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 ) ) ) )