Metamath Proof Explorer

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 β€˜ 𝐷 )
Assertion blopn ( ( 𝐷 ∈ ( ∞Met β€˜ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) β†’ ( 𝑃 ( ball β€˜ 𝐷 ) 𝑅 ) ∈ 𝐽 )

Proof

Step Hyp Ref Expression
1 mopni.1 ⊒ 𝐽 = ( MetOpen β€˜ 𝐷 )
2 1 blssopn ⊒ ( 𝐷 ∈ ( ∞Met β€˜ 𝑋 ) β†’ ran ( ball β€˜ 𝐷 ) βŠ† 𝐽 )
3 2 3ad2ant1 ⊒ ( ( 𝐷 ∈ ( ∞Met β€˜ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) β†’ ran ( ball β€˜ 𝐷 ) βŠ† 𝐽 )
4 blelrn ⊒ ( ( 𝐷 ∈ ( ∞Met β€˜ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) β†’ ( 𝑃 ( ball β€˜ 𝐷 ) 𝑅 ) ∈ ran ( ball β€˜ 𝐷 ) )
5 3 4 sseldd ⊒ ( ( 𝐷 ∈ ( ∞Met β€˜ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) β†’ ( 𝑃 ( ball β€˜ 𝐷 ) 𝑅 ) ∈ 𝐽 )