Description: The third argument passed to evalSub is in the domain. (Contributed by SN, 15-Dec-2023)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | selvcllem2.u | |- U = ( I mPoly R ) |
|
| selvcllem2.t | |- T = ( J mPoly U ) |
||
| selvcllem2.c | |- C = ( algSc ` T ) |
||
| selvcllem2.d | |- D = ( C o. ( algSc ` U ) ) |
||
| selvcllem2.i | |- ( ph -> I e. V ) |
||
| selvcllem2.j | |- ( ph -> J e. W ) |
||
| selvcllem2.r | |- ( ph -> R e. CRing ) |
||
| Assertion | selvcllem3 | |- ( ph -> ran D e. ( SubRing ` T ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | selvcllem2.u | |- U = ( I mPoly R ) |
|
| 2 | selvcllem2.t | |- T = ( J mPoly U ) |
|
| 3 | selvcllem2.c | |- C = ( algSc ` T ) |
|
| 4 | selvcllem2.d | |- D = ( C o. ( algSc ` U ) ) |
|
| 5 | selvcllem2.i | |- ( ph -> I e. V ) |
|
| 6 | selvcllem2.j | |- ( ph -> J e. W ) |
|
| 7 | selvcllem2.r | |- ( ph -> R e. CRing ) |
|
| 8 | 1 2 3 4 5 6 7 | selvcllem2 | |- ( ph -> D e. ( R RingHom T ) ) |
| 9 | rnrhmsubrg | |- ( D e. ( R RingHom T ) -> ran D e. ( SubRing ` T ) ) |
|
| 10 | 8 9 | syl | |- ( ph -> ran D e. ( SubRing ` T ) ) |