| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
| 2 |
|
ax-icn |
⊢ i ∈ ℂ |
| 3 |
|
simpl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → 𝐴 ∈ dom arctan ) |
| 4 |
|
atandm2 |
⊢ ( 𝐴 ∈ dom arctan ↔ ( 𝐴 ∈ ℂ ∧ ( 1 − ( i · 𝐴 ) ) ≠ 0 ∧ ( 1 + ( i · 𝐴 ) ) ≠ 0 ) ) |
| 5 |
3 4
|
sylib |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( 𝐴 ∈ ℂ ∧ ( 1 − ( i · 𝐴 ) ) ≠ 0 ∧ ( 1 + ( i · 𝐴 ) ) ≠ 0 ) ) |
| 6 |
5
|
simp1d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → 𝐴 ∈ ℂ ) |
| 7 |
|
mulcl |
⊢ ( ( i ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( i · 𝐴 ) ∈ ℂ ) |
| 8 |
2 6 7
|
sylancr |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( i · 𝐴 ) ∈ ℂ ) |
| 9 |
|
addcl |
⊢ ( ( 1 ∈ ℂ ∧ ( i · 𝐴 ) ∈ ℂ ) → ( 1 + ( i · 𝐴 ) ) ∈ ℂ ) |
| 10 |
1 8 9
|
sylancr |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( 1 + ( i · 𝐴 ) ) ∈ ℂ ) |
| 11 |
5
|
simp3d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( 1 + ( i · 𝐴 ) ) ≠ 0 ) |
| 12 |
10 11
|
logcld |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ∈ ℂ ) |
| 13 |
|
subcl |
⊢ ( ( 1 ∈ ℂ ∧ ( i · 𝐴 ) ∈ ℂ ) → ( 1 − ( i · 𝐴 ) ) ∈ ℂ ) |
| 14 |
1 8 13
|
sylancr |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( 1 − ( i · 𝐴 ) ) ∈ ℂ ) |
| 15 |
5
|
simp2d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( 1 − ( i · 𝐴 ) ) ≠ 0 ) |
| 16 |
14 15
|
logcld |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ∈ ℂ ) |
| 17 |
12 16
|
imsubd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) − ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) = ( ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) − ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) ) |
| 18 |
2
|
a1i |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → i ∈ ℂ ) |
| 19 |
18 6 18
|
subdid |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( i · ( 𝐴 − i ) ) = ( ( i · 𝐴 ) − ( i · i ) ) ) |
| 20 |
|
ixi |
⊢ ( i · i ) = - 1 |
| 21 |
20
|
oveq2i |
⊢ ( ( i · 𝐴 ) − ( i · i ) ) = ( ( i · 𝐴 ) − - 1 ) |
| 22 |
|
subneg |
⊢ ( ( ( i · 𝐴 ) ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( i · 𝐴 ) − - 1 ) = ( ( i · 𝐴 ) + 1 ) ) |
| 23 |
8 1 22
|
sylancl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( i · 𝐴 ) − - 1 ) = ( ( i · 𝐴 ) + 1 ) ) |
| 24 |
21 23
|
eqtrid |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( i · 𝐴 ) − ( i · i ) ) = ( ( i · 𝐴 ) + 1 ) ) |
| 25 |
|
addcom |
⊢ ( ( ( i · 𝐴 ) ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( i · 𝐴 ) + 1 ) = ( 1 + ( i · 𝐴 ) ) ) |
| 26 |
8 1 25
|
sylancl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( i · 𝐴 ) + 1 ) = ( 1 + ( i · 𝐴 ) ) ) |
| 27 |
19 24 26
|
3eqtrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( i · ( 𝐴 − i ) ) = ( 1 + ( i · 𝐴 ) ) ) |
| 28 |
27
|
fveq2d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( log ‘ ( i · ( 𝐴 − i ) ) ) = ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) |
| 29 |
|
subcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ i ∈ ℂ ) → ( 𝐴 − i ) ∈ ℂ ) |
| 30 |
6 2 29
|
sylancl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( 𝐴 − i ) ∈ ℂ ) |
| 31 |
|
resub |
⊢ ( ( 𝐴 ∈ ℂ ∧ i ∈ ℂ ) → ( ℜ ‘ ( 𝐴 − i ) ) = ( ( ℜ ‘ 𝐴 ) − ( ℜ ‘ i ) ) ) |
| 32 |
6 2 31
|
sylancl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℜ ‘ ( 𝐴 − i ) ) = ( ( ℜ ‘ 𝐴 ) − ( ℜ ‘ i ) ) ) |
| 33 |
|
rei |
⊢ ( ℜ ‘ i ) = 0 |
| 34 |
33
|
oveq2i |
⊢ ( ( ℜ ‘ 𝐴 ) − ( ℜ ‘ i ) ) = ( ( ℜ ‘ 𝐴 ) − 0 ) |
| 35 |
6
|
recld |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℜ ‘ 𝐴 ) ∈ ℝ ) |
| 36 |
35
|
recnd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℜ ‘ 𝐴 ) ∈ ℂ ) |
| 37 |
36
|
subid1d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℜ ‘ 𝐴 ) − 0 ) = ( ℜ ‘ 𝐴 ) ) |
| 38 |
34 37
|
eqtrid |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℜ ‘ 𝐴 ) − ( ℜ ‘ i ) ) = ( ℜ ‘ 𝐴 ) ) |
| 39 |
32 38
|
eqtrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℜ ‘ ( 𝐴 − i ) ) = ( ℜ ‘ 𝐴 ) ) |
| 40 |
|
gt0ne0 |
⊢ ( ( ( ℜ ‘ 𝐴 ) ∈ ℝ ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℜ ‘ 𝐴 ) ≠ 0 ) |
| 41 |
35 40
|
sylancom |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℜ ‘ 𝐴 ) ≠ 0 ) |
| 42 |
39 41
|
eqnetrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℜ ‘ ( 𝐴 − i ) ) ≠ 0 ) |
| 43 |
|
fveq2 |
⊢ ( ( 𝐴 − i ) = 0 → ( ℜ ‘ ( 𝐴 − i ) ) = ( ℜ ‘ 0 ) ) |
| 44 |
|
re0 |
⊢ ( ℜ ‘ 0 ) = 0 |
| 45 |
43 44
|
eqtrdi |
⊢ ( ( 𝐴 − i ) = 0 → ( ℜ ‘ ( 𝐴 − i ) ) = 0 ) |
| 46 |
45
|
necon3i |
⊢ ( ( ℜ ‘ ( 𝐴 − i ) ) ≠ 0 → ( 𝐴 − i ) ≠ 0 ) |
| 47 |
42 46
|
syl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( 𝐴 − i ) ≠ 0 ) |
| 48 |
|
simpr |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → 0 < ( ℜ ‘ 𝐴 ) ) |
| 49 |
|
0re |
⊢ 0 ∈ ℝ |
| 50 |
|
ltle |
⊢ ( ( 0 ∈ ℝ ∧ ( ℜ ‘ 𝐴 ) ∈ ℝ ) → ( 0 < ( ℜ ‘ 𝐴 ) → 0 ≤ ( ℜ ‘ 𝐴 ) ) ) |
| 51 |
49 35 50
|
sylancr |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( 0 < ( ℜ ‘ 𝐴 ) → 0 ≤ ( ℜ ‘ 𝐴 ) ) ) |
| 52 |
48 51
|
mpd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → 0 ≤ ( ℜ ‘ 𝐴 ) ) |
| 53 |
52 39
|
breqtrrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → 0 ≤ ( ℜ ‘ ( 𝐴 − i ) ) ) |
| 54 |
|
logimul |
⊢ ( ( ( 𝐴 − i ) ∈ ℂ ∧ ( 𝐴 − i ) ≠ 0 ∧ 0 ≤ ( ℜ ‘ ( 𝐴 − i ) ) ) → ( log ‘ ( i · ( 𝐴 − i ) ) ) = ( ( log ‘ ( 𝐴 − i ) ) + ( i · ( π / 2 ) ) ) ) |
| 55 |
30 47 53 54
|
syl3anc |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( log ‘ ( i · ( 𝐴 − i ) ) ) = ( ( log ‘ ( 𝐴 − i ) ) + ( i · ( π / 2 ) ) ) ) |
| 56 |
28 55
|
eqtr3d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( log ‘ ( 1 + ( i · 𝐴 ) ) ) = ( ( log ‘ ( 𝐴 − i ) ) + ( i · ( π / 2 ) ) ) ) |
| 57 |
56
|
fveq2d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) = ( ℑ ‘ ( ( log ‘ ( 𝐴 − i ) ) + ( i · ( π / 2 ) ) ) ) ) |
| 58 |
30 47
|
logcld |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( log ‘ ( 𝐴 − i ) ) ∈ ℂ ) |
| 59 |
|
halfpire |
⊢ ( π / 2 ) ∈ ℝ |
| 60 |
59
|
recni |
⊢ ( π / 2 ) ∈ ℂ |
| 61 |
2 60
|
mulcli |
⊢ ( i · ( π / 2 ) ) ∈ ℂ |
| 62 |
|
imadd |
⊢ ( ( ( log ‘ ( 𝐴 − i ) ) ∈ ℂ ∧ ( i · ( π / 2 ) ) ∈ ℂ ) → ( ℑ ‘ ( ( log ‘ ( 𝐴 − i ) ) + ( i · ( π / 2 ) ) ) ) = ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) + ( ℑ ‘ ( i · ( π / 2 ) ) ) ) ) |
| 63 |
58 61 62
|
sylancl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( ( log ‘ ( 𝐴 − i ) ) + ( i · ( π / 2 ) ) ) ) = ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) + ( ℑ ‘ ( i · ( π / 2 ) ) ) ) ) |
| 64 |
|
reim |
⊢ ( ( π / 2 ) ∈ ℂ → ( ℜ ‘ ( π / 2 ) ) = ( ℑ ‘ ( i · ( π / 2 ) ) ) ) |
| 65 |
60 64
|
ax-mp |
⊢ ( ℜ ‘ ( π / 2 ) ) = ( ℑ ‘ ( i · ( π / 2 ) ) ) |
| 66 |
|
rere |
⊢ ( ( π / 2 ) ∈ ℝ → ( ℜ ‘ ( π / 2 ) ) = ( π / 2 ) ) |
| 67 |
59 66
|
ax-mp |
⊢ ( ℜ ‘ ( π / 2 ) ) = ( π / 2 ) |
| 68 |
65 67
|
eqtr3i |
⊢ ( ℑ ‘ ( i · ( π / 2 ) ) ) = ( π / 2 ) |
| 69 |
68
|
oveq2i |
⊢ ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) + ( ℑ ‘ ( i · ( π / 2 ) ) ) ) = ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) + ( π / 2 ) ) |
| 70 |
63 69
|
eqtrdi |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( ( log ‘ ( 𝐴 − i ) ) + ( i · ( π / 2 ) ) ) ) = ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) + ( π / 2 ) ) ) |
| 71 |
57 70
|
eqtrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) = ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) + ( π / 2 ) ) ) |
| 72 |
|
addcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ i ∈ ℂ ) → ( 𝐴 + i ) ∈ ℂ ) |
| 73 |
6 2 72
|
sylancl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( 𝐴 + i ) ∈ ℂ ) |
| 74 |
|
mulcl |
⊢ ( ( i ∈ ℂ ∧ ( 𝐴 + i ) ∈ ℂ ) → ( i · ( 𝐴 + i ) ) ∈ ℂ ) |
| 75 |
2 73 74
|
sylancr |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( i · ( 𝐴 + i ) ) ∈ ℂ ) |
| 76 |
|
reim |
⊢ ( ( 𝐴 + i ) ∈ ℂ → ( ℜ ‘ ( 𝐴 + i ) ) = ( ℑ ‘ ( i · ( 𝐴 + i ) ) ) ) |
| 77 |
73 76
|
syl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℜ ‘ ( 𝐴 + i ) ) = ( ℑ ‘ ( i · ( 𝐴 + i ) ) ) ) |
| 78 |
|
readd |
⊢ ( ( 𝐴 ∈ ℂ ∧ i ∈ ℂ ) → ( ℜ ‘ ( 𝐴 + i ) ) = ( ( ℜ ‘ 𝐴 ) + ( ℜ ‘ i ) ) ) |
| 79 |
6 2 78
|
sylancl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℜ ‘ ( 𝐴 + i ) ) = ( ( ℜ ‘ 𝐴 ) + ( ℜ ‘ i ) ) ) |
| 80 |
33
|
oveq2i |
⊢ ( ( ℜ ‘ 𝐴 ) + ( ℜ ‘ i ) ) = ( ( ℜ ‘ 𝐴 ) + 0 ) |
| 81 |
36
|
addridd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℜ ‘ 𝐴 ) + 0 ) = ( ℜ ‘ 𝐴 ) ) |
| 82 |
80 81
|
eqtrid |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℜ ‘ 𝐴 ) + ( ℜ ‘ i ) ) = ( ℜ ‘ 𝐴 ) ) |
| 83 |
79 82
|
eqtrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℜ ‘ ( 𝐴 + i ) ) = ( ℜ ‘ 𝐴 ) ) |
| 84 |
77 83
|
eqtr3d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( i · ( 𝐴 + i ) ) ) = ( ℜ ‘ 𝐴 ) ) |
| 85 |
48 84
|
breqtrrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → 0 < ( ℑ ‘ ( i · ( 𝐴 + i ) ) ) ) |
| 86 |
|
logneg2 |
⊢ ( ( ( i · ( 𝐴 + i ) ) ∈ ℂ ∧ 0 < ( ℑ ‘ ( i · ( 𝐴 + i ) ) ) ) → ( log ‘ - ( i · ( 𝐴 + i ) ) ) = ( ( log ‘ ( i · ( 𝐴 + i ) ) ) − ( i · π ) ) ) |
| 87 |
75 85 86
|
syl2anc |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( log ‘ - ( i · ( 𝐴 + i ) ) ) = ( ( log ‘ ( i · ( 𝐴 + i ) ) ) − ( i · π ) ) ) |
| 88 |
18 6 18
|
adddid |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( i · ( 𝐴 + i ) ) = ( ( i · 𝐴 ) + ( i · i ) ) ) |
| 89 |
20
|
oveq2i |
⊢ ( ( i · 𝐴 ) + ( i · i ) ) = ( ( i · 𝐴 ) + - 1 ) |
| 90 |
|
negsub |
⊢ ( ( ( i · 𝐴 ) ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( i · 𝐴 ) + - 1 ) = ( ( i · 𝐴 ) − 1 ) ) |
| 91 |
8 1 90
|
sylancl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( i · 𝐴 ) + - 1 ) = ( ( i · 𝐴 ) − 1 ) ) |
| 92 |
89 91
|
eqtrid |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( i · 𝐴 ) + ( i · i ) ) = ( ( i · 𝐴 ) − 1 ) ) |
| 93 |
88 92
|
eqtrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( i · ( 𝐴 + i ) ) = ( ( i · 𝐴 ) − 1 ) ) |
| 94 |
93
|
negeqd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → - ( i · ( 𝐴 + i ) ) = - ( ( i · 𝐴 ) − 1 ) ) |
| 95 |
|
negsubdi2 |
⊢ ( ( ( i · 𝐴 ) ∈ ℂ ∧ 1 ∈ ℂ ) → - ( ( i · 𝐴 ) − 1 ) = ( 1 − ( i · 𝐴 ) ) ) |
| 96 |
8 1 95
|
sylancl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → - ( ( i · 𝐴 ) − 1 ) = ( 1 − ( i · 𝐴 ) ) ) |
| 97 |
94 96
|
eqtrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → - ( i · ( 𝐴 + i ) ) = ( 1 − ( i · 𝐴 ) ) ) |
| 98 |
97
|
fveq2d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( log ‘ - ( i · ( 𝐴 + i ) ) ) = ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) |
| 99 |
83 41
|
eqnetrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℜ ‘ ( 𝐴 + i ) ) ≠ 0 ) |
| 100 |
|
fveq2 |
⊢ ( ( 𝐴 + i ) = 0 → ( ℜ ‘ ( 𝐴 + i ) ) = ( ℜ ‘ 0 ) ) |
| 101 |
100 44
|
eqtrdi |
⊢ ( ( 𝐴 + i ) = 0 → ( ℜ ‘ ( 𝐴 + i ) ) = 0 ) |
| 102 |
101
|
necon3i |
⊢ ( ( ℜ ‘ ( 𝐴 + i ) ) ≠ 0 → ( 𝐴 + i ) ≠ 0 ) |
| 103 |
99 102
|
syl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( 𝐴 + i ) ≠ 0 ) |
| 104 |
73 103
|
logcld |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( log ‘ ( 𝐴 + i ) ) ∈ ℂ ) |
| 105 |
61
|
a1i |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( i · ( π / 2 ) ) ∈ ℂ ) |
| 106 |
|
picn |
⊢ π ∈ ℂ |
| 107 |
2 106
|
mulcli |
⊢ ( i · π ) ∈ ℂ |
| 108 |
107
|
a1i |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( i · π ) ∈ ℂ ) |
| 109 |
52 83
|
breqtrrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → 0 ≤ ( ℜ ‘ ( 𝐴 + i ) ) ) |
| 110 |
|
logimul |
⊢ ( ( ( 𝐴 + i ) ∈ ℂ ∧ ( 𝐴 + i ) ≠ 0 ∧ 0 ≤ ( ℜ ‘ ( 𝐴 + i ) ) ) → ( log ‘ ( i · ( 𝐴 + i ) ) ) = ( ( log ‘ ( 𝐴 + i ) ) + ( i · ( π / 2 ) ) ) ) |
| 111 |
73 103 109 110
|
syl3anc |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( log ‘ ( i · ( 𝐴 + i ) ) ) = ( ( log ‘ ( 𝐴 + i ) ) + ( i · ( π / 2 ) ) ) ) |
| 112 |
111
|
oveq1d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( log ‘ ( i · ( 𝐴 + i ) ) ) − ( i · π ) ) = ( ( ( log ‘ ( 𝐴 + i ) ) + ( i · ( π / 2 ) ) ) − ( i · π ) ) ) |
| 113 |
104 105 108 112
|
assraddsubd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( log ‘ ( i · ( 𝐴 + i ) ) ) − ( i · π ) ) = ( ( log ‘ ( 𝐴 + i ) ) + ( ( i · ( π / 2 ) ) − ( i · π ) ) ) ) |
| 114 |
87 98 113
|
3eqtr3d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( log ‘ ( 1 − ( i · 𝐴 ) ) ) = ( ( log ‘ ( 𝐴 + i ) ) + ( ( i · ( π / 2 ) ) − ( i · π ) ) ) ) |
| 115 |
114
|
fveq2d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) = ( ℑ ‘ ( ( log ‘ ( 𝐴 + i ) ) + ( ( i · ( π / 2 ) ) − ( i · π ) ) ) ) ) |
| 116 |
61 107
|
subcli |
⊢ ( ( i · ( π / 2 ) ) − ( i · π ) ) ∈ ℂ |
| 117 |
|
imadd |
⊢ ( ( ( log ‘ ( 𝐴 + i ) ) ∈ ℂ ∧ ( ( i · ( π / 2 ) ) − ( i · π ) ) ∈ ℂ ) → ( ℑ ‘ ( ( log ‘ ( 𝐴 + i ) ) + ( ( i · ( π / 2 ) ) − ( i · π ) ) ) ) = ( ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) + ( ℑ ‘ ( ( i · ( π / 2 ) ) − ( i · π ) ) ) ) ) |
| 118 |
104 116 117
|
sylancl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( ( log ‘ ( 𝐴 + i ) ) + ( ( i · ( π / 2 ) ) − ( i · π ) ) ) ) = ( ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) + ( ℑ ‘ ( ( i · ( π / 2 ) ) − ( i · π ) ) ) ) ) |
| 119 |
|
imsub |
⊢ ( ( ( i · ( π / 2 ) ) ∈ ℂ ∧ ( i · π ) ∈ ℂ ) → ( ℑ ‘ ( ( i · ( π / 2 ) ) − ( i · π ) ) ) = ( ( ℑ ‘ ( i · ( π / 2 ) ) ) − ( ℑ ‘ ( i · π ) ) ) ) |
| 120 |
61 107 119
|
mp2an |
⊢ ( ℑ ‘ ( ( i · ( π / 2 ) ) − ( i · π ) ) ) = ( ( ℑ ‘ ( i · ( π / 2 ) ) ) − ( ℑ ‘ ( i · π ) ) ) |
| 121 |
|
reim |
⊢ ( π ∈ ℂ → ( ℜ ‘ π ) = ( ℑ ‘ ( i · π ) ) ) |
| 122 |
106 121
|
ax-mp |
⊢ ( ℜ ‘ π ) = ( ℑ ‘ ( i · π ) ) |
| 123 |
|
pire |
⊢ π ∈ ℝ |
| 124 |
|
rere |
⊢ ( π ∈ ℝ → ( ℜ ‘ π ) = π ) |
| 125 |
123 124
|
ax-mp |
⊢ ( ℜ ‘ π ) = π |
| 126 |
122 125
|
eqtr3i |
⊢ ( ℑ ‘ ( i · π ) ) = π |
| 127 |
68 126
|
oveq12i |
⊢ ( ( ℑ ‘ ( i · ( π / 2 ) ) ) − ( ℑ ‘ ( i · π ) ) ) = ( ( π / 2 ) − π ) |
| 128 |
60
|
negcli |
⊢ - ( π / 2 ) ∈ ℂ |
| 129 |
106 60
|
negsubi |
⊢ ( π + - ( π / 2 ) ) = ( π − ( π / 2 ) ) |
| 130 |
|
pidiv2halves |
⊢ ( ( π / 2 ) + ( π / 2 ) ) = π |
| 131 |
106 60 60 130
|
subaddrii |
⊢ ( π − ( π / 2 ) ) = ( π / 2 ) |
| 132 |
129 131
|
eqtri |
⊢ ( π + - ( π / 2 ) ) = ( π / 2 ) |
| 133 |
60 106 128 132
|
subaddrii |
⊢ ( ( π / 2 ) − π ) = - ( π / 2 ) |
| 134 |
120 127 133
|
3eqtri |
⊢ ( ℑ ‘ ( ( i · ( π / 2 ) ) − ( i · π ) ) ) = - ( π / 2 ) |
| 135 |
134
|
oveq2i |
⊢ ( ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) + ( ℑ ‘ ( ( i · ( π / 2 ) ) − ( i · π ) ) ) ) = ( ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) + - ( π / 2 ) ) |
| 136 |
118 135
|
eqtrdi |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( ( log ‘ ( 𝐴 + i ) ) + ( ( i · ( π / 2 ) ) − ( i · π ) ) ) ) = ( ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) + - ( π / 2 ) ) ) |
| 137 |
115 136
|
eqtrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) = ( ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) + - ( π / 2 ) ) ) |
| 138 |
71 137
|
oveq12d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) − ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) = ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) + ( π / 2 ) ) − ( ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) + - ( π / 2 ) ) ) ) |
| 139 |
58
|
imcld |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) ∈ ℝ ) |
| 140 |
139
|
recnd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) ∈ ℂ ) |
| 141 |
60
|
a1i |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( π / 2 ) ∈ ℂ ) |
| 142 |
104
|
imcld |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ∈ ℝ ) |
| 143 |
142
|
recnd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ∈ ℂ ) |
| 144 |
128
|
a1i |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → - ( π / 2 ) ∈ ℂ ) |
| 145 |
140 141 143 144
|
addsub4d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) + ( π / 2 ) ) − ( ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) + - ( π / 2 ) ) ) = ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) + ( ( π / 2 ) − - ( π / 2 ) ) ) ) |
| 146 |
60 60
|
subnegi |
⊢ ( ( π / 2 ) − - ( π / 2 ) ) = ( ( π / 2 ) + ( π / 2 ) ) |
| 147 |
146 130
|
eqtri |
⊢ ( ( π / 2 ) − - ( π / 2 ) ) = π |
| 148 |
147
|
oveq2i |
⊢ ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) + ( ( π / 2 ) − - ( π / 2 ) ) ) = ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) + π ) |
| 149 |
145 148
|
eqtrdi |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) + ( π / 2 ) ) − ( ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) + - ( π / 2 ) ) ) = ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) + π ) ) |
| 150 |
17 138 149
|
3eqtrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) − ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) = ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) + π ) ) |
| 151 |
139 142
|
resubcld |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) ∈ ℝ ) |
| 152 |
|
readdcl |
⊢ ( ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) ∈ ℝ ∧ π ∈ ℝ ) → ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) + π ) ∈ ℝ ) |
| 153 |
151 123 152
|
sylancl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) + π ) ∈ ℝ ) |
| 154 |
123
|
renegcli |
⊢ - π ∈ ℝ |
| 155 |
154
|
recni |
⊢ - π ∈ ℂ |
| 156 |
155 106
|
negsubi |
⊢ ( - π + - π ) = ( - π − π ) |
| 157 |
154
|
a1i |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → - π ∈ ℝ ) |
| 158 |
142
|
renegcld |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → - ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ∈ ℝ ) |
| 159 |
30 47
|
logimcld |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( - π < ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) ∧ ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) ≤ π ) ) |
| 160 |
159
|
simpld |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → - π < ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) ) |
| 161 |
73 103
|
logimcld |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( - π < ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ∧ ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ≤ π ) ) |
| 162 |
161
|
simprd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ≤ π ) |
| 163 |
|
leneg |
⊢ ( ( ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ∈ ℝ ∧ π ∈ ℝ ) → ( ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ≤ π ↔ - π ≤ - ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) ) |
| 164 |
142 123 163
|
sylancl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ≤ π ↔ - π ≤ - ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) ) |
| 165 |
162 164
|
mpbid |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → - π ≤ - ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) |
| 166 |
157 157 139 158 160 165
|
ltleaddd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( - π + - π ) < ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) + - ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) ) |
| 167 |
140 143
|
negsubd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) + - ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) = ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) ) |
| 168 |
166 167
|
breqtrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( - π + - π ) < ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) ) |
| 169 |
156 168
|
eqbrtrrid |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( - π − π ) < ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) ) |
| 170 |
123
|
a1i |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → π ∈ ℝ ) |
| 171 |
157 170 151
|
ltsubaddd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( - π − π ) < ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) ↔ - π < ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) + π ) ) ) |
| 172 |
169 171
|
mpbid |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → - π < ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) + π ) ) |
| 173 |
|
0red |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → 0 ∈ ℝ ) |
| 174 |
6
|
imcld |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ 𝐴 ) ∈ ℝ ) |
| 175 |
|
peano2rem |
⊢ ( ( ℑ ‘ 𝐴 ) ∈ ℝ → ( ( ℑ ‘ 𝐴 ) − 1 ) ∈ ℝ ) |
| 176 |
174 175
|
syl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ 𝐴 ) − 1 ) ∈ ℝ ) |
| 177 |
|
peano2re |
⊢ ( ( ℑ ‘ 𝐴 ) ∈ ℝ → ( ( ℑ ‘ 𝐴 ) + 1 ) ∈ ℝ ) |
| 178 |
174 177
|
syl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ 𝐴 ) + 1 ) ∈ ℝ ) |
| 179 |
174
|
ltm1d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ 𝐴 ) − 1 ) < ( ℑ ‘ 𝐴 ) ) |
| 180 |
174
|
ltp1d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ 𝐴 ) < ( ( ℑ ‘ 𝐴 ) + 1 ) ) |
| 181 |
176 174 178 179 180
|
lttrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ 𝐴 ) − 1 ) < ( ( ℑ ‘ 𝐴 ) + 1 ) ) |
| 182 |
|
ltdiv1 |
⊢ ( ( ( ( ℑ ‘ 𝐴 ) − 1 ) ∈ ℝ ∧ ( ( ℑ ‘ 𝐴 ) + 1 ) ∈ ℝ ∧ ( ( ℜ ‘ 𝐴 ) ∈ ℝ ∧ 0 < ( ℜ ‘ 𝐴 ) ) ) → ( ( ( ℑ ‘ 𝐴 ) − 1 ) < ( ( ℑ ‘ 𝐴 ) + 1 ) ↔ ( ( ( ℑ ‘ 𝐴 ) − 1 ) / ( ℜ ‘ 𝐴 ) ) < ( ( ( ℑ ‘ 𝐴 ) + 1 ) / ( ℜ ‘ 𝐴 ) ) ) ) |
| 183 |
176 178 35 48 182
|
syl112anc |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ( ℑ ‘ 𝐴 ) − 1 ) < ( ( ℑ ‘ 𝐴 ) + 1 ) ↔ ( ( ( ℑ ‘ 𝐴 ) − 1 ) / ( ℜ ‘ 𝐴 ) ) < ( ( ( ℑ ‘ 𝐴 ) + 1 ) / ( ℜ ‘ 𝐴 ) ) ) ) |
| 184 |
181 183
|
mpbid |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ( ℑ ‘ 𝐴 ) − 1 ) / ( ℜ ‘ 𝐴 ) ) < ( ( ( ℑ ‘ 𝐴 ) + 1 ) / ( ℜ ‘ 𝐴 ) ) ) |
| 185 |
|
imsub |
⊢ ( ( 𝐴 ∈ ℂ ∧ i ∈ ℂ ) → ( ℑ ‘ ( 𝐴 − i ) ) = ( ( ℑ ‘ 𝐴 ) − ( ℑ ‘ i ) ) ) |
| 186 |
6 2 185
|
sylancl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( 𝐴 − i ) ) = ( ( ℑ ‘ 𝐴 ) − ( ℑ ‘ i ) ) ) |
| 187 |
|
imi |
⊢ ( ℑ ‘ i ) = 1 |
| 188 |
187
|
oveq2i |
⊢ ( ( ℑ ‘ 𝐴 ) − ( ℑ ‘ i ) ) = ( ( ℑ ‘ 𝐴 ) − 1 ) |
| 189 |
186 188
|
eqtrdi |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( 𝐴 − i ) ) = ( ( ℑ ‘ 𝐴 ) − 1 ) ) |
| 190 |
189 39
|
oveq12d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ ( 𝐴 − i ) ) / ( ℜ ‘ ( 𝐴 − i ) ) ) = ( ( ( ℑ ‘ 𝐴 ) − 1 ) / ( ℜ ‘ 𝐴 ) ) ) |
| 191 |
|
imadd |
⊢ ( ( 𝐴 ∈ ℂ ∧ i ∈ ℂ ) → ( ℑ ‘ ( 𝐴 + i ) ) = ( ( ℑ ‘ 𝐴 ) + ( ℑ ‘ i ) ) ) |
| 192 |
6 2 191
|
sylancl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( 𝐴 + i ) ) = ( ( ℑ ‘ 𝐴 ) + ( ℑ ‘ i ) ) ) |
| 193 |
187
|
oveq2i |
⊢ ( ( ℑ ‘ 𝐴 ) + ( ℑ ‘ i ) ) = ( ( ℑ ‘ 𝐴 ) + 1 ) |
| 194 |
192 193
|
eqtrdi |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( 𝐴 + i ) ) = ( ( ℑ ‘ 𝐴 ) + 1 ) ) |
| 195 |
194 83
|
oveq12d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ ( 𝐴 + i ) ) / ( ℜ ‘ ( 𝐴 + i ) ) ) = ( ( ( ℑ ‘ 𝐴 ) + 1 ) / ( ℜ ‘ 𝐴 ) ) ) |
| 196 |
184 190 195
|
3brtr4d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ ( 𝐴 − i ) ) / ( ℜ ‘ ( 𝐴 − i ) ) ) < ( ( ℑ ‘ ( 𝐴 + i ) ) / ( ℜ ‘ ( 𝐴 + i ) ) ) ) |
| 197 |
|
tanarg |
⊢ ( ( ( 𝐴 − i ) ∈ ℂ ∧ ( ℜ ‘ ( 𝐴 − i ) ) ≠ 0 ) → ( tan ‘ ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) ) = ( ( ℑ ‘ ( 𝐴 − i ) ) / ( ℜ ‘ ( 𝐴 − i ) ) ) ) |
| 198 |
30 42 197
|
syl2anc |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( tan ‘ ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) ) = ( ( ℑ ‘ ( 𝐴 − i ) ) / ( ℜ ‘ ( 𝐴 − i ) ) ) ) |
| 199 |
|
tanarg |
⊢ ( ( ( 𝐴 + i ) ∈ ℂ ∧ ( ℜ ‘ ( 𝐴 + i ) ) ≠ 0 ) → ( tan ‘ ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) = ( ( ℑ ‘ ( 𝐴 + i ) ) / ( ℜ ‘ ( 𝐴 + i ) ) ) ) |
| 200 |
73 99 199
|
syl2anc |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( tan ‘ ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) = ( ( ℑ ‘ ( 𝐴 + i ) ) / ( ℜ ‘ ( 𝐴 + i ) ) ) ) |
| 201 |
196 198 200
|
3brtr4d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( tan ‘ ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) ) < ( tan ‘ ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) ) |
| 202 |
48 39
|
breqtrrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → 0 < ( ℜ ‘ ( 𝐴 − i ) ) ) |
| 203 |
|
argregt0 |
⊢ ( ( ( 𝐴 − i ) ∈ ℂ ∧ 0 < ( ℜ ‘ ( 𝐴 − i ) ) ) → ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ) |
| 204 |
30 202 203
|
syl2anc |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ) |
| 205 |
48 83
|
breqtrrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → 0 < ( ℜ ‘ ( 𝐴 + i ) ) ) |
| 206 |
|
argregt0 |
⊢ ( ( ( 𝐴 + i ) ∈ ℂ ∧ 0 < ( ℜ ‘ ( 𝐴 + i ) ) ) → ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ) |
| 207 |
73 205 206
|
syl2anc |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ) |
| 208 |
|
tanord |
⊢ ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ∧ ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ) → ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) < ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ↔ ( tan ‘ ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) ) < ( tan ‘ ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) ) ) |
| 209 |
204 207 208
|
syl2anc |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) < ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ↔ ( tan ‘ ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) ) < ( tan ‘ ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) ) ) |
| 210 |
201 209
|
mpbird |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) < ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) |
| 211 |
143
|
addlidd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( 0 + ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) = ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) |
| 212 |
210 211
|
breqtrrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) < ( 0 + ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) ) |
| 213 |
139 142 173
|
ltsubaddd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) < 0 ↔ ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) < ( 0 + ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) ) ) |
| 214 |
212 213
|
mpbird |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) < 0 ) |
| 215 |
151 173 170 214
|
ltadd1dd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) + π ) < ( 0 + π ) ) |
| 216 |
106
|
addlidi |
⊢ ( 0 + π ) = π |
| 217 |
215 216
|
breqtrdi |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) + π ) < π ) |
| 218 |
154
|
rexri |
⊢ - π ∈ ℝ* |
| 219 |
123
|
rexri |
⊢ π ∈ ℝ* |
| 220 |
|
elioo2 |
⊢ ( ( - π ∈ ℝ* ∧ π ∈ ℝ* ) → ( ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) + π ) ∈ ( - π (,) π ) ↔ ( ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) + π ) ∈ ℝ ∧ - π < ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) + π ) ∧ ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) + π ) < π ) ) ) |
| 221 |
218 219 220
|
mp2an |
⊢ ( ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) + π ) ∈ ( - π (,) π ) ↔ ( ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) + π ) ∈ ℝ ∧ - π < ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) + π ) ∧ ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) + π ) < π ) ) |
| 222 |
153 172 217 221
|
syl3anbrc |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ( ℑ ‘ ( log ‘ ( 𝐴 − i ) ) ) − ( ℑ ‘ ( log ‘ ( 𝐴 + i ) ) ) ) + π ) ∈ ( - π (,) π ) ) |
| 223 |
150 222
|
eqeltrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) − ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) ∈ ( - π (,) π ) ) |