Step |
Hyp |
Ref |
Expression |
1 |
|
reccl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 1 / 𝐴 ) ∈ ℂ ) |
2 |
|
recne0 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 1 / 𝐴 ) ≠ 0 ) |
3 |
|
eflog |
⊢ ( ( ( 1 / 𝐴 ) ∈ ℂ ∧ ( 1 / 𝐴 ) ≠ 0 ) → ( exp ‘ ( log ‘ ( 1 / 𝐴 ) ) ) = ( 1 / 𝐴 ) ) |
4 |
1 2 3
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( exp ‘ ( log ‘ ( 1 / 𝐴 ) ) ) = ( 1 / 𝐴 ) ) |
5 |
4
|
eqcomd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 1 / 𝐴 ) = ( exp ‘ ( log ‘ ( 1 / 𝐴 ) ) ) ) |
6 |
5
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 1 / ( 1 / 𝐴 ) ) = ( 1 / ( exp ‘ ( log ‘ ( 1 / 𝐴 ) ) ) ) ) |
7 |
|
eflog |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( exp ‘ ( log ‘ 𝐴 ) ) = 𝐴 ) |
8 |
|
recrec |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 1 / ( 1 / 𝐴 ) ) = 𝐴 ) |
9 |
7 8
|
eqtr4d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( exp ‘ ( log ‘ 𝐴 ) ) = ( 1 / ( 1 / 𝐴 ) ) ) |
10 |
1 2
|
logcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( log ‘ ( 1 / 𝐴 ) ) ∈ ℂ ) |
11 |
|
efneg |
⊢ ( ( log ‘ ( 1 / 𝐴 ) ) ∈ ℂ → ( exp ‘ - ( log ‘ ( 1 / 𝐴 ) ) ) = ( 1 / ( exp ‘ ( log ‘ ( 1 / 𝐴 ) ) ) ) ) |
12 |
10 11
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( exp ‘ - ( log ‘ ( 1 / 𝐴 ) ) ) = ( 1 / ( exp ‘ ( log ‘ ( 1 / 𝐴 ) ) ) ) ) |
13 |
6 9 12
|
3eqtr4d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( exp ‘ ( log ‘ 𝐴 ) ) = ( exp ‘ - ( log ‘ ( 1 / 𝐴 ) ) ) ) |
14 |
13
|
3adant3 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ ( ℑ ‘ ( log ‘ 𝐴 ) ) ≠ π ) → ( exp ‘ ( log ‘ 𝐴 ) ) = ( exp ‘ - ( log ‘ ( 1 / 𝐴 ) ) ) ) |
15 |
14
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ ( ℑ ‘ ( log ‘ 𝐴 ) ) ≠ π ) → ( log ‘ ( exp ‘ ( log ‘ 𝐴 ) ) ) = ( log ‘ ( exp ‘ - ( log ‘ ( 1 / 𝐴 ) ) ) ) ) |
16 |
|
logrncl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( log ‘ 𝐴 ) ∈ ran log ) |
17 |
16
|
3adant3 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ ( ℑ ‘ ( log ‘ 𝐴 ) ) ≠ π ) → ( log ‘ 𝐴 ) ∈ ran log ) |
18 |
|
logef |
⊢ ( ( log ‘ 𝐴 ) ∈ ran log → ( log ‘ ( exp ‘ ( log ‘ 𝐴 ) ) ) = ( log ‘ 𝐴 ) ) |
19 |
17 18
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ ( ℑ ‘ ( log ‘ 𝐴 ) ) ≠ π ) → ( log ‘ ( exp ‘ ( log ‘ 𝐴 ) ) ) = ( log ‘ 𝐴 ) ) |
20 |
|
df-ne |
⊢ ( ( ℑ ‘ ( log ‘ 𝐴 ) ) ≠ π ↔ ¬ ( ℑ ‘ ( log ‘ 𝐴 ) ) = π ) |
21 |
|
lognegb |
⊢ ( ( ( 1 / 𝐴 ) ∈ ℂ ∧ ( 1 / 𝐴 ) ≠ 0 ) → ( - ( 1 / 𝐴 ) ∈ ℝ+ ↔ ( ℑ ‘ ( log ‘ ( 1 / 𝐴 ) ) ) = π ) ) |
22 |
1 2 21
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( - ( 1 / 𝐴 ) ∈ ℝ+ ↔ ( ℑ ‘ ( log ‘ ( 1 / 𝐴 ) ) ) = π ) ) |
23 |
22
|
biimprd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( ℑ ‘ ( log ‘ ( 1 / 𝐴 ) ) ) = π → - ( 1 / 𝐴 ) ∈ ℝ+ ) ) |
24 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
25 |
|
divneg2 |
⊢ ( ( 1 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → - ( 1 / 𝐴 ) = ( 1 / - 𝐴 ) ) |
26 |
24 25
|
mp3an1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → - ( 1 / 𝐴 ) = ( 1 / - 𝐴 ) ) |
27 |
26
|
eleq1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( - ( 1 / 𝐴 ) ∈ ℝ+ ↔ ( 1 / - 𝐴 ) ∈ ℝ+ ) ) |
28 |
23 27
|
sylibd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( ℑ ‘ ( log ‘ ( 1 / 𝐴 ) ) ) = π → ( 1 / - 𝐴 ) ∈ ℝ+ ) ) |
29 |
|
negcl |
⊢ ( 𝐴 ∈ ℂ → - 𝐴 ∈ ℂ ) |
30 |
|
negeq0 |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 = 0 ↔ - 𝐴 = 0 ) ) |
31 |
30
|
necon3bid |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ≠ 0 ↔ - 𝐴 ≠ 0 ) ) |
32 |
31
|
biimpa |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → - 𝐴 ≠ 0 ) |
33 |
|
rpreccl |
⊢ ( ( 1 / - 𝐴 ) ∈ ℝ+ → ( 1 / ( 1 / - 𝐴 ) ) ∈ ℝ+ ) |
34 |
|
recrec |
⊢ ( ( - 𝐴 ∈ ℂ ∧ - 𝐴 ≠ 0 ) → ( 1 / ( 1 / - 𝐴 ) ) = - 𝐴 ) |
35 |
34
|
eleq1d |
⊢ ( ( - 𝐴 ∈ ℂ ∧ - 𝐴 ≠ 0 ) → ( ( 1 / ( 1 / - 𝐴 ) ) ∈ ℝ+ ↔ - 𝐴 ∈ ℝ+ ) ) |
36 |
33 35
|
syl5ib |
⊢ ( ( - 𝐴 ∈ ℂ ∧ - 𝐴 ≠ 0 ) → ( ( 1 / - 𝐴 ) ∈ ℝ+ → - 𝐴 ∈ ℝ+ ) ) |
37 |
29 32 36
|
syl2an2r |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( 1 / - 𝐴 ) ∈ ℝ+ → - 𝐴 ∈ ℝ+ ) ) |
38 |
28 37
|
syld |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( ℑ ‘ ( log ‘ ( 1 / 𝐴 ) ) ) = π → - 𝐴 ∈ ℝ+ ) ) |
39 |
|
lognegb |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( - 𝐴 ∈ ℝ+ ↔ ( ℑ ‘ ( log ‘ 𝐴 ) ) = π ) ) |
40 |
38 39
|
sylibd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( ℑ ‘ ( log ‘ ( 1 / 𝐴 ) ) ) = π → ( ℑ ‘ ( log ‘ 𝐴 ) ) = π ) ) |
41 |
40
|
con3d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ¬ ( ℑ ‘ ( log ‘ 𝐴 ) ) = π → ¬ ( ℑ ‘ ( log ‘ ( 1 / 𝐴 ) ) ) = π ) ) |
42 |
41
|
3impia |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ ¬ ( ℑ ‘ ( log ‘ 𝐴 ) ) = π ) → ¬ ( ℑ ‘ ( log ‘ ( 1 / 𝐴 ) ) ) = π ) |
43 |
20 42
|
syl3an3b |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ ( ℑ ‘ ( log ‘ 𝐴 ) ) ≠ π ) → ¬ ( ℑ ‘ ( log ‘ ( 1 / 𝐴 ) ) ) = π ) |
44 |
|
logrncl |
⊢ ( ( ( 1 / 𝐴 ) ∈ ℂ ∧ ( 1 / 𝐴 ) ≠ 0 ) → ( log ‘ ( 1 / 𝐴 ) ) ∈ ran log ) |
45 |
1 2 44
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( log ‘ ( 1 / 𝐴 ) ) ∈ ran log ) |
46 |
|
logreclem |
⊢ ( ( ( log ‘ ( 1 / 𝐴 ) ) ∈ ran log ∧ ¬ ( ℑ ‘ ( log ‘ ( 1 / 𝐴 ) ) ) = π ) → - ( log ‘ ( 1 / 𝐴 ) ) ∈ ran log ) |
47 |
45 46
|
stoic3 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ ¬ ( ℑ ‘ ( log ‘ ( 1 / 𝐴 ) ) ) = π ) → - ( log ‘ ( 1 / 𝐴 ) ) ∈ ran log ) |
48 |
43 47
|
syld3an3 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ ( ℑ ‘ ( log ‘ 𝐴 ) ) ≠ π ) → - ( log ‘ ( 1 / 𝐴 ) ) ∈ ran log ) |
49 |
|
logef |
⊢ ( - ( log ‘ ( 1 / 𝐴 ) ) ∈ ran log → ( log ‘ ( exp ‘ - ( log ‘ ( 1 / 𝐴 ) ) ) ) = - ( log ‘ ( 1 / 𝐴 ) ) ) |
50 |
48 49
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ ( ℑ ‘ ( log ‘ 𝐴 ) ) ≠ π ) → ( log ‘ ( exp ‘ - ( log ‘ ( 1 / 𝐴 ) ) ) ) = - ( log ‘ ( 1 / 𝐴 ) ) ) |
51 |
15 19 50
|
3eqtr3d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ ( ℑ ‘ ( log ‘ 𝐴 ) ) ≠ π ) → ( log ‘ 𝐴 ) = - ( log ‘ ( 1 / 𝐴 ) ) ) |