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 e. ( PsMet ` X ) /\ P e. X /\ R e. RR+ ) -> P e. ( P ( ball ` D ) R ) )

Proof

Step	Hyp	Ref	Expression
1		rpxr	|- ( R e. RR+ -> R e. RR* )
2		rpgt0	|- ( R e. RR+ -> 0 < R )
3	1 2	jca	|- ( R e. RR+ -> ( R e. RR* /\ 0 < R ) )
4		xblcntrps	|- ( ( D e. ( PsMet ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> P e. ( P ( ball ` D ) R ) )
5	3 4	syl3an3	|- ( ( D e. ( PsMet ` X ) /\ P e. X /\ R e. RR+ ) -> P e. ( P ( ball ` D ) R ) )