Metamath Proof Explorer

Theorem mstri

Description: Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of Gleason p. 223. (Contributed by Mario Carneiro, 2-Oct-2015)

Ref Expression
Hypotheses mscl.x 
|- X = ( Base ` M )
mscl.d 
|- D = ( dist ` M )
Assertion mstri 
|- ( ( M e. MetSp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D B ) <_ ( ( A D C ) + ( C D B ) ) )

Proof

Step Hyp Ref Expression
1 mscl.x 
 |-  X = ( Base ` M )
2 mscl.d 
 |-  D = ( dist ` M )
3 1 2 msmet2 
 |-  ( M e. MetSp -> ( D |` ( X X. X ) ) e. ( Met ` X ) )
4 mettri 
 |-  ( ( ( D |` ( X X. X ) ) e. ( Met ` X ) /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A ( D |` ( X X. X ) ) B ) <_ ( ( A ( D |` ( X X. X ) ) C ) + ( C ( D |` ( X X. X ) ) B ) ) )
5 3 4 sylan 
 |-  ( ( M e. MetSp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A ( D |` ( X X. X ) ) B ) <_ ( ( A ( D |` ( X X. X ) ) C ) + ( C ( D |` ( X X. X ) ) B ) ) )
6 simpr1 
 |-  ( ( M e. MetSp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> A e. X )
7 simpr2 
 |-  ( ( M e. MetSp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> B e. X )
8 6 7 ovresd 
 |-  ( ( M e. MetSp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A ( D |` ( X X. X ) ) B ) = ( A D B ) )
9 simpr3 
 |-  ( ( M e. MetSp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> C e. X )
10 6 9 ovresd 
 |-  ( ( M e. MetSp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A ( D |` ( X X. X ) ) C ) = ( A D C ) )
11 9 7 ovresd 
 |-  ( ( M e. MetSp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( C ( D |` ( X X. X ) ) B ) = ( C D B ) )
12 10 11 oveq12d 
 |-  ( ( M e. MetSp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A ( D |` ( X X. X ) ) C ) + ( C ( D |` ( X X. X ) ) B ) ) = ( ( A D C ) + ( C D B ) ) )
13 5 8 12 3brtr3d 
 |-  ( ( M e. MetSp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D B ) <_ ( ( A D C ) + ( C D B ) ) )