Theorem xblcntrps

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	xblcntrps	⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ ( 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ) → 𝑃 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		simp2	⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ ( 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ) → 𝑃 ∈ 𝑋 )
2		psmet0	⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ) → ( 𝑃 𝐷 𝑃 ) = 0 )
3	2	3adant3	⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ ( 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ) → ( 𝑃 𝐷 𝑃 ) = 0 )
4		simp3r	⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ ( 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ) → 0 < 𝑅 )
5	3 4	eqbrtrd	⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ ( 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ) → ( 𝑃 𝐷 𝑃 ) < 𝑅 )
6		elblps	⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ* ) → ( 𝑃 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ↔ ( 𝑃 ∈ 𝑋 ∧ ( 𝑃 𝐷 𝑃 ) < 𝑅 ) ) )
7	6	3adant3r	⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ ( 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ) → ( 𝑃 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ↔ ( 𝑃 ∈ 𝑋 ∧ ( 𝑃 𝐷 𝑃 ) < 𝑅 ) ) )
8	1 5 7	mpbir2and	⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ ( 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ) → 𝑃 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) )