Description: Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 4-Oct-2015) (Revised by Thierry Arnoux, 16-Jun-2019) (Revised by AV, 29-Oct-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | srapart.a | |- ( ph -> A = ( ( subringAlg ` W ) ` S ) ) |
|
| srapart.s | |- ( ph -> S C_ ( Base ` W ) ) |
||
| Assertion | sramulr | |- ( ph -> ( .r ` W ) = ( .r ` A ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | srapart.a | |- ( ph -> A = ( ( subringAlg ` W ) ` S ) ) |
|
| 2 | srapart.s | |- ( ph -> S C_ ( Base ` W ) ) |
|
| 3 | mulrid | |- .r = Slot ( .r ` ndx ) |
|
| 4 | scandxnmulrndx | |- ( Scalar ` ndx ) =/= ( .r ` ndx ) |
|
| 5 | vscandxnmulrndx | |- ( .s ` ndx ) =/= ( .r ` ndx ) |
|
| 6 | ipndxnmulrndx | |- ( .i ` ndx ) =/= ( .r ` ndx ) |
|
| 7 | 1 2 3 4 5 6 | sralem | |- ( ph -> ( .r ` W ) = ( .r ` A ) ) |