Definition df-sinh 50667

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5190, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 50670 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 50673 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 50664	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 11126	. . 3 class ℂ
4		ci 11130	. . . . . 6 class i
5	2	cv 1569	. . . . . 6 class 𝑥
6		cmul 11133	. . . . . 6 class ·
7	4, 5, 6	co 7417	. . . . 5 class (i · 𝑥)
8		csin 16155	. . . . 5 class sin
9	7, 8	cfv 6537	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11899	. . . 4 class /
11	9, 4, 10	co 7417	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5190	. 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 used by:  sinhval-named  50670