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Metamath Proof Explorer

Theorem psmettri

Description: Triangle inequality for the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 11-Feb-2018)

Ref Expression
Assertion psmettri 
|- ( ( D e. ( PsMet ` X ) /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D B ) <_ ( ( A D C ) +e ( C D B ) ) )

Proof

Step Hyp Ref Expression
1 simpl 
 |-  ( ( D e. ( PsMet ` X ) /\ ( A e. X /\ B e. X /\ C e. X ) ) -> D e. ( PsMet ` X ) )
2 simpr3 
 |-  ( ( D e. ( PsMet ` X ) /\ ( A e. X /\ B e. X /\ C e. X ) ) -> C e. X )
3 simpr1 
 |-  ( ( D e. ( PsMet ` X ) /\ ( A e. X /\ B e. X /\ C e. X ) ) -> A e. X )
4 simpr2 
 |-  ( ( D e. ( PsMet ` X ) /\ ( A e. X /\ B e. X /\ C e. X ) ) -> B e. X )
5 psmettri2 
 |-  ( ( D e. ( PsMet ` X ) /\ ( C e. X /\ A e. X /\ B e. X ) ) -> ( A D B ) <_ ( ( C D A ) +e ( C D B ) ) )
6 1 2 3 4 5 syl13anc 
 |-  ( ( D e. ( PsMet ` X ) /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D B ) <_ ( ( C D A ) +e ( C D B ) ) )
7 psmetsym 
 |-  ( ( D e. ( PsMet ` X ) /\ C e. X /\ A e. X ) -> ( C D A ) = ( A D C ) )
8 1 2 3 7 syl3anc 
 |-  ( ( D e. ( PsMet ` X ) /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( C D A ) = ( A D C ) )
9 8 oveq1d 
 |-  ( ( D e. ( PsMet ` X ) /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( C D A ) +e ( C D B ) ) = ( ( A D C ) +e ( C D B ) ) )
10 6 9 breqtrd 
 |-  ( ( D e. ( PsMet ` X ) /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D B ) <_ ( ( A D C ) +e ( C D B ) ) )