Definition df-sinh 46679

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5164, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 46682 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 46685 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 46676	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 10919	. . 3 class ℂ
4		ci 10923	. . . . . 6 class i
5	2	cv 1538	. . . . . 6 class 𝑥
6		cmul 10926	. . . . . 6 class ·
7	4, 5, 6	co 7307	. . . . 5 class (i · 𝑥)
8		csin 15822	. . . . 5 class sin
9	7, 8	cfv 6458	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11682	. . . 4 class /
11	9, 4, 10	co 7307	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5164	. 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
13	1, 12	wceq 1539	1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))

This definition is referenced by:  sinhval-named  46682