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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-sinh | Structured version Visualization version GIF version |
Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt 5137, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). 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.) |
Ref | Expression |
---|---|
df-sinh | ⊢ sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csinh 44757 | . 2 class sinh | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cc 10523 | . . 3 class ℂ | |
4 | ci 10527 | . . . . . 6 class i | |
5 | 2 | cv 1527 | . . . . . 6 class 𝑥 |
6 | cmul 10530 | . . . . . 6 class · | |
7 | 4, 5, 6 | co 7145 | . . . . 5 class (i · 𝑥) |
8 | csin 15405 | . . . . 5 class sin | |
9 | 7, 8 | cfv 6348 | . . . 4 class (sin‘(i · 𝑥)) |
10 | cdiv 11285 | . . . 4 class / | |
11 | 9, 4, 10 | co 7145 | . . 3 class ((sin‘(i · 𝑥)) / i) |
12 | 2, 3, 11 | cmpt 5137 | . 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i)) |
13 | 1, 12 | wceq 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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