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Theorem efiatan2 23830
Description: Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 3-Apr-2015.)
Assertion
Ref Expression
efiatan2  |-  ( A  e.  dom arctan  ->  ( exp `  ( _i  x.  (arctan `  A ) ) )  =  ( ( 1  +  ( _i  x.  A ) )  / 
( sqr `  (
1  +  ( A ^ 2 ) ) ) ) )

Proof of Theorem efiatan2
StepHypRef Expression
1 ax-icn 9599 . . . . 5  |-  _i  e.  CC
2 atancl 23794 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  e.  CC )
3 mulcl 9624 . . . . 5  |-  ( ( _i  e.  CC  /\  (arctan `  A )  e.  CC )  ->  (
_i  x.  (arctan `  A
) )  e.  CC )
41, 2, 3sylancr 667 . . . 4  |-  ( A  e.  dom arctan  ->  ( _i  x.  (arctan `  A
) )  e.  CC )
5 efcl 14125 . . . 4  |-  ( ( _i  x.  (arctan `  A ) )  e.  CC  ->  ( exp `  ( _i  x.  (arctan `  A ) ) )  e.  CC )
64, 5syl 17 . . 3  |-  ( A  e.  dom arctan  ->  ( exp `  ( _i  x.  (arctan `  A ) ) )  e.  CC )
7 ax-1cn 9598 . . . . 5  |-  1  e.  CC
8 atandm2 23790 . . . . . . 7  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
98simp1bi 1020 . . . . . 6  |-  ( A  e.  dom arctan  ->  A  e.  CC )
109sqcld 12414 . . . . 5  |-  ( A  e.  dom arctan  ->  ( A ^ 2 )  e.  CC )
11 addcl 9622 . . . . 5  |-  ( ( 1  e.  CC  /\  ( A ^ 2 )  e.  CC )  -> 
( 1  +  ( A ^ 2 ) )  e.  CC )
127, 10, 11sylancr 667 . . . 4  |-  ( A  e.  dom arctan  ->  ( 1  +  ( A ^
2 ) )  e.  CC )
1312sqrtcld 13487 . . 3  |-  ( A  e.  dom arctan  ->  ( sqr `  ( 1  +  ( A ^ 2 ) ) )  e.  CC )
1412sqsqrtd 13489 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( sqr `  ( 1  +  ( A ^
2 ) ) ) ^ 2 )  =  ( 1  +  ( A ^ 2 ) ) )
15 atandm4 23792 . . . . . 6  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  +  ( A ^
2 ) )  =/=  0 ) )
1615simprbi 465 . . . . 5  |-  ( A  e.  dom arctan  ->  ( 1  +  ( A ^
2 ) )  =/=  0 )
1714, 16eqnetrd 2717 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( sqr `  ( 1  +  ( A ^
2 ) ) ) ^ 2 )  =/=  0 )
18 sqne0 12341 . . . . 5  |-  ( ( sqr `  ( 1  +  ( A ^
2 ) ) )  e.  CC  ->  (
( ( sqr `  (
1  +  ( A ^ 2 ) ) ) ^ 2 )  =/=  0  <->  ( sqr `  ( 1  +  ( A ^ 2 ) ) )  =/=  0
) )
1913, 18syl 17 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( ( sqr `  (
1  +  ( A ^ 2 ) ) ) ^ 2 )  =/=  0  <->  ( sqr `  ( 1  +  ( A ^ 2 ) ) )  =/=  0
) )
2017, 19mpbid 213 . . 3  |-  ( A  e.  dom arctan  ->  ( sqr `  ( 1  +  ( A ^ 2 ) ) )  =/=  0
)
216, 13, 20divcan4d 10390 . 2  |-  ( A  e.  dom arctan  ->  ( ( ( exp `  (
_i  x.  (arctan `  A
) ) )  x.  ( sqr `  (
1  +  ( A ^ 2 ) ) ) )  /  ( sqr `  ( 1  +  ( A ^ 2 ) ) ) )  =  ( exp `  (
_i  x.  (arctan `  A
) ) ) )
22 halfcn 10830 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
2312, 16logcld 23507 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  +  ( A ^ 2 ) ) )  e.  CC )
24 mulcl 9624 . . . . . . 7  |-  ( ( ( 1  /  2
)  e.  CC  /\  ( log `  ( 1  +  ( A ^
2 ) ) )  e.  CC )  -> 
( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) )  e.  CC )
2522, 23, 24sylancr 667 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( A ^ 2 ) ) ) )  e.  CC )
26 efadd 14136 . . . . . 6  |-  ( ( ( _i  x.  (arctan `  A ) )  e.  CC  /\  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( A ^ 2 ) ) ) )  e.  CC )  ->  ( exp `  (
( _i  x.  (arctan `  A ) )  +  ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) ) )  =  ( ( exp `  (
_i  x.  (arctan `  A
) ) )  x.  ( exp `  (
( 1  /  2
)  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) ) ) )
274, 25, 26syl2anc 665 . . . . 5  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( _i  x.  (arctan `  A ) )  +  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( A ^ 2 ) ) ) ) ) )  =  ( ( exp `  ( _i  x.  (arctan `  A ) ) )  x.  ( exp `  (
( 1  /  2
)  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) ) ) )
28 2cn 10681 . . . . . . . . . . . 12  |-  2  e.  CC
2928a1i 11 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  2  e.  CC )
30 mulcl 9624 . . . . . . . . . . . . . 14  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
311, 9, 30sylancr 667 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  ( _i  x.  A )  e.  CC )
32 addcl 9622 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
337, 31, 32sylancr 667 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  e.  CC )
348simp3bi 1022 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  =/=  0 )
3533, 34logcld 23507 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )
3629, 35, 4subdid 10075 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  (
_i  x.  (arctan `  A
) ) ) )  =  ( ( 2  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) )  -  (
2  x.  ( _i  x.  (arctan `  A
) ) ) ) )
37 atanval 23797 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
3837oveq2d 6318 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  x.  (arctan `  A
) )  =  ( ( 2  x.  _i )  x.  ( (
_i  /  2 )  x.  ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) ) ) )
391a1i 11 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  _i  e.  CC )
4029, 39, 2mulassd 9667 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  x.  (arctan `  A
) )  =  ( 2  x.  ( _i  x.  (arctan `  A
) ) ) )
41 halfcl 10839 . . . . . . . . . . . . . . . . . 18  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
421, 41ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  ( _i 
/  2 )  e.  CC
4328, 1, 42mulassi 9653 . . . . . . . . . . . . . . . 16  |-  ( ( 2  x.  _i )  x.  ( _i  / 
2 ) )  =  ( 2  x.  (
_i  x.  ( _i  /  2 ) ) )
4428, 1, 42mul12i 9829 . . . . . . . . . . . . . . . 16  |-  ( 2  x.  ( _i  x.  ( _i  /  2
) ) )  =  ( _i  x.  (
2  x.  ( _i 
/  2 ) ) )
45 2ne0 10703 . . . . . . . . . . . . . . . . . . 19  |-  2  =/=  0
461, 28, 45divcan2i 10351 . . . . . . . . . . . . . . . . . 18  |-  ( 2  x.  ( _i  / 
2 ) )  =  _i
4746oveq2i 6313 . . . . . . . . . . . . . . . . 17  |-  ( _i  x.  ( 2  x.  ( _i  /  2
) ) )  =  ( _i  x.  _i )
48 ixi 10242 . . . . . . . . . . . . . . . . 17  |-  ( _i  x.  _i )  = 
-u 1
4947, 48eqtri 2451 . . . . . . . . . . . . . . . 16  |-  ( _i  x.  ( 2  x.  ( _i  /  2
) ) )  = 
-u 1
5043, 44, 493eqtri 2455 . . . . . . . . . . . . . . 15  |-  ( ( 2  x.  _i )  x.  ( _i  / 
2 ) )  = 
-u 1
5150oveq1i 6312 . . . . . . . . . . . . . 14  |-  ( ( ( 2  x.  _i )  x.  ( _i  /  2 ) )  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )  =  ( -u 1  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )
52 subcl 9875 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  -  ( _i  x.  A
) )  e.  CC )
537, 31, 52sylancr 667 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  e.  CC )
548simp2bi 1021 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  =/=  0 )
5553, 54logcld 23507 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  -  (
_i  x.  A )
) )  e.  CC )
5655, 35subcld 9987 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
5756mulm1d 10071 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  ( -u
1  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) )  = 
-u ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )
5851, 57syl5eq 2475 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  x.  ( _i  /  2 ) )  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )  =  -u ( ( log `  ( 1  -  (
_i  x.  A )
) )  -  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )
59 2mulicn 10837 . . . . . . . . . . . . . . 15  |-  ( 2  x.  _i )  e.  CC
6059a1i 11 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  ( 2  x.  _i )  e.  CC )
6142a1i 11 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  ( _i 
/  2 )  e.  CC )
6260, 61, 56mulassd 9667 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  x.  ( _i  /  2 ) )  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )  =  ( ( 2  x.  _i )  x.  ( ( _i  / 
2 )  x.  (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) ) )
6355, 35negsubdi2d 10003 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  -u (
( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( ( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) )
6458, 62, 633eqtr3d 2471 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  x.  ( ( _i 
/  2 )  x.  ( ( log `  (
1  -  ( _i  x.  A ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) ) )  =  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
6538, 40, 643eqtr3d 2471 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( _i  x.  (arctan `  A ) ) )  =  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
6665oveq2d 6318 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  -  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  =  ( ( 2  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) )  -  (
( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )
67 mulcl 9624 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )  -> 
( 2  x.  ( log `  ( 1  +  ( _i  x.  A
) ) ) )  e.  CC )
6828, 35, 67sylancr 667 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
6968, 35, 55subsubd 10015 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  -  ( ( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) )  =  ( ( ( 2  x.  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  +  ( log `  ( 1  -  (
_i  x.  A )
) ) ) )
70352timesd 10856 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) )  =  ( ( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )
7170oveq1d 6317 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( ( ( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )
7235, 35pncand 9988 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  ( ( ( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  +  ( _i  x.  A ) ) ) )
7371, 72eqtrd 2463 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  +  ( _i  x.  A ) ) ) )
7473oveq1d 6317 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  ( log `  ( 1  +  ( _i  x.  A
) ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) )  +  ( log `  ( 1  -  ( _i  x.  A ) ) ) )  =  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  +  ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
75 atanlogadd 23827 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  +  ( log `  (
1  -  ( _i  x.  A ) ) ) )  e.  ran  log )
76 logef 23518 . . . . . . . . . . . . 13  |-  ( ( ( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  -  (
_i  x.  A )
) ) )  e. 
ran  log  ->  ( log `  ( exp `  (
( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )  =  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  +  ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
7775, 76syl 17 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( log `  ( exp `  (
( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )  =  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  +  ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
78 efadd 14136 . . . . . . . . . . . . . . 15  |-  ( ( ( log `  (
1  +  ( _i  x.  A ) ) )  e.  CC  /\  ( log `  ( 1  -  ( _i  x.  A ) ) )  e.  CC )  -> 
( exp `  (
( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  -  (
_i  x.  A )
) ) ) )  =  ( ( exp `  ( log `  (
1  +  ( _i  x.  A ) ) ) )  x.  ( exp `  ( log `  (
1  -  ( _i  x.  A ) ) ) ) ) )
7935, 55, 78syl2anc 665 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  -  (
_i  x.  A )
) ) ) )  =  ( ( exp `  ( log `  (
1  +  ( _i  x.  A ) ) ) )  x.  ( exp `  ( log `  (
1  -  ( _i  x.  A ) ) ) ) ) )
80 eflog 23513 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1  +  ( _i  x.  A ) )  e.  CC  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 )  ->  ( exp `  ( log `  ( 1  +  ( _i  x.  A
) ) ) )  =  ( 1  +  ( _i  x.  A
) ) )
8133, 34, 80syl2anc 665 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( exp `  ( log `  (
1  +  ( _i  x.  A ) ) ) )  =  ( 1  +  ( _i  x.  A ) ) )
82 eflog 23513 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1  -  (
_i  x.  A )
)  e.  CC  /\  ( 1  -  (
_i  x.  A )
)  =/=  0 )  ->  ( exp `  ( log `  ( 1  -  ( _i  x.  A
) ) ) )  =  ( 1  -  ( _i  x.  A
) ) )
8353, 54, 82syl2anc 665 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( exp `  ( log `  (
1  -  ( _i  x.  A ) ) ) )  =  ( 1  -  ( _i  x.  A ) ) )
8481, 83oveq12d 6320 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  x.  ( exp `  ( log `  ( 1  -  ( _i  x.  A
) ) ) ) )  =  ( ( 1  +  ( _i  x.  A ) )  x.  ( 1  -  ( _i  x.  A
) ) ) )
85 sq1 12369 . . . . . . . . . . . . . . . . 17  |-  ( 1 ^ 2 )  =  1
8685a1i 11 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom arctan  ->  ( 1 ^ 2 )  =  1 )
87 sqmul 12338 . . . . . . . . . . . . . . . . . 18  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( ( _i  x.  A ) ^ 2 )  =  ( ( _i ^ 2 )  x.  ( A ^
2 ) ) )
881, 9, 87sylancr 667 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  A ) ^ 2 )  =  ( ( _i ^
2 )  x.  ( A ^ 2 ) ) )
89 i2 12375 . . . . . . . . . . . . . . . . . . 19  |-  ( _i
^ 2 )  = 
-u 1
9089oveq1i 6312 . . . . . . . . . . . . . . . . . 18  |-  ( ( _i ^ 2 )  x.  ( A ^
2 ) )  =  ( -u 1  x.  ( A ^ 2 ) )
9110mulm1d 10071 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  dom arctan  ->  ( -u
1  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
9290, 91syl5eq 2475 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  dom arctan  ->  ( ( _i ^ 2 )  x.  ( A ^
2 ) )  = 
-u ( A ^
2 ) )
9388, 92eqtrd 2463 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  A ) ^ 2 )  = 
-u ( A ^
2 ) )
9486, 93oveq12d 6320 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( ( 1 ^ 2 )  -  ( ( _i  x.  A ) ^
2 ) )  =  ( 1  -  -u ( A ^ 2 ) ) )
95 subsq 12382 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( ( 1 ^ 2 )  -  ( ( _i  x.  A ) ^ 2 ) )  =  ( ( 1  +  ( _i  x.  A ) )  x.  ( 1  -  ( _i  x.  A ) ) ) )
967, 31, 95sylancr 667 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( ( 1 ^ 2 )  -  ( ( _i  x.  A ) ^
2 ) )  =  ( ( 1  +  ( _i  x.  A
) )  x.  (
1  -  ( _i  x.  A ) ) ) )
97 subneg 9924 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  CC  /\  ( A ^ 2 )  e.  CC )  -> 
( 1  -  -u ( A ^ 2 ) )  =  ( 1  +  ( A ^ 2 ) ) )
987, 10, 97sylancr 667 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( 1  -  -u ( A ^
2 ) )  =  ( 1  +  ( A ^ 2 ) ) )
9994, 96, 983eqtr3d 2471 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( _i  x.  A ) )  x.  ( 1  -  ( _i  x.  A
) ) )  =  ( 1  +  ( A ^ 2 ) ) )
10079, 84, 993eqtrd 2467 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  -  (
_i  x.  A )
) ) ) )  =  ( 1  +  ( A ^ 2 ) ) )
101100fveq2d 5882 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( log `  ( exp `  (
( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )  =  ( log `  ( 1  +  ( A ^ 2 ) ) ) )
10277, 101eqtr3d 2465 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  +  ( log `  (
1  -  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  +  ( A ^
2 ) ) ) )
10369, 74, 1023eqtrd 2467 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  -  ( ( log `  (
1  +  ( _i  x.  A ) ) )  -  ( log `  ( 1  -  (
_i  x.  A )
) ) ) )  =  ( log `  (
1  +  ( A ^ 2 ) ) ) )
10436, 66, 1033eqtrd 2467 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  (
_i  x.  (arctan `  A
) ) ) )  =  ( log `  (
1  +  ( A ^ 2 ) ) ) )
105104oveq1d 6317 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( _i  x.  (arctan `  A ) ) ) )  /  2
)  =  ( ( log `  ( 1  +  ( A ^
2 ) ) )  /  2 ) )
10635, 4subcld 9987 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( _i  x.  (arctan `  A ) ) )  e.  CC )
10745a1i 11 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  2  =/=  0 )
108106, 29, 107divcan3d 10389 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( _i  x.  (arctan `  A ) ) ) )  /  2
)  =  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( _i  x.  (arctan `  A ) ) ) )
10923, 29, 107divrec2d 10388 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( A ^
2 ) ) )  /  2 )  =  ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) )
110105, 108, 1093eqtr3d 2471 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( _i  x.  (arctan `  A ) ) )  =  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( A ^ 2 ) ) ) ) )
11135, 4, 25subaddd 10005 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( ( log `  (
1  +  ( _i  x.  A ) ) )  -  ( _i  x.  (arctan `  A
) ) )  =  ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) )  <-> 
( ( _i  x.  (arctan `  A ) )  +  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( A ^ 2 ) ) ) ) )  =  ( log `  (
1  +  ( _i  x.  A ) ) ) ) )
112110, 111mpbid 213 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  (arctan `  A ) )  +  ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) )  =  ( log `  ( 1  +  ( _i  x.  A ) ) ) )
113112fveq2d 5882 . . . . 5  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( _i  x.  (arctan `  A ) )  +  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( A ^ 2 ) ) ) ) ) )  =  ( exp `  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )
11427, 113eqtr3d 2465 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( _i  x.  (arctan `  A
) ) )  x.  ( exp `  (
( 1  /  2
)  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) ) )  =  ( exp `  ( log `  (
1  +  ( _i  x.  A ) ) ) ) )
11522a1i 11 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 1  /  2 )  e.  CC )
11612, 16, 115cxpefd 23644 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( A ^ 2 ) )  ^c  ( 1  /  2 ) )  =  ( exp `  (
( 1  /  2
)  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) ) )
117 cxpsqrt 23635 . . . . . . 7  |-  ( ( 1  +  ( A ^ 2 ) )  e.  CC  ->  (
( 1  +  ( A ^ 2 ) )  ^c  ( 1  /  2 ) )  =  ( sqr `  ( 1  +  ( A ^ 2 ) ) ) )
11812, 117syl 17 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( A ^ 2 ) )  ^c  ( 1  /  2 ) )  =  ( sqr `  (
1  +  ( A ^ 2 ) ) ) )
119116, 118eqtr3d 2465 . . . . 5  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) )  =  ( sqr `  ( 1  +  ( A ^ 2 ) ) ) )
120119oveq2d 6318 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( _i  x.  (arctan `  A
) ) )  x.  ( exp `  (
( 1  /  2
)  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) ) )  =  ( ( exp `  ( _i  x.  (arctan `  A
) ) )  x.  ( sqr `  (
1  +  ( A ^ 2 ) ) ) ) )
121114, 120, 813eqtr3d 2471 . . 3  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( _i  x.  (arctan `  A
) ) )  x.  ( sqr `  (
1  +  ( A ^ 2 ) ) ) )  =  ( 1  +  ( _i  x.  A ) ) )
122121oveq1d 6317 . 2  |-  ( A  e.  dom arctan  ->  ( ( ( exp `  (
_i  x.  (arctan `  A
) ) )  x.  ( sqr `  (
1  +  ( A ^ 2 ) ) ) )  /  ( sqr `  ( 1  +  ( A ^ 2 ) ) ) )  =  ( ( 1  +  ( _i  x.  A ) )  / 
( sqr `  (
1  +  ( A ^ 2 ) ) ) ) )
12321, 122eqtr3d 2465 1  |-  ( A  e.  dom arctan  ->  ( exp `  ( _i  x.  (arctan `  A ) ) )  =  ( ( 1  +  ( _i  x.  A ) )  / 
( sqr `  (
1  +  ( A ^ 2 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1868    =/= wne 2618   dom cdm 4850   ran crn 4851   ` cfv 5598  (class class class)co 6302   CCcc 9538   0cc0 9540   1c1 9541   _ici 9542    + caddc 9543    x. cmul 9545    - cmin 9861   -ucneg 9862    / cdiv 10270   2c2 10660   ^cexp 12272   sqrcsqrt 13285   expce 14102   logclog 23491    ^c ccxp 23492  arctancatan 23777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618  ax-addf 9619  ax-mulf 9620
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-supp 6923  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7887  df-fi 7928  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-5 10672  df-6 10673  df-7 10674  df-8 10675  df-9 10676  df-10 10677  df-n0 10871  df-z 10939  df-dec 11053  df-uz 11161  df-q 11266  df-rp 11304  df-xneg 11410  df-xadd 11411  df-xmul 11412  df-ioo 11640  df-ioc 11641  df-ico 11642  df-icc 11643  df-fz 11786  df-fzo 11917  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13119  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-limsup 13514  df-clim 13540  df-rlim 13541  df-sum 13741  df-ef 14109  df-sin 14111  df-cos 14112  df-pi 14114  df-struct 15111  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-mulr 15192  df-starv 15193  df-sca 15194  df-vsca 15195  df-ip 15196  df-tset 15197  df-ple 15198  df-ds 15200  df-unif 15201  df-hom 15202  df-cco 15203  df-rest 15309  df-topn 15310  df-0g 15328  df-gsum 15329  df-topgen 15330  df-pt 15331  df-prds 15334  df-xrs 15388  df-qtop 15394  df-imas 15395  df-xps 15398  df-mre 15480  df-mrc 15481  df-acs 15483  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-submnd 16571  df-mulg 16664  df-cntz 16959  df-cmn 17420  df-psmet 18950  df-xmet 18951  df-met 18952  df-bl 18953  df-mopn 18954  df-fbas 18955  df-fg 18956  df-cnfld 18959  df-top 19908  df-bases 19909  df-topon 19910  df-topsp 19911  df-cld 20021  df-ntr 20022  df-cls 20023  df-nei 20101  df-lp 20139  df-perf 20140  df-cn 20230  df-cnp 20231  df-haus 20318  df-tx 20564  df-hmeo 20757  df-fil 20848  df-fm 20940  df-flim 20941  df-flf 20942  df-xms 21322  df-ms 21323  df-tms 21324  df-cncf 21897  df-limc 22808  df-dv 22809  df-log 23493  df-cxp 23494  df-atan 23780
This theorem is referenced by:  cosatan  23834
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