Definition df-tan 12163

Description: Define the tangent function. We define it this way for cmpt 4145, which requires the form ( x e. A |-> B ). (Contributed by Mario Carneiro, 14-Mar-2014.)

Assertion

Ref	Expression
df-tan	|- tan = ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( ( sin ` x ) / ( cos ` x ) ) )

Detailed syntax breakdown of Definition df-tan

Step	Hyp	Ref	Expression
1		ctan 12157	. 2 class tan
2		vx	. . 3 setvar x
3		ccos 12156	. . . . 5 class cos
4	3	ccnv 4718	. . . 4 class `' cos
5		cc 7997	. . . . 5 class CC
6		cc0 7999	. . . . . 6 class 0
7	6	csn 3666	. . . . 5 class { 0 }
8	5, 7	cdif 3194	. . . 4 class ( CC \ { 0 } )
9	4, 8	cima 4722	. . 3 class ( `' cos " ( CC \ { 0 } ) )
10	2	cv 1394	. . . . 5 class x
11		csin 12155	. . . . 5 class sin
12	10, 11	cfv 5318	. . . 4 class ( sin ` x )
13	10, 3	cfv 5318	. . . 4 class ( cos ` x )
14		cdiv 8819	. . . 4 class /
15	12, 13, 14	co 6001	. . 3 class ( ( sin ` x ) / ( cos ` x ) )
16	2, 9, 15	cmpt 4145	. 2 class ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( ( sin ` x ) / ( cos ` x ) ) )
17	1, 16	wceq 1395	1 wff tan = ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( ( sin ` x ) / ( cos ` x ) ) )

This definition is referenced by:  tanvalap  12219