Definition df-tan 16151

Description: Define the tangent function. We define it this way for cmpt 5194, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). (Contributed by Mario Carneiro, 14-Mar-2014.)

Assertion

Ref	Expression
df-tan	⊢ tan = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))

Detailed syntax breakdown of Definition df-tan

Step	Hyp	Ref	Expression
1		ctan 16145	. 2 class tan
2		vx	. . 3 setvar 𝑥
3		ccos 16144	. . . . 5 class cos
4	3	ccnv 5662	. . . 4 class ◡cos
5		cc 11117	. . . . 5 class ℂ
6		cc0 11119	. . . . . 6 class 0
7	6	csn 4591	. . . . 5 class {0}
8	5, 7	cdif 3903	. . . 4 class (ℂ ∖ {0})
9	4, 8	cima 5666	. . 3 class (◡cos “ (ℂ ∖ {0}))
10	2	cv 1569	. . . . 5 class 𝑥
11		csin 16143	. . . . 5 class sin
12	10, 11	cfv 6540	. . . 4 class (sin‘𝑥)
13	10, 3	cfv 6540	. . . 4 class (cos‘𝑥)
14		cdiv 11890	. . . 4 class /
15	12, 13, 14	co 7419	. . 3 class ((sin‘𝑥) / (cos‘𝑥))
16	2, 9, 15	cmpt 5194	. 2 class (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
17	1, 16	wceq 1570	1 wff tan = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))

This definition is used by:  tanval  16210  dvtan  38382  tannpoly  47704