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 ) ) )