Definition df-sinh 49722

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5188, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 49725 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 49728 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 49719	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 11066	. . 3 class ℂ
4		ci 11070	. . . . . 6 class i
5	2	cv 1539	. . . . . 6 class 𝑥
6		cmul 11073	. . . . . 6 class ·
7	4, 5, 6	co 7387	. . . . 5 class (i · 𝑥)
8		csin 16029	. . . . 5 class sin
9	7, 8	cfv 6511	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11835	. . . 4 class /
11	9, 4, 10	co 7387	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5188	. 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
13	1, 12	wceq 1540	1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))

This definition is referenced by:  sinhval-named  49725