Description: The scalar ring of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015) (Revised by Mario Carneiro, 30-Aug-2015) (Revised by AV, 1-Nov-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | opsrbas.s | |- S = ( I mPwSer R ) |
|
| opsrbas.o | |- O = ( ( I ordPwSer R ) ` T ) |
||
| opsrbas.t | |- ( ph -> T C_ ( I X. I ) ) |
||
| opsrsca.i | |- ( ph -> I e. V ) |
||
| opsrsca.r | |- ( ph -> R e. W ) |
||
| Assertion | opsrsca | |- ( ph -> R = ( Scalar ` O ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opsrbas.s | |- S = ( I mPwSer R ) |
|
| 2 | opsrbas.o | |- O = ( ( I ordPwSer R ) ` T ) |
|
| 3 | opsrbas.t | |- ( ph -> T C_ ( I X. I ) ) |
|
| 4 | opsrsca.i | |- ( ph -> I e. V ) |
|
| 5 | opsrsca.r | |- ( ph -> R e. W ) |
|
| 6 | 1 4 5 | psrsca | |- ( ph -> R = ( Scalar ` S ) ) |
| 7 | scaid | |- Scalar = Slot ( Scalar ` ndx ) |
|
| 8 | plendxnscandx | |- ( le ` ndx ) =/= ( Scalar ` ndx ) |
|
| 9 | 8 | necomi | |- ( Scalar ` ndx ) =/= ( le ` ndx ) |
| 10 | 1 2 3 7 9 | opsrbaslem | |- ( ph -> ( Scalar ` S ) = ( Scalar ` O ) ) |
| 11 | 6 10 | eqtrd | |- ( ph -> R = ( Scalar ` O ) ) |