Step |
Hyp |
Ref |
Expression |
1 |
|
mplsubg.s |
|- S = ( I mPwSer R ) |
2 |
|
mplsubg.p |
|- P = ( I mPoly R ) |
3 |
|
mplsubg.u |
|- U = ( Base ` P ) |
4 |
|
mplsubg.i |
|- ( ph -> I e. W ) |
5 |
|
eqid |
|- ( Base ` S ) = ( Base ` S ) |
6 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
7 |
2 1 5 6 3
|
mplbas |
|- U = { g e. ( Base ` S ) | g finSupp ( 0g ` R ) } |
8 |
1 5
|
psrelbasfun |
|- ( g e. ( Base ` S ) -> Fun g ) |
9 |
8
|
adantl |
|- ( ( ph /\ g e. ( Base ` S ) ) -> Fun g ) |
10 |
|
simpr |
|- ( ( ph /\ g e. ( Base ` S ) ) -> g e. ( Base ` S ) ) |
11 |
|
fvexd |
|- ( ( ph /\ g e. ( Base ` S ) ) -> ( 0g ` R ) e. _V ) |
12 |
|
funisfsupp |
|- ( ( Fun g /\ g e. ( Base ` S ) /\ ( 0g ` R ) e. _V ) -> ( g finSupp ( 0g ` R ) <-> ( g supp ( 0g ` R ) ) e. Fin ) ) |
13 |
9 10 11 12
|
syl3anc |
|- ( ( ph /\ g e. ( Base ` S ) ) -> ( g finSupp ( 0g ` R ) <-> ( g supp ( 0g ` R ) ) e. Fin ) ) |
14 |
13
|
rabbidva |
|- ( ph -> { g e. ( Base ` S ) | g finSupp ( 0g ` R ) } = { g e. ( Base ` S ) | ( g supp ( 0g ` R ) ) e. Fin } ) |
15 |
7 14
|
syl5eq |
|- ( ph -> U = { g e. ( Base ` S ) | ( g supp ( 0g ` R ) ) e. Fin } ) |