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Metamath Proof Explorer

Theorem isxms

Description: Express the predicate " <. X , D >. is an extended metric space" with underlying set X and distance function D . (Contributed by Mario Carneiro, 2-Sep-2015)

Ref Expression
Hypotheses isms.j 
|- J = ( TopOpen ` K )
isms.x 
|- X = ( Base ` K )
isms.d 
|- D = ( ( dist ` K ) |` ( X X. X ) )
Assertion isxms 
|- ( K e. *MetSp <-> ( K e. TopSp /\ 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 fveq2 
 |-  ( f = K -> ( TopOpen ` f ) = ( TopOpen ` K ) )
5 4 1 eqtr4di 
 |-  ( f = K -> ( TopOpen ` f ) = J )
6 fveq2 
 |-  ( f = K -> ( dist ` f ) = ( dist ` K ) )
7 fveq2 
 |-  ( f = K -> ( Base ` f ) = ( Base ` K ) )
8 7 2 eqtr4di 
 |-  ( f = K -> ( Base ` f ) = X )
9 8 sqxpeqd 
 |-  ( f = K -> ( ( Base ` f ) X. ( Base ` f ) ) = ( X X. X ) )
10 6 9 reseq12d 
 |-  ( f = K -> ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) = ( ( dist ` K ) |` ( X X. X ) ) )
11 10 3 eqtr4di 
 |-  ( f = K -> ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) = D )
12 11 fveq2d 
 |-  ( f = K -> ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) = ( MetOpen ` D ) )
13 5 12 eqeq12d 
 |-  ( f = K -> ( ( TopOpen ` f ) = ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) <-> J = ( MetOpen ` D ) ) )
14 df-xms 
 |-  *MetSp = { f e. TopSp | ( TopOpen ` f ) = ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) }
15 13 14 elrab2 
 |-  ( K e. *MetSp <-> ( K e. TopSp /\ J = ( MetOpen ` D ) ) )