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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 5231, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). See sinhval-named 48967 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 48970 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 48961 | . 2 class sinh | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cc 11151 | . . 3 class ℂ | |
4 | ci 11155 | . . . . . 6 class i | |
5 | 2 | cv 1536 | . . . . . 6 class 𝑥 |
6 | cmul 11158 | . . . . . 6 class · | |
7 | 4, 5, 6 | co 7431 | . . . . 5 class (i · 𝑥) |
8 | csin 16096 | . . . . 5 class sin | |
9 | 7, 8 | cfv 6563 | . . . 4 class (sin‘(i · 𝑥)) |
10 | cdiv 11918 | . . . 4 class / | |
11 | 9, 4, 10 | co 7431 | . . 3 class ((sin‘(i · 𝑥)) / i) |
12 | 2, 3, 11 | cmpt 5231 | . 2 class (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i)) |
13 | 1, 12 | wceq 1537 | 1 wff sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i)) |
Colors of variables: wff setvar class |
This definition is referenced by: sinhval-named 48967 |
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