Step |
Hyp |
Ref |
Expression |
1 |
|
0re |
|- 0 e. RR |
2 |
|
atandm2 |
|- ( A e. dom arctan <-> ( A e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) ) |
3 |
2
|
simp1bi |
|- ( A e. dom arctan -> A e. CC ) |
4 |
3
|
recld |
|- ( A e. dom arctan -> ( Re ` A ) e. RR ) |
5 |
|
leloe |
|- ( ( 0 e. RR /\ ( Re ` A ) e. RR ) -> ( 0 <_ ( Re ` A ) <-> ( 0 < ( Re ` A ) \/ 0 = ( Re ` A ) ) ) ) |
6 |
1 4 5
|
sylancr |
|- ( A e. dom arctan -> ( 0 <_ ( Re ` A ) <-> ( 0 < ( Re ` A ) \/ 0 = ( Re ` A ) ) ) ) |
7 |
6
|
biimpa |
|- ( ( A e. dom arctan /\ 0 <_ ( Re ` A ) ) -> ( 0 < ( Re ` A ) \/ 0 = ( Re ` A ) ) ) |
8 |
|
ax-1cn |
|- 1 e. CC |
9 |
|
ax-icn |
|- _i e. CC |
10 |
|
mulcl |
|- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC ) |
11 |
9 3 10
|
sylancr |
|- ( A e. dom arctan -> ( _i x. A ) e. CC ) |
12 |
|
addcl |
|- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 + ( _i x. A ) ) e. CC ) |
13 |
8 11 12
|
sylancr |
|- ( A e. dom arctan -> ( 1 + ( _i x. A ) ) e. CC ) |
14 |
2
|
simp3bi |
|- ( A e. dom arctan -> ( 1 + ( _i x. A ) ) =/= 0 ) |
15 |
13 14
|
logcld |
|- ( A e. dom arctan -> ( log ` ( 1 + ( _i x. A ) ) ) e. CC ) |
16 |
|
subcl |
|- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 - ( _i x. A ) ) e. CC ) |
17 |
8 11 16
|
sylancr |
|- ( A e. dom arctan -> ( 1 - ( _i x. A ) ) e. CC ) |
18 |
2
|
simp2bi |
|- ( A e. dom arctan -> ( 1 - ( _i x. A ) ) =/= 0 ) |
19 |
17 18
|
logcld |
|- ( A e. dom arctan -> ( log ` ( 1 - ( _i x. A ) ) ) e. CC ) |
20 |
15 19
|
addcld |
|- ( A e. dom arctan -> ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) e. CC ) |
21 |
20
|
adantr |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) e. CC ) |
22 |
|
pire |
|- _pi e. RR |
23 |
22
|
renegcli |
|- -u _pi e. RR |
24 |
23
|
a1i |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> -u _pi e. RR ) |
25 |
19
|
adantr |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( log ` ( 1 - ( _i x. A ) ) ) e. CC ) |
26 |
25
|
imcld |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) e. RR ) |
27 |
15
|
adantr |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( log ` ( 1 + ( _i x. A ) ) ) e. CC ) |
28 |
27
|
imcld |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) e. RR ) |
29 |
28 26
|
readdcld |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) + ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) ) e. RR ) |
30 |
17
|
adantr |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( 1 - ( _i x. A ) ) e. CC ) |
31 |
|
im1 |
|- ( Im ` 1 ) = 0 |
32 |
31
|
oveq1i |
|- ( ( Im ` 1 ) - ( Im ` ( _i x. A ) ) ) = ( 0 - ( Im ` ( _i x. A ) ) ) |
33 |
|
df-neg |
|- -u ( Im ` ( _i x. A ) ) = ( 0 - ( Im ` ( _i x. A ) ) ) |
34 |
32 33
|
eqtr4i |
|- ( ( Im ` 1 ) - ( Im ` ( _i x. A ) ) ) = -u ( Im ` ( _i x. A ) ) |
35 |
11
|
adantr |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( _i x. A ) e. CC ) |
36 |
|
imsub |
|- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( Im ` ( 1 - ( _i x. A ) ) ) = ( ( Im ` 1 ) - ( Im ` ( _i x. A ) ) ) ) |
37 |
8 35 36
|
sylancr |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( Im ` ( 1 - ( _i x. A ) ) ) = ( ( Im ` 1 ) - ( Im ` ( _i x. A ) ) ) ) |
38 |
3
|
adantr |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> A e. CC ) |
39 |
|
reim |
|- ( A e. CC -> ( Re ` A ) = ( Im ` ( _i x. A ) ) ) |
40 |
38 39
|
syl |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( Re ` A ) = ( Im ` ( _i x. A ) ) ) |
41 |
40
|
negeqd |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> -u ( Re ` A ) = -u ( Im ` ( _i x. A ) ) ) |
42 |
34 37 41
|
3eqtr4a |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( Im ` ( 1 - ( _i x. A ) ) ) = -u ( Re ` A ) ) |
43 |
4
|
lt0neg2d |
|- ( A e. dom arctan -> ( 0 < ( Re ` A ) <-> -u ( Re ` A ) < 0 ) ) |
44 |
43
|
biimpa |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> -u ( Re ` A ) < 0 ) |
45 |
42 44
|
eqbrtrd |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( Im ` ( 1 - ( _i x. A ) ) ) < 0 ) |
46 |
|
argimlt0 |
|- ( ( ( 1 - ( _i x. A ) ) e. CC /\ ( Im ` ( 1 - ( _i x. A ) ) ) < 0 ) -> ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) e. ( -u _pi (,) 0 ) ) |
47 |
30 45 46
|
syl2anc |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) e. ( -u _pi (,) 0 ) ) |
48 |
|
eliooord |
|- ( ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) e. ( -u _pi (,) 0 ) -> ( -u _pi < ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) /\ ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) < 0 ) ) |
49 |
47 48
|
syl |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( -u _pi < ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) /\ ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) < 0 ) ) |
50 |
49
|
simpld |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> -u _pi < ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) ) |
51 |
13
|
adantr |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( 1 + ( _i x. A ) ) e. CC ) |
52 |
|
simpr |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> 0 < ( Re ` A ) ) |
53 |
|
imadd |
|- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( Im ` ( 1 + ( _i x. A ) ) ) = ( ( Im ` 1 ) + ( Im ` ( _i x. A ) ) ) ) |
54 |
8 35 53
|
sylancr |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( Im ` ( 1 + ( _i x. A ) ) ) = ( ( Im ` 1 ) + ( Im ` ( _i x. A ) ) ) ) |
55 |
40
|
oveq2d |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( ( Im ` 1 ) + ( Re ` A ) ) = ( ( Im ` 1 ) + ( Im ` ( _i x. A ) ) ) ) |
56 |
31
|
oveq1i |
|- ( ( Im ` 1 ) + ( Re ` A ) ) = ( 0 + ( Re ` A ) ) |
57 |
38
|
recld |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( Re ` A ) e. RR ) |
58 |
57
|
recnd |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( Re ` A ) e. CC ) |
59 |
58
|
addid2d |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( 0 + ( Re ` A ) ) = ( Re ` A ) ) |
60 |
56 59
|
syl5eq |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( ( Im ` 1 ) + ( Re ` A ) ) = ( Re ` A ) ) |
61 |
54 55 60
|
3eqtr2d |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( Im ` ( 1 + ( _i x. A ) ) ) = ( Re ` A ) ) |
62 |
52 61
|
breqtrrd |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> 0 < ( Im ` ( 1 + ( _i x. A ) ) ) ) |
63 |
|
argimgt0 |
|- ( ( ( 1 + ( _i x. A ) ) e. CC /\ 0 < ( Im ` ( 1 + ( _i x. A ) ) ) ) -> ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) e. ( 0 (,) _pi ) ) |
64 |
51 62 63
|
syl2anc |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) e. ( 0 (,) _pi ) ) |
65 |
|
eliooord |
|- ( ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) e. ( 0 (,) _pi ) -> ( 0 < ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) /\ ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) < _pi ) ) |
66 |
64 65
|
syl |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( 0 < ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) /\ ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) < _pi ) ) |
67 |
66
|
simpld |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> 0 < ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) ) |
68 |
28 26
|
ltaddpos2d |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( 0 < ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) <-> ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) < ( ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) + ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) ) |
69 |
67 68
|
mpbid |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) < ( ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) + ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) |
70 |
24 26 29 50 69
|
lttrd |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> -u _pi < ( ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) + ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) |
71 |
27 25
|
imaddd |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( Im ` ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) + ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) |
72 |
70 71
|
breqtrrd |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> -u _pi < ( Im ` ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) |
73 |
22
|
a1i |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> _pi e. RR ) |
74 |
|
0red |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> 0 e. RR ) |
75 |
49
|
simprd |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) < 0 ) |
76 |
26 74 28 75
|
ltadd2dd |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) + ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) ) < ( ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) + 0 ) ) |
77 |
28
|
recnd |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) e. CC ) |
78 |
77
|
addid1d |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) + 0 ) = ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) ) |
79 |
76 78
|
breqtrd |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) + ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) ) < ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) ) |
80 |
66
|
simprd |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) < _pi ) |
81 |
29 28 73 79 80
|
lttrd |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) + ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) ) < _pi ) |
82 |
29 73 81
|
ltled |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( ( Im ` ( log ` ( 1 + ( _i x. A ) ) ) ) + ( Im ` ( log ` ( 1 - ( _i x. A ) ) ) ) ) <_ _pi ) |
83 |
71 82
|
eqbrtrd |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( Im ` ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) ) <_ _pi ) |
84 |
|
ellogrn |
|- ( ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) e. ran log <-> ( ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) e. CC /\ -u _pi < ( Im ` ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) ) /\ ( Im ` ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) ) <_ _pi ) ) |
85 |
21 72 83 84
|
syl3anbrc |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) e. ran log ) |
86 |
|
0red |
|- ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) -> 0 e. RR ) |
87 |
11
|
adantr |
|- ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) -> ( _i x. A ) e. CC ) |
88 |
|
simpr |
|- ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) -> 0 = ( Re ` A ) ) |
89 |
3
|
adantr |
|- ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) -> A e. CC ) |
90 |
89 39
|
syl |
|- ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) -> ( Re ` A ) = ( Im ` ( _i x. A ) ) ) |
91 |
88 90
|
eqtr2d |
|- ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) -> ( Im ` ( _i x. A ) ) = 0 ) |
92 |
87 91
|
reim0bd |
|- ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) -> ( _i x. A ) e. RR ) |
93 |
15 19
|
addcomd |
|- ( A e. dom arctan -> ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) = ( ( log ` ( 1 - ( _i x. A ) ) ) + ( log ` ( 1 + ( _i x. A ) ) ) ) ) |
94 |
93
|
ad2antrr |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ 0 <_ ( _i x. A ) ) -> ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) = ( ( log ` ( 1 - ( _i x. A ) ) ) + ( log ` ( 1 + ( _i x. A ) ) ) ) ) |
95 |
|
logrncl |
|- ( ( ( 1 - ( _i x. A ) ) e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 ) -> ( log ` ( 1 - ( _i x. A ) ) ) e. ran log ) |
96 |
17 18 95
|
syl2anc |
|- ( A e. dom arctan -> ( log ` ( 1 - ( _i x. A ) ) ) e. ran log ) |
97 |
96
|
ad2antrr |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ 0 <_ ( _i x. A ) ) -> ( log ` ( 1 - ( _i x. A ) ) ) e. ran log ) |
98 |
|
1re |
|- 1 e. RR |
99 |
92
|
adantr |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ 0 <_ ( _i x. A ) ) -> ( _i x. A ) e. RR ) |
100 |
|
readdcl |
|- ( ( 1 e. RR /\ ( _i x. A ) e. RR ) -> ( 1 + ( _i x. A ) ) e. RR ) |
101 |
98 99 100
|
sylancr |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ 0 <_ ( _i x. A ) ) -> ( 1 + ( _i x. A ) ) e. RR ) |
102 |
|
0red |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ 0 <_ ( _i x. A ) ) -> 0 e. RR ) |
103 |
|
1red |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ 0 <_ ( _i x. A ) ) -> 1 e. RR ) |
104 |
|
0lt1 |
|- 0 < 1 |
105 |
104
|
a1i |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ 0 <_ ( _i x. A ) ) -> 0 < 1 ) |
106 |
|
addge01 |
|- ( ( 1 e. RR /\ ( _i x. A ) e. RR ) -> ( 0 <_ ( _i x. A ) <-> 1 <_ ( 1 + ( _i x. A ) ) ) ) |
107 |
98 92 106
|
sylancr |
|- ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) -> ( 0 <_ ( _i x. A ) <-> 1 <_ ( 1 + ( _i x. A ) ) ) ) |
108 |
107
|
biimpa |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ 0 <_ ( _i x. A ) ) -> 1 <_ ( 1 + ( _i x. A ) ) ) |
109 |
102 103 101 105 108
|
ltletrd |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ 0 <_ ( _i x. A ) ) -> 0 < ( 1 + ( _i x. A ) ) ) |
110 |
101 109
|
elrpd |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ 0 <_ ( _i x. A ) ) -> ( 1 + ( _i x. A ) ) e. RR+ ) |
111 |
110
|
relogcld |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ 0 <_ ( _i x. A ) ) -> ( log ` ( 1 + ( _i x. A ) ) ) e. RR ) |
112 |
|
logrnaddcl |
|- ( ( ( log ` ( 1 - ( _i x. A ) ) ) e. ran log /\ ( log ` ( 1 + ( _i x. A ) ) ) e. RR ) -> ( ( log ` ( 1 - ( _i x. A ) ) ) + ( log ` ( 1 + ( _i x. A ) ) ) ) e. ran log ) |
113 |
97 111 112
|
syl2anc |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ 0 <_ ( _i x. A ) ) -> ( ( log ` ( 1 - ( _i x. A ) ) ) + ( log ` ( 1 + ( _i x. A ) ) ) ) e. ran log ) |
114 |
94 113
|
eqeltrd |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ 0 <_ ( _i x. A ) ) -> ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) e. ran log ) |
115 |
|
logrncl |
|- ( ( ( 1 + ( _i x. A ) ) e. CC /\ ( 1 + ( _i x. A ) ) =/= 0 ) -> ( log ` ( 1 + ( _i x. A ) ) ) e. ran log ) |
116 |
13 14 115
|
syl2anc |
|- ( A e. dom arctan -> ( log ` ( 1 + ( _i x. A ) ) ) e. ran log ) |
117 |
116
|
ad2antrr |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ ( _i x. A ) <_ 0 ) -> ( log ` ( 1 + ( _i x. A ) ) ) e. ran log ) |
118 |
92
|
adantr |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ ( _i x. A ) <_ 0 ) -> ( _i x. A ) e. RR ) |
119 |
|
resubcl |
|- ( ( 1 e. RR /\ ( _i x. A ) e. RR ) -> ( 1 - ( _i x. A ) ) e. RR ) |
120 |
98 118 119
|
sylancr |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ ( _i x. A ) <_ 0 ) -> ( 1 - ( _i x. A ) ) e. RR ) |
121 |
|
0red |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ ( _i x. A ) <_ 0 ) -> 0 e. RR ) |
122 |
|
1red |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ ( _i x. A ) <_ 0 ) -> 1 e. RR ) |
123 |
104
|
a1i |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ ( _i x. A ) <_ 0 ) -> 0 < 1 ) |
124 |
|
1m0e1 |
|- ( 1 - 0 ) = 1 |
125 |
|
1red |
|- ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) -> 1 e. RR ) |
126 |
92 86 125
|
lesub2d |
|- ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) -> ( ( _i x. A ) <_ 0 <-> ( 1 - 0 ) <_ ( 1 - ( _i x. A ) ) ) ) |
127 |
126
|
biimpa |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ ( _i x. A ) <_ 0 ) -> ( 1 - 0 ) <_ ( 1 - ( _i x. A ) ) ) |
128 |
124 127
|
eqbrtrrid |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ ( _i x. A ) <_ 0 ) -> 1 <_ ( 1 - ( _i x. A ) ) ) |
129 |
121 122 120 123 128
|
ltletrd |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ ( _i x. A ) <_ 0 ) -> 0 < ( 1 - ( _i x. A ) ) ) |
130 |
120 129
|
elrpd |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ ( _i x. A ) <_ 0 ) -> ( 1 - ( _i x. A ) ) e. RR+ ) |
131 |
130
|
relogcld |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ ( _i x. A ) <_ 0 ) -> ( log ` ( 1 - ( _i x. A ) ) ) e. RR ) |
132 |
|
logrnaddcl |
|- ( ( ( log ` ( 1 + ( _i x. A ) ) ) e. ran log /\ ( log ` ( 1 - ( _i x. A ) ) ) e. RR ) -> ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) e. ran log ) |
133 |
117 131 132
|
syl2anc |
|- ( ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) /\ ( _i x. A ) <_ 0 ) -> ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) e. ran log ) |
134 |
86 92 114 133
|
lecasei |
|- ( ( A e. dom arctan /\ 0 = ( Re ` A ) ) -> ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) e. ran log ) |
135 |
85 134
|
jaodan |
|- ( ( A e. dom arctan /\ ( 0 < ( Re ` A ) \/ 0 = ( Re ` A ) ) ) -> ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) e. ran log ) |
136 |
7 135
|
syldan |
|- ( ( A e. dom arctan /\ 0 <_ ( Re ` A ) ) -> ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) e. ran log ) |