Definition df-sinh 50552

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5197, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 50555 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 50558 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 50549	. 2 class sinh
2		vx	. . 3 setvar 𝑥
3		cc 11116	. . 3 class ℂ
4		ci 11120	. . . . . 6 class i
5	2	cv 1569	. . . . . 6 class 𝑥
6		cmul 11123	. . . . . 6 class ·
7	4, 5, 6	co 7423	. . . . 5 class (i · 𝑥)
8		csin 16142	. . . . 5 class sin
9	7, 8	cfv 6543	. . . 4 class (sin‘(i · 𝑥))
10		cdiv 11889	. . . 4 class /
11	9, 4, 10	co 7423	. . 3 class ((sin‘(i · 𝑥)) / i)
12	2, 3, 11	cmpt 5197	. 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  50555