Theorem asinval 24609

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.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . 5 |- ( x = A -> ( _i x. x ) = ( _i x. A ) )
2		oveq1 6657	. . . . . . 7 |- ( x = A -> ( x ^ 2 ) = ( A ^ 2 ) )
3	2	oveq2d 6666	. . . . . 6 |- ( x = A -> ( 1 - ( x ^ 2 ) ) = ( 1 - ( A ^ 2 ) ) )
4	3	fveq2d 6195	. . . . 5 |- ( x = A -> ( sqr ` ( 1 - ( x ^ 2 ) ) ) = ( sqr ` ( 1 - ( A ^ 2 ) ) ) )
5	1, 4	oveq12d 6668	. . . 4 |- ( x = A -> ( ( _i x. x ) + ( sqr ` ( 1 - ( x ^ 2 ) ) ) ) = ( ( _i x. A ) + ( sqr ` ( 1 - ( A ^ 2 ) ) ) ) )
6	5	fveq2d 6195	. . 3 |- ( x = A -> ( log ` ( ( _i x. x ) + ( sqr ` ( 1 - ( x ^ 2 ) ) ) ) ) = ( log ` ( ( _i x. A ) + ( sqr ` ( 1 - ( A ^ 2 ) ) ) ) ) )
7	6	oveq2d 6666	. 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 24592	. 2 |- arcsin = ( x e. CC |-> ( -u _i x. ( log ` ( ( _i x. x ) + ( sqr ` ( 1 - ( x ^ 2 ) ) ) ) ) ) )
9		ovex 6678	. 2 |- ( -u _i x. ( log ` ( ( _i x. A ) + ( sqr ` ( 1 - ( A ^ 2 ) ) ) ) ) ) e. _V
10	7, 8, 9	fvmpt 6282	1 |- ( A e. CC -> (arcsin ` A ) = ( -u _i x. ( log ` ( ( _i x. A ) + ( sqr ` ( 1 - ( A ^ 2 ) ) ) ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   _ici 9938   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   2c2 11070   ^cexp 12860   sqrcsqrt 13973   logclog 24301  arcsincasin 24589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-asin 24592

This theorem is referenced by:  asinneg  24613  efiasin  24615  asinsin  24619  asin1  24621  asinbnd  24626  areacirclem4  33503