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 |
|
ressmplbas2.w |
|- W = ( I mPwSer H ) |
8 |
|
ressmplbas2.c |
|- C = ( Base ` W ) |
9 |
|
ressmplbas2.k |
|- K = ( Base ` S ) |
10 |
|
eqid |
|- ( I mPwSer R ) = ( I mPwSer R ) |
11 |
10 2 7 8
|
subrgpsr |
|- ( ( I e. V /\ T e. ( SubRing ` R ) ) -> C e. ( SubRing ` ( I mPwSer R ) ) ) |
12 |
5 6 11
|
syl2anc |
|- ( ph -> C e. ( SubRing ` ( I mPwSer R ) ) ) |
13 |
|
eqid |
|- ( Base ` ( I mPwSer R ) ) = ( Base ` ( I mPwSer R ) ) |
14 |
13
|
subrgss |
|- ( C e. ( SubRing ` ( I mPwSer R ) ) -> C C_ ( Base ` ( I mPwSer R ) ) ) |
15 |
12 14
|
syl |
|- ( ph -> C C_ ( Base ` ( I mPwSer R ) ) ) |
16 |
|
df-ss |
|- ( C C_ ( Base ` ( I mPwSer R ) ) <-> ( C i^i ( Base ` ( I mPwSer R ) ) ) = C ) |
17 |
15 16
|
sylib |
|- ( ph -> ( C i^i ( Base ` ( I mPwSer R ) ) ) = C ) |
18 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
19 |
2 18
|
subrg0 |
|- ( T e. ( SubRing ` R ) -> ( 0g ` R ) = ( 0g ` H ) ) |
20 |
6 19
|
syl |
|- ( ph -> ( 0g ` R ) = ( 0g ` H ) ) |
21 |
20
|
breq2d |
|- ( ph -> ( f finSupp ( 0g ` R ) <-> f finSupp ( 0g ` H ) ) ) |
22 |
21
|
abbidv |
|- ( ph -> { f | f finSupp ( 0g ` R ) } = { f | f finSupp ( 0g ` H ) } ) |
23 |
17 22
|
ineq12d |
|- ( ph -> ( ( C i^i ( Base ` ( I mPwSer R ) ) ) i^i { f | f finSupp ( 0g ` R ) } ) = ( C i^i { f | f finSupp ( 0g ` H ) } ) ) |
24 |
23
|
eqcomd |
|- ( ph -> ( C i^i { f | f finSupp ( 0g ` H ) } ) = ( ( C i^i ( Base ` ( I mPwSer R ) ) ) i^i { f | f finSupp ( 0g ` R ) } ) ) |
25 |
|
eqid |
|- ( 0g ` H ) = ( 0g ` H ) |
26 |
3 7 8 25 4
|
mplbas |
|- B = { f e. C | f finSupp ( 0g ` H ) } |
27 |
|
dfrab3 |
|- { f e. C | f finSupp ( 0g ` H ) } = ( C i^i { f | f finSupp ( 0g ` H ) } ) |
28 |
26 27
|
eqtri |
|- B = ( C i^i { f | f finSupp ( 0g ` H ) } ) |
29 |
1 10 13 18 9
|
mplbas |
|- K = { f e. ( Base ` ( I mPwSer R ) ) | f finSupp ( 0g ` R ) } |
30 |
|
dfrab3 |
|- { f e. ( Base ` ( I mPwSer R ) ) | f finSupp ( 0g ` R ) } = ( ( Base ` ( I mPwSer R ) ) i^i { f | f finSupp ( 0g ` R ) } ) |
31 |
29 30
|
eqtri |
|- K = ( ( Base ` ( I mPwSer R ) ) i^i { f | f finSupp ( 0g ` R ) } ) |
32 |
31
|
ineq2i |
|- ( C i^i K ) = ( C i^i ( ( Base ` ( I mPwSer R ) ) i^i { f | f finSupp ( 0g ` R ) } ) ) |
33 |
|
inass |
|- ( ( C i^i ( Base ` ( I mPwSer R ) ) ) i^i { f | f finSupp ( 0g ` R ) } ) = ( C i^i ( ( Base ` ( I mPwSer R ) ) i^i { f | f finSupp ( 0g ` R ) } ) ) |
34 |
32 33
|
eqtr4i |
|- ( C i^i K ) = ( ( C i^i ( Base ` ( I mPwSer R ) ) ) i^i { f | f finSupp ( 0g ` R ) } ) |
35 |
24 28 34
|
3eqtr4g |
|- ( ph -> B = ( C i^i K ) ) |