Theorem mettri2

Description: Triangle inequality for the distance function of a metric space. (Contributed by NM, 30-Aug-2006) (Revised by Mario Carneiro, 20-Aug-2015)

	Ref	Expression
Assertion	mettri2	⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 𝐷 𝐵 ) ≤ ( ( 𝐶 𝐷 𝐴 ) + ( 𝐶 𝐷 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		metxmet	⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) )
2		xmettri2	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 𝐷 𝐵 ) ≤ ( ( 𝐶 𝐷 𝐴 ) +𝑒 ( 𝐶 𝐷 𝐵 ) ) )
3	1 2	sylan	⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 𝐷 𝐵 ) ≤ ( ( 𝐶 𝐷 𝐴 ) +𝑒 ( 𝐶 𝐷 𝐵 ) ) )
4		metcl	⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐶 𝐷 𝐴 ) ∈ ℝ )
5	4	3adant3r3	⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐶 𝐷 𝐴 ) ∈ ℝ )
6		metcl	⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐶 𝐷 𝐵 ) ∈ ℝ )
7	6	3adant3r2	⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐶 𝐷 𝐵 ) ∈ ℝ )
8	5 7	rexaddd	⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( ( 𝐶 𝐷 𝐴 ) +𝑒 ( 𝐶 𝐷 𝐵 ) ) = ( ( 𝐶 𝐷 𝐴 ) + ( 𝐶 𝐷 𝐵 ) ) )
9	3 8	breqtrd	⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 𝐷 𝐵 ) ≤ ( ( 𝐶 𝐷 𝐴 ) + ( 𝐶 𝐷 𝐵 ) ) )