Description: Vector addition on a subspace in terms of vector addition on the parent space. (Contributed by NM, 28-Jan-2008) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | sspg.y | |- Y = ( BaseSet ` W ) |
|
| sspg.g | |- G = ( +v ` U ) |
||
| sspg.f | |- F = ( +v ` W ) |
||
| sspg.h | |- H = ( SubSp ` U ) |
||
| Assertion | sspgval | |- ( ( ( U e. NrmCVec /\ W e. H ) /\ ( A e. Y /\ B e. Y ) ) -> ( A F B ) = ( A G B ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sspg.y | |- Y = ( BaseSet ` W ) |
|
| 2 | sspg.g | |- G = ( +v ` U ) |
|
| 3 | sspg.f | |- F = ( +v ` W ) |
|
| 4 | sspg.h | |- H = ( SubSp ` U ) |
|
| 5 | 1 2 3 4 | sspg | |- ( ( U e. NrmCVec /\ W e. H ) -> F = ( G |` ( Y X. Y ) ) ) |
| 6 | 5 | oveqd | |- ( ( U e. NrmCVec /\ W e. H ) -> ( A F B ) = ( A ( G |` ( Y X. Y ) ) B ) ) |
| 7 | ovres | |- ( ( A e. Y /\ B e. Y ) -> ( A ( G |` ( Y X. Y ) ) B ) = ( A G B ) ) |
|
| 8 | 6 7 | sylan9eq | |- ( ( ( U e. NrmCVec /\ W e. H ) /\ ( A e. Y /\ B e. Y ) ) -> ( A F B ) = ( A G B ) ) |