Theorem xmsge0

Description: The distance function in an extended metric space is nonnegative. (Contributed by Mario Carneiro, 4-Oct-2015)

	Ref	Expression
Hypotheses	mscl.x	|- X = ( Base ` M )
	mscl.d	|- D = ( dist ` M )
Assertion	xmsge0	|- ( ( M e. *MetSp /\ A e. X /\ B e. X ) -> 0 <_ ( A D B ) )

Proof

Step	Hyp	Ref	Expression
1		mscl.x	|- X = ( Base ` M )
2		mscl.d	|- D = ( dist ` M )
3	1 2	xmsxmet2	|- ( M e. *MetSp -> ( D |` ( X X. X ) ) e. ( *Met ` X ) )
4		xmetge0	|- ( ( ( D |` ( X X. X ) ) e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> 0 <_ ( A ( D |` ( X X. X ) ) B ) )
5	3 4	syl3an1	|- ( ( M e. *MetSp /\ A e. X /\ B e. X ) -> 0 <_ ( A ( D |` ( X X. X ) ) B ) )
6		ovres	|- ( ( A e. X /\ B e. X ) -> ( A ( D |` ( X X. X ) ) B ) = ( A D B ) )
7	6	3adant1	|- ( ( M e. *MetSp /\ A e. X /\ B e. X ) -> ( A ( D |` ( X X. X ) ) B ) = ( A D B ) )
8	5 7	breqtrd	|- ( ( M e. *MetSp /\ A e. X /\ B e. X ) -> 0 <_ ( A D B ) )