Definition df-sinh 47778

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5232, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 47781 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 47784 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 47775	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 11108	. . 3 class ℂ
4		ci 11112	. . . . . 6 class i
5	2	cv 1541	. . . . . 6 class 𝑥
6		cmul 11115	. . . . . 6 class ·
7	4, 5, 6	co 7409	. . . . 5 class (i · 𝑥)
8		csin 16007	. . . . 5 class sin
9	7, 8	cfv 6544	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11871	. . . 4 class /
11	9, 4, 10	co 7409	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5232	. 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  47781