Metamath Proof Explorer


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 ) )