Metamath Proof Explorer


Theorem mplsubglem2

Description: Lemma for mplsubg and mpllss . (Contributed by AV, 16-Jul-2019)

Ref Expression
Hypotheses mplsubg.s
|- S = ( I mPwSer R )
mplsubg.p
|- P = ( I mPoly R )
mplsubg.u
|- U = ( Base ` P )
mplsubg.i
|- ( ph -> I e. W )
Assertion mplsubglem2
|- ( ph -> U = { g e. ( Base ` S ) | ( g supp ( 0g ` R ) ) e. Fin } )

Proof

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 } )