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Definition df-sinh 47618
Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5227, which requires the form (𝑥𝐴𝐵). See sinhval-named 47621 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 47624 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 47615 . 2 class sinh
2 vx . . 3 setvar 𝑥
3 cc 11095 . . 3 class
4 ci 11099 . . . . . 6 class i
52cv 1541 . . . . . 6 class 𝑥
6 cmul 11102 . . . . . 6 class ·
74, 5, 6co 7396 . . . . 5 class (i · 𝑥)
8 csin 15994 . . . . 5 class sin
97, 8cfv 6535 . . . 4 class (sin‘(i · 𝑥))
10 cdiv 11858 . . . 4 class /
119, 4, 10co 7396 . . 3 class ((sin‘(i · 𝑥)) / i)
122, 3, 11cmpt 5227 . 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
131, 12wceq 1542 1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
Colors of variables: wff setvar class
This definition is referenced by:  sinhval-named  47621
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