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Theorem asinval 20178
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 5866 . . . . 5  |-  ( x  =  A  ->  (
_i  x.  x )  =  ( _i  x.  A ) )
2 oveq1 5865 . . . . . . 7  |-  ( x  =  A  ->  (
x ^ 2 )  =  ( A ^
2 ) )
32oveq2d 5874 . . . . . 6  |-  ( x  =  A  ->  (
1  -  ( x ^ 2 ) )  =  ( 1  -  ( A ^ 2 ) ) )
43fveq2d 5529 . . . . 5  |-  ( x  =  A  ->  ( sqr `  ( 1  -  ( x ^ 2 ) ) )  =  ( sqr `  (
1  -  ( A ^ 2 ) ) ) )
51, 4oveq12d 5876 . . . 4  |-  ( x  =  A  ->  (
( _i  x.  x
)  +  ( sqr `  ( 1  -  (
x ^ 2 ) ) ) )  =  ( ( _i  x.  A )  +  ( sqr `  ( 1  -  ( A ^
2 ) ) ) ) )
65fveq2d 5529 . . 3  |-  ( x  =  A  ->  ( log `  ( ( _i  x.  x )  +  ( sqr `  (
1  -  ( x ^ 2 ) ) ) ) )  =  ( log `  (
( _i  x.  A
)  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ) ) )
76oveq2d 5874 . 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 20161 . 2  |- arcsin  =  ( x  e.  CC  |->  (
-u _i  x.  ( log `  ( ( _i  x.  x )  +  ( sqr `  (
1  -  ( x ^ 2 ) ) ) ) ) ) )
9 ovex 5883 . 2  |-  ( -u _i  x.  ( log `  (
( _i  x.  A
)  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ) ) )  e.  _V
107, 8, 9fvmpt 5602 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 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038   2c2 9795   ^cexp 11104   sqrcsqr 11718   logclog 19912  arcsincasin 20158
This theorem is referenced by:  asinneg  20182  efiasin  20184  asinsin  20188  asin1  20190  asinbnd  20195  dvreasin  24923  areacirclem5  24929
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-asin 20161
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