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Metamath Proof Explorer

Theorem isms

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 isms 
|- ( K e. MetSp <-> ( K e. *MetSp /\ D e. ( Met ` X ) ) )

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 -> ( dist ` f ) = ( dist ` K ) )
5 fveq2 
 |-  ( f = K -> ( Base ` f ) = ( Base ` K ) )
6 5 2 eqtr4di 
 |-  ( f = K -> ( Base ` f ) = X )
7 6 sqxpeqd 
 |-  ( f = K -> ( ( Base ` f ) X. ( Base ` f ) ) = ( X X. X ) )
8 4 7 reseq12d 
 |-  ( f = K -> ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) = ( ( dist ` K ) |` ( X X. X ) ) )
9 8 3 eqtr4di 
 |-  ( f = K -> ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) = D )
10 6 fveq2d 
 |-  ( f = K -> ( Met ` ( Base ` f ) ) = ( Met ` X ) )
11 9 10 eleq12d 
 |-  ( f = K -> ( ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) e. ( Met ` ( Base ` f ) ) <-> D e. ( Met ` X ) ) )
12 df-ms 
 |-  MetSp = { f e. *MetSp | ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) e. ( Met ` ( Base ` f ) ) }
13 11 12 elrab2 
 |-  ( K e. MetSp <-> ( K e. *MetSp /\ D e. ( Met ` X ) ) )