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 ) |