Description: The inner product on a subspace in terms of the inner product on the parent space. (Contributed by NM, 28-Jan-2008) (Revised by AV, 19-Oct-2021)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | ssipeq.x | |- X = ( W |`s U ) |
|
| ssipeq.i | |- ., = ( .i ` W ) |
||
| ssipeq.p | |- P = ( .i ` X ) |
||
| ssipeq.s | |- S = ( LSubSp ` W ) |
||
| Assertion | phssipval | |- ( ( ( W e. PreHil /\ U e. S ) /\ ( A e. U /\ B e. U ) ) -> ( A P B ) = ( A ., B ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssipeq.x | |- X = ( W |`s U ) |
|
| 2 | ssipeq.i | |- ., = ( .i ` W ) |
|
| 3 | ssipeq.p | |- P = ( .i ` X ) |
|
| 4 | ssipeq.s | |- S = ( LSubSp ` W ) |
|
| 5 | 1 2 3 | ssipeq | |- ( U e. S -> P = ., ) |
| 6 | 5 | oveqd | |- ( U e. S -> ( A P B ) = ( A ., B ) ) |
| 7 | 6 | ad2antlr | |- ( ( ( W e. PreHil /\ U e. S ) /\ ( A e. U /\ B e. U ) ) -> ( A P B ) = ( A ., B ) ) |