Definition df-sinh 50488

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5193, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 50491 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 50494 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 50485	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 11099	. . 3 class ℂ
4		ci 11103	. . . . . 6 class i
5	2	cv 1569	. . . . . 6 class 𝑥
6		cmul 11106	. . . . . 6 class ·
7	4, 5, 6	co 7412	. . . . 5 class (i · 𝑥)
8		csin 16118	. . . . 5 class sin
9	7, 8	cfv 6538	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11872	. . . 4 class /
11	9, 4, 10	co 7412	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5193	. 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
13	1, 12	wceq 1570	1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))

This definition is referenced by:  sinhval-named  50491