Metamath Proof Explorer


Theorem mpllsslem

Description: If A is an ideal of subsets (a nonempty collection closed under subset and binary union) of the set D of finite bags (the primary applications being A = Fin and A = ~P B for some B ), then the set of all power series whose coefficient functions are supported on an element of A is a linear subspace of the set of all power series. (Contributed by Mario Carneiro, 12-Jan-2015) (Revised by AV, 16-Jul-2019)

Ref Expression
Hypotheses mplsubglem.s
|- S = ( I mPwSer R )
mplsubglem.b
|- B = ( Base ` S )
mplsubglem.z
|- .0. = ( 0g ` R )
mplsubglem.d
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
mplsubglem.i
|- ( ph -> I e. W )
mplsubglem.0
|- ( ph -> (/) e. A )
mplsubglem.a
|- ( ( ph /\ ( x e. A /\ y e. A ) ) -> ( x u. y ) e. A )
mplsubglem.y
|- ( ( ph /\ ( x e. A /\ y C_ x ) ) -> y e. A )
mplsubglem.u
|- ( ph -> U = { g e. B | ( g supp .0. ) e. A } )
mpllsslem.r
|- ( ph -> R e. Ring )
Assertion mpllsslem
|- ( ph -> U e. ( LSubSp ` S ) )

Proof

Step Hyp Ref Expression
1 mplsubglem.s
 |-  S = ( I mPwSer R )
2 mplsubglem.b
 |-  B = ( Base ` S )
3 mplsubglem.z
 |-  .0. = ( 0g ` R )
4 mplsubglem.d
 |-  D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
5 mplsubglem.i
 |-  ( ph -> I e. W )
6 mplsubglem.0
 |-  ( ph -> (/) e. A )
7 mplsubglem.a
 |-  ( ( ph /\ ( x e. A /\ y e. A ) ) -> ( x u. y ) e. A )
8 mplsubglem.y
 |-  ( ( ph /\ ( x e. A /\ y C_ x ) ) -> y e. A )
9 mplsubglem.u
 |-  ( ph -> U = { g e. B | ( g supp .0. ) e. A } )
10 mpllsslem.r
 |-  ( ph -> R e. Ring )
11 1 5 10 psrsca
 |-  ( ph -> R = ( Scalar ` S ) )
12 eqidd
 |-  ( ph -> ( Base ` R ) = ( Base ` R ) )
13 2 a1i
 |-  ( ph -> B = ( Base ` S ) )
14 eqidd
 |-  ( ph -> ( +g ` S ) = ( +g ` S ) )
15 eqidd
 |-  ( ph -> ( .s ` S ) = ( .s ` S ) )
16 eqidd
 |-  ( ph -> ( LSubSp ` S ) = ( LSubSp ` S ) )
17 ringgrp
 |-  ( R e. Ring -> R e. Grp )
18 10 17 syl
 |-  ( ph -> R e. Grp )
19 1 2 3 4 5 6 7 8 9 18 mplsubglem
 |-  ( ph -> U e. ( SubGrp ` S ) )
20 2 subgss
 |-  ( U e. ( SubGrp ` S ) -> U C_ B )
21 19 20 syl
 |-  ( ph -> U C_ B )
22 eqid
 |-  ( 0g ` S ) = ( 0g ` S )
23 22 subg0cl
 |-  ( U e. ( SubGrp ` S ) -> ( 0g ` S ) e. U )
24 ne0i
 |-  ( ( 0g ` S ) e. U -> U =/= (/) )
25 19 23 24 3syl
 |-  ( ph -> U =/= (/) )
26 19 adantr
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U /\ w e. U ) ) -> U e. ( SubGrp ` S ) )
27 eqid
 |-  ( .s ` S ) = ( .s ` S )
28 eqid
 |-  ( Base ` R ) = ( Base ` R )
29 10 adantr
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> R e. Ring )
30 simprl
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> u e. ( Base ` R ) )
31 simprr
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> v e. U )
32 9 adantr
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> U = { g e. B | ( g supp .0. ) e. A } )
33 32 eleq2d
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( v e. U <-> v e. { g e. B | ( g supp .0. ) e. A } ) )
34 oveq1
 |-  ( g = v -> ( g supp .0. ) = ( v supp .0. ) )
35 34 eleq1d
 |-  ( g = v -> ( ( g supp .0. ) e. A <-> ( v supp .0. ) e. A ) )
36 35 elrab
 |-  ( v e. { g e. B | ( g supp .0. ) e. A } <-> ( v e. B /\ ( v supp .0. ) e. A ) )
37 33 36 bitrdi
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( v e. U <-> ( v e. B /\ ( v supp .0. ) e. A ) ) )
38 31 37 mpbid
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( v e. B /\ ( v supp .0. ) e. A ) )
39 38 simpld
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> v e. B )
40 1 27 28 2 29 30 39 psrvscacl
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( u ( .s ` S ) v ) e. B )
41 ovex
 |-  ( ( u ( .s ` S ) v ) supp .0. ) e. _V
42 41 a1i
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( ( u ( .s ` S ) v ) supp .0. ) e. _V )
43 sseq2
 |-  ( x = ( v supp .0. ) -> ( y C_ x <-> y C_ ( v supp .0. ) ) )
44 43 imbi1d
 |-  ( x = ( v supp .0. ) -> ( ( y C_ x -> y e. A ) <-> ( y C_ ( v supp .0. ) -> y e. A ) ) )
45 44 albidv
 |-  ( x = ( v supp .0. ) -> ( A. y ( y C_ x -> y e. A ) <-> A. y ( y C_ ( v supp .0. ) -> y e. A ) ) )
46 8 expr
 |-  ( ( ph /\ x e. A ) -> ( y C_ x -> y e. A ) )
47 46 alrimiv
 |-  ( ( ph /\ x e. A ) -> A. y ( y C_ x -> y e. A ) )
48 47 ralrimiva
 |-  ( ph -> A. x e. A A. y ( y C_ x -> y e. A ) )
49 48 adantr
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> A. x e. A A. y ( y C_ x -> y e. A ) )
50 38 simprd
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( v supp .0. ) e. A )
51 45 49 50 rspcdva
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> A. y ( y C_ ( v supp .0. ) -> y e. A ) )
52 1 28 4 2 40 psrelbas
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( u ( .s ` S ) v ) : D --> ( Base ` R ) )
53 eqid
 |-  ( .r ` R ) = ( .r ` R )
54 30 adantr
 |-  ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> u e. ( Base ` R ) )
55 39 adantr
 |-  ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> v e. B )
56 eldifi
 |-  ( k e. ( D \ ( v supp .0. ) ) -> k e. D )
57 56 adantl
 |-  ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> k e. D )
58 1 27 28 2 53 4 54 55 57 psrvscaval
 |-  ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> ( ( u ( .s ` S ) v ) ` k ) = ( u ( .r ` R ) ( v ` k ) ) )
59 1 28 4 2 39 psrelbas
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> v : D --> ( Base ` R ) )
60 ssidd
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( v supp .0. ) C_ ( v supp .0. ) )
61 ovex
 |-  ( NN0 ^m I ) e. _V
62 4 61 rabex2
 |-  D e. _V
63 62 a1i
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> D e. _V )
64 3 fvexi
 |-  .0. e. _V
65 64 a1i
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> .0. e. _V )
66 59 60 63 65 suppssr
 |-  ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> ( v ` k ) = .0. )
67 66 oveq2d
 |-  ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> ( u ( .r ` R ) ( v ` k ) ) = ( u ( .r ` R ) .0. ) )
68 28 53 3 ringrz
 |-  ( ( R e. Ring /\ u e. ( Base ` R ) ) -> ( u ( .r ` R ) .0. ) = .0. )
69 10 30 68 syl2an2r
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( u ( .r ` R ) .0. ) = .0. )
70 69 adantr
 |-  ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> ( u ( .r ` R ) .0. ) = .0. )
71 58 67 70 3eqtrd
 |-  ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> ( ( u ( .s ` S ) v ) ` k ) = .0. )
72 52 71 suppss
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( ( u ( .s ` S ) v ) supp .0. ) C_ ( v supp .0. ) )
73 sseq1
 |-  ( y = ( ( u ( .s ` S ) v ) supp .0. ) -> ( y C_ ( v supp .0. ) <-> ( ( u ( .s ` S ) v ) supp .0. ) C_ ( v supp .0. ) ) )
74 eleq1
 |-  ( y = ( ( u ( .s ` S ) v ) supp .0. ) -> ( y e. A <-> ( ( u ( .s ` S ) v ) supp .0. ) e. A ) )
75 73 74 imbi12d
 |-  ( y = ( ( u ( .s ` S ) v ) supp .0. ) -> ( ( y C_ ( v supp .0. ) -> y e. A ) <-> ( ( ( u ( .s ` S ) v ) supp .0. ) C_ ( v supp .0. ) -> ( ( u ( .s ` S ) v ) supp .0. ) e. A ) ) )
76 75 spcgv
 |-  ( ( ( u ( .s ` S ) v ) supp .0. ) e. _V -> ( A. y ( y C_ ( v supp .0. ) -> y e. A ) -> ( ( ( u ( .s ` S ) v ) supp .0. ) C_ ( v supp .0. ) -> ( ( u ( .s ` S ) v ) supp .0. ) e. A ) ) )
77 42 51 72 76 syl3c
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( ( u ( .s ` S ) v ) supp .0. ) e. A )
78 32 eleq2d
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( ( u ( .s ` S ) v ) e. U <-> ( u ( .s ` S ) v ) e. { g e. B | ( g supp .0. ) e. A } ) )
79 oveq1
 |-  ( g = ( u ( .s ` S ) v ) -> ( g supp .0. ) = ( ( u ( .s ` S ) v ) supp .0. ) )
80 79 eleq1d
 |-  ( g = ( u ( .s ` S ) v ) -> ( ( g supp .0. ) e. A <-> ( ( u ( .s ` S ) v ) supp .0. ) e. A ) )
81 80 elrab
 |-  ( ( u ( .s ` S ) v ) e. { g e. B | ( g supp .0. ) e. A } <-> ( ( u ( .s ` S ) v ) e. B /\ ( ( u ( .s ` S ) v ) supp .0. ) e. A ) )
82 78 81 bitrdi
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( ( u ( .s ` S ) v ) e. U <-> ( ( u ( .s ` S ) v ) e. B /\ ( ( u ( .s ` S ) v ) supp .0. ) e. A ) ) )
83 40 77 82 mpbir2and
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( u ( .s ` S ) v ) e. U )
84 83 3adantr3
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U /\ w e. U ) ) -> ( u ( .s ` S ) v ) e. U )
85 simpr3
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U /\ w e. U ) ) -> w e. U )
86 eqid
 |-  ( +g ` S ) = ( +g ` S )
87 86 subgcl
 |-  ( ( U e. ( SubGrp ` S ) /\ ( u ( .s ` S ) v ) e. U /\ w e. U ) -> ( ( u ( .s ` S ) v ) ( +g ` S ) w ) e. U )
88 26 84 85 87 syl3anc
 |-  ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U /\ w e. U ) ) -> ( ( u ( .s ` S ) v ) ( +g ` S ) w ) e. U )
89 11 12 13 14 15 16 21 25 88 islssd
 |-  ( ph -> U e. ( LSubSp ` S ) )