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Mirrors > Home > ILE Home > Th. List > df-tan | GIF version |
Description: Define the tangent function. We define it this way for cmpt 4040, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). (Contributed by Mario Carneiro, 14-Mar-2014.) |
Ref | Expression |
---|---|
df-tan | ⊢ tan = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ctan 11581 | . 2 class tan | |
2 | vx | . . 3 setvar 𝑥 | |
3 | ccos 11580 | . . . . 5 class cos | |
4 | 3 | ccnv 4600 | . . . 4 class ◡cos |
5 | cc 7745 | . . . . 5 class ℂ | |
6 | cc0 7747 | . . . . . 6 class 0 | |
7 | 6 | csn 3573 | . . . . 5 class {0} |
8 | 5, 7 | cdif 3111 | . . . 4 class (ℂ ∖ {0}) |
9 | 4, 8 | cima 4604 | . . 3 class (◡cos “ (ℂ ∖ {0})) |
10 | 2 | cv 1341 | . . . . 5 class 𝑥 |
11 | csin 11579 | . . . . 5 class sin | |
12 | 10, 11 | cfv 5185 | . . . 4 class (sin‘𝑥) |
13 | 10, 3 | cfv 5185 | . . . 4 class (cos‘𝑥) |
14 | cdiv 8562 | . . . 4 class / | |
15 | 12, 13, 14 | co 5839 | . . 3 class ((sin‘𝑥) / (cos‘𝑥)) |
16 | 2, 9, 15 | cmpt 4040 | . 2 class (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥))) |
17 | 1, 16 | wceq 1342 | 1 wff tan = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥))) |
Colors of variables: wff set class |
This definition is referenced by: tanvalap 11643 |
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