Theorem cphnvc

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

	Ref	Expression
Assertion	cphnvc	⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec )

Proof

Step	Hyp	Ref	Expression
1		cphnlm	⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod )
2		cphlvec	⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec )
3		isnvc	⊢ ( 𝑊 ∈ NrmVec ↔ ( 𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec ) )
4	1 2 3	sylanbrc	⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec )