Theorem xmscl

Description: Closure of the distance function of an extended metric space. (Contributed by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Hypotheses	mscl.x	⊢ 𝑋 = ( Base ‘ 𝑀 )
	mscl.d	⊢ 𝐷 = ( dist ‘ 𝑀 )
Assertion	xmscl	⊢ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) ∈ ℝ* )

Proof

Step	Hyp	Ref	Expression
1		mscl.x	⊢ 𝑋 = ( Base ‘ 𝑀 )
2		mscl.d	⊢ 𝐷 = ( dist ‘ 𝑀 )
3		ovres	⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = ( 𝐴 𝐷 𝐵 ) )
4	3	3adant1	⊢ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = ( 𝐴 𝐷 𝐵 ) )
5	1 2	xmsxmet2	⊢ ( 𝑀 ∈ ∞MetSp → ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) ∈ ( ∞Met ‘ 𝑋 ) )
6		xmetcl	⊢ ( ( ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) ∈ ℝ* )
7	5 6	syl3an1	⊢ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) ∈ ℝ* )
8	4 7	eqeltrrd	⊢ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) ∈ ℝ* )