| Step |
Hyp |
Ref |
Expression |
| 1 |
|
fveq2 |
⊢ ( 𝐴 = 0 → ( ℜ ‘ 𝐴 ) = ( ℜ ‘ 0 ) ) |
| 2 |
|
re0 |
⊢ ( ℜ ‘ 0 ) = 0 |
| 3 |
1 2
|
eqtrdi |
⊢ ( 𝐴 = 0 → ( ℜ ‘ 𝐴 ) = 0 ) |
| 4 |
3
|
necon3i |
⊢ ( ( ℜ ‘ 𝐴 ) ≠ 0 → 𝐴 ≠ 0 ) |
| 5 |
|
logcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( log ‘ 𝐴 ) ∈ ℂ ) |
| 6 |
4 5
|
sylan2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( log ‘ 𝐴 ) ∈ ℂ ) |
| 7 |
6
|
imcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ℑ ‘ ( log ‘ 𝐴 ) ) ∈ ℝ ) |
| 8 |
7
|
recnd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ℑ ‘ ( log ‘ 𝐴 ) ) ∈ ℂ ) |
| 9 |
|
sqcl |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ 2 ) ∈ ℂ ) |
| 10 |
9
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( 𝐴 ↑ 2 ) ∈ ℂ ) |
| 11 |
|
abscl |
⊢ ( 𝐴 ∈ ℂ → ( abs ‘ 𝐴 ) ∈ ℝ ) |
| 12 |
11
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( abs ‘ 𝐴 ) ∈ ℝ ) |
| 13 |
12
|
recnd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( abs ‘ 𝐴 ) ∈ ℂ ) |
| 14 |
13
|
sqcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( abs ‘ 𝐴 ) ↑ 2 ) ∈ ℂ ) |
| 15 |
|
absrpcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( abs ‘ 𝐴 ) ∈ ℝ+ ) |
| 16 |
4 15
|
sylan2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( abs ‘ 𝐴 ) ∈ ℝ+ ) |
| 17 |
16
|
rpne0d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( abs ‘ 𝐴 ) ≠ 0 ) |
| 18 |
|
sqne0 |
⊢ ( ( abs ‘ 𝐴 ) ∈ ℂ → ( ( ( abs ‘ 𝐴 ) ↑ 2 ) ≠ 0 ↔ ( abs ‘ 𝐴 ) ≠ 0 ) ) |
| 19 |
13 18
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( abs ‘ 𝐴 ) ↑ 2 ) ≠ 0 ↔ ( abs ‘ 𝐴 ) ≠ 0 ) ) |
| 20 |
17 19
|
mpbird |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( abs ‘ 𝐴 ) ↑ 2 ) ≠ 0 ) |
| 21 |
10 14 14 20
|
divdird |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) = ( ( ( 𝐴 ↑ 2 ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) + ( ( ( abs ‘ 𝐴 ) ↑ 2 ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) ) |
| 22 |
|
ax-icn |
⊢ i ∈ ℂ |
| 23 |
|
mulcl |
⊢ ( ( i ∈ ℂ ∧ ( ℑ ‘ ( log ‘ 𝐴 ) ) ∈ ℂ ) → ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ∈ ℂ ) |
| 24 |
22 8 23
|
sylancr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ∈ ℂ ) |
| 25 |
|
2z |
⊢ 2 ∈ ℤ |
| 26 |
|
efexp |
⊢ ( ( ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ∈ ℂ ∧ 2 ∈ ℤ ) → ( exp ‘ ( 2 · ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) ) = ( ( exp ‘ ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) ↑ 2 ) ) |
| 27 |
24 25 26
|
sylancl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( exp ‘ ( 2 · ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) ) = ( ( exp ‘ ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) ↑ 2 ) ) |
| 28 |
|
efiarg |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( exp ‘ ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) = ( 𝐴 / ( abs ‘ 𝐴 ) ) ) |
| 29 |
4 28
|
sylan2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( exp ‘ ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) = ( 𝐴 / ( abs ‘ 𝐴 ) ) ) |
| 30 |
29
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( exp ‘ ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) ↑ 2 ) = ( ( 𝐴 / ( abs ‘ 𝐴 ) ) ↑ 2 ) ) |
| 31 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → 𝐴 ∈ ℂ ) |
| 32 |
31 13 17
|
sqdivd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 𝐴 / ( abs ‘ 𝐴 ) ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) |
| 33 |
27 30 32
|
3eqtrrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 𝐴 ↑ 2 ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) = ( exp ‘ ( 2 · ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) ) ) |
| 34 |
14 20
|
dividd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( abs ‘ 𝐴 ) ↑ 2 ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) = 1 ) |
| 35 |
33 34
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( 𝐴 ↑ 2 ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) + ( ( ( abs ‘ 𝐴 ) ↑ 2 ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) = ( ( exp ‘ ( 2 · ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) ) + 1 ) ) |
| 36 |
21 35
|
eqtr2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( exp ‘ ( 2 · ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) ) + 1 ) = ( ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) |
| 37 |
10 14
|
addcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ∈ ℂ ) |
| 38 |
22
|
a1i |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → i ∈ ℂ ) |
| 39 |
|
2cn |
⊢ 2 ∈ ℂ |
| 40 |
|
recl |
⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℝ ) |
| 41 |
40
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ℜ ‘ 𝐴 ) ∈ ℝ ) |
| 42 |
41
|
recnd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ℜ ‘ 𝐴 ) ∈ ℂ ) |
| 43 |
42
|
sqcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ∈ ℂ ) |
| 44 |
|
mulcl |
⊢ ( ( 2 ∈ ℂ ∧ ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ∈ ℂ ) → ( 2 · ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) ∈ ℂ ) |
| 45 |
39 43 44
|
sylancr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( 2 · ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) ∈ ℂ ) |
| 46 |
39
|
a1i |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → 2 ∈ ℂ ) |
| 47 |
|
imcl |
⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℝ ) |
| 48 |
47
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ℑ ‘ 𝐴 ) ∈ ℝ ) |
| 49 |
48
|
recnd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ℑ ‘ 𝐴 ) ∈ ℂ ) |
| 50 |
42 49
|
mulcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ∈ ℂ ) |
| 51 |
38 46 50
|
mul12d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( i · ( 2 · ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) = ( 2 · ( i · ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) ) |
| 52 |
38 42 49
|
mul12d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( i · ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) = ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) |
| 53 |
52
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( 2 · ( i · ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) = ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) ) |
| 54 |
51 53
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( i · ( 2 · ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) = ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) ) |
| 55 |
|
mulcl |
⊢ ( ( i ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) ∈ ℂ ) → ( i · ( ℑ ‘ 𝐴 ) ) ∈ ℂ ) |
| 56 |
22 49 55
|
sylancr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( i · ( ℑ ‘ 𝐴 ) ) ∈ ℂ ) |
| 57 |
42 56
|
mulcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ∈ ℂ ) |
| 58 |
|
mulcl |
⊢ ( ( 2 ∈ ℂ ∧ ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ∈ ℂ ) → ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) ∈ ℂ ) |
| 59 |
39 57 58
|
sylancr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) ∈ ℂ ) |
| 60 |
54 59
|
eqeltrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( i · ( 2 · ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) ∈ ℂ ) |
| 61 |
38 45 60
|
adddid |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( i · ( ( 2 · ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) + ( i · ( 2 · ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) ) ) = ( ( i · ( 2 · ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) ) + ( i · ( i · ( 2 · ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) ) ) ) |
| 62 |
|
mulcl |
⊢ ( ( ( ℜ ‘ 𝐴 ) ∈ ℂ ∧ i ∈ ℂ ) → ( ( ℜ ‘ 𝐴 ) · i ) ∈ ℂ ) |
| 63 |
42 22 62
|
sylancl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ℜ ‘ 𝐴 ) · i ) ∈ ℂ ) |
| 64 |
46 63 42
|
mulassd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) · ( ℜ ‘ 𝐴 ) ) = ( 2 · ( ( ( ℜ ‘ 𝐴 ) · i ) · ( ℜ ‘ 𝐴 ) ) ) ) |
| 65 |
42
|
sqvald |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ℜ ‘ 𝐴 ) ↑ 2 ) = ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐴 ) ) ) |
| 66 |
65
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) · i ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐴 ) ) · i ) ) |
| 67 |
|
mulcom |
⊢ ( ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ∈ ℂ ∧ i ∈ ℂ ) → ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) · i ) = ( i · ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) ) |
| 68 |
43 22 67
|
sylancl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) · i ) = ( i · ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) ) |
| 69 |
42 42 38
|
mul32d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐴 ) ) · i ) = ( ( ( ℜ ‘ 𝐴 ) · i ) · ( ℜ ‘ 𝐴 ) ) ) |
| 70 |
66 68 69
|
3eqtr3d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( i · ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) = ( ( ( ℜ ‘ 𝐴 ) · i ) · ( ℜ ‘ 𝐴 ) ) ) |
| 71 |
70
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( 2 · ( i · ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) ) = ( 2 · ( ( ( ℜ ‘ 𝐴 ) · i ) · ( ℜ ‘ 𝐴 ) ) ) ) |
| 72 |
46 38 43
|
mul12d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( 2 · ( i · ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) ) = ( i · ( 2 · ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) ) ) |
| 73 |
64 71 72
|
3eqtr2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) · ( ℜ ‘ 𝐴 ) ) = ( i · ( 2 · ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) ) ) |
| 74 |
|
ixi |
⊢ ( i · i ) = - 1 |
| 75 |
74
|
oveq1i |
⊢ ( ( i · i ) · ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℜ ‘ 𝐴 ) ) ) = ( - 1 · ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℜ ‘ 𝐴 ) ) ) |
| 76 |
|
mulcl |
⊢ ( ( 2 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) ∈ ℂ ) → ( 2 · ( ℑ ‘ 𝐴 ) ) ∈ ℂ ) |
| 77 |
39 49 76
|
sylancr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( 2 · ( ℑ ‘ 𝐴 ) ) ∈ ℂ ) |
| 78 |
77 42
|
mulcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℜ ‘ 𝐴 ) ) ∈ ℂ ) |
| 79 |
38 38 78
|
mulassd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( i · i ) · ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℜ ‘ 𝐴 ) ) ) = ( i · ( i · ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℜ ‘ 𝐴 ) ) ) ) ) |
| 80 |
75 79
|
eqtr3id |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( - 1 · ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℜ ‘ 𝐴 ) ) ) = ( i · ( i · ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℜ ‘ 𝐴 ) ) ) ) ) |
| 81 |
78
|
mulm1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( - 1 · ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℜ ‘ 𝐴 ) ) ) = - ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℜ ‘ 𝐴 ) ) ) |
| 82 |
46 49 42
|
mulassd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℜ ‘ 𝐴 ) ) = ( 2 · ( ( ℑ ‘ 𝐴 ) · ( ℜ ‘ 𝐴 ) ) ) ) |
| 83 |
49 42
|
mulcomd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ℑ ‘ 𝐴 ) · ( ℜ ‘ 𝐴 ) ) = ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) |
| 84 |
83
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( 2 · ( ( ℑ ‘ 𝐴 ) · ( ℜ ‘ 𝐴 ) ) ) = ( 2 · ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) |
| 85 |
82 84
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℜ ‘ 𝐴 ) ) = ( 2 · ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) |
| 86 |
85
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( i · ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℜ ‘ 𝐴 ) ) ) = ( i · ( 2 · ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) ) |
| 87 |
86
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( i · ( i · ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℜ ‘ 𝐴 ) ) ) ) = ( i · ( i · ( 2 · ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) ) ) |
| 88 |
80 81 87
|
3eqtr3d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → - ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℜ ‘ 𝐴 ) ) = ( i · ( i · ( 2 · ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) ) ) |
| 89 |
73 88
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) · ( ℜ ‘ 𝐴 ) ) + - ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℜ ‘ 𝐴 ) ) ) = ( ( i · ( 2 · ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) ) + ( i · ( i · ( 2 · ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) ) ) ) |
| 90 |
|
mulcl |
⊢ ( ( 2 ∈ ℂ ∧ ( ( ℜ ‘ 𝐴 ) · i ) ∈ ℂ ) → ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) ∈ ℂ ) |
| 91 |
39 63 90
|
sylancr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) ∈ ℂ ) |
| 92 |
91 42
|
mulcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) · ( ℜ ‘ 𝐴 ) ) ∈ ℂ ) |
| 93 |
92 78
|
negsubd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) · ( ℜ ‘ 𝐴 ) ) + - ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℜ ‘ 𝐴 ) ) ) = ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) · ( ℜ ‘ 𝐴 ) ) − ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℜ ‘ 𝐴 ) ) ) ) |
| 94 |
61 89 93
|
3eqtr2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( i · ( ( 2 · ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) + ( i · ( 2 · ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) ) ) = ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) · ( ℜ ‘ 𝐴 ) ) − ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℜ ‘ 𝐴 ) ) ) ) |
| 95 |
49
|
sqcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ∈ ℂ ) |
| 96 |
59 95
|
subcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) − ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ∈ ℂ ) |
| 97 |
43 96 43 95
|
add4d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) − ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) + ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) = ( ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) + ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) − ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) ) |
| 98 |
|
replim |
⊢ ( 𝐴 ∈ ℂ → 𝐴 = ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) ) |
| 99 |
98
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → 𝐴 = ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) ) |
| 100 |
99
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( 𝐴 ↑ 2 ) = ( ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) ↑ 2 ) ) |
| 101 |
|
binom2 |
⊢ ( ( ( ℜ ‘ 𝐴 ) ∈ ℂ ∧ ( i · ( ℑ ‘ 𝐴 ) ) ∈ ℂ ) → ( ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) ↑ 2 ) = ( ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) ) + ( ( i · ( ℑ ‘ 𝐴 ) ) ↑ 2 ) ) ) |
| 102 |
42 56 101
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) ↑ 2 ) = ( ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) ) + ( ( i · ( ℑ ‘ 𝐴 ) ) ↑ 2 ) ) ) |
| 103 |
|
sqmul |
⊢ ( ( i ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) ∈ ℂ ) → ( ( i · ( ℑ ‘ 𝐴 ) ) ↑ 2 ) = ( ( i ↑ 2 ) · ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |
| 104 |
22 49 103
|
sylancr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( i · ( ℑ ‘ 𝐴 ) ) ↑ 2 ) = ( ( i ↑ 2 ) · ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |
| 105 |
|
i2 |
⊢ ( i ↑ 2 ) = - 1 |
| 106 |
105
|
oveq1i |
⊢ ( ( i ↑ 2 ) · ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) = ( - 1 · ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) |
| 107 |
104 106
|
eqtrdi |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( i · ( ℑ ‘ 𝐴 ) ) ↑ 2 ) = ( - 1 · ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |
| 108 |
95
|
mulm1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( - 1 · ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) = - ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) |
| 109 |
107 108
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( i · ( ℑ ‘ 𝐴 ) ) ↑ 2 ) = - ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) |
| 110 |
109
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) ) + ( ( i · ( ℑ ‘ 𝐴 ) ) ↑ 2 ) ) = ( ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) ) + - ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |
| 111 |
43 59
|
addcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) ) ∈ ℂ ) |
| 112 |
111 95
|
negsubd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) ) + - ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) = ( ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) ) − ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |
| 113 |
102 110 112
|
3eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) ↑ 2 ) = ( ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) ) − ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |
| 114 |
43 59 95
|
addsubassd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) ) − ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) − ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) ) |
| 115 |
100 113 114
|
3eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( 𝐴 ↑ 2 ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) − ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) ) |
| 116 |
|
absvalsq2 |
⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |
| 117 |
116
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |
| 118 |
115 117
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) = ( ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) − ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) + ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) ) |
| 119 |
43
|
2timesd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( 2 · ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) ) |
| 120 |
59 95
|
npcand |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) − ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) = ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) ) |
| 121 |
53 51 120
|
3eqtr4d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( i · ( 2 · ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) = ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) − ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |
| 122 |
119 121
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 2 · ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) + ( i · ( 2 · ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) ) = ( ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) + ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) − ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) ) |
| 123 |
97 118 122
|
3eqtr4d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) = ( ( 2 · ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) + ( i · ( 2 · ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) ) ) |
| 124 |
123
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( i · ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) = ( i · ( ( 2 · ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) + ( i · ( 2 · ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) ) ) ) |
| 125 |
91 77 42
|
subdird |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) − ( 2 · ( ℑ ‘ 𝐴 ) ) ) · ( ℜ ‘ 𝐴 ) ) = ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) · ( ℜ ‘ 𝐴 ) ) − ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℜ ‘ 𝐴 ) ) ) ) |
| 126 |
94 124 125
|
3eqtr4d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( i · ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) = ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) − ( 2 · ( ℑ ‘ 𝐴 ) ) ) · ( ℜ ‘ 𝐴 ) ) ) |
| 127 |
91 77
|
subcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) − ( 2 · ( ℑ ‘ 𝐴 ) ) ) ∈ ℂ ) |
| 128 |
|
mulcom |
⊢ ( ( ( ℜ ‘ 𝐴 ) ∈ ℂ ∧ i ∈ ℂ ) → ( ( ℜ ‘ 𝐴 ) · i ) = ( i · ( ℜ ‘ 𝐴 ) ) ) |
| 129 |
42 22 128
|
sylancl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ℜ ‘ 𝐴 ) · i ) = ( i · ( ℜ ‘ 𝐴 ) ) ) |
| 130 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ℜ ‘ 𝐴 ) ≠ 0 ) |
| 131 |
|
eleq1 |
⊢ ( ( i · ( ℜ ‘ 𝐴 ) ) = ( ℑ ‘ 𝐴 ) → ( ( i · ( ℜ ‘ 𝐴 ) ) ∈ ℝ ↔ ( ℑ ‘ 𝐴 ) ∈ ℝ ) ) |
| 132 |
48 131
|
syl5ibrcom |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( i · ( ℜ ‘ 𝐴 ) ) = ( ℑ ‘ 𝐴 ) → ( i · ( ℜ ‘ 𝐴 ) ) ∈ ℝ ) ) |
| 133 |
|
rimul |
⊢ ( ( ( ℜ ‘ 𝐴 ) ∈ ℝ ∧ ( i · ( ℜ ‘ 𝐴 ) ) ∈ ℝ ) → ( ℜ ‘ 𝐴 ) = 0 ) |
| 134 |
41 132 133
|
syl6an |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( i · ( ℜ ‘ 𝐴 ) ) = ( ℑ ‘ 𝐴 ) → ( ℜ ‘ 𝐴 ) = 0 ) ) |
| 135 |
134
|
necon3d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ℜ ‘ 𝐴 ) ≠ 0 → ( i · ( ℜ ‘ 𝐴 ) ) ≠ ( ℑ ‘ 𝐴 ) ) ) |
| 136 |
130 135
|
mpd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( i · ( ℜ ‘ 𝐴 ) ) ≠ ( ℑ ‘ 𝐴 ) ) |
| 137 |
129 136
|
eqnetrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ℜ ‘ 𝐴 ) · i ) ≠ ( ℑ ‘ 𝐴 ) ) |
| 138 |
91 77
|
subeq0ad |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) − ( 2 · ( ℑ ‘ 𝐴 ) ) ) = 0 ↔ ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) = ( 2 · ( ℑ ‘ 𝐴 ) ) ) ) |
| 139 |
|
2ne0 |
⊢ 2 ≠ 0 |
| 140 |
139
|
a1i |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → 2 ≠ 0 ) |
| 141 |
63 49 46 140
|
mulcand |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) = ( 2 · ( ℑ ‘ 𝐴 ) ) ↔ ( ( ℜ ‘ 𝐴 ) · i ) = ( ℑ ‘ 𝐴 ) ) ) |
| 142 |
138 141
|
bitrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) − ( 2 · ( ℑ ‘ 𝐴 ) ) ) = 0 ↔ ( ( ℜ ‘ 𝐴 ) · i ) = ( ℑ ‘ 𝐴 ) ) ) |
| 143 |
142
|
necon3bid |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) − ( 2 · ( ℑ ‘ 𝐴 ) ) ) ≠ 0 ↔ ( ( ℜ ‘ 𝐴 ) · i ) ≠ ( ℑ ‘ 𝐴 ) ) ) |
| 144 |
137 143
|
mpbird |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) − ( 2 · ( ℑ ‘ 𝐴 ) ) ) ≠ 0 ) |
| 145 |
127 42 144 130
|
mulne0d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) − ( 2 · ( ℑ ‘ 𝐴 ) ) ) · ( ℜ ‘ 𝐴 ) ) ≠ 0 ) |
| 146 |
126 145
|
eqnetrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( i · ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) ≠ 0 ) |
| 147 |
|
oveq2 |
⊢ ( ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) = 0 → ( i · ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) = ( i · 0 ) ) |
| 148 |
|
it0e0 |
⊢ ( i · 0 ) = 0 |
| 149 |
147 148
|
eqtrdi |
⊢ ( ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) = 0 → ( i · ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) = 0 ) |
| 150 |
149
|
necon3i |
⊢ ( ( i · ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) ≠ 0 → ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ≠ 0 ) |
| 151 |
146 150
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ≠ 0 ) |
| 152 |
37 14 151 20
|
divne0d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ≠ 0 ) |
| 153 |
36 152
|
eqnetrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( exp ‘ ( 2 · ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) ) + 1 ) ≠ 0 ) |
| 154 |
|
tanval3 |
⊢ ( ( ( ℑ ‘ ( log ‘ 𝐴 ) ) ∈ ℂ ∧ ( ( exp ‘ ( 2 · ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) ) + 1 ) ≠ 0 ) → ( tan ‘ ( ℑ ‘ ( log ‘ 𝐴 ) ) ) = ( ( ( exp ‘ ( 2 · ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) ) − 1 ) / ( i · ( ( exp ‘ ( 2 · ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) ) + 1 ) ) ) ) |
| 155 |
8 153 154
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( tan ‘ ( ℑ ‘ ( log ‘ 𝐴 ) ) ) = ( ( ( exp ‘ ( 2 · ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) ) − 1 ) / ( i · ( ( exp ‘ ( 2 · ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) ) + 1 ) ) ) ) |
| 156 |
10 14 14 20
|
divsubdird |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( 𝐴 ↑ 2 ) − ( ( abs ‘ 𝐴 ) ↑ 2 ) ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) = ( ( ( 𝐴 ↑ 2 ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) − ( ( ( abs ‘ 𝐴 ) ↑ 2 ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) ) |
| 157 |
33 34
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( 𝐴 ↑ 2 ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) − ( ( ( abs ‘ 𝐴 ) ↑ 2 ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) = ( ( exp ‘ ( 2 · ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) ) − 1 ) ) |
| 158 |
156 157
|
eqtr2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( exp ‘ ( 2 · ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) ) − 1 ) = ( ( ( 𝐴 ↑ 2 ) − ( ( abs ‘ 𝐴 ) ↑ 2 ) ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) |
| 159 |
36
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( i · ( ( exp ‘ ( 2 · ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) ) + 1 ) ) = ( i · ( ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) ) |
| 160 |
38 37 14 20
|
divassd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( i · ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) = ( i · ( ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) ) |
| 161 |
159 160
|
eqtr4d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( i · ( ( exp ‘ ( 2 · ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) ) + 1 ) ) = ( ( i · ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) |
| 162 |
158 161
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( exp ‘ ( 2 · ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) ) − 1 ) / ( i · ( ( exp ‘ ( 2 · ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) ) + 1 ) ) ) = ( ( ( ( 𝐴 ↑ 2 ) − ( ( abs ‘ 𝐴 ) ↑ 2 ) ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) / ( ( i · ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) ) |
| 163 |
10 14
|
subcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 𝐴 ↑ 2 ) − ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ∈ ℂ ) |
| 164 |
|
mulcl |
⊢ ( ( i ∈ ℂ ∧ ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ∈ ℂ ) → ( i · ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) ∈ ℂ ) |
| 165 |
22 37 164
|
sylancr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( i · ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) ∈ ℂ ) |
| 166 |
163 165 14 146 20
|
divcan7d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( ( 𝐴 ↑ 2 ) − ( ( abs ‘ 𝐴 ) ↑ 2 ) ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) / ( ( i · ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) = ( ( ( 𝐴 ↑ 2 ) − ( ( abs ‘ 𝐴 ) ↑ 2 ) ) / ( i · ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) ) ) |
| 167 |
115 117
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 𝐴 ↑ 2 ) − ( ( abs ‘ 𝐴 ) ↑ 2 ) ) = ( ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) − ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) − ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) ) |
| 168 |
43 96 95
|
pnpcand |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) − ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) − ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) = ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) − ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) − ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |
| 169 |
59 95 95
|
subsub4d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) − ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) − ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) = ( ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) − ( ( ( ℑ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) ) |
| 170 |
95
|
2timesd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( 2 · ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) = ( ( ( ℑ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |
| 171 |
170
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) − ( 2 · ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) = ( ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) − ( ( ( ℑ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) ) |
| 172 |
46 63 49
|
mulassd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) · ( ℑ ‘ 𝐴 ) ) = ( 2 · ( ( ( ℜ ‘ 𝐴 ) · i ) · ( ℑ ‘ 𝐴 ) ) ) ) |
| 173 |
42 38 49
|
mulassd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( ℜ ‘ 𝐴 ) · i ) · ( ℑ ‘ 𝐴 ) ) = ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) |
| 174 |
173
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( 2 · ( ( ( ℜ ‘ 𝐴 ) · i ) · ( ℑ ‘ 𝐴 ) ) ) = ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) ) |
| 175 |
172 174
|
eqtr2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) = ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) · ( ℑ ‘ 𝐴 ) ) ) |
| 176 |
49
|
sqvald |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ℑ ‘ 𝐴 ) ↑ 2 ) = ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) |
| 177 |
176
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( 2 · ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) = ( 2 · ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) |
| 178 |
46 49 49
|
mulassd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℑ ‘ 𝐴 ) ) = ( 2 · ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) |
| 179 |
177 178
|
eqtr4d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( 2 · ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) = ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℑ ‘ 𝐴 ) ) ) |
| 180 |
175 179
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) − ( 2 · ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) = ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) · ( ℑ ‘ 𝐴 ) ) − ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℑ ‘ 𝐴 ) ) ) ) |
| 181 |
91 77 49
|
subdird |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) − ( 2 · ( ℑ ‘ 𝐴 ) ) ) · ( ℑ ‘ 𝐴 ) ) = ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) · ( ℑ ‘ 𝐴 ) ) − ( ( 2 · ( ℑ ‘ 𝐴 ) ) · ( ℑ ‘ 𝐴 ) ) ) ) |
| 182 |
180 181
|
eqtr4d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) − ( 2 · ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) = ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) − ( 2 · ( ℑ ‘ 𝐴 ) ) ) · ( ℑ ‘ 𝐴 ) ) ) |
| 183 |
169 171 182
|
3eqtr2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · ( i · ( ℑ ‘ 𝐴 ) ) ) ) − ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) − ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) = ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) − ( 2 · ( ℑ ‘ 𝐴 ) ) ) · ( ℑ ‘ 𝐴 ) ) ) |
| 184 |
167 168 183
|
3eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( 𝐴 ↑ 2 ) − ( ( abs ‘ 𝐴 ) ↑ 2 ) ) = ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) − ( 2 · ( ℑ ‘ 𝐴 ) ) ) · ( ℑ ‘ 𝐴 ) ) ) |
| 185 |
184 126
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( 𝐴 ↑ 2 ) − ( ( abs ‘ 𝐴 ) ↑ 2 ) ) / ( i · ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) ) = ( ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) − ( 2 · ( ℑ ‘ 𝐴 ) ) ) · ( ℑ ‘ 𝐴 ) ) / ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) − ( 2 · ( ℑ ‘ 𝐴 ) ) ) · ( ℜ ‘ 𝐴 ) ) ) ) |
| 186 |
49 42 127 130 144
|
divcan5d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) − ( 2 · ( ℑ ‘ 𝐴 ) ) ) · ( ℑ ‘ 𝐴 ) ) / ( ( ( 2 · ( ( ℜ ‘ 𝐴 ) · i ) ) − ( 2 · ( ℑ ‘ 𝐴 ) ) ) · ( ℜ ‘ 𝐴 ) ) ) = ( ( ℑ ‘ 𝐴 ) / ( ℜ ‘ 𝐴 ) ) ) |
| 187 |
166 185 186
|
3eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( ( ( ( 𝐴 ↑ 2 ) − ( ( abs ‘ 𝐴 ) ↑ 2 ) ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) / ( ( i · ( ( 𝐴 ↑ 2 ) + ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) = ( ( ℑ ‘ 𝐴 ) / ( ℜ ‘ 𝐴 ) ) ) |
| 188 |
155 162 187
|
3eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ≠ 0 ) → ( tan ‘ ( ℑ ‘ ( log ‘ 𝐴 ) ) ) = ( ( ℑ ‘ 𝐴 ) / ( ℜ ‘ 𝐴 ) ) ) |