Definition df-sinh 50362

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5186, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 50365 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 50368 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 50359	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 11086	. . 3 class ℂ
4		ci 11090	. . . . . 6 class i
5	2	cv 1562	. . . . . 6 class 𝑥
6		cmul 11093	. . . . . 6 class ·
7	4, 5, 6	co 7400	. . . . 5 class (i · 𝑥)
8		csin 16107	. . . . 5 class sin
9	7, 8	cfv 6525	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11859	. . . 4 class /
11	9, 4, 10	co 7400	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5186	. 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
13	1, 12	wceq 1563	1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))

This definition is referenced by:  sinhval-named  50365