Theorem blcom

Description: Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014)

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

Proof

Step	Hyp	Ref	Expression
1		elbl2	|- ( ( ( D e. ( *Met ` X ) /\ R e. RR* ) /\ ( P e. X /\ A e. X ) ) -> ( A e. ( P ( ball ` D ) R ) <-> ( P D A ) < R ) )
2		elbl3	|- ( ( ( D e. ( *Met ` X ) /\ R e. RR* ) /\ ( A e. X /\ P e. X ) ) -> ( P e. ( A ( ball ` D ) R ) <-> ( P D A ) < R ) )
3	2	ancom2s	|- ( ( ( D e. ( *Met ` X ) /\ R e. RR* ) /\ ( P e. X /\ A e. X ) ) -> ( P e. ( A ( ball ` D ) R ) <-> ( P D A ) < R ) )
4	1 3	bitr4d	|- ( ( ( D e. ( *Met ` X ) /\ R e. RR* ) /\ ( P e. X /\ A e. X ) ) -> ( A e. ( P ( ball ` D ) R ) <-> P e. ( A ( ball ` D ) R ) ) )