Definition df-sin 12419

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

Assertion

Ref	Expression
df-sin	⊢ sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))

Detailed syntax breakdown of Definition df-sin

Step	Hyp	Ref	Expression
1		csin 12413	. 2 class sin
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 12411	. . . . . 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		cmin 8497	. . . . 5 class −
14	9, 12, 13	co 6085	. . . 4 class ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥)))
15		c2 9356	. . . . 5 class 2
16	15, 4, 6	co 6085	. . . 4 class (2 · i)
17		cdiv 9003	. . . 4 class /
18	14, 16, 17	co 6085	. . 3 class (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i))
19	2, 3, 18	cmpt 4192	. 2 class (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))
20	1, 19	wceq 1402	1 wff sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))

This definition is used by:  sinval  12471  sinf  12473  sincn  15872