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Metamath Proof Explorer

Theorem blcntrps

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 ) )