Proof of Theorem atantan
Step | Hyp | Ref
| Expression |
1 | | cosne0 25418 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (cos‘𝐴) ≠ 0) |
2 | | atandmtan 25803 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧
(cos‘𝐴) ≠ 0)
→ (tan‘𝐴) ∈
dom arctan) |
3 | 1, 2 | syldan 594 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (tan‘𝐴) ∈ dom arctan) |
4 | | atanval 25767 |
. . 3
⊢
((tan‘𝐴)
∈ dom arctan → (arctan‘(tan‘𝐴)) = ((i / 2) · ((log‘(1
− (i · (tan‘𝐴)))) − (log‘(1 + (i ·
(tan‘𝐴))))))) |
5 | 3, 4 | syl 17 |
. 2
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (arctan‘(tan‘𝐴)) = ((i / 2) · ((log‘(1
− (i · (tan‘𝐴)))) − (log‘(1 + (i ·
(tan‘𝐴))))))) |
6 | | ax-1cn 10787 |
. . . . . . 7
⊢ 1 ∈
ℂ |
7 | | ax-icn 10788 |
. . . . . . . 8
⊢ i ∈
ℂ |
8 | | tancl 15690 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧
(cos‘𝐴) ≠ 0)
→ (tan‘𝐴) ∈
ℂ) |
9 | 1, 8 | syldan 594 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (tan‘𝐴) ∈ ℂ) |
10 | | mulcl 10813 |
. . . . . . . 8
⊢ ((i
∈ ℂ ∧ (tan‘𝐴) ∈ ℂ) → (i ·
(tan‘𝐴)) ∈
ℂ) |
11 | 7, 9, 10 | sylancr 590 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (i · (tan‘𝐴)) ∈ ℂ) |
12 | | addcl 10811 |
. . . . . . 7
⊢ ((1
∈ ℂ ∧ (i · (tan‘𝐴)) ∈ ℂ) → (1 + (i ·
(tan‘𝐴))) ∈
ℂ) |
13 | 6, 11, 12 | sylancr 590 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (1 + (i · (tan‘𝐴))) ∈
ℂ) |
14 | | atandm2 25760 |
. . . . . . . 8
⊢
((tan‘𝐴)
∈ dom arctan ↔ ((tan‘𝐴) ∈ ℂ ∧ (1 − (i
· (tan‘𝐴)))
≠ 0 ∧ (1 + (i · (tan‘𝐴))) ≠ 0)) |
15 | 3, 14 | sylib 221 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((tan‘𝐴) ∈ ℂ ∧ (1 − (i
· (tan‘𝐴)))
≠ 0 ∧ (1 + (i · (tan‘𝐴))) ≠ 0)) |
16 | 15 | simp3d 1146 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (1 + (i · (tan‘𝐴))) ≠ 0) |
17 | 13, 16 | logcld 25459 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (log‘(1 + (i · (tan‘𝐴)))) ∈
ℂ) |
18 | | subcl 11077 |
. . . . . . 7
⊢ ((1
∈ ℂ ∧ (i · (tan‘𝐴)) ∈ ℂ) → (1 − (i
· (tan‘𝐴)))
∈ ℂ) |
19 | 6, 11, 18 | sylancr 590 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (1 − (i · (tan‘𝐴))) ∈
ℂ) |
20 | 15 | simp2d 1145 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (1 − (i · (tan‘𝐴))) ≠ 0) |
21 | 19, 20 | logcld 25459 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (log‘(1 − (i ·
(tan‘𝐴)))) ∈
ℂ) |
22 | 17, 21 | negsubdi2d 11205 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → -((log‘(1 + (i ·
(tan‘𝐴)))) −
(log‘(1 − (i · (tan‘𝐴))))) = ((log‘(1 − (i ·
(tan‘𝐴)))) −
(log‘(1 + (i · (tan‘𝐴)))))) |
23 | | efsub 15661 |
. . . . . . . . 9
⊢
(((log‘(1 + (i · (tan‘𝐴)))) ∈ ℂ ∧ (log‘(1
− (i · (tan‘𝐴)))) ∈ ℂ) →
(exp‘((log‘(1 + (i · (tan‘𝐴)))) − (log‘(1 − (i
· (tan‘𝐴))))))
= ((exp‘(log‘(1 + (i · (tan‘𝐴))))) / (exp‘(log‘(1 − (i
· (tan‘𝐴))))))) |
24 | 17, 21, 23 | syl2anc 587 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (exp‘((log‘(1 + (i ·
(tan‘𝐴)))) −
(log‘(1 − (i · (tan‘𝐴)))))) = ((exp‘(log‘(1 + (i
· (tan‘𝐴)))))
/ (exp‘(log‘(1 − (i · (tan‘𝐴))))))) |
25 | | coscl 15688 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ ℂ →
(cos‘𝐴) ∈
ℂ) |
26 | 25 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (cos‘𝐴) ∈ ℂ) |
27 | | sincl 15687 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ ℂ →
(sin‘𝐴) ∈
ℂ) |
28 | 27 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (sin‘𝐴) ∈ ℂ) |
29 | | mulcl 10813 |
. . . . . . . . . . . . 13
⊢ ((i
∈ ℂ ∧ (sin‘𝐴) ∈ ℂ) → (i ·
(sin‘𝐴)) ∈
ℂ) |
30 | 7, 28, 29 | sylancr 590 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (i · (sin‘𝐴)) ∈ ℂ) |
31 | 26, 30, 26, 1 | divdird 11646 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (((cos‘𝐴) + (i · (sin‘𝐴))) / (cos‘𝐴)) = (((cos‘𝐴) / (cos‘𝐴)) + ((i · (sin‘𝐴)) / (cos‘𝐴)))) |
32 | 26, 1 | dividd 11606 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((cos‘𝐴) / (cos‘𝐴)) = 1) |
33 | 7 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → i ∈ ℂ) |
34 | 33, 28, 26, 1 | divassd 11643 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((i · (sin‘𝐴)) / (cos‘𝐴)) = (i · ((sin‘𝐴) / (cos‘𝐴)))) |
35 | | tanval 15689 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ℂ ∧
(cos‘𝐴) ≠ 0)
→ (tan‘𝐴) =
((sin‘𝐴) /
(cos‘𝐴))) |
36 | 1, 35 | syldan 594 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴))) |
37 | 36 | oveq2d 7229 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (i · (tan‘𝐴)) = (i · ((sin‘𝐴) / (cos‘𝐴)))) |
38 | 34, 37 | eqtr4d 2780 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((i · (sin‘𝐴)) / (cos‘𝐴)) = (i · (tan‘𝐴))) |
39 | 32, 38 | oveq12d 7231 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (((cos‘𝐴) / (cos‘𝐴)) + ((i · (sin‘𝐴)) / (cos‘𝐴))) = (1 + (i ·
(tan‘𝐴)))) |
40 | 31, 39 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (((cos‘𝐴) + (i · (sin‘𝐴))) / (cos‘𝐴)) = (1 + (i · (tan‘𝐴)))) |
41 | | efival 15713 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ ℂ →
(exp‘(i · 𝐴))
= ((cos‘𝐴) + (i
· (sin‘𝐴)))) |
42 | 41 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (exp‘(i · 𝐴)) = ((cos‘𝐴) + (i · (sin‘𝐴)))) |
43 | 42 | oveq1d 7228 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((exp‘(i · 𝐴)) / (cos‘𝐴)) = (((cos‘𝐴) + (i · (sin‘𝐴))) / (cos‘𝐴))) |
44 | | eflog 25465 |
. . . . . . . . . . 11
⊢ (((1 + (i
· (tan‘𝐴)))
∈ ℂ ∧ (1 + (i · (tan‘𝐴))) ≠ 0) → (exp‘(log‘(1
+ (i · (tan‘𝐴))))) = (1 + (i · (tan‘𝐴)))) |
45 | 13, 16, 44 | syl2anc 587 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (exp‘(log‘(1 + (i ·
(tan‘𝐴))))) = (1 + (i
· (tan‘𝐴)))) |
46 | 40, 43, 45 | 3eqtr4d 2787 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((exp‘(i · 𝐴)) / (cos‘𝐴)) = (exp‘(log‘(1 + (i ·
(tan‘𝐴)))))) |
47 | 26, 30, 26, 1 | divsubdird 11647 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (((cos‘𝐴) − (i · (sin‘𝐴))) / (cos‘𝐴)) = (((cos‘𝐴) / (cos‘𝐴)) − ((i · (sin‘𝐴)) / (cos‘𝐴)))) |
48 | 32, 38 | oveq12d 7231 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (((cos‘𝐴) / (cos‘𝐴)) − ((i · (sin‘𝐴)) / (cos‘𝐴))) = (1 − (i ·
(tan‘𝐴)))) |
49 | 47, 48 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (((cos‘𝐴) − (i · (sin‘𝐴))) / (cos‘𝐴)) = (1 − (i ·
(tan‘𝐴)))) |
50 | | negcl 11078 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ∈ ℂ → -𝐴 ∈
ℂ) |
51 | 50 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → -𝐴 ∈ ℂ) |
52 | | efival 15713 |
. . . . . . . . . . . . . 14
⊢ (-𝐴 ∈ ℂ →
(exp‘(i · -𝐴))
= ((cos‘-𝐴) + (i
· (sin‘-𝐴)))) |
53 | 51, 52 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (exp‘(i · -𝐴)) = ((cos‘-𝐴) + (i · (sin‘-𝐴)))) |
54 | | cosneg 15708 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ∈ ℂ →
(cos‘-𝐴) =
(cos‘𝐴)) |
55 | 54 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (cos‘-𝐴) = (cos‘𝐴)) |
56 | | sinneg 15707 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐴 ∈ ℂ →
(sin‘-𝐴) =
-(sin‘𝐴)) |
57 | 56 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (sin‘-𝐴) = -(sin‘𝐴)) |
58 | 57 | oveq2d 7229 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (i · (sin‘-𝐴)) = (i · -(sin‘𝐴))) |
59 | | mulneg2 11269 |
. . . . . . . . . . . . . . . 16
⊢ ((i
∈ ℂ ∧ (sin‘𝐴) ∈ ℂ) → (i ·
-(sin‘𝐴)) = -(i
· (sin‘𝐴))) |
60 | 7, 28, 59 | sylancr 590 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (i · -(sin‘𝐴)) = -(i · (sin‘𝐴))) |
61 | 58, 60 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (i · (sin‘-𝐴)) = -(i · (sin‘𝐴))) |
62 | 55, 61 | oveq12d 7231 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((cos‘-𝐴) + (i · (sin‘-𝐴))) = ((cos‘𝐴) + -(i ·
(sin‘𝐴)))) |
63 | 53, 62 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (exp‘(i · -𝐴)) = ((cos‘𝐴) + -(i · (sin‘𝐴)))) |
64 | | simpl 486 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → 𝐴 ∈ ℂ) |
65 | | mulneg2 11269 |
. . . . . . . . . . . . . 14
⊢ ((i
∈ ℂ ∧ 𝐴
∈ ℂ) → (i · -𝐴) = -(i · 𝐴)) |
66 | 7, 64, 65 | sylancr 590 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (i · -𝐴) = -(i · 𝐴)) |
67 | 66 | fveq2d 6721 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (exp‘(i · -𝐴)) = (exp‘-(i · 𝐴))) |
68 | 26, 30 | negsubd 11195 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((cos‘𝐴) + -(i · (sin‘𝐴))) = ((cos‘𝐴) − (i ·
(sin‘𝐴)))) |
69 | 63, 67, 68 | 3eqtr3d 2785 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (exp‘-(i · 𝐴)) = ((cos‘𝐴) − (i · (sin‘𝐴)))) |
70 | 69 | oveq1d 7228 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((exp‘-(i · 𝐴)) / (cos‘𝐴)) = (((cos‘𝐴) − (i · (sin‘𝐴))) / (cos‘𝐴))) |
71 | | eflog 25465 |
. . . . . . . . . . 11
⊢ (((1
− (i · (tan‘𝐴))) ∈ ℂ ∧ (1 − (i
· (tan‘𝐴)))
≠ 0) → (exp‘(log‘(1 − (i · (tan‘𝐴))))) = (1 − (i ·
(tan‘𝐴)))) |
72 | 19, 20, 71 | syl2anc 587 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (exp‘(log‘(1 − (i ·
(tan‘𝐴))))) = (1
− (i · (tan‘𝐴)))) |
73 | 49, 70, 72 | 3eqtr4d 2787 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((exp‘-(i · 𝐴)) / (cos‘𝐴)) = (exp‘(log‘(1 − (i
· (tan‘𝐴)))))) |
74 | 46, 73 | oveq12d 7231 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (((exp‘(i · 𝐴)) / (cos‘𝐴)) / ((exp‘-(i · 𝐴)) / (cos‘𝐴))) = ((exp‘(log‘(1
+ (i · (tan‘𝐴))))) / (exp‘(log‘(1 − (i
· (tan‘𝐴))))))) |
75 | | mulcl 10813 |
. . . . . . . . . . . 12
⊢ ((i
∈ ℂ ∧ 𝐴
∈ ℂ) → (i · 𝐴) ∈ ℂ) |
76 | 7, 64, 75 | sylancr 590 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (i · 𝐴) ∈ ℂ) |
77 | | efcl 15644 |
. . . . . . . . . . 11
⊢ ((i
· 𝐴) ∈ ℂ
→ (exp‘(i · 𝐴)) ∈ ℂ) |
78 | 76, 77 | syl 17 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (exp‘(i · 𝐴)) ∈ ℂ) |
79 | 76 | negcld 11176 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → -(i · 𝐴) ∈ ℂ) |
80 | | efcl 15644 |
. . . . . . . . . . 11
⊢ (-(i
· 𝐴) ∈ ℂ
→ (exp‘-(i · 𝐴)) ∈ ℂ) |
81 | 79, 80 | syl 17 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (exp‘-(i · 𝐴)) ∈ ℂ) |
82 | | efne0 15658 |
. . . . . . . . . . 11
⊢ (-(i
· 𝐴) ∈ ℂ
→ (exp‘-(i · 𝐴)) ≠ 0) |
83 | 79, 82 | syl 17 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (exp‘-(i · 𝐴)) ≠ 0) |
84 | 78, 81, 26, 83, 1 | divcan7d 11636 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (((exp‘(i · 𝐴)) / (cos‘𝐴)) / ((exp‘-(i · 𝐴)) / (cos‘𝐴))) = ((exp‘(i ·
𝐴)) / (exp‘-(i
· 𝐴)))) |
85 | | efsub 15661 |
. . . . . . . . . 10
⊢ (((i
· 𝐴) ∈ ℂ
∧ -(i · 𝐴)
∈ ℂ) → (exp‘((i · 𝐴) − -(i · 𝐴))) = ((exp‘(i · 𝐴)) / (exp‘-(i ·
𝐴)))) |
86 | 76, 79, 85 | syl2anc 587 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (exp‘((i · 𝐴) − -(i · 𝐴))) = ((exp‘(i · 𝐴)) / (exp‘-(i ·
𝐴)))) |
87 | 76, 76 | subnegd 11196 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((i · 𝐴) − -(i · 𝐴)) = ((i · 𝐴) + (i · 𝐴))) |
88 | 76 | 2timesd 12073 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (2 · (i · 𝐴)) = ((i · 𝐴) + (i · 𝐴))) |
89 | 87, 88 | eqtr4d 2780 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((i · 𝐴) − -(i · 𝐴)) = (2 · (i · 𝐴))) |
90 | 89 | fveq2d 6721 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (exp‘((i · 𝐴) − -(i · 𝐴))) = (exp‘(2 · (i ·
𝐴)))) |
91 | 84, 86, 90 | 3eqtr2d 2783 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (((exp‘(i · 𝐴)) / (cos‘𝐴)) / ((exp‘-(i · 𝐴)) / (cos‘𝐴))) = (exp‘(2 · (i
· 𝐴)))) |
92 | 24, 74, 91 | 3eqtr2d 2783 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (exp‘((log‘(1 + (i ·
(tan‘𝐴)))) −
(log‘(1 − (i · (tan‘𝐴)))))) = (exp‘(2 · (i ·
𝐴)))) |
93 | 92 | fveq2d 6721 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (log‘(exp‘((log‘(1 + (i
· (tan‘𝐴))))
− (log‘(1 − (i · (tan‘𝐴))))))) = (log‘(exp‘(2 ·
(i · 𝐴))))) |
94 | 64 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) → 𝐴 ∈ ℂ) |
95 | 94 | renegd 14772 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) → (ℜ‘-𝐴) = -(ℜ‘𝐴)) |
96 | 94 | recld 14757 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) → (ℜ‘𝐴) ∈
ℝ) |
97 | 96 | renegcld 11259 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) → -(ℜ‘𝐴) ∈
ℝ) |
98 | | simpr 488 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) → (ℜ‘𝐴) < 0) |
99 | 96 | lt0neg1d 11401 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) → ((ℜ‘𝐴) < 0 ↔ 0 <
-(ℜ‘𝐴))) |
100 | 98, 99 | mpbid 235 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) → 0 < -(ℜ‘𝐴)) |
101 | | eliooord 12994 |
. . . . . . . . . . . . . . . . . . 19
⊢
((ℜ‘𝐴)
∈ (-(π / 2)(,)(π / 2)) → (-(π / 2) < (ℜ‘𝐴) ∧ (ℜ‘𝐴) < (π /
2))) |
102 | 101 | adantl 485 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (-(π / 2) < (ℜ‘𝐴) ∧ (ℜ‘𝐴) < (π /
2))) |
103 | 102 | simpld 498 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → -(π / 2) < (ℜ‘𝐴)) |
104 | 103 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) → -(π / 2) <
(ℜ‘𝐴)) |
105 | | halfpire 25354 |
. . . . . . . . . . . . . . . . 17
⊢ (π /
2) ∈ ℝ |
106 | | ltnegcon1 11333 |
. . . . . . . . . . . . . . . . 17
⊢ (((π /
2) ∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ) → (-(π / 2) <
(ℜ‘𝐴) ↔
-(ℜ‘𝐴) <
(π / 2))) |
107 | 105, 96, 106 | sylancr 590 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) → (-(π / 2) <
(ℜ‘𝐴) ↔
-(ℜ‘𝐴) <
(π / 2))) |
108 | 104, 107 | mpbid 235 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) → -(ℜ‘𝐴) < (π /
2)) |
109 | | 0xr 10880 |
. . . . . . . . . . . . . . . 16
⊢ 0 ∈
ℝ* |
110 | 105 | rexri 10891 |
. . . . . . . . . . . . . . . 16
⊢ (π /
2) ∈ ℝ* |
111 | | elioo2 12976 |
. . . . . . . . . . . . . . . 16
⊢ ((0
∈ ℝ* ∧ (π / 2) ∈ ℝ*) →
(-(ℜ‘𝐴) ∈
(0(,)(π / 2)) ↔ (-(ℜ‘𝐴) ∈ ℝ ∧ 0 <
-(ℜ‘𝐴) ∧
-(ℜ‘𝐴) <
(π / 2)))) |
112 | 109, 110,
111 | mp2an 692 |
. . . . . . . . . . . . . . 15
⊢
(-(ℜ‘𝐴)
∈ (0(,)(π / 2)) ↔ (-(ℜ‘𝐴) ∈ ℝ ∧ 0 <
-(ℜ‘𝐴) ∧
-(ℜ‘𝐴) <
(π / 2))) |
113 | 97, 100, 108, 112 | syl3anbrc 1345 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) → -(ℜ‘𝐴) ∈ (0(,)(π /
2))) |
114 | 95, 113 | eqeltrd 2838 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) → (ℜ‘-𝐴) ∈ (0(,)(π /
2))) |
115 | | tanregt0 25428 |
. . . . . . . . . . . . 13
⊢ ((-𝐴 ∈ ℂ ∧
(ℜ‘-𝐴) ∈
(0(,)(π / 2))) → 0 < (ℜ‘(tan‘-𝐴))) |
116 | 51, 114, 115 | syl2an2r 685 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) → 0 <
(ℜ‘(tan‘-𝐴))) |
117 | | tanneg 15709 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ∈ ℂ ∧
(cos‘𝐴) ≠ 0)
→ (tan‘-𝐴) =
-(tan‘𝐴)) |
118 | 1, 117 | syldan 594 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (tan‘-𝐴) = -(tan‘𝐴)) |
119 | 118 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) → (tan‘-𝐴) = -(tan‘𝐴)) |
120 | 119 | fveq2d 6721 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) →
(ℜ‘(tan‘-𝐴)) = (ℜ‘-(tan‘𝐴))) |
121 | 9 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) → (tan‘𝐴) ∈
ℂ) |
122 | 121 | renegd 14772 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) →
(ℜ‘-(tan‘𝐴)) = -(ℜ‘(tan‘𝐴))) |
123 | 120, 122 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) →
(ℜ‘(tan‘-𝐴)) = -(ℜ‘(tan‘𝐴))) |
124 | 116, 123 | breqtrd 5079 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) → 0 <
-(ℜ‘(tan‘𝐴))) |
125 | 9 | recld 14757 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (ℜ‘(tan‘𝐴)) ∈ ℝ) |
126 | 125 | adantr 484 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) →
(ℜ‘(tan‘𝐴)) ∈ ℝ) |
127 | 126 | lt0neg1d 11401 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) →
((ℜ‘(tan‘𝐴)) < 0 ↔ 0 <
-(ℜ‘(tan‘𝐴)))) |
128 | 124, 127 | mpbird 260 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) →
(ℜ‘(tan‘𝐴)) < 0) |
129 | 128 | lt0ne0d 11397 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) →
(ℜ‘(tan‘𝐴)) ≠ 0) |
130 | | atanlogsub 25799 |
. . . . . . . . 9
⊢
(((tan‘𝐴)
∈ dom arctan ∧ (ℜ‘(tan‘𝐴)) ≠ 0) → ((log‘(1 + (i
· (tan‘𝐴))))
− (log‘(1 − (i · (tan‘𝐴))))) ∈ ran log) |
131 | 3, 129, 130 | syl2an2r 685 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) < 0) → ((log‘(1 + (i
· (tan‘𝐴))))
− (log‘(1 − (i · (tan‘𝐴))))) ∈ ran log) |
132 | | 1re 10833 |
. . . . . . . . . . . . 13
⊢ 1 ∈
ℝ |
133 | | ioossre 12996 |
. . . . . . . . . . . . . 14
⊢ (-1(,)1)
⊆ ℝ |
134 | 7 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → i ∈
ℂ) |
135 | 11 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (i · (tan‘𝐴)) ∈
ℂ) |
136 | | ine0 11267 |
. . . . . . . . . . . . . . . . 17
⊢ i ≠
0 |
137 | 136 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → i ≠ 0) |
138 | | ixi 11461 |
. . . . . . . . . . . . . . . . . . 19
⊢ (i
· i) = -1 |
139 | 138 | oveq1i 7223 |
. . . . . . . . . . . . . . . . . 18
⊢ ((i
· i) · (tan‘𝐴)) = (-1 · (tan‘𝐴)) |
140 | 9 | adantr 484 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (tan‘𝐴) ∈ ℂ) |
141 | 140 | mulm1d 11284 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (-1 · (tan‘𝐴)) = -(tan‘𝐴)) |
142 | 118 | adantr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (tan‘-𝐴) = -(tan‘𝐴)) |
143 | 141, 142 | eqtr4d 2780 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (-1 · (tan‘𝐴)) = (tan‘-𝐴)) |
144 | 139, 143 | syl5eq 2790 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → ((i · i) ·
(tan‘𝐴)) =
(tan‘-𝐴)) |
145 | 134, 134,
140 | mulassd 10856 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → ((i · i) ·
(tan‘𝐴)) = (i
· (i · (tan‘𝐴)))) |
146 | 138 | oveq1i 7223 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((i
· i) · 𝐴) =
(-1 · 𝐴) |
147 | 64 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → 𝐴 ∈ ℂ) |
148 | 147 | mulm1d 11284 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (-1 · 𝐴) = -𝐴) |
149 | 146, 148 | syl5eq 2790 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → ((i · i) · 𝐴) = -𝐴) |
150 | 134, 134,
147 | mulassd 10856 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → ((i · i) · 𝐴) = (i · (i ·
𝐴))) |
151 | 149, 150 | eqtr3d 2779 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → -𝐴 = (i · (i · 𝐴))) |
152 | 151 | fveq2d 6721 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (tan‘-𝐴) = (tan‘(i · (i · 𝐴)))) |
153 | 144, 145,
152 | 3eqtr3d 2785 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (i · (i ·
(tan‘𝐴))) =
(tan‘(i · (i · 𝐴)))) |
154 | 134, 135,
137, 153 | mvllmuld 11664 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (i · (tan‘𝐴)) = ((tan‘(i · (i
· 𝐴))) /
i)) |
155 | 76 | adantr 484 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (i · 𝐴) ∈ ℂ) |
156 | | reim 14672 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐴 ∈ ℂ →
(ℜ‘𝐴) =
(ℑ‘(i · 𝐴))) |
157 | 156 | adantr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (ℜ‘𝐴) = (ℑ‘(i · 𝐴))) |
158 | 157 | eqeq1d 2739 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((ℜ‘𝐴) = 0 ↔ (ℑ‘(i ·
𝐴)) = 0)) |
159 | 158 | biimpa 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (ℑ‘(i ·
𝐴)) = 0) |
160 | 155, 159 | reim0bd 14763 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (i · 𝐴) ∈ ℝ) |
161 | | tanhbnd 15722 |
. . . . . . . . . . . . . . . 16
⊢ ((i
· 𝐴) ∈ ℝ
→ ((tan‘(i · (i · 𝐴))) / i) ∈ (-1(,)1)) |
162 | 160, 161 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → ((tan‘(i · (i
· 𝐴))) / i) ∈
(-1(,)1)) |
163 | 154, 162 | eqeltrd 2838 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (i · (tan‘𝐴)) ∈
(-1(,)1)) |
164 | 133, 163 | sseldi 3899 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (i · (tan‘𝐴)) ∈
ℝ) |
165 | | readdcl 10812 |
. . . . . . . . . . . . 13
⊢ ((1
∈ ℝ ∧ (i · (tan‘𝐴)) ∈ ℝ) → (1 + (i ·
(tan‘𝐴))) ∈
ℝ) |
166 | 132, 164,
165 | sylancr 590 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (1 + (i ·
(tan‘𝐴))) ∈
ℝ) |
167 | | df-neg 11065 |
. . . . . . . . . . . . . 14
⊢ -1 = (0
− 1) |
168 | | eliooord 12994 |
. . . . . . . . . . . . . . . 16
⊢ ((i
· (tan‘𝐴))
∈ (-1(,)1) → (-1 < (i · (tan‘𝐴)) ∧ (i · (tan‘𝐴)) < 1)) |
169 | 163, 168 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (-1 < (i ·
(tan‘𝐴)) ∧ (i
· (tan‘𝐴))
< 1)) |
170 | 169 | simpld 498 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → -1 < (i ·
(tan‘𝐴))) |
171 | 167, 170 | eqbrtrrid 5089 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (0 − 1) < (i ·
(tan‘𝐴))) |
172 | | 0red 10836 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → 0 ∈
ℝ) |
173 | 132 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → 1 ∈
ℝ) |
174 | 172, 173,
164 | ltsubadd2d 11430 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → ((0 − 1) < (i
· (tan‘𝐴))
↔ 0 < (1 + (i · (tan‘𝐴))))) |
175 | 171, 174 | mpbid 235 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → 0 < (1 + (i ·
(tan‘𝐴)))) |
176 | 166, 175 | elrpd 12625 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (1 + (i ·
(tan‘𝐴))) ∈
ℝ+) |
177 | 176 | relogcld 25511 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (log‘(1 + (i ·
(tan‘𝐴)))) ∈
ℝ) |
178 | 169 | simprd 499 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (i · (tan‘𝐴)) < 1) |
179 | | difrp 12624 |
. . . . . . . . . . . . 13
⊢ (((i
· (tan‘𝐴))
∈ ℝ ∧ 1 ∈ ℝ) → ((i · (tan‘𝐴)) < 1 ↔ (1 − (i
· (tan‘𝐴)))
∈ ℝ+)) |
180 | 164, 132,
179 | sylancl 589 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → ((i · (tan‘𝐴)) < 1 ↔ (1 − (i
· (tan‘𝐴)))
∈ ℝ+)) |
181 | 178, 180 | mpbid 235 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (1 − (i ·
(tan‘𝐴))) ∈
ℝ+) |
182 | 181 | relogcld 25511 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → (log‘(1 − (i
· (tan‘𝐴))))
∈ ℝ) |
183 | 177, 182 | resubcld 11260 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → ((log‘(1 + (i ·
(tan‘𝐴)))) −
(log‘(1 − (i · (tan‘𝐴))))) ∈ ℝ) |
184 | | relogrn 25450 |
. . . . . . . . 9
⊢
(((log‘(1 + (i · (tan‘𝐴)))) − (log‘(1 − (i
· (tan‘𝐴)))))
∈ ℝ → ((log‘(1 + (i · (tan‘𝐴)))) − (log‘(1 − (i
· (tan‘𝐴)))))
∈ ran log) |
185 | 183, 184 | syl 17 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ (ℜ‘𝐴) = 0) → ((log‘(1 + (i ·
(tan‘𝐴)))) −
(log‘(1 − (i · (tan‘𝐴))))) ∈ ran log) |
186 | 64 | adantr 484 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ 0 < (ℜ‘𝐴)) → 𝐴 ∈ ℂ) |
187 | 186 | recld 14757 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ 0 < (ℜ‘𝐴)) → (ℜ‘𝐴) ∈ ℝ) |
188 | | simpr 488 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ 0 < (ℜ‘𝐴)) → 0 < (ℜ‘𝐴)) |
189 | 102 | simprd 499 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (ℜ‘𝐴) < (π / 2)) |
190 | 189 | adantr 484 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ 0 < (ℜ‘𝐴)) → (ℜ‘𝐴) < (π / 2)) |
191 | | elioo2 12976 |
. . . . . . . . . . . . 13
⊢ ((0
∈ ℝ* ∧ (π / 2) ∈ ℝ*) →
((ℜ‘𝐴) ∈
(0(,)(π / 2)) ↔ ((ℜ‘𝐴) ∈ ℝ ∧ 0 <
(ℜ‘𝐴) ∧
(ℜ‘𝐴) < (π
/ 2)))) |
192 | 109, 110,
191 | mp2an 692 |
. . . . . . . . . . . 12
⊢
((ℜ‘𝐴)
∈ (0(,)(π / 2)) ↔ ((ℜ‘𝐴) ∈ ℝ ∧ 0 <
(ℜ‘𝐴) ∧
(ℜ‘𝐴) < (π
/ 2))) |
193 | 187, 188,
190, 192 | syl3anbrc 1345 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ 0 < (ℜ‘𝐴)) → (ℜ‘𝐴) ∈ (0(,)(π / 2))) |
194 | | tanregt0 25428 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(0(,)(π / 2))) → 0 < (ℜ‘(tan‘𝐴))) |
195 | 64, 193, 194 | syl2an2r 685 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ 0 < (ℜ‘𝐴)) → 0 <
(ℜ‘(tan‘𝐴))) |
196 | 195 | gt0ne0d 11396 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ 0 < (ℜ‘𝐴)) → (ℜ‘(tan‘𝐴)) ≠ 0) |
197 | 3, 196, 130 | syl2an2r 685 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) ∧ 0 < (ℜ‘𝐴)) → ((log‘(1 + (i ·
(tan‘𝐴)))) −
(log‘(1 − (i · (tan‘𝐴))))) ∈ ran log) |
198 | | recl 14673 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ℂ →
(ℜ‘𝐴) ∈
ℝ) |
199 | 198 | adantr 484 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (ℜ‘𝐴) ∈ ℝ) |
200 | | 0re 10835 |
. . . . . . . . 9
⊢ 0 ∈
ℝ |
201 | | lttri4 10917 |
. . . . . . . . 9
⊢
(((ℜ‘𝐴)
∈ ℝ ∧ 0 ∈ ℝ) → ((ℜ‘𝐴) < 0 ∨ (ℜ‘𝐴) = 0 ∨ 0 <
(ℜ‘𝐴))) |
202 | 199, 200,
201 | sylancl 589 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((ℜ‘𝐴) < 0 ∨ (ℜ‘𝐴) = 0 ∨ 0 <
(ℜ‘𝐴))) |
203 | 131, 185,
197, 202 | mpjao3dan 1433 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((log‘(1 + (i ·
(tan‘𝐴)))) −
(log‘(1 − (i · (tan‘𝐴))))) ∈ ran log) |
204 | | logef 25470 |
. . . . . . 7
⊢
(((log‘(1 + (i · (tan‘𝐴)))) − (log‘(1 − (i
· (tan‘𝐴)))))
∈ ran log → (log‘(exp‘((log‘(1 + (i ·
(tan‘𝐴)))) −
(log‘(1 − (i · (tan‘𝐴))))))) = ((log‘(1 + (i ·
(tan‘𝐴)))) −
(log‘(1 − (i · (tan‘𝐴)))))) |
205 | 203, 204 | syl 17 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (log‘(exp‘((log‘(1 + (i
· (tan‘𝐴))))
− (log‘(1 − (i · (tan‘𝐴))))))) = ((log‘(1 + (i ·
(tan‘𝐴)))) −
(log‘(1 − (i · (tan‘𝐴)))))) |
206 | | 2cn 11905 |
. . . . . . . . 9
⊢ 2 ∈
ℂ |
207 | | mulcl 10813 |
. . . . . . . . 9
⊢ ((2
∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (2 · (i
· 𝐴)) ∈
ℂ) |
208 | 206, 76, 207 | sylancr 590 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (2 · (i · 𝐴)) ∈ ℂ) |
209 | | picn 25349 |
. . . . . . . . . . . 12
⊢ π
∈ ℂ |
210 | | 2ne0 11934 |
. . . . . . . . . . . 12
⊢ 2 ≠
0 |
211 | | divneg 11524 |
. . . . . . . . . . . 12
⊢ ((π
∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -(π / 2) =
(-π / 2)) |
212 | 209, 206,
210, 211 | mp3an 1463 |
. . . . . . . . . . 11
⊢ -(π /
2) = (-π / 2) |
213 | 212, 103 | eqbrtrrid 5089 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (-π / 2) < (ℜ‘𝐴)) |
214 | | pire 25348 |
. . . . . . . . . . . . 13
⊢ π
∈ ℝ |
215 | 214 | renegcli 11139 |
. . . . . . . . . . . 12
⊢ -π
∈ ℝ |
216 | 215 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → -π ∈ ℝ) |
217 | | 2re 11904 |
. . . . . . . . . . . 12
⊢ 2 ∈
ℝ |
218 | 217 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → 2 ∈ ℝ) |
219 | | 2pos 11933 |
. . . . . . . . . . . 12
⊢ 0 <
2 |
220 | 219 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → 0 < 2) |
221 | | ltdivmul 11707 |
. . . . . . . . . . 11
⊢ ((-π
∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ ∧ (2 ∈ ℝ
∧ 0 < 2)) → ((-π / 2) < (ℜ‘𝐴) ↔ -π < (2 ·
(ℜ‘𝐴)))) |
222 | 216, 199,
218, 220, 221 | syl112anc 1376 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((-π / 2) < (ℜ‘𝐴) ↔ -π < (2 ·
(ℜ‘𝐴)))) |
223 | 213, 222 | mpbid 235 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → -π < (2 · (ℜ‘𝐴))) |
224 | | immul2 14700 |
. . . . . . . . . . 11
⊢ ((2
∈ ℝ ∧ (i · 𝐴) ∈ ℂ) → (ℑ‘(2
· (i · 𝐴))) =
(2 · (ℑ‘(i · 𝐴)))) |
225 | 217, 76, 224 | sylancr 590 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (ℑ‘(2 · (i · 𝐴))) = (2 ·
(ℑ‘(i · 𝐴)))) |
226 | 157 | oveq2d 7229 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (2 · (ℜ‘𝐴)) = (2 · (ℑ‘(i ·
𝐴)))) |
227 | 225, 226 | eqtr4d 2780 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (ℑ‘(2 · (i · 𝐴))) = (2 ·
(ℜ‘𝐴))) |
228 | 223, 227 | breqtrrd 5081 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → -π < (ℑ‘(2 · (i
· 𝐴)))) |
229 | | remulcl 10814 |
. . . . . . . . . . 11
⊢ ((2
∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ) → (2 ·
(ℜ‘𝐴)) ∈
ℝ) |
230 | 217, 199,
229 | sylancr 590 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (2 · (ℜ‘𝐴)) ∈ ℝ) |
231 | 214 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → π ∈ ℝ) |
232 | | ltmuldiv2 11706 |
. . . . . . . . . . . 12
⊢
(((ℜ‘𝐴)
∈ ℝ ∧ π ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 <
2)) → ((2 · (ℜ‘𝐴)) < π ↔ (ℜ‘𝐴) < (π /
2))) |
233 | 199, 231,
218, 220, 232 | syl112anc 1376 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((2 · (ℜ‘𝐴)) < π ↔ (ℜ‘𝐴) < (π /
2))) |
234 | 189, 233 | mpbird 260 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (2 · (ℜ‘𝐴)) < π) |
235 | 230, 231,
234 | ltled 10980 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (2 · (ℜ‘𝐴)) ≤ π) |
236 | 227, 235 | eqbrtrd 5075 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (ℑ‘(2 · (i · 𝐴))) ≤ π) |
237 | | ellogrn 25448 |
. . . . . . . 8
⊢ ((2
· (i · 𝐴))
∈ ran log ↔ ((2 · (i · 𝐴)) ∈ ℂ ∧ -π <
(ℑ‘(2 · (i · 𝐴))) ∧ (ℑ‘(2 · (i
· 𝐴))) ≤
π)) |
238 | 208, 228,
236, 237 | syl3anbrc 1345 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (2 · (i · 𝐴)) ∈ ran log) |
239 | | logef 25470 |
. . . . . . 7
⊢ ((2
· (i · 𝐴))
∈ ran log → (log‘(exp‘(2 · (i · 𝐴)))) = (2 · (i ·
𝐴))) |
240 | 238, 239 | syl 17 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (log‘(exp‘(2 · (i
· 𝐴)))) = (2
· (i · 𝐴))) |
241 | 93, 205, 240 | 3eqtr3d 2785 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((log‘(1 + (i ·
(tan‘𝐴)))) −
(log‘(1 − (i · (tan‘𝐴))))) = (2 · (i · 𝐴))) |
242 | 241 | negeqd 11072 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → -((log‘(1 + (i ·
(tan‘𝐴)))) −
(log‘(1 − (i · (tan‘𝐴))))) = -(2 · (i · 𝐴))) |
243 | 22, 242 | eqtr3d 2779 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((log‘(1 − (i ·
(tan‘𝐴)))) −
(log‘(1 + (i · (tan‘𝐴))))) = -(2 · (i · 𝐴))) |
244 | 243 | oveq2d 7229 |
. 2
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((i / 2) · ((log‘(1 − (i
· (tan‘𝐴))))
− (log‘(1 + (i · (tan‘𝐴)))))) = ((i / 2) · -(2 · (i
· 𝐴)))) |
245 | | halfcl 12055 |
. . . . 5
⊢ (i ∈
ℂ → (i / 2) ∈ ℂ) |
246 | 7, 245 | mp1i 13 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (i / 2) ∈ ℂ) |
247 | 206 | a1i 11 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → 2 ∈ ℂ) |
248 | 246, 247,
79 | mulassd 10856 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (((i / 2) · 2) · -(i ·
𝐴)) = ((i / 2) · (2
· -(i · 𝐴)))) |
249 | 7, 206, 210 | divcan1i 11576 |
. . . . 5
⊢ ((i / 2)
· 2) = i |
250 | 249 | oveq1i 7223 |
. . . 4
⊢ (((i / 2)
· 2) · -(i · 𝐴)) = (i · -(i · 𝐴)) |
251 | 33, 33, 51 | mulassd 10856 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((i · i) · -𝐴) = (i · (i · -𝐴))) |
252 | 138 | oveq1i 7223 |
. . . . . 6
⊢ ((i
· i) · -𝐴) =
(-1 · -𝐴) |
253 | | mul2neg 11271 |
. . . . . . . 8
⊢ ((1
∈ ℂ ∧ 𝐴
∈ ℂ) → (-1 · -𝐴) = (1 · 𝐴)) |
254 | 6, 64, 253 | sylancr 590 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (-1 · -𝐴) = (1 · 𝐴)) |
255 | | mulid2 10832 |
. . . . . . . 8
⊢ (𝐴 ∈ ℂ → (1
· 𝐴) = 𝐴) |
256 | 255 | adantr 484 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (1 · 𝐴) = 𝐴) |
257 | 254, 256 | eqtrd 2777 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (-1 · -𝐴) = 𝐴) |
258 | 252, 257 | syl5eq 2790 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((i · i) · -𝐴) = 𝐴) |
259 | 66 | oveq2d 7229 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (i · (i · -𝐴)) = (i · -(i · 𝐴))) |
260 | 251, 258,
259 | 3eqtr3rd 2786 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (i · -(i · 𝐴)) = 𝐴) |
261 | 250, 260 | syl5eq 2790 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (((i / 2) · 2) · -(i ·
𝐴)) = 𝐴) |
262 | | mulneg2 11269 |
. . . . 5
⊢ ((2
∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (2 · -(i
· 𝐴)) = -(2 ·
(i · 𝐴))) |
263 | 206, 76, 262 | sylancr 590 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (2 · -(i · 𝐴)) = -(2 · (i · 𝐴))) |
264 | 263 | oveq2d 7229 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((i / 2) · (2 · -(i ·
𝐴))) = ((i / 2) ·
-(2 · (i · 𝐴)))) |
265 | 248, 261,
264 | 3eqtr3rd 2786 |
. 2
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → ((i / 2) · -(2 · (i ·
𝐴))) = 𝐴) |
266 | 5, 244, 265 | 3eqtrd 2781 |
1
⊢ ((𝐴 ∈ ℂ ∧
(ℜ‘𝐴) ∈
(-(π / 2)(,)(π / 2))) → (arctan‘(tan‘𝐴)) = 𝐴) |