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Theorem atancj 20752
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 20749 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 445 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  A  e.  CC )
2 simpr 449 . . . 4  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Re `  A
)  =/=  0 )
3 fveq2 5730 . . . . . 6  |-  ( A  =  -u _i  ->  (
Re `  A )  =  ( Re `  -u _i ) )
4 ax-icn 9051 . . . . . . . 8  |-  _i  e.  CC
54renegi 11987 . . . . . . 7  |-  ( Re
`  -u _i )  = 
-u ( Re `  _i )
6 rei 11963 . . . . . . . 8  |-  ( Re
`  _i )  =  0
76negeqi 9301 . . . . . . 7  |-  -u (
Re `  _i )  =  -u 0
8 neg0 9349 . . . . . . 7  |-  -u 0  =  0
95, 7, 83eqtri 2462 . . . . . 6  |-  ( Re
`  -u _i )  =  0
103, 9syl6eq 2486 . . . . 5  |-  ( A  =  -u _i  ->  (
Re `  A )  =  0 )
1110necon3i 2645 . . . 4  |-  ( ( Re `  A )  =/=  0  ->  A  =/=  -u _i )
122, 11syl 16 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  A  =/=  -u _i )
13 fveq2 5730 . . . . . 6  |-  ( A  =  _i  ->  (
Re `  A )  =  ( Re `  _i ) )
1413, 6syl6eq 2486 . . . . 5  |-  ( A  =  _i  ->  (
Re `  A )  =  0 )
1514necon3i 2645 . . . 4  |-  ( ( Re `  A )  =/=  0  ->  A  =/=  _i )
162, 15syl 16 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  A  =/=  _i )
17 atandm 20718 . . 3  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
181, 12, 16, 17syl3anbrc 1139 . 2  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  A  e.  dom arctan )
19 halfcl 10195 . . . . . 6  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
204, 19ax-mp 8 . . . . 5  |-  ( _i 
/  2 )  e.  CC
21 ax-1cn 9050 . . . . . . . 8  |-  1  e.  CC
22 mulcl 9076 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
234, 1, 22sylancr 646 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( _i  x.  A
)  e.  CC )
24 subcl 9307 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  -  ( _i  x.  A
) )  e.  CC )
2521, 23, 24sylancr 646 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  -  (
_i  x.  A )
)  e.  CC )
26 atandm2 20719 . . . . . . . . 9  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
2718, 26sylib 190 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( A  e.  CC  /\  ( 1  -  (
_i  x.  A )
)  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
2827simp2d 971 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  -  (
_i  x.  A )
)  =/=  0 )
2925, 28logcld 20470 . . . . . 6  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
1  -  ( _i  x.  A ) ) )  e.  CC )
30 addcl 9074 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
3121, 23, 30sylancr 646 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  +  ( _i  x.  A ) )  e.  CC )
3227simp3d 972 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  +  ( _i  x.  A ) )  =/=  0 )
3331, 32logcld 20470 . . . . . 6  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
1  +  ( _i  x.  A ) ) )  e.  CC )
3429, 33subcld 9413 . . . . 5  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  e.  CC )
35 cjmul 11949 . . . . 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 646 . . . 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 10085 . . . . . . . 8  |-  2  =/=  0
38 2cn 10072 . . . . . . . . 9  |-  2  e.  CC
394, 38cjdivi 11998 . . . . . . . 8  |-  ( 2  =/=  0  ->  (
* `  ( _i  /  2 ) )  =  ( ( * `  _i )  /  (
* `  2 )
) )
4037, 39ax-mp 8 . . . . . . 7  |-  ( * `
 ( _i  / 
2 ) )  =  ( ( * `  _i )  /  (
* `  2 )
)
41 divneg 9711 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
_i  /  2 )  =  ( -u _i  /  2 ) )
424, 38, 37, 41mp3an 1280 . . . . . . . 8  |-  -u (
_i  /  2 )  =  ( -u _i  /  2 )
43 cji 11966 . . . . . . . . 9  |-  ( * `
 _i )  = 
-u _i
44 2re 10071 . . . . . . . . . 10  |-  2  e.  RR
45 cjre 11946 . . . . . . . . . 10  |-  ( 2  e.  RR  ->  (
* `  2 )  =  2 )
4644, 45ax-mp 8 . . . . . . . . 9  |-  ( * `
 2 )  =  2
4743, 46oveq12i 6095 . . . . . . . 8  |-  ( ( * `  _i )  /  ( * ` 
2 ) )  =  ( -u _i  / 
2 )
4842, 47eqtr4i 2461 . . . . . . 7  |-  -u (
_i  /  2 )  =  ( ( * `
 _i )  / 
( * `  2
) )
4940, 48eqtr4i 2461 . . . . . 6  |-  ( * `
 ( _i  / 
2 ) )  = 
-u ( _i  / 
2 )
5049oveq1i 6093 . . . . 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 12003 . . . . . 6  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )  e.  CC )
52 mulneg12 9474 . . . . . 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 646 . . . . 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 2482 . . . 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 11956 . . . . . . . . 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 644 . . . . . . . 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 11942 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( Im `  ( 1  -  (
_i  x.  A )
) )  =  ( ( Im `  1
)  -  ( Im
`  ( _i  x.  A ) ) ) )
5821, 23, 57sylancr 646 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  -  ( _i  x.  A ) ) )  =  ( ( Im `  1 )  -  ( Im `  ( _i  x.  A
) ) ) )
59 reim 11916 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  ->  (
Re `  A )  =  ( Im `  ( _i  x.  A
) ) )
6059adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Re `  A
)  =  ( Im
`  ( _i  x.  A ) ) )
6160oveq2d 6099 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( Im ` 
1 )  -  (
Re `  A )
)  =  ( ( Im `  1 )  -  ( Im `  ( _i  x.  A
) ) ) )
6258, 61eqtr4d 2473 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  -  ( _i  x.  A ) ) )  =  ( ( Im `  1 )  -  ( Re `  A ) ) )
63 df-neg 9296 . . . . . . . . . . . . . 14  |-  -u (
Re `  A )  =  ( 0  -  ( Re `  A
) )
64 im1 11962 . . . . . . . . . . . . . . 15  |-  ( Im
`  1 )  =  0
6564oveq1i 6093 . . . . . . . . . . . . . 14  |-  ( ( Im `  1 )  -  ( Re `  A ) )  =  ( 0  -  (
Re `  A )
)
6663, 65eqtr4i 2461 . . . . . . . . . . . . 13  |-  -u (
Re `  A )  =  ( ( Im
`  1 )  -  ( Re `  A ) )
6762, 66syl6eqr 2488 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  -  ( _i  x.  A ) ) )  =  -u (
Re `  A )
)
68 recl 11917 . . . . . . . . . . . . . . 15  |-  ( A  e.  CC  ->  (
Re `  A )  e.  RR )
6968adantr 453 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Re `  A
)  e.  RR )
7069recnd 9116 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Re `  A
)  e.  CC )
7170, 2negne0d 9411 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  ->  -u ( Re `  A
)  =/=  0 )
7267, 71eqnetrd 2621 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  -  ( _i  x.  A ) ) )  =/=  0 )
73 logcj 20503 . . . . . . . . . . 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 644 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
* `  ( 1  -  ( _i  x.  A ) ) ) )  =  ( * `
 ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
75 cjsub 11956 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( * `  ( 1  -  (
_i  x.  A )
) )  =  ( ( * `  1
)  -  ( * `
 ( _i  x.  A ) ) ) )
7621, 23, 75sylancr 646 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
1  -  ( _i  x.  A ) ) )  =  ( ( * `  1 )  -  ( * `  ( _i  x.  A
) ) ) )
77 1re 9092 . . . . . . . . . . . . . 14  |-  1  e.  RR
78 cjre 11946 . . . . . . . . . . . . . 14  |-  ( 1  e.  RR  ->  (
* `  1 )  =  1 )
7977, 78mp1i 12 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  1
)  =  1 )
80 cjmul 11949 . . . . . . . . . . . . . . 15  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( * `  (
_i  x.  A )
)  =  ( ( * `  _i )  x.  ( * `  A ) ) )
814, 1, 80sylancr 646 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
_i  x.  A )
)  =  ( ( * `  _i )  x.  ( * `  A ) ) )
8243oveq1i 6093 . . . . . . . . . . . . . . 15  |-  ( ( * `  _i )  x.  ( * `  A ) )  =  ( -u _i  x.  ( * `  A
) )
83 cjcl 11912 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
8483adantr 453 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  A
)  e.  CC )
85 mulneg1 9472 . . . . . . . . . . . . . . . 16  |-  ( ( _i  e.  CC  /\  ( * `  A
)  e.  CC )  ->  ( -u _i  x.  ( * `  A
) )  =  -u ( _i  x.  (
* `  A )
) )
864, 84, 85sylancr 646 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( -u _i  x.  (
* `  A )
)  =  -u (
_i  x.  ( * `  A ) ) )
8782, 86syl5eq 2482 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( * `  _i )  x.  (
* `  A )
)  =  -u (
_i  x.  ( * `  A ) ) )
8881, 87eqtrd 2470 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
_i  x.  A )
)  =  -u (
_i  x.  ( * `  A ) ) )
8979, 88oveq12d 6101 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( * ` 
1 )  -  (
* `  ( _i  x.  A ) ) )  =  ( 1  - 
-u ( _i  x.  ( * `  A
) ) ) )
90 mulcl 9076 . . . . . . . . . . . . . 14  |-  ( ( _i  e.  CC  /\  ( * `  A
)  e.  CC )  ->  ( _i  x.  ( * `  A
) )  e.  CC )
914, 84, 90sylancr 646 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( _i  x.  (
* `  A )
)  e.  CC )
92 subneg 9352 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  (
* `  A )
)  e.  CC )  ->  ( 1  - 
-u ( _i  x.  ( * `  A
) ) )  =  ( 1  +  ( _i  x.  ( * `
 A ) ) ) )
9321, 91, 92sylancr 646 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  -  -u (
_i  x.  ( * `  A ) ) )  =  ( 1  +  ( _i  x.  (
* `  A )
) ) )
9476, 89, 933eqtrd 2474 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
1  -  ( _i  x.  A ) ) )  =  ( 1  +  ( _i  x.  ( * `  A
) ) ) )
9594fveq2d 5734 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
* `  ( 1  -  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  +  ( _i  x.  ( * `
 A ) ) ) ) )
9674, 95eqtr3d 2472 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  ( log `  ( 1  -  ( _i  x.  A
) ) ) )  =  ( log `  (
1  +  ( _i  x.  ( * `  A ) ) ) ) )
97 imadd 11941 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( Im `  ( 1  +  ( _i  x.  A ) ) )  =  ( ( Im `  1
)  +  ( Im
`  ( _i  x.  A ) ) ) )
9821, 23, 97sylancr 646 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  +  ( _i  x.  A ) ) )  =  ( ( Im `  1 )  +  ( Im `  ( _i  x.  A
) ) ) )
9960oveq2d 6099 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 0  +  ( Re `  A ) )  =  ( 0  +  ( Im `  ( _i  x.  A
) ) ) )
10064oveq1i 6093 . . . . . . . . . . . . . 14  |-  ( ( Im `  1 )  +  ( Im `  ( _i  x.  A
) ) )  =  ( 0  +  ( Im `  ( _i  x.  A ) ) )
10199, 100syl6eqr 2488 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 0  +  ( Re `  A ) )  =  ( ( Im `  1 )  +  ( Im `  ( _i  x.  A
) ) ) )
10270addid2d 9269 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 0  +  ( Re `  A ) )  =  ( Re
`  A ) )
10398, 101, 1023eqtr2d 2476 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  +  ( _i  x.  A ) ) )  =  ( Re
`  A ) )
104103, 2eqnetrd 2621 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( Im `  (
1  +  ( _i  x.  A ) ) )  =/=  0 )
105 logcj 20503 . . . . . . . . . . 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 644 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
* `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( * `
 ( log `  (
1  +  ( _i  x.  A ) ) ) ) )
107 cjadd 11948 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( * `  ( 1  +  ( _i  x.  A ) ) )  =  ( ( * `  1
)  +  ( * `
 ( _i  x.  A ) ) ) )
10821, 23, 107sylancr 646 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
1  +  ( _i  x.  A ) ) )  =  ( ( * `  1 )  +  ( * `  ( _i  x.  A
) ) ) )
10979, 88oveq12d 6101 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( ( * ` 
1 )  +  ( * `  ( _i  x.  A ) ) )  =  ( 1  +  -u ( _i  x.  ( * `  A
) ) ) )
110 negsub 9351 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  (
* `  A )
)  e.  CC )  ->  ( 1  + 
-u ( _i  x.  ( * `  A
) ) )  =  ( 1  -  (
_i  x.  ( * `  A ) ) ) )
11121, 91, 110sylancr 646 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  +  -u ( _i  x.  (
* `  A )
) )  =  ( 1  -  ( _i  x.  ( * `  A ) ) ) )
112108, 109, 1113eqtrd 2474 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (
1  +  ( _i  x.  A ) ) )  =  ( 1  -  ( _i  x.  ( * `  A
) ) ) )
113112fveq2d 5734 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
* `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  -  (
_i  x.  ( * `  A ) ) ) ) )
114106, 113eqtr3d 2472 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  ( log `  ( 1  +  ( _i  x.  A
) ) ) )  =  ( log `  (
1  -  ( _i  x.  ( * `  A ) ) ) ) )
11596, 114oveq12d 6101 . . . . . . . 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 2470 . . . . . . 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 9302 . . . . . 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 9074 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  ( _i  x.  (
* `  A )
)  e.  CC )  ->  ( 1  +  ( _i  x.  (
* `  A )
) )  e.  CC )
11921, 91, 118sylancr 646 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  +  ( _i  x.  ( * `
 A ) ) )  e.  CC )
120 atandmcj 20751 . . . . . . . . . 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 20719 . . . . . . . . . 10  |-  ( ( * `  A )  e.  dom arctan  <->  ( ( * `
 A )  e.  CC  /\  ( 1  -  ( _i  x.  ( * `  A
) ) )  =/=  0  /\  ( 1  +  ( _i  x.  ( * `  A
) ) )  =/=  0 ) )
123122simp3bi 975 . . . . . . . . 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 20470 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
1  +  ( _i  x.  ( * `  A ) ) ) )  e.  CC )
126 subcl 9307 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  ( _i  x.  (
* `  A )
)  e.  CC )  ->  ( 1  -  ( _i  x.  (
* `  A )
) )  e.  CC )
12721, 91, 126sylancr 646 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( 1  -  (
_i  x.  ( * `  A ) ) )  e.  CC )
128122simp2bi 974 . . . . . . . . 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 20470 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( log `  (
1  -  ( _i  x.  ( * `  A ) ) ) )  e.  CC )
131125, 130negsubdi2d 9429 . . . . . 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 2470 . . . . 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 6099 . . . 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 2474 . . 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 20726 . . . . 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 5734 . . 3  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (arctan `  A ) )  =  ( * `  (
( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) ) )
138 atanval 20726 . . . 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 2480 . 2  |-  ( ( A  e.  CC  /\  ( Re `  A )  =/=  0 )  -> 
( * `  (arctan `  A ) )  =  (arctan `  ( * `  A ) ) )
14118, 140jca 520 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   dom cdm 4880   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993   _ici 8994    + caddc 8995    x. cmul 8997    - cmin 9293   -ucneg 9294    / cdiv 9679   2c2 10051   *ccj 11903   Recre 11904   Imcim 11905   logclog 20454  arctancatan 20706
This theorem is referenced by:  atanrecl  20753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ioo 10922  df-ioc 10923  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-fac 11569  df-bc 11596  df-hash 11621  df-shft 11884  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-limsup 12267  df-clim 12284  df-rlim 12285  df-sum 12482  df-ef 12672  df-sin 12674  df-cos 12675  df-pi 12677  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-mulg 14817  df-cntz 15118  df-cmn 15416  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-fbas 16701  df-fg 16702  df-cnfld 16706  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cld 17085  df-ntr 17086  df-cls 17087  df-nei 17164  df-lp 17202  df-perf 17203  df-cn 17293  df-cnp 17294  df-haus 17381  df-tx 17596  df-hmeo 17789  df-fil 17880  df-fm 17972  df-flim 17973  df-flf 17974  df-xms 18352  df-ms 18353  df-tms 18354  df-cncf 18910  df-limc 19755  df-dv 19756  df-log 20456  df-atan 20709
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