Definition df-sinh 50994

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5185, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 50997 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 51000 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 50991	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 11201	. . 3 class ℂ
4		ci 11205	. . . . . 6 class i
5	2	cv 1568	. . . . . 6 class 𝑥
6		cmul 11208	. . . . . 6 class ·
7	4, 5, 6	co 7420	. . . . 5 class (i · 𝑥)
8		csin 16234	. . . . 5 class sin
9	7, 8	cfv 6538	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11976	. . . 4 class /
11	9, 4, 10	co 7420	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5185	. 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
13	1, 12	wceq 1569	1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))

This definition is used by:  sinhval-named  50997