Step |
Hyp |
Ref |
Expression |
1 |
|
atanval |
|- ( A e. dom arctan -> ( arctan ` A ) = ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) |
2 |
1
|
oveq2d |
|- ( A e. dom arctan -> ( ( 2 x. _i ) x. ( arctan ` A ) ) = ( ( 2 x. _i ) x. ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) ) |
3 |
|
2cn |
|- 2 e. CC |
4 |
3
|
a1i |
|- ( A e. dom arctan -> 2 e. CC ) |
5 |
|
ax-icn |
|- _i e. CC |
6 |
5
|
a1i |
|- ( A e. dom arctan -> _i e. CC ) |
7 |
|
atancl |
|- ( A e. dom arctan -> ( arctan ` A ) e. CC ) |
8 |
4 6 7
|
mulassd |
|- ( A e. dom arctan -> ( ( 2 x. _i ) x. ( arctan ` A ) ) = ( 2 x. ( _i x. ( arctan ` A ) ) ) ) |
9 |
|
halfcl |
|- ( _i e. CC -> ( _i / 2 ) e. CC ) |
10 |
5 9
|
ax-mp |
|- ( _i / 2 ) e. CC |
11 |
3 5 10
|
mulassi |
|- ( ( 2 x. _i ) x. ( _i / 2 ) ) = ( 2 x. ( _i x. ( _i / 2 ) ) ) |
12 |
3 5 10
|
mul12i |
|- ( 2 x. ( _i x. ( _i / 2 ) ) ) = ( _i x. ( 2 x. ( _i / 2 ) ) ) |
13 |
|
2ne0 |
|- 2 =/= 0 |
14 |
5 3 13
|
divcan2i |
|- ( 2 x. ( _i / 2 ) ) = _i |
15 |
14
|
oveq2i |
|- ( _i x. ( 2 x. ( _i / 2 ) ) ) = ( _i x. _i ) |
16 |
|
ixi |
|- ( _i x. _i ) = -u 1 |
17 |
15 16
|
eqtri |
|- ( _i x. ( 2 x. ( _i / 2 ) ) ) = -u 1 |
18 |
11 12 17
|
3eqtri |
|- ( ( 2 x. _i ) x. ( _i / 2 ) ) = -u 1 |
19 |
18
|
oveq1i |
|- ( ( ( 2 x. _i ) x. ( _i / 2 ) ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) = ( -u 1 x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) |
20 |
|
ax-1cn |
|- 1 e. CC |
21 |
|
atandm2 |
|- ( A e. dom arctan <-> ( A e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) ) |
22 |
21
|
simp1bi |
|- ( A e. dom arctan -> A e. CC ) |
23 |
|
mulcl |
|- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC ) |
24 |
5 22 23
|
sylancr |
|- ( A e. dom arctan -> ( _i x. A ) e. CC ) |
25 |
|
subcl |
|- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 - ( _i x. A ) ) e. CC ) |
26 |
20 24 25
|
sylancr |
|- ( A e. dom arctan -> ( 1 - ( _i x. A ) ) e. CC ) |
27 |
21
|
simp2bi |
|- ( A e. dom arctan -> ( 1 - ( _i x. A ) ) =/= 0 ) |
28 |
26 27
|
logcld |
|- ( A e. dom arctan -> ( log ` ( 1 - ( _i x. A ) ) ) e. CC ) |
29 |
|
addcl |
|- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 + ( _i x. A ) ) e. CC ) |
30 |
20 24 29
|
sylancr |
|- ( A e. dom arctan -> ( 1 + ( _i x. A ) ) e. CC ) |
31 |
21
|
simp3bi |
|- ( A e. dom arctan -> ( 1 + ( _i x. A ) ) =/= 0 ) |
32 |
30 31
|
logcld |
|- ( A e. dom arctan -> ( log ` ( 1 + ( _i x. A ) ) ) e. CC ) |
33 |
28 32
|
subcld |
|- ( A e. dom arctan -> ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) e. CC ) |
34 |
33
|
mulm1d |
|- ( A e. dom arctan -> ( -u 1 x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) = -u ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) |
35 |
19 34
|
syl5eq |
|- ( A e. dom arctan -> ( ( ( 2 x. _i ) x. ( _i / 2 ) ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) = -u ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) |
36 |
|
2mulicn |
|- ( 2 x. _i ) e. CC |
37 |
36
|
a1i |
|- ( A e. dom arctan -> ( 2 x. _i ) e. CC ) |
38 |
10
|
a1i |
|- ( A e. dom arctan -> ( _i / 2 ) e. CC ) |
39 |
37 38 33
|
mulassd |
|- ( A e. dom arctan -> ( ( ( 2 x. _i ) x. ( _i / 2 ) ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) = ( ( 2 x. _i ) x. ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) ) |
40 |
28 32
|
negsubdi2d |
|- ( A e. dom arctan -> -u ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) = ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) |
41 |
35 39 40
|
3eqtr3d |
|- ( A e. dom arctan -> ( ( 2 x. _i ) x. ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) = ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) |
42 |
2 8 41
|
3eqtr3d |
|- ( A e. dom arctan -> ( 2 x. ( _i x. ( arctan ` A ) ) ) = ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) |
43 |
42
|
fveq2d |
|- ( A e. dom arctan -> ( exp ` ( 2 x. ( _i x. ( arctan ` A ) ) ) ) = ( exp ` ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) |
44 |
|
efsub |
|- ( ( ( log ` ( 1 + ( _i x. A ) ) ) e. CC /\ ( log ` ( 1 - ( _i x. A ) ) ) e. CC ) -> ( exp ` ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( ( exp ` ( log ` ( 1 + ( _i x. A ) ) ) ) / ( exp ` ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) |
45 |
32 28 44
|
syl2anc |
|- ( A e. dom arctan -> ( exp ` ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( ( exp ` ( log ` ( 1 + ( _i x. A ) ) ) ) / ( exp ` ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) |
46 |
|
eflog |
|- ( ( ( 1 + ( _i x. A ) ) e. CC /\ ( 1 + ( _i x. A ) ) =/= 0 ) -> ( exp ` ( log ` ( 1 + ( _i x. A ) ) ) ) = ( 1 + ( _i x. A ) ) ) |
47 |
30 31 46
|
syl2anc |
|- ( A e. dom arctan -> ( exp ` ( log ` ( 1 + ( _i x. A ) ) ) ) = ( 1 + ( _i x. A ) ) ) |
48 |
|
eflog |
|- ( ( ( 1 - ( _i x. A ) ) e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 ) -> ( exp ` ( log ` ( 1 - ( _i x. A ) ) ) ) = ( 1 - ( _i x. A ) ) ) |
49 |
26 27 48
|
syl2anc |
|- ( A e. dom arctan -> ( exp ` ( log ` ( 1 - ( _i x. A ) ) ) ) = ( 1 - ( _i x. A ) ) ) |
50 |
47 49
|
oveq12d |
|- ( A e. dom arctan -> ( ( exp ` ( log ` ( 1 + ( _i x. A ) ) ) ) / ( exp ` ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( ( 1 + ( _i x. A ) ) / ( 1 - ( _i x. A ) ) ) ) |
51 |
|
negsub |
|- ( ( _i e. CC /\ A e. CC ) -> ( _i + -u A ) = ( _i - A ) ) |
52 |
5 22 51
|
sylancr |
|- ( A e. dom arctan -> ( _i + -u A ) = ( _i - A ) ) |
53 |
6
|
mulid1d |
|- ( A e. dom arctan -> ( _i x. 1 ) = _i ) |
54 |
16
|
oveq1i |
|- ( ( _i x. _i ) x. A ) = ( -u 1 x. A ) |
55 |
6 6 22
|
mulassd |
|- ( A e. dom arctan -> ( ( _i x. _i ) x. A ) = ( _i x. ( _i x. A ) ) ) |
56 |
22
|
mulm1d |
|- ( A e. dom arctan -> ( -u 1 x. A ) = -u A ) |
57 |
54 55 56
|
3eqtr3a |
|- ( A e. dom arctan -> ( _i x. ( _i x. A ) ) = -u A ) |
58 |
53 57
|
oveq12d |
|- ( A e. dom arctan -> ( ( _i x. 1 ) + ( _i x. ( _i x. A ) ) ) = ( _i + -u A ) ) |
59 |
6 22 6
|
pnpcan2d |
|- ( A e. dom arctan -> ( ( _i + _i ) - ( A + _i ) ) = ( _i - A ) ) |
60 |
52 58 59
|
3eqtr4d |
|- ( A e. dom arctan -> ( ( _i x. 1 ) + ( _i x. ( _i x. A ) ) ) = ( ( _i + _i ) - ( A + _i ) ) ) |
61 |
20
|
a1i |
|- ( A e. dom arctan -> 1 e. CC ) |
62 |
6 61 24
|
adddid |
|- ( A e. dom arctan -> ( _i x. ( 1 + ( _i x. A ) ) ) = ( ( _i x. 1 ) + ( _i x. ( _i x. A ) ) ) ) |
63 |
6
|
2timesd |
|- ( A e. dom arctan -> ( 2 x. _i ) = ( _i + _i ) ) |
64 |
63
|
oveq1d |
|- ( A e. dom arctan -> ( ( 2 x. _i ) - ( A + _i ) ) = ( ( _i + _i ) - ( A + _i ) ) ) |
65 |
60 62 64
|
3eqtr4d |
|- ( A e. dom arctan -> ( _i x. ( 1 + ( _i x. A ) ) ) = ( ( 2 x. _i ) - ( A + _i ) ) ) |
66 |
6 61 24
|
subdid |
|- ( A e. dom arctan -> ( _i x. ( 1 - ( _i x. A ) ) ) = ( ( _i x. 1 ) - ( _i x. ( _i x. A ) ) ) ) |
67 |
53 57
|
oveq12d |
|- ( A e. dom arctan -> ( ( _i x. 1 ) - ( _i x. ( _i x. A ) ) ) = ( _i - -u A ) ) |
68 |
|
subneg |
|- ( ( _i e. CC /\ A e. CC ) -> ( _i - -u A ) = ( _i + A ) ) |
69 |
5 22 68
|
sylancr |
|- ( A e. dom arctan -> ( _i - -u A ) = ( _i + A ) ) |
70 |
67 69
|
eqtrd |
|- ( A e. dom arctan -> ( ( _i x. 1 ) - ( _i x. ( _i x. A ) ) ) = ( _i + A ) ) |
71 |
|
addcom |
|- ( ( _i e. CC /\ A e. CC ) -> ( _i + A ) = ( A + _i ) ) |
72 |
5 22 71
|
sylancr |
|- ( A e. dom arctan -> ( _i + A ) = ( A + _i ) ) |
73 |
66 70 72
|
3eqtrd |
|- ( A e. dom arctan -> ( _i x. ( 1 - ( _i x. A ) ) ) = ( A + _i ) ) |
74 |
65 73
|
oveq12d |
|- ( A e. dom arctan -> ( ( _i x. ( 1 + ( _i x. A ) ) ) / ( _i x. ( 1 - ( _i x. A ) ) ) ) = ( ( ( 2 x. _i ) - ( A + _i ) ) / ( A + _i ) ) ) |
75 |
|
ine0 |
|- _i =/= 0 |
76 |
75
|
a1i |
|- ( A e. dom arctan -> _i =/= 0 ) |
77 |
30 26 6 27 76
|
divcan5d |
|- ( A e. dom arctan -> ( ( _i x. ( 1 + ( _i x. A ) ) ) / ( _i x. ( 1 - ( _i x. A ) ) ) ) = ( ( 1 + ( _i x. A ) ) / ( 1 - ( _i x. A ) ) ) ) |
78 |
|
addcl |
|- ( ( A e. CC /\ _i e. CC ) -> ( A + _i ) e. CC ) |
79 |
22 5 78
|
sylancl |
|- ( A e. dom arctan -> ( A + _i ) e. CC ) |
80 |
|
subneg |
|- ( ( A e. CC /\ _i e. CC ) -> ( A - -u _i ) = ( A + _i ) ) |
81 |
22 5 80
|
sylancl |
|- ( A e. dom arctan -> ( A - -u _i ) = ( A + _i ) ) |
82 |
|
atandm |
|- ( A e. dom arctan <-> ( A e. CC /\ A =/= -u _i /\ A =/= _i ) ) |
83 |
82
|
simp2bi |
|- ( A e. dom arctan -> A =/= -u _i ) |
84 |
|
negicn |
|- -u _i e. CC |
85 |
|
subeq0 |
|- ( ( A e. CC /\ -u _i e. CC ) -> ( ( A - -u _i ) = 0 <-> A = -u _i ) ) |
86 |
85
|
necon3bid |
|- ( ( A e. CC /\ -u _i e. CC ) -> ( ( A - -u _i ) =/= 0 <-> A =/= -u _i ) ) |
87 |
22 84 86
|
sylancl |
|- ( A e. dom arctan -> ( ( A - -u _i ) =/= 0 <-> A =/= -u _i ) ) |
88 |
83 87
|
mpbird |
|- ( A e. dom arctan -> ( A - -u _i ) =/= 0 ) |
89 |
81 88
|
eqnetrrd |
|- ( A e. dom arctan -> ( A + _i ) =/= 0 ) |
90 |
37 79 79 89
|
divsubdird |
|- ( A e. dom arctan -> ( ( ( 2 x. _i ) - ( A + _i ) ) / ( A + _i ) ) = ( ( ( 2 x. _i ) / ( A + _i ) ) - ( ( A + _i ) / ( A + _i ) ) ) ) |
91 |
74 77 90
|
3eqtr3d |
|- ( A e. dom arctan -> ( ( 1 + ( _i x. A ) ) / ( 1 - ( _i x. A ) ) ) = ( ( ( 2 x. _i ) / ( A + _i ) ) - ( ( A + _i ) / ( A + _i ) ) ) ) |
92 |
79 89
|
dividd |
|- ( A e. dom arctan -> ( ( A + _i ) / ( A + _i ) ) = 1 ) |
93 |
92
|
oveq2d |
|- ( A e. dom arctan -> ( ( ( 2 x. _i ) / ( A + _i ) ) - ( ( A + _i ) / ( A + _i ) ) ) = ( ( ( 2 x. _i ) / ( A + _i ) ) - 1 ) ) |
94 |
50 91 93
|
3eqtrd |
|- ( A e. dom arctan -> ( ( exp ` ( log ` ( 1 + ( _i x. A ) ) ) ) / ( exp ` ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( ( ( 2 x. _i ) / ( A + _i ) ) - 1 ) ) |
95 |
43 45 94
|
3eqtrd |
|- ( A e. dom arctan -> ( exp ` ( 2 x. ( _i x. ( arctan ` A ) ) ) ) = ( ( ( 2 x. _i ) / ( A + _i ) ) - 1 ) ) |