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Metamath Proof Explorer

Theorem resubdrg

Description: The real numbers form a division subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014) (Revised by Thierry Arnoux, 30-Jun-2019)

Ref Expression
Assertion resubdrg ( ℝ ∈ ( SubRing ‘ ℂfld ) ∧ ℝfld ∈ DivRing )

Proof

Step Hyp Ref Expression
1 recn ⊢ ( 𝑥 ∈ ℝ → 𝑥 ∈ ℂ )
2 readdcl ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 + 𝑦 ) ∈ ℝ )
3 renegcl ⊢ ( 𝑥 ∈ ℝ → - 𝑥 ∈ ℝ )
4 1re ⊢ 1 ∈ ℝ
5 remulcl ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 · 𝑦 ) ∈ ℝ )
6 rereccl ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑥 ≠ 0 ) → ( 1 / 𝑥 ) ∈ ℝ )
7 1 2 3 4 5 6 cnsubdrglem ⊢ ( ℝ ∈ ( SubRing ‘ ℂfld ) ∧ ( ℂfld ↾s ℝ ) ∈ DivRing )
8 df-refld ⊢ ℝfld = ( ℂfld ↾s ℝ )
9 8 eleq1i ⊢ ( ℝfld ∈ DivRing ↔ ( ℂfld ↾s ℝ ) ∈ DivRing )
10 9 anbi2i ⊢ ( ( ℝ ∈ ( SubRing ‘ ℂfld ) ∧ ℝfld ∈ DivRing ) ↔ ( ℝ ∈ ( SubRing ‘ ℂfld ) ∧ ( ℂfld ↾s ℝ ) ∈ DivRing ) )
11 7 10 mpbir ⊢ ( ℝ ∈ ( SubRing ‘ ℂfld ) ∧ ℝfld ∈ DivRing )