Theorem sinval 14852

Description: Value of the sine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.)

Assertion

Ref	Expression
sinval	|- ( A e. CC -> ( sin ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )

Proof of Theorem sinval

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	1	fveq2d 6195	. . . 4 |- ( x = A -> ( exp ` ( _i x. x ) ) = ( exp ` ( _i x. A ) ) )
3		oveq2 6658	. . . . 5 |- ( x = A -> ( -u _i x. x ) = ( -u _i x. A ) )
4	3	fveq2d 6195	. . . 4 |- ( x = A -> ( exp ` ( -u _i x. x ) ) = ( exp ` ( -u _i x. A ) ) )
5	2, 4	oveq12d 6668	. . 3 |- ( x = A -> ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) = ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) )
6	5	oveq1d 6665	. 2 |- ( x = A -> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
7		df-sin 14800	. 2 |- sin = ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) )
8		ovex 6678	. 2 |- ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) e. _V
9	6, 7, 8	fvmpt 6282	1 |- ( A e. CC -> ( sin ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   _ici 9938   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   expce 14792   sincsin 14794

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-sin 14800

This theorem is referenced by:  tanval2  14863  resinval  14865  sinneg  14876  efival  14882  sinhval  14884  sinadd  14894  dvsincos  23744  sinper  24233  sineq0  24273  efeq1  24275  sinasin  24616  sineq0ALT  39173