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Metamath Proof Explorer

Theorem sspm

Description: Vector subtraction on a subspace is a restriction of vector subtraction on the parent space. (Contributed by NM, 28-Jan-2008) (New usage is discouraged.)

Ref Expression
Hypotheses sspm.y 
|- Y = ( BaseSet ` W )
sspm.m 
|- M = ( -v ` U )
sspm.l 
|- L = ( -v ` W )
sspm.h 
|- H = ( SubSp ` U )
Assertion sspm 
|- ( ( U e. NrmCVec /\ W e. H ) -> L = ( M |` ( Y X. Y ) ) )

Proof

Step Hyp Ref Expression
1 sspm.y 
 |-  Y = ( BaseSet ` W )
2 sspm.m 
 |-  M = ( -v ` U )
3 sspm.l 
 |-  L = ( -v ` W )
4 sspm.h 
 |-  H = ( SubSp ` U )
5 1 2 3 4 sspmval 
 |-  ( ( ( U e. NrmCVec /\ W e. H ) /\ ( x e. Y /\ y e. Y ) ) -> ( x L y ) = ( x M y ) )
6 1 3 nvmf 
 |-  ( W e. NrmCVec -> L : ( Y X. Y ) --> Y )
7 eqid 
 |-  ( BaseSet ` U ) = ( BaseSet ` U )
8 7 2 nvmf 
 |-  ( U e. NrmCVec -> M : ( ( BaseSet ` U ) X. ( BaseSet ` U ) ) --> ( BaseSet ` U ) )
9 1 4 5 6 8 sspmlem 
 |-  ( ( U e. NrmCVec /\ W e. H ) -> L = ( M |` ( Y X. Y ) ) )