Theorem cncdrg

Description: The only complete subfields of the complex numbers are RR and CC . (Contributed by Mario Carneiro, 15-Oct-2015)

	Ref	Expression
Hypothesis	resscdrg.1	⊢ 𝐹 = ( ℂfld ↾s 𝐾 )
Assertion	cncdrg	⊢ ( ( 𝐾 ∈ ( SubRing ‘ ℂfld ) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp ) → 𝐾 ∈ { ℝ , ℂ } )

Proof

Step	Hyp	Ref	Expression
1		resscdrg.1	⊢ 𝐹 = ( ℂfld ↾s 𝐾 )
2		simp1	⊢ ( ( 𝐾 ∈ ( SubRing ‘ ℂfld ) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp ) → 𝐾 ∈ ( SubRing ‘ ℂfld ) )
3	1	resscdrg	⊢ ( ( 𝐾 ∈ ( SubRing ‘ ℂfld ) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp ) → ℝ ⊆ 𝐾 )
4		cnsubrg	⊢ ( ( 𝐾 ∈ ( SubRing ‘ ℂfld ) ∧ ℝ ⊆ 𝐾 ) → 𝐾 ∈ { ℝ , ℂ } )
5	2 3 4	syl2anc	⊢ ( ( 𝐾 ∈ ( SubRing ‘ ℂfld ) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp ) → 𝐾 ∈ { ℝ , ℂ } )