Definition df-sinh 46387

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5161, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 46390 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 46393 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 46384	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 10853	. . 3 class ℂ
4		ci 10857	. . . . . 6 class i
5	2	cv 1540	. . . . . 6 class 𝑥
6		cmul 10860	. . . . . 6 class ·
7	4, 5, 6	co 7268	. . . . 5 class (i · 𝑥)
8		csin 15754	. . . . 5 class sin
9	7, 8	cfv 6430	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11615	. . . 4 class /
11	9, 4, 10	co 7268	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5161	. 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
13	1, 12	wceq 1541	1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))

This definition is referenced by:  sinhval-named  46390