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Definition df-sinh 46106
Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5135, which requires the form (𝑥𝐴𝐵). See sinhval-named 46109 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 46112 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
StepHypRef Expression
1 csinh 46103 . 2 class sinh
2 vx . . 3 setvar 𝑥
3 cc 10727 . . 3 class
4 ci 10731 . . . . . 6 class i
52cv 1542 . . . . . 6 class 𝑥
6 cmul 10734 . . . . . 6 class ·
74, 5, 6co 7213 . . . . 5 class (i · 𝑥)
8 csin 15625 . . . . 5 class sin
97, 8cfv 6380 . . . 4 class (sin‘(i · 𝑥))
10 cdiv 11489 . . . 4 class /
119, 4, 10co 7213 . . 3 class ((sin‘(i · 𝑥)) / i)
122, 3, 11cmpt 5135 . 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
131, 12wceq 1543 1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
Colors of variables: wff setvar class
This definition is referenced by:  sinhval-named  46109
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