Theorem msmet

Description: The distance function, suitably truncated, is a metric on X . (Contributed by Mario Carneiro, 12-Nov-2013)

	Ref	Expression
Hypotheses	msf.x	|- X = ( Base ` M )
	msf.d	|- D = ( ( dist ` M ) |` ( X X. X ) )
Assertion	msmet	|- ( M e. MetSp -> D e. ( Met ` X ) )

Proof

Step	Hyp	Ref	Expression
1		msf.x	|- X = ( Base ` M )
2		msf.d	|- D = ( ( dist ` M ) |` ( X X. X ) )
3		eqid	|- ( TopOpen ` M ) = ( TopOpen ` M )
4	3 1 2	isms2	|- ( M e. MetSp <-> ( D e. ( Met ` X ) /\ ( TopOpen ` M ) = ( MetOpen ` D ) ) )
5	4	simplbi	|- ( M e. MetSp -> D e. ( Met ` X ) )