Step | Hyp | Ref
| Expression |
1 | | simpr 485 |
. . . . . . . . 9
⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) →
(♯‘𝑊) ∈
ℙ) |
2 | 1 | adantr 481 |
. . . . . . . 8
⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (♯‘𝑊) ∈ ℙ) |
3 | | simp1 1135 |
. . . . . . . . 9
⊢ ((𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊) → 𝐿 ∈ ℤ) |
4 | 3 | adantl 482 |
. . . . . . . 8
⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → 𝐿 ∈ ℤ) |
5 | | simpr2 1194 |
. . . . . . . 8
⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝐿 mod (♯‘𝑊)) ≠ 0) |
6 | 2, 4, 5 | 3jca 1127 |
. . . . . . 7
⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → ((♯‘𝑊) ∈ ℙ ∧ 𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0)) |
7 | 6 | adantr 481 |
. . . . . 6
⊢ ((((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → ((♯‘𝑊) ∈ ℙ ∧ 𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0)) |
8 | | simpr 485 |
. . . . . 6
⊢ ((((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → 𝑖 ∈ (0..^(♯‘𝑊))) |
9 | | modprmn0modprm0 16506 |
. . . . . 6
⊢
(((♯‘𝑊)
∈ ℙ ∧ 𝐿
∈ ℤ ∧ (𝐿 mod
(♯‘𝑊)) ≠ 0)
→ (𝑖 ∈
(0..^(♯‘𝑊))
→ ∃𝑗 ∈
(0..^(♯‘𝑊))((𝑖 + (𝑗 · 𝐿)) mod (♯‘𝑊)) = 0)) |
10 | 7, 8, 9 | sylc 65 |
. . . . 5
⊢ ((((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → ∃𝑗 ∈ (0..^(♯‘𝑊))((𝑖 + (𝑗 · 𝐿)) mod (♯‘𝑊)) = 0) |
11 | | oveq1 7278 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑗 → (𝑘 · 𝐿) = (𝑗 · 𝐿)) |
12 | 11 | oveq2d 7287 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑗 → (𝑖 + (𝑘 · 𝐿)) = (𝑖 + (𝑗 · 𝐿))) |
13 | 12 | fvoveq1d 7293 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑗 → (𝑊‘((𝑖 + (𝑘 · 𝐿)) mod (♯‘𝑊))) = (𝑊‘((𝑖 + (𝑗 · 𝐿)) mod (♯‘𝑊)))) |
14 | 13 | eqeq2d 2751 |
. . . . . . . . 9
⊢ (𝑘 = 𝑗 → ((𝑊‘𝑖) = (𝑊‘((𝑖 + (𝑘 · 𝐿)) mod (♯‘𝑊))) ↔ (𝑊‘𝑖) = (𝑊‘((𝑖 + (𝑗 · 𝐿)) mod (♯‘𝑊))))) |
15 | | simpl 483 |
. . . . . . . . . . . . 13
⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → 𝑊 ∈ Word 𝑉) |
16 | 15, 3 | anim12i 613 |
. . . . . . . . . . . 12
⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ)) |
17 | 16 | adantr 481 |
. . . . . . . . . . 11
⊢ ((((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → (𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ)) |
18 | 17 | adantl 482 |
. . . . . . . . . 10
⊢ (((𝑗 ∈
(0..^(♯‘𝑊))
∧ ((𝑖 + (𝑗 · 𝐿)) mod (♯‘𝑊)) = 0) ∧ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(♯‘𝑊)))) → (𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ)) |
19 | | simpr3 1195 |
. . . . . . . . . . . 12
⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝑊 cyclShift 𝐿) = 𝑊) |
20 | 19 | anim1i 615 |
. . . . . . . . . . 11
⊢ ((((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝑖 ∈ (0..^(♯‘𝑊)))) |
21 | 20 | adantl 482 |
. . . . . . . . . 10
⊢ (((𝑗 ∈
(0..^(♯‘𝑊))
∧ ((𝑖 + (𝑗 · 𝐿)) mod (♯‘𝑊)) = 0) ∧ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(♯‘𝑊)))) → ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝑖 ∈ (0..^(♯‘𝑊)))) |
22 | | cshweqrep 14532 |
. . . . . . . . . 10
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) → (((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → ∀𝑘 ∈ ℕ0 (𝑊‘𝑖) = (𝑊‘((𝑖 + (𝑘 · 𝐿)) mod (♯‘𝑊))))) |
23 | 18, 21, 22 | sylc 65 |
. . . . . . . . 9
⊢ (((𝑗 ∈
(0..^(♯‘𝑊))
∧ ((𝑖 + (𝑗 · 𝐿)) mod (♯‘𝑊)) = 0) ∧ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(♯‘𝑊)))) → ∀𝑘 ∈ ℕ0 (𝑊‘𝑖) = (𝑊‘((𝑖 + (𝑘 · 𝐿)) mod (♯‘𝑊)))) |
24 | | elfzonn0 13430 |
. . . . . . . . . 10
⊢ (𝑗 ∈
(0..^(♯‘𝑊))
→ 𝑗 ∈
ℕ0) |
25 | 24 | ad2antrr 723 |
. . . . . . . . 9
⊢ (((𝑗 ∈
(0..^(♯‘𝑊))
∧ ((𝑖 + (𝑗 · 𝐿)) mod (♯‘𝑊)) = 0) ∧ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(♯‘𝑊)))) → 𝑗 ∈ ℕ0) |
26 | 14, 23, 25 | rspcdva 3563 |
. . . . . . . 8
⊢ (((𝑗 ∈
(0..^(♯‘𝑊))
∧ ((𝑖 + (𝑗 · 𝐿)) mod (♯‘𝑊)) = 0) ∧ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(♯‘𝑊)))) → (𝑊‘𝑖) = (𝑊‘((𝑖 + (𝑗 · 𝐿)) mod (♯‘𝑊)))) |
27 | | fveq2 6771 |
. . . . . . . . . 10
⊢ (((𝑖 + (𝑗 · 𝐿)) mod (♯‘𝑊)) = 0 → (𝑊‘((𝑖 + (𝑗 · 𝐿)) mod (♯‘𝑊))) = (𝑊‘0)) |
28 | 27 | adantl 482 |
. . . . . . . . 9
⊢ ((𝑗 ∈
(0..^(♯‘𝑊))
∧ ((𝑖 + (𝑗 · 𝐿)) mod (♯‘𝑊)) = 0) → (𝑊‘((𝑖 + (𝑗 · 𝐿)) mod (♯‘𝑊))) = (𝑊‘0)) |
29 | 28 | adantr 481 |
. . . . . . . 8
⊢ (((𝑗 ∈
(0..^(♯‘𝑊))
∧ ((𝑖 + (𝑗 · 𝐿)) mod (♯‘𝑊)) = 0) ∧ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(♯‘𝑊)))) → (𝑊‘((𝑖 + (𝑗 · 𝐿)) mod (♯‘𝑊))) = (𝑊‘0)) |
30 | 26, 29 | eqtrd 2780 |
. . . . . . 7
⊢ (((𝑗 ∈
(0..^(♯‘𝑊))
∧ ((𝑖 + (𝑗 · 𝐿)) mod (♯‘𝑊)) = 0) ∧ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(♯‘𝑊)))) → (𝑊‘𝑖) = (𝑊‘0)) |
31 | 30 | ex 413 |
. . . . . 6
⊢ ((𝑗 ∈
(0..^(♯‘𝑊))
∧ ((𝑖 + (𝑗 · 𝐿)) mod (♯‘𝑊)) = 0) → ((((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑖) = (𝑊‘0))) |
32 | 31 | rexlimiva 3212 |
. . . . 5
⊢
(∃𝑗 ∈
(0..^(♯‘𝑊))((𝑖 + (𝑗 · 𝐿)) mod (♯‘𝑊)) = 0 → ((((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑖) = (𝑊‘0))) |
33 | 10, 32 | mpcom 38 |
. . . 4
⊢ ((((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑖) = (𝑊‘0)) |
34 | 33 | ralrimiva 3110 |
. . 3
⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)) |
35 | | repswsymballbi 14491 |
. . . 4
⊢ (𝑊 ∈ Word 𝑉 → (𝑊 = ((𝑊‘0) repeatS (♯‘𝑊)) ↔ ∀𝑖 ∈
(0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))) |
36 | 35 | ad2antrr 723 |
. . 3
⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝑊 = ((𝑊‘0) repeatS (♯‘𝑊)) ↔ ∀𝑖 ∈
(0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))) |
37 | 34, 36 | mpbird 256 |
. 2
⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))) |
38 | 37 | ex 413 |
1
⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → ((𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊) → 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊)))) |