Definition df-sec 45683

Description: Define the secant function. We define it this way for cmpt 5112, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). The sec function is defined in ISO 80000-2:2009(E) operation 2-13.6 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 5112. (Contributed by David A. Wheeler, 14-Mar-2014.)

Assertion

Ref	Expression
df-sec	⊢ sec = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 / (cos‘𝑥)))

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-sec

Step	Hyp	Ref	Expression
1		csec 45680	. 2 class sec
2		vx	. . 3 setvar 𝑥
3		vy	. . . . . . 7 setvar 𝑦
4	3	cv 1537	. . . . . 6 class 𝑦
5		ccos 15466	. . . . . 6 class cos
6	4, 5	cfv 6335	. . . . 5 class (cos‘𝑦)
7		cc0 10575	. . . . 5 class 0
8	6, 7	wne 2951	. . . 4 wff (cos‘𝑦) ≠ 0
9		cc 10573	. . . 4 class ℂ
10	8, 3, 9	crab 3074	. . 3 class {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0}
11		c1 10576	. . . 4 class 1
12	2	cv 1537	. . . . 5 class 𝑥
13	12, 5	cfv 6335	. . . 4 class (cos‘𝑥)
14		cdiv 11335	. . . 4 class /
15	11, 13, 14	co 7150	. . 3 class (1 / (cos‘𝑥))
16	2, 10, 15	cmpt 5112	. 2 class (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 / (cos‘𝑥)))
17	1, 16	wceq 1538	1 wff sec = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 / (cos‘𝑥)))

This definition is referenced by:  secval  45686