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Theorem efiatan2 20762
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 9054 . . . . 5  |-  _i  e.  CC
2 atancl 20726 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  e.  CC )
3 mulcl 9079 . . . . 5  |-  ( ( _i  e.  CC  /\  (arctan `  A )  e.  CC )  ->  (
_i  x.  (arctan `  A
) )  e.  CC )
41, 2, 3sylancr 646 . . . 4  |-  ( A  e.  dom arctan  ->  ( _i  x.  (arctan `  A
) )  e.  CC )
5 efcl 12690 . . . 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 9053 . . . . 5  |-  1  e.  CC
8 atandm2 20722 . . . . . . 7  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
98simp1bi 973 . . . . . 6  |-  ( A  e.  dom arctan  ->  A  e.  CC )
109sqcld 11526 . . . . 5  |-  ( A  e.  dom arctan  ->  ( A ^ 2 )  e.  CC )
11 addcl 9077 . . . . 5  |-  ( ( 1  e.  CC  /\  ( A ^ 2 )  e.  CC )  -> 
( 1  +  ( A ^ 2 ) )  e.  CC )
127, 10, 11sylancr 646 . . . 4  |-  ( A  e.  dom arctan  ->  ( 1  +  ( A ^
2 ) )  e.  CC )
1312sqrcld 12244 . . 3  |-  ( A  e.  dom arctan  ->  ( sqr `  ( 1  +  ( A ^ 2 ) ) )  e.  CC )
1412sqsqrd 12246 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( sqr `  ( 1  +  ( A ^
2 ) ) ) ^ 2 )  =  ( 1  +  ( A ^ 2 ) ) )
15 atandm4 20724 . . . . . 6  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  +  ( A ^
2 ) )  =/=  0 ) )
1615simprbi 452 . . . . 5  |-  ( A  e.  dom arctan  ->  ( 1  +  ( A ^
2 ) )  =/=  0 )
1714, 16eqnetrd 2621 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( sqr `  ( 1  +  ( A ^
2 ) ) ) ^ 2 )  =/=  0 )
18 sqne0 11453 . . . . 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 203 . . 3  |-  ( A  e.  dom arctan  ->  ( sqr `  ( 1  +  ( A ^ 2 ) ) )  =/=  0
)
216, 13, 20divcan4d 9801 . 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 2cn 10075 . . . . . . . 8  |-  2  e.  CC
23 2ne0 10088 . . . . . . . 8  |-  2  =/=  0
2422, 23reccli 9749 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
2512, 16logcld 20473 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  +  ( A ^ 2 ) ) )  e.  CC )
26 mulcl 9079 . . . . . . 7  |-  ( ( ( 1  /  2
)  e.  CC  /\  ( log `  ( 1  +  ( A ^
2 ) ) )  e.  CC )  -> 
( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) )  e.  CC )
2724, 25, 26sylancr 646 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( A ^ 2 ) ) ) )  e.  CC )
28 efadd 12701 . . . . . 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 ) ) ) ) ) ) )
294, 27, 28syl2anc 644 . . . . 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 ) ) ) ) ) ) )
3022a1i 11 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  2  e.  CC )
31 mulcl 9079 . . . . . . . . . . . . . 14  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
321, 9, 31sylancr 646 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  ( _i  x.  A )  e.  CC )
33 addcl 9077 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
347, 32, 33sylancr 646 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  e.  CC )
358simp3bi 975 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  =/=  0 )
3634, 35logcld 20473 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )
3730, 36, 4subdid 9494 . . . . . . . . . 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
) ) ) ) )
38 atanval 20729 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
3938oveq2d 6100 . . . . . . . . . . . 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
) ) ) ) ) ) )
401a1i 11 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  _i  e.  CC )
4130, 40, 2mulassd 9116 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  x.  (arctan `  A
) )  =  ( 2  x.  ( _i  x.  (arctan `  A
) ) ) )
42 halfcl 10198 . . . . . . . . . . . . . . . . . 18  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
431, 42ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  ( _i 
/  2 )  e.  CC
4422, 1, 43mulassi 9104 . . . . . . . . . . . . . . . 16  |-  ( ( 2  x.  _i )  x.  ( _i  / 
2 ) )  =  ( 2  x.  (
_i  x.  ( _i  /  2 ) ) )
4522, 1, 43mul12i 9266 . . . . . . . . . . . . . . . 16  |-  ( 2  x.  ( _i  x.  ( _i  /  2
) ) )  =  ( _i  x.  (
2  x.  ( _i 
/  2 ) ) )
461, 22, 23divcan2i 9762 . . . . . . . . . . . . . . . . . 18  |-  ( 2  x.  ( _i  / 
2 ) )  =  _i
4746oveq2i 6095 . . . . . . . . . . . . . . . . 17  |-  ( _i  x.  ( 2  x.  ( _i  /  2
) ) )  =  ( _i  x.  _i )
48 ixi 9656 . . . . . . . . . . . . . . . . 17  |-  ( _i  x.  _i )  = 
-u 1
4947, 48eqtri 2458 . . . . . . . . . . . . . . . 16  |-  ( _i  x.  ( 2  x.  ( _i  /  2
) ) )  = 
-u 1
5044, 45, 493eqtri 2462 . . . . . . . . . . . . . . 15  |-  ( ( 2  x.  _i )  x.  ( _i  / 
2 ) )  = 
-u 1
5150oveq1i 6094 . . . . . . . . . . . . . 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 9310 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  -  ( _i  x.  A
) )  e.  CC )
537, 32, 52sylancr 646 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  e.  CC )
548simp2bi 974 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  =/=  0 )
5553, 54logcld 20473 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  -  (
_i  x.  A )
) )  e.  CC )
5655, 36subcld 9416 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
5756mulm1d 9490 . . . . . . . . . . . . . 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 2482 . . . . . . . . . . . . 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
) ) ) ) )
5922, 1mulcli 9100 . . . . . . . . . . . . . . 15  |-  ( 2  x.  _i )  e.  CC
6059a1i 11 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  ( 2  x.  _i )  e.  CC )
6143a1i 11 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  ( _i 
/  2 )  e.  CC )
6260, 61, 56mulassd 9116 . . . . . . . . . . . . 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, 36negsubdi2d 9432 . . . . . . . . . . . . 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 2478 . . . . . . . . . . . 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 ) ) ) ) )
6539, 41, 643eqtr3d 2478 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( _i  x.  (arctan `  A ) ) )  =  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
6665oveq2d 6100 . . . . . . . . . 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 9079 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )  -> 
( 2  x.  ( log `  ( 1  +  ( _i  x.  A
) ) ) )  e.  CC )
6822, 36, 67sylancr 646 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
6968, 36, 55subsubd 9444 . . . . . . . . . . 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 )
) ) ) )
70362timesd 10215 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) )  =  ( ( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )
7170oveq1d 6099 . . . . . . . . . . . . 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 ) ) ) ) )
7236, 36pncand 9417 . . . . . . . . . . . . 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 2470 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  +  ( _i  x.  A ) ) ) )
7473oveq1d 6099 . . . . . . . . . . 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 20759 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  +  ( log `  (
1  -  ( _i  x.  A ) ) ) )  e.  ran  log )
76 logef 20481 . . . . . . . . . . . . 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 12701 . . . . . . . . . . . . . . 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 ) ) ) ) ) )
7936, 55, 78syl2anc 644 . . . . . . . . . . . . . 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 20479 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1  +  ( _i  x.  A ) )  e.  CC  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 )  ->  ( exp `  ( log `  ( 1  +  ( _i  x.  A
) ) ) )  =  ( 1  +  ( _i  x.  A
) ) )
8134, 35, 80syl2anc 644 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( exp `  ( log `  (
1  +  ( _i  x.  A ) ) ) )  =  ( 1  +  ( _i  x.  A ) ) )
82 eflog 20479 . . . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( exp `  ( log `  (
1  -  ( _i  x.  A ) ) ) )  =  ( 1  -  ( _i  x.  A ) ) )
8481, 83oveq12d 6102 . . . . . . . . . . . . . 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 11481 . . . . . . . . . . . . . . . . 17  |-  ( 1 ^ 2 )  =  1
8685a1i 11 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom arctan  ->  ( 1 ^ 2 )  =  1 )
87 sqmul 11450 . . . . . . . . . . . . . . . . . 18  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( ( _i  x.  A ) ^ 2 )  =  ( ( _i ^ 2 )  x.  ( A ^
2 ) ) )
881, 9, 87sylancr 646 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  A ) ^ 2 )  =  ( ( _i ^
2 )  x.  ( A ^ 2 ) ) )
89 i2 11486 . . . . . . . . . . . . . . . . . . 19  |-  ( _i
^ 2 )  = 
-u 1
9089oveq1i 6094 . . . . . . . . . . . . . . . . . 18  |-  ( ( _i ^ 2 )  x.  ( A ^
2 ) )  =  ( -u 1  x.  ( A ^ 2 ) )
9110mulm1d 9490 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  dom arctan  ->  ( -u
1  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
9290, 91syl5eq 2482 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  dom arctan  ->  ( ( _i ^ 2 )  x.  ( A ^
2 ) )  = 
-u ( A ^
2 ) )
9388, 92eqtrd 2470 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  A ) ^ 2 )  = 
-u ( A ^
2 ) )
9486, 93oveq12d 6102 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( ( 1 ^ 2 )  -  ( ( _i  x.  A ) ^
2 ) )  =  ( 1  -  -u ( A ^ 2 ) ) )
95 subsq 11493 . . . . . . . . . . . . . . . 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, 32, 95sylancr 646 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( ( 1 ^ 2 )  -  ( ( _i  x.  A ) ^
2 ) )  =  ( ( 1  +  ( _i  x.  A
) )  x.  (
1  -  ( _i  x.  A ) ) ) )
97 subneg 9355 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  CC  /\  ( A ^ 2 )  e.  CC )  -> 
( 1  -  -u ( A ^ 2 ) )  =  ( 1  +  ( A ^ 2 ) ) )
987, 10, 97sylancr 646 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( 1  -  -u ( A ^
2 ) )  =  ( 1  +  ( A ^ 2 ) ) )
9994, 96, 983eqtr3d 2478 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( _i  x.  A ) )  x.  ( 1  -  ( _i  x.  A
) ) )  =  ( 1  +  ( A ^ 2 ) ) )
10079, 84, 993eqtrd 2474 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  -  (
_i  x.  A )
) ) ) )  =  ( 1  +  ( A ^ 2 ) ) )
101100fveq2d 5735 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( log `  ( exp `  (
( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )  =  ( log `  ( 1  +  ( A ^ 2 ) ) ) )
10277, 101eqtr3d 2472 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  +  ( log `  (
1  -  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  +  ( A ^
2 ) ) ) )
10369, 74, 1023eqtrd 2474 . . . . . . . . . 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 ) ) ) )
10437, 66, 1033eqtrd 2474 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  (
_i  x.  (arctan `  A
) ) ) )  =  ( log `  (
1  +  ( A ^ 2 ) ) ) )
105104oveq1d 6099 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( _i  x.  (arctan `  A ) ) ) )  /  2
)  =  ( ( log `  ( 1  +  ( A ^
2 ) ) )  /  2 ) )
10636, 4subcld 9416 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( _i  x.  (arctan `  A ) ) )  e.  CC )
10723a1i 11 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  2  =/=  0 )
108106, 30, 107divcan3d 9800 . . . . . . . 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 ) ) ) )
10925, 30, 107divrec2d 9799 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( A ^
2 ) ) )  /  2 )  =  ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) )
110105, 108, 1093eqtr3d 2478 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( _i  x.  (arctan `  A ) ) )  =  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( A ^ 2 ) ) ) ) )
11136, 4, 27subaddd 9434 . . . . . . 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 203 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  (arctan `  A ) )  +  ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) )  =  ( log `  ( 1  +  ( _i  x.  A ) ) ) )
113112fveq2d 5735 . . . . 5  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( _i  x.  (arctan `  A ) )  +  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( A ^ 2 ) ) ) ) ) )  =  ( exp `  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )
11429, 113eqtr3d 2472 . . . 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 ) ) ) ) )
11524a1i 11 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 1  /  2 )  e.  CC )
11612, 16, 115cxpefd 20608 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( A ^ 2 ) )  ^ c  ( 1  /  2 ) )  =  ( exp `  (
( 1  /  2
)  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) ) )
117 cxpsqr 20599 . . . . . . 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 2472 . . . . 5  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) )  =  ( sqr `  ( 1  +  ( A ^ 2 ) ) ) )
120119oveq2d 6100 . . . 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 2478 . . 3  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( _i  x.  (arctan `  A
) ) )  x.  ( sqr `  (
1  +  ( A ^ 2 ) ) ) )  =  ( 1  +  ( _i  x.  A ) ) )
122121oveq1d 6099 . 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 2472 1  |-  ( A  e.  dom arctan  ->  ( exp `  ( _i  x.  (arctan `  A ) ) )  =  ( ( 1  +  ( _i  x.  A ) )  / 
( sqr `  (
1  +  ( A ^ 2 ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726    =/= wne 2601   dom cdm 4881   ran crn 4882   ` cfv 5457  (class class class)co 6084   CCcc 8993   0cc0 8995   1c1 8996   _ici 8997    + caddc 8998    x. cmul 9000    - cmin 9296   -ucneg 9297    / cdiv 9682   2c2 10054   ^cexp 11387   sqrcsqr 12043   expce 12669   logclog 20457    ^ c ccxp 20458  arctancatan 20709
This theorem is referenced by:  cosatan  20766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
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 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-ioc 10926  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-fl 11207  df-mod 11256  df-seq 11329  df-exp 11388  df-fac 11572  df-bc 11599  df-hash 11624  df-shft 11887  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-limsup 12270  df-clim 12287  df-rlim 12288  df-sum 12485  df-ef 12675  df-sin 12677  df-cos 12678  df-pi 12680  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-submnd 14744  df-mulg 14820  df-cntz 15121  df-cmn 15419  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-fbas 16704  df-fg 16705  df-cnfld 16709  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-lp 17205  df-perf 17206  df-cn 17296  df-cnp 17297  df-haus 17384  df-tx 17599  df-hmeo 17792  df-fil 17883  df-fm 17975  df-flim 17976  df-flf 17977  df-xms 18355  df-ms 18356  df-tms 18357  df-cncf 18913  df-limc 19758  df-dv 19759  df-log 20459  df-cxp 20460  df-atan 20712
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