Definition df-sinh 50208

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5166, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 50211 for a simple way to evaluate it. We define this function by dividing by i, which uses fewer operations than many conventional definitions (and thus is more convenient to use in set.mm). See sinh-conventional 50214 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.)

Assertion

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

Detailed syntax breakdown of Definition df-sinh

Step	Hyp	Ref	Expression
1		csinh 50205	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 11036	. . 3 class ℂ
4		ci 11040	. . . . . 6 class i
5	2	cv 1541	. . . . . 6 class 𝑥
6		cmul 11043	. . . . . 6 class ·
7	4, 5, 6	co 7367	. . . . 5 class (i · 𝑥)
8		csin 16028	. . . . 5 class sin
9	7, 8	cfv 6498	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11807	. . . 4 class /
11	9, 4, 10	co 7367	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5166	. 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
13	1, 12	wceq 1542	1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))

This definition is referenced by:  sinhval-named  50211