Definition df-sinh 49599

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5196, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 49602 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 49605 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 49596	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 11084	. . 3 class ℂ
4		ci 11088	. . . . . 6 class i
5	2	cv 1539	. . . . . 6 class 𝑥
6		cmul 11091	. . . . . 6 class ·
7	4, 5, 6	co 7394	. . . . 5 class (i · 𝑥)
8		csin 16036	. . . . 5 class sin
9	7, 8	cfv 6519	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11851	. . . 4 class /
11	9, 4, 10	co 7394	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5196	. 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
13	1, 12	wceq 1540	1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))

This definition is referenced by:  sinhval-named  49602