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 ) ∧ 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		xblcntr	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ ( 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ) → 𝑃 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) )
5	3 4	syl3an3	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+ ) → 𝑃 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) )