MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efiatan2 Unicode version

Theorem efiatan2 20229
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 8812 . . . . 5  |-  _i  e.  CC
2 atancl 20193 . . . . 5  |-  ( A  e.  dom arctan  ->  (arctan `  A )  e.  CC )
3 mulcl 8837 . . . . 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 12380 . . . 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 8811 . . . . 5  |-  1  e.  CC
8 atandm2 20189 . . . . . . 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 11259 . . . . 5  |-  ( A  e.  dom arctan  ->  ( A ^ 2 )  e.  CC )
11 addcl 8835 . . . . 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 11935 . . 3  |-  ( A  e.  dom arctan  ->  ( sqr `  ( 1  +  ( A ^ 2 ) ) )  e.  CC )
1412sqsqrd 11937 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( sqr `  ( 1  +  ( A ^
2 ) ) ) ^ 2 )  =  ( 1  +  ( A ^ 2 ) ) )
15 atandm4 20191 . . . . . 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 2477 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( sqr `  ( 1  +  ( A ^
2 ) ) ) ^ 2 )  =/=  0 )
18 sqne0 11186 . . . . 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 9558 . 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 9832 . . . . . . . 8  |-  2  e.  CC
23 2ne0 9845 . . . . . . . 8  |-  2  =/=  0
2422, 23reccli 9506 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
25 logcl 19942 . . . . . . . 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 8837 . . . . . . 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 12391 . . . . . 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 8837 . . . . . . . . . . . . . 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 8835 . . . . . . . . . . . . 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 19942 . . . . . . . . . . . 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 9251 . . . . . . . . . 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 20196 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  (arctan `  A )  =  ( ( _i  /  2
)  x.  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) ) ) )
4140oveq2d 5890 . . . . . . . . . . . 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 8874 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  x.  (arctan `  A
) )  =  ( 2  x.  ( _i  x.  (arctan `  A
) ) ) )
44 halfcl 9953 . . . . . . . . . . . . . . . . . 18  |-  ( _i  e.  CC  ->  (
_i  /  2 )  e.  CC )
451, 44ax-mp 8 . . . . . . . . . . . . . . . . 17  |-  ( _i 
/  2 )  e.  CC
4622, 1, 45mulassi 8862 . . . . . . . . . . . . . . . 16  |-  ( ( 2  x.  _i )  x.  ( _i  / 
2 ) )  =  ( 2  x.  (
_i  x.  ( _i  /  2 ) ) )
4722, 1, 45mul12i 9023 . . . . . . . . . . . . . . . 16  |-  ( 2  x.  ( _i  x.  ( _i  /  2
) ) )  =  ( _i  x.  (
2  x.  ( _i 
/  2 ) ) )
481, 22, 23divcan2i 9519 . . . . . . . . . . . . . . . . . 18  |-  ( 2  x.  ( _i  / 
2 ) )  =  _i
4948oveq2i 5885 . . . . . . . . . . . . . . . . 17  |-  ( _i  x.  ( 2  x.  ( _i  /  2
) ) )  =  ( _i  x.  _i )
50 ixi 9413 . . . . . . . . . . . . . . . . 17  |-  ( _i  x.  _i )  = 
-u 1
5149, 50eqtri 2316 . . . . . . . . . . . . . . . 16  |-  ( _i  x.  ( 2  x.  ( _i  /  2
) ) )  = 
-u 1
5246, 47, 513eqtri 2320 . . . . . . . . . . . . . . 15  |-  ( ( 2  x.  _i )  x.  ( _i  / 
2 ) )  = 
-u 1
5352oveq1i 5884 . . . . . . . . . . . . . 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 9067 . . . . . . . . . . . . . . . . . 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 19942 . . . . . . . . . . . . . . . . 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 9173 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  -  ( _i  x.  A ) ) )  -  ( log `  (
1  +  ( _i  x.  A ) ) ) )  e.  CC )
6059mulm1d 9247 . . . . . . . . . . . . . 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 2340 . . . . . . . . . . . . 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 8858 . . . . . . . . . . . . . . 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 8874 . . . . . . . . . . . . 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 9189 . . . . . . . . . . . . 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 2336 . . . . . . . . . . . 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 2336 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( _i  x.  (arctan `  A ) ) )  =  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( log `  (
1  -  ( _i  x.  A ) ) ) ) )
6968oveq2d 5890 . . . . . . . . . 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 8837 . . . . . . . . . . . . 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 9201 . . . . . . . . . . 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 9970 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( log `  (
1  +  ( _i  x.  A ) ) ) )  =  ( ( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  +  ( _i  x.  A ) ) ) ) )
7473oveq1d 5889 . . . . . . . . . . . . 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 9174 . . . . . . . . . . . . 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 2328 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  -  ( log `  ( 1  +  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  +  ( _i  x.  A ) ) ) )
7776oveq1d 5889 . . . . . . . . . . 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 20226 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  +  ( log `  (
1  -  ( _i  x.  A ) ) ) )  e.  ran  log )
79 logef 19951 . . . . . . . . . . . . 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 12391 . . . . . . . . . . . . . . 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 19949 . . . . . . . . . . . . . . . 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 19949 . . . . . . . . . . . . . . . 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 5892 . . . . . . . . . . . . . 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 11214 . . . . . . . . . . . . . . . . 17  |-  ( 1 ^ 2 )  =  1
8988a1i 10 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom arctan  ->  ( 1 ^ 2 )  =  1 )
90 sqmul 11183 . . . . . . . . . . . . . . . . . 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 11219 . . . . . . . . . . . . . . . . . . 19  |-  ( _i
^ 2 )  = 
-u 1
9392oveq1i 5884 . . . . . . . . . . . . . . . . . 18  |-  ( ( _i ^ 2 )  x.  ( A ^
2 ) )  =  ( -u 1  x.  ( A ^ 2 ) )
9410mulm1d 9247 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  dom arctan  ->  ( -u
1  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
9593, 94syl5eq 2340 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  dom arctan  ->  ( ( _i ^ 2 )  x.  ( A ^
2 ) )  = 
-u ( A ^
2 ) )
9691, 95eqtrd 2328 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  A ) ^ 2 )  = 
-u ( A ^
2 ) )
9789, 96oveq12d 5892 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom arctan  ->  ( ( 1 ^ 2 )  -  ( ( _i  x.  A ) ^
2 ) )  =  ( 1  -  -u ( A ^ 2 ) ) )
98 subsq 11226 . . . . . . . . . . . . . . . 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 9112 . . . . . . . . . . . . . . . 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 2336 . . . . . . . . . . . . . 14  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( _i  x.  A ) )  x.  ( 1  -  ( _i  x.  A
) ) )  =  ( 1  +  ( A ^ 2 ) ) )
10382, 87, 1023eqtrd 2332 . . . . . . . . . . . . 13  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  -  (
_i  x.  A )
) ) ) )  =  ( 1  +  ( A ^ 2 ) ) )
104103fveq2d 5545 . . . . . . . . . . . 12  |-  ( A  e.  dom arctan  ->  ( log `  ( exp `  (
( log `  (
1  +  ( _i  x.  A ) ) )  +  ( log `  ( 1  -  (
_i  x.  A )
) ) ) ) )  =  ( log `  ( 1  +  ( A ^ 2 ) ) ) )
10580, 104eqtr3d 2330 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  +  ( log `  (
1  -  ( _i  x.  A ) ) ) )  =  ( log `  ( 1  +  ( A ^
2 ) ) ) )
10672, 77, 1053eqtrd 2332 . . . . . . . . . 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 2332 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  (
_i  x.  (arctan `  A
) ) ) )  =  ( log `  (
1  +  ( A ^ 2 ) ) ) )
108107oveq1d 5889 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( _i  x.  (arctan `  A ) ) ) )  /  2
)  =  ( ( log `  ( 1  +  ( A ^
2 ) ) )  /  2 ) )
10938, 4subcld 9173 . . . . . . . . 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 9557 . . . . . . . 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 9556 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( A ^
2 ) ) )  /  2 )  =  ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) )
113108, 111, 1123eqtr3d 2336 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( log `  ( 1  +  ( _i  x.  A ) ) )  -  ( _i  x.  (arctan `  A ) ) )  =  ( ( 1  /  2 )  x.  ( log `  (
1  +  ( A ^ 2 ) ) ) ) )
11438, 4, 28subaddd 9191 . . . . . . 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 5545 . . . . 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 2330 . . . 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 20075 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 1  +  ( A ^ 2 ) )  ^ c  ( 1  /  2 ) )  =  ( exp `  (
( 1  /  2
)  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) ) )
120 cxpsqr 20066 . . . . . . 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 2330 . . . . 5  |-  ( A  e.  dom arctan  ->  ( exp `  ( ( 1  / 
2 )  x.  ( log `  ( 1  +  ( A ^ 2 ) ) ) ) )  =  ( sqr `  ( 1  +  ( A ^ 2 ) ) ) )
123122oveq2d 5890 . . . 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 2336 . . 3  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( _i  x.  (arctan `  A
) ) )  x.  ( sqr `  (
1  +  ( A ^ 2 ) ) ) )  =  ( 1  +  ( _i  x.  A ) ) )
125124oveq1d 5889 . 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 2330 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 1632    e. wcel 1696    =/= wne 2459   dom cdm 4705   ran crn 4706   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754   _ici 8755    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054    / cdiv 9439   2c2 9811   ^cexp 11120   sqrcsqr 11734   expce 12359   logclog 19928    ^ c ccxp 19929  arctancatan 20176
This theorem is referenced by:  cosatan  20233
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-cxp 19931  df-atan 20179
  Copyright terms: Public domain W3C validator