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Metamath Proof Explorer

Theorem blnei

Description: A ball around a point is a neighborhood of the point. (Contributed by NM, 8-Nov-2007) (Revised by Mario Carneiro, 24-Aug-2015)

Ref Expression
Hypothesis mopni.1 ⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
Assertion blnei ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+ ) → ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) )

Proof

Step Hyp Ref Expression
1 mopni.1 ⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
2 1 mopntop ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ Top )
3 2 3ad2ant1 ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+ ) → 𝐽 ∈ Top )
4 rpxr ⊢ ( 𝑅 ∈ ℝ+ → 𝑅 ∈ ℝ* )
5 1 blopn ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) → ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ∈ 𝐽 )
6 4 5 syl3an3 ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+ ) → ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ∈ 𝐽 )
7 blcntr ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+ ) → 𝑃 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) )
8 opnneip ⊢ ( ( 𝐽 ∈ Top ∧ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ∈ 𝐽 ∧ 𝑃 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ) → ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) )
9 3 6 7 8 syl3anc ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+ ) → ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) )