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Definition df-sinh 49252
Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5225, which requires the form (𝑥𝐴𝐵). See sinhval-named 49255 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 49258 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 49249 . 2 class sinh
2 vx . . 3 setvar 𝑥
3 cc 11153 . . 3 class
4 ci 11157 . . . . . 6 class i
52cv 1539 . . . . . 6 class 𝑥
6 cmul 11160 . . . . . 6 class ·
74, 5, 6co 7431 . . . . 5 class (i · 𝑥)
8 csin 16099 . . . . 5 class sin
97, 8cfv 6561 . . . 4 class (sin‘(i · 𝑥))
10 cdiv 11920 . . . 4 class /
119, 4, 10co 7431 . . 3 class ((sin‘(i · 𝑥)) / i)
122, 3, 11cmpt 5225 . 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
131, 12wceq 1540 1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
Colors of variables: wff setvar class
This definition is referenced by:  sinhval-named  49255
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