Theorem lssnvc

Description: A subspace of a normed vector space is a normed vector space. (Contributed by Mario Carneiro, 4-Oct-2015)

	Ref	Expression
Hypotheses	lssnlm.x	|- X = ( W |`s U )
	lssnlm.s	|- S = ( LSubSp ` W )
Assertion	lssnvc	|- ( ( W e. NrmVec /\ U e. S ) -> X e. NrmVec )

Proof

Step	Hyp	Ref	Expression
1		lssnlm.x	|- X = ( W |`s U )
2		lssnlm.s	|- S = ( LSubSp ` W )
3		nvcnlm	|- ( W e. NrmVec -> W e. NrmMod )
4	1 2	lssnlm	|- ( ( W e. NrmMod /\ U e. S ) -> X e. NrmMod )
5	3 4	sylan	|- ( ( W e. NrmVec /\ U e. S ) -> X e. NrmMod )
6		eqid	|- ( Scalar ` W ) = ( Scalar ` W )
7	1 6	resssca	|- ( U e. S -> ( Scalar ` W ) = ( Scalar ` X ) )
8	7	adantl	|- ( ( W e. NrmVec /\ U e. S ) -> ( Scalar ` W ) = ( Scalar ` X ) )
9		nvclvec	|- ( W e. NrmVec -> W e. LVec )
10	6	lvecdrng	|- ( W e. LVec -> ( Scalar ` W ) e. DivRing )
11	9 10	syl	|- ( W e. NrmVec -> ( Scalar ` W ) e. DivRing )
12	11	adantr	|- ( ( W e. NrmVec /\ U e. S ) -> ( Scalar ` W ) e. DivRing )
13	8 12	eqeltrrd	|- ( ( W e. NrmVec /\ U e. S ) -> ( Scalar ` X ) e. DivRing )
14		eqid	|- ( Scalar ` X ) = ( Scalar ` X )
15	14	isnvc2	|- ( X e. NrmVec <-> ( X e. NrmMod /\ ( Scalar ` X ) e. DivRing ) )
16	5 13 15	sylanbrc	|- ( ( W e. NrmVec /\ U e. S ) -> X e. NrmVec )