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Definition df-sinh 42215
Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 4637, which requires the form (𝑥𝐴𝐵). See sinhval-named 42218 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 metamath). See sinh-conventional 42221 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 42212 . 2 class sinh
2 vx . . 3 setvar 𝑥
3 cc 9790 . . 3 class
4 ci 9794 . . . . . 6 class i
52cv 1473 . . . . . 6 class 𝑥
6 cmul 9797 . . . . . 6 class ·
74, 5, 6co 6526 . . . . 5 class (i · 𝑥)
8 csin 14581 . . . . 5 class sin
97, 8cfv 5789 . . . 4 class (sin‘(i · 𝑥))
10 cdiv 10535 . . . 4 class /
119, 4, 10co 6526 . . 3 class ((sin‘(i · 𝑥)) / i)
122, 3, 11cmpt 4637 . 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
131, 12wceq 1474 1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
Colors of variables: wff setvar class
This definition is referenced by:  sinhval-named  42218
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