Theorem hlress

Description: The scalar field of a subcomplex Hilbert space contains RR . (Contributed by Mario Carneiro, 8-Oct-2015)

	Ref	Expression
Hypotheses	hlress.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
	hlress.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
Assertion	hlress	⊢ ( 𝑊 ∈ ℂHil → ℝ ⊆ 𝐾 )

Proof

Step	Hyp	Ref	Expression
1		hlress.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2		hlress.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
3	1 2	hlprlem	⊢ ( 𝑊 ∈ ℂHil → ( 𝐾 ∈ ( SubRing ‘ ℂfld ) ∧ ( ℂfld ↾s 𝐾 ) ∈ DivRing ∧ ( ℂfld ↾s 𝐾 ) ∈ CMetSp ) )
4		eqid	⊢ ( ℂfld ↾s 𝐾 ) = ( ℂfld ↾s 𝐾 )
5	4	resscdrg	⊢ ( ( 𝐾 ∈ ( SubRing ‘ ℂfld ) ∧ ( ℂfld ↾s 𝐾 ) ∈ DivRing ∧ ( ℂfld ↾s 𝐾 ) ∈ CMetSp ) → ℝ ⊆ 𝐾 )
6	3 5	syl	⊢ ( 𝑊 ∈ ℂHil → ℝ ⊆ 𝐾 )