Definition df-sinh 50230

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5160, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 50233 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 50236 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 50227	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 11034	. . 3 class ℂ
4		ci 11038	. . . . . 6 class i
5	2	cv 1546	. . . . . 6 class 𝑥
6		cmul 11041	. . . . . 6 class ·
7	4, 5, 6	co 7363	. . . . 5 class (i · 𝑥)
8		csin 16026	. . . . 5 class sin
9	7, 8	cfv 6492	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11805	. . . 4 class /
11	9, 4, 10	co 7363	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5160	. 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
13	1, 12	wceq 1547	1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))

This definition is referenced by:  sinhval-named  50233