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Metamath Proof Explorer

Theorem nvvcop

Description: A normed complex vector space is a vector space. (Contributed by NM, 5-Jun-2008) (Revised by Mario Carneiro, 1-May-2015) (New usage is discouraged.)

Ref Expression
Assertion nvvcop 
|- ( <. W , N >. e. NrmCVec -> W e. CVecOLD )

Proof

Step Hyp Ref Expression
1 nvss 
 |-  NrmCVec C_ ( CVecOLD X. _V )
2 1 sseli 
 |-  ( <. W , N >. e. NrmCVec -> <. W , N >. e. ( CVecOLD X. _V ) )
3 opelxp1 
 |-  ( <. W , N >. e. ( CVecOLD X. _V ) -> W e. CVecOLD )
4 2 3 syl 
 |-  ( <. W , N >. e. NrmCVec -> W e. CVecOLD )