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Theorem asinval 20194
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 5882 . . . . 5  |-  ( x  =  A  ->  (
_i  x.  x )  =  ( _i  x.  A ) )
2 oveq1 5881 . . . . . . 7  |-  ( x  =  A  ->  (
x ^ 2 )  =  ( A ^
2 ) )
32oveq2d 5890 . . . . . 6  |-  ( x  =  A  ->  (
1  -  ( x ^ 2 ) )  =  ( 1  -  ( A ^ 2 ) ) )
43fveq2d 5545 . . . . 5  |-  ( x  =  A  ->  ( sqr `  ( 1  -  ( x ^ 2 ) ) )  =  ( sqr `  (
1  -  ( A ^ 2 ) ) ) )
51, 4oveq12d 5892 . . . 4  |-  ( x  =  A  ->  (
( _i  x.  x
)  +  ( sqr `  ( 1  -  (
x ^ 2 ) ) ) )  =  ( ( _i  x.  A )  +  ( sqr `  ( 1  -  ( A ^
2 ) ) ) ) )
65fveq2d 5545 . . 3  |-  ( x  =  A  ->  ( log `  ( ( _i  x.  x )  +  ( sqr `  (
1  -  ( x ^ 2 ) ) ) ) )  =  ( log `  (
( _i  x.  A
)  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ) ) )
76oveq2d 5890 . 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 20177 . 2  |- arcsin  =  ( x  e.  CC  |->  (
-u _i  x.  ( log `  ( ( _i  x.  x )  +  ( sqr `  (
1  -  ( x ^ 2 ) ) ) ) ) ) )
9 ovex 5899 . 2  |-  ( -u _i  x.  ( log `  (
( _i  x.  A
)  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ) ) )  e.  _V
107, 8, 9fvmpt 5618 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 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754   _ici 8755    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054   2c2 9811   ^cexp 11120   sqrcsqr 11734   logclog 19928  arcsincasin 20174
This theorem is referenced by:  asinneg  20198  efiasin  20200  asinsin  20204  asin1  20206  asinbnd  20211  dvreasin  25026  areacirclem5  25032
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-asin 20177
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