Definition df-acos 25452

Description: Define the arccosine function. See also remarks for df-asin 25451. Since we define arccos in terms of arcsin, it shares the same branch points and cuts, namely (-∞, -1) ∪ (1, +∞). (Contributed by Mario Carneiro, 31-Mar-2015.)

Assertion

Ref	Expression
df-acos	⊢ arccos = (𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥)))

Detailed syntax breakdown of Definition df-acos

Step	Hyp	Ref	Expression
1		cacos 25449	. 2 class arccos
2		vx	. . 3 setvar 𝑥
3		cc 10524	. . 3 class ℂ
4		cpi 15412	. . . . 5 class π
5		c2 11680	. . . . 5 class 2
6		cdiv 11286	. . . . 5 class /
7	4, 5, 6	co 7135	. . . 4 class (π / 2)
8	2	cv 1537	. . . . 5 class 𝑥
9		casin 25448	. . . . 5 class arcsin
10	8, 9	cfv 6324	. . . 4 class (arcsin‘𝑥)
11		cmin 10859	. . . 4 class −
12	7, 10, 11	co 7135	. . 3 class ((π / 2) − (arcsin‘𝑥))
13	2, 3, 12	cmpt 5110	. 2 class (𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥)))
14	1, 13	wceq 1538	1 wff arccos = (𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥)))

This definition is referenced by:  acosf  25460  acosval  25469  dvacos  35142