Metamath Proof Explorer

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 e. CPreHil -> W e. NrmVec )

Proof

Step Hyp Ref Expression
1 cphnlm 
 |-  ( W e. CPreHil -> W e. NrmMod )
2 cphlvec 
 |-  ( W e. CPreHil -> W e. LVec )
3 isnvc 
 |-  ( W e. NrmVec <-> ( W e. NrmMod /\ W e. LVec ) )
4 1 2 3 sylanbrc 
 |-  ( W e. CPreHil -> W e. NrmVec )