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Metamath Proof Explorer

Theorem distspace

Description: A set X together with a (distance) function D which is a pseudometric is adistance space (according to E. Deza, M.M. Deza: "Dictionary of Distances", Elsevier, 2006), i.e. a (base) set X equipped with adistance D , which is a mapping of two elements of the base set to the (extended) reals and which is nonnegative, symmetric and equal to 0 if the two elements are equal. (Contributed by AV, 15-Oct-2021) (Revised by AV, 5-Jul-2022)

Ref Expression
Assertion distspace 
|- ( ( D e. ( PsMet ` X ) /\ A e. X /\ B e. X ) -> ( ( D : ( X X. X ) --> RR* /\ ( A D A ) = 0 ) /\ ( 0 <_ ( A D B ) /\ ( A D B ) = ( B D A ) ) ) )

Proof

Step Hyp Ref Expression
1 psmetf 
 |-  ( D e. ( PsMet ` X ) -> D : ( X X. X ) --> RR* )
2 1 3ad2ant1 
 |-  ( ( D e. ( PsMet ` X ) /\ A e. X /\ B e. X ) -> D : ( X X. X ) --> RR* )
3 psmet0 
 |-  ( ( D e. ( PsMet ` X ) /\ A e. X ) -> ( A D A ) = 0 )
4 3 3adant3 
 |-  ( ( D e. ( PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D A ) = 0 )
5 2 4 jca 
 |-  ( ( D e. ( PsMet ` X ) /\ A e. X /\ B e. X ) -> ( D : ( X X. X ) --> RR* /\ ( A D A ) = 0 ) )
6 psmetge0 
 |-  ( ( D e. ( PsMet ` X ) /\ A e. X /\ B e. X ) -> 0 <_ ( A D B ) )
7 psmetsym 
 |-  ( ( D e. ( PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) = ( B D A ) )
8 5 6 7 jca32 
 |-  ( ( D e. ( PsMet ` X ) /\ A e. X /\ B e. X ) -> ( ( D : ( X X. X ) --> RR* /\ ( A D A ) = 0 ) /\ ( 0 <_ ( A D B ) /\ ( A D B ) = ( B D A ) ) ) )