Step |
Hyp |
Ref |
Expression |
1 |
|
rpre |
|- ( A e. RR+ -> A e. RR ) |
2 |
|
atanrecl |
|- ( A e. RR -> ( arctan ` A ) e. RR ) |
3 |
1 2
|
syl |
|- ( A e. RR+ -> ( arctan ` A ) e. RR ) |
4 |
|
picn |
|- _pi e. CC |
5 |
|
2cn |
|- 2 e. CC |
6 |
|
2ne0 |
|- 2 =/= 0 |
7 |
|
divneg |
|- ( ( _pi e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> -u ( _pi / 2 ) = ( -u _pi / 2 ) ) |
8 |
4 5 6 7
|
mp3an |
|- -u ( _pi / 2 ) = ( -u _pi / 2 ) |
9 |
|
ax-1cn |
|- 1 e. CC |
10 |
|
ax-icn |
|- _i e. CC |
11 |
1
|
recnd |
|- ( A e. RR+ -> A e. CC ) |
12 |
|
mulcl |
|- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC ) |
13 |
10 11 12
|
sylancr |
|- ( A e. RR+ -> ( _i x. A ) e. CC ) |
14 |
|
addcl |
|- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 + ( _i x. A ) ) e. CC ) |
15 |
9 13 14
|
sylancr |
|- ( A e. RR+ -> ( 1 + ( _i x. A ) ) e. CC ) |
16 |
|
atanre |
|- ( A e. RR -> A e. dom arctan ) |
17 |
1 16
|
syl |
|- ( A e. RR+ -> A e. dom arctan ) |
18 |
|
atandm2 |
|- ( A e. dom arctan <-> ( A e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) ) |
19 |
17 18
|
sylib |
|- ( A e. RR+ -> ( A e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) ) |
20 |
19
|
simp3d |
|- ( A e. RR+ -> ( 1 + ( _i x. A ) ) =/= 0 ) |
21 |
15 20
|
logcld |
|- ( A e. RR+ -> ( log ` ( 1 + ( _i x. A ) ) ) e. CC ) |
22 |
|
subcl |
|- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 - ( _i x. A ) ) e. CC ) |
23 |
9 13 22
|
sylancr |
|- ( A e. RR+ -> ( 1 - ( _i x. A ) ) e. CC ) |
24 |
19
|
simp2d |
|- ( A e. RR+ -> ( 1 - ( _i x. A ) ) =/= 0 ) |
25 |
23 24
|
logcld |
|- ( A e. RR+ -> ( log ` ( 1 - ( _i x. A ) ) ) e. CC ) |
26 |
21 25
|
subcld |
|- ( A e. RR+ -> ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) e. CC ) |
27 |
|
imre |
|- ( ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) e. CC -> ( Im ` ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( Re ` ( -u _i x. ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) ) |
28 |
26 27
|
syl |
|- ( A e. RR+ -> ( Im ` ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( Re ` ( -u _i x. ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) ) |
29 |
|
atanval |
|- ( A e. dom arctan -> ( arctan ` A ) = ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) |
30 |
17 29
|
syl |
|- ( A e. RR+ -> ( arctan ` A ) = ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) |
31 |
30
|
oveq2d |
|- ( A e. RR+ -> ( 2 x. ( arctan ` A ) ) = ( 2 x. ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) ) |
32 |
10 5 6
|
divcan2i |
|- ( 2 x. ( _i / 2 ) ) = _i |
33 |
32
|
oveq1i |
|- ( ( 2 x. ( _i / 2 ) ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) = ( _i x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) |
34 |
|
2re |
|- 2 e. RR |
35 |
34
|
a1i |
|- ( A e. RR+ -> 2 e. RR ) |
36 |
35
|
recnd |
|- ( A e. RR+ -> 2 e. CC ) |
37 |
|
halfcl |
|- ( _i e. CC -> ( _i / 2 ) e. CC ) |
38 |
10 37
|
mp1i |
|- ( A e. RR+ -> ( _i / 2 ) e. CC ) |
39 |
25 21
|
subcld |
|- ( A e. RR+ -> ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) e. CC ) |
40 |
36 38 39
|
mulassd |
|- ( A e. RR+ -> ( ( 2 x. ( _i / 2 ) ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) = ( 2 x. ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) ) |
41 |
33 40
|
syl5eqr |
|- ( A e. RR+ -> ( _i x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) = ( 2 x. ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) ) |
42 |
31 41
|
eqtr4d |
|- ( A e. RR+ -> ( 2 x. ( arctan ` A ) ) = ( _i x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) |
43 |
21 25
|
negsubdi2d |
|- ( A e. RR+ -> -u ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) = ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) |
44 |
43
|
oveq2d |
|- ( A e. RR+ -> ( _i x. -u ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( _i x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) |
45 |
42 44
|
eqtr4d |
|- ( A e. RR+ -> ( 2 x. ( arctan ` A ) ) = ( _i x. -u ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) |
46 |
|
mulneg12 |
|- ( ( _i e. CC /\ ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) e. CC ) -> ( -u _i x. ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( _i x. -u ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) |
47 |
10 26 46
|
sylancr |
|- ( A e. RR+ -> ( -u _i x. ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( _i x. -u ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) |
48 |
45 47
|
eqtr4d |
|- ( A e. RR+ -> ( 2 x. ( arctan ` A ) ) = ( -u _i x. ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) |
49 |
48
|
fveq2d |
|- ( A e. RR+ -> ( Re ` ( 2 x. ( arctan ` A ) ) ) = ( Re ` ( -u _i x. ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) ) |
50 |
|
remulcl |
|- ( ( 2 e. RR /\ ( arctan ` A ) e. RR ) -> ( 2 x. ( arctan ` A ) ) e. RR ) |
51 |
34 3 50
|
sylancr |
|- ( A e. RR+ -> ( 2 x. ( arctan ` A ) ) e. RR ) |
52 |
51
|
rered |
|- ( A e. RR+ -> ( Re ` ( 2 x. ( arctan ` A ) ) ) = ( 2 x. ( arctan ` A ) ) ) |
53 |
28 49 52
|
3eqtr2rd |
|- ( A e. RR+ -> ( 2 x. ( arctan ` A ) ) = ( Im ` ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) |
54 |
|
rpgt0 |
|- ( A e. RR+ -> 0 < A ) |
55 |
1
|
rered |
|- ( A e. RR+ -> ( Re ` A ) = A ) |
56 |
54 55
|
breqtrrd |
|- ( A e. RR+ -> 0 < ( Re ` A ) ) |
57 |
|
atanlogsublem |
|- ( ( A e. dom arctan /\ 0 < ( Re ` A ) ) -> ( Im ` ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) e. ( -u _pi (,) _pi ) ) |
58 |
17 56 57
|
syl2anc |
|- ( A e. RR+ -> ( Im ` ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) e. ( -u _pi (,) _pi ) ) |
59 |
53 58
|
eqeltrd |
|- ( A e. RR+ -> ( 2 x. ( arctan ` A ) ) e. ( -u _pi (,) _pi ) ) |
60 |
|
eliooord |
|- ( ( 2 x. ( arctan ` A ) ) e. ( -u _pi (,) _pi ) -> ( -u _pi < ( 2 x. ( arctan ` A ) ) /\ ( 2 x. ( arctan ` A ) ) < _pi ) ) |
61 |
59 60
|
syl |
|- ( A e. RR+ -> ( -u _pi < ( 2 x. ( arctan ` A ) ) /\ ( 2 x. ( arctan ` A ) ) < _pi ) ) |
62 |
61
|
simpld |
|- ( A e. RR+ -> -u _pi < ( 2 x. ( arctan ` A ) ) ) |
63 |
|
pire |
|- _pi e. RR |
64 |
63
|
renegcli |
|- -u _pi e. RR |
65 |
64
|
a1i |
|- ( A e. RR+ -> -u _pi e. RR ) |
66 |
|
2pos |
|- 0 < 2 |
67 |
66
|
a1i |
|- ( A e. RR+ -> 0 < 2 ) |
68 |
|
ltdivmul |
|- ( ( -u _pi e. RR /\ ( arctan ` A ) e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( -u _pi / 2 ) < ( arctan ` A ) <-> -u _pi < ( 2 x. ( arctan ` A ) ) ) ) |
69 |
65 3 35 67 68
|
syl112anc |
|- ( A e. RR+ -> ( ( -u _pi / 2 ) < ( arctan ` A ) <-> -u _pi < ( 2 x. ( arctan ` A ) ) ) ) |
70 |
62 69
|
mpbird |
|- ( A e. RR+ -> ( -u _pi / 2 ) < ( arctan ` A ) ) |
71 |
8 70
|
eqbrtrid |
|- ( A e. RR+ -> -u ( _pi / 2 ) < ( arctan ` A ) ) |
72 |
61
|
simprd |
|- ( A e. RR+ -> ( 2 x. ( arctan ` A ) ) < _pi ) |
73 |
63
|
a1i |
|- ( A e. RR+ -> _pi e. RR ) |
74 |
|
ltmuldiv2 |
|- ( ( ( arctan ` A ) e. RR /\ _pi e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( 2 x. ( arctan ` A ) ) < _pi <-> ( arctan ` A ) < ( _pi / 2 ) ) ) |
75 |
3 73 35 67 74
|
syl112anc |
|- ( A e. RR+ -> ( ( 2 x. ( arctan ` A ) ) < _pi <-> ( arctan ` A ) < ( _pi / 2 ) ) ) |
76 |
72 75
|
mpbid |
|- ( A e. RR+ -> ( arctan ` A ) < ( _pi / 2 ) ) |
77 |
|
halfpire |
|- ( _pi / 2 ) e. RR |
78 |
77
|
renegcli |
|- -u ( _pi / 2 ) e. RR |
79 |
78
|
rexri |
|- -u ( _pi / 2 ) e. RR* |
80 |
77
|
rexri |
|- ( _pi / 2 ) e. RR* |
81 |
|
elioo2 |
|- ( ( -u ( _pi / 2 ) e. RR* /\ ( _pi / 2 ) e. RR* ) -> ( ( arctan ` A ) e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) <-> ( ( arctan ` A ) e. RR /\ -u ( _pi / 2 ) < ( arctan ` A ) /\ ( arctan ` A ) < ( _pi / 2 ) ) ) ) |
82 |
79 80 81
|
mp2an |
|- ( ( arctan ` A ) e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) <-> ( ( arctan ` A ) e. RR /\ -u ( _pi / 2 ) < ( arctan ` A ) /\ ( arctan ` A ) < ( _pi / 2 ) ) ) |
83 |
3 71 76 82
|
syl3anbrc |
|- ( A e. RR+ -> ( arctan ` A ) e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) ) |