Theorem hlcph

Description: Every subcomplex Hilbert space is a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015)

	Ref	Expression
Assertion	hlcph	⊢ ( 𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil )

Proof

Step	Hyp	Ref	Expression
1		ishl	⊢ ( 𝑊 ∈ ℂHil ↔ ( 𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil ) )
2	1	simprbi	⊢ ( 𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil )