Definition df-cos 12418

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

Assertion

Ref	Expression
df-cos	⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 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 ℂ
4		ci 8181	. . . . . . 7 class i
5	2	cv 1401	. . . . . . 7 class 𝑥
6		cmul 8184	. . . . . . 7 class ·
7	4, 5, 6	co 6085	. . . . . 6 class (i · 𝑥)
8		ce 12409	. . . . . 6 class exp
9	7, 8	cfv 5377	. . . . 5 class (exp‘(i · 𝑥))
10	4	cneg 8498	. . . . . . 7 class -i
11	10, 5, 6	co 6085	. . . . . 6 class (-i · 𝑥)
12	11, 8	cfv 5377	. . . . 5 class (exp‘(-i · 𝑥))
13		caddc 8182	. . . . 5 class +
14	9, 12, 13	co 6085	. . . 4 class ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))
15		c2 9355	. . . 4 class 2
16		cdiv 9002	. . . 4 class /
17	14, 15, 16	co 6085	. . 3 class (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)
18	2, 3, 17	cmpt 4192	. 2 class (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
19	1, 18	wceq 1402	1 wff cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))

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