Metamath Proof Explorer

Theorem cmslssbn

Description: A complete linear subspace of a normed vector space is a Banach space. We furthermore have to assume that the field of scalars is complete since this is a requirement in the current definition of Banach spaces df-bn . (Contributed by AV, 8-Oct-2022)

Ref Expression
Hypotheses cmslssbn.x 
|- X = ( W |`s U )
cmslssbn.s 
|- S = ( LSubSp ` W )
Assertion cmslssbn 
|- ( ( ( W e. NrmVec /\ ( Scalar ` W ) e. CMetSp ) /\ ( X e. CMetSp /\ U e. S ) ) -> X e. Ban )

Proof

Step Hyp Ref Expression
1 cmslssbn.x 
 |-  X = ( W |`s U )
2 cmslssbn.s 
 |-  S = ( LSubSp ` W )
3 1 2 lssnvc 
 |-  ( ( W e. NrmVec /\ U e. S ) -> X e. NrmVec )
4 3 ad2ant2rl 
 |-  ( ( ( W e. NrmVec /\ ( Scalar ` W ) e. CMetSp ) /\ ( X e. CMetSp /\ U e. S ) ) -> X e. NrmVec )
5 simprl 
 |-  ( ( ( W e. NrmVec /\ ( Scalar ` W ) e. CMetSp ) /\ ( X e. CMetSp /\ U e. S ) ) -> X e. CMetSp )
6 eqid 
 |-  ( Scalar ` W ) = ( Scalar ` W )
7 1 6 resssca 
 |-  ( U e. S -> ( Scalar ` W ) = ( Scalar ` X ) )
8 7 ad2antll 
 |-  ( ( W e. NrmVec /\ ( X e. CMetSp /\ U e. S ) ) -> ( Scalar ` W ) = ( Scalar ` X ) )
9 8 eleq1d 
 |-  ( ( W e. NrmVec /\ ( X e. CMetSp /\ U e. S ) ) -> ( ( Scalar ` W ) e. CMetSp <-> ( Scalar ` X ) e. CMetSp ) )
10 9 biimpd 
 |-  ( ( W e. NrmVec /\ ( X e. CMetSp /\ U e. S ) ) -> ( ( Scalar ` W ) e. CMetSp -> ( Scalar ` X ) e. CMetSp ) )
11 10 impancom 
 |-  ( ( W e. NrmVec /\ ( Scalar ` W ) e. CMetSp ) -> ( ( X e. CMetSp /\ U e. S ) -> ( Scalar ` X ) e. CMetSp ) )
12 11 imp 
 |-  ( ( ( W e. NrmVec /\ ( Scalar ` W ) e. CMetSp ) /\ ( X e. CMetSp /\ U e. S ) ) -> ( Scalar ` X ) e. CMetSp )
13 eqid 
 |-  ( Scalar ` X ) = ( Scalar ` X )
14 13 isbn 
 |-  ( X e. Ban <-> ( X e. NrmVec /\ X e. CMetSp /\ ( Scalar ` X ) e. CMetSp ) )
15 4 5 12 14 syl3anbrc 
 |-  ( ( ( W e. NrmVec /\ ( Scalar ` W ) e. CMetSp ) /\ ( X e. CMetSp /\ U e. S ) ) -> X e. Ban )