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Theorem asinval 20589
Description: Value of the arcsin function. (Contributed by Mario Carneiro, 31-Mar-2015.)
Assertion
Ref Expression
asinval  |-  ( A  e.  CC  ->  (arcsin `  A )  =  (
-u _i  x.  ( log `  ( ( _i  x.  A )  +  ( sqr `  (
1  -  ( A ^ 2 ) ) ) ) ) ) )

Proof of Theorem asinval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oveq2 6028 . . . . 5  |-  ( x  =  A  ->  (
_i  x.  x )  =  ( _i  x.  A ) )
2 oveq1 6027 . . . . . . 7  |-  ( x  =  A  ->  (
x ^ 2 )  =  ( A ^
2 ) )
32oveq2d 6036 . . . . . 6  |-  ( x  =  A  ->  (
1  -  ( x ^ 2 ) )  =  ( 1  -  ( A ^ 2 ) ) )
43fveq2d 5672 . . . . 5  |-  ( x  =  A  ->  ( sqr `  ( 1  -  ( x ^ 2 ) ) )  =  ( sqr `  (
1  -  ( A ^ 2 ) ) ) )
51, 4oveq12d 6038 . . . 4  |-  ( x  =  A  ->  (
( _i  x.  x
)  +  ( sqr `  ( 1  -  (
x ^ 2 ) ) ) )  =  ( ( _i  x.  A )  +  ( sqr `  ( 1  -  ( A ^
2 ) ) ) ) )
65fveq2d 5672 . . 3  |-  ( x  =  A  ->  ( log `  ( ( _i  x.  x )  +  ( sqr `  (
1  -  ( x ^ 2 ) ) ) ) )  =  ( log `  (
( _i  x.  A
)  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ) ) )
76oveq2d 6036 . 2  |-  ( x  =  A  ->  ( -u _i  x.  ( log `  ( ( _i  x.  x )  +  ( sqr `  ( 1  -  ( x ^
2 ) ) ) ) ) )  =  ( -u _i  x.  ( log `  ( ( _i  x.  A )  +  ( sqr `  (
1  -  ( A ^ 2 ) ) ) ) ) ) )
8 df-asin 20572 . 2  |- arcsin  =  ( x  e.  CC  |->  (
-u _i  x.  ( log `  ( ( _i  x.  x )  +  ( sqr `  (
1  -  ( x ^ 2 ) ) ) ) ) ) )
9 ovex 6045 . 2  |-  ( -u _i  x.  ( log `  (
( _i  x.  A
)  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ) ) )  e.  _V
107, 8, 9fvmpt 5745 1  |-  ( A  e.  CC  ->  (arcsin `  A )  =  (
-u _i  x.  ( log `  ( ( _i  x.  A )  +  ( sqr `  (
1  -  ( A ^ 2 ) ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   ` cfv 5394  (class class class)co 6020   CCcc 8921   1c1 8924   _ici 8925    + caddc 8926    x. cmul 8928    - cmin 9223   -ucneg 9224   2c2 9981   ^cexp 11309   sqrcsqr 11965   logclog 20319  arcsincasin 20569
This theorem is referenced by:  asinneg  20593  efiasin  20595  asinsin  20599  asin1  20601  asinbnd  20606  dvreasin  25980  areacirclem5  25986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-asin 20572
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