Definition df-csc 42486

Description: Define the cosecant function. We define it this way for cmpt 4729, 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.)

Assertion

Ref	Expression
df-csc	|- csc = ( x e. { y e. CC | ( sin ` y ) =/= 0 } |-> ( 1 / ( sin ` x ) ) )

Distinct variable group:   x, y

Detailed syntax breakdown of Definition df-csc

Step	Hyp	Ref	Expression
1		ccsc 42483	. 2 class csc
2		vx	. . 3 setvar x
3		vy	. . . . . . 7 setvar y
4	3	cv 1482	. . . . . 6 class y
5		csin 14794	. . . . . 6 class sin
6	4, 5	cfv 5888	. . . . 5 class ( sin ` y )
7		cc0 9936	. . . . 5 class 0
8	6, 7	wne 2794	. . . 4 wff ( sin ` y ) =/= 0
9		cc 9934	. . . 4 class CC
10	8, 3, 9	crab 2916	. . 3 class { y e. CC | ( sin ` y ) =/= 0 }
11		c1 9937	. . . 4 class 1
12	2	cv 1482	. . . . 5 class x
13	12, 5	cfv 5888	. . . 4 class ( sin ` x )
14		cdiv 10684	. . . 4 class /
15	11, 13, 14	co 6650	. . 3 class ( 1 / ( sin ` x ) )
16	2, 10, 15	cmpt 4729	. 2 class ( x e. { y e. CC | ( sin ` y ) =/= 0 } |-> ( 1 / ( sin ` x ) ) )
17	1, 16	wceq 1483	1 wff csc = ( x e. { y e. CC | ( sin ` y ) =/= 0 } |-> ( 1 / ( sin ` x ) ) )

This definition is referenced by:  cscval  42489