Definition df-sin 13667

Description: Define the sine function. (Contributed by NM, 14-Mar-2005.)

Assertion

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

Detailed syntax breakdown of Definition df-sin

Step	Hyp	Ref	Expression
1		csin 13661	. 2 class sin
2		vx	. . 3 setvar x
3		cc 9490	. . 3 class CC
4		ci 9494	. . . . . . 7 class _i
5	2	cv 1378	. . . . . . 7 class x
6		cmul 9497	. . . . . . 7 class x.
7	4, 5, 6	co 6284	. . . . . 6 class ( _i x. x )
8		ce 13659	. . . . . 6 class exp
9	7, 8	cfv 5588	. . . . 5 class ( exp ` ( _i x. x ) )
10	4	cneg 9806	. . . . . . 7 class -u _i
11	10, 5, 6	co 6284	. . . . . 6 class ( -u _i x. x )
12	11, 8	cfv 5588	. . . . 5 class ( exp ` ( -u _i x. x ) )
13		cmin 9805	. . . . 5 class -
14	9, 12, 13	co 6284	. . . 4 class ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) )
15		c2 10585	. . . . 5 class 2
16	15, 4, 6	co 6284	. . . 4 class ( 2 x. _i )
17		cdiv 10206	. . . 4 class /
18	14, 16, 17	co 6284	. . 3 class ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) )
19	2, 3, 18	cmpt 4505	. 2 class ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) )
20	1, 19	wceq 1379	1 wff sin = ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) )

This definition is referenced by:  sinval  13718  sinf  13720  dvsincos  22145  sincn  22601