Metamath Proof Explorer

Theorem xblcntr

Description: A ball contains its center. (Contributed by NM, 2-Sep-2006) (Revised by Mario Carneiro, 12-Nov-2013)

Ref Expression
Assertion xblcntr 
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> P e. ( P ( ball ` D ) R ) )

Proof

Step Hyp Ref Expression
1 simp2 
 |-  ( ( D e. ( *Met ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> P e. X )
2 xmet0 
 |-  ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( P D P ) = 0 )
3 2 3adant3 
 |-  ( ( D e. ( *Met ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> ( P D P ) = 0 )
4 simp3r 
 |-  ( ( D e. ( *Met ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> 0 < R )
5 3 4 eqbrtrd 
 |-  ( ( D e. ( *Met ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> ( P D P ) < R )
6 elbl 
 |-  ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P e. ( P ( ball ` D ) R ) <-> ( P e. X /\ ( P D P ) < R ) ) )
7 6 3adant3r 
 |-  ( ( D e. ( *Met ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> ( P e. ( P ( ball ` D ) R ) <-> ( P e. X /\ ( P D P ) < R ) ) )
8 1 5 7 mpbir2and 
 |-  ( ( D e. ( *Met ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> P e. ( P ( ball ` D ) R ) )