Database
BASIC TOPOLOGY
Metric spaces
Metric space balls
blcntrps
Metamath Proof Explorer
Description: A ball contains its center. (Contributed by NM , 2-Sep-2006) (Revised by Mario Carneiro , 12-Nov-2013) (Revised by Thierry Arnoux , 11-Mar-2018)
Ref
Expression
Assertion
blcntrps
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ 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
xblcntrps
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ ( 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ) → 𝑃 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) )
5
3 4
syl3an3
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+ ) → 𝑃 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) )