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Theorem atancj 22310
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 22307 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 457 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  A  e.  CC )
2 simpr 461 . . . 4  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Re `  A
)  =/=  0 )
3 fveq2 5696 . . . . . 6  |-  ( A  =  -u _i  ->  (
Re `  A )  =  ( Re `  -u _i ) )
4 ax-icn 9346 . . . . . . . 8  |-  _i  e.  CC
54renegi 12674 . . . . . . 7  |-  ( Re
`  -u _i )  = 
-u ( Re `  _i )
6 rei 12650 . . . . . . . 8  |-  ( Re
`  _i )  =  0
76negeqi 9608 . . . . . . 7  |-  -u (
Re `  _i )  =  -u 0
8 neg0 9660 . . . . . . 7  |-  -u 0  =  0
95, 7, 83eqtri 2467 . . . . . 6  |-  ( Re
`  -u _i )  =  0
103, 9syl6eq 2491 . . . . 5  |-  ( A  =  -u _i  ->  (
Re `  A )  =  0 )
1110necon3i 2655 . . . 4  |-  ( ( Re `  A )  =/=  0  ->  A  =/=  -u _i )
122, 11syl 16 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  A  =/=  -u _i )
13 fveq2 5696 . . . . . 6  |-  ( A  =  _i  ->  (
Re `  A )  =  ( Re `  _i ) )
1413, 6syl6eq 2491 . . . . 5  |-  ( A  =  _i  ->  (
Re `  A )  =  0 )
1514necon3i 2655 . . . 4  |-  ( ( Re `  A )  =/=  0  ->  A  =/=  _i )
162, 15syl 16 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  A  =/=  _i )
17 atandm 22276 . . 3  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
181, 12, 16, 17syl3anbrc 1172 . 2  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  A  e.  dom arctan )
19 halfcl 10555 . . . . . 6  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
204, 19ax-mp 5 . . . . 5  |-  ( _i 
/  2 )  e.  CC
21 ax-1cn 9345 . . . . . . . 8  |-  1  e.  CC
22 mulcl 9371 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
234, 1, 22sylancr 663 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( _i  x.  A
)  e.  CC )
24 subcl 9614 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  -  ( _i  x.  A
) )  e.  CC )
2521, 23, 24sylancr 663 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  -  (
_i  x.  A )
)  e.  CC )
26 atandm2 22277 . . . . . . . . 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 1001 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  -  (
_i  x.  A )
)  =/=  0 )
2925, 28logcld 22027 . . . . . 6  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
1  -  ( _i  x.  A ) ) )  e.  CC )
30 addcl 9369 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
3121, 23, 30sylancr 663 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  +  ( _i  x.  A ) )  e.  CC )
3227simp3d 1002 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  +  ( _i  x.  A ) )  =/=  0 )
3331, 32logcld 22027 . . . . . 6  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
1  +  ( _i  x.  A ) ) )  e.  CC )
3429, 33subcld 9724 . . . . 5  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  e.  CC )
35 cjmul 12636 . . . . 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 663 . . . 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 10419 . . . . . . . 8  |-  2  =/=  0
38 2cn 10397 . . . . . . . . 9  |-  2  e.  CC
394, 38cjdivi 12685 . . . . . . . 8  |-  ( 2  =/=  0  ->  (
* `  ( _i  /  2 ) )  =  ( ( * `  _i )  /  (
* `  2 )
) )
4037, 39ax-mp 5 . . . . . . 7  |-  ( * `
 ( _i  / 
2 ) )  =  ( ( * `  _i )  /  (
* `  2 )
)
41 divneg 10031 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
_i  /  2 )  =  ( -u _i  /  2 ) )
424, 38, 37, 41mp3an 1314 . . . . . . . 8  |-  -u (
_i  /  2 )  =  ( -u _i  /  2 )
43 cji 12653 . . . . . . . . 9  |-  ( * `
 _i )  = 
-u _i
44 2re 10396 . . . . . . . . . 10  |-  2  e.  RR
45 cjre 12633 . . . . . . . . . 10  |-  ( 2  e.  RR  ->  (
* `  2 )  =  2 )
4644, 45ax-mp 5 . . . . . . . . 9  |-  ( * `
 2 )  =  2
4743, 46oveq12i 6108 . . . . . . . 8  |-  ( ( * `  _i )  /  ( * ` 
2 ) )  =  ( -u _i  / 
2 )
4842, 47eqtr4i 2466 . . . . . . 7  |-  -u (
_i  /  2 )  =  ( ( * `
 _i )  / 
( * `  2
) )
4940, 48eqtr4i 2466 . . . . . 6  |-  ( * `
 ( _i  / 
2 ) )  = 
-u ( _i  / 
2 )
5049oveq1i 6106 . . . . 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 12690 . . . . . 6  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )  e.  CC )
52 mulneg12 9788 . . . . . 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 663 . . . . 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 2487 . . . 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 12643 . . . . . . . . 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 661 . . . . . . . 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 12629 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( Im `  ( 1  -  (
_i  x.  A )
) )  =  ( ( Im `  1
)  -  ( Im
`  ( _i  x.  A ) ) ) )
5821, 23, 57sylancr 663 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  -  ( _i  x.  A ) ) )  =  ( ( Im `  1 )  -  ( Im `  ( _i  x.  A
) ) ) )
59 reim 12603 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  ->  (
Re `  A )  =  ( Im `  ( _i  x.  A
) ) )
6059adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Re `  A
)  =  ( Im
`  ( _i  x.  A ) ) )
6160oveq2d 6112 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( Im ` 
1 )  -  (
Re `  A )
)  =  ( ( Im `  1 )  -  ( Im `  ( _i  x.  A
) ) ) )
6258, 61eqtr4d 2478 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  -  ( _i  x.  A ) ) )  =  ( ( Im `  1 )  -  ( Re `  A ) ) )
63 df-neg 9603 . . . . . . . . . . . . . 14  |-  -u (
Re `  A )  =  ( 0  -  ( Re `  A
) )
64 im1 12649 . . . . . . . . . . . . . . 15  |-  ( Im
`  1 )  =  0
6564oveq1i 6106 . . . . . . . . . . . . . 14  |-  ( ( Im `  1 )  -  ( Re `  A ) )  =  ( 0  -  (
Re `  A )
)
6663, 65eqtr4i 2466 . . . . . . . . . . . . 13  |-  -u (
Re `  A )  =  ( ( Im
`  1 )  -  ( Re `  A ) )
6762, 66syl6eqr 2493 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  -  ( _i  x.  A ) ) )  =  -u (
Re `  A )
)
68 recl 12604 . . . . . . . . . . . . . . 15  |-  ( A  e.  CC  ->  (
Re `  A )  e.  RR )
6968adantr 465 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Re `  A
)  e.  RR )
7069recnd 9417 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Re `  A
)  e.  CC )
7170, 2negne0d 9722 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  -u ( Re `  A
)  =/=  0 )
7267, 71eqnetrd 2631 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  -  ( _i  x.  A ) ) )  =/=  0 )
73 logcj 22060 . . . . . . . . . . 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 661 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
* `  ( 1  -  ( _i  x.  A ) ) ) )  =  ( * `
 ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
75 cjsub 12643 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( * `  ( 1  -  (
_i  x.  A )
) )  =  ( ( * `  1
)  -  ( * `
 ( _i  x.  A ) ) ) )
7621, 23, 75sylancr 663 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
1  -  ( _i  x.  A ) ) )  =  ( ( * `  1 )  -  ( * `  ( _i  x.  A
) ) ) )
77 1re 9390 . . . . . . . . . . . . . 14  |-  1  e.  RR
78 cjre 12633 . . . . . . . . . . . . . 14  |-  ( 1  e.  RR  ->  (
* `  1 )  =  1 )
7977, 78mp1i 12 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  1
)  =  1 )
80 cjmul 12636 . . . . . . . . . . . . . . 15  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( * `  (
_i  x.  A )
)  =  ( ( * `  _i )  x.  ( * `  A ) ) )
814, 1, 80sylancr 663 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
_i  x.  A )
)  =  ( ( * `  _i )  x.  ( * `  A ) ) )
8243oveq1i 6106 . . . . . . . . . . . . . . 15  |-  ( ( * `  _i )  x.  ( * `  A ) )  =  ( -u _i  x.  ( * `  A
) )
83 cjcl 12599 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
8483adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  A
)  e.  CC )
85 mulneg1 9786 . . . . . . . . . . . . . . . 16  |-  ( ( _i  e.  CC  /\  ( * `  A
)  e.  CC )  ->  ( -u _i  x.  ( * `  A
) )  =  -u ( _i  x.  (
* `  A )
) )
864, 84, 85sylancr 663 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( -u _i  x.  (
* `  A )
)  =  -u (
_i  x.  ( * `  A ) ) )
8782, 86syl5eq 2487 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( * `  _i )  x.  (
* `  A )
)  =  -u (
_i  x.  ( * `  A ) ) )
8881, 87eqtrd 2475 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
_i  x.  A )
)  =  -u (
_i  x.  ( * `  A ) ) )
8979, 88oveq12d 6114 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( * ` 
1 )  -  (
* `  ( _i  x.  A ) ) )  =  ( 1  - 
-u ( _i  x.  ( * `  A
) ) ) )
90 mulcl 9371 . . . . . . . . . . . . . 14  |-  ( ( _i  e.  CC  /\  ( * `  A
)  e.  CC )  ->  ( _i  x.  ( * `  A
) )  e.  CC )
914, 84, 90sylancr 663 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( _i  x.  (
* `  A )
)  e.  CC )
92 subneg 9663 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  (
* `  A )
)  e.  CC )  ->  ( 1  - 
-u ( _i  x.  ( * `  A
) ) )  =  ( 1  +  ( _i  x.  ( * `
 A ) ) ) )
9321, 91, 92sylancr 663 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  -  -u (
_i  x.  ( * `  A ) ) )  =  ( 1  +  ( _i  x.  (
* `  A )
) ) )
9476, 89, 933eqtrd 2479 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
1  -  ( _i  x.  A ) ) )  =  ( 1  +  ( _i  x.  ( * `  A
) ) ) )
9594fveq2d 5700 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
* `  ( 1  -  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  +  ( _i  x.  ( * `
 A ) ) ) ) )
9674, 95eqtr3d 2477 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  ( log `  ( 1  -  ( _i  x.  A
) ) ) )  =  ( log `  (
1  +  ( _i  x.  ( * `  A ) ) ) ) )
97 imadd 12628 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( Im `  ( 1  +  ( _i  x.  A ) ) )  =  ( ( Im `  1
)  +  ( Im
`  ( _i  x.  A ) ) ) )
9821, 23, 97sylancr 663 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  +  ( _i  x.  A ) ) )  =  ( ( Im `  1 )  +  ( Im `  ( _i  x.  A
) ) ) )
9960oveq2d 6112 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 0  +  ( Re `  A ) )  =  ( 0  +  ( Im `  ( _i  x.  A
) ) ) )
10064oveq1i 6106 . . . . . . . . . . . . . 14  |-  ( ( Im `  1 )  +  ( Im `  ( _i  x.  A
) ) )  =  ( 0  +  ( Im `  ( _i  x.  A ) ) )
10199, 100syl6eqr 2493 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 0  +  ( Re `  A ) )  =  ( ( Im `  1 )  +  ( Im `  ( _i  x.  A
) ) ) )
10270addid2d 9575 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 0  +  ( Re `  A ) )  =  ( Re
`  A ) )
10398, 101, 1023eqtr2d 2481 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  +  ( _i  x.  A ) ) )  =  ( Re
`  A ) )
104103, 2eqnetrd 2631 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  +  ( _i  x.  A ) ) )  =/=  0 )
105 logcj 22060 . . . . . . . . . . 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 661 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
* `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( * `
 ( log `  (
1  +  ( _i  x.  A ) ) ) ) )
107 cjadd 12635 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( * `  ( 1  +  ( _i  x.  A ) ) )  =  ( ( * `  1
)  +  ( * `
 ( _i  x.  A ) ) ) )
10821, 23, 107sylancr 663 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
1  +  ( _i  x.  A ) ) )  =  ( ( * `  1 )  +  ( * `  ( _i  x.  A
) ) ) )
10979, 88oveq12d 6114 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( * ` 
1 )  +  ( * `  ( _i  x.  A ) ) )  =  ( 1  +  -u ( _i  x.  ( * `  A
) ) ) )
110 negsub 9662 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  (
* `  A )
)  e.  CC )  ->  ( 1  + 
-u ( _i  x.  ( * `  A
) ) )  =  ( 1  -  (
_i  x.  ( * `  A ) ) ) )
11121, 91, 110sylancr 663 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  +  -u ( _i  x.  (
* `  A )
) )  =  ( 1  -  ( _i  x.  ( * `  A ) ) ) )
112108, 109, 1113eqtrd 2479 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
1  +  ( _i  x.  A ) ) )  =  ( 1  -  ( _i  x.  ( * `  A
) ) ) )
113112fveq2d 5700 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
* `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  -  (
_i  x.  ( * `  A ) ) ) ) )
114106, 113eqtr3d 2477 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  ( log `  ( 1  +  ( _i  x.  A
) ) ) )  =  ( log `  (
1  -  ( _i  x.  ( * `  A ) ) ) ) )
11596, 114oveq12d 6114 . . . . . . . 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 2475 . . . . . . 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 9609 . . . . . 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 9369 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  ( _i  x.  (
* `  A )
)  e.  CC )  ->  ( 1  +  ( _i  x.  (
* `  A )
) )  e.  CC )
11921, 91, 118sylancr 663 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  +  ( _i  x.  ( * `
 A ) ) )  e.  CC )
120 atandmcj 22309 . . . . . . . . . 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 22277 . . . . . . . . . 10  |-  ( ( * `  A )  e.  dom arctan  <->  ( ( * `
 A )  e.  CC  /\  ( 1  -  ( _i  x.  ( * `  A
) ) )  =/=  0  /\  ( 1  +  ( _i  x.  ( * `  A
) ) )  =/=  0 ) )
123122simp3bi 1005 . . . . . . . . 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 22027 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
1  +  ( _i  x.  ( * `  A ) ) ) )  e.  CC )
126 subcl 9614 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  ( _i  x.  (
* `  A )
)  e.  CC )  ->  ( 1  -  ( _i  x.  (
* `  A )
) )  e.  CC )
12721, 91, 126sylancr 663 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  -  (
_i  x.  ( * `  A ) ) )  e.  CC )
128122simp2bi 1004 . . . . . . . . 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 22027 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
1  -  ( _i  x.  ( * `  A ) ) ) )  e.  CC )
131125, 130negsubdi2d 9740 . . . . . 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 2475 . . . . 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 6112 . . . 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 2479 . . 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 22284 . . . . 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 5700 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (arctan `  A ) )  =  ( * `  (
( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) ) )
138 atanval 22284 . . . 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 2485 . 2  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (arctan `  A ) )  =  (arctan `  ( * `  A ) ) )
14118, 140jca 532 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   dom cdm 4845   ` cfv 5423  (class class class)co 6096   CCcc 9285   RRcr 9286   0cc0 9287   1c1 9288   _ici 9289    + caddc 9290    x. cmul 9292    - cmin 9600   -ucneg 9601    / cdiv 9998   2c2 10376   *ccj 12590   Recre 12591   Imcim 12592   logclog 22011  arctancatan 22264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365  ax-addf 9366  ax-mulf 9367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-fi 7666  df-sup 7696  df-oi 7729  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-ioo 11309  df-ioc 11310  df-ico 11311  df-icc 11312  df-fz 11443  df-fzo 11554  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-fac 12057  df-bc 12084  df-hash 12109  df-shft 12561  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-limsup 12954  df-clim 12971  df-rlim 12972  df-sum 13169  df-ef 13358  df-sin 13360  df-cos 13361  df-pi 13363  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-hom 14267  df-cco 14268  df-rest 14366  df-topn 14367  df-0g 14385  df-gsum 14386  df-topgen 14387  df-pt 14388  df-prds 14391  df-xrs 14445  df-qtop 14450  df-imas 14451  df-xps 14453  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-mulg 15553  df-cntz 15840  df-cmn 16284  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-fbas 17819  df-fg 17820  df-cnfld 17824  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-cld 18628  df-ntr 18629  df-cls 18630  df-nei 18707  df-lp 18745  df-perf 18746  df-cn 18836  df-cnp 18837  df-haus 18924  df-tx 19140  df-hmeo 19333  df-fil 19424  df-fm 19516  df-flim 19517  df-flf 19518  df-xms 19900  df-ms 19901  df-tms 19902  df-cncf 20459  df-limc 21346  df-dv 21347  df-log 22013  df-atan 22267
This theorem is referenced by:  atanrecl  22311
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