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 |
|
mpllss.r |
|- ( ph -> R e. Ring ) |
6 |
|
eqid |
|- ( Base ` S ) = ( Base ` S ) |
7 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
8 |
|
eqid |
|- { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } |
9 |
|
0fin |
|- (/) e. Fin |
10 |
9
|
a1i |
|- ( ph -> (/) e. Fin ) |
11 |
|
unfi |
|- ( ( x e. Fin /\ y e. Fin ) -> ( x u. y ) e. Fin ) |
12 |
11
|
adantl |
|- ( ( ph /\ ( x e. Fin /\ y e. Fin ) ) -> ( x u. y ) e. Fin ) |
13 |
|
ssfi |
|- ( ( x e. Fin /\ y C_ x ) -> y e. Fin ) |
14 |
13
|
adantl |
|- ( ( ph /\ ( x e. Fin /\ y C_ x ) ) -> y e. Fin ) |
15 |
1 2 3 4
|
mplsubglem2 |
|- ( ph -> U = { g e. ( Base ` S ) | ( g supp ( 0g ` R ) ) e. Fin } ) |
16 |
1 6 7 8 4 10 12 14 15 5
|
mpllsslem |
|- ( ph -> U e. ( LSubSp ` S ) ) |