Definition df-sinh 47298

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5193, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 47301 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 47304 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 47295	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 11058	. . 3 class ℂ
4		ci 11062	. . . . . 6 class i
5	2	cv 1540	. . . . . 6 class 𝑥
6		cmul 11065	. . . . . 6 class ·
7	4, 5, 6	co 7362	. . . . 5 class (i · 𝑥)
8		csin 15957	. . . . 5 class sin
9	7, 8	cfv 6501	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11821	. . . 4 class /
11	9, 4, 10	co 7362	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5193	. 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  47301