Step |
Hyp |
Ref |
Expression |
1 |
|
mplgrp.p |
|- P = ( I mPoly R ) |
2 |
|
drngring |
|- ( R e. DivRing -> R e. Ring ) |
3 |
1
|
mpllmod |
|- ( ( I e. V /\ R e. Ring ) -> P e. LMod ) |
4 |
2 3
|
sylan2 |
|- ( ( I e. V /\ R e. DivRing ) -> P e. LMod ) |
5 |
|
simpl |
|- ( ( I e. V /\ R e. DivRing ) -> I e. V ) |
6 |
|
simpr |
|- ( ( I e. V /\ R e. DivRing ) -> R e. DivRing ) |
7 |
1 5 6
|
mplsca |
|- ( ( I e. V /\ R e. DivRing ) -> R = ( Scalar ` P ) ) |
8 |
7 6
|
eqeltrrd |
|- ( ( I e. V /\ R e. DivRing ) -> ( Scalar ` P ) e. DivRing ) |
9 |
|
eqid |
|- ( Scalar ` P ) = ( Scalar ` P ) |
10 |
9
|
islvec |
|- ( P e. LVec <-> ( P e. LMod /\ ( Scalar ` P ) e. DivRing ) ) |
11 |
4 8 10
|
sylanbrc |
|- ( ( I e. V /\ R e. DivRing ) -> P e. LVec ) |