Definition df-sinh 49858

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5174, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 49861 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 49864 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 49855	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 11011	. . 3 class ℂ
4		ci 11015	. . . . . 6 class i
5	2	cv 1540	. . . . . 6 class 𝑥
6		cmul 11018	. . . . . 6 class ·
7	4, 5, 6	co 7352	. . . . 5 class (i · 𝑥)
8		csin 15972	. . . . 5 class sin
9	7, 8	cfv 6486	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11781	. . . 4 class /
11	9, 4, 10	co 7352	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5174	. 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
13	1, 12	wceq 1541	1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))

This definition is referenced by:  sinhval-named  49861