Definition df-sinh 50318

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5180, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 50321 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 50324 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 50315	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 11068	. . 3 class ℂ
4		ci 11072	. . . . . 6 class i
5	2	cv 1558	. . . . . 6 class 𝑥
6		cmul 11075	. . . . . 6 class ·
7	4, 5, 6	co 7392	. . . . 5 class (i · 𝑥)
8		csin 16076	. . . . 5 class sin
9	7, 8	cfv 6517	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11841	. . . 4 class /
11	9, 4, 10	co 7392	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5180	. 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
13	1, 12	wceq 1559	1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))

This definition is referenced by:  sinhval-named  50321