Theorem cphsca

Description: A subcomplex pre-Hilbert space is a vector space over a subfield of CCfld . (Contributed by Mario Carneiro, 8-Oct-2015)

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

Proof

Step	Hyp	Ref	Expression
1		cphsca.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2		cphsca.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
3		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
4		eqid	⊢ ( ·𝑖 ‘ 𝑊 ) = ( ·𝑖 ‘ 𝑊 )
5		eqid	⊢ ( norm ‘ 𝑊 ) = ( norm ‘ 𝑊 )
6	3 4 5 1 2	iscph	⊢ ( 𝑊 ∈ ℂPreHil ↔ ( ( 𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = ( ℂfld ↾s 𝐾 ) ) ∧ ( √ “ ( 𝐾 ∩ ( 0 [,) +∞ ) ) ) ⊆ 𝐾 ∧ ( norm ‘ 𝑊 ) = ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) ) )
7	6	simp1bi	⊢ ( 𝑊 ∈ ℂPreHil → ( 𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = ( ℂfld ↾s 𝐾 ) ) )
8	7	simp3d	⊢ ( 𝑊 ∈ ℂPreHil → 𝐹 = ( ℂfld ↾s 𝐾 ) )