Metamath Proof Explorer

Theorem mstri2

Description: Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015)

Ref Expression
Hypotheses mscl.x ⊢ X = Base M
mscl.d ⊢ D = dist ⁡ M
Assertion mstri2 ⊢ M ∈ MetSp ∧ C ∈ X ∧ A ∈ X ∧ B ∈ X → A D B ≤ C D A + 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 ∈ MetSp → D ↾ X × X ∈ Met ⁡ X
4 mettri2 ⊢ D ↾ X × X ∈ Met ⁡ X ∧ C ∈ X ∧ A ∈ X ∧ B ∈ X → A D ↾ X × X B ≤ C D ↾ X × X A + C D ↾ X × X B
5 3 4 sylan ⊢ M ∈ MetSp ∧ C ∈ X ∧ A ∈ X ∧ B ∈ X → A D ↾ X × X B ≤ C D ↾ X × X A + C D ↾ X × X B
6 simpr2 ⊢ M ∈ MetSp ∧ C ∈ X ∧ A ∈ X ∧ B ∈ X → A ∈ X
7 simpr3 ⊢ M ∈ MetSp ∧ C ∈ X ∧ A ∈ X ∧ B ∈ X → B ∈ X
8 6 7 ovresd ⊢ M ∈ MetSp ∧ C ∈ X ∧ A ∈ X ∧ B ∈ X → A D ↾ X × X B = A D B
9 simpr1 ⊢ M ∈ MetSp ∧ C ∈ X ∧ A ∈ X ∧ B ∈ X → C ∈ X
10 9 6 ovresd ⊢ M ∈ MetSp ∧ C ∈ X ∧ A ∈ X ∧ B ∈ X → C D ↾ X × X A = C D A
11 9 7 ovresd ⊢ M ∈ MetSp ∧ C ∈ X ∧ A ∈ X ∧ B ∈ X → C D ↾ X × X B = C D B
12 10 11 oveq12d ⊢ M ∈ MetSp ∧ C ∈ X ∧ A ∈ X ∧ B ∈ X → C D ↾ X × X A + C D ↾ X × X B = C D A + C D B
13 5 8 12 3brtr3d ⊢ M ∈ MetSp ∧ C ∈ X ∧ A ∈ X ∧ B ∈ X → A D B ≤ C D A + C D B