Theorem cphphl

Description: A subcomplex pre-Hilbert space is a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015)

	Ref	Expression
Assertion	cphphl	⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
2		eqid	⊢ ( ·𝑖 ‘ 𝑊 ) = ( ·𝑖 ‘ 𝑊 )
3		eqid	⊢ ( norm ‘ 𝑊 ) = ( norm ‘ 𝑊 )
4		eqid	⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
5		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) )
6	1 2 3 4 5	iscph	⊢ ( 𝑊 ∈ ℂPreHil ↔ ( ( 𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ ( Scalar ‘ 𝑊 ) = ( ℂfld ↾s ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ) ∧ ( √ “ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∩ ( 0 [,) +∞ ) ) ) ⊆ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( norm ‘ 𝑊 ) = ( 𝑥 ∈ ( Base ‘ 𝑊 ) ↦ ( √ ‘ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) ) )
7	6	simp1bi	⊢ ( 𝑊 ∈ ℂPreHil → ( 𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ ( Scalar ‘ 𝑊 ) = ( ℂfld ↾s ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ) )
8	7	simp1d	⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil )