Definition df-csc 42285

Description: Define the cosecant function. We define it this way for cmpt 4643, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). The csc function is defined in ISO 80000-2:2009(E) operation 2-13.7 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)

Assertion

Ref	Expression
df-csc	⊢ csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥)))

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-csc

Step	Hyp	Ref	Expression
1		ccsc 42282	. 2 class csc
2		vx	. . 3 setvar 𝑥
3		vy	. . . . . . 7 setvar 𝑦
4	3	cv 1474	. . . . . 6 class 𝑦
5		csin 14633	. . . . . 6 class sin
6	4, 5	cfv 5804	. . . . 5 class (sin‘𝑦)
7		cc0 9815	. . . . 5 class 0
8	6, 7	wne 2780	. . . 4 wff (sin‘𝑦) ≠ 0
9		cc 9813	. . . 4 class ℂ
10	8, 3, 9	crab 2900	. . 3 class {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0}
11		c1 9816	. . . 4 class 1
12	2	cv 1474	. . . . 5 class 𝑥
13	12, 5	cfv 5804	. . . 4 class (sin‘𝑥)
14		cdiv 10563	. . . 4 class /
15	11, 13, 14	co 6549	. . 3 class (1 / (sin‘𝑥))
16	2, 10, 15	cmpt 4643	. 2 class (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥)))
17	1, 16	wceq 1475	1 wff csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥)))

This definition is referenced by:  cscval  42288