Step |
Hyp |
Ref |
Expression |
1 |
|
mplgrp.p |
|- P = ( I mPoly R ) |
2 |
|
eqid |
|- ( I mPwSer R ) = ( I mPwSer R ) |
3 |
|
simpl |
|- ( ( I e. V /\ R e. Ring ) -> I e. V ) |
4 |
|
simpr |
|- ( ( I e. V /\ R e. Ring ) -> R e. Ring ) |
5 |
2 3 4
|
psrlmod |
|- ( ( I e. V /\ R e. Ring ) -> ( I mPwSer R ) e. LMod ) |
6 |
|
eqid |
|- ( Base ` P ) = ( Base ` P ) |
7 |
2 1 6 3 4
|
mpllss |
|- ( ( I e. V /\ R e. Ring ) -> ( Base ` P ) e. ( LSubSp ` ( I mPwSer R ) ) ) |
8 |
1 2 6
|
mplval2 |
|- P = ( ( I mPwSer R ) |`s ( Base ` P ) ) |
9 |
|
eqid |
|- ( LSubSp ` ( I mPwSer R ) ) = ( LSubSp ` ( I mPwSer R ) ) |
10 |
8 9
|
lsslmod |
|- ( ( ( I mPwSer R ) e. LMod /\ ( Base ` P ) e. ( LSubSp ` ( I mPwSer R ) ) ) -> P e. LMod ) |
11 |
5 7 10
|
syl2anc |
|- ( ( I e. V /\ R e. Ring ) -> P e. LMod ) |