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Metamath Proof Explorer

Theorem nvex

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

Ref Expression
Assertion nvex 
|- ( <. <. G , S >. , N >. e. NrmCVec -> ( G e. _V /\ S e. _V /\ N e. _V ) )

Proof

Step Hyp Ref Expression
1 nvvcop 
 |-  ( <. <. G , S >. , N >. e. NrmCVec -> <. G , S >. e. CVecOLD )
2 vcex 
 |-  ( <. G , S >. e. CVecOLD -> ( G e. _V /\ S e. _V ) )
3 1 2 syl 
 |-  ( <. <. G , S >. , N >. e. NrmCVec -> ( G e. _V /\ S e. _V ) )
4 nvss 
 |-  NrmCVec C_ ( CVecOLD X. _V )
5 4 sseli 
 |-  ( <. <. G , S >. , N >. e. NrmCVec -> <. <. G , S >. , N >. e. ( CVecOLD X. _V ) )
6 opelxp2 
 |-  ( <. <. G , S >. , N >. e. ( CVecOLD X. _V ) -> N e. _V )
7 5 6 syl 
 |-  ( <. <. G , S >. , N >. e. NrmCVec -> N e. _V )
8 df-3an 
 |-  ( ( G e. _V /\ S e. _V /\ N e. _V ) <-> ( ( G e. _V /\ S e. _V ) /\ N e. _V ) )
9 3 7 8 sylanbrc 
 |-  ( <. <. G , S >. , N >. e. NrmCVec -> ( G e. _V /\ S e. _V /\ N e. _V ) )