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Metamath Proof Explorer

Theorem isms2

Description: Express the predicate " <. X , D >. is a metric space" with underlying set X and distance function D . (Contributed by NM, 27-Aug-2006) (Revised by Mario Carneiro, 24-Aug-2015)

Ref Expression
Hypotheses isms.j 
|- J = ( TopOpen ` K )
isms.x 
|- X = ( Base ` K )
isms.d 
|- D = ( ( dist ` K ) |` ( X X. X ) )
Assertion isms2 
|- ( K e. MetSp <-> ( D e. ( Met ` X ) /\ J = ( MetOpen ` D ) ) )

Proof

Step Hyp Ref Expression
1 isms.j 
 |-  J = ( TopOpen ` K )
2 isms.x 
 |-  X = ( Base ` K )
3 isms.d 
 |-  D = ( ( dist ` K ) |` ( X X. X ) )
4 1 2 3 isxms2 
 |-  ( K e. *MetSp <-> ( D e. ( *Met ` X ) /\ J = ( MetOpen ` D ) ) )
5 4 anbi1i 
 |-  ( ( K e. *MetSp /\ D e. ( Met ` X ) ) <-> ( ( D e. ( *Met ` X ) /\ J = ( MetOpen ` D ) ) /\ D e. ( Met ` X ) ) )
6 1 2 3 isms 
 |-  ( K e. MetSp <-> ( K e. *MetSp /\ D e. ( Met ` X ) ) )
7 metxmet 
 |-  ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
8 7 pm4.71ri 
 |-  ( D e. ( Met ` X ) <-> ( D e. ( *Met ` X ) /\ D e. ( Met ` X ) ) )
9 8 anbi1i 
 |-  ( ( D e. ( Met ` X ) /\ J = ( MetOpen ` D ) ) <-> ( ( D e. ( *Met ` X ) /\ D e. ( Met ` X ) ) /\ J = ( MetOpen ` D ) ) )
10 an32 
 |-  ( ( ( D e. ( *Met ` X ) /\ D e. ( Met ` X ) ) /\ J = ( MetOpen ` D ) ) <-> ( ( D e. ( *Met ` X ) /\ J = ( MetOpen ` D ) ) /\ D e. ( Met ` X ) ) )
11 9 10 bitri 
 |-  ( ( D e. ( Met ` X ) /\ J = ( MetOpen ` D ) ) <-> ( ( D e. ( *Met ` X ) /\ J = ( MetOpen ` D ) ) /\ D e. ( Met ` X ) ) )
12 5 6 11 3bitr4i 
 |-  ( K e. MetSp <-> ( D e. ( Met ` X ) /\ J = ( MetOpen ` D ) ) )