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Theorem atancj 23441
Description: The arctangent function distributes under conjugation. (The condition that  Re ( A )  =/=  0 is necessary because the branch cuts are chosen so that the negative imaginary line "agrees with" neighboring values with negative real part, while the positive imaginary line agrees with values with positive real part. This makes atanneg 23438 true unconditionally but messes up conjugation symmetry, and it is impossible to have both in a single-valued function. The claim is true on the imaginary line between  -u 1 and  1, though.) (Contributed by Mario Carneiro, 31-Mar-2015.)
Assertion
Ref Expression
atancj  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( A  e.  dom arctan  /\  ( * `  (arctan `  A ) )  =  (arctan `  ( * `  A ) ) ) )

Proof of Theorem atancj
StepHypRef Expression
1 simpl 455 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  A  e.  CC )
2 simpr 459 . . . 4  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Re `  A
)  =/=  0 )
3 fveq2 5848 . . . . . 6  |-  ( A  =  -u _i  ->  (
Re `  A )  =  ( Re `  -u _i ) )
4 ax-icn 9540 . . . . . . . 8  |-  _i  e.  CC
54renegi 13098 . . . . . . 7  |-  ( Re
`  -u _i )  = 
-u ( Re `  _i )
6 rei 13074 . . . . . . . 8  |-  ( Re
`  _i )  =  0
76negeqi 9804 . . . . . . 7  |-  -u (
Re `  _i )  =  -u 0
8 neg0 9856 . . . . . . 7  |-  -u 0  =  0
95, 7, 83eqtri 2487 . . . . . 6  |-  ( Re
`  -u _i )  =  0
103, 9syl6eq 2511 . . . . 5  |-  ( A  =  -u _i  ->  (
Re `  A )  =  0 )
1110necon3i 2694 . . . 4  |-  ( ( Re `  A )  =/=  0  ->  A  =/=  -u _i )
122, 11syl 16 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  A  =/=  -u _i )
13 fveq2 5848 . . . . . 6  |-  ( A  =  _i  ->  (
Re `  A )  =  ( Re `  _i ) )
1413, 6syl6eq 2511 . . . . 5  |-  ( A  =  _i  ->  (
Re `  A )  =  0 )
1514necon3i 2694 . . . 4  |-  ( ( Re `  A )  =/=  0  ->  A  =/=  _i )
162, 15syl 16 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  A  =/=  _i )
17 atandm 23407 . . 3  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
181, 12, 16, 17syl3anbrc 1178 . 2  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  A  e.  dom arctan )
19 halfcl 10760 . . . . . 6  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
204, 19ax-mp 5 . . . . 5  |-  ( _i 
/  2 )  e.  CC
21 ax-1cn 9539 . . . . . . . 8  |-  1  e.  CC
22 mulcl 9565 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
234, 1, 22sylancr 661 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( _i  x.  A
)  e.  CC )
24 subcl 9810 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  -  ( _i  x.  A
) )  e.  CC )
2521, 23, 24sylancr 661 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  -  (
_i  x.  A )
)  e.  CC )
26 atandm2 23408 . . . . . . . . 9  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
2718, 26sylib 196 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( A  e.  CC  /\  ( 1  -  (
_i  x.  A )
)  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
2827simp2d 1007 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  -  (
_i  x.  A )
)  =/=  0 )
2925, 28logcld 23127 . . . . . 6  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
1  -  ( _i  x.  A ) ) )  e.  CC )
30 addcl 9563 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
3121, 23, 30sylancr 661 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  +  ( _i  x.  A ) )  e.  CC )
3227simp3d 1008 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  +  ( _i  x.  A ) )  =/=  0 )
3331, 32logcld 23127 . . . . . 6  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
1  +  ( _i  x.  A ) ) )  e.  CC )
3429, 33subcld 9922 . . . . 5  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  e.  CC )
35 cjmul 13060 . . . . 5  |-  ( ( ( _i  /  2
)  e.  CC  /\  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  e.  CC )  ->  (
* `  ( (
_i  /  2 )  x.  ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) ) )  =  ( ( * `  (
_i  /  2 ) )  x.  ( * `
 ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) ) ) )
3620, 34, 35sylancr 661 . . . 4  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )  =  ( ( * `
 ( _i  / 
2 ) )  x.  ( * `  (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) ) )
37 2ne0 10624 . . . . . . . 8  |-  2  =/=  0
38 2cn 10602 . . . . . . . . 9  |-  2  e.  CC
394, 38cjdivi 13109 . . . . . . . 8  |-  ( 2  =/=  0  ->  (
* `  ( _i  /  2 ) )  =  ( ( * `  _i )  /  (
* `  2 )
) )
4037, 39ax-mp 5 . . . . . . 7  |-  ( * `
 ( _i  / 
2 ) )  =  ( ( * `  _i )  /  (
* `  2 )
)
41 divneg 10235 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
_i  /  2 )  =  ( -u _i  /  2 ) )
424, 38, 37, 41mp3an 1322 . . . . . . . 8  |-  -u (
_i  /  2 )  =  ( -u _i  /  2 )
43 cji 13077 . . . . . . . . 9  |-  ( * `
 _i )  = 
-u _i
44 2re 10601 . . . . . . . . . 10  |-  2  e.  RR
45 cjre 13057 . . . . . . . . . 10  |-  ( 2  e.  RR  ->  (
* `  2 )  =  2 )
4644, 45ax-mp 5 . . . . . . . . 9  |-  ( * `
 2 )  =  2
4743, 46oveq12i 6282 . . . . . . . 8  |-  ( ( * `  _i )  /  ( * ` 
2 ) )  =  ( -u _i  / 
2 )
4842, 47eqtr4i 2486 . . . . . . 7  |-  -u (
_i  /  2 )  =  ( ( * `
 _i )  / 
( * `  2
) )
4940, 48eqtr4i 2486 . . . . . 6  |-  ( * `
 ( _i  / 
2 ) )  = 
-u ( _i  / 
2 )
5049oveq1i 6280 . . . . 5  |-  ( ( * `  ( _i 
/  2 ) )  x.  ( * `  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) )  =  ( -u ( _i  /  2
)  x.  ( * `
 ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) ) )
5134cjcld 13114 . . . . . 6  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )  e.  CC )
52 mulneg12 9991 . . . . . 6  |-  ( ( ( _i  /  2
)  e.  CC  /\  ( * `  (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )  e.  CC )  -> 
( -u ( _i  / 
2 )  x.  (
* `  ( ( log `  ( 1  -  ( _i  x.  A
) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) )  =  ( ( _i  / 
2 )  x.  -u (
* `  ( ( log `  ( 1  -  ( _i  x.  A
) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) ) )
5320, 51, 52sylancr 661 . . . . 5  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( -u ( _i  / 
2 )  x.  (
* `  ( ( log `  ( 1  -  ( _i  x.  A
) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) )  =  ( ( _i  / 
2 )  x.  -u (
* `  ( ( log `  ( 1  -  ( _i  x.  A
) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) ) )
5450, 53syl5eq 2507 . . . 4  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( * `  ( _i  /  2
) )  x.  (
* `  ( ( log `  ( 1  -  ( _i  x.  A
) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) )  =  ( ( _i  / 
2 )  x.  -u (
* `  ( ( log `  ( 1  -  ( _i  x.  A
) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) ) )
55 cjsub 13067 . . . . . . . . 9  |-  ( ( ( log `  (
1  -  ( _i  x.  A ) ) )  e.  CC  /\  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )  -> 
( * `  (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )  =  ( ( * `
 ( log `  (
1  -  ( _i  x.  A ) ) ) )  -  (
* `  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) )
5629, 33, 55syl2anc 659 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )  =  ( ( * `
 ( log `  (
1  -  ( _i  x.  A ) ) ) )  -  (
* `  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) )
57 imsub 13053 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( Im `  ( 1  -  (
_i  x.  A )
) )  =  ( ( Im `  1
)  -  ( Im
`  ( _i  x.  A ) ) ) )
5821, 23, 57sylancr 661 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  -  ( _i  x.  A ) ) )  =  ( ( Im `  1 )  -  ( Im `  ( _i  x.  A
) ) ) )
59 reim 13027 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  ->  (
Re `  A )  =  ( Im `  ( _i  x.  A
) ) )
6059adantr 463 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Re `  A
)  =  ( Im
`  ( _i  x.  A ) ) )
6160oveq2d 6286 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( Im ` 
1 )  -  (
Re `  A )
)  =  ( ( Im `  1 )  -  ( Im `  ( _i  x.  A
) ) ) )
6258, 61eqtr4d 2498 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  -  ( _i  x.  A ) ) )  =  ( ( Im `  1 )  -  ( Re `  A ) ) )
63 df-neg 9799 . . . . . . . . . . . . . 14  |-  -u (
Re `  A )  =  ( 0  -  ( Re `  A
) )
64 im1 13073 . . . . . . . . . . . . . . 15  |-  ( Im
`  1 )  =  0
6564oveq1i 6280 . . . . . . . . . . . . . 14  |-  ( ( Im `  1 )  -  ( Re `  A ) )  =  ( 0  -  (
Re `  A )
)
6663, 65eqtr4i 2486 . . . . . . . . . . . . 13  |-  -u (
Re `  A )  =  ( ( Im
`  1 )  -  ( Re `  A ) )
6762, 66syl6eqr 2513 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  -  ( _i  x.  A ) ) )  =  -u (
Re `  A )
)
68 recl 13028 . . . . . . . . . . . . . . 15  |-  ( A  e.  CC  ->  (
Re `  A )  e.  RR )
6968adantr 463 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Re `  A
)  e.  RR )
7069recnd 9611 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Re `  A
)  e.  CC )
7170, 2negne0d 9920 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  -u ( Re `  A
)  =/=  0 )
7267, 71eqnetrd 2747 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  -  ( _i  x.  A ) ) )  =/=  0 )
73 logcj 23162 . . . . . . . . . . 11  |-  ( ( ( 1  -  (
_i  x.  A )
)  e.  CC  /\  ( Im `  ( 1  -  ( _i  x.  A ) ) )  =/=  0 )  -> 
( log `  (
* `  ( 1  -  ( _i  x.  A ) ) ) )  =  ( * `
 ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
7425, 72, 73syl2anc 659 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
* `  ( 1  -  ( _i  x.  A ) ) ) )  =  ( * `
 ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
75 cjsub 13067 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( * `  ( 1  -  (
_i  x.  A )
) )  =  ( ( * `  1
)  -  ( * `
 ( _i  x.  A ) ) ) )
7621, 23, 75sylancr 661 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
1  -  ( _i  x.  A ) ) )  =  ( ( * `  1 )  -  ( * `  ( _i  x.  A
) ) ) )
77 1re 9584 . . . . . . . . . . . . . 14  |-  1  e.  RR
78 cjre 13057 . . . . . . . . . . . . . 14  |-  ( 1  e.  RR  ->  (
* `  1 )  =  1 )
7977, 78mp1i 12 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  1
)  =  1 )
80 cjmul 13060 . . . . . . . . . . . . . . 15  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( * `  (
_i  x.  A )
)  =  ( ( * `  _i )  x.  ( * `  A ) ) )
814, 1, 80sylancr 661 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
_i  x.  A )
)  =  ( ( * `  _i )  x.  ( * `  A ) ) )
8243oveq1i 6280 . . . . . . . . . . . . . . 15  |-  ( ( * `  _i )  x.  ( * `  A ) )  =  ( -u _i  x.  ( * `  A
) )
83 cjcl 13023 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
8483adantr 463 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  A
)  e.  CC )
85 mulneg1 9989 . . . . . . . . . . . . . . . 16  |-  ( ( _i  e.  CC  /\  ( * `  A
)  e.  CC )  ->  ( -u _i  x.  ( * `  A
) )  =  -u ( _i  x.  (
* `  A )
) )
864, 84, 85sylancr 661 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( -u _i  x.  (
* `  A )
)  =  -u (
_i  x.  ( * `  A ) ) )
8782, 86syl5eq 2507 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( * `  _i )  x.  (
* `  A )
)  =  -u (
_i  x.  ( * `  A ) ) )
8881, 87eqtrd 2495 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
_i  x.  A )
)  =  -u (
_i  x.  ( * `  A ) ) )
8979, 88oveq12d 6288 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( * ` 
1 )  -  (
* `  ( _i  x.  A ) ) )  =  ( 1  - 
-u ( _i  x.  ( * `  A
) ) ) )
90 mulcl 9565 . . . . . . . . . . . . . 14  |-  ( ( _i  e.  CC  /\  ( * `  A
)  e.  CC )  ->  ( _i  x.  ( * `  A
) )  e.  CC )
914, 84, 90sylancr 661 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( _i  x.  (
* `  A )
)  e.  CC )
92 subneg 9859 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  (
* `  A )
)  e.  CC )  ->  ( 1  - 
-u ( _i  x.  ( * `  A
) ) )  =  ( 1  +  ( _i  x.  ( * `
 A ) ) ) )
9321, 91, 92sylancr 661 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  -  -u (
_i  x.  ( * `  A ) ) )  =  ( 1  +  ( _i  x.  (
* `  A )
) ) )
9476, 89, 933eqtrd 2499 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
1  -  ( _i  x.  A ) ) )  =  ( 1  +  ( _i  x.  ( * `  A
) ) ) )
9594fveq2d 5852 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
* `  ( 1  -  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  +  ( _i  x.  ( * `
 A ) ) ) ) )
9674, 95eqtr3d 2497 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  ( log `  ( 1  -  ( _i  x.  A
) ) ) )  =  ( log `  (
1  +  ( _i  x.  ( * `  A ) ) ) ) )
97 imadd 13052 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( Im `  ( 1  +  ( _i  x.  A ) ) )  =  ( ( Im `  1
)  +  ( Im
`  ( _i  x.  A ) ) ) )
9821, 23, 97sylancr 661 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  +  ( _i  x.  A ) ) )  =  ( ( Im `  1 )  +  ( Im `  ( _i  x.  A
) ) ) )
9960oveq2d 6286 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 0  +  ( Re `  A ) )  =  ( 0  +  ( Im `  ( _i  x.  A
) ) ) )
10064oveq1i 6280 . . . . . . . . . . . . . 14  |-  ( ( Im `  1 )  +  ( Im `  ( _i  x.  A
) ) )  =  ( 0  +  ( Im `  ( _i  x.  A ) ) )
10199, 100syl6eqr 2513 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 0  +  ( Re `  A ) )  =  ( ( Im `  1 )  +  ( Im `  ( _i  x.  A
) ) ) )
10270addid2d 9770 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 0  +  ( Re `  A ) )  =  ( Re
`  A ) )
10398, 101, 1023eqtr2d 2501 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  +  ( _i  x.  A ) ) )  =  ( Re
`  A ) )
104103, 2eqnetrd 2747 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  +  ( _i  x.  A ) ) )  =/=  0 )
105 logcj 23162 . . . . . . . . . . 11  |-  ( ( ( 1  +  ( _i  x.  A ) )  e.  CC  /\  ( Im `  ( 1  +  ( _i  x.  A ) ) )  =/=  0 )  -> 
( log `  (
* `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( * `
 ( log `  (
1  +  ( _i  x.  A ) ) ) ) )
10631, 104, 105syl2anc 659 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
* `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( * `
 ( log `  (
1  +  ( _i  x.  A ) ) ) ) )
107 cjadd 13059 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( * `  ( 1  +  ( _i  x.  A ) ) )  =  ( ( * `  1
)  +  ( * `
 ( _i  x.  A ) ) ) )
10821, 23, 107sylancr 661 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
1  +  ( _i  x.  A ) ) )  =  ( ( * `  1 )  +  ( * `  ( _i  x.  A
) ) ) )
10979, 88oveq12d 6288 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( * ` 
1 )  +  ( * `  ( _i  x.  A ) ) )  =  ( 1  +  -u ( _i  x.  ( * `  A
) ) ) )
110 negsub 9858 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  (
* `  A )
)  e.  CC )  ->  ( 1  + 
-u ( _i  x.  ( * `  A
) ) )  =  ( 1  -  (
_i  x.  ( * `  A ) ) ) )
11121, 91, 110sylancr 661 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  +  -u ( _i  x.  (
* `  A )
) )  =  ( 1  -  ( _i  x.  ( * `  A ) ) ) )
112108, 109, 1113eqtrd 2499 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
1  +  ( _i  x.  A ) ) )  =  ( 1  -  ( _i  x.  ( * `  A
) ) ) )
113112fveq2d 5852 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
* `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  -  (
_i  x.  ( * `  A ) ) ) ) )
114106, 113eqtr3d 2497 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  ( log `  ( 1  +  ( _i  x.  A
) ) ) )  =  ( log `  (
1  -  ( _i  x.  ( * `  A ) ) ) ) )
11596, 114oveq12d 6288 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( * `  ( log `  ( 1  -  ( _i  x.  A ) ) ) )  -  ( * `
 ( log `  (
1  +  ( _i  x.  A ) ) ) ) )  =  ( ( log `  (
1  +  ( _i  x.  ( * `  A ) ) ) )  -  ( log `  ( 1  -  (
_i  x.  ( * `  A ) ) ) ) ) )
11656, 115eqtrd 2495 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )  =  ( ( log `  ( 1  +  ( _i  x.  ( * `
 A ) ) ) )  -  ( log `  ( 1  -  ( _i  x.  (
* `  A )
) ) ) ) )
117116negeqd 9805 . . . . . 6  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  -u ( * `  (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )  =  -u ( ( log `  ( 1  +  ( _i  x.  ( * `
 A ) ) ) )  -  ( log `  ( 1  -  ( _i  x.  (
* `  A )
) ) ) ) )
118 addcl 9563 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  ( _i  x.  (
* `  A )
)  e.  CC )  ->  ( 1  +  ( _i  x.  (
* `  A )
) )  e.  CC )
11921, 91, 118sylancr 661 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  +  ( _i  x.  ( * `
 A ) ) )  e.  CC )
120 atandmcj 23440 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( * `
 A )  e. 
dom arctan )
12118, 120syl 16 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  A
)  e.  dom arctan )
122 atandm2 23408 . . . . . . . . . 10  |-  ( ( * `  A )  e.  dom arctan  <->  ( ( * `
 A )  e.  CC  /\  ( 1  -  ( _i  x.  ( * `  A
) ) )  =/=  0  /\  ( 1  +  ( _i  x.  ( * `  A
) ) )  =/=  0 ) )
123122simp3bi 1011 . . . . . . . . 9  |-  ( ( * `  A )  e.  dom arctan  ->  ( 1  +  ( _i  x.  ( * `  A
) ) )  =/=  0 )
124121, 123syl 16 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  +  ( _i  x.  ( * `
 A ) ) )  =/=  0 )
125119, 124logcld 23127 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
1  +  ( _i  x.  ( * `  A ) ) ) )  e.  CC )
126 subcl 9810 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  ( _i  x.  (
* `  A )
)  e.  CC )  ->  ( 1  -  ( _i  x.  (
* `  A )
) )  e.  CC )
12721, 91, 126sylancr 661 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  -  (
_i  x.  ( * `  A ) ) )  e.  CC )
128122simp2bi 1010 . . . . . . . . 9  |-  ( ( * `  A )  e.  dom arctan  ->  ( 1  -  ( _i  x.  ( * `  A
) ) )  =/=  0 )
129121, 128syl 16 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  -  (
_i  x.  ( * `  A ) ) )  =/=  0 )
130127, 129logcld 23127 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
1  -  ( _i  x.  ( * `  A ) ) ) )  e.  CC )
131125, 130negsubdi2d 9938 . . . . . 6  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  -u ( ( log `  (
1  +  ( _i  x.  ( * `  A ) ) ) )  -  ( log `  ( 1  -  (
_i  x.  ( * `  A ) ) ) ) )  =  ( ( log `  (
1  -  ( _i  x.  ( * `  A ) ) ) )  -  ( log `  ( 1  +  ( _i  x.  ( * `
 A ) ) ) ) ) )
132117, 131eqtrd 2495 . . . . 5  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  -u ( * `  (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )  =  ( ( log `  ( 1  -  (
_i  x.  ( * `  A ) ) ) )  -  ( log `  ( 1  +  ( _i  x.  ( * `
 A ) ) ) ) ) )
133132oveq2d 6286 . . . 4  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( _i  / 
2 )  x.  -u (
* `  ( ( log `  ( 1  -  ( _i  x.  A
) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) )  =  ( ( _i  / 
2 )  x.  (
( log `  (
1  -  ( _i  x.  ( * `  A ) ) ) )  -  ( log `  ( 1  +  ( _i  x.  ( * `
 A ) ) ) ) ) ) )
13436, 54, 1333eqtrd 2499 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )  =  ( ( _i 
/  2 )  x.  ( ( log `  (
1  -  ( _i  x.  ( * `  A ) ) ) )  -  ( log `  ( 1  +  ( _i  x.  ( * `
 A ) ) ) ) ) ) )
135 atanval 23415 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
13618, 135syl 16 . . . 4  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
(arctan `  A )  =  ( ( _i 
/  2 )  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) )
137136fveq2d 5852 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (arctan `  A ) )  =  ( * `  (
( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) ) )
138 atanval 23415 . . . 4  |-  ( ( * `  A )  e.  dom arctan  ->  (arctan `  ( * `  A
) )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  ( * `  A
) ) ) )  -  ( log `  (
1  +  ( _i  x.  ( * `  A ) ) ) ) ) ) )
139121, 138syl 16 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
(arctan `  ( * `  A ) )  =  ( ( _i  / 
2 )  x.  (
( log `  (
1  -  ( _i  x.  ( * `  A ) ) ) )  -  ( log `  ( 1  +  ( _i  x.  ( * `
 A ) ) ) ) ) ) )
140134, 137, 1393eqtr4d 2505 . 2  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (arctan `  A ) )  =  (arctan `  ( * `  A ) ) )
14118, 140jca 530 1  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( A  e.  dom arctan  /\  ( * `  (arctan `  A ) )  =  (arctan `  ( * `  A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   dom cdm 4988   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482   _ici 9483    + caddc 9484    x. cmul 9486    - cmin 9796   -ucneg 9797    / cdiv 10202   2c2 10581   *ccj 13014   Recre 13015   Imcim 13016   logclog 23111  arctancatan 23395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12093  df-exp 12152  df-fac 12339  df-bc 12366  df-hash 12391  df-shft 12985  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-limsup 13379  df-clim 13396  df-rlim 13397  df-sum 13594  df-ef 13888  df-sin 13890  df-cos 13891  df-pi 13893  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-starv 14802  df-sca 14803  df-vsca 14804  df-ip 14805  df-tset 14806  df-ple 14807  df-ds 14809  df-unif 14810  df-hom 14811  df-cco 14812  df-rest 14915  df-topn 14916  df-0g 14934  df-gsum 14935  df-topgen 14936  df-pt 14937  df-prds 14940  df-xrs 14994  df-qtop 14999  df-imas 15000  df-xps 15002  df-mre 15078  df-mrc 15079  df-acs 15081  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-mulg 16262  df-cntz 16557  df-cmn 17002  df-psmet 18609  df-xmet 18610  df-met 18611  df-bl 18612  df-mopn 18613  df-fbas 18614  df-fg 18615  df-cnfld 18619  df-top 19569  df-bases 19571  df-topon 19572  df-topsp 19573  df-cld 19690  df-ntr 19691  df-cls 19692  df-nei 19769  df-lp 19807  df-perf 19808  df-cn 19898  df-cnp 19899  df-haus 19986  df-tx 20232  df-hmeo 20425  df-fil 20516  df-fm 20608  df-flim 20609  df-flf 20610  df-xms 20992  df-ms 20993  df-tms 20994  df-cncf 21551  df-limc 22439  df-dv 22440  df-log 23113  df-atan 23398
This theorem is referenced by:  atanrecl  23442
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