| Step | Hyp | Ref
| Expression |
| 1 | | oveq2 7439 |
. . . . 5
⊢ (𝑥 = 1 → (2 · 𝑥) = (2 ·
1)) |
| 2 | 1 | oveq2d 7447 |
. . . 4
⊢ (𝑥 = 1 → (1...(2 ·
𝑥)) = (1...(2 ·
1))) |
| 3 | 2 | sumeq1d 15736 |
. . 3
⊢ (𝑥 = 1 → Σ𝑛 ∈ (1...(2 · 𝑥))(cos‘(𝑛 · π)) = Σ𝑛 ∈ (1...(2 · 1))(cos‘(𝑛 ·
π))) |
| 4 | 3 | eqeq1d 2739 |
. 2
⊢ (𝑥 = 1 → (Σ𝑛 ∈ (1...(2 · 𝑥))(cos‘(𝑛 · π)) = 0 ↔ Σ𝑛 ∈ (1...(2 ·
1))(cos‘(𝑛 ·
π)) = 0)) |
| 5 | | oveq2 7439 |
. . . . 5
⊢ (𝑥 = 𝑦 → (2 · 𝑥) = (2 · 𝑦)) |
| 6 | 5 | oveq2d 7447 |
. . . 4
⊢ (𝑥 = 𝑦 → (1...(2 · 𝑥)) = (1...(2 · 𝑦))) |
| 7 | 6 | sumeq1d 15736 |
. . 3
⊢ (𝑥 = 𝑦 → Σ𝑛 ∈ (1...(2 · 𝑥))(cos‘(𝑛 · π)) = Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π))) |
| 8 | 7 | eqeq1d 2739 |
. 2
⊢ (𝑥 = 𝑦 → (Σ𝑛 ∈ (1...(2 · 𝑥))(cos‘(𝑛 · π)) = 0 ↔ Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) = 0)) |
| 9 | | oveq2 7439 |
. . . . 5
⊢ (𝑥 = (𝑦 + 1) → (2 · 𝑥) = (2 · (𝑦 + 1))) |
| 10 | 9 | oveq2d 7447 |
. . . 4
⊢ (𝑥 = (𝑦 + 1) → (1...(2 · 𝑥)) = (1...(2 · (𝑦 + 1)))) |
| 11 | 10 | sumeq1d 15736 |
. . 3
⊢ (𝑥 = (𝑦 + 1) → Σ𝑛 ∈ (1...(2 · 𝑥))(cos‘(𝑛 · π)) = Σ𝑛 ∈ (1...(2 · (𝑦 + 1)))(cos‘(𝑛 · π))) |
| 12 | 11 | eqeq1d 2739 |
. 2
⊢ (𝑥 = (𝑦 + 1) → (Σ𝑛 ∈ (1...(2 · 𝑥))(cos‘(𝑛 · π)) = 0 ↔ Σ𝑛 ∈ (1...(2 · (𝑦 + 1)))(cos‘(𝑛 · π)) =
0)) |
| 13 | | oveq2 7439 |
. . . . 5
⊢ (𝑥 = 𝐾 → (2 · 𝑥) = (2 · 𝐾)) |
| 14 | 13 | oveq2d 7447 |
. . . 4
⊢ (𝑥 = 𝐾 → (1...(2 · 𝑥)) = (1...(2 · 𝐾))) |
| 15 | 14 | sumeq1d 15736 |
. . 3
⊢ (𝑥 = 𝐾 → Σ𝑛 ∈ (1...(2 · 𝑥))(cos‘(𝑛 · π)) = Σ𝑛 ∈ (1...(2 · 𝐾))(cos‘(𝑛 · π))) |
| 16 | 15 | eqeq1d 2739 |
. 2
⊢ (𝑥 = 𝐾 → (Σ𝑛 ∈ (1...(2 · 𝑥))(cos‘(𝑛 · π)) = 0 ↔ Σ𝑛 ∈ (1...(2 · 𝐾))(cos‘(𝑛 · π)) = 0)) |
| 17 | | ax-1cn 11213 |
. . . . . 6
⊢ 1 ∈
ℂ |
| 18 | 17 | 2timesi 12404 |
. . . . 5
⊢ (2
· 1) = (1 + 1) |
| 19 | 18 | oveq2i 7442 |
. . . 4
⊢ (1...(2
· 1)) = (1...(1 + 1)) |
| 20 | 19 | sumeq1i 15733 |
. . 3
⊢
Σ𝑛 ∈
(1...(2 · 1))(cos‘(𝑛 · π)) = Σ𝑛 ∈ (1...(1 + 1))(cos‘(𝑛 ·
π)) |
| 21 | | 1z 12647 |
. . . . . . . 8
⊢ 1 ∈
ℤ |
| 22 | | uzid 12893 |
. . . . . . . 8
⊢ (1 ∈
ℤ → 1 ∈ (ℤ≥‘1)) |
| 23 | 21, 22 | ax-mp 5 |
. . . . . . 7
⊢ 1 ∈
(ℤ≥‘1) |
| 24 | 23 | a1i 11 |
. . . . . 6
⊢ (⊤
→ 1 ∈ (ℤ≥‘1)) |
| 25 | | elfzelz 13564 |
. . . . . . . . . 10
⊢ (𝑛 ∈ (1...(1 + 1)) →
𝑛 ∈
ℤ) |
| 26 | 25 | zcnd 12723 |
. . . . . . . . 9
⊢ (𝑛 ∈ (1...(1 + 1)) →
𝑛 ∈
ℂ) |
| 27 | 26 | adantl 481 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑛
∈ (1...(1 + 1))) → 𝑛 ∈ ℂ) |
| 28 | | picn 26501 |
. . . . . . . . 9
⊢ π
∈ ℂ |
| 29 | 28 | a1i 11 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑛
∈ (1...(1 + 1))) → π ∈ ℂ) |
| 30 | 27, 29 | mulcld 11281 |
. . . . . . 7
⊢
((⊤ ∧ 𝑛
∈ (1...(1 + 1))) → (𝑛 · π) ∈
ℂ) |
| 31 | 30 | coscld 16167 |
. . . . . 6
⊢
((⊤ ∧ 𝑛
∈ (1...(1 + 1))) → (cos‘(𝑛 · π)) ∈
ℂ) |
| 32 | | id 22 |
. . . . . . . 8
⊢ (𝑛 = (1 + 1) → 𝑛 = (1 + 1)) |
| 33 | | 1p1e2 12391 |
. . . . . . . 8
⊢ (1 + 1) =
2 |
| 34 | 32, 33 | eqtrdi 2793 |
. . . . . . 7
⊢ (𝑛 = (1 + 1) → 𝑛 = 2) |
| 35 | 34 | fvoveq1d 7453 |
. . . . . 6
⊢ (𝑛 = (1 + 1) →
(cos‘(𝑛 ·
π)) = (cos‘(2 · π))) |
| 36 | 24, 31, 35 | fsump1 15792 |
. . . . 5
⊢ (⊤
→ Σ𝑛 ∈
(1...(1 + 1))(cos‘(𝑛
· π)) = (Σ𝑛
∈ (1...1)(cos‘(𝑛
· π)) + (cos‘(2 · π)))) |
| 37 | 36 | mptru 1547 |
. . . 4
⊢
Σ𝑛 ∈
(1...(1 + 1))(cos‘(𝑛
· π)) = (Σ𝑛
∈ (1...1)(cos‘(𝑛
· π)) + (cos‘(2 · π))) |
| 38 | | coscl 16163 |
. . . . . . . 8
⊢ (π
∈ ℂ → (cos‘π) ∈ ℂ) |
| 39 | 28, 38 | ax-mp 5 |
. . . . . . 7
⊢
(cos‘π) ∈ ℂ |
| 40 | | oveq1 7438 |
. . . . . . . . . 10
⊢ (𝑛 = 1 → (𝑛 · π) = (1 ·
π)) |
| 41 | 28 | mullidi 11266 |
. . . . . . . . . 10
⊢ (1
· π) = π |
| 42 | 40, 41 | eqtrdi 2793 |
. . . . . . . . 9
⊢ (𝑛 = 1 → (𝑛 · π) = π) |
| 43 | 42 | fveq2d 6910 |
. . . . . . . 8
⊢ (𝑛 = 1 → (cos‘(𝑛 · π)) =
(cos‘π)) |
| 44 | 43 | fsum1 15783 |
. . . . . . 7
⊢ ((1
∈ ℤ ∧ (cos‘π) ∈ ℂ) → Σ𝑛 ∈ (1...1)(cos‘(𝑛 · π)) =
(cos‘π)) |
| 45 | 21, 39, 44 | mp2an 692 |
. . . . . 6
⊢
Σ𝑛 ∈
(1...1)(cos‘(𝑛
· π)) = (cos‘π) |
| 46 | | cospi 26514 |
. . . . . 6
⊢
(cos‘π) = -1 |
| 47 | 45, 46 | eqtri 2765 |
. . . . 5
⊢
Σ𝑛 ∈
(1...1)(cos‘(𝑛
· π)) = -1 |
| 48 | | cos2pi 26518 |
. . . . 5
⊢
(cos‘(2 · π)) = 1 |
| 49 | 47, 48 | oveq12i 7443 |
. . . 4
⊢
(Σ𝑛 ∈
(1...1)(cos‘(𝑛
· π)) + (cos‘(2 · π))) = (-1 + 1) |
| 50 | | neg1cn 12380 |
. . . . 5
⊢ -1 ∈
ℂ |
| 51 | | 1pneg1e0 12385 |
. . . . 5
⊢ (1 + -1)
= 0 |
| 52 | 17, 50, 51 | addcomli 11453 |
. . . 4
⊢ (-1 + 1)
= 0 |
| 53 | 37, 49, 52 | 3eqtri 2769 |
. . 3
⊢
Σ𝑛 ∈
(1...(1 + 1))(cos‘(𝑛
· π)) = 0 |
| 54 | 20, 53 | eqtri 2765 |
. 2
⊢
Σ𝑛 ∈
(1...(2 · 1))(cos‘(𝑛 · π)) = 0 |
| 55 | 18 | oveq2i 7442 |
. . . . . . . 8
⊢ ((2
· 𝑦) + (2 ·
1)) = ((2 · 𝑦) + (1
+ 1)) |
| 56 | | 2cnd 12344 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℕ → 2 ∈
ℂ) |
| 57 | | nncn 12274 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℕ → 𝑦 ∈
ℂ) |
| 58 | 17 | a1i 11 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℕ → 1 ∈
ℂ) |
| 59 | 56, 57, 58 | adddid 11285 |
. . . . . . . 8
⊢ (𝑦 ∈ ℕ → (2
· (𝑦 + 1)) = ((2
· 𝑦) + (2 ·
1))) |
| 60 | 56, 57 | mulcld 11281 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℕ → (2
· 𝑦) ∈
ℂ) |
| 61 | 60, 58, 58 | addassd 11283 |
. . . . . . . 8
⊢ (𝑦 ∈ ℕ → (((2
· 𝑦) + 1) + 1) = ((2
· 𝑦) + (1 +
1))) |
| 62 | 55, 59, 61 | 3eqtr4a 2803 |
. . . . . . 7
⊢ (𝑦 ∈ ℕ → (2
· (𝑦 + 1)) = (((2
· 𝑦) + 1) +
1)) |
| 63 | 62 | oveq2d 7447 |
. . . . . 6
⊢ (𝑦 ∈ ℕ → (1...(2
· (𝑦 + 1))) =
(1...(((2 · 𝑦) + 1)
+ 1))) |
| 64 | 63 | sumeq1d 15736 |
. . . . 5
⊢ (𝑦 ∈ ℕ →
Σ𝑛 ∈ (1...(2
· (𝑦 +
1)))(cos‘(𝑛 ·
π)) = Σ𝑛 ∈
(1...(((2 · 𝑦) + 1)
+ 1))(cos‘(𝑛 ·
π))) |
| 65 | 64 | adantr 480 |
. . . 4
⊢ ((𝑦 ∈ ℕ ∧
Σ𝑛 ∈ (1...(2
· 𝑦))(cos‘(𝑛 · π)) = 0) → Σ𝑛 ∈ (1...(2 · (𝑦 + 1)))(cos‘(𝑛 · π)) = Σ𝑛 ∈ (1...(((2 · 𝑦) + 1) + 1))(cos‘(𝑛 ·
π))) |
| 66 | | 1red 11262 |
. . . . . . . 8
⊢ (𝑦 ∈ ℕ → 1 ∈
ℝ) |
| 67 | | 2re 12340 |
. . . . . . . . . . 11
⊢ 2 ∈
ℝ |
| 68 | 67 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℕ → 2 ∈
ℝ) |
| 69 | | nnre 12273 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℕ → 𝑦 ∈
ℝ) |
| 70 | 68, 69 | remulcld 11291 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℕ → (2
· 𝑦) ∈
ℝ) |
| 71 | 70, 66 | readdcld 11290 |
. . . . . . . 8
⊢ (𝑦 ∈ ℕ → ((2
· 𝑦) + 1) ∈
ℝ) |
| 72 | | 2rp 13039 |
. . . . . . . . . . 11
⊢ 2 ∈
ℝ+ |
| 73 | 72 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℕ → 2 ∈
ℝ+) |
| 74 | | nnrp 13046 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℕ → 𝑦 ∈
ℝ+) |
| 75 | 73, 74 | rpmulcld 13093 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℕ → (2
· 𝑦) ∈
ℝ+) |
| 76 | 66, 75 | ltaddrp2d 13111 |
. . . . . . . 8
⊢ (𝑦 ∈ ℕ → 1 <
((2 · 𝑦) +
1)) |
| 77 | 66, 71, 76 | ltled 11409 |
. . . . . . 7
⊢ (𝑦 ∈ ℕ → 1 ≤
((2 · 𝑦) +
1)) |
| 78 | | 2z 12649 |
. . . . . . . . . . 11
⊢ 2 ∈
ℤ |
| 79 | 78 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℕ → 2 ∈
ℤ) |
| 80 | | nnz 12634 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℕ → 𝑦 ∈
ℤ) |
| 81 | 79, 80 | zmulcld 12728 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℕ → (2
· 𝑦) ∈
ℤ) |
| 82 | 81 | peano2zd 12725 |
. . . . . . . 8
⊢ (𝑦 ∈ ℕ → ((2
· 𝑦) + 1) ∈
ℤ) |
| 83 | | eluz 12892 |
. . . . . . . 8
⊢ ((1
∈ ℤ ∧ ((2 · 𝑦) + 1) ∈ ℤ) → (((2 ·
𝑦) + 1) ∈
(ℤ≥‘1) ↔ 1 ≤ ((2 · 𝑦) + 1))) |
| 84 | 21, 82, 83 | sylancr 587 |
. . . . . . 7
⊢ (𝑦 ∈ ℕ → (((2
· 𝑦) + 1) ∈
(ℤ≥‘1) ↔ 1 ≤ ((2 · 𝑦) + 1))) |
| 85 | 77, 84 | mpbird 257 |
. . . . . 6
⊢ (𝑦 ∈ ℕ → ((2
· 𝑦) + 1) ∈
(ℤ≥‘1)) |
| 86 | | elfzelz 13564 |
. . . . . . . . . 10
⊢ (𝑛 ∈ (1...(((2 · 𝑦) + 1) + 1)) → 𝑛 ∈
ℤ) |
| 87 | 86 | zcnd 12723 |
. . . . . . . . 9
⊢ (𝑛 ∈ (1...(((2 · 𝑦) + 1) + 1)) → 𝑛 ∈
ℂ) |
| 88 | 28 | a1i 11 |
. . . . . . . . 9
⊢ (𝑛 ∈ (1...(((2 · 𝑦) + 1) + 1)) → π ∈
ℂ) |
| 89 | 87, 88 | mulcld 11281 |
. . . . . . . 8
⊢ (𝑛 ∈ (1...(((2 · 𝑦) + 1) + 1)) → (𝑛 · π) ∈
ℂ) |
| 90 | 89 | coscld 16167 |
. . . . . . 7
⊢ (𝑛 ∈ (1...(((2 · 𝑦) + 1) + 1)) →
(cos‘(𝑛 ·
π)) ∈ ℂ) |
| 91 | 90 | adantl 481 |
. . . . . 6
⊢ ((𝑦 ∈ ℕ ∧ 𝑛 ∈ (1...(((2 · 𝑦) + 1) + 1))) →
(cos‘(𝑛 ·
π)) ∈ ℂ) |
| 92 | | fvoveq1 7454 |
. . . . . 6
⊢ (𝑛 = (((2 · 𝑦) + 1) + 1) →
(cos‘(𝑛 ·
π)) = (cos‘((((2 · 𝑦) + 1) + 1) · π))) |
| 93 | 85, 91, 92 | fsump1 15792 |
. . . . 5
⊢ (𝑦 ∈ ℕ →
Σ𝑛 ∈ (1...(((2
· 𝑦) + 1) +
1))(cos‘(𝑛 ·
π)) = (Σ𝑛 ∈
(1...((2 · 𝑦) +
1))(cos‘(𝑛 ·
π)) + (cos‘((((2 · 𝑦) + 1) + 1) ·
π)))) |
| 94 | 93 | adantr 480 |
. . . 4
⊢ ((𝑦 ∈ ℕ ∧
Σ𝑛 ∈ (1...(2
· 𝑦))(cos‘(𝑛 · π)) = 0) → Σ𝑛 ∈ (1...(((2 · 𝑦) + 1) + 1))(cos‘(𝑛 · π)) = (Σ𝑛 ∈ (1...((2 · 𝑦) + 1))(cos‘(𝑛 · π)) +
(cos‘((((2 · 𝑦) + 1) + 1) ·
π)))) |
| 95 | | 1lt2 12437 |
. . . . . . . . . . . 12
⊢ 1 <
2 |
| 96 | 95 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ℕ → 1 <
2) |
| 97 | | 2t1e2 12429 |
. . . . . . . . . . . 12
⊢ (2
· 1) = 2 |
| 98 | | nnge1 12294 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℕ → 1 ≤
𝑦) |
| 99 | 66, 69, 73 | lemul2d 13121 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℕ → (1 ≤
𝑦 ↔ (2 · 1)
≤ (2 · 𝑦))) |
| 100 | 98, 99 | mpbid 232 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℕ → (2
· 1) ≤ (2 · 𝑦)) |
| 101 | 97, 100 | eqbrtrrid 5179 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ℕ → 2 ≤ (2
· 𝑦)) |
| 102 | 66, 68, 70, 96, 101 | ltletrd 11421 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℕ → 1 < (2
· 𝑦)) |
| 103 | 66, 70, 102 | ltled 11409 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℕ → 1 ≤ (2
· 𝑦)) |
| 104 | | eluz 12892 |
. . . . . . . . . 10
⊢ ((1
∈ ℤ ∧ (2 · 𝑦) ∈ ℤ) → ((2 · 𝑦) ∈
(ℤ≥‘1) ↔ 1 ≤ (2 · 𝑦))) |
| 105 | 21, 81, 104 | sylancr 587 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℕ → ((2
· 𝑦) ∈
(ℤ≥‘1) ↔ 1 ≤ (2 · 𝑦))) |
| 106 | 103, 105 | mpbird 257 |
. . . . . . . 8
⊢ (𝑦 ∈ ℕ → (2
· 𝑦) ∈
(ℤ≥‘1)) |
| 107 | | elfzelz 13564 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ (1...((2 · 𝑦) + 1)) → 𝑛 ∈
ℤ) |
| 108 | 107 | zcnd 12723 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ (1...((2 · 𝑦) + 1)) → 𝑛 ∈
ℂ) |
| 109 | 28 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ (1...((2 · 𝑦) + 1)) → π ∈
ℂ) |
| 110 | 108, 109 | mulcld 11281 |
. . . . . . . . . 10
⊢ (𝑛 ∈ (1...((2 · 𝑦) + 1)) → (𝑛 · π) ∈
ℂ) |
| 111 | 110 | coscld 16167 |
. . . . . . . . 9
⊢ (𝑛 ∈ (1...((2 · 𝑦) + 1)) → (cos‘(𝑛 · π)) ∈
ℂ) |
| 112 | 111 | adantl 481 |
. . . . . . . 8
⊢ ((𝑦 ∈ ℕ ∧ 𝑛 ∈ (1...((2 · 𝑦) + 1))) →
(cos‘(𝑛 ·
π)) ∈ ℂ) |
| 113 | | fvoveq1 7454 |
. . . . . . . 8
⊢ (𝑛 = ((2 · 𝑦) + 1) → (cos‘(𝑛 · π)) =
(cos‘(((2 · 𝑦)
+ 1) · π))) |
| 114 | 106, 112,
113 | fsump1 15792 |
. . . . . . 7
⊢ (𝑦 ∈ ℕ →
Σ𝑛 ∈ (1...((2
· 𝑦) +
1))(cos‘(𝑛 ·
π)) = (Σ𝑛 ∈
(1...(2 · 𝑦))(cos‘(𝑛 · π)) + (cos‘(((2 ·
𝑦) + 1) ·
π)))) |
| 115 | 33, 97 | eqtr4i 2768 |
. . . . . . . . . . . 12
⊢ (1 + 1) =
(2 · 1) |
| 116 | 115 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ℕ → (1 + 1) =
(2 · 1)) |
| 117 | 116 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℕ → ((2
· 𝑦) + (1 + 1)) =
((2 · 𝑦) + (2
· 1))) |
| 118 | 117, 61, 59 | 3eqtr4d 2787 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℕ → (((2
· 𝑦) + 1) + 1) = (2
· (𝑦 +
1))) |
| 119 | 118 | fvoveq1d 7453 |
. . . . . . . 8
⊢ (𝑦 ∈ ℕ →
(cos‘((((2 · 𝑦) + 1) + 1) · π)) = (cos‘((2
· (𝑦 + 1)) ·
π))) |
| 120 | 57, 58 | addcld 11280 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈
ℂ) |
| 121 | 28 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℕ → π
∈ ℂ) |
| 122 | 56, 120, 121 | mulassd 11284 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℕ → ((2
· (𝑦 + 1)) ·
π) = (2 · ((𝑦 +
1) · π))) |
| 123 | 122 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ℕ → (((2
· (𝑦 + 1)) ·
π) / (2 · π)) = ((2 · ((𝑦 + 1) · π)) / (2 ·
π))) |
| 124 | 120, 121 | mulcld 11281 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) · π) ∈
ℂ) |
| 125 | | 0re 11263 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
ℝ |
| 126 | | pipos 26502 |
. . . . . . . . . . . . . 14
⊢ 0 <
π |
| 127 | 125, 126 | gtneii 11373 |
. . . . . . . . . . . . 13
⊢ π ≠
0 |
| 128 | 127 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℕ → π ≠
0) |
| 129 | 73 | rpne0d 13082 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℕ → 2 ≠
0) |
| 130 | 124, 121,
56, 128, 129 | divcan5d 12069 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ℕ → ((2
· ((𝑦 + 1) ·
π)) / (2 · π)) = (((𝑦 + 1) · π) /
π)) |
| 131 | 120, 121,
128 | divcan4d 12049 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ℕ → (((𝑦 + 1) · π) / π) =
(𝑦 + 1)) |
| 132 | 123, 130,
131 | 3eqtrd 2781 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℕ → (((2
· (𝑦 + 1)) ·
π) / (2 · π)) = (𝑦 + 1)) |
| 133 | 80 | peano2zd 12725 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈
ℤ) |
| 134 | 132, 133 | eqeltrd 2841 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℕ → (((2
· (𝑦 + 1)) ·
π) / (2 · π)) ∈ ℤ) |
| 135 | | peano2cn 11433 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℂ → (𝑦 + 1) ∈
ℂ) |
| 136 | 57, 135 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈
ℂ) |
| 137 | 56, 136 | mulcld 11281 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ℕ → (2
· (𝑦 + 1)) ∈
ℂ) |
| 138 | 137, 121 | mulcld 11281 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℕ → ((2
· (𝑦 + 1)) ·
π) ∈ ℂ) |
| 139 | | coseq1 26567 |
. . . . . . . . . 10
⊢ (((2
· (𝑦 + 1)) ·
π) ∈ ℂ → ((cos‘((2 · (𝑦 + 1)) · π)) = 1 ↔ (((2
· (𝑦 + 1)) ·
π) / (2 · π)) ∈ ℤ)) |
| 140 | 138, 139 | syl 17 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℕ →
((cos‘((2 · (𝑦
+ 1)) · π)) = 1 ↔ (((2 · (𝑦 + 1)) · π) / (2 · π))
∈ ℤ)) |
| 141 | 134, 140 | mpbird 257 |
. . . . . . . 8
⊢ (𝑦 ∈ ℕ →
(cos‘((2 · (𝑦
+ 1)) · π)) = 1) |
| 142 | 119, 141 | eqtrd 2777 |
. . . . . . 7
⊢ (𝑦 ∈ ℕ →
(cos‘((((2 · 𝑦) + 1) + 1) · π)) =
1) |
| 143 | 114, 142 | oveq12d 7449 |
. . . . . 6
⊢ (𝑦 ∈ ℕ →
(Σ𝑛 ∈ (1...((2
· 𝑦) +
1))(cos‘(𝑛 ·
π)) + (cos‘((((2 · 𝑦) + 1) + 1) · π))) = ((Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) + (cos‘(((2 ·
𝑦) + 1) · π))) +
1)) |
| 144 | 143 | adantr 480 |
. . . . 5
⊢ ((𝑦 ∈ ℕ ∧
Σ𝑛 ∈ (1...(2
· 𝑦))(cos‘(𝑛 · π)) = 0) → (Σ𝑛 ∈ (1...((2 · 𝑦) + 1))(cos‘(𝑛 · π)) +
(cos‘((((2 · 𝑦) + 1) + 1) · π))) = ((Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) + (cos‘(((2 ·
𝑦) + 1) · π))) +
1)) |
| 145 | | simpr 484 |
. . . . . . . 8
⊢ ((𝑦 ∈ ℕ ∧
Σ𝑛 ∈ (1...(2
· 𝑦))(cos‘(𝑛 · π)) = 0) → Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) = 0) |
| 146 | 60, 58, 121 | adddird 11286 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℕ → (((2
· 𝑦) + 1) ·
π) = (((2 · 𝑦)
· π) + (1 · π))) |
| 147 | 60, 121 | mulcld 11281 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℕ → ((2
· 𝑦) · π)
∈ ℂ) |
| 148 | 41, 121 | eqeltrid 2845 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℕ → (1
· π) ∈ ℂ) |
| 149 | 147, 148 | addcomd 11463 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℕ → (((2
· 𝑦) · π)
+ (1 · π)) = ((1 · π) + ((2 · 𝑦) · π))) |
| 150 | 41 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℕ → (1
· π) = π) |
| 151 | 56, 57 | mulcomd 11282 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ ℕ → (2
· 𝑦) = (𝑦 · 2)) |
| 152 | 151 | oveq1d 7446 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ ℕ → ((2
· 𝑦) · π)
= ((𝑦 · 2) ·
π)) |
| 153 | 57, 56, 121 | mulassd 11284 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ ℕ → ((𝑦 · 2) · π) =
(𝑦 · (2 ·
π))) |
| 154 | 152, 153 | eqtrd 2777 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℕ → ((2
· 𝑦) · π)
= (𝑦 · (2 ·
π))) |
| 155 | 150, 154 | oveq12d 7449 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℕ → ((1
· π) + ((2 · 𝑦) · π)) = (π + (𝑦 · (2 ·
π)))) |
| 156 | 146, 149,
155 | 3eqtrd 2781 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ℕ → (((2
· 𝑦) + 1) ·
π) = (π + (𝑦 ·
(2 · π)))) |
| 157 | 156 | fveq2d 6910 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℕ →
(cos‘(((2 · 𝑦)
+ 1) · π)) = (cos‘(π + (𝑦 · (2 ·
π))))) |
| 158 | | cosper 26524 |
. . . . . . . . . . 11
⊢ ((π
∈ ℂ ∧ 𝑦
∈ ℤ) → (cos‘(π + (𝑦 · (2 · π)))) =
(cos‘π)) |
| 159 | 28, 80, 158 | sylancr 587 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℕ →
(cos‘(π + (𝑦
· (2 · π)))) = (cos‘π)) |
| 160 | 46 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℕ →
(cos‘π) = -1) |
| 161 | 157, 159,
160 | 3eqtrd 2781 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℕ →
(cos‘(((2 · 𝑦)
+ 1) · π)) = -1) |
| 162 | 161 | adantr 480 |
. . . . . . . 8
⊢ ((𝑦 ∈ ℕ ∧
Σ𝑛 ∈ (1...(2
· 𝑦))(cos‘(𝑛 · π)) = 0) → (cos‘(((2
· 𝑦) + 1) ·
π)) = -1) |
| 163 | 145, 162 | oveq12d 7449 |
. . . . . . 7
⊢ ((𝑦 ∈ ℕ ∧
Σ𝑛 ∈ (1...(2
· 𝑦))(cos‘(𝑛 · π)) = 0) → (Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) + (cos‘(((2 ·
𝑦) + 1) · π))) =
(0 + -1)) |
| 164 | 163 | oveq1d 7446 |
. . . . . 6
⊢ ((𝑦 ∈ ℕ ∧
Σ𝑛 ∈ (1...(2
· 𝑦))(cos‘(𝑛 · π)) = 0) → ((Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) + (cos‘(((2 ·
𝑦) + 1) · π))) +
1) = ((0 + -1) + 1)) |
| 165 | 50 | addlidi 11449 |
. . . . . . . 8
⊢ (0 + -1)
= -1 |
| 166 | 165 | oveq1i 7441 |
. . . . . . 7
⊢ ((0 + -1)
+ 1) = (-1 + 1) |
| 167 | 166, 52 | eqtri 2765 |
. . . . . 6
⊢ ((0 + -1)
+ 1) = 0 |
| 168 | 164, 167 | eqtrdi 2793 |
. . . . 5
⊢ ((𝑦 ∈ ℕ ∧
Σ𝑛 ∈ (1...(2
· 𝑦))(cos‘(𝑛 · π)) = 0) → ((Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) + (cos‘(((2 ·
𝑦) + 1) · π))) +
1) = 0) |
| 169 | 144, 168 | eqtrd 2777 |
. . . 4
⊢ ((𝑦 ∈ ℕ ∧
Σ𝑛 ∈ (1...(2
· 𝑦))(cos‘(𝑛 · π)) = 0) → (Σ𝑛 ∈ (1...((2 · 𝑦) + 1))(cos‘(𝑛 · π)) +
(cos‘((((2 · 𝑦) + 1) + 1) · π))) =
0) |
| 170 | 65, 94, 169 | 3eqtrd 2781 |
. . 3
⊢ ((𝑦 ∈ ℕ ∧
Σ𝑛 ∈ (1...(2
· 𝑦))(cos‘(𝑛 · π)) = 0) → Σ𝑛 ∈ (1...(2 · (𝑦 + 1)))(cos‘(𝑛 · π)) =
0) |
| 171 | 170 | ex 412 |
. 2
⊢ (𝑦 ∈ ℕ →
(Σ𝑛 ∈ (1...(2
· 𝑦))(cos‘(𝑛 · π)) = 0 → Σ𝑛 ∈ (1...(2 · (𝑦 + 1)))(cos‘(𝑛 · π)) =
0)) |
| 172 | 4, 8, 12, 16, 54, 171 | nnind 12284 |
1
⊢ (𝐾 ∈ ℕ →
Σ𝑛 ∈ (1...(2
· 𝐾))(cos‘(𝑛 · π)) = 0) |