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 ) )