Definition df-cot 44865

Description: Define the cotangent function. We define it this way for cmpt 5146, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). The cot function is defined in ISO 80000-2:2009(E) operation 2-13.5 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 5146 (Contributed by David A. Wheeler, 14-Mar-2014.)

Assertion

Ref	Expression
df-cot	⊢ cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-cot

Step	Hyp	Ref	Expression
1		ccot 44862	. 2 class cot
2		vx	. . 3 setvar 𝑥
3		vy	. . . . . . 7 setvar 𝑦
4	3	cv 1536	. . . . . 6 class 𝑦
5		csin 15417	. . . . . 6 class sin
6	4, 5	cfv 6355	. . . . 5 class (sin‘𝑦)
7		cc0 10537	. . . . 5 class 0
8	6, 7	wne 3016	. . . 4 wff (sin‘𝑦) ≠ 0
9		cc 10535	. . . 4 class ℂ
10	8, 3, 9	crab 3142	. . 3 class {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0}
11	2	cv 1536	. . . . 5 class 𝑥
12		ccos 15418	. . . . 5 class cos
13	11, 12	cfv 6355	. . . 4 class (cos‘𝑥)
14	11, 5	cfv 6355	. . . 4 class (sin‘𝑥)
15		cdiv 11297	. . . 4 class /
16	13, 14, 15	co 7156	. . 3 class ((cos‘𝑥) / (sin‘𝑥))
17	2, 10, 16	cmpt 5146	. 2 class (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
18	1, 17	wceq 1537	1 wff cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))

This definition is referenced by:  cotval  44868