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Theorem mpllsslem 17893
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.)
Hypotheses
Ref Expression
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
Ref Expression
mpllsslem  |-  ( ph  ->  U  e.  ( LSubSp `  S ) )
Distinct variable groups:    f, g, x, y,  .0.    A, f, g, x, y    B, f, g    D, g    f, I    ph, x, y    S, f, g, y
Allowed substitution hints:    ph( f, g)    B( x, y)    D( x, y, f)    R( x, y, f, g)    S( x)    U( x, y, f, g)    I( x, y, g)    W( x, y, f, g)

Proof of Theorem mpllsslem
Dummy variables  k  u  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mplsubglem.s . . 3  |-  S  =  ( I mPwSer  R )
2 mplsubglem.i . . 3  |-  ( ph  ->  I  e.  W )
3 mpllsslem.r . . 3  |-  ( ph  ->  R  e.  Ring )
41, 2, 3psrsca 17841 . 2  |-  ( ph  ->  R  =  (Scalar `  S ) )
5 eqidd 2468 . 2  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  R ) )
6 mplsubglem.b . . 3  |-  B  =  ( Base `  S
)
76a1i 11 . 2  |-  ( ph  ->  B  =  ( Base `  S ) )
8 eqidd 2468 . 2  |-  ( ph  ->  ( +g  `  S
)  =  ( +g  `  S ) )
9 eqidd 2468 . 2  |-  ( ph  ->  ( .s `  S
)  =  ( .s
`  S ) )
10 eqidd 2468 . 2  |-  ( ph  ->  ( LSubSp `  S )  =  ( LSubSp `  S
) )
11 mplsubglem.z . . . 4  |-  .0.  =  ( 0g `  R )
12 mplsubglem.d . . . 4  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
13 mplsubglem.0 . . . 4  |-  ( ph  -> 
(/)  e.  A )
14 mplsubglem.a . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  u.  y
)  e.  A )
15 mplsubglem.y . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  C_  x ) )  -> 
y  e.  A )
16 mplsubglem.u . . . 4  |-  ( ph  ->  U  =  { g  e.  B  |  ( g supp  .0.  )  e.  A } )
17 rnggrp 17005 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Grp )
183, 17syl 16 . . . 4  |-  ( ph  ->  R  e.  Grp )
191, 6, 11, 12, 2, 13, 14, 15, 16, 18mplsubglem 17892 . . 3  |-  ( ph  ->  U  e.  (SubGrp `  S ) )
206subgss 16007 . . 3  |-  ( U  e.  (SubGrp `  S
)  ->  U  C_  B
)
2119, 20syl 16 . 2  |-  ( ph  ->  U  C_  B )
22 eqid 2467 . . . 4  |-  ( 0g
`  S )  =  ( 0g `  S
)
2322subg0cl 16014 . . 3  |-  ( U  e.  (SubGrp `  S
)  ->  ( 0g `  S )  e.  U
)
24 ne0i 3791 . . 3  |-  ( ( 0g `  S )  e.  U  ->  U  =/=  (/) )
2519, 23, 243syl 20 . 2  |-  ( ph  ->  U  =/=  (/) )
2619adantr 465 . . 3  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U  /\  w  e.  U
) )  ->  U  e.  (SubGrp `  S )
)
27 eqid 2467 . . . . . 6  |-  ( .s
`  S )  =  ( .s `  S
)
28 eqid 2467 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
293adantr 465 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  R  e.  Ring )
30 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  u  e.  ( Base `  R ) )
31 simprr 756 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
v  e.  U )
3216adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  U  =  { g  e.  B  |  (
g supp  .0.  )  e.  A } )
3332eleq2d 2537 . . . . . . . . 9  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( v  e.  U  <->  v  e.  { g  e.  B  |  ( g supp 
.0.  )  e.  A } ) )
34 oveq1 6291 . . . . . . . . . . 11  |-  ( g  =  v  ->  (
g supp  .0.  )  =  ( v supp  .0.  )
)
3534eleq1d 2536 . . . . . . . . . 10  |-  ( g  =  v  ->  (
( g supp  .0.  )  e.  A  <->  ( v supp  .0.  )  e.  A )
)
3635elrab 3261 . . . . . . . . 9  |-  ( v  e.  { g  e.  B  |  ( g supp 
.0.  )  e.  A } 
<->  ( v  e.  B  /\  ( v supp  .0.  )  e.  A ) )
3733, 36syl6bb 261 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( v  e.  U  <->  ( v  e.  B  /\  ( v supp  .0.  )  e.  A ) ) )
3831, 37mpbid 210 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( v  e.  B  /\  ( v supp  .0.  )  e.  A ) )
3938simpld 459 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
v  e.  B )
401, 27, 28, 6, 29, 30, 39psrvscacl 17845 . . . . 5  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .s
`  S ) v )  e.  B )
41 ovex 6309 . . . . . . 7  |-  ( ( u ( .s `  S ) v ) supp 
.0.  )  e.  _V
4241a1i 11 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( ( u ( .s `  S ) v ) supp  .0.  )  e.  _V )
4338simprd 463 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( v supp  .0.  )  e.  A )
4415expr 615 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (
y  C_  x  ->  y  e.  A ) )
4544alrimiv 1695 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  A. y
( y  C_  x  ->  y  e.  A ) )
4645ralrimiva 2878 . . . . . . . 8  |-  ( ph  ->  A. x  e.  A  A. y ( y  C_  x  ->  y  e.  A
) )
4746adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  A. x  e.  A  A. y ( y  C_  x  ->  y  e.  A
) )
48 sseq2 3526 . . . . . . . . . 10  |-  ( x  =  ( v supp  .0.  )  ->  ( y  C_  x 
<->  y  C_  ( v supp  .0.  ) ) )
4948imbi1d 317 . . . . . . . . 9  |-  ( x  =  ( v supp  .0.  )  ->  ( ( y 
C_  x  ->  y  e.  A )  <->  ( y  C_  ( v supp  .0.  )  ->  y  e.  A ) ) )
5049albidv 1689 . . . . . . . 8  |-  ( x  =  ( v supp  .0.  )  ->  ( A. y
( y  C_  x  ->  y  e.  A )  <->  A. y ( y  C_  ( v supp  .0.  )  ->  y  e.  A ) ) )
5150rspcv 3210 . . . . . . 7  |-  ( ( v supp  .0.  )  e.  A  ->  ( A. x  e.  A  A. y
( y  C_  x  ->  y  e.  A )  ->  A. y ( y 
C_  ( v supp  .0.  )  ->  y  e.  A
) ) )
5243, 47, 51sylc 60 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  A. y ( y  C_  ( v supp  .0.  )  ->  y  e.  A ) )
531, 28, 12, 6, 40psrelbas 17831 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .s
`  S ) v ) : D --> ( Base `  R ) )
54 eqid 2467 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
5530adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( v supp  .0.  )
) )  ->  u  e.  ( Base `  R
) )
5639adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( v supp  .0.  )
) )  ->  v  e.  B )
57 eldifi 3626 . . . . . . . . . 10  |-  ( k  e.  ( D  \ 
( v supp  .0.  )
)  ->  k  e.  D )
5857adantl 466 . . . . . . . . 9  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( v supp  .0.  )
) )  ->  k  e.  D )
591, 27, 28, 6, 54, 12, 55, 56, 58psrvscaval 17844 . . . . . . . 8  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( v supp  .0.  )
) )  ->  (
( u ( .s
`  S ) v ) `  k )  =  ( u ( .r `  R ) ( v `  k
) ) )
601, 28, 12, 6, 39psrelbas 17831 . . . . . . . . . 10  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
v : D --> ( Base `  R ) )
61 ssid 3523 . . . . . . . . . . 11  |-  ( v supp 
.0.  )  C_  (
v supp  .0.  )
6261a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( v supp  .0.  )  C_  ( v supp  .0.  )
)
63 ovex 6309 . . . . . . . . . . . 12  |-  ( NN0 
^m  I )  e. 
_V
6412, 63rabex2 4600 . . . . . . . . . . 11  |-  D  e. 
_V
6564a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  D  e.  _V )
66 fvex 5876 . . . . . . . . . . . 12  |-  ( 0g
`  R )  e. 
_V
6711, 66eqeltri 2551 . . . . . . . . . . 11  |-  .0.  e.  _V
6867a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  .0.  e.  _V )
6960, 62, 65, 68suppssr 6931 . . . . . . . . 9  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( v supp  .0.  )
) )  ->  (
v `  k )  =  .0.  )
7069oveq2d 6300 . . . . . . . 8  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( v supp  .0.  )
) )  ->  (
u ( .r `  R ) ( v `
 k ) )  =  ( u ( .r `  R )  .0.  ) )
7128, 54, 11rngrz 17037 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  u  e.  ( Base `  R
) )  ->  (
u ( .r `  R )  .0.  )  =  .0.  )
7229, 30, 71syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .r
`  R )  .0.  )  =  .0.  )
7372adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( v supp  .0.  )
) )  ->  (
u ( .r `  R )  .0.  )  =  .0.  )
7459, 70, 733eqtrd 2512 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( v supp  .0.  )
) )  ->  (
( u ( .s
`  S ) v ) `  k )  =  .0.  )
7553, 74suppss 6930 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( ( u ( .s `  S ) v ) supp  .0.  )  C_  ( v supp  .0.  )
)
76 sseq1 3525 . . . . . . . 8  |-  ( y  =  ( ( u ( .s `  S
) v ) supp  .0.  )  ->  ( y  C_  ( v supp  .0.  )  <->  ( ( u ( .s
`  S ) v ) supp  .0.  )  C_  ( v supp  .0.  )
) )
77 eleq1 2539 . . . . . . . 8  |-  ( y  =  ( ( u ( .s `  S
) v ) supp  .0.  )  ->  ( y  e.  A  <->  ( ( u ( .s `  S
) v ) supp  .0.  )  e.  A )
)
7876, 77imbi12d 320 . . . . . . 7  |-  ( 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 ) ) )
7978spcgv 3198 . . . . . 6  |-  ( ( ( 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 )
) )
8042, 52, 75, 79syl3c 61 . . . . 5  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( ( u ( .s `  S ) v ) supp  .0.  )  e.  A )
8132eleq2d 2537 . . . . . 6  |-  ( (
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 } ) )
82 oveq1 6291 . . . . . . . 8  |-  ( g  =  ( u ( .s `  S ) v )  ->  (
g supp  .0.  )  =  ( ( u ( .s `  S ) v ) supp  .0.  )
)
8382eleq1d 2536 . . . . . . 7  |-  ( g  =  ( u ( .s `  S ) v )  ->  (
( g supp  .0.  )  e.  A  <->  ( ( u ( .s `  S
) v ) supp  .0.  )  e.  A )
)
8483elrab 3261 . . . . . 6  |-  ( ( 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 ) )
8581, 84syl6bb 261 . . . . 5  |-  ( (
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 ) ) )
8640, 80, 85mpbir2and 920 . . . 4  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .s
`  S ) v )  e.  U )
87863adantr3 1157 . . 3  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U  /\  w  e.  U
) )  ->  (
u ( .s `  S ) v )  e.  U )
88 simpr3 1004 . . 3  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U  /\  w  e.  U
) )  ->  w  e.  U )
89 eqid 2467 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
9089subgcl 16016 . . 3  |-  ( ( U  e.  (SubGrp `  S )  /\  (
u ( .s `  S ) v )  e.  U  /\  w  e.  U )  ->  (
( u ( .s
`  S ) v ) ( +g  `  S
) w )  e.  U )
9126, 87, 88, 90syl3anc 1228 . 2  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U  /\  w  e.  U
) )  ->  (
( u ( .s
`  S ) v ) ( +g  `  S
) w )  e.  U )
924, 5, 7, 8, 9, 10, 21, 25, 91islssd 17382 1  |-  ( ph  ->  U  e.  ( LSubSp `  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973   A.wal 1377    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   {crab 2818   _Vcvv 3113    \ cdif 3473    u. cun 3474    C_ wss 3476   (/)c0 3785   `'ccnv 4998   "cima 5002   ` cfv 5588  (class class class)co 6284   supp csupp 6901    ^m cmap 7420   Fincfn 7516   NNcn 10536   NN0cn0 10795   Basecbs 14490   +g cplusg 14555   .rcmulr 14556   .scvsca 14559   0gc0g 14695   Grpcgrp 15727  SubGrpcsubg 16000   Ringcrg 17000   LSubSpclss 17378   mPwSer cmps 17799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-sca 14571  df-vsca 14572  df-tset 14574  df-0g 14697  df-mnd 15732  df-grp 15867  df-minusg 15868  df-subg 16003  df-mgp 16944  df-rng 17002  df-lss 17379  df-psr 17804
This theorem is referenced by:  mpllss  17898
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