Description: The scalar product operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013) (Revised by Mario Carneiro, 29-Aug-2015)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | lmodstr.w | |- W = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , F >. } u. { <. ( .s ` ndx ) , .x. >. } ) |
|
| Assertion | lmodvsca | |- ( .x. e. X -> .x. = ( .s ` W ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lmodstr.w | |- W = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , F >. } u. { <. ( .s ` ndx ) , .x. >. } ) |
|
| 2 | 1 | lmodstr | |- W Struct <. 1 , 6 >. |
| 3 | vscaid | |- .s = Slot ( .s ` ndx ) |
|
| 4 | ssun2 | |- { <. ( .s ` ndx ) , .x. >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , F >. } u. { <. ( .s ` ndx ) , .x. >. } ) |
|
| 5 | 4 1 | sseqtrri | |- { <. ( .s ` ndx ) , .x. >. } C_ W |
| 6 | 2 3 5 | strfv | |- ( .x. e. X -> .x. = ( .s ` W ) ) |