Theorem cphsubrg

Description: The scalar field of a subcomplex pre-Hilbert space is a subring of CCfld . (Contributed by Mario Carneiro, 8-Oct-2015)

	Ref	Expression
Hypotheses	cphsca.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
	cphsca.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
Assertion	cphsubrg	⊢ ( 𝑊 ∈ ℂPreHil → 𝐾 ∈ ( SubRing ‘ ℂfld ) )

Proof

Step	Hyp	Ref	Expression
1		cphsca.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2		cphsca.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
3	1 2	cphsca	⊢ ( 𝑊 ∈ ℂPreHil → 𝐹 = ( ℂfld ↾s 𝐾 ) )
4		cphlvec	⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec )
5	1	lvecdrng	⊢ ( 𝑊 ∈ LVec → 𝐹 ∈ DivRing )
6	4 5	syl	⊢ ( 𝑊 ∈ ℂPreHil → 𝐹 ∈ DivRing )
7	2 3 6	cphsubrglem	⊢ ( 𝑊 ∈ ℂPreHil → ( 𝐹 = ( ℂfld ↾s 𝐾 ) ∧ 𝐾 = ( 𝐾 ∩ ℂ ) ∧ 𝐾 ∈ ( SubRing ‘ ℂfld ) ) )
8	7	simp3d	⊢ ( 𝑊 ∈ ℂPreHil → 𝐾 ∈ ( SubRing ‘ ℂfld ) )