Metamath Proof Explorer


Theorem mettri3

Description: Triangle inequality for the distance function of a metric space. (Contributed by NM, 13-Mar-2007)

Ref Expression
Assertion mettri3 D Met X A X B X C X A D B A D C + B D C

Proof

Step Hyp Ref Expression
1 mettri D Met X A X B X C X A D B A D C + C D B
2 metsym D Met X B X C X B D C = C D B
3 2 3adant3r1 D Met X A X B X C X B D C = C D B
4 3 oveq2d D Met X A X B X C X A D C + B D C = A D C + C D B
5 1 4 breqtrrd D Met X A X B X C X A D B A D C + B D C