Definition df-sinh 49715

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5183, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 49718 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 49721 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 49712	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 11042	. . 3 class ℂ
4		ci 11046	. . . . . 6 class i
5	2	cv 1539	. . . . . 6 class 𝑥
6		cmul 11049	. . . . . 6 class ·
7	4, 5, 6	co 7369	. . . . 5 class (i · 𝑥)
8		csin 16005	. . . . 5 class sin
9	7, 8	cfv 6499	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11811	. . . 4 class /
11	9, 4, 10	co 7369	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5183	. 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  49718