Metamath Proof Explorer

Theorem xmssym

Description: The distance function in an extended metric space is symmetric. (Contributed by Mario Carneiro, 2-Oct-2015)

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

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 xmetsym 
 |-  ( ( ( D |` ( X X. X ) ) e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A ( D |` ( X X. X ) ) B ) = ( B ( D |` ( X X. X ) ) A ) )
5 3 4 syl3an1 
 |-  ( ( M e. *MetSp /\ A e. X /\ B e. X ) -> ( A ( D |` ( X X. X ) ) B ) = ( B ( D |` ( X X. X ) ) A ) )
6 simp2 
 |-  ( ( M e. *MetSp /\ A e. X /\ B e. X ) -> A e. X )
7 simp3 
 |-  ( ( M e. *MetSp /\ A e. X /\ B e. X ) -> B e. X )
8 6 7 ovresd 
 |-  ( ( M e. *MetSp /\ A e. X /\ B e. X ) -> ( A ( D |` ( X X. X ) ) B ) = ( A D B ) )
9 7 6 ovresd 
 |-  ( ( M e. *MetSp /\ A e. X /\ B e. X ) -> ( B ( D |` ( X X. X ) ) A ) = ( B D A ) )
10 5 8 9 3eqtr3d 
 |-  ( ( M e. *MetSp /\ A e. X /\ B e. X ) -> ( A D B ) = ( B D A ) )