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 | df-sca | |- Scalar = Slot 5 |
|
| 4 | 5nn | |- 5 e. NN |
|
| 5 | 2re | |- 2 e. RR |
|
| 6 | 2lt5 | |- 2 < 5 |
|
| 7 | 5 6 | gtneii | |- 5 =/= 2 |
| 8 | 1 3 4 7 | mgplem | |- ( Scalar ` R ) = ( Scalar ` M ) |
| 9 | 2 8 | eqtri | |- S = ( Scalar ` M ) |