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Theorem asinval 20714
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 6081 . . . . 5  |-  ( x  =  A  ->  (
_i  x.  x )  =  ( _i  x.  A ) )
2 oveq1 6080 . . . . . . 7  |-  ( x  =  A  ->  (
x ^ 2 )  =  ( A ^
2 ) )
32oveq2d 6089 . . . . . 6  |-  ( x  =  A  ->  (
1  -  ( x ^ 2 ) )  =  ( 1  -  ( A ^ 2 ) ) )
43fveq2d 5724 . . . . 5  |-  ( x  =  A  ->  ( sqr `  ( 1  -  ( x ^ 2 ) ) )  =  ( sqr `  (
1  -  ( A ^ 2 ) ) ) )
51, 4oveq12d 6091 . . . 4  |-  ( x  =  A  ->  (
( _i  x.  x
)  +  ( sqr `  ( 1  -  (
x ^ 2 ) ) ) )  =  ( ( _i  x.  A )  +  ( sqr `  ( 1  -  ( A ^
2 ) ) ) ) )
65fveq2d 5724 . . 3  |-  ( x  =  A  ->  ( log `  ( ( _i  x.  x )  +  ( sqr `  (
1  -  ( x ^ 2 ) ) ) ) )  =  ( log `  (
( _i  x.  A
)  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ) ) )
76oveq2d 6089 . 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 20697 . 2  |- arcsin  =  ( x  e.  CC  |->  (
-u _i  x.  ( log `  ( ( _i  x.  x )  +  ( sqr `  (
1  -  ( x ^ 2 ) ) ) ) ) ) )
9 ovex 6098 . 2  |-  ( -u _i  x.  ( log `  (
( _i  x.  A
)  +  ( sqr `  ( 1  -  ( A ^ 2 ) ) ) ) ) )  e.  _V
107, 8, 9fvmpt 5798 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 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   CCcc 8980   1c1 8983   _ici 8984    + caddc 8985    x. cmul 8987    - cmin 9283   -ucneg 9284   2c2 10041   ^cexp 11374   sqrcsqr 12030   logclog 20444  arcsincasin 20694
This theorem is referenced by:  asinneg  20718  efiasin  20720  asinsin  20724  asin1  20726  asinbnd  20731  dvreasin  26280  areacirclem5  26286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-asin 20697
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