Definition df-sinh 47618

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5227, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 47621 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 47624 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 47615	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 11095	. . 3 class ℂ
4		ci 11099	. . . . . 6 class i
5	2	cv 1541	. . . . . 6 class 𝑥
6		cmul 11102	. . . . . 6 class ·
7	4, 5, 6	co 7396	. . . . 5 class (i · 𝑥)
8		csin 15994	. . . . 5 class sin
9	7, 8	cfv 6535	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11858	. . . 4 class /
11	9, 4, 10	co 7396	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5227	. 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
13	1, 12	wceq 1542	1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))

This definition is referenced by:  sinhval-named  47621