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 ) |
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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 ) ) |
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8 | 6 7 | sylan9eq | |- ( ( ( U e. NrmCVec /\ W e. H ) /\ ( A e. CC /\ B e. Y ) ) -> ( A R B ) = ( A S B ) ) |