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 = ( MetOpen ‘ 𝐷 )
Assertion	blopn	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) → ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ∈ 𝐽 )

Proof

Step	Hyp	Ref	Expression
1		mopni.1	⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
2	1	blssopn	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ran ( ball ‘ 𝐷 ) ⊆ 𝐽 )
3	2	3ad2ant1	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) → ran ( ball ‘ 𝐷 ) ⊆ 𝐽 )
4		blelrn	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) → ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ∈ ran ( ball ‘ 𝐷 ) )
5	3 4	sseldd	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) → ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ∈ 𝐽 )