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 )