Definition df-sinh 49764

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5172, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 49767 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 49770 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 49761	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 11001	. . 3 class ℂ
4		ci 11005	. . . . . 6 class i
5	2	cv 1540	. . . . . 6 class 𝑥
6		cmul 11008	. . . . . 6 class ·
7	4, 5, 6	co 7346	. . . . 5 class (i · 𝑥)
8		csin 15967	. . . . 5 class sin
9	7, 8	cfv 6481	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11771	. . . 4 class /
11	9, 4, 10	co 7346	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5172	. 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
13	1, 12	wceq 1541	1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))

This definition is referenced by:  sinhval-named  49767