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 ) ) )