Step |
Hyp |
Ref |
Expression |
1 |
|
atansopn.d |
|- D = ( CC \ ( -oo (,] 0 ) ) |
2 |
|
atansopn.s |
|- S = { y e. CC | ( 1 + ( y ^ 2 ) ) e. D } |
3 |
|
cnelprrecn |
|- CC e. { RR , CC } |
4 |
3
|
a1i |
|- ( T. -> CC e. { RR , CC } ) |
5 |
|
ax-1cn |
|- 1 e. CC |
6 |
|
ax-icn |
|- _i e. CC |
7 |
1 2
|
atansssdm |
|- S C_ dom arctan |
8 |
|
simpr |
|- ( ( T. /\ x e. S ) -> x e. S ) |
9 |
7 8
|
sselid |
|- ( ( T. /\ x e. S ) -> x e. dom arctan ) |
10 |
|
atandm2 |
|- ( x e. dom arctan <-> ( x e. CC /\ ( 1 - ( _i x. x ) ) =/= 0 /\ ( 1 + ( _i x. x ) ) =/= 0 ) ) |
11 |
9 10
|
sylib |
|- ( ( T. /\ x e. S ) -> ( x e. CC /\ ( 1 - ( _i x. x ) ) =/= 0 /\ ( 1 + ( _i x. x ) ) =/= 0 ) ) |
12 |
11
|
simp1d |
|- ( ( T. /\ x e. S ) -> x e. CC ) |
13 |
|
mulcl |
|- ( ( _i e. CC /\ x e. CC ) -> ( _i x. x ) e. CC ) |
14 |
6 12 13
|
sylancr |
|- ( ( T. /\ x e. S ) -> ( _i x. x ) e. CC ) |
15 |
|
subcl |
|- ( ( 1 e. CC /\ ( _i x. x ) e. CC ) -> ( 1 - ( _i x. x ) ) e. CC ) |
16 |
5 14 15
|
sylancr |
|- ( ( T. /\ x e. S ) -> ( 1 - ( _i x. x ) ) e. CC ) |
17 |
11
|
simp2d |
|- ( ( T. /\ x e. S ) -> ( 1 - ( _i x. x ) ) =/= 0 ) |
18 |
16 17
|
logcld |
|- ( ( T. /\ x e. S ) -> ( log ` ( 1 - ( _i x. x ) ) ) e. CC ) |
19 |
|
addcl |
|- ( ( 1 e. CC /\ ( _i x. x ) e. CC ) -> ( 1 + ( _i x. x ) ) e. CC ) |
20 |
5 14 19
|
sylancr |
|- ( ( T. /\ x e. S ) -> ( 1 + ( _i x. x ) ) e. CC ) |
21 |
11
|
simp3d |
|- ( ( T. /\ x e. S ) -> ( 1 + ( _i x. x ) ) =/= 0 ) |
22 |
20 21
|
logcld |
|- ( ( T. /\ x e. S ) -> ( log ` ( 1 + ( _i x. x ) ) ) e. CC ) |
23 |
18 22
|
subcld |
|- ( ( T. /\ x e. S ) -> ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) e. CC ) |
24 |
|
ovexd |
|- ( ( T. /\ x e. S ) -> ( ( 2 / _i ) / ( 1 + ( x ^ 2 ) ) ) e. _V ) |
25 |
|
ovexd |
|- ( ( T. /\ x e. S ) -> ( 1 / ( x + _i ) ) e. _V ) |
26 |
1 2
|
atans2 |
|- ( x e. S <-> ( x e. CC /\ ( 1 - ( _i x. x ) ) e. D /\ ( 1 + ( _i x. x ) ) e. D ) ) |
27 |
26
|
simp2bi |
|- ( x e. S -> ( 1 - ( _i x. x ) ) e. D ) |
28 |
27
|
adantl |
|- ( ( T. /\ x e. S ) -> ( 1 - ( _i x. x ) ) e. D ) |
29 |
|
negex |
|- -u _i e. _V |
30 |
29
|
a1i |
|- ( ( T. /\ x e. S ) -> -u _i e. _V ) |
31 |
1
|
logdmss |
|- D C_ ( CC \ { 0 } ) |
32 |
|
simpr |
|- ( ( T. /\ y e. D ) -> y e. D ) |
33 |
31 32
|
sselid |
|- ( ( T. /\ y e. D ) -> y e. ( CC \ { 0 } ) ) |
34 |
|
logf1o |
|- log : ( CC \ { 0 } ) -1-1-onto-> ran log |
35 |
|
f1of |
|- ( log : ( CC \ { 0 } ) -1-1-onto-> ran log -> log : ( CC \ { 0 } ) --> ran log ) |
36 |
34 35
|
ax-mp |
|- log : ( CC \ { 0 } ) --> ran log |
37 |
36
|
ffvelrni |
|- ( y e. ( CC \ { 0 } ) -> ( log ` y ) e. ran log ) |
38 |
|
logrncn |
|- ( ( log ` y ) e. ran log -> ( log ` y ) e. CC ) |
39 |
33 37 38
|
3syl |
|- ( ( T. /\ y e. D ) -> ( log ` y ) e. CC ) |
40 |
|
ovexd |
|- ( ( T. /\ y e. D ) -> ( 1 / y ) e. _V ) |
41 |
6
|
a1i |
|- ( T. -> _i e. CC ) |
42 |
41 13
|
sylan |
|- ( ( T. /\ x e. CC ) -> ( _i x. x ) e. CC ) |
43 |
5 42 15
|
sylancr |
|- ( ( T. /\ x e. CC ) -> ( 1 - ( _i x. x ) ) e. CC ) |
44 |
29
|
a1i |
|- ( ( T. /\ x e. CC ) -> -u _i e. _V ) |
45 |
|
1cnd |
|- ( ( T. /\ x e. CC ) -> 1 e. CC ) |
46 |
|
0cnd |
|- ( ( T. /\ x e. CC ) -> 0 e. CC ) |
47 |
|
1cnd |
|- ( T. -> 1 e. CC ) |
48 |
4 47
|
dvmptc |
|- ( T. -> ( CC _D ( x e. CC |-> 1 ) ) = ( x e. CC |-> 0 ) ) |
49 |
6
|
a1i |
|- ( ( T. /\ x e. CC ) -> _i e. CC ) |
50 |
|
simpr |
|- ( ( T. /\ x e. CC ) -> x e. CC ) |
51 |
4
|
dvmptid |
|- ( T. -> ( CC _D ( x e. CC |-> x ) ) = ( x e. CC |-> 1 ) ) |
52 |
4 50 45 51 41
|
dvmptcmul |
|- ( T. -> ( CC _D ( x e. CC |-> ( _i x. x ) ) ) = ( x e. CC |-> ( _i x. 1 ) ) ) |
53 |
6
|
mulid1i |
|- ( _i x. 1 ) = _i |
54 |
53
|
mpteq2i |
|- ( x e. CC |-> ( _i x. 1 ) ) = ( x e. CC |-> _i ) |
55 |
52 54
|
eqtrdi |
|- ( T. -> ( CC _D ( x e. CC |-> ( _i x. x ) ) ) = ( x e. CC |-> _i ) ) |
56 |
4 45 46 48 42 49 55
|
dvmptsub |
|- ( T. -> ( CC _D ( x e. CC |-> ( 1 - ( _i x. x ) ) ) ) = ( x e. CC |-> ( 0 - _i ) ) ) |
57 |
|
df-neg |
|- -u _i = ( 0 - _i ) |
58 |
57
|
mpteq2i |
|- ( x e. CC |-> -u _i ) = ( x e. CC |-> ( 0 - _i ) ) |
59 |
56 58
|
eqtr4di |
|- ( T. -> ( CC _D ( x e. CC |-> ( 1 - ( _i x. x ) ) ) ) = ( x e. CC |-> -u _i ) ) |
60 |
2
|
ssrab3 |
|- S C_ CC |
61 |
60
|
a1i |
|- ( T. -> S C_ CC ) |
62 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
63 |
62
|
cnfldtopon |
|- ( TopOpen ` CCfld ) e. ( TopOn ` CC ) |
64 |
63
|
toponrestid |
|- ( TopOpen ` CCfld ) = ( ( TopOpen ` CCfld ) |`t CC ) |
65 |
1 2
|
atansopn |
|- S e. ( TopOpen ` CCfld ) |
66 |
65
|
a1i |
|- ( T. -> S e. ( TopOpen ` CCfld ) ) |
67 |
4 43 44 59 61 64 62 66
|
dvmptres |
|- ( T. -> ( CC _D ( x e. S |-> ( 1 - ( _i x. x ) ) ) ) = ( x e. S |-> -u _i ) ) |
68 |
|
fssres |
|- ( ( log : ( CC \ { 0 } ) --> ran log /\ D C_ ( CC \ { 0 } ) ) -> ( log |` D ) : D --> ran log ) |
69 |
36 31 68
|
mp2an |
|- ( log |` D ) : D --> ran log |
70 |
69
|
a1i |
|- ( T. -> ( log |` D ) : D --> ran log ) |
71 |
70
|
feqmptd |
|- ( T. -> ( log |` D ) = ( y e. D |-> ( ( log |` D ) ` y ) ) ) |
72 |
|
fvres |
|- ( y e. D -> ( ( log |` D ) ` y ) = ( log ` y ) ) |
73 |
72
|
mpteq2ia |
|- ( y e. D |-> ( ( log |` D ) ` y ) ) = ( y e. D |-> ( log ` y ) ) |
74 |
71 73
|
eqtr2di |
|- ( T. -> ( y e. D |-> ( log ` y ) ) = ( log |` D ) ) |
75 |
74
|
oveq2d |
|- ( T. -> ( CC _D ( y e. D |-> ( log ` y ) ) ) = ( CC _D ( log |` D ) ) ) |
76 |
1
|
dvlog |
|- ( CC _D ( log |` D ) ) = ( y e. D |-> ( 1 / y ) ) |
77 |
75 76
|
eqtrdi |
|- ( T. -> ( CC _D ( y e. D |-> ( log ` y ) ) ) = ( y e. D |-> ( 1 / y ) ) ) |
78 |
|
fveq2 |
|- ( y = ( 1 - ( _i x. x ) ) -> ( log ` y ) = ( log ` ( 1 - ( _i x. x ) ) ) ) |
79 |
|
oveq2 |
|- ( y = ( 1 - ( _i x. x ) ) -> ( 1 / y ) = ( 1 / ( 1 - ( _i x. x ) ) ) ) |
80 |
4 4 28 30 39 40 67 77 78 79
|
dvmptco |
|- ( T. -> ( CC _D ( x e. S |-> ( log ` ( 1 - ( _i x. x ) ) ) ) ) = ( x e. S |-> ( ( 1 / ( 1 - ( _i x. x ) ) ) x. -u _i ) ) ) |
81 |
|
irec |
|- ( 1 / _i ) = -u _i |
82 |
81
|
oveq2i |
|- ( ( 1 / ( 1 - ( _i x. x ) ) ) x. ( 1 / _i ) ) = ( ( 1 / ( 1 - ( _i x. x ) ) ) x. -u _i ) |
83 |
6
|
a1i |
|- ( ( T. /\ x e. S ) -> _i e. CC ) |
84 |
|
ine0 |
|- _i =/= 0 |
85 |
84
|
a1i |
|- ( ( T. /\ x e. S ) -> _i =/= 0 ) |
86 |
16 83 17 85
|
recdiv2d |
|- ( ( T. /\ x e. S ) -> ( ( 1 / ( 1 - ( _i x. x ) ) ) / _i ) = ( 1 / ( ( 1 - ( _i x. x ) ) x. _i ) ) ) |
87 |
16 17
|
reccld |
|- ( ( T. /\ x e. S ) -> ( 1 / ( 1 - ( _i x. x ) ) ) e. CC ) |
88 |
87 83 85
|
divrecd |
|- ( ( T. /\ x e. S ) -> ( ( 1 / ( 1 - ( _i x. x ) ) ) / _i ) = ( ( 1 / ( 1 - ( _i x. x ) ) ) x. ( 1 / _i ) ) ) |
89 |
|
1cnd |
|- ( ( T. /\ x e. S ) -> 1 e. CC ) |
90 |
89 14 83
|
subdird |
|- ( ( T. /\ x e. S ) -> ( ( 1 - ( _i x. x ) ) x. _i ) = ( ( 1 x. _i ) - ( ( _i x. x ) x. _i ) ) ) |
91 |
6
|
mulid2i |
|- ( 1 x. _i ) = _i |
92 |
91
|
a1i |
|- ( ( T. /\ x e. S ) -> ( 1 x. _i ) = _i ) |
93 |
83 12 83
|
mul32d |
|- ( ( T. /\ x e. S ) -> ( ( _i x. x ) x. _i ) = ( ( _i x. _i ) x. x ) ) |
94 |
|
ixi |
|- ( _i x. _i ) = -u 1 |
95 |
94
|
oveq1i |
|- ( ( _i x. _i ) x. x ) = ( -u 1 x. x ) |
96 |
12
|
mulm1d |
|- ( ( T. /\ x e. S ) -> ( -u 1 x. x ) = -u x ) |
97 |
95 96
|
syl5eq |
|- ( ( T. /\ x e. S ) -> ( ( _i x. _i ) x. x ) = -u x ) |
98 |
93 97
|
eqtrd |
|- ( ( T. /\ x e. S ) -> ( ( _i x. x ) x. _i ) = -u x ) |
99 |
92 98
|
oveq12d |
|- ( ( T. /\ x e. S ) -> ( ( 1 x. _i ) - ( ( _i x. x ) x. _i ) ) = ( _i - -u x ) ) |
100 |
|
subneg |
|- ( ( _i e. CC /\ x e. CC ) -> ( _i - -u x ) = ( _i + x ) ) |
101 |
6 12 100
|
sylancr |
|- ( ( T. /\ x e. S ) -> ( _i - -u x ) = ( _i + x ) ) |
102 |
90 99 101
|
3eqtrd |
|- ( ( T. /\ x e. S ) -> ( ( 1 - ( _i x. x ) ) x. _i ) = ( _i + x ) ) |
103 |
83 12 102
|
comraddd |
|- ( ( T. /\ x e. S ) -> ( ( 1 - ( _i x. x ) ) x. _i ) = ( x + _i ) ) |
104 |
103
|
oveq2d |
|- ( ( T. /\ x e. S ) -> ( 1 / ( ( 1 - ( _i x. x ) ) x. _i ) ) = ( 1 / ( x + _i ) ) ) |
105 |
86 88 104
|
3eqtr3d |
|- ( ( T. /\ x e. S ) -> ( ( 1 / ( 1 - ( _i x. x ) ) ) x. ( 1 / _i ) ) = ( 1 / ( x + _i ) ) ) |
106 |
82 105
|
eqtr3id |
|- ( ( T. /\ x e. S ) -> ( ( 1 / ( 1 - ( _i x. x ) ) ) x. -u _i ) = ( 1 / ( x + _i ) ) ) |
107 |
106
|
mpteq2dva |
|- ( T. -> ( x e. S |-> ( ( 1 / ( 1 - ( _i x. x ) ) ) x. -u _i ) ) = ( x e. S |-> ( 1 / ( x + _i ) ) ) ) |
108 |
80 107
|
eqtrd |
|- ( T. -> ( CC _D ( x e. S |-> ( log ` ( 1 - ( _i x. x ) ) ) ) ) = ( x e. S |-> ( 1 / ( x + _i ) ) ) ) |
109 |
|
ovexd |
|- ( ( T. /\ x e. S ) -> ( 1 / ( x - _i ) ) e. _V ) |
110 |
26
|
simp3bi |
|- ( x e. S -> ( 1 + ( _i x. x ) ) e. D ) |
111 |
110
|
adantl |
|- ( ( T. /\ x e. S ) -> ( 1 + ( _i x. x ) ) e. D ) |
112 |
5 42 19
|
sylancr |
|- ( ( T. /\ x e. CC ) -> ( 1 + ( _i x. x ) ) e. CC ) |
113 |
4 45 46 48 42 49 55
|
dvmptadd |
|- ( T. -> ( CC _D ( x e. CC |-> ( 1 + ( _i x. x ) ) ) ) = ( x e. CC |-> ( 0 + _i ) ) ) |
114 |
6
|
addid2i |
|- ( 0 + _i ) = _i |
115 |
114
|
mpteq2i |
|- ( x e. CC |-> ( 0 + _i ) ) = ( x e. CC |-> _i ) |
116 |
113 115
|
eqtrdi |
|- ( T. -> ( CC _D ( x e. CC |-> ( 1 + ( _i x. x ) ) ) ) = ( x e. CC |-> _i ) ) |
117 |
4 112 49 116 61 64 62 66
|
dvmptres |
|- ( T. -> ( CC _D ( x e. S |-> ( 1 + ( _i x. x ) ) ) ) = ( x e. S |-> _i ) ) |
118 |
|
fveq2 |
|- ( y = ( 1 + ( _i x. x ) ) -> ( log ` y ) = ( log ` ( 1 + ( _i x. x ) ) ) ) |
119 |
|
oveq2 |
|- ( y = ( 1 + ( _i x. x ) ) -> ( 1 / y ) = ( 1 / ( 1 + ( _i x. x ) ) ) ) |
120 |
4 4 111 83 39 40 117 77 118 119
|
dvmptco |
|- ( T. -> ( CC _D ( x e. S |-> ( log ` ( 1 + ( _i x. x ) ) ) ) ) = ( x e. S |-> ( ( 1 / ( 1 + ( _i x. x ) ) ) x. _i ) ) ) |
121 |
89 20 83 21 85
|
divdiv2d |
|- ( ( T. /\ x e. S ) -> ( 1 / ( ( 1 + ( _i x. x ) ) / _i ) ) = ( ( 1 x. _i ) / ( 1 + ( _i x. x ) ) ) ) |
122 |
89 14 83 85
|
divdird |
|- ( ( T. /\ x e. S ) -> ( ( 1 + ( _i x. x ) ) / _i ) = ( ( 1 / _i ) + ( ( _i x. x ) / _i ) ) ) |
123 |
81
|
a1i |
|- ( ( T. /\ x e. S ) -> ( 1 / _i ) = -u _i ) |
124 |
12 83 85
|
divcan3d |
|- ( ( T. /\ x e. S ) -> ( ( _i x. x ) / _i ) = x ) |
125 |
123 124
|
oveq12d |
|- ( ( T. /\ x e. S ) -> ( ( 1 / _i ) + ( ( _i x. x ) / _i ) ) = ( -u _i + x ) ) |
126 |
|
negicn |
|- -u _i e. CC |
127 |
|
addcom |
|- ( ( -u _i e. CC /\ x e. CC ) -> ( -u _i + x ) = ( x + -u _i ) ) |
128 |
126 12 127
|
sylancr |
|- ( ( T. /\ x e. S ) -> ( -u _i + x ) = ( x + -u _i ) ) |
129 |
|
negsub |
|- ( ( x e. CC /\ _i e. CC ) -> ( x + -u _i ) = ( x - _i ) ) |
130 |
12 6 129
|
sylancl |
|- ( ( T. /\ x e. S ) -> ( x + -u _i ) = ( x - _i ) ) |
131 |
128 130
|
eqtrd |
|- ( ( T. /\ x e. S ) -> ( -u _i + x ) = ( x - _i ) ) |
132 |
122 125 131
|
3eqtrd |
|- ( ( T. /\ x e. S ) -> ( ( 1 + ( _i x. x ) ) / _i ) = ( x - _i ) ) |
133 |
132
|
oveq2d |
|- ( ( T. /\ x e. S ) -> ( 1 / ( ( 1 + ( _i x. x ) ) / _i ) ) = ( 1 / ( x - _i ) ) ) |
134 |
89 83 20 21
|
div23d |
|- ( ( T. /\ x e. S ) -> ( ( 1 x. _i ) / ( 1 + ( _i x. x ) ) ) = ( ( 1 / ( 1 + ( _i x. x ) ) ) x. _i ) ) |
135 |
121 133 134
|
3eqtr3rd |
|- ( ( T. /\ x e. S ) -> ( ( 1 / ( 1 + ( _i x. x ) ) ) x. _i ) = ( 1 / ( x - _i ) ) ) |
136 |
135
|
mpteq2dva |
|- ( T. -> ( x e. S |-> ( ( 1 / ( 1 + ( _i x. x ) ) ) x. _i ) ) = ( x e. S |-> ( 1 / ( x - _i ) ) ) ) |
137 |
120 136
|
eqtrd |
|- ( T. -> ( CC _D ( x e. S |-> ( log ` ( 1 + ( _i x. x ) ) ) ) ) = ( x e. S |-> ( 1 / ( x - _i ) ) ) ) |
138 |
4 18 25 108 22 109 137
|
dvmptsub |
|- ( T. -> ( CC _D ( x e. S |-> ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) ) = ( x e. S |-> ( ( 1 / ( x + _i ) ) - ( 1 / ( x - _i ) ) ) ) ) |
139 |
|
subcl |
|- ( ( x e. CC /\ _i e. CC ) -> ( x - _i ) e. CC ) |
140 |
12 6 139
|
sylancl |
|- ( ( T. /\ x e. S ) -> ( x - _i ) e. CC ) |
141 |
|
addcl |
|- ( ( x e. CC /\ _i e. CC ) -> ( x + _i ) e. CC ) |
142 |
12 6 141
|
sylancl |
|- ( ( T. /\ x e. S ) -> ( x + _i ) e. CC ) |
143 |
12
|
sqcld |
|- ( ( T. /\ x e. S ) -> ( x ^ 2 ) e. CC ) |
144 |
|
addcl |
|- ( ( 1 e. CC /\ ( x ^ 2 ) e. CC ) -> ( 1 + ( x ^ 2 ) ) e. CC ) |
145 |
5 143 144
|
sylancr |
|- ( ( T. /\ x e. S ) -> ( 1 + ( x ^ 2 ) ) e. CC ) |
146 |
|
atandm4 |
|- ( x e. dom arctan <-> ( x e. CC /\ ( 1 + ( x ^ 2 ) ) =/= 0 ) ) |
147 |
146
|
simprbi |
|- ( x e. dom arctan -> ( 1 + ( x ^ 2 ) ) =/= 0 ) |
148 |
9 147
|
syl |
|- ( ( T. /\ x e. S ) -> ( 1 + ( x ^ 2 ) ) =/= 0 ) |
149 |
140 142 145 148
|
divsubdird |
|- ( ( T. /\ x e. S ) -> ( ( ( x - _i ) - ( x + _i ) ) / ( 1 + ( x ^ 2 ) ) ) = ( ( ( x - _i ) / ( 1 + ( x ^ 2 ) ) ) - ( ( x + _i ) / ( 1 + ( x ^ 2 ) ) ) ) ) |
150 |
130
|
oveq1d |
|- ( ( T. /\ x e. S ) -> ( ( x + -u _i ) - ( x + _i ) ) = ( ( x - _i ) - ( x + _i ) ) ) |
151 |
126
|
a1i |
|- ( ( T. /\ x e. S ) -> -u _i e. CC ) |
152 |
12 151 83
|
pnpcand |
|- ( ( T. /\ x e. S ) -> ( ( x + -u _i ) - ( x + _i ) ) = ( -u _i - _i ) ) |
153 |
150 152
|
eqtr3d |
|- ( ( T. /\ x e. S ) -> ( ( x - _i ) - ( x + _i ) ) = ( -u _i - _i ) ) |
154 |
|
2cn |
|- 2 e. CC |
155 |
154 6 84
|
divreci |
|- ( 2 / _i ) = ( 2 x. ( 1 / _i ) ) |
156 |
81
|
oveq2i |
|- ( 2 x. ( 1 / _i ) ) = ( 2 x. -u _i ) |
157 |
155 156
|
eqtri |
|- ( 2 / _i ) = ( 2 x. -u _i ) |
158 |
126
|
2timesi |
|- ( 2 x. -u _i ) = ( -u _i + -u _i ) |
159 |
126 6
|
negsubi |
|- ( -u _i + -u _i ) = ( -u _i - _i ) |
160 |
157 158 159
|
3eqtri |
|- ( 2 / _i ) = ( -u _i - _i ) |
161 |
153 160
|
eqtr4di |
|- ( ( T. /\ x e. S ) -> ( ( x - _i ) - ( x + _i ) ) = ( 2 / _i ) ) |
162 |
161
|
oveq1d |
|- ( ( T. /\ x e. S ) -> ( ( ( x - _i ) - ( x + _i ) ) / ( 1 + ( x ^ 2 ) ) ) = ( ( 2 / _i ) / ( 1 + ( x ^ 2 ) ) ) ) |
163 |
140
|
mulid1d |
|- ( ( T. /\ x e. S ) -> ( ( x - _i ) x. 1 ) = ( x - _i ) ) |
164 |
140 142
|
mulcomd |
|- ( ( T. /\ x e. S ) -> ( ( x - _i ) x. ( x + _i ) ) = ( ( x + _i ) x. ( x - _i ) ) ) |
165 |
|
i2 |
|- ( _i ^ 2 ) = -u 1 |
166 |
165
|
oveq2i |
|- ( ( x ^ 2 ) - ( _i ^ 2 ) ) = ( ( x ^ 2 ) - -u 1 ) |
167 |
|
subneg |
|- ( ( ( x ^ 2 ) e. CC /\ 1 e. CC ) -> ( ( x ^ 2 ) - -u 1 ) = ( ( x ^ 2 ) + 1 ) ) |
168 |
143 5 167
|
sylancl |
|- ( ( T. /\ x e. S ) -> ( ( x ^ 2 ) - -u 1 ) = ( ( x ^ 2 ) + 1 ) ) |
169 |
166 168
|
syl5eq |
|- ( ( T. /\ x e. S ) -> ( ( x ^ 2 ) - ( _i ^ 2 ) ) = ( ( x ^ 2 ) + 1 ) ) |
170 |
|
subsq |
|- ( ( x e. CC /\ _i e. CC ) -> ( ( x ^ 2 ) - ( _i ^ 2 ) ) = ( ( x + _i ) x. ( x - _i ) ) ) |
171 |
12 6 170
|
sylancl |
|- ( ( T. /\ x e. S ) -> ( ( x ^ 2 ) - ( _i ^ 2 ) ) = ( ( x + _i ) x. ( x - _i ) ) ) |
172 |
|
addcom |
|- ( ( ( x ^ 2 ) e. CC /\ 1 e. CC ) -> ( ( x ^ 2 ) + 1 ) = ( 1 + ( x ^ 2 ) ) ) |
173 |
143 5 172
|
sylancl |
|- ( ( T. /\ x e. S ) -> ( ( x ^ 2 ) + 1 ) = ( 1 + ( x ^ 2 ) ) ) |
174 |
169 171 173
|
3eqtr3d |
|- ( ( T. /\ x e. S ) -> ( ( x + _i ) x. ( x - _i ) ) = ( 1 + ( x ^ 2 ) ) ) |
175 |
164 174
|
eqtrd |
|- ( ( T. /\ x e. S ) -> ( ( x - _i ) x. ( x + _i ) ) = ( 1 + ( x ^ 2 ) ) ) |
176 |
163 175
|
oveq12d |
|- ( ( T. /\ x e. S ) -> ( ( ( x - _i ) x. 1 ) / ( ( x - _i ) x. ( x + _i ) ) ) = ( ( x - _i ) / ( 1 + ( x ^ 2 ) ) ) ) |
177 |
|
subneg |
|- ( ( x e. CC /\ _i e. CC ) -> ( x - -u _i ) = ( x + _i ) ) |
178 |
12 6 177
|
sylancl |
|- ( ( T. /\ x e. S ) -> ( x - -u _i ) = ( x + _i ) ) |
179 |
|
atandm |
|- ( x e. dom arctan <-> ( x e. CC /\ x =/= -u _i /\ x =/= _i ) ) |
180 |
9 179
|
sylib |
|- ( ( T. /\ x e. S ) -> ( x e. CC /\ x =/= -u _i /\ x =/= _i ) ) |
181 |
180
|
simp2d |
|- ( ( T. /\ x e. S ) -> x =/= -u _i ) |
182 |
|
subeq0 |
|- ( ( x e. CC /\ -u _i e. CC ) -> ( ( x - -u _i ) = 0 <-> x = -u _i ) ) |
183 |
182
|
necon3bid |
|- ( ( x e. CC /\ -u _i e. CC ) -> ( ( x - -u _i ) =/= 0 <-> x =/= -u _i ) ) |
184 |
12 126 183
|
sylancl |
|- ( ( T. /\ x e. S ) -> ( ( x - -u _i ) =/= 0 <-> x =/= -u _i ) ) |
185 |
181 184
|
mpbird |
|- ( ( T. /\ x e. S ) -> ( x - -u _i ) =/= 0 ) |
186 |
178 185
|
eqnetrrd |
|- ( ( T. /\ x e. S ) -> ( x + _i ) =/= 0 ) |
187 |
180
|
simp3d |
|- ( ( T. /\ x e. S ) -> x =/= _i ) |
188 |
|
subeq0 |
|- ( ( x e. CC /\ _i e. CC ) -> ( ( x - _i ) = 0 <-> x = _i ) ) |
189 |
188
|
necon3bid |
|- ( ( x e. CC /\ _i e. CC ) -> ( ( x - _i ) =/= 0 <-> x =/= _i ) ) |
190 |
12 6 189
|
sylancl |
|- ( ( T. /\ x e. S ) -> ( ( x - _i ) =/= 0 <-> x =/= _i ) ) |
191 |
187 190
|
mpbird |
|- ( ( T. /\ x e. S ) -> ( x - _i ) =/= 0 ) |
192 |
89 142 140 186 191
|
divcan5d |
|- ( ( T. /\ x e. S ) -> ( ( ( x - _i ) x. 1 ) / ( ( x - _i ) x. ( x + _i ) ) ) = ( 1 / ( x + _i ) ) ) |
193 |
176 192
|
eqtr3d |
|- ( ( T. /\ x e. S ) -> ( ( x - _i ) / ( 1 + ( x ^ 2 ) ) ) = ( 1 / ( x + _i ) ) ) |
194 |
142
|
mulid1d |
|- ( ( T. /\ x e. S ) -> ( ( x + _i ) x. 1 ) = ( x + _i ) ) |
195 |
194 174
|
oveq12d |
|- ( ( T. /\ x e. S ) -> ( ( ( x + _i ) x. 1 ) / ( ( x + _i ) x. ( x - _i ) ) ) = ( ( x + _i ) / ( 1 + ( x ^ 2 ) ) ) ) |
196 |
89 140 142 191 186
|
divcan5d |
|- ( ( T. /\ x e. S ) -> ( ( ( x + _i ) x. 1 ) / ( ( x + _i ) x. ( x - _i ) ) ) = ( 1 / ( x - _i ) ) ) |
197 |
195 196
|
eqtr3d |
|- ( ( T. /\ x e. S ) -> ( ( x + _i ) / ( 1 + ( x ^ 2 ) ) ) = ( 1 / ( x - _i ) ) ) |
198 |
193 197
|
oveq12d |
|- ( ( T. /\ x e. S ) -> ( ( ( x - _i ) / ( 1 + ( x ^ 2 ) ) ) - ( ( x + _i ) / ( 1 + ( x ^ 2 ) ) ) ) = ( ( 1 / ( x + _i ) ) - ( 1 / ( x - _i ) ) ) ) |
199 |
149 162 198
|
3eqtr3rd |
|- ( ( T. /\ x e. S ) -> ( ( 1 / ( x + _i ) ) - ( 1 / ( x - _i ) ) ) = ( ( 2 / _i ) / ( 1 + ( x ^ 2 ) ) ) ) |
200 |
199
|
mpteq2dva |
|- ( T. -> ( x e. S |-> ( ( 1 / ( x + _i ) ) - ( 1 / ( x - _i ) ) ) ) = ( x e. S |-> ( ( 2 / _i ) / ( 1 + ( x ^ 2 ) ) ) ) ) |
201 |
138 200
|
eqtrd |
|- ( T. -> ( CC _D ( x e. S |-> ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) ) = ( x e. S |-> ( ( 2 / _i ) / ( 1 + ( x ^ 2 ) ) ) ) ) |
202 |
|
halfcl |
|- ( _i e. CC -> ( _i / 2 ) e. CC ) |
203 |
6 202
|
mp1i |
|- ( T. -> ( _i / 2 ) e. CC ) |
204 |
4 23 24 201 203
|
dvmptcmul |
|- ( T. -> ( CC _D ( x e. S |-> ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) ) ) = ( x e. S |-> ( ( _i / 2 ) x. ( ( 2 / _i ) / ( 1 + ( x ^ 2 ) ) ) ) ) ) |
205 |
|
df-atan |
|- arctan = ( x e. ( CC \ { -u _i , _i } ) |-> ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) ) |
206 |
205
|
reseq1i |
|- ( arctan |` S ) = ( ( x e. ( CC \ { -u _i , _i } ) |-> ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) ) |` S ) |
207 |
|
atanf |
|- arctan : ( CC \ { -u _i , _i } ) --> CC |
208 |
207
|
fdmi |
|- dom arctan = ( CC \ { -u _i , _i } ) |
209 |
7 208
|
sseqtri |
|- S C_ ( CC \ { -u _i , _i } ) |
210 |
|
resmpt |
|- ( S C_ ( CC \ { -u _i , _i } ) -> ( ( x e. ( CC \ { -u _i , _i } ) |-> ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) ) |` S ) = ( x e. S |-> ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) ) ) |
211 |
209 210
|
ax-mp |
|- ( ( x e. ( CC \ { -u _i , _i } ) |-> ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) ) |` S ) = ( x e. S |-> ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) ) |
212 |
206 211
|
eqtri |
|- ( arctan |` S ) = ( x e. S |-> ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) ) |
213 |
212
|
a1i |
|- ( T. -> ( arctan |` S ) = ( x e. S |-> ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) ) ) |
214 |
213
|
oveq2d |
|- ( T. -> ( CC _D ( arctan |` S ) ) = ( CC _D ( x e. S |-> ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. x ) ) ) - ( log ` ( 1 + ( _i x. x ) ) ) ) ) ) ) ) |
215 |
|
2ne0 |
|- 2 =/= 0 |
216 |
|
divcan6 |
|- ( ( ( _i e. CC /\ _i =/= 0 ) /\ ( 2 e. CC /\ 2 =/= 0 ) ) -> ( ( _i / 2 ) x. ( 2 / _i ) ) = 1 ) |
217 |
6 84 154 215 216
|
mp4an |
|- ( ( _i / 2 ) x. ( 2 / _i ) ) = 1 |
218 |
217
|
oveq1i |
|- ( ( ( _i / 2 ) x. ( 2 / _i ) ) / ( 1 + ( x ^ 2 ) ) ) = ( 1 / ( 1 + ( x ^ 2 ) ) ) |
219 |
6 202
|
mp1i |
|- ( ( T. /\ x e. S ) -> ( _i / 2 ) e. CC ) |
220 |
154 6 84
|
divcli |
|- ( 2 / _i ) e. CC |
221 |
220
|
a1i |
|- ( ( T. /\ x e. S ) -> ( 2 / _i ) e. CC ) |
222 |
219 221 145 148
|
divassd |
|- ( ( T. /\ x e. S ) -> ( ( ( _i / 2 ) x. ( 2 / _i ) ) / ( 1 + ( x ^ 2 ) ) ) = ( ( _i / 2 ) x. ( ( 2 / _i ) / ( 1 + ( x ^ 2 ) ) ) ) ) |
223 |
218 222
|
eqtr3id |
|- ( ( T. /\ x e. S ) -> ( 1 / ( 1 + ( x ^ 2 ) ) ) = ( ( _i / 2 ) x. ( ( 2 / _i ) / ( 1 + ( x ^ 2 ) ) ) ) ) |
224 |
223
|
mpteq2dva |
|- ( T. -> ( x e. S |-> ( 1 / ( 1 + ( x ^ 2 ) ) ) ) = ( x e. S |-> ( ( _i / 2 ) x. ( ( 2 / _i ) / ( 1 + ( x ^ 2 ) ) ) ) ) ) |
225 |
204 214 224
|
3eqtr4d |
|- ( T. -> ( CC _D ( arctan |` S ) ) = ( x e. S |-> ( 1 / ( 1 + ( x ^ 2 ) ) ) ) ) |
226 |
225
|
mptru |
|- ( CC _D ( arctan |` S ) ) = ( x e. S |-> ( 1 / ( 1 + ( x ^ 2 ) ) ) ) |