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Database BASIC TOPOLOGY Metric spaces Open sets of a metric space msf
Theorem msf
Description: The distance function of a metric space is a function into the real
numbers. (Contributed by NM , 30-Aug-2006) (Revised by Mario Carneiro , 12-Nov-2013)
Ref
Expression
Hypotheses
msf.x ⊢ 𝑋 = ( Base ‘ 𝑀 )
msf.d ⊢ 𝐷 = ( ( dist ‘ 𝑀 ) ↾ ( 𝑋 × 𝑋 ) )
Assertion
msf ⊢ ( 𝑀 ∈ MetSp → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ )
Proof
Step
Hyp
Ref
Expression
1
msf.x ⊢ 𝑋 = ( Base ‘ 𝑀 )
2
msf.d ⊢ 𝐷 = ( ( dist ‘ 𝑀 ) ↾ ( 𝑋 × 𝑋 ) )
3
1 2
msmet ⊢ ( 𝑀 ∈ MetSp → 𝐷 ∈ ( Met ‘ 𝑋 ) )
4
metf ⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ )
5
3 4
syl ⊢ ( 𝑀 ∈ MetSp → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ )