Step |
Hyp |
Ref |
Expression |
1 |
|
subrgmpl.s |
|- S = ( I mPoly R ) |
2 |
|
subrgmpl.h |
|- H = ( R |`s T ) |
3 |
|
subrgmpl.u |
|- U = ( I mPoly H ) |
4 |
|
subrgmpl.b |
|- B = ( Base ` U ) |
5 |
|
simpl |
|- ( ( I e. V /\ T e. ( SubRing ` R ) ) -> I e. V ) |
6 |
|
simpr |
|- ( ( I e. V /\ T e. ( SubRing ` R ) ) -> T e. ( SubRing ` R ) ) |
7 |
|
eqid |
|- ( I mPwSer H ) = ( I mPwSer H ) |
8 |
|
eqid |
|- ( Base ` ( I mPwSer H ) ) = ( Base ` ( I mPwSer H ) ) |
9 |
|
eqid |
|- ( Base ` S ) = ( Base ` S ) |
10 |
1 2 3 4 5 6 7 8 9
|
ressmplbas2 |
|- ( ( I e. V /\ T e. ( SubRing ` R ) ) -> B = ( ( Base ` ( I mPwSer H ) ) i^i ( Base ` S ) ) ) |
11 |
|
eqid |
|- ( I mPwSer R ) = ( I mPwSer R ) |
12 |
11 2 7 8
|
subrgpsr |
|- ( ( I e. V /\ T e. ( SubRing ` R ) ) -> ( Base ` ( I mPwSer H ) ) e. ( SubRing ` ( I mPwSer R ) ) ) |
13 |
|
subrgrcl |
|- ( T e. ( SubRing ` R ) -> R e. Ring ) |
14 |
13
|
adantl |
|- ( ( I e. V /\ T e. ( SubRing ` R ) ) -> R e. Ring ) |
15 |
11 1 9 5 14
|
mplsubrg |
|- ( ( I e. V /\ T e. ( SubRing ` R ) ) -> ( Base ` S ) e. ( SubRing ` ( I mPwSer R ) ) ) |
16 |
|
subrgin |
|- ( ( ( Base ` ( I mPwSer H ) ) e. ( SubRing ` ( I mPwSer R ) ) /\ ( Base ` S ) e. ( SubRing ` ( I mPwSer R ) ) ) -> ( ( Base ` ( I mPwSer H ) ) i^i ( Base ` S ) ) e. ( SubRing ` ( I mPwSer R ) ) ) |
17 |
12 15 16
|
syl2anc |
|- ( ( I e. V /\ T e. ( SubRing ` R ) ) -> ( ( Base ` ( I mPwSer H ) ) i^i ( Base ` S ) ) e. ( SubRing ` ( I mPwSer R ) ) ) |
18 |
10 17
|
eqeltrd |
|- ( ( I e. V /\ T e. ( SubRing ` R ) ) -> B e. ( SubRing ` ( I mPwSer R ) ) ) |
19 |
|
inss2 |
|- ( ( Base ` ( I mPwSer H ) ) i^i ( Base ` S ) ) C_ ( Base ` S ) |
20 |
10 19
|
eqsstrdi |
|- ( ( I e. V /\ T e. ( SubRing ` R ) ) -> B C_ ( Base ` S ) ) |
21 |
1 11 9
|
mplval2 |
|- S = ( ( I mPwSer R ) |`s ( Base ` S ) ) |
22 |
21
|
subsubrg |
|- ( ( Base ` S ) e. ( SubRing ` ( I mPwSer R ) ) -> ( B e. ( SubRing ` S ) <-> ( B e. ( SubRing ` ( I mPwSer R ) ) /\ B C_ ( Base ` S ) ) ) ) |
23 |
15 22
|
syl |
|- ( ( I e. V /\ T e. ( SubRing ` R ) ) -> ( B e. ( SubRing ` S ) <-> ( B e. ( SubRing ` ( I mPwSer R ) ) /\ B C_ ( Base ` S ) ) ) ) |
24 |
18 20 23
|
mpbir2and |
|- ( ( I e. V /\ T e. ( SubRing ` R ) ) -> B e. ( SubRing ` S ) ) |