Description: The multiplication monoid has the same (if any) scalars as the original ring. Mostly to simplify pwsmgp . (Contributed by Mario Carneiro, 12-Mar-2015) (Revised by Mario Carneiro, 5-May-2015)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | mgpbas.1 | |- M = ( mulGrp ` R ) |
|
| mgpsca.s | |- S = ( Scalar ` R ) |
||
| Assertion | mgpsca | |- S = ( Scalar ` M ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mgpbas.1 | |- M = ( mulGrp ` R ) |
|
| 2 | mgpsca.s | |- S = ( Scalar ` R ) |
|
| 3 | eqid | |- ( .r ` R ) = ( .r ` R ) |
|
| 4 | 1 3 | mgpval | |- M = ( R sSet <. ( +g ` ndx ) , ( .r ` R ) >. ) |
| 5 | scaid | |- Scalar = Slot ( Scalar ` ndx ) |
|
| 6 | scandxnplusgndx | |- ( Scalar ` ndx ) =/= ( +g ` ndx ) |
|
| 7 | 4 5 6 | setsplusg | |- ( Scalar ` R ) = ( Scalar ` M ) |
| 8 | 2 7 | eqtri | |- S = ( Scalar ` M ) |