| 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 5 6 7
|
selvcllem1 |
|- ( ph -> T e. AssAlg ) |
| 9 |
|
eqid |
|- ( Scalar ` T ) = ( Scalar ` T ) |
| 10 |
3 9
|
asclrhm |
|- ( T e. AssAlg -> C e. ( ( Scalar ` T ) RingHom T ) ) |
| 11 |
8 10
|
syl |
|- ( ph -> C e. ( ( Scalar ` T ) RingHom T ) ) |
| 12 |
1
|
mplassa |
|- ( ( I e. V /\ R e. CRing ) -> U e. AssAlg ) |
| 13 |
5 7 12
|
syl2anc |
|- ( ph -> U e. AssAlg ) |
| 14 |
2 6 13
|
mplsca |
|- ( ph -> U = ( Scalar ` T ) ) |
| 15 |
14
|
oveq1d |
|- ( ph -> ( U RingHom T ) = ( ( Scalar ` T ) RingHom T ) ) |
| 16 |
11 15
|
eleqtrrd |
|- ( ph -> C e. ( U RingHom T ) ) |
| 17 |
|
eqid |
|- ( algSc ` U ) = ( algSc ` U ) |
| 18 |
|
eqid |
|- ( Scalar ` U ) = ( Scalar ` U ) |
| 19 |
17 18
|
asclrhm |
|- ( U e. AssAlg -> ( algSc ` U ) e. ( ( Scalar ` U ) RingHom U ) ) |
| 20 |
13 19
|
syl |
|- ( ph -> ( algSc ` U ) e. ( ( Scalar ` U ) RingHom U ) ) |
| 21 |
|
rhmco |
|- ( ( C e. ( U RingHom T ) /\ ( algSc ` U ) e. ( ( Scalar ` U ) RingHom U ) ) -> ( C o. ( algSc ` U ) ) e. ( ( Scalar ` U ) RingHom T ) ) |
| 22 |
16 20 21
|
syl2anc |
|- ( ph -> ( C o. ( algSc ` U ) ) e. ( ( Scalar ` U ) RingHom T ) ) |
| 23 |
1 5 7
|
mplsca |
|- ( ph -> R = ( Scalar ` U ) ) |
| 24 |
23
|
oveq1d |
|- ( ph -> ( R RingHom T ) = ( ( Scalar ` U ) RingHom T ) ) |
| 25 |
22 24
|
eleqtrrd |
|- ( ph -> ( C o. ( algSc ` U ) ) e. ( R RingHom T ) ) |
| 26 |
4 25
|
eqeltrid |
|- ( ph -> D e. ( R RingHom T ) ) |