Theorem xmettri2

Description: Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015)

	Ref	Expression
Assertion	xmettri2	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 𝐷 𝐵 ) ≤ ( ( 𝐶 𝐷 𝐴 ) +𝑒 ( 𝐶 𝐷 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elfvdm	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝑋 ∈ dom ∞Met )
2		isxmet	⊢ ( 𝑋 ∈ dom ∞Met → ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ↔ ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( ( 𝑥 𝐷 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ∧ ∀ 𝑧 ∈ 𝑋 ( 𝑥 𝐷 𝑦 ) ≤ ( ( 𝑧 𝐷 𝑥 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) ) ) ) )
3	1 2	syl	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ↔ ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( ( 𝑥 𝐷 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ∧ ∀ 𝑧 ∈ 𝑋 ( 𝑥 𝐷 𝑦 ) ≤ ( ( 𝑧 𝐷 𝑥 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) ) ) ) )
4	3	ibi	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( ( 𝑥 𝐷 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ∧ ∀ 𝑧 ∈ 𝑋 ( 𝑥 𝐷 𝑦 ) ≤ ( ( 𝑧 𝐷 𝑥 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) ) ) )
5		simpr	⊢ ( ( ( ( 𝑥 𝐷 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ∧ ∀ 𝑧 ∈ 𝑋 ( 𝑥 𝐷 𝑦 ) ≤ ( ( 𝑧 𝐷 𝑥 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) ) → ∀ 𝑧 ∈ 𝑋 ( 𝑥 𝐷 𝑦 ) ≤ ( ( 𝑧 𝐷 𝑥 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) )
6	5	2ralimi	⊢ ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( ( 𝑥 𝐷 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ∧ ∀ 𝑧 ∈ 𝑋 ( 𝑥 𝐷 𝑦 ) ≤ ( ( 𝑧 𝐷 𝑥 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) ) → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( 𝑥 𝐷 𝑦 ) ≤ ( ( 𝑧 𝐷 𝑥 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) )
7	4 6	simpl2im	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( 𝑥 𝐷 𝑦 ) ≤ ( ( 𝑧 𝐷 𝑥 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) )
8		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐷 𝑦 ) = ( 𝐴 𝐷 𝑦 ) )
9		oveq2	⊢ ( 𝑥 = 𝐴 → ( 𝑧 𝐷 𝑥 ) = ( 𝑧 𝐷 𝐴 ) )
10	9	oveq1d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑧 𝐷 𝑥 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) = ( ( 𝑧 𝐷 𝐴 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) )
11	8 10	breq12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 𝐷 𝑦 ) ≤ ( ( 𝑧 𝐷 𝑥 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) ↔ ( 𝐴 𝐷 𝑦 ) ≤ ( ( 𝑧 𝐷 𝐴 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) ) )
12		oveq2	⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝐷 𝑦 ) = ( 𝐴 𝐷 𝐵 ) )
13		oveq2	⊢ ( 𝑦 = 𝐵 → ( 𝑧 𝐷 𝑦 ) = ( 𝑧 𝐷 𝐵 ) )
14	13	oveq2d	⊢ ( 𝑦 = 𝐵 → ( ( 𝑧 𝐷 𝐴 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) = ( ( 𝑧 𝐷 𝐴 ) +𝑒 ( 𝑧 𝐷 𝐵 ) ) )
15	12 14	breq12d	⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 𝐷 𝑦 ) ≤ ( ( 𝑧 𝐷 𝐴 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) ↔ ( 𝐴 𝐷 𝐵 ) ≤ ( ( 𝑧 𝐷 𝐴 ) +𝑒 ( 𝑧 𝐷 𝐵 ) ) ) )
16		oveq1	⊢ ( 𝑧 = 𝐶 → ( 𝑧 𝐷 𝐴 ) = ( 𝐶 𝐷 𝐴 ) )
17		oveq1	⊢ ( 𝑧 = 𝐶 → ( 𝑧 𝐷 𝐵 ) = ( 𝐶 𝐷 𝐵 ) )
18	16 17	oveq12d	⊢ ( 𝑧 = 𝐶 → ( ( 𝑧 𝐷 𝐴 ) +𝑒 ( 𝑧 𝐷 𝐵 ) ) = ( ( 𝐶 𝐷 𝐴 ) +𝑒 ( 𝐶 𝐷 𝐵 ) ) )
19	18	breq2d	⊢ ( 𝑧 = 𝐶 → ( ( 𝐴 𝐷 𝐵 ) ≤ ( ( 𝑧 𝐷 𝐴 ) +𝑒 ( 𝑧 𝐷 𝐵 ) ) ↔ ( 𝐴 𝐷 𝐵 ) ≤ ( ( 𝐶 𝐷 𝐴 ) +𝑒 ( 𝐶 𝐷 𝐵 ) ) ) )
20	11 15 19	rspc3v	⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( 𝑥 𝐷 𝑦 ) ≤ ( ( 𝑧 𝐷 𝑥 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) → ( 𝐴 𝐷 𝐵 ) ≤ ( ( 𝐶 𝐷 𝐴 ) +𝑒 ( 𝐶 𝐷 𝐵 ) ) ) )
21	7 20	syl5	⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) ≤ ( ( 𝐶 𝐷 𝐴 ) +𝑒 ( 𝐶 𝐷 𝐵 ) ) ) )
22	21	3comr	⊢ ( ( 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) ≤ ( ( 𝐶 𝐷 𝐴 ) +𝑒 ( 𝐶 𝐷 𝐵 ) ) ) )
23	22	impcom	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 𝐷 𝐵 ) ≤ ( ( 𝐶 𝐷 𝐴 ) +𝑒 ( 𝐶 𝐷 𝐵 ) ) )