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Metamath Proof Explorer

Theorem mscl

Description: Closure of the distance function of a metric space. (Contributed by NM, 30-Aug-2006) (Revised by Mario Carneiro, 2-Oct-2015)

Ref Expression
Hypotheses mscl.x 
|- X = ( Base ` M )
mscl.d 
|- D = ( dist ` M )
Assertion mscl 
|- ( ( M e. MetSp /\ A e. X /\ B e. X ) -> ( A D B ) e. RR )

Proof

Step Hyp Ref Expression
1 mscl.x 
 |-  X = ( Base ` M )
2 mscl.d 
 |-  D = ( dist ` M )
3 ovres 
 |-  ( ( A e. X /\ B e. X ) -> ( A ( D |` ( X X. X ) ) B ) = ( A D B ) )
4 3 3adant1 
 |-  ( ( M e. MetSp /\ A e. X /\ B e. X ) -> ( A ( D |` ( X X. X ) ) B ) = ( A D B ) )
5 1 2 msmet2 
 |-  ( M e. MetSp -> ( D |` ( X X. X ) ) e. ( Met ` X ) )
6 metcl 
 |-  ( ( ( D |` ( X X. X ) ) e. ( Met ` X ) /\ A e. X /\ B e. X ) -> ( A ( D |` ( X X. X ) ) B ) e. RR )
7 5 6 syl3an1 
 |-  ( ( M e. MetSp /\ A e. X /\ B e. X ) -> ( A ( D |` ( X X. X ) ) B ) e. RR )
8 4 7 eqeltrrd 
 |-  ( ( M e. MetSp /\ A e. X /\ B e. X ) -> ( A D B ) e. RR )