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Metamath Proof Explorer

Theorem xmetgt0

Description: The distance function of an extended metric space is positive for unequal points. (Contributed by Mario Carneiro, 20-Aug-2015)

Ref Expression
Assertion xmetgt0 
|- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A =/= B <-> 0 < ( A D B ) ) )

Proof

Step Hyp Ref Expression
1 xmetge0 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> 0 <_ ( A D B ) )
2 1 biantrud 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) <_ 0 <-> ( ( A D B ) <_ 0 /\ 0 <_ ( A D B ) ) ) )
3 xmetcl 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR* )
4 0xr 
 |-  0 e. RR*
5 xrletri3 
 |-  ( ( ( A D B ) e. RR* /\ 0 e. RR* ) -> ( ( A D B ) = 0 <-> ( ( A D B ) <_ 0 /\ 0 <_ ( A D B ) ) ) )
6 3 4 5 sylancl 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) = 0 <-> ( ( A D B ) <_ 0 /\ 0 <_ ( A D B ) ) ) )
7 2 6 bitr4d 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) <_ 0 <-> ( A D B ) = 0 ) )
8 xrlenlt 
 |-  ( ( ( A D B ) e. RR* /\ 0 e. RR* ) -> ( ( A D B ) <_ 0 <-> -. 0 < ( A D B ) ) )
9 3 4 8 sylancl 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) <_ 0 <-> -. 0 < ( A D B ) ) )
10 xmeteq0 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) = 0 <-> A = B ) )
11 7 9 10 3bitr3d 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( -. 0 < ( A D B ) <-> A = B ) )
12 11 necon1abid 
 |-  ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A =/= B <-> 0 < ( A D B ) ) )