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
|
syl5eq |
⊢ ( ( 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
|
syl5eq |
⊢ ( ( 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
|
addid1d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℜ ‘ 𝐴 ) + 0 ) = ( ℜ ‘ 𝐴 ) ) |
82 |
80 81
|
syl5eq |
⊢ ( ( 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
|
syl5eq |
⊢ ( ( 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
|
addid2d |
⊢ ( ( 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
|
addid2i |
⊢ ( 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 ) ) |