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Definition df-sinh 44760
Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5137, which requires the form (𝑥𝐴𝐵). See sinhval-named 44763 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 44766 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 44757 . 2 class sinh
2 vx . . 3 setvar 𝑥
3 cc 10523 . . 3 class
4 ci 10527 . . . . . 6 class i
52cv 1527 . . . . . 6 class 𝑥
6 cmul 10530 . . . . . 6 class ·
74, 5, 6co 7145 . . . . 5 class (i · 𝑥)
8 csin 15405 . . . . 5 class sin
97, 8cfv 6348 . . . 4 class (sin‘(i · 𝑥))
10 cdiv 11285 . . . 4 class /
119, 4, 10co 7145 . . 3 class ((sin‘(i · 𝑥)) / i)
122, 3, 11cmpt 5137 . 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
131, 12wceq 1528 1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
Colors of variables: wff setvar class
This definition is referenced by:  sinhval-named  44763
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