Definition df-cot 42487

Description: Define the cotangent function. We define it this way for cmpt 4729, which requires the form ( x e. 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 (Contributed by David A. Wheeler, 14-Mar-2014.)

Assertion

Ref	Expression
df-cot	|- cot = ( x e. { y e. CC | ( 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 42484	. 2 class cot
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	2	cv 1482	. . . . 5 class x
12		ccos 14795	. . . . 5 class cos
13	11, 12	cfv 5888	. . . 4 class ( cos ` x )
14	11, 5	cfv 5888	. . . 4 class ( sin ` x )
15		cdiv 10684	. . . 4 class /
16	13, 14, 15	co 6650	. . 3 class ( ( cos ` x ) / ( sin ` x ) )
17	2, 10, 16	cmpt 4729	. 2 class ( x e. { y e. CC | ( sin ` y ) =/= 0 } |-> ( ( cos ` x ) / ( sin ` x ) ) )
18	1, 17	wceq 1483	1 wff cot = ( x e. { y e. CC | ( sin ` y ) =/= 0 } |-> ( ( cos ` x ) / ( sin ` x ) ) )

This definition is referenced by:  cotval  42490