Description: The scalar ring of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015) (Revised by Mario Carneiro, 30-Aug-2015)
| 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 | df-sca | |- Scalar = Slot 5 |
|
| 8 | 5nn | |- 5 e. NN |
|
| 9 | 5lt10 | |- 5 < ; 1 0 |
|
| 10 | 1 2 3 7 8 9 | opsrbaslem | |- ( ph -> ( Scalar ` S ) = ( Scalar ` O ) ) |
| 11 | 6 10 | eqtrd | |- ( ph -> R = ( Scalar ` O ) ) |