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Metamath Proof Explorer

Theorem recmet

Description: The real numbers are a complete metric space. (Contributed by Jeff Madsen, 2-Sep-2009) (Proof shortened by Mario Carneiro, 12-Sep-2015)

Ref Expression
Assertion recmet ⊢ abs ∘ − ↾ ℝ 2 ∈ CMet ⁡ ℝ

Proof

Step Hyp Ref Expression
1 eqid ⊢ TopOpen ⁡ ℂ fld = TopOpen ⁡ ℂ fld
2 1 recld2 ⊢ ℝ ∈ Clsd ⁡ TopOpen ⁡ ℂ fld
3 eqid ⊢ abs ∘ − = abs ∘ −
4 3 cncmet ⊢ abs ∘ − ∈ CMet ⁡ ℂ
5 1 cnfldtopn ⊢ TopOpen ⁡ ℂ fld = MetOpen ⁡ abs ∘ −
6 5 cmetss ⊢ abs ∘ − ∈ CMet ⁡ ℂ → abs ∘ − ↾ ℝ 2 ∈ CMet ⁡ ℝ ↔ ℝ ∈ Clsd ⁡ TopOpen ⁡ ℂ fld
7 4 6 ax-mp ⊢ abs ∘ − ↾ ℝ 2 ∈ CMet ⁡ ℝ ↔ ℝ ∈ Clsd ⁡ TopOpen ⁡ ℂ fld
8 2 7 mpbir ⊢ abs ∘ − ↾ ℝ 2 ∈ CMet ⁡ ℝ