Theorem mstri3

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	mstri3	⊢ ( ( 𝑀 ∈ MetSp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝐷 𝐵 ) ≤ ( ( 𝐴 𝐷 𝐶 ) + ( 𝐵 𝐷 𝐶 ) ) )

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		mettri3	⊢ ( ( ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) ∈ ( Met ‘ 𝑋 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) ≤ ( ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐶 ) + ( 𝐵 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐶 ) ) )
5	3 4	sylan	⊢ ( ( 𝑀 ∈ MetSp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) ≤ ( ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐶 ) + ( 𝐵 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐶 ) ) )
6		simpr1	⊢ ( ( 𝑀 ∈ MetSp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → 𝐴 ∈ 𝑋 )
7		simpr2	⊢ ( ( 𝑀 ∈ MetSp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → 𝐵 ∈ 𝑋 )
8	6 7	ovresd	⊢ ( ( 𝑀 ∈ MetSp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = ( 𝐴 𝐷 𝐵 ) )
9		simpr3	⊢ ( ( 𝑀 ∈ MetSp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → 𝐶 ∈ 𝑋 )
10	6 9	ovresd	⊢ ( ( 𝑀 ∈ MetSp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐶 ) = ( 𝐴 𝐷 𝐶 ) )
11	7 9	ovresd	⊢ ( ( 𝑀 ∈ MetSp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐵 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐶 ) = ( 𝐵 𝐷 𝐶 ) )
12	10 11	oveq12d	⊢ ( ( 𝑀 ∈ MetSp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐶 ) + ( 𝐵 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐶 ) ) = ( ( 𝐴 𝐷 𝐶 ) + ( 𝐵 𝐷 𝐶 ) ) )
13	5 8 12	3brtr3d	⊢ ( ( 𝑀 ∈ MetSp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝐷 𝐵 ) ≤ ( ( 𝐴 𝐷 𝐶 ) + ( 𝐵 𝐷 𝐶 ) ) )