Proof of Theorem atantayl2
| Step | Hyp | Ref
| Expression |
| 1 | | atantayl2.1 |
. . . 4
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, ((-1↑((𝑛 − 1) / 2)) ·
((𝐴↑𝑛) / 𝑛)))) |
| 2 | | ax-icn 11214 |
. . . . . . . . . . . . . . . 16
⊢ i ∈
ℂ |
| 3 | 2 | negcli 11577 |
. . . . . . . . . . . . . . 15
⊢ -i ∈
ℂ |
| 4 | 3 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → -i ∈
ℂ) |
| 5 | | nnnn0 12533 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℕ0) |
| 6 | 5 | ad2antlr 727 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → 𝑛 ∈
ℕ0) |
| 7 | 4, 6 | expcld 14186 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) →
(-i↑𝑛) ∈
ℂ) |
| 8 | | sqneg 14156 |
. . . . . . . . . . . . . . . . 17
⊢ (i ∈
ℂ → (-i↑2) = (i↑2)) |
| 9 | 2, 8 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢
(-i↑2) = (i↑2) |
| 10 | 9 | oveq1i 7441 |
. . . . . . . . . . . . . . 15
⊢
((-i↑2)↑(𝑛
/ 2)) = ((i↑2)↑(𝑛
/ 2)) |
| 11 | | ine0 11698 |
. . . . . . . . . . . . . . . . . 18
⊢ i ≠
0 |
| 12 | 2, 11 | negne0i 11584 |
. . . . . . . . . . . . . . . . 17
⊢ -i ≠
0 |
| 13 | 12 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → -i ≠
0) |
| 14 | | 2z 12649 |
. . . . . . . . . . . . . . . . 17
⊢ 2 ∈
ℤ |
| 15 | 14 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → 2 ∈
ℤ) |
| 16 | | 2ne0 12370 |
. . . . . . . . . . . . . . . . . 18
⊢ 2 ≠
0 |
| 17 | | nnz 12634 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℤ) |
| 18 | 17 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) →
𝑛 ∈
ℤ) |
| 19 | | dvdsval2 16293 |
. . . . . . . . . . . . . . . . . 18
⊢ ((2
∈ ℤ ∧ 2 ≠ 0 ∧ 𝑛 ∈ ℤ) → (2 ∥ 𝑛 ↔ (𝑛 / 2) ∈ ℤ)) |
| 20 | 14, 16, 18, 19 | mp3an12i 1467 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) → (2
∥ 𝑛 ↔ (𝑛 / 2) ∈
ℤ)) |
| 21 | 20 | biimpa 476 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → (𝑛 / 2) ∈
ℤ) |
| 22 | | expmulz 14149 |
. . . . . . . . . . . . . . . 16
⊢ (((-i
∈ ℂ ∧ -i ≠ 0) ∧ (2 ∈ ℤ ∧ (𝑛 / 2) ∈ ℤ)) →
(-i↑(2 · (𝑛 /
2))) = ((-i↑2)↑(𝑛
/ 2))) |
| 23 | 4, 13, 15, 21, 22 | syl22anc 839 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) →
(-i↑(2 · (𝑛 /
2))) = ((-i↑2)↑(𝑛
/ 2))) |
| 24 | 2 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → i ∈
ℂ) |
| 25 | 11 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → i ≠
0) |
| 26 | | expmulz 14149 |
. . . . . . . . . . . . . . . 16
⊢ (((i
∈ ℂ ∧ i ≠ 0) ∧ (2 ∈ ℤ ∧ (𝑛 / 2) ∈ ℤ)) →
(i↑(2 · (𝑛 /
2))) = ((i↑2)↑(𝑛
/ 2))) |
| 27 | 24, 25, 15, 21, 26 | syl22anc 839 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) →
(i↑(2 · (𝑛 /
2))) = ((i↑2)↑(𝑛
/ 2))) |
| 28 | 10, 23, 27 | 3eqtr4a 2803 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) →
(-i↑(2 · (𝑛 /
2))) = (i↑(2 · (𝑛 / 2)))) |
| 29 | | nncn 12274 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℂ) |
| 30 | 29 | ad2antlr 727 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → 𝑛 ∈
ℂ) |
| 31 | | 2cnd 12344 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → 2 ∈
ℂ) |
| 32 | 16 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → 2 ≠
0) |
| 33 | 30, 31, 32 | divcan2d 12045 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → (2
· (𝑛 / 2)) = 𝑛) |
| 34 | 33 | oveq2d 7447 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) →
(-i↑(2 · (𝑛 /
2))) = (-i↑𝑛)) |
| 35 | 33 | oveq2d 7447 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) →
(i↑(2 · (𝑛 /
2))) = (i↑𝑛)) |
| 36 | 28, 34, 35 | 3eqtr3d 2785 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) →
(-i↑𝑛) = (i↑𝑛)) |
| 37 | 7, 36 | subeq0bd 11689 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) →
((-i↑𝑛) −
(i↑𝑛)) =
0) |
| 38 | 37 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → (i
· ((-i↑𝑛)
− (i↑𝑛))) = (i
· 0)) |
| 39 | | it0e0 12488 |
. . . . . . . . . . 11
⊢ (i
· 0) = 0 |
| 40 | 38, 39 | eqtrdi 2793 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → (i
· ((-i↑𝑛)
− (i↑𝑛))) =
0) |
| 41 | 40 | oveq1d 7446 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → ((i
· ((-i↑𝑛)
− (i↑𝑛))) / 2) =
(0 / 2)) |
| 42 | | 2cn 12341 |
. . . . . . . . . 10
⊢ 2 ∈
ℂ |
| 43 | 42, 16 | div0i 12001 |
. . . . . . . . 9
⊢ (0 / 2) =
0 |
| 44 | 41, 43 | eqtrdi 2793 |
. . . . . . . 8
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → ((i
· ((-i↑𝑛)
− (i↑𝑛))) / 2) =
0) |
| 45 | 44 | oveq1d 7446 |
. . . . . . 7
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → (((i
· ((-i↑𝑛)
− (i↑𝑛))) / 2)
· ((𝐴↑𝑛) / 𝑛)) = (0 · ((𝐴↑𝑛) / 𝑛))) |
| 46 | | simplll 775 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → 𝐴 ∈
ℂ) |
| 47 | 46, 6 | expcld 14186 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → (𝐴↑𝑛) ∈ ℂ) |
| 48 | | nnne0 12300 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0) |
| 49 | 48 | ad2antlr 727 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → 𝑛 ≠ 0) |
| 50 | 47, 30, 49 | divcld 12043 |
. . . . . . . 8
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → ((𝐴↑𝑛) / 𝑛) ∈ ℂ) |
| 51 | 50 | mul02d 11459 |
. . . . . . 7
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → (0
· ((𝐴↑𝑛) / 𝑛)) = 0) |
| 52 | 45, 51 | eqtr2d 2778 |
. . . . . 6
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧ 2
∥ 𝑛) → 0 = (((i
· ((-i↑𝑛)
− (i↑𝑛))) / 2)
· ((𝐴↑𝑛) / 𝑛))) |
| 53 | | 2cnd 12344 |
. . . . . . . 8
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) → 2
∈ ℂ) |
| 54 | | ax-1cn 11213 |
. . . . . . . . . . 11
⊢ 1 ∈
ℂ |
| 55 | 54 | negcli 11577 |
. . . . . . . . . 10
⊢ -1 ∈
ℂ |
| 56 | 55 | a1i 11 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
-1 ∈ ℂ) |
| 57 | | neg1ne0 12382 |
. . . . . . . . . 10
⊢ -1 ≠
0 |
| 58 | 57 | a1i 11 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
-1 ≠ 0) |
| 59 | 29 | ad2antlr 727 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
𝑛 ∈
ℂ) |
| 60 | | peano2cn 11433 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℂ → (𝑛 + 1) ∈
ℂ) |
| 61 | 59, 60 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(𝑛 + 1) ∈
ℂ) |
| 62 | 16 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) → 2
≠ 0) |
| 63 | 61, 53, 53, 62 | divsubdird 12082 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(((𝑛 + 1) − 2) / 2) =
(((𝑛 + 1) / 2) − (2 /
2))) |
| 64 | | 2div2e1 12407 |
. . . . . . . . . . . . 13
⊢ (2 / 2) =
1 |
| 65 | 64 | oveq2i 7442 |
. . . . . . . . . . . 12
⊢ (((𝑛 + 1) / 2) − (2 / 2)) =
(((𝑛 + 1) / 2) −
1) |
| 66 | 63, 65 | eqtrdi 2793 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(((𝑛 + 1) − 2) / 2) =
(((𝑛 + 1) / 2) −
1)) |
| 67 | | df-2 12329 |
. . . . . . . . . . . . . 14
⊢ 2 = (1 +
1) |
| 68 | 67 | oveq2i 7442 |
. . . . . . . . . . . . 13
⊢ ((𝑛 + 1) − 2) = ((𝑛 + 1) − (1 +
1)) |
| 69 | 54 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) → 1
∈ ℂ) |
| 70 | 59, 69, 69 | pnpcan2d 11658 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
((𝑛 + 1) − (1 + 1)) =
(𝑛 −
1)) |
| 71 | 68, 70 | eqtrid 2789 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
((𝑛 + 1) − 2) =
(𝑛 −
1)) |
| 72 | 71 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(((𝑛 + 1) − 2) / 2) =
((𝑛 − 1) /
2)) |
| 73 | 66, 72 | eqtr3d 2779 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(((𝑛 + 1) / 2) − 1) =
((𝑛 − 1) /
2)) |
| 74 | 20 | notbid 318 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) →
(¬ 2 ∥ 𝑛 ↔
¬ (𝑛 / 2) ∈
ℤ)) |
| 75 | | zeo 12704 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℤ → ((𝑛 / 2) ∈ ℤ ∨
((𝑛 + 1) / 2) ∈
ℤ)) |
| 76 | 18, 75 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) →
((𝑛 / 2) ∈ ℤ
∨ ((𝑛 + 1) / 2) ∈
ℤ)) |
| 77 | 76 | ord 865 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) →
(¬ (𝑛 / 2) ∈
ℤ → ((𝑛 + 1) /
2) ∈ ℤ)) |
| 78 | 74, 77 | sylbid 240 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) →
(¬ 2 ∥ 𝑛 →
((𝑛 + 1) / 2) ∈
ℤ)) |
| 79 | 78 | imp 406 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
((𝑛 + 1) / 2) ∈
ℤ) |
| 80 | | peano2zm 12660 |
. . . . . . . . . . 11
⊢ (((𝑛 + 1) / 2) ∈ ℤ →
(((𝑛 + 1) / 2) − 1)
∈ ℤ) |
| 81 | 79, 80 | syl 17 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(((𝑛 + 1) / 2) − 1)
∈ ℤ) |
| 82 | 73, 81 | eqeltrrd 2842 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
((𝑛 − 1) / 2) ∈
ℤ) |
| 83 | 56, 58, 82 | expclzd 14191 |
. . . . . . . 8
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(-1↑((𝑛 − 1) /
2)) ∈ ℂ) |
| 84 | 83 | 2timesd 12509 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(2 · (-1↑((𝑛
− 1) / 2))) = ((-1↑((𝑛 − 1) / 2)) + (-1↑((𝑛 − 1) /
2)))) |
| 85 | | subcl 11507 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑛 ∈ ℂ ∧ 1 ∈
ℂ) → (𝑛 −
1) ∈ ℂ) |
| 86 | 59, 54, 85 | sylancl 586 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(𝑛 − 1) ∈
ℂ) |
| 87 | 86, 53, 62 | divcan2d 12045 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(2 · ((𝑛 − 1)
/ 2)) = (𝑛 −
1)) |
| 88 | 87 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(-i↑(2 · ((𝑛
− 1) / 2))) = (-i↑(𝑛 − 1))) |
| 89 | 3 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
-i ∈ ℂ) |
| 90 | 12 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
-i ≠ 0) |
| 91 | 17 | ad2antlr 727 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
𝑛 ∈
ℤ) |
| 92 | 89, 90, 91 | expm1d 14196 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(-i↑(𝑛 − 1)) =
((-i↑𝑛) /
-i)) |
| 93 | 88, 92 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(-i↑(2 · ((𝑛
− 1) / 2))) = ((-i↑𝑛) / -i)) |
| 94 | 14 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) → 2
∈ ℤ) |
| 95 | | expmulz 14149 |
. . . . . . . . . . . . 13
⊢ (((-i
∈ ℂ ∧ -i ≠ 0) ∧ (2 ∈ ℤ ∧ ((𝑛 − 1) / 2) ∈
ℤ)) → (-i↑(2 · ((𝑛 − 1) / 2))) =
((-i↑2)↑((𝑛
− 1) / 2))) |
| 96 | 89, 90, 94, 82, 95 | syl22anc 839 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(-i↑(2 · ((𝑛
− 1) / 2))) = ((-i↑2)↑((𝑛 − 1) / 2))) |
| 97 | 5 | ad2antlr 727 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
𝑛 ∈
ℕ0) |
| 98 | | expcl 14120 |
. . . . . . . . . . . . . 14
⊢ ((-i
∈ ℂ ∧ 𝑛
∈ ℕ0) → (-i↑𝑛) ∈ ℂ) |
| 99 | 3, 97, 98 | sylancr 587 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(-i↑𝑛) ∈
ℂ) |
| 100 | 99, 89, 90 | divrec2d 12047 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
((-i↑𝑛) / -i) = ((1 /
-i) · (-i↑𝑛))) |
| 101 | 93, 96, 100 | 3eqtr3d 2785 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
((-i↑2)↑((𝑛
− 1) / 2)) = ((1 / -i) · (-i↑𝑛))) |
| 102 | | i2 14241 |
. . . . . . . . . . . . 13
⊢
(i↑2) = -1 |
| 103 | 9, 102 | eqtri 2765 |
. . . . . . . . . . . 12
⊢
(-i↑2) = -1 |
| 104 | 103 | oveq1i 7441 |
. . . . . . . . . . 11
⊢
((-i↑2)↑((𝑛 − 1) / 2)) = (-1↑((𝑛 − 1) /
2)) |
| 105 | | irec 14240 |
. . . . . . . . . . . . . 14
⊢ (1 / i) =
-i |
| 106 | 105 | negeqi 11501 |
. . . . . . . . . . . . 13
⊢ -(1 / i)
= --i |
| 107 | | divneg2 11991 |
. . . . . . . . . . . . . 14
⊢ ((1
∈ ℂ ∧ i ∈ ℂ ∧ i ≠ 0) → -(1 / i) = (1 /
-i)) |
| 108 | 54, 2, 11, 107 | mp3an 1463 |
. . . . . . . . . . . . 13
⊢ -(1 / i)
= (1 / -i) |
| 109 | 2 | negnegi 11579 |
. . . . . . . . . . . . 13
⊢ --i =
i |
| 110 | 106, 108,
109 | 3eqtr3i 2773 |
. . . . . . . . . . . 12
⊢ (1 / -i)
= i |
| 111 | 110 | oveq1i 7441 |
. . . . . . . . . . 11
⊢ ((1 / -i)
· (-i↑𝑛)) = (i
· (-i↑𝑛)) |
| 112 | 101, 104,
111 | 3eqtr3g 2800 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(-1↑((𝑛 − 1) /
2)) = (i · (-i↑𝑛))) |
| 113 | 87 | oveq2d 7447 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(i↑(2 · ((𝑛
− 1) / 2))) = (i↑(𝑛 − 1))) |
| 114 | 2 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) → i
∈ ℂ) |
| 115 | 11 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) → i
≠ 0) |
| 116 | 114, 115,
91 | expm1d 14196 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(i↑(𝑛 − 1)) =
((i↑𝑛) /
i)) |
| 117 | 113, 116 | eqtrd 2777 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(i↑(2 · ((𝑛
− 1) / 2))) = ((i↑𝑛) / i)) |
| 118 | | expmulz 14149 |
. . . . . . . . . . . . . 14
⊢ (((i
∈ ℂ ∧ i ≠ 0) ∧ (2 ∈ ℤ ∧ ((𝑛 − 1) / 2) ∈
ℤ)) → (i↑(2 · ((𝑛 − 1) / 2))) = ((i↑2)↑((𝑛 − 1) /
2))) |
| 119 | 114, 115,
94, 82, 118 | syl22anc 839 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(i↑(2 · ((𝑛
− 1) / 2))) = ((i↑2)↑((𝑛 − 1) / 2))) |
| 120 | | expcl 14120 |
. . . . . . . . . . . . . . 15
⊢ ((i
∈ ℂ ∧ 𝑛
∈ ℕ0) → (i↑𝑛) ∈ ℂ) |
| 121 | 2, 97, 120 | sylancr 587 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(i↑𝑛) ∈
ℂ) |
| 122 | 121, 114,
115 | divrec2d 12047 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
((i↑𝑛) / i) = ((1 / i)
· (i↑𝑛))) |
| 123 | 117, 119,
122 | 3eqtr3d 2785 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
((i↑2)↑((𝑛
− 1) / 2)) = ((1 / i) · (i↑𝑛))) |
| 124 | 102 | oveq1i 7441 |
. . . . . . . . . . . 12
⊢
((i↑2)↑((𝑛
− 1) / 2)) = (-1↑((𝑛 − 1) / 2)) |
| 125 | 105 | oveq1i 7441 |
. . . . . . . . . . . 12
⊢ ((1 / i)
· (i↑𝑛)) = (-i
· (i↑𝑛)) |
| 126 | 123, 124,
125 | 3eqtr3g 2800 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(-1↑((𝑛 − 1) /
2)) = (-i · (i↑𝑛))) |
| 127 | | mulneg1 11699 |
. . . . . . . . . . . 12
⊢ ((i
∈ ℂ ∧ (i↑𝑛) ∈ ℂ) → (-i ·
(i↑𝑛)) = -(i ·
(i↑𝑛))) |
| 128 | 2, 121, 127 | sylancr 587 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(-i · (i↑𝑛)) =
-(i · (i↑𝑛))) |
| 129 | 126, 128 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(-1↑((𝑛 − 1) /
2)) = -(i · (i↑𝑛))) |
| 130 | 112, 129 | oveq12d 7449 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
((-1↑((𝑛 − 1) /
2)) + (-1↑((𝑛 −
1) / 2))) = ((i · (-i↑𝑛)) + -(i · (i↑𝑛)))) |
| 131 | | mulcl 11239 |
. . . . . . . . . . . 12
⊢ ((i
∈ ℂ ∧ (-i↑𝑛) ∈ ℂ) → (i ·
(-i↑𝑛)) ∈
ℂ) |
| 132 | 2, 99, 131 | sylancr 587 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(i · (-i↑𝑛))
∈ ℂ) |
| 133 | | mulcl 11239 |
. . . . . . . . . . . 12
⊢ ((i
∈ ℂ ∧ (i↑𝑛) ∈ ℂ) → (i ·
(i↑𝑛)) ∈
ℂ) |
| 134 | 2, 121, 133 | sylancr 587 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(i · (i↑𝑛))
∈ ℂ) |
| 135 | 132, 134 | negsubd 11626 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
((i · (-i↑𝑛)) +
-(i · (i↑𝑛))) =
((i · (-i↑𝑛))
− (i · (i↑𝑛)))) |
| 136 | 114, 99, 121 | subdid 11719 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(i · ((-i↑𝑛)
− (i↑𝑛))) = ((i
· (-i↑𝑛))
− (i · (i↑𝑛)))) |
| 137 | 135, 136 | eqtr4d 2780 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
((i · (-i↑𝑛)) +
-(i · (i↑𝑛))) =
(i · ((-i↑𝑛)
− (i↑𝑛)))) |
| 138 | 84, 130, 137 | 3eqtrd 2781 |
. . . . . . . 8
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(2 · (-1↑((𝑛
− 1) / 2))) = (i · ((-i↑𝑛) − (i↑𝑛)))) |
| 139 | 53, 83, 62, 138 | mvllmuld 12099 |
. . . . . . 7
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
(-1↑((𝑛 − 1) /
2)) = ((i · ((-i↑𝑛) − (i↑𝑛))) / 2)) |
| 140 | 139 | oveq1d 7446 |
. . . . . 6
⊢ ((((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) ∧
¬ 2 ∥ 𝑛) →
((-1↑((𝑛 − 1) /
2)) · ((𝐴↑𝑛) / 𝑛)) = (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛))) |
| 141 | 52, 140 | ifeqda 4562 |
. . . . 5
⊢ (((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1) ∧
𝑛 ∈ ℕ) →
if(2 ∥ 𝑛, 0,
((-1↑((𝑛 − 1) /
2)) · ((𝐴↑𝑛) / 𝑛))) = (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛))) |
| 142 | 141 | mpteq2dva 5242 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1)
→ (𝑛 ∈ ℕ
↦ if(2 ∥ 𝑛, 0,
((-1↑((𝑛 − 1) /
2)) · ((𝐴↑𝑛) / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((i ·
((-i↑𝑛) −
(i↑𝑛))) / 2) ·
((𝐴↑𝑛) / 𝑛)))) |
| 143 | 1, 142 | eqtrid 2789 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1)
→ 𝐹 = (𝑛 ∈ ℕ ↦ (((i
· ((-i↑𝑛)
− (i↑𝑛))) / 2)
· ((𝐴↑𝑛) / 𝑛)))) |
| 144 | 143 | seqeq3d 14050 |
. 2
⊢ ((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1)
→ seq1( + , 𝐹) = seq1(
+ , (𝑛 ∈ ℕ
↦ (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛))))) |
| 145 | | eqid 2737 |
. . 3
⊢ (𝑛 ∈ ℕ ↦ (((i
· ((-i↑𝑛)
− (i↑𝑛))) / 2)
· ((𝐴↑𝑛) / 𝑛))) = (𝑛 ∈ ℕ ↦ (((i ·
((-i↑𝑛) −
(i↑𝑛))) / 2) ·
((𝐴↑𝑛) / 𝑛))) |
| 146 | 145 | atantayl 26980 |
. 2
⊢ ((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1)
→ seq1( + , (𝑛 ∈
ℕ ↦ (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛)))) ⇝ (arctan‘𝐴)) |
| 147 | 144, 146 | eqbrtrd 5165 |
1
⊢ ((𝐴 ∈ ℂ ∧
(abs‘𝐴) < 1)
→ seq1( + , 𝐹) ⇝
(arctan‘𝐴)) |