Description: Scalar multiplication on a subspace in terms of scalar multiplication on the parent space. (Contributed by NM, 28-Jan-2008) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | ssps.y | |- Y = ( BaseSet ` W ) |
|
| ssps.s | |- S = ( .sOLD ` U ) |
||
| ssps.r | |- R = ( .sOLD ` W ) |
||
| ssps.h | |- H = ( SubSp ` U ) |
||
| Assertion | sspsval | |- ( ( ( U e. NrmCVec /\ W e. H ) /\ ( A e. CC /\ B e. Y ) ) -> ( A R B ) = ( A S B ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssps.y | |- Y = ( BaseSet ` W ) |
|
| 2 | ssps.s | |- S = ( .sOLD ` U ) |
|
| 3 | ssps.r | |- R = ( .sOLD ` W ) |
|
| 4 | ssps.h | |- H = ( SubSp ` U ) |
|
| 5 | 1 2 3 4 | ssps | |- ( ( U e. NrmCVec /\ W e. H ) -> R = ( S |` ( CC X. Y ) ) ) |
| 6 | 5 | oveqd | |- ( ( U e. NrmCVec /\ W e. H ) -> ( A R B ) = ( A ( S |` ( CC X. Y ) ) B ) ) |
| 7 | ovres | |- ( ( A e. CC /\ B e. Y ) -> ( A ( S |` ( CC X. Y ) ) B ) = ( A S B ) ) |
|
| 8 | 6 7 | sylan9eq | |- ( ( ( U e. NrmCVec /\ W e. H ) /\ ( A e. CC /\ B e. Y ) ) -> ( A R B ) = ( A S B ) ) |