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Theorem efiatan2 20213
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 8796 . . . . 5  |-  _i  e.  CC
2 atancl 20177 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  e.  CC )
3 mulcl 8821 . . . . 5  |-  ( ( _i  e.  CC  /\  (arctan `  A )  e.  CC )  ->  (
_i  x.  (arctan `  A
) )  e.  CC )
41, 2, 3sylancr 644 . . . 4  |-  ( A  e.  dom arctan  ->  ( _i  x.  (arctan `  A
) )  e.  CC )
5 efcl 12364 . . . 4  |-  ( ( _i  x.  (arctan `  A ) )  e.  CC  ->  ( exp `  ( _i  x.  (arctan `  A ) ) )  e.  CC )
64, 5syl 15 . . 3  |-  ( A  e.  dom arctan  ->  ( exp `  ( _i  x.  (arctan `  A ) ) )  e.  CC )
7 ax-1cn 8795 . . . . 5  |-  1  e.  CC
8 atandm2 20173 . . . . . . 7  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
98simp1bi 970 . . . . . 6  |-  ( A  e.  dom arctan  ->  A  e.  CC )
109sqcld 11243 . . . . 5  |-  ( A  e.  dom arctan  ->  ( A ^ 2 )  e.  CC )
11 addcl 8819 . . . . 5  |-  ( ( 1  e.  CC  /\  ( A ^ 2 )  e.  CC )  -> 
( 1  +  ( A ^ 2 ) )  e.  CC )
127, 10, 11sylancr 644 . . . 4  |-  ( A  e.  dom arctan  ->  ( 1  +  ( A ^
2 ) )  e.  CC )
1312sqrcld 11919 . . 3  |-  ( A  e.  dom arctan  ->  ( sqr `  ( 1  +  ( A ^ 2 ) ) )  e.  CC )
1412sqsqrd 11921 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( sqr `  ( 1  +  ( A ^
2 ) ) ) ^ 2 )  =  ( 1  +  ( A ^ 2 ) ) )
15 atandm4 20175 . . . . . 6  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  +  ( A ^
2 ) )  =/=  0 ) )
1615simprbi 450 . . . . 5  |-  ( A  e.  dom arctan  ->  ( 1  +  ( A ^
2 ) )  =/=  0 )
1714, 16eqnetrd 2464 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( sqr `  ( 1  +  ( A ^
2 ) ) ) ^ 2 )  =/=  0 )
18 sqne0 11170 . . . . 5  |-  ( ( sqr `  ( 1  +  ( A ^
2 ) ) )  e.  CC  ->  (
( ( sqr `  (
1  +  ( A ^ 2 ) ) ) ^ 2 )  =/=  0  <->  ( sqr `  ( 1  +  ( A ^ 2 ) ) )  =/=  0
) )
1913, 18syl 15 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( ( sqr `  (
1  +  ( A ^ 2 ) ) ) ^ 2 )  =/=  0  <->  ( sqr `  ( 1  +  ( A ^ 2 ) ) )  =/=  0
) )
2017, 19mpbid 201 . . 3  |-  ( A  e.  dom arctan  ->  ( sqr `  ( 1  +  ( A ^ 2 ) ) )  =/=  0
)
216, 13, 20divcan4d 9542 . 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 9816 . . . . . . . 8  |-  2  e.  CC
23 2ne0 9829 . . . . . . . 8  |-  2  =/=  0
2422, 23reccli 9490 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
25 logcl 19926 . . . . . . . 8  |-  ( ( ( 1  +  ( A ^ 2 ) )  e.  CC  /\  ( 1  +  ( A ^ 2 ) )  =/=  0 )  ->  ( log `  (
1  +  ( A ^ 2 ) ) )  e.  CC )
2612, 16, 25syl2anc 642 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  +  ( A ^ 2 ) ) )  e.  CC )
27 mulcl 8821 . . . . . . 7  |-  ( ( ( 1  /  2
)  e.  CC  /\  ( log `  ( 1  +  ( A ^
2 ) ) )  e.  CC )  -> 
( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) )  e.  CC )
2824, 26, 27sylancr 644 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( A ^ 2 ) ) ) )  e.  CC )
29 efadd 12375 . . . . . 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 ) ) ) ) ) ) )
304, 28, 29syl2anc 642 . . . . 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 ) ) ) ) ) ) )
3122a1i 10 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  2  e.  CC )
32 mulcl 8821 . . . . . . . . . . . . . 14  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
331, 9, 32sylancr 644 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  ( _i  x.  A )  e.  CC )
34 addcl 8819 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
357, 33, 34sylancr 644 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  e.  CC )
368simp3bi 972 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( 1  +  ( _i  x.  A ) )  =/=  0 )
37 logcl 19926 . . . . . . . . . . . 12  |-  ( ( ( 1  +  ( _i  x.  A ) )  e.  CC  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 )  ->  ( log `  (
1  +  ( _i  x.  A ) ) )  e.  CC )
3835, 36, 37syl2anc 642 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )
3931, 38, 4subdid 9235 . . . . . . . . . 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
) ) ) ) )
40 atanval 20180 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
4140oveq2d 5874 . . . . . . . . . . . 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
) ) ) ) ) ) )
421a1i 10 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  _i  e.  CC )
4331, 42, 2mulassd 8858 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  x.  (arctan `  A
) )  =  ( 2  x.  ( _i  x.  (arctan `  A
) ) ) )
44 halfcl 9937 . . . . . . . . . . . . . . . . . 18  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
451, 44ax-mp 8 . . . . . . . . . . . . . . . . 17  |-  ( _i 
/  2 )  e.  CC
4622, 1, 45mulassi 8846 . . . . . . . . . . . . . . . 16  |-  ( ( 2  x.  _i )  x.  ( _i  / 
2 ) )  =  ( 2  x.  (
_i  x.  ( _i  /  2 ) ) )
4722, 1, 45mul12i 9007 . . . . . . . . . . . . . . . 16  |-  ( 2  x.  ( _i  x.  ( _i  /  2
) ) )  =  ( _i  x.  (
2  x.  ( _i 
/  2 ) ) )
481, 22, 23divcan2i 9503 . . . . . . . . . . . . . . . . . 18  |-  ( 2  x.  ( _i  / 
2 ) )  =  _i
4948oveq2i 5869 . . . . . . . . . . . . . . . . 17  |-  ( _i  x.  ( 2  x.  ( _i  /  2
) ) )  =  ( _i  x.  _i )
50 ixi 9397 . . . . . . . . . . . . . . . . 17  |-  ( _i  x.  _i )  = 
-u 1
5149, 50eqtri 2303 . . . . . . . . . . . . . . . 16  |-  ( _i  x.  ( 2  x.  ( _i  /  2
) ) )  = 
-u 1
5246, 47, 513eqtri 2307 . . . . . . . . . . . . . . 15  |-  ( ( 2  x.  _i )  x.  ( _i  / 
2 ) )  = 
-u 1
5352oveq1i 5868 . . . . . . . . . . . . . 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 ) ) ) ) )
54 subcl 9051 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  -  ( _i  x.  A
) )  e.  CC )
557, 33, 54sylancr 644 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  e.  CC )
568simp2bi 971 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  dom arctan  ->  ( 1  -  ( _i  x.  A ) )  =/=  0 )
57 logcl 19926 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 1  -  (
_i  x.  A )
)  e.  CC  /\  ( 1  -  (
_i  x.  A )
)  =/=  0 )  ->  ( log `  (
1  -  ( _i  x.  A ) ) )  e.  CC )
5855, 56, 57syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom arctan  ->  ( log `  ( 1  -  (
_i  x.  A )
) )  e.  CC )
5958, 38subcld 9157 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
6059mulm1d 9231 . . . . . . . . . . . . . 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
) ) ) ) )
6153, 60syl5eq 2327 . . . . . . . . . . . . 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
) ) ) ) )
6222, 1mulcli 8842 . . . . . . . . . . . . . . 15  |-  ( 2  x.  _i )  e.  CC
6362a1i 10 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  ( 2  x.  _i )  e.  CC )
6445a1i 10 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  ( _i 
/  2 )  e.  CC )
6563, 64, 59mulassd 8858 . . . . . . . . . . . . 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 ) ) ) ) ) ) )
6658, 38negsubdi2d 9173 . . . . . . . . . . . . 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 )
) ) ) )
6761, 65, 663eqtr3d 2323 . . . . . . . . . . . 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 ) ) ) ) )
6841, 43, 673eqtr3d 2323 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( _i  x.  (arctan `  A ) ) )  =  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
6968oveq2d 5874 . . . . . . . . . 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 )
) ) ) ) )
70 mulcl 8821 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  ( log `  ( 1  +  ( _i  x.  A ) ) )  e.  CC )  -> 
( 2  x.  ( log `  ( 1  +  ( _i  x.  A
) ) ) )  e.  CC )
7122, 38, 70sylancr 644 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
7271, 38, 58subsubd 9185 . . . . . . . . . . 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 )
) ) ) )
73382timesd 9954 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) )  =  ( ( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )
7473oveq1d 5873 . . . . . . . . . . . . 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 ) ) ) ) )
7538, 38pncand 9158 . . . . . . . . . . . . 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 ) ) ) )
7674, 75eqtrd 2315 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  +  ( _i  x.  A ) ) ) )
7776oveq1d 5873 . . . . . . . . . . 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 ) ) ) ) )
78 atanlogadd 20210 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  +  ( log `  (
1  -  ( _i  x.  A ) ) ) )  e.  ran  log )
79 logef 19935 . . . . . . . . . . . . 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 ) ) ) ) )
8078, 79syl 15 . . . . . . . . . . . 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 ) ) ) ) )
81 efadd 12375 . . . . . . . . . . . . . . 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 ) ) ) ) ) )
8238, 58, 81syl2anc 642 . . . . . . . . . . . . . 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 ) ) ) ) ) )
83 eflog 19933 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1  +  ( _i  x.  A ) )  e.  CC  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 )  ->  ( exp `  ( log `  ( 1  +  ( _i  x.  A
) ) ) )  =  ( 1  +  ( _i  x.  A
) ) )
8435, 36, 83syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( exp `  ( log `  (
1  +  ( _i  x.  A ) ) ) )  =  ( 1  +  ( _i  x.  A ) ) )
85 eflog 19933 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1  -  (
_i  x.  A )
)  e.  CC  /\  ( 1  -  (
_i  x.  A )
)  =/=  0 )  ->  ( exp `  ( log `  ( 1  -  ( _i  x.  A
) ) ) )  =  ( 1  -  ( _i  x.  A
) ) )
8655, 56, 85syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( exp `  ( log `  (
1  -  ( _i  x.  A ) ) ) )  =  ( 1  -  ( _i  x.  A ) ) )
8784, 86oveq12d 5876 . . . . . . . . . . . . . 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
) ) ) )
88 sq1 11198 . . . . . . . . . . . . . . . . 17  |-  ( 1 ^ 2 )  =  1
8988a1i 10 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom arctan  ->  ( 1 ^ 2 )  =  1 )
90 sqmul 11167 . . . . . . . . . . . . . . . . . 18  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( ( _i  x.  A ) ^ 2 )  =  ( ( _i ^ 2 )  x.  ( A ^
2 ) ) )
911, 9, 90sylancr 644 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  A ) ^ 2 )  =  ( ( _i ^
2 )  x.  ( A ^ 2 ) ) )
92 i2 11203 . . . . . . . . . . . . . . . . . . 19  |-  ( _i
^ 2 )  = 
-u 1
9392oveq1i 5868 . . . . . . . . . . . . . . . . . 18  |-  ( ( _i ^ 2 )  x.  ( A ^
2 ) )  =  ( -u 1  x.  ( A ^ 2 ) )
9410mulm1d 9231 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  dom arctan  ->  ( -u
1  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
9593, 94syl5eq 2327 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  dom arctan  ->  ( ( _i ^ 2 )  x.  ( A ^
2 ) )  = 
-u ( A ^
2 ) )
9691, 95eqtrd 2315 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  A ) ^ 2 )  = 
-u ( A ^
2 ) )
9789, 96oveq12d 5876 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( ( 1 ^ 2 )  -  ( ( _i  x.  A ) ^
2 ) )  =  ( 1  -  -u ( A ^ 2 ) ) )
98 subsq 11210 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( ( 1 ^ 2 )  -  ( ( _i  x.  A ) ^ 2 ) )  =  ( ( 1  +  ( _i  x.  A ) )  x.  ( 1  -  ( _i  x.  A ) ) ) )
997, 33, 98sylancr 644 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( ( 1 ^ 2 )  -  ( ( _i  x.  A ) ^
2 ) )  =  ( ( 1  +  ( _i  x.  A
) )  x.  (
1  -  ( _i  x.  A ) ) ) )
100 subneg 9096 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  CC  /\  ( A ^ 2 )  e.  CC )  -> 
( 1  -  -u ( A ^ 2 ) )  =  ( 1  +  ( A ^ 2 ) ) )
1017, 10, 100sylancr 644 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( 1  -  -u ( A ^
2 ) )  =  ( 1  +  ( A ^ 2 ) ) )
10297, 99, 1013eqtr3d 2323 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( _i  x.  A ) )  x.  ( 1  -  ( _i  x.  A
) ) )  =  ( 1  +  ( A ^ 2 ) ) )
10382, 87, 1023eqtrd 2319 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  -  (
_i  x.  A )
) ) ) )  =  ( 1  +  ( A ^ 2 ) ) )
104103fveq2d 5529 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( log `  ( exp `  (
( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )  =  ( log `  ( 1  +  ( A ^ 2 ) ) ) )
10580, 104eqtr3d 2317 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  +  ( log `  (
1  -  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  +  ( A ^
2 ) ) ) )
10672, 77, 1053eqtrd 2319 . . . . . . . . . 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 ) ) ) )
10739, 69, 1063eqtrd 2319 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  (
_i  x.  (arctan `  A
) ) ) )  =  ( log `  (
1  +  ( A ^ 2 ) ) ) )
108107oveq1d 5873 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( _i  x.  (arctan `  A ) ) ) )  /  2
)  =  ( ( log `  ( 1  +  ( A ^
2 ) ) )  /  2 ) )
10938, 4subcld 9157 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( _i  x.  (arctan `  A ) ) )  e.  CC )
11023a1i 10 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  2  =/=  0 )
111109, 31, 110divcan3d 9541 . . . . . . . 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 ) ) ) )
11226, 31, 110divrec2d 9540 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( A ^
2 ) ) )  /  2 )  =  ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) )
113108, 111, 1123eqtr3d 2323 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( _i  x.  (arctan `  A ) ) )  =  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( A ^ 2 ) ) ) ) )
11438, 4, 28subaddd 9175 . . . . . . 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 ) ) ) ) )
115113, 114mpbid 201 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  (arctan `  A ) )  +  ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) )  =  ( log `  ( 1  +  ( _i  x.  A ) ) ) )
116115fveq2d 5529 . . . . 5  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( _i  x.  (arctan `  A ) )  +  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( A ^ 2 ) ) ) ) ) )  =  ( exp `  ( log `  ( 1  +  ( _i  x.  A
) ) ) ) )
11730, 116eqtr3d 2317 . . . 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 ) ) ) ) )
11824a1i 10 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 1  /  2 )  e.  CC )
11912, 16, 118cxpefd 20059 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( A ^ 2 ) )  ^ c  ( 1  /  2 ) )  =  ( exp `  (
( 1  /  2
)  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) ) )
120 cxpsqr 20050 . . . . . . 7  |-  ( ( 1  +  ( A ^ 2 ) )  e.  CC  ->  (
( 1  +  ( A ^ 2 ) )  ^ c  ( 1  /  2 ) )  =  ( sqr `  ( 1  +  ( A ^ 2 ) ) ) )
12112, 120syl 15 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( A ^ 2 ) )  ^ c  ( 1  /  2 ) )  =  ( sqr `  (
1  +  ( A ^ 2 ) ) ) )
122119, 121eqtr3d 2317 . . . . 5  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) )  =  ( sqr `  ( 1  +  ( A ^ 2 ) ) ) )
123122oveq2d 5874 . . . 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 ) ) ) ) )
124117, 123, 843eqtr3d 2323 . . 3  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( _i  x.  (arctan `  A
) ) )  x.  ( sqr `  (
1  +  ( A ^ 2 ) ) ) )  =  ( 1  +  ( _i  x.  A ) ) )
125124oveq1d 5873 . 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 ) ) ) ) )
12621, 125eqtr3d 2317 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 176    = wceq 1623    e. wcel 1684    =/= wne 2446   dom cdm 4689   ran crn 4690   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   2c2 9795   ^cexp 11104   sqrcsqr 11718   expce 12343   logclog 19912    ^ c ccxp 19913  arctancatan 20160
This theorem is referenced by:  cosatan  20217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-cxp 19915  df-atan 20163
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