Definition df-sinh 50086

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5181, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 50089 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 50092 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 50083	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 11036	. . 3 class ℂ
4		ci 11040	. . . . . 6 class i
5	2	cv 1541	. . . . . 6 class 𝑥
6		cmul 11043	. . . . . 6 class ·
7	4, 5, 6	co 7368	. . . . 5 class (i · 𝑥)
8		csin 15998	. . . . 5 class sin
9	7, 8	cfv 6500	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11806	. . . 4 class /
11	9, 4, 10	co 7368	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5181	. 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
13	1, 12	wceq 1542	1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))

This definition is referenced by:  sinhval-named  50089