Definition df-cos 12418

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

Assertion

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

Detailed syntax breakdown of Definition df-cos

Step	Hyp	Ref	Expression
1		ccos 12412	. 2 class cos
2		vx	. . 3 setvar x
3		cc 8177	. . 3 class CC
4		ci 8181	. . . . . . 7 class _i
5	2	cv 1401	. . . . . . 7 class x
6		cmul 8184	. . . . . . 7 class x.
7	4, 5, 6	co 6085	. . . . . 6 class ( _i x. x )
8		ce 12409	. . . . . 6 class exp
9	7, 8	cfv 5377	. . . . 5 class ( exp ` ( _i x. x ) )
10	4	cneg 8498	. . . . . . 7 class -u _i
11	10, 5, 6	co 6085	. . . . . 6 class ( -u _i x. x )
12	11, 8	cfv 5377	. . . . 5 class ( exp ` ( -u _i x. x ) )
13		caddc 8182	. . . . 5 class +
14	9, 12, 13	co 6085	. . . 4 class ( ( exp ` ( _i x. x ) ) + ( exp ` ( -u _i x. x ) ) )
15		c2 9355	. . . 4 class 2
16		cdiv 9002	. . . 4 class /
17	14, 15, 16	co 6085	. . 3 class ( ( ( exp ` ( _i x. x ) ) + ( exp ` ( -u _i x. x ) ) ) / 2 )
18	2, 3, 17	cmpt 4192	. 2 class ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) + ( exp ` ( -u _i x. x ) ) ) / 2 ) )
19	1, 18	wceq 1402	1 wff cos = ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) + ( exp ` ( -u _i x. x ) ) ) / 2 ) )

This definition is used by:  cosval  12470  cosf  12472  coscn  15871