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Metamath Proof Explorer

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 )