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Theorem atancj 22969
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 22966 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 5864 . . . . . 6  |-  ( A  =  -u _i  ->  (
Re `  A )  =  ( Re `  -u _i ) )
4 ax-icn 9547 . . . . . . . 8  |-  _i  e.  CC
54renegi 12972 . . . . . . 7  |-  ( Re
`  -u _i )  = 
-u ( Re `  _i )
6 rei 12948 . . . . . . . 8  |-  ( Re
`  _i )  =  0
76negeqi 9809 . . . . . . 7  |-  -u (
Re `  _i )  =  -u 0
8 neg0 9861 . . . . . . 7  |-  -u 0  =  0
95, 7, 83eqtri 2500 . . . . . 6  |-  ( Re
`  -u _i )  =  0
103, 9syl6eq 2524 . . . . 5  |-  ( A  =  -u _i  ->  (
Re `  A )  =  0 )
1110necon3i 2707 . . . 4  |-  ( ( Re `  A )  =/=  0  ->  A  =/=  -u _i )
122, 11syl 16 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  A  =/=  -u _i )
13 fveq2 5864 . . . . . 6  |-  ( A  =  _i  ->  (
Re `  A )  =  ( Re `  _i ) )
1413, 6syl6eq 2524 . . . . 5  |-  ( A  =  _i  ->  (
Re `  A )  =  0 )
1514necon3i 2707 . . . 4  |-  ( ( Re `  A )  =/=  0  ->  A  =/=  _i )
162, 15syl 16 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  A  =/=  _i )
17 atandm 22935 . . 3  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
181, 12, 16, 17syl3anbrc 1180 . 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 9546 . . . . . . . 8  |-  1  e.  CC
22 mulcl 9572 . . . . . . . . 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 9815 . . . . . . . 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 22936 . . . . . . . . 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 1009 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  -  (
_i  x.  A )
)  =/=  0 )
2925, 28logcld 22686 . . . . . 6  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
1  -  ( _i  x.  A ) ) )  e.  CC )
30 addcl 9570 . . . . . . . 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 1010 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  +  ( _i  x.  A ) )  =/=  0 )
3331, 32logcld 22686 . . . . . 6  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
1  +  ( _i  x.  A ) ) )  e.  CC )
3429, 33subcld 9926 . . . . 5  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  e.  CC )
35 cjmul 12934 . . . . 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 10624 . . . . . . . 8  |-  2  =/=  0
38 2cn 10602 . . . . . . . . 9  |-  2  e.  CC
394, 38cjdivi 12983 . . . . . . . 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 1324 . . . . . . . 8  |-  -u (
_i  /  2 )  =  ( -u _i  /  2 )
43 cji 12951 . . . . . . . . 9  |-  ( * `
 _i )  = 
-u _i
44 2re 10601 . . . . . . . . . 10  |-  2  e.  RR
45 cjre 12931 . . . . . . . . . 10  |-  ( 2  e.  RR  ->  (
* `  2 )  =  2 )
4644, 45ax-mp 5 . . . . . . . . 9  |-  ( * `
 2 )  =  2
4743, 46oveq12i 6294 . . . . . . . 8  |-  ( ( * `  _i )  /  ( * ` 
2 ) )  =  ( -u _i  / 
2 )
4842, 47eqtr4i 2499 . . . . . . 7  |-  -u (
_i  /  2 )  =  ( ( * `
 _i )  / 
( * `  2
) )
4940, 48eqtr4i 2499 . . . . . 6  |-  ( * `
 ( _i  / 
2 ) )  = 
-u ( _i  / 
2 )
5049oveq1i 6292 . . . . 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 12988 . . . . . 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 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 2520 . . . 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 12941 . . . . . . . . 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 12927 . . . . . . . . . . . . . . 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 12901 . . . . . . . . . . . . . . . 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 6298 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( Im ` 
1 )  -  (
Re `  A )
)  =  ( ( Im `  1 )  -  ( Im `  ( _i  x.  A
) ) ) )
6258, 61eqtr4d 2511 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  -  ( _i  x.  A ) ) )  =  ( ( Im `  1 )  -  ( Re `  A ) ) )
63 df-neg 9804 . . . . . . . . . . . . . 14  |-  -u (
Re `  A )  =  ( 0  -  ( Re `  A
) )
64 im1 12947 . . . . . . . . . . . . . . 15  |-  ( Im
`  1 )  =  0
6564oveq1i 6292 . . . . . . . . . . . . . 14  |-  ( ( Im `  1 )  -  ( Re `  A ) )  =  ( 0  -  (
Re `  A )
)
6663, 65eqtr4i 2499 . . . . . . . . . . . . 13  |-  -u (
Re `  A )  =  ( ( Im
`  1 )  -  ( Re `  A ) )
6762, 66syl6eqr 2526 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  -  ( _i  x.  A ) ) )  =  -u (
Re `  A )
)
68 recl 12902 . . . . . . . . . . . . . . 15  |-  ( A  e.  CC  ->  (
Re `  A )  e.  RR )
6968adantr 465 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Re `  A
)  e.  RR )
7069recnd 9618 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Re `  A
)  e.  CC )
7170, 2negne0d 9924 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  -u ( Re `  A
)  =/=  0 )
7267, 71eqnetrd 2760 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  -  ( _i  x.  A ) ) )  =/=  0 )
73 logcj 22719 . . . . . . . . . . 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 12941 . . . . . . . . . . . . 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 9591 . . . . . . . . . . . . . 14  |-  1  e.  RR
78 cjre 12931 . . . . . . . . . . . . . 14  |-  ( 1  e.  RR  ->  (
* `  1 )  =  1 )
7977, 78mp1i 12 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  1
)  =  1 )
80 cjmul 12934 . . . . . . . . . . . . . . 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 6292 . . . . . . . . . . . . . . 15  |-  ( ( * `  _i )  x.  ( * `  A ) )  =  ( -u _i  x.  ( * `  A
) )
83 cjcl 12897 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
8483adantr 465 . . . . . . . . . . . . . . . 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 663 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( -u _i  x.  (
* `  A )
)  =  -u (
_i  x.  ( * `  A ) ) )
8782, 86syl5eq 2520 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( * `  _i )  x.  (
* `  A )
)  =  -u (
_i  x.  ( * `  A ) ) )
8881, 87eqtrd 2508 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
_i  x.  A )
)  =  -u (
_i  x.  ( * `  A ) ) )
8979, 88oveq12d 6300 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( * ` 
1 )  -  (
* `  ( _i  x.  A ) ) )  =  ( 1  - 
-u ( _i  x.  ( * `  A
) ) ) )
90 mulcl 9572 . . . . . . . . . . . . . 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 9864 . . . . . . . . . . . . 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 2512 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
1  -  ( _i  x.  A ) ) )  =  ( 1  +  ( _i  x.  ( * `  A
) ) ) )
9594fveq2d 5868 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
* `  ( 1  -  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  +  ( _i  x.  ( * `
 A ) ) ) ) )
9674, 95eqtr3d 2510 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  ( log `  ( 1  -  ( _i  x.  A
) ) ) )  =  ( log `  (
1  +  ( _i  x.  ( * `  A ) ) ) ) )
97 imadd 12926 . . . . . . . . . . . . . 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 6298 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 0  +  ( Re `  A ) )  =  ( 0  +  ( Im `  ( _i  x.  A
) ) ) )
10064oveq1i 6292 . . . . . . . . . . . . . 14  |-  ( ( Im `  1 )  +  ( Im `  ( _i  x.  A
) ) )  =  ( 0  +  ( Im `  ( _i  x.  A ) ) )
10199, 100syl6eqr 2526 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 0  +  ( Re `  A ) )  =  ( ( Im `  1 )  +  ( Im `  ( _i  x.  A
) ) ) )
10270addid2d 9776 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 0  +  ( Re `  A ) )  =  ( Re
`  A ) )
10398, 101, 1023eqtr2d 2514 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  +  ( _i  x.  A ) ) )  =  ( Re
`  A ) )
104103, 2eqnetrd 2760 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  +  ( _i  x.  A ) ) )  =/=  0 )
105 logcj 22719 . . . . . . . . . . 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 12933 . . . . . . . . . . . . 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 6300 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( * ` 
1 )  +  ( * `  ( _i  x.  A ) ) )  =  ( 1  +  -u ( _i  x.  ( * `  A
) ) ) )
110 negsub 9863 . . . . . . . . . . . . 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 2512 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
1  +  ( _i  x.  A ) ) )  =  ( 1  -  ( _i  x.  ( * `  A
) ) ) )
113112fveq2d 5868 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
* `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  -  (
_i  x.  ( * `  A ) ) ) ) )
114106, 113eqtr3d 2510 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  ( log `  ( 1  +  ( _i  x.  A
) ) ) )  =  ( log `  (
1  -  ( _i  x.  ( * `  A ) ) ) ) )
11596, 114oveq12d 6300 . . . . . . . 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 2508 . . . . . . 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 9810 . . . . . 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 9570 . . . . . . . . 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 22968 . . . . . . . . . 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 22936 . . . . . . . . . 10  |-  ( ( * `  A )  e.  dom arctan  <->  ( ( * `
 A )  e.  CC  /\  ( 1  -  ( _i  x.  ( * `  A
) ) )  =/=  0  /\  ( 1  +  ( _i  x.  ( * `  A
) ) )  =/=  0 ) )
123122simp3bi 1013 . . . . . . . . 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 22686 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
1  +  ( _i  x.  ( * `  A ) ) ) )  e.  CC )
126 subcl 9815 . . . . . . . . 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 1012 . . . . . . . . 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 22686 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
1  -  ( _i  x.  ( * `  A ) ) ) )  e.  CC )
131125, 130negsubdi2d 9942 . . . . . 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 2508 . . . . 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 6298 . . . 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 2512 . . 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 22943 . . . . 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 5868 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (arctan `  A ) )  =  ( * `  (
( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) ) )
138 atanval 22943 . . . 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 2518 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   dom cdm 4999   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489   _ici 9490    + caddc 9491    x. cmul 9493    - cmin 9801   -ucneg 9802    / cdiv 10202   2c2 10581   *ccj 12888   Recre 12889   Imcim 12890   logclog 22670  arctancatan 22923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  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 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ioc 11530  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-fac 12318  df-bc 12345  df-hash 12370  df-shft 12859  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-limsup 13253  df-clim 13270  df-rlim 13271  df-sum 13468  df-ef 13661  df-sin 13663  df-cos 13664  df-pi 13666  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-lp 19403  df-perf 19404  df-cn 19494  df-cnp 19495  df-haus 19582  df-tx 19798  df-hmeo 19991  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-xms 20558  df-ms 20559  df-tms 20560  df-cncf 21117  df-limc 22005  df-dv 22006  df-log 22672  df-atan 22926
This theorem is referenced by:  atanrecl  22970
  Copyright terms: Public domain W3C validator