Definition df-csc

Description: Define the cosecant function. We define it this way for cmpt , which requires the form ( x e. 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)

	Ref	Expression
Assertion	df-csc	$$⊢\csc =\left(x\in \left\{y\in ℂ|\sin y\ne 0\right\}⟼\frac{1}{\sin x}\right)$$

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		ccsc	$$class\csc$$
1		vx	$$setvarx$$
2		vy	$$setvary$$
3		cc	$$classℂ$$
4		csin	$$class\sin$$
5	2	cv	$$setvary$$
6	5 4	cfv	$$class\sin y$$
7		cc0	$$class0$$
8	6 7	wne	$$wff\sin y\ne 0$$
9	8 2 3	crab	$$class\left\{y\in ℂ|\sin y\ne 0\right\}$$
10		c1	$$class1$$
11		cdiv	$$class÷$$
12	1	cv	$$setvarx$$
13	12 4	cfv	$$class\sin x$$
14	10 13 11	co	$$class\left(\frac{1}{\sin x}\right)$$
15	1 9 14	cmpt	$$class\left(x\in \left\{y\in ℂ|\sin y\ne 0\right\}⟼\frac{1}{\sin x}\right)$$
16	0 15	wceq	$$wff\csc =\left(x\in \left\{y\in ℂ|\sin y\ne 0\right\}⟼\frac{1}{\sin x}\right)$$