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Theorem atancj 20206
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 20203 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 443 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  A  e.  CC )
2 simpr 447 . . . 4  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Re `  A
)  =/=  0 )
3 fveq2 5525 . . . . . 6  |-  ( A  =  -u _i  ->  (
Re `  A )  =  ( Re `  -u _i ) )
4 ax-icn 8796 . . . . . . . 8  |-  _i  e.  CC
54renegi 11665 . . . . . . 7  |-  ( Re
`  -u _i )  = 
-u ( Re `  _i )
6 rei 11641 . . . . . . . 8  |-  ( Re
`  _i )  =  0
76negeqi 9045 . . . . . . 7  |-  -u (
Re `  _i )  =  -u 0
8 neg0 9093 . . . . . . 7  |-  -u 0  =  0
95, 7, 83eqtri 2307 . . . . . 6  |-  ( Re
`  -u _i )  =  0
103, 9syl6eq 2331 . . . . 5  |-  ( A  =  -u _i  ->  (
Re `  A )  =  0 )
1110necon3i 2485 . . . 4  |-  ( ( Re `  A )  =/=  0  ->  A  =/=  -u _i )
122, 11syl 15 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  A  =/=  -u _i )
13 fveq2 5525 . . . . . 6  |-  ( A  =  _i  ->  (
Re `  A )  =  ( Re `  _i ) )
1413, 6syl6eq 2331 . . . . 5  |-  ( A  =  _i  ->  (
Re `  A )  =  0 )
1514necon3i 2485 . . . 4  |-  ( ( Re `  A )  =/=  0  ->  A  =/=  _i )
162, 15syl 15 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  A  =/=  _i )
17 atandm 20172 . . 3  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
181, 12, 16, 17syl3anbrc 1136 . 2  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  A  e.  dom arctan )
19 halfcl 9937 . . . . . 6  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
204, 19ax-mp 8 . . . . 5  |-  ( _i 
/  2 )  e.  CC
21 ax-1cn 8795 . . . . . . . 8  |-  1  e.  CC
22 mulcl 8821 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
234, 1, 22sylancr 644 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( _i  x.  A
)  e.  CC )
24 subcl 9051 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  -  ( _i  x.  A
) )  e.  CC )
2521, 23, 24sylancr 644 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  -  (
_i  x.  A )
)  e.  CC )
26 atandm2 20173 . . . . . . . . 9  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
2718, 26sylib 188 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( A  e.  CC  /\  ( 1  -  (
_i  x.  A )
)  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
2827simp2d 968 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  -  (
_i  x.  A )
)  =/=  0 )
29 logcl 19926 . . . . . . 7  |-  ( ( ( 1  -  (
_i  x.  A )
)  e.  CC  /\  ( 1  -  (
_i  x.  A )
)  =/=  0 )  ->  ( log `  (
1  -  ( _i  x.  A ) ) )  e.  CC )
3025, 28, 29syl2anc 642 . . . . . 6  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
1  -  ( _i  x.  A ) ) )  e.  CC )
31 addcl 8819 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
3221, 23, 31sylancr 644 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  +  ( _i  x.  A ) )  e.  CC )
3327simp3d 969 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  +  ( _i  x.  A ) )  =/=  0 )
34 logcl 19926 . . . . . . 7  |-  ( ( ( 1  +  ( _i  x.  A ) )  e.  CC  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 )  ->  ( log `  (
1  +  ( _i  x.  A ) ) )  e.  CC )
3532, 33, 34syl2anc 642 . . . . . 6  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
1  +  ( _i  x.  A ) ) )  e.  CC )
3630, 35subcld 9157 . . . . 5  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  e.  CC )
37 cjmul 11627 . . . . 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
) ) ) ) ) ) )
3820, 36, 37sylancr 644 . . . 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 ) ) ) ) ) ) )
39 2ne0 9829 . . . . . . . 8  |-  2  =/=  0
40 2cn 9816 . . . . . . . . 9  |-  2  e.  CC
414, 40cjdivi 11676 . . . . . . . 8  |-  ( 2  =/=  0  ->  (
* `  ( _i  /  2 ) )  =  ( ( * `  _i )  /  (
* `  2 )
) )
4239, 41ax-mp 8 . . . . . . 7  |-  ( * `
 ( _i  / 
2 ) )  =  ( ( * `  _i )  /  (
* `  2 )
)
43 divneg 9455 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
_i  /  2 )  =  ( -u _i  /  2 ) )
444, 40, 39, 43mp3an 1277 . . . . . . . 8  |-  -u (
_i  /  2 )  =  ( -u _i  /  2 )
45 cji 11644 . . . . . . . . 9  |-  ( * `
 _i )  = 
-u _i
46 2re 9815 . . . . . . . . . 10  |-  2  e.  RR
47 cjre 11624 . . . . . . . . . 10  |-  ( 2  e.  RR  ->  (
* `  2 )  =  2 )
4846, 47ax-mp 8 . . . . . . . . 9  |-  ( * `
 2 )  =  2
4945, 48oveq12i 5870 . . . . . . . 8  |-  ( ( * `  _i )  /  ( * ` 
2 ) )  =  ( -u _i  / 
2 )
5044, 49eqtr4i 2306 . . . . . . 7  |-  -u (
_i  /  2 )  =  ( ( * `
 _i )  / 
( * `  2
) )
5142, 50eqtr4i 2306 . . . . . 6  |-  ( * `
 ( _i  / 
2 ) )  = 
-u ( _i  / 
2 )
5251oveq1i 5868 . . . . 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
) ) ) ) ) )
5336cjcld 11681 . . . . . 6  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )  e.  CC )
54 mulneg12 9218 . . . . . 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 ) ) ) ) ) ) )
5520, 53, 54sylancr 644 . . . . 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 ) ) ) ) ) ) )
5652, 55syl5eq 2327 . . . 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 ) ) ) ) ) ) )
57 cjsub 11634 . . . . . . . . 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 ) ) ) ) ) )
5830, 35, 57syl2anc 642 . . . . . . . 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 ) ) ) ) ) )
59 imsub 11620 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( Im `  ( 1  -  (
_i  x.  A )
) )  =  ( ( Im `  1
)  -  ( Im
`  ( _i  x.  A ) ) ) )
6021, 23, 59sylancr 644 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  -  ( _i  x.  A ) ) )  =  ( ( Im `  1 )  -  ( Im `  ( _i  x.  A
) ) ) )
61 reim 11594 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  ->  (
Re `  A )  =  ( Im `  ( _i  x.  A
) ) )
6261adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Re `  A
)  =  ( Im
`  ( _i  x.  A ) ) )
6362oveq2d 5874 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( Im ` 
1 )  -  (
Re `  A )
)  =  ( ( Im `  1 )  -  ( Im `  ( _i  x.  A
) ) ) )
6460, 63eqtr4d 2318 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  -  ( _i  x.  A ) ) )  =  ( ( Im `  1 )  -  ( Re `  A ) ) )
65 df-neg 9040 . . . . . . . . . . . . . 14  |-  -u (
Re `  A )  =  ( 0  -  ( Re `  A
) )
66 im1 11640 . . . . . . . . . . . . . . 15  |-  ( Im
`  1 )  =  0
6766oveq1i 5868 . . . . . . . . . . . . . 14  |-  ( ( Im `  1 )  -  ( Re `  A ) )  =  ( 0  -  (
Re `  A )
)
6865, 67eqtr4i 2306 . . . . . . . . . . . . 13  |-  -u (
Re `  A )  =  ( ( Im
`  1 )  -  ( Re `  A ) )
6964, 68syl6eqr 2333 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  -  ( _i  x.  A ) ) )  =  -u (
Re `  A )
)
70 recl 11595 . . . . . . . . . . . . . . 15  |-  ( A  e.  CC  ->  (
Re `  A )  e.  RR )
7170adantr 451 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Re `  A
)  e.  RR )
7271recnd 8861 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Re `  A
)  e.  CC )
7372, 2negne0d 9155 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  -u ( Re `  A
)  =/=  0 )
7469, 73eqnetrd 2464 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  -  ( _i  x.  A ) ) )  =/=  0 )
75 logcj 19960 . . . . . . . . . . 11  |-  ( ( ( 1  -  (
_i  x.  A )
)  e.  CC  /\  ( Im `  ( 1  -  ( _i  x.  A ) ) )  =/=  0 )  -> 
( log `  (
* `  ( 1  -  ( _i  x.  A ) ) ) )  =  ( * `
 ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
7625, 74, 75syl2anc 642 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
* `  ( 1  -  ( _i  x.  A ) ) ) )  =  ( * `
 ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
77 cjsub 11634 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( * `  ( 1  -  (
_i  x.  A )
) )  =  ( ( * `  1
)  -  ( * `
 ( _i  x.  A ) ) ) )
7821, 23, 77sylancr 644 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
1  -  ( _i  x.  A ) ) )  =  ( ( * `  1 )  -  ( * `  ( _i  x.  A
) ) ) )
79 1re 8837 . . . . . . . . . . . . . 14  |-  1  e.  RR
80 cjre 11624 . . . . . . . . . . . . . 14  |-  ( 1  e.  RR  ->  (
* `  1 )  =  1 )
8179, 80mp1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  1
)  =  1 )
82 cjmul 11627 . . . . . . . . . . . . . . 15  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( * `  (
_i  x.  A )
)  =  ( ( * `  _i )  x.  ( * `  A ) ) )
834, 1, 82sylancr 644 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
_i  x.  A )
)  =  ( ( * `  _i )  x.  ( * `  A ) ) )
8445oveq1i 5868 . . . . . . . . . . . . . . 15  |-  ( ( * `  _i )  x.  ( * `  A ) )  =  ( -u _i  x.  ( * `  A
) )
85 cjcl 11590 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
8685adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  A
)  e.  CC )
87 mulneg1 9216 . . . . . . . . . . . . . . . 16  |-  ( ( _i  e.  CC  /\  ( * `  A
)  e.  CC )  ->  ( -u _i  x.  ( * `  A
) )  =  -u ( _i  x.  (
* `  A )
) )
884, 86, 87sylancr 644 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( -u _i  x.  (
* `  A )
)  =  -u (
_i  x.  ( * `  A ) ) )
8984, 88syl5eq 2327 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( * `  _i )  x.  (
* `  A )
)  =  -u (
_i  x.  ( * `  A ) ) )
9083, 89eqtrd 2315 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
_i  x.  A )
)  =  -u (
_i  x.  ( * `  A ) ) )
9181, 90oveq12d 5876 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( * ` 
1 )  -  (
* `  ( _i  x.  A ) ) )  =  ( 1  - 
-u ( _i  x.  ( * `  A
) ) ) )
92 mulcl 8821 . . . . . . . . . . . . . 14  |-  ( ( _i  e.  CC  /\  ( * `  A
)  e.  CC )  ->  ( _i  x.  ( * `  A
) )  e.  CC )
934, 86, 92sylancr 644 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( _i  x.  (
* `  A )
)  e.  CC )
94 subneg 9096 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  (
* `  A )
)  e.  CC )  ->  ( 1  - 
-u ( _i  x.  ( * `  A
) ) )  =  ( 1  +  ( _i  x.  ( * `
 A ) ) ) )
9521, 93, 94sylancr 644 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  -  -u (
_i  x.  ( * `  A ) ) )  =  ( 1  +  ( _i  x.  (
* `  A )
) ) )
9678, 91, 953eqtrd 2319 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
1  -  ( _i  x.  A ) ) )  =  ( 1  +  ( _i  x.  ( * `  A
) ) ) )
9796fveq2d 5529 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
* `  ( 1  -  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  +  ( _i  x.  ( * `
 A ) ) ) ) )
9876, 97eqtr3d 2317 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  ( log `  ( 1  -  ( _i  x.  A
) ) ) )  =  ( log `  (
1  +  ( _i  x.  ( * `  A ) ) ) ) )
99 imadd 11619 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( Im `  ( 1  +  ( _i  x.  A ) ) )  =  ( ( Im `  1
)  +  ( Im
`  ( _i  x.  A ) ) ) )
10021, 23, 99sylancr 644 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  +  ( _i  x.  A ) ) )  =  ( ( Im `  1 )  +  ( Im `  ( _i  x.  A
) ) ) )
10162oveq2d 5874 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 0  +  ( Re `  A ) )  =  ( 0  +  ( Im `  ( _i  x.  A
) ) ) )
10266oveq1i 5868 . . . . . . . . . . . . . 14  |-  ( ( Im `  1 )  +  ( Im `  ( _i  x.  A
) ) )  =  ( 0  +  ( Im `  ( _i  x.  A ) ) )
103101, 102syl6eqr 2333 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 0  +  ( Re `  A ) )  =  ( ( Im `  1 )  +  ( Im `  ( _i  x.  A
) ) ) )
10472addid2d 9013 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 0  +  ( Re `  A ) )  =  ( Re
`  A ) )
105100, 103, 1043eqtr2d 2321 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  +  ( _i  x.  A ) ) )  =  ( Re
`  A ) )
106105, 2eqnetrd 2464 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  +  ( _i  x.  A ) ) )  =/=  0 )
107 logcj 19960 . . . . . . . . . . 11  |-  ( ( ( 1  +  ( _i  x.  A ) )  e.  CC  /\  ( Im `  ( 1  +  ( _i  x.  A ) ) )  =/=  0 )  -> 
( log `  (
* `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( * `
 ( log `  (
1  +  ( _i  x.  A ) ) ) ) )
10832, 106, 107syl2anc 642 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
* `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( * `
 ( log `  (
1  +  ( _i  x.  A ) ) ) ) )
109 cjadd 11626 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( * `  ( 1  +  ( _i  x.  A ) ) )  =  ( ( * `  1
)  +  ( * `
 ( _i  x.  A ) ) ) )
11021, 23, 109sylancr 644 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
1  +  ( _i  x.  A ) ) )  =  ( ( * `  1 )  +  ( * `  ( _i  x.  A
) ) ) )
11181, 90oveq12d 5876 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( * ` 
1 )  +  ( * `  ( _i  x.  A ) ) )  =  ( 1  +  -u ( _i  x.  ( * `  A
) ) ) )
112 negsub 9095 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  (
* `  A )
)  e.  CC )  ->  ( 1  + 
-u ( _i  x.  ( * `  A
) ) )  =  ( 1  -  (
_i  x.  ( * `  A ) ) ) )
11321, 93, 112sylancr 644 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  +  -u ( _i  x.  (
* `  A )
) )  =  ( 1  -  ( _i  x.  ( * `  A ) ) ) )
114110, 111, 1133eqtrd 2319 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
1  +  ( _i  x.  A ) ) )  =  ( 1  -  ( _i  x.  ( * `  A
) ) ) )
115114fveq2d 5529 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
* `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  -  (
_i  x.  ( * `  A ) ) ) ) )
116108, 115eqtr3d 2317 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  ( log `  ( 1  +  ( _i  x.  A
) ) ) )  =  ( log `  (
1  -  ( _i  x.  ( * `  A ) ) ) ) )
11798, 116oveq12d 5876 . . . . . . . 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 ) ) ) ) ) )
11858, 117eqtrd 2315 . . . . . . 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 )
) ) ) ) )
119118negeqd 9046 . . . . . 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 )
) ) ) ) )
120 addcl 8819 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  ( _i  x.  (
* `  A )
)  e.  CC )  ->  ( 1  +  ( _i  x.  (
* `  A )
) )  e.  CC )
12121, 93, 120sylancr 644 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  +  ( _i  x.  ( * `
 A ) ) )  e.  CC )
122 atandmcj 20205 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( * `
 A )  e. 
dom arctan )
12318, 122syl 15 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  A
)  e.  dom arctan )
124 atandm2 20173 . . . . . . . . . 10  |-  ( ( * `  A )  e.  dom arctan  <->  ( ( * `
 A )  e.  CC  /\  ( 1  -  ( _i  x.  ( * `  A
) ) )  =/=  0  /\  ( 1  +  ( _i  x.  ( * `  A
) ) )  =/=  0 ) )
125124simp3bi 972 . . . . . . . . 9  |-  ( ( * `  A )  e.  dom arctan  ->  ( 1  +  ( _i  x.  ( * `  A
) ) )  =/=  0 )
126123, 125syl 15 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  +  ( _i  x.  ( * `
 A ) ) )  =/=  0 )
127 logcl 19926 . . . . . . . 8  |-  ( ( ( 1  +  ( _i  x.  ( * `
 A ) ) )  e.  CC  /\  ( 1  +  ( _i  x.  ( * `
 A ) ) )  =/=  0 )  ->  ( log `  (
1  +  ( _i  x.  ( * `  A ) ) ) )  e.  CC )
128121, 126, 127syl2anc 642 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
1  +  ( _i  x.  ( * `  A ) ) ) )  e.  CC )
129 subcl 9051 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  ( _i  x.  (
* `  A )
)  e.  CC )  ->  ( 1  -  ( _i  x.  (
* `  A )
) )  e.  CC )
13021, 93, 129sylancr 644 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  -  (
_i  x.  ( * `  A ) ) )  e.  CC )
131124simp2bi 971 . . . . . . . . 9  |-  ( ( * `  A )  e.  dom arctan  ->  ( 1  -  ( _i  x.  ( * `  A
) ) )  =/=  0 )
132123, 131syl 15 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  -  (
_i  x.  ( * `  A ) ) )  =/=  0 )
133 logcl 19926 . . . . . . . 8  |-  ( ( ( 1  -  (
_i  x.  ( * `  A ) ) )  e.  CC  /\  (
1  -  ( _i  x.  ( * `  A ) ) )  =/=  0 )  -> 
( log `  (
1  -  ( _i  x.  ( * `  A ) ) ) )  e.  CC )
134130, 132, 133syl2anc 642 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
1  -  ( _i  x.  ( * `  A ) ) ) )  e.  CC )
135128, 134negsubdi2d 9173 . . . . . 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 ) ) ) ) ) )
136119, 135eqtrd 2315 . . . . 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 ) ) ) ) ) )
137136oveq2d 5874 . . . 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 ) ) ) ) ) ) )
13838, 56, 1373eqtrd 2319 . . 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 ) ) ) ) ) ) )
139 atanval 20180 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
14018, 139syl 15 . . . 4  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
(arctan `  A )  =  ( ( _i 
/  2 )  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) )
141140fveq2d 5529 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (arctan `  A ) )  =  ( * `  (
( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) ) )
142 atanval 20180 . . . 4  |-  ( ( * `  A )  e.  dom arctan  ->  (arctan `  ( * `  A
) )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  ( * `  A
) ) ) )  -  ( log `  (
1  +  ( _i  x.  ( * `  A ) ) ) ) ) ) )
143123, 142syl 15 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
(arctan `  ( * `  A ) )  =  ( ( _i  / 
2 )  x.  (
( log `  (
1  -  ( _i  x.  ( * `  A ) ) ) )  -  ( log `  ( 1  +  ( _i  x.  ( * `
 A ) ) ) ) ) ) )
144138, 141, 1433eqtr4d 2325 . 2  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (arctan `  A ) )  =  (arctan `  ( * `  A ) ) )
14518, 144jca 518 1  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( A  e.  dom arctan  /\  ( * `  (arctan `  A ) )  =  (arctan `  ( * `  A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   dom cdm 4689   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   2c2 9795   *ccj 11581   Recre 11582   Imcim 11583   logclog 19912  arctancatan 20160
This theorem is referenced by:  atanrecl  20207
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-atan 20163
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