Step |
Hyp |
Ref |
Expression |
1 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → ( ℑ ‘ 𝐴 ) < 0 ) |
2 |
1
|
lt0ne0d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → ( ℑ ‘ 𝐴 ) ≠ 0 ) |
3 |
|
fveq2 |
⊢ ( 𝐴 = 0 → ( ℑ ‘ 𝐴 ) = ( ℑ ‘ 0 ) ) |
4 |
|
im0 |
⊢ ( ℑ ‘ 0 ) = 0 |
5 |
3 4
|
eqtrdi |
⊢ ( 𝐴 = 0 → ( ℑ ‘ 𝐴 ) = 0 ) |
6 |
5
|
necon3i |
⊢ ( ( ℑ ‘ 𝐴 ) ≠ 0 → 𝐴 ≠ 0 ) |
7 |
2 6
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → 𝐴 ≠ 0 ) |
8 |
|
logcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( log ‘ 𝐴 ) ∈ ℂ ) |
9 |
7 8
|
syldan |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → ( log ‘ 𝐴 ) ∈ ℂ ) |
10 |
9
|
imcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → ( ℑ ‘ ( log ‘ 𝐴 ) ) ∈ ℝ ) |
11 |
|
logcj |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) ≠ 0 ) → ( log ‘ ( ∗ ‘ 𝐴 ) ) = ( ∗ ‘ ( log ‘ 𝐴 ) ) ) |
12 |
2 11
|
syldan |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → ( log ‘ ( ∗ ‘ 𝐴 ) ) = ( ∗ ‘ ( log ‘ 𝐴 ) ) ) |
13 |
12
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → ( ℑ ‘ ( log ‘ ( ∗ ‘ 𝐴 ) ) ) = ( ℑ ‘ ( ∗ ‘ ( log ‘ 𝐴 ) ) ) ) |
14 |
9
|
imcjd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → ( ℑ ‘ ( ∗ ‘ ( log ‘ 𝐴 ) ) ) = - ( ℑ ‘ ( log ‘ 𝐴 ) ) ) |
15 |
13 14
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → ( ℑ ‘ ( log ‘ ( ∗ ‘ 𝐴 ) ) ) = - ( ℑ ‘ ( log ‘ 𝐴 ) ) ) |
16 |
|
cjcl |
⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ 𝐴 ) ∈ ℂ ) |
17 |
|
imcl |
⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℝ ) |
18 |
17
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → ( ℑ ‘ 𝐴 ) ∈ ℝ ) |
19 |
18
|
lt0neg1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → ( ( ℑ ‘ 𝐴 ) < 0 ↔ 0 < - ( ℑ ‘ 𝐴 ) ) ) |
20 |
1 19
|
mpbid |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → 0 < - ( ℑ ‘ 𝐴 ) ) |
21 |
|
imcj |
⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ ( ∗ ‘ 𝐴 ) ) = - ( ℑ ‘ 𝐴 ) ) |
22 |
21
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → ( ℑ ‘ ( ∗ ‘ 𝐴 ) ) = - ( ℑ ‘ 𝐴 ) ) |
23 |
20 22
|
breqtrrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → 0 < ( ℑ ‘ ( ∗ ‘ 𝐴 ) ) ) |
24 |
|
argimgt0 |
⊢ ( ( ( ∗ ‘ 𝐴 ) ∈ ℂ ∧ 0 < ( ℑ ‘ ( ∗ ‘ 𝐴 ) ) ) → ( ℑ ‘ ( log ‘ ( ∗ ‘ 𝐴 ) ) ) ∈ ( 0 (,) π ) ) |
25 |
16 23 24
|
syl2an2r |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → ( ℑ ‘ ( log ‘ ( ∗ ‘ 𝐴 ) ) ) ∈ ( 0 (,) π ) ) |
26 |
|
eliooord |
⊢ ( ( ℑ ‘ ( log ‘ ( ∗ ‘ 𝐴 ) ) ) ∈ ( 0 (,) π ) → ( 0 < ( ℑ ‘ ( log ‘ ( ∗ ‘ 𝐴 ) ) ) ∧ ( ℑ ‘ ( log ‘ ( ∗ ‘ 𝐴 ) ) ) < π ) ) |
27 |
25 26
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → ( 0 < ( ℑ ‘ ( log ‘ ( ∗ ‘ 𝐴 ) ) ) ∧ ( ℑ ‘ ( log ‘ ( ∗ ‘ 𝐴 ) ) ) < π ) ) |
28 |
27
|
simprd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → ( ℑ ‘ ( log ‘ ( ∗ ‘ 𝐴 ) ) ) < π ) |
29 |
15 28
|
eqbrtrrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → - ( ℑ ‘ ( log ‘ 𝐴 ) ) < π ) |
30 |
|
pire |
⊢ π ∈ ℝ |
31 |
|
ltnegcon1 |
⊢ ( ( ( ℑ ‘ ( log ‘ 𝐴 ) ) ∈ ℝ ∧ π ∈ ℝ ) → ( - ( ℑ ‘ ( log ‘ 𝐴 ) ) < π ↔ - π < ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) |
32 |
10 30 31
|
sylancl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → ( - ( ℑ ‘ ( log ‘ 𝐴 ) ) < π ↔ - π < ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) |
33 |
29 32
|
mpbid |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → - π < ( ℑ ‘ ( log ‘ 𝐴 ) ) ) |
34 |
27
|
simpld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → 0 < ( ℑ ‘ ( log ‘ ( ∗ ‘ 𝐴 ) ) ) ) |
35 |
34 15
|
breqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → 0 < - ( ℑ ‘ ( log ‘ 𝐴 ) ) ) |
36 |
10
|
lt0neg1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → ( ( ℑ ‘ ( log ‘ 𝐴 ) ) < 0 ↔ 0 < - ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) |
37 |
35 36
|
mpbird |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → ( ℑ ‘ ( log ‘ 𝐴 ) ) < 0 ) |
38 |
30
|
renegcli |
⊢ - π ∈ ℝ |
39 |
38
|
rexri |
⊢ - π ∈ ℝ* |
40 |
|
0xr |
⊢ 0 ∈ ℝ* |
41 |
|
elioo2 |
⊢ ( ( - π ∈ ℝ* ∧ 0 ∈ ℝ* ) → ( ( ℑ ‘ ( log ‘ 𝐴 ) ) ∈ ( - π (,) 0 ) ↔ ( ( ℑ ‘ ( log ‘ 𝐴 ) ) ∈ ℝ ∧ - π < ( ℑ ‘ ( log ‘ 𝐴 ) ) ∧ ( ℑ ‘ ( log ‘ 𝐴 ) ) < 0 ) ) ) |
42 |
39 40 41
|
mp2an |
⊢ ( ( ℑ ‘ ( log ‘ 𝐴 ) ) ∈ ( - π (,) 0 ) ↔ ( ( ℑ ‘ ( log ‘ 𝐴 ) ) ∈ ℝ ∧ - π < ( ℑ ‘ ( log ‘ 𝐴 ) ) ∧ ( ℑ ‘ ( log ‘ 𝐴 ) ) < 0 ) ) |
43 |
10 33 37 42
|
syl3anbrc |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → ( ℑ ‘ ( log ‘ 𝐴 ) ) ∈ ( - π (,) 0 ) ) |