Description: The multiplication operation 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 ) ) |
||
| Assertion | opsrmulr | |- ( ph -> ( .r ` S ) = ( .r ` 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 | mulridx | |- .r = Slot ( .r ` ndx ) |
|
| 5 | plendxnmulrndx | |- ( le ` ndx ) =/= ( .r ` ndx ) |
|
| 6 | 5 | necomi | |- ( .r ` ndx ) =/= ( le ` ndx ) |
| 7 | 1 2 3 4 6 | opsrbaslem | |- ( ph -> ( .r ` S ) = ( .r ` O ) ) |