Theorem acosf

Description: Domain and range of the arccos function. (Contributed by Mario Carneiro, 31-Mar-2015)

	Ref	Expression
Assertion	acosf	$$⊢arccos:ℂ⟶ℂ$$

Proof

Step	Hyp	Ref	Expression
1		df-acos	$$⊢arccos=\left(x\in ℂ⟼\left(\frac{\mathrm{\pi }}{2}\right)-arcsin\left(x\right)\right)$$
2		picn	$$⊢\mathrm{\pi }\in ℂ$$
3		halfcl	$$⊢\mathrm{\pi }\in ℂ\to \frac{\mathrm{\pi }}{2}\in ℂ$$
4	2 3	ax-mp	$$⊢\frac{\mathrm{\pi }}{2}\in ℂ$$
5		asincl	$$⊢x\in ℂ\to arcsin\left(x\right)\in ℂ$$
6		subcl	$$⊢\left(\frac{\mathrm{\pi }}{2}\in ℂ\wedge arcsin\left(x\right)\in ℂ\right)\to \left(\frac{\mathrm{\pi }}{2}\right)-arcsin\left(x\right)\in ℂ$$
7	4 5 6	sylancr	$$⊢x\in ℂ\to \left(\frac{\mathrm{\pi }}{2}\right)-arcsin\left(x\right)\in ℂ$$
8	1 7	fmpti	$$⊢arccos:ℂ⟶ℂ$$