| Step |
Hyp |
Ref |
Expression |
| 1 |
|
1cnd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → 1 ∈ ℂ ) |
| 2 |
|
simpl1 |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → 𝐴 ∈ ℂ ) |
| 3 |
1 2
|
negsubd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( 1 + - 𝐴 ) = ( 1 − 𝐴 ) ) |
| 4 |
|
1rp |
⊢ 1 ∈ ℝ+ |
| 5 |
4
|
a1i |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → 1 ∈ ℝ+ ) |
| 6 |
|
simpl3 |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ¬ 1 = 𝐴 ) |
| 7 |
|
simpl2 |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( abs ‘ 𝐴 ) = 1 ) |
| 8 |
1 2 1
|
sub32d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( ( 1 − 𝐴 ) − 1 ) = ( ( 1 − 1 ) − 𝐴 ) ) |
| 9 |
|
1m1e0 |
⊢ ( 1 − 1 ) = 0 |
| 10 |
9
|
oveq1i |
⊢ ( ( 1 − 1 ) − 𝐴 ) = ( 0 − 𝐴 ) |
| 11 |
|
df-neg |
⊢ - 𝐴 = ( 0 − 𝐴 ) |
| 12 |
10 11
|
eqtr4i |
⊢ ( ( 1 − 1 ) − 𝐴 ) = - 𝐴 |
| 13 |
8 12
|
eqtrdi |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( ( 1 − 𝐴 ) − 1 ) = - 𝐴 ) |
| 14 |
|
1cnd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → 1 ∈ ℂ ) |
| 15 |
|
simp1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → 𝐴 ∈ ℂ ) |
| 16 |
14 15
|
subcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( 1 − 𝐴 ) ∈ ℂ ) |
| 17 |
16
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( 1 − 𝐴 ) ∈ ℂ ) |
| 18 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
| 19 |
|
subeq0 |
⊢ ( ( 1 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( ( 1 − 𝐴 ) = 0 ↔ 1 = 𝐴 ) ) |
| 20 |
18 19
|
mpan |
⊢ ( 𝐴 ∈ ℂ → ( ( 1 − 𝐴 ) = 0 ↔ 1 = 𝐴 ) ) |
| 21 |
20
|
biimpd |
⊢ ( 𝐴 ∈ ℂ → ( ( 1 − 𝐴 ) = 0 → 1 = 𝐴 ) ) |
| 22 |
21
|
con3dimp |
⊢ ( ( 𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴 ) → ¬ ( 1 − 𝐴 ) = 0 ) |
| 23 |
22
|
neqned |
⊢ ( ( 𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴 ) → ( 1 − 𝐴 ) ≠ 0 ) |
| 24 |
23
|
3adant2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( 1 − 𝐴 ) ≠ 0 ) |
| 25 |
24
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( 1 − 𝐴 ) ≠ 0 ) |
| 26 |
17 25
|
recrecd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( 1 / ( 1 / ( 1 − 𝐴 ) ) ) = ( 1 − 𝐴 ) ) |
| 27 |
14 16 24
|
div2negd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( - 1 / - ( 1 − 𝐴 ) ) = ( 1 / ( 1 − 𝐴 ) ) ) |
| 28 |
27
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( - 1 / - ( 1 − 𝐴 ) ) = ( 1 / ( 1 − 𝐴 ) ) ) |
| 29 |
15
|
negcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → - 𝐴 ∈ ℂ ) |
| 30 |
29 16 24
|
cjdivd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( ∗ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = ( ( ∗ ‘ - 𝐴 ) / ( ∗ ‘ ( 1 − 𝐴 ) ) ) ) |
| 31 |
15
|
cjnegd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( ∗ ‘ - 𝐴 ) = - ( ∗ ‘ 𝐴 ) ) |
| 32 |
|
fveq2 |
⊢ ( 𝐴 = 0 → ( abs ‘ 𝐴 ) = ( abs ‘ 0 ) ) |
| 33 |
|
abs0 |
⊢ ( abs ‘ 0 ) = 0 |
| 34 |
32 33
|
eqtrdi |
⊢ ( 𝐴 = 0 → ( abs ‘ 𝐴 ) = 0 ) |
| 35 |
|
eqtr2 |
⊢ ( ( ( abs ‘ 𝐴 ) = 1 ∧ ( abs ‘ 𝐴 ) = 0 ) → 1 = 0 ) |
| 36 |
34 35
|
sylan2 |
⊢ ( ( ( abs ‘ 𝐴 ) = 1 ∧ 𝐴 = 0 ) → 1 = 0 ) |
| 37 |
|
ax-1ne0 |
⊢ 1 ≠ 0 |
| 38 |
|
neneq |
⊢ ( 1 ≠ 0 → ¬ 1 = 0 ) |
| 39 |
37 38
|
mp1i |
⊢ ( ( ( abs ‘ 𝐴 ) = 1 ∧ 𝐴 = 0 ) → ¬ 1 = 0 ) |
| 40 |
36 39
|
pm2.65da |
⊢ ( ( abs ‘ 𝐴 ) = 1 → ¬ 𝐴 = 0 ) |
| 41 |
40
|
adantl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ) → ¬ 𝐴 = 0 ) |
| 42 |
|
df-ne |
⊢ ( 𝐴 ≠ 0 ↔ ¬ 𝐴 = 0 ) |
| 43 |
|
oveq1 |
⊢ ( ( abs ‘ 𝐴 ) = 1 → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( 1 ↑ 2 ) ) |
| 44 |
|
sq1 |
⊢ ( 1 ↑ 2 ) = 1 |
| 45 |
43 44
|
eqtrdi |
⊢ ( ( abs ‘ 𝐴 ) = 1 → ( ( abs ‘ 𝐴 ) ↑ 2 ) = 1 ) |
| 46 |
45
|
adantl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ) → ( ( abs ‘ 𝐴 ) ↑ 2 ) = 1 ) |
| 47 |
|
absvalsq |
⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) |
| 48 |
47
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ) → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) |
| 49 |
46 48
|
eqtr3d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ) → 1 = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) |
| 50 |
49
|
3adant3 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ 𝐴 ≠ 0 ) → 1 = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) |
| 51 |
50
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ 𝐴 ≠ 0 ) → ( 1 / 𝐴 ) = ( ( 𝐴 · ( ∗ ‘ 𝐴 ) ) / 𝐴 ) ) |
| 52 |
|
simp1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ 𝐴 ≠ 0 ) → 𝐴 ∈ ℂ ) |
| 53 |
52
|
cjcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ 𝐴 ≠ 0 ) → ( ∗ ‘ 𝐴 ) ∈ ℂ ) |
| 54 |
|
simp3 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ 𝐴 ≠ 0 ) → 𝐴 ≠ 0 ) |
| 55 |
53 52 54
|
divcan3d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ 𝐴 ≠ 0 ) → ( ( 𝐴 · ( ∗ ‘ 𝐴 ) ) / 𝐴 ) = ( ∗ ‘ 𝐴 ) ) |
| 56 |
51 55
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ 𝐴 ≠ 0 ) → ( 1 / 𝐴 ) = ( ∗ ‘ 𝐴 ) ) |
| 57 |
42 56
|
syl3an3br |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 𝐴 = 0 ) → ( 1 / 𝐴 ) = ( ∗ ‘ 𝐴 ) ) |
| 58 |
41 57
|
mpd3an3 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ) → ( 1 / 𝐴 ) = ( ∗ ‘ 𝐴 ) ) |
| 59 |
58
|
eqcomd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ) → ( ∗ ‘ 𝐴 ) = ( 1 / 𝐴 ) ) |
| 60 |
59
|
3adant3 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( ∗ ‘ 𝐴 ) = ( 1 / 𝐴 ) ) |
| 61 |
60
|
negeqd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → - ( ∗ ‘ 𝐴 ) = - ( 1 / 𝐴 ) ) |
| 62 |
31 61
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( ∗ ‘ - 𝐴 ) = - ( 1 / 𝐴 ) ) |
| 63 |
62
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( ( ∗ ‘ - 𝐴 ) / ( ∗ ‘ ( 1 − 𝐴 ) ) ) = ( - ( 1 / 𝐴 ) / ( ∗ ‘ ( 1 − 𝐴 ) ) ) ) |
| 64 |
|
cjsub |
⊢ ( ( 1 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( ∗ ‘ ( 1 − 𝐴 ) ) = ( ( ∗ ‘ 1 ) − ( ∗ ‘ 𝐴 ) ) ) |
| 65 |
18 64
|
mpan |
⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ ( 1 − 𝐴 ) ) = ( ( ∗ ‘ 1 ) − ( ∗ ‘ 𝐴 ) ) ) |
| 66 |
|
1red |
⊢ ( 𝐴 ∈ ℂ → 1 ∈ ℝ ) |
| 67 |
66
|
cjred |
⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ 1 ) = 1 ) |
| 68 |
67
|
oveq1d |
⊢ ( 𝐴 ∈ ℂ → ( ( ∗ ‘ 1 ) − ( ∗ ‘ 𝐴 ) ) = ( 1 − ( ∗ ‘ 𝐴 ) ) ) |
| 69 |
65 68
|
eqtrd |
⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ ( 1 − 𝐴 ) ) = ( 1 − ( ∗ ‘ 𝐴 ) ) ) |
| 70 |
69
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ) → ( ∗ ‘ ( 1 − 𝐴 ) ) = ( 1 − ( ∗ ‘ 𝐴 ) ) ) |
| 71 |
59
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ) → ( 1 − ( ∗ ‘ 𝐴 ) ) = ( 1 − ( 1 / 𝐴 ) ) ) |
| 72 |
70 71
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ) → ( ∗ ‘ ( 1 − 𝐴 ) ) = ( 1 − ( 1 / 𝐴 ) ) ) |
| 73 |
72
|
3adant3 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( ∗ ‘ ( 1 − 𝐴 ) ) = ( 1 − ( 1 / 𝐴 ) ) ) |
| 74 |
73
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( - ( 1 / 𝐴 ) / ( ∗ ‘ ( 1 − 𝐴 ) ) ) = ( - ( 1 / 𝐴 ) / ( 1 − ( 1 / 𝐴 ) ) ) ) |
| 75 |
30 63 74
|
3eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( ∗ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = ( - ( 1 / 𝐴 ) / ( 1 − ( 1 / 𝐴 ) ) ) ) |
| 76 |
40
|
3ad2ant2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ¬ 𝐴 = 0 ) |
| 77 |
76
|
neqned |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → 𝐴 ≠ 0 ) |
| 78 |
|
1cnd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → 1 ∈ ℂ ) |
| 79 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → 𝐴 ∈ ℂ ) |
| 80 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → 𝐴 ≠ 0 ) |
| 81 |
78 79 80
|
divnegd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → - ( 1 / 𝐴 ) = ( - 1 / 𝐴 ) ) |
| 82 |
81
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( - ( 1 / 𝐴 ) / ( 1 − ( 1 / 𝐴 ) ) ) = ( ( - 1 / 𝐴 ) / ( 1 − ( 1 / 𝐴 ) ) ) ) |
| 83 |
15 77 82
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( - ( 1 / 𝐴 ) / ( 1 − ( 1 / 𝐴 ) ) ) = ( ( - 1 / 𝐴 ) / ( 1 − ( 1 / 𝐴 ) ) ) ) |
| 84 |
14
|
negcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → - 1 ∈ ℂ ) |
| 85 |
84 15 77
|
divcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( - 1 / 𝐴 ) ∈ ℂ ) |
| 86 |
15 77
|
reccld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( 1 / 𝐴 ) ∈ ℂ ) |
| 87 |
14 86
|
subcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( 1 − ( 1 / 𝐴 ) ) ∈ ℂ ) |
| 88 |
16 24
|
cjne0d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( ∗ ‘ ( 1 − 𝐴 ) ) ≠ 0 ) |
| 89 |
73 88
|
eqnetrrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( 1 − ( 1 / 𝐴 ) ) ≠ 0 ) |
| 90 |
85 87 15 89 77
|
divcan5d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( ( 𝐴 · ( - 1 / 𝐴 ) ) / ( 𝐴 · ( 1 − ( 1 / 𝐴 ) ) ) ) = ( ( - 1 / 𝐴 ) / ( 1 − ( 1 / 𝐴 ) ) ) ) |
| 91 |
84 15 77
|
divcan2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( 𝐴 · ( - 1 / 𝐴 ) ) = - 1 ) |
| 92 |
15 14 86
|
subdid |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( 𝐴 · ( 1 − ( 1 / 𝐴 ) ) ) = ( ( 𝐴 · 1 ) − ( 𝐴 · ( 1 / 𝐴 ) ) ) ) |
| 93 |
15
|
mulid1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( 𝐴 · 1 ) = 𝐴 ) |
| 94 |
15 77
|
recidd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( 𝐴 · ( 1 / 𝐴 ) ) = 1 ) |
| 95 |
93 94
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( ( 𝐴 · 1 ) − ( 𝐴 · ( 1 / 𝐴 ) ) ) = ( 𝐴 − 1 ) ) |
| 96 |
92 95
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( 𝐴 · ( 1 − ( 1 / 𝐴 ) ) ) = ( 𝐴 − 1 ) ) |
| 97 |
91 96
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( ( 𝐴 · ( - 1 / 𝐴 ) ) / ( 𝐴 · ( 1 − ( 1 / 𝐴 ) ) ) ) = ( - 1 / ( 𝐴 − 1 ) ) ) |
| 98 |
83 90 97
|
3eqtr2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( - ( 1 / 𝐴 ) / ( 1 − ( 1 / 𝐴 ) ) ) = ( - 1 / ( 𝐴 − 1 ) ) ) |
| 99 |
|
subcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 1 ∈ ℂ ) → ( 𝐴 − 1 ) ∈ ℂ ) |
| 100 |
99
|
negnegd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 1 ∈ ℂ ) → - - ( 𝐴 − 1 ) = ( 𝐴 − 1 ) ) |
| 101 |
|
negsubdi2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 1 ∈ ℂ ) → - ( 𝐴 − 1 ) = ( 1 − 𝐴 ) ) |
| 102 |
101
|
negeqd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 1 ∈ ℂ ) → - - ( 𝐴 − 1 ) = - ( 1 − 𝐴 ) ) |
| 103 |
100 102
|
eqtr3d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 1 ∈ ℂ ) → ( 𝐴 − 1 ) = - ( 1 − 𝐴 ) ) |
| 104 |
15 14 103
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( 𝐴 − 1 ) = - ( 1 − 𝐴 ) ) |
| 105 |
104
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( - 1 / ( 𝐴 − 1 ) ) = ( - 1 / - ( 1 − 𝐴 ) ) ) |
| 106 |
75 98 105
|
3eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( ∗ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = ( - 1 / - ( 1 − 𝐴 ) ) ) |
| 107 |
106
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( ∗ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = ( - 1 / - ( 1 − 𝐴 ) ) ) |
| 108 |
29 16 24
|
divcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( - 𝐴 / ( 1 − 𝐴 ) ) ∈ ℂ ) |
| 109 |
108
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( - 𝐴 / ( 1 − 𝐴 ) ) ∈ ℂ ) |
| 110 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) |
| 111 |
109 110
|
reim0bd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( - 𝐴 / ( 1 − 𝐴 ) ) ∈ ℝ ) |
| 112 |
111
|
cjred |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( ∗ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = ( - 𝐴 / ( 1 − 𝐴 ) ) ) |
| 113 |
112 111
|
eqeltrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( ∗ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ∈ ℝ ) |
| 114 |
107 113
|
eqeltrrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( - 1 / - ( 1 − 𝐴 ) ) ∈ ℝ ) |
| 115 |
28 114
|
eqeltrrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( 1 / ( 1 − 𝐴 ) ) ∈ ℝ ) |
| 116 |
16 24
|
recne0d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( 1 / ( 1 − 𝐴 ) ) ≠ 0 ) |
| 117 |
116
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( 1 / ( 1 − 𝐴 ) ) ≠ 0 ) |
| 118 |
115 117
|
rereccld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( 1 / ( 1 / ( 1 − 𝐴 ) ) ) ∈ ℝ ) |
| 119 |
26 118
|
eqeltrrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( 1 − 𝐴 ) ∈ ℝ ) |
| 120 |
|
1red |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → 1 ∈ ℝ ) |
| 121 |
119 120
|
resubcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( ( 1 − 𝐴 ) − 1 ) ∈ ℝ ) |
| 122 |
13 121
|
eqeltrrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → - 𝐴 ∈ ℝ ) |
| 123 |
2 122
|
negrebd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → 𝐴 ∈ ℝ ) |
| 124 |
123
|
absord |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( ( abs ‘ 𝐴 ) = 𝐴 ∨ ( abs ‘ 𝐴 ) = - 𝐴 ) ) |
| 125 |
|
eqeq1 |
⊢ ( ( abs ‘ 𝐴 ) = 1 → ( ( abs ‘ 𝐴 ) = 𝐴 ↔ 1 = 𝐴 ) ) |
| 126 |
125
|
biimpd |
⊢ ( ( abs ‘ 𝐴 ) = 1 → ( ( abs ‘ 𝐴 ) = 𝐴 → 1 = 𝐴 ) ) |
| 127 |
|
eqeq1 |
⊢ ( ( abs ‘ 𝐴 ) = 1 → ( ( abs ‘ 𝐴 ) = - 𝐴 ↔ 1 = - 𝐴 ) ) |
| 128 |
127
|
biimpd |
⊢ ( ( abs ‘ 𝐴 ) = 1 → ( ( abs ‘ 𝐴 ) = - 𝐴 → 1 = - 𝐴 ) ) |
| 129 |
126 128
|
orim12d |
⊢ ( ( abs ‘ 𝐴 ) = 1 → ( ( ( abs ‘ 𝐴 ) = 𝐴 ∨ ( abs ‘ 𝐴 ) = - 𝐴 ) → ( 1 = 𝐴 ∨ 1 = - 𝐴 ) ) ) |
| 130 |
7 124 129
|
sylc |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( 1 = 𝐴 ∨ 1 = - 𝐴 ) ) |
| 131 |
130
|
ord |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( ¬ 1 = 𝐴 → 1 = - 𝐴 ) ) |
| 132 |
6 131
|
mpd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → 1 = - 𝐴 ) |
| 133 |
132 5
|
eqeltrrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → - 𝐴 ∈ ℝ+ ) |
| 134 |
5 133
|
rpaddcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( 1 + - 𝐴 ) ∈ ℝ+ ) |
| 135 |
3 134
|
eqeltrrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( 1 − 𝐴 ) ∈ ℝ+ ) |
| 136 |
135
|
relogcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( log ‘ ( 1 − 𝐴 ) ) ∈ ℝ ) |
| 137 |
136
|
reim0d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( ℑ ‘ ( log ‘ ( 1 − 𝐴 ) ) ) = 0 ) |
| 138 |
133 135
|
rpdivcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( - 𝐴 / ( 1 − 𝐴 ) ) ∈ ℝ+ ) |
| 139 |
138
|
relogcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( log ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ∈ ℝ ) |
| 140 |
139
|
reim0d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( ℑ ‘ ( log ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ) = 0 ) |
| 141 |
137 140
|
eqtr4d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) = 0 ) → ( ℑ ‘ ( log ‘ ( 1 − 𝐴 ) ) ) = ( ℑ ‘ ( log ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ) ) |
| 142 |
16 24
|
logcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( log ‘ ( 1 − 𝐴 ) ) ∈ ℂ ) |
| 143 |
142
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ≠ 0 ) → ( log ‘ ( 1 − 𝐴 ) ) ∈ ℂ ) |
| 144 |
143
|
imcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ≠ 0 ) → ( ℑ ‘ ( log ‘ ( 1 − 𝐴 ) ) ) ∈ ℝ ) |
| 145 |
144
|
recnd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ≠ 0 ) → ( ℑ ‘ ( log ‘ ( 1 − 𝐴 ) ) ) ∈ ℂ ) |
| 146 |
108
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ≠ 0 ) → ( - 𝐴 / ( 1 − 𝐴 ) ) ∈ ℂ ) |
| 147 |
15 77
|
negne0d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → - 𝐴 ≠ 0 ) |
| 148 |
29 16 147 24
|
divne0d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( - 𝐴 / ( 1 − 𝐴 ) ) ≠ 0 ) |
| 149 |
148
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ≠ 0 ) → ( - 𝐴 / ( 1 − 𝐴 ) ) ≠ 0 ) |
| 150 |
146 149
|
logcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ≠ 0 ) → ( log ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ∈ ℂ ) |
| 151 |
150
|
imcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ≠ 0 ) → ( ℑ ‘ ( log ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ) ∈ ℝ ) |
| 152 |
151
|
recnd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ≠ 0 ) → ( ℑ ‘ ( log ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ) ∈ ℂ ) |
| 153 |
106
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( log ‘ ( ∗ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ) = ( log ‘ ( - 1 / - ( 1 − 𝐴 ) ) ) ) |
| 154 |
153
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ≠ 0 ) → ( log ‘ ( ∗ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ) = ( log ‘ ( - 1 / - ( 1 − 𝐴 ) ) ) ) |
| 155 |
|
logcj |
⊢ ( ( ( - 𝐴 / ( 1 − 𝐴 ) ) ∈ ℂ ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ≠ 0 ) → ( log ‘ ( ∗ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ) = ( ∗ ‘ ( log ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ) ) |
| 156 |
108 155
|
sylan |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ≠ 0 ) → ( log ‘ ( ∗ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ) = ( ∗ ‘ ( log ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ) ) |
| 157 |
16 24
|
reccld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( 1 / ( 1 − 𝐴 ) ) ∈ ℂ ) |
| 158 |
157 116
|
logcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( log ‘ ( 1 / ( 1 − 𝐴 ) ) ) ∈ ℂ ) |
| 159 |
158
|
negnegd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → - - ( log ‘ ( 1 / ( 1 − 𝐴 ) ) ) = ( log ‘ ( 1 / ( 1 − 𝐴 ) ) ) ) |
| 160 |
|
isosctrlem1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( ℑ ‘ ( log ‘ ( 1 − 𝐴 ) ) ) ≠ π ) |
| 161 |
|
logrec |
⊢ ( ( ( 1 − 𝐴 ) ∈ ℂ ∧ ( 1 − 𝐴 ) ≠ 0 ∧ ( ℑ ‘ ( log ‘ ( 1 − 𝐴 ) ) ) ≠ π ) → ( log ‘ ( 1 − 𝐴 ) ) = - ( log ‘ ( 1 / ( 1 − 𝐴 ) ) ) ) |
| 162 |
16 24 160 161
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( log ‘ ( 1 − 𝐴 ) ) = - ( log ‘ ( 1 / ( 1 − 𝐴 ) ) ) ) |
| 163 |
162
|
negeqd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → - ( log ‘ ( 1 − 𝐴 ) ) = - - ( log ‘ ( 1 / ( 1 − 𝐴 ) ) ) ) |
| 164 |
27
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( log ‘ ( - 1 / - ( 1 − 𝐴 ) ) ) = ( log ‘ ( 1 / ( 1 − 𝐴 ) ) ) ) |
| 165 |
159 163 164
|
3eqtr4rd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( log ‘ ( - 1 / - ( 1 − 𝐴 ) ) ) = - ( log ‘ ( 1 − 𝐴 ) ) ) |
| 166 |
165
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ≠ 0 ) → ( log ‘ ( - 1 / - ( 1 − 𝐴 ) ) ) = - ( log ‘ ( 1 − 𝐴 ) ) ) |
| 167 |
154 156 166
|
3eqtr3rd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ≠ 0 ) → - ( log ‘ ( 1 − 𝐴 ) ) = ( ∗ ‘ ( log ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ) ) |
| 168 |
167
|
fveq2d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ≠ 0 ) → ( ℑ ‘ - ( log ‘ ( 1 − 𝐴 ) ) ) = ( ℑ ‘ ( ∗ ‘ ( log ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ) ) ) |
| 169 |
143
|
imnegd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ≠ 0 ) → ( ℑ ‘ - ( log ‘ ( 1 − 𝐴 ) ) ) = - ( ℑ ‘ ( log ‘ ( 1 − 𝐴 ) ) ) ) |
| 170 |
150
|
imcjd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ≠ 0 ) → ( ℑ ‘ ( ∗ ‘ ( log ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ) ) = - ( ℑ ‘ ( log ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ) ) |
| 171 |
168 169 170
|
3eqtr3d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ≠ 0 ) → - ( ℑ ‘ ( log ‘ ( 1 − 𝐴 ) ) ) = - ( ℑ ‘ ( log ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ) ) |
| 172 |
145 152 171
|
neg11d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) ∧ ( ℑ ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ≠ 0 ) → ( ℑ ‘ ( log ‘ ( 1 − 𝐴 ) ) ) = ( ℑ ‘ ( log ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ) ) |
| 173 |
141 172
|
pm2.61dane |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) = 1 ∧ ¬ 1 = 𝐴 ) → ( ℑ ‘ ( log ‘ ( 1 − 𝐴 ) ) ) = ( ℑ ‘ ( log ‘ ( - 𝐴 / ( 1 − 𝐴 ) ) ) ) ) |