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Theorem efiatan2 23448
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 9540 . . . . 5  |-  _i  e.  CC
2 atancl 23412 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  e.  CC )
3 mulcl 9565 . . . . 5  |-  ( ( _i  e.  CC  /\  (arctan `  A )  e.  CC )  ->  (
_i  x.  (arctan `  A
) )  e.  CC )
41, 2, 3sylancr 661 . . . 4  |-  ( A  e.  dom arctan  ->  ( _i  x.  (arctan `  A
) )  e.  CC )
5 efcl 13903 . . . 4  |-  ( ( _i  x.  (arctan `  A ) )  e.  CC  ->  ( exp `  ( _i  x.  (arctan `  A ) ) )  e.  CC )
64, 5syl 16 . . 3  |-  ( A  e.  dom arctan  ->  ( exp `  ( _i  x.  (arctan `  A ) ) )  e.  CC )
7 ax-1cn 9539 . . . . 5  |-  1  e.  CC
8 atandm2 23408 . . . . . . 7  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
98simp1bi 1009 . . . . . 6  |-  ( A  e.  dom arctan  ->  A  e.  CC )
109sqcld 12293 . . . . 5  |-  ( A  e.  dom arctan  ->  ( A ^ 2 )  e.  CC )
11 addcl 9563 . . . . 5  |-  ( ( 1  e.  CC  /\  ( A ^ 2 )  e.  CC )  -> 
( 1  +  ( A ^ 2 ) )  e.  CC )
127, 10, 11sylancr 661 . . . 4  |-  ( A  e.  dom arctan  ->  ( 1  +  ( A ^
2 ) )  e.  CC )
1312sqrtcld 13353 . . 3  |-  ( A  e.  dom arctan  ->  ( sqr `  ( 1  +  ( A ^ 2 ) ) )  e.  CC )
1412sqsqrtd 13355 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( sqr `  ( 1  +  ( A ^
2 ) ) ) ^ 2 )  =  ( 1  +  ( A ^ 2 ) ) )
15 atandm4 23410 . . . . . 6  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  +  ( A ^
2 ) )  =/=  0 ) )
1615simprbi 462 . . . . 5  |-  ( A  e.  dom arctan  ->  ( 1  +  ( A ^
2 ) )  =/=  0 )
1714, 16eqnetrd 2747 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( sqr `  ( 1  +  ( A ^
2 ) ) ) ^ 2 )  =/=  0 )
18 sqne0 12219 . . . . 5  |-  ( ( sqr `  ( 1  +  ( A ^
2 ) ) )  e.  CC  ->  (
( ( sqr `  (
1  +  ( A ^ 2 ) ) ) ^ 2 )  =/=  0  <->  ( sqr `  ( 1  +  ( A ^ 2 ) ) )  =/=  0
) )
1913, 18syl 16 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( ( sqr `  (
1  +  ( A ^ 2 ) ) ) ^ 2 )  =/=  0  <->  ( sqr `  ( 1  +  ( A ^ 2 ) ) )  =/=  0
) )
2017, 19mpbid 210 . . 3  |-  ( A  e.  dom arctan  ->  ( sqr `  ( 1  +  ( A ^ 2 ) ) )  =/=  0
)
216, 13, 20divcan4d 10322 . 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 10751 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
2312, 16logcld 23127 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  +  ( A ^ 2 ) ) )  e.  CC )
24 mulcl 9565 . . . . . . 7  |-  ( ( ( 1  /  2
)  e.  CC  /\  ( log `  ( 1  +  ( A ^
2 ) ) )  e.  CC )  -> 
( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) )  e.  CC )
2522, 23, 24sylancr 661 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( A ^ 2 ) ) ) )  e.  CC )
26 efadd 13914 . . . . . 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 659 . . . . 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 10602 . . . . . . . . . . . 12  |-  2  e.  CC
2928a1i 11 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  2  e.  CC )
30 mulcl 9565 . . . . . . . . . . . . . 14  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
311, 9, 30sylancr 661 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  ( _i  x.  A )  e.  CC )
32 addcl 9563 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
337, 31, 32sylancr 661 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  e.  CC )
348simp3bi 1011 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  =/=  0 )
3533, 34logcld 23127 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )
3629, 35, 4subdid 10008 . . . . . . . . . 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 23415 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
3837oveq2d 6286 . . . . . . . . . . . 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 9608 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  x.  (arctan `  A
) )  =  ( 2  x.  ( _i  x.  (arctan `  A
) ) ) )
41 halfcl 10760 . . . . . . . . . . . . . . . . . 18  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
421, 41ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  ( _i 
/  2 )  e.  CC
4328, 1, 42mulassi 9594 . . . . . . . . . . . . . . . 16  |-  ( ( 2  x.  _i )  x.  ( _i  / 
2 ) )  =  ( 2  x.  (
_i  x.  ( _i  /  2 ) ) )
4428, 1, 42mul12i 9764 . . . . . . . . . . . . . . . 16  |-  ( 2  x.  ( _i  x.  ( _i  /  2
) ) )  =  ( _i  x.  (
2  x.  ( _i 
/  2 ) ) )
45 2ne0 10624 . . . . . . . . . . . . . . . . . . 19  |-  2  =/=  0
461, 28, 45divcan2i 10283 . . . . . . . . . . . . . . . . . 18  |-  ( 2  x.  ( _i  / 
2 ) )  =  _i
4746oveq2i 6281 . . . . . . . . . . . . . . . . 17  |-  ( _i  x.  ( 2  x.  ( _i  /  2
) ) )  =  ( _i  x.  _i )
48 ixi 10174 . . . . . . . . . . . . . . . . 17  |-  ( _i  x.  _i )  = 
-u 1
4947, 48eqtri 2483 . . . . . . . . . . . . . . . 16  |-  ( _i  x.  ( 2  x.  ( _i  /  2
) ) )  = 
-u 1
5043, 44, 493eqtri 2487 . . . . . . . . . . . . . . 15  |-  ( ( 2  x.  _i )  x.  ( _i  / 
2 ) )  = 
-u 1
5150oveq1i 6280 . . . . . . . . . . . . . 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 9810 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  -  ( _i  x.  A
) )  e.  CC )
537, 31, 52sylancr 661 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  e.  CC )
548simp2bi 1010 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  =/=  0 )
5553, 54logcld 23127 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  -  (
_i  x.  A )
) )  e.  CC )
5655, 35subcld 9922 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
5756mulm1d 10004 . . . . . . . . . . . . . 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 2507 . . . . . . . . . . . . 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 10758 . . . . . . . . . . . . . . 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 9608 . . . . . . . . . . . . 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 9938 . . . . . . . . . . . . 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 2503 . . . . . . . . . . . 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 2503 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( _i  x.  (arctan `  A ) ) )  =  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
6665oveq2d 6286 . . . . . . . . . 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 9565 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )  -> 
( 2  x.  ( log `  ( 1  +  ( _i  x.  A
) ) ) )  e.  CC )
6828, 35, 67sylancr 661 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
6968, 35, 55subsubd 9950 . . . . . . . . . . 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 10777 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) )  =  ( ( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )
7170oveq1d 6285 . . . . . . . . . . . . 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 9923 . . . . . . . . . . . . 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 2495 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  +  ( _i  x.  A ) ) ) )
7473oveq1d 6285 . . . . . . . . . . 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 23445 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  +  ( log `  (
1  -  ( _i  x.  A ) ) ) )  e.  ran  log )
76 logef 23138 . . . . . . . . . . . . 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 16 . . . . . . . . . . . 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 13914 . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . 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 23133 . . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( exp `  ( log `  (
1  +  ( _i  x.  A ) ) ) )  =  ( 1  +  ( _i  x.  A ) ) )
82 eflog 23133 . . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( exp `  ( log `  (
1  -  ( _i  x.  A ) ) ) )  =  ( 1  -  ( _i  x.  A ) ) )
8481, 83oveq12d 6288 . . . . . . . . . . . . . 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 12247 . . . . . . . . . . . . . . . . 17  |-  ( 1 ^ 2 )  =  1
8685a1i 11 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom arctan  ->  ( 1 ^ 2 )  =  1 )
87 sqmul 12216 . . . . . . . . . . . . . . . . . 18  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( ( _i  x.  A ) ^ 2 )  =  ( ( _i ^ 2 )  x.  ( A ^
2 ) ) )
881, 9, 87sylancr 661 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  A ) ^ 2 )  =  ( ( _i ^
2 )  x.  ( A ^ 2 ) ) )
89 i2 12253 . . . . . . . . . . . . . . . . . . 19  |-  ( _i
^ 2 )  = 
-u 1
9089oveq1i 6280 . . . . . . . . . . . . . . . . . 18  |-  ( ( _i ^ 2 )  x.  ( A ^
2 ) )  =  ( -u 1  x.  ( A ^ 2 ) )
9110mulm1d 10004 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  dom arctan  ->  ( -u
1  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
9290, 91syl5eq 2507 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  dom arctan  ->  ( ( _i ^ 2 )  x.  ( A ^
2 ) )  = 
-u ( A ^
2 ) )
9388, 92eqtrd 2495 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  A ) ^ 2 )  = 
-u ( A ^
2 ) )
9486, 93oveq12d 6288 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( ( 1 ^ 2 )  -  ( ( _i  x.  A ) ^
2 ) )  =  ( 1  -  -u ( A ^ 2 ) ) )
95 subsq 12260 . . . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( ( 1 ^ 2 )  -  ( ( _i  x.  A ) ^
2 ) )  =  ( ( 1  +  ( _i  x.  A
) )  x.  (
1  -  ( _i  x.  A ) ) ) )
97 subneg 9859 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  CC  /\  ( A ^ 2 )  e.  CC )  -> 
( 1  -  -u ( A ^ 2 ) )  =  ( 1  +  ( A ^ 2 ) ) )
987, 10, 97sylancr 661 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( 1  -  -u ( A ^
2 ) )  =  ( 1  +  ( A ^ 2 ) ) )
9994, 96, 983eqtr3d 2503 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( _i  x.  A ) )  x.  ( 1  -  ( _i  x.  A
) ) )  =  ( 1  +  ( A ^ 2 ) ) )
10079, 84, 993eqtrd 2499 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  -  (
_i  x.  A )
) ) ) )  =  ( 1  +  ( A ^ 2 ) ) )
101100fveq2d 5852 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( log `  ( exp `  (
( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )  =  ( log `  ( 1  +  ( A ^ 2 ) ) ) )
10277, 101eqtr3d 2497 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  +  ( log `  (
1  -  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  +  ( A ^
2 ) ) ) )
10369, 74, 1023eqtrd 2499 . . . . . . . . . 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 2499 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  (
_i  x.  (arctan `  A
) ) ) )  =  ( log `  (
1  +  ( A ^ 2 ) ) ) )
105104oveq1d 6285 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( _i  x.  (arctan `  A ) ) ) )  /  2
)  =  ( ( log `  ( 1  +  ( A ^
2 ) ) )  /  2 ) )
10635, 4subcld 9922 . . . . . . . . 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 10321 . . . . . . . 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 10320 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( A ^
2 ) ) )  /  2 )  =  ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) )
110105, 108, 1093eqtr3d 2503 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( _i  x.  (arctan `  A ) ) )  =  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( A ^ 2 ) ) ) ) )
11135, 4, 25subaddd 9940 . . . . . . 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 210 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  (arctan `  A ) )  +  ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) )  =  ( log `  ( 1  +  ( _i  x.  A ) ) ) )
113112fveq2d 5852 . . . . 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 2497 . . . 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 23264 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( A ^ 2 ) )  ^c  ( 1  /  2 ) )  =  ( exp `  (
( 1  /  2
)  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) ) )
117 cxpsqrt 23255 . . . . . . 7  |-  ( ( 1  +  ( A ^ 2 ) )  e.  CC  ->  (
( 1  +  ( A ^ 2 ) )  ^c  ( 1  /  2 ) )  =  ( sqr `  ( 1  +  ( A ^ 2 ) ) ) )
11812, 117syl 16 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( A ^ 2 ) )  ^c  ( 1  /  2 ) )  =  ( sqr `  (
1  +  ( A ^ 2 ) ) ) )
119116, 118eqtr3d 2497 . . . . 5  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) )  =  ( sqr `  ( 1  +  ( A ^ 2 ) ) ) )
120119oveq2d 6286 . . . 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 2503 . . 3  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( _i  x.  (arctan `  A
) ) )  x.  ( sqr `  (
1  +  ( A ^ 2 ) ) ) )  =  ( 1  +  ( _i  x.  A ) ) )
122121oveq1d 6285 . 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 2497 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 184    = wceq 1398    e. wcel 1823    =/= wne 2649   dom cdm 4988   ran crn 4989   ` cfv 5570  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482   _ici 9483    + caddc 9484    x. cmul 9486    - cmin 9796   -ucneg 9797    / cdiv 10202   2c2 10581   ^cexp 12151   sqrcsqrt 13151   expce 13882   logclog 23111    ^c ccxp 23112  arctancatan 23395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  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 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12093  df-exp 12152  df-fac 12339  df-bc 12366  df-hash 12391  df-shft 12985  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-limsup 13379  df-clim 13396  df-rlim 13397  df-sum 13594  df-ef 13888  df-sin 13890  df-cos 13891  df-pi 13893  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-starv 14802  df-sca 14803  df-vsca 14804  df-ip 14805  df-tset 14806  df-ple 14807  df-ds 14809  df-unif 14810  df-hom 14811  df-cco 14812  df-rest 14915  df-topn 14916  df-0g 14934  df-gsum 14935  df-topgen 14936  df-pt 14937  df-prds 14940  df-xrs 14994  df-qtop 14999  df-imas 15000  df-xps 15002  df-mre 15078  df-mrc 15079  df-acs 15081  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-mulg 16262  df-cntz 16557  df-cmn 17002  df-psmet 18609  df-xmet 18610  df-met 18611  df-bl 18612  df-mopn 18613  df-fbas 18614  df-fg 18615  df-cnfld 18619  df-top 19569  df-bases 19571  df-topon 19572  df-topsp 19573  df-cld 19690  df-ntr 19691  df-cls 19692  df-nei 19769  df-lp 19807  df-perf 19808  df-cn 19898  df-cnp 19899  df-haus 19986  df-tx 20232  df-hmeo 20425  df-fil 20516  df-fm 20608  df-flim 20609  df-flf 20610  df-xms 20992  df-ms 20993  df-tms 20994  df-cncf 21551  df-limc 22439  df-dv 22440  df-log 23113  df-cxp 23114  df-atan 23398
This theorem is referenced by:  cosatan  23452
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