Theorem xmseq0

Description: The distance between two points in an extended metric space is zero iff the two points are identical. (Contributed by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Hypotheses	mscl.x	|- X = ( Base ` M )
	mscl.d	|- D = ( dist ` M )
Assertion	xmseq0	|- ( ( M e. *MetSp /\ A e. X /\ B e. X ) -> ( ( A D B ) = 0 <-> A = B ) )

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	4	eqeq1d	|- ( ( M e. *MetSp /\ A e. X /\ B e. X ) -> ( ( A ( D |` ( X X. X ) ) B ) = 0 <-> ( A D B ) = 0 ) )
6	1 2	xmsxmet2	|- ( M e. *MetSp -> ( D |` ( X X. X ) ) e. ( *Met ` X ) )
7		xmeteq0	|- ( ( ( D |` ( X X. X ) ) e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( A ( D |` ( X X. X ) ) B ) = 0 <-> A = B ) )
8	6 7	syl3an1	|- ( ( M e. *MetSp /\ A e. X /\ B e. X ) -> ( ( A ( D |` ( X X. X ) ) B ) = 0 <-> A = B ) )
9	5 8	bitr3d	|- ( ( M e. *MetSp /\ A e. X /\ B e. X ) -> ( ( A D B ) = 0 <-> A = B ) )