Definition df-sinh 46106

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5135, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 46109 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 46112 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 46103	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 10727	. . 3 class ℂ
4		ci 10731	. . . . . 6 class i
5	2	cv 1542	. . . . . 6 class 𝑥
6		cmul 10734	. . . . . 6 class ·
7	4, 5, 6	co 7213	. . . . 5 class (i · 𝑥)
8		csin 15625	. . . . 5 class sin
9	7, 8	cfv 6380	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11489	. . . 4 class /
11	9, 4, 10	co 7213	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5135	. 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
13	1, 12	wceq 1543	1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))

This definition is referenced by:  sinhval-named  46109