Metamath Proof Explorer

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 e. ( *Met ` 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 xblcntr 
 |-  ( ( D e. ( *Met ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> P e. ( P ( ball ` D ) R ) )
5 3 4 syl3an3 
 |-  ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR+ ) -> P e. ( P ( ball ` D ) R ) )