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Metamath Proof Explorer

Theorem setsmsds

Description: The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015) (Proof shortened by AV, 11-Nov-2024)

Ref Expression
Hypotheses setsms.x 
|- ( ph -> X = ( Base ` M ) )
setsms.d 
|- ( ph -> D = ( ( dist ` M ) |` ( X X. X ) ) )
setsms.k 
|- ( ph -> K = ( M sSet <. ( TopSet ` ndx ) , ( MetOpen ` D ) >. ) )
Assertion setsmsds 
|- ( ph -> ( dist ` M ) = ( dist ` K ) )

Proof

Step Hyp Ref Expression
1 setsms.x 
 |-  ( ph -> X = ( Base ` M ) )
2 setsms.d 
 |-  ( ph -> D = ( ( dist ` M ) |` ( X X. X ) ) )
3 setsms.k 
 |-  ( ph -> K = ( M sSet <. ( TopSet ` ndx ) , ( MetOpen ` D ) >. ) )
4 dsid 
 |-  dist = Slot ( dist ` ndx )
5 dsndxntsetndx 
 |-  ( dist ` ndx ) =/= ( TopSet ` ndx )
6 4 5 setsnid 
 |-  ( dist ` M ) = ( dist ` ( M sSet <. ( TopSet ` ndx ) , ( MetOpen ` D ) >. ) )
7 3 fveq2d 
 |-  ( ph -> ( dist ` K ) = ( dist ` ( M sSet <. ( TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
8 6 7 eqtr4id 
 |-  ( ph -> ( dist ` M ) = ( dist ` K ) )