 | 130, 162,
165 | syl2anc 693 |
. . . . . . . . . . 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)) |
167 | 166 | ex 450 |
. . . . . . . . . 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))) |
168 | 85, 124, 167 | pm2.61ii 177 |
. . . . . . . . 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)) |
169 | 168 | ex 450 |
. . . . . . . 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))) |
170 | 169 | rexlimdva 3031 |
. . . . . . 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))) |
171 | 170 | ex 450 |
. . . . . 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)))) |
172 | 171 | com23 86 |
. . . . 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)))) |
173 | 172 | ex 450 |
. . . 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))))) |
174 | 173 | com24 95 |
. . 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))))) |
175 | 174 | 3imp 1256 |
. 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))) |
176 | 175 | 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))) |