Theorem elbl2

Description: Membership in a ball. (Contributed by NM, 9-Mar-2007)

	Ref	Expression
Assertion	elbl2	|- ( ( ( D e. ( *Met ` X ) /\ R e. RR* ) /\ ( P e. X /\ A e. X ) ) -> ( A e. ( P ( ball ` D ) R ) <-> ( P D A ) < R ) )

Proof

Step	Hyp	Ref	Expression
1		simprr	|- ( ( ( D e. ( *Met ` X ) /\ R e. RR* ) /\ ( P e. X /\ A e. X ) ) -> A e. X )
2		elbl	|- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( A e. ( P ( ball ` D ) R ) <-> ( A e. X /\ ( P D A ) < R ) ) )
3	2	3expa	|- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ R e. RR* ) -> ( A e. ( P ( ball ` D ) R ) <-> ( A e. X /\ ( P D A ) < R ) ) )
4	3	an32s	|- ( ( ( D e. ( *Met ` X ) /\ R e. RR* ) /\ P e. X ) -> ( A e. ( P ( ball ` D ) R ) <-> ( A e. X /\ ( P D A ) < R ) ) )
5	4	adantrr	|- ( ( ( D e. ( *Met ` X ) /\ R e. RR* ) /\ ( P e. X /\ A e. X ) ) -> ( A e. ( P ( ball ` D ) R ) <-> ( A e. X /\ ( P D A ) < R ) ) )
6	1 5	mpbirand	|- ( ( ( D e. ( *Met ` X ) /\ R e. RR* ) /\ ( P e. X /\ A e. X ) ) -> ( A e. ( P ( ball ` D ) R ) <-> ( P D A ) < R ) )