Step |
Hyp |
Ref |
Expression |
1 |
|
ressmpl.s |
|- S = ( I mPoly R ) |
2 |
|
ressmpl.h |
|- H = ( R |`s T ) |
3 |
|
ressmpl.u |
|- U = ( I mPoly H ) |
4 |
|
ressmpl.b |
|- B = ( Base ` U ) |
5 |
|
ressmpl.1 |
|- ( ph -> I e. V ) |
6 |
|
ressmpl.2 |
|- ( ph -> T e. ( SubRing ` R ) ) |
7 |
|
ressmpl.p |
|- P = ( S |`s B ) |
8 |
|
eqid |
|- ( I mPwSer H ) = ( I mPwSer H ) |
9 |
|
eqid |
|- ( Base ` ( I mPwSer H ) ) = ( Base ` ( I mPwSer H ) ) |
10 |
|
eqid |
|- ( Base ` S ) = ( Base ` S ) |
11 |
1 2 3 4 5 6 8 9 10
|
ressmplbas2 |
|- ( ph -> B = ( ( Base ` ( I mPwSer H ) ) i^i ( Base ` S ) ) ) |
12 |
|
inss2 |
|- ( ( Base ` ( I mPwSer H ) ) i^i ( Base ` S ) ) C_ ( Base ` S ) |
13 |
11 12
|
eqsstrdi |
|- ( ph -> B C_ ( Base ` S ) ) |
14 |
7 10
|
ressbas2 |
|- ( B C_ ( Base ` S ) -> B = ( Base ` P ) ) |
15 |
13 14
|
syl |
|- ( ph -> B = ( Base ` P ) ) |