Definition df-sinh 48825

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5249, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 48828 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 48831 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 48822	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 11182	. . 3 class ℂ
4		ci 11186	. . . . . 6 class i
5	2	cv 1536	. . . . . 6 class 𝑥
6		cmul 11189	. . . . . 6 class ·
7	4, 5, 6	co 7448	. . . . 5 class (i · 𝑥)
8		csin 16111	. . . . 5 class sin
9	7, 8	cfv 6573	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11947	. . . 4 class /
11	9, 4, 10	co 7448	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5249	. 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
13	1, 12	wceq 1537	1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))

This definition is referenced by:  sinhval-named  48828