Definition df-tan 16252

Description: Define the tangent function. We define it this way for cmpt 5189, 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 16246	. 2 class tan
2		vx	. . 3 setvar 𝑥
3		ccos 16245	. . . . 5 class cos
4	3	ccnv 5654	. . . 4 class ◡cos
5		cc 11211	. . . . 5 class ℂ
6		cc0 11213	. . . . . 6 class 0
7	6	csn 4588	. . . . 5 class {0}
8	5, 7	cdif 4048	. . . 4 class (ℂ ∖ {0})
9	4, 8	cima 5658	. . 3 class (◡cos “ (ℂ ∖ {0}))
10	2	cv 1568	. . . . 5 class 𝑥
11		csin 16244	. . . . 5 class sin
12	10, 11	cfv 6547	. . . 4 class (sin‘𝑥)
13	10, 3	cfv 6547	. . . 4 class (cos‘𝑥)
14		cdiv 11986	. . . 4 class /
15	12, 13, 14	co 7430	. . 3 class ((sin‘𝑥) / (cos‘𝑥))
16	2, 9, 15	cmpt 5189	. 2 class (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
17	1, 16	wceq 1569	1 wff tan = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))

This definition is used by:  tanval  16311  dvtan  38734  tannpoly  48085