Metamath Proof Explorer


Theorem blcntr

Description: A ball contains its center. (Contributed by NM, 2-Sep-2006) (Revised by Mario Carneiro, 12-Nov-2013)

Ref Expression
Assertion blcntr ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃𝑋𝑅 ∈ ℝ+ ) → 𝑃 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) )

Proof

Step Hyp Ref Expression
1 rpxr ( 𝑅 ∈ ℝ+𝑅 ∈ ℝ* )
2 rpgt0 ( 𝑅 ∈ ℝ+ → 0 < 𝑅 )
3 1 2 jca ( 𝑅 ∈ ℝ+ → ( 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) )
4 xblcntr ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃𝑋 ∧ ( 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ) → 𝑃 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) )
5 3 4 syl3an3 ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃𝑋𝑅 ∈ ℝ+ ) → 𝑃 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) )