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 ) ) |
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 ) ) |