Theorem blopn

Description: A ball of a metric space is an open set. (Contributed by NM, 9-Mar-2007) (Revised by Mario Carneiro, 12-Nov-2013)

	Ref	Expression
Hypothesis	mopni.1	$$⊢J=\mathrm{MetOpen}\left(D\right)$$
Assertion	blopn	$$⊢\left(D\in \mathrm{\infty Met}\left(X\right)\wedge P\in X\wedge R\in {ℝ}^{*}\right)\to P\mathrm{ball}\left(D\right)R\in J$$

Proof

Step	Hyp	Ref	Expression
1		mopni.1	$$⊢J=\mathrm{MetOpen}\left(D\right)$$
2	1	blssopn	$$⊢D\in \mathrm{\infty Met}\left(X\right)\to \mathrm{ran}\mathrm{ball}\left(D\right)\subseteq J$$
3	2	3ad2ant1	$$⊢\left(D\in \mathrm{\infty Met}\left(X\right)\wedge P\in X\wedge R\in {ℝ}^{*}\right)\to \mathrm{ran}\mathrm{ball}\left(D\right)\subseteq J$$
4		blelrn	$$⊢\left(D\in \mathrm{\infty Met}\left(X\right)\wedge P\in X\wedge R\in {ℝ}^{*}\right)\to P\mathrm{ball}\left(D\right)R\in \mathrm{ran}\mathrm{ball}\left(D\right)$$
5	3 4	sseldd	$$⊢\left(D\in \mathrm{\infty Met}\left(X\right)\wedge P\in X\wedge R\in {ℝ}^{*}\right)\to P\mathrm{ball}\left(D\right)R\in J$$