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 | |
|
mscl.d | |
||
Assertion | mstri | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mscl.x | |
|
2 | mscl.d | |
|
3 | 1 2 | msmet2 | |
4 | mettri | |
|
5 | 3 4 | sylan | |
6 | simpr1 | |
|
7 | simpr2 | |
|
8 | 6 7 | ovresd | |
9 | simpr3 | |
|
10 | 6 9 | ovresd | |
11 | 9 7 | ovresd | |
12 | 10 11 | oveq12d | |
13 | 5 8 12 | 3brtr3d | |