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Theorem mpllsslem 17516
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 17465 . 2  |-  ( ph  ->  R  =  (Scalar `  S ) )
5 eqidd 2444 . 2  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  R ) )
6 mplsubglem.b . . 3  |-  B  =  ( Base `  S
)
76a1i 11 . 2  |-  ( ph  ->  B  =  ( Base `  S ) )
8 eqidd 2444 . 2  |-  ( ph  ->  ( +g  `  S
)  =  ( +g  `  S ) )
9 eqidd 2444 . 2  |-  ( ph  ->  ( .s `  S
)  =  ( .s
`  S ) )
10 eqidd 2444 . 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 16655 . . . . 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 17515 . . 3  |-  ( ph  ->  U  e.  (SubGrp `  S ) )
206subgss 15687 . . 3  |-  ( U  e.  (SubGrp `  S
)  ->  U  C_  B
)
2119, 20syl 16 . 2  |-  ( ph  ->  U  C_  B )
22 eqid 2443 . . . 4  |-  ( 0g
`  S )  =  ( 0g `  S
)
2322subg0cl 15694 . . 3  |-  ( U  e.  (SubGrp `  S
)  ->  ( 0g `  S )  e.  U
)
24 ne0i 3648 . . 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 2443 . . . . . 6  |-  ( .s
`  S )  =  ( .s `  S
)
28 eqid 2443 . . . . . 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 2510 . . . . . . . . 9  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( v  e.  U  <->  v  e.  { g  e.  B  |  ( g supp 
.0.  )  e.  A } ) )
34 oveq1 6103 . . . . . . . . . . 11  |-  ( g  =  v  ->  (
g supp  .0.  )  =  ( v supp  .0.  )
)
3534eleq1d 2509 . . . . . . . . . 10  |-  ( g  =  v  ->  (
( g supp  .0.  )  e.  A  <->  ( v supp  .0.  )  e.  A )
)
3635elrab 3122 . . . . . . . . 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 17469 . . . . 5  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .s
`  S ) v )  e.  B )
41 ovex 6121 . . . . . . 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 1685 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  A. y
( y  C_  x  ->  y  e.  A ) )
4645ralrimiva 2804 . . . . . . . 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 3383 . . . . . . . . . 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 1679 . . . . . . . 8  |-  ( x  =  ( v supp  .0.  )  ->  ( A. y
( y  C_  x  ->  y  e.  A )  <->  A. y ( y  C_  ( v supp  .0.  )  ->  y  e.  A ) ) )
5150rspcv 3074 . . . . . . 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 17455 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .s
`  S ) v ) : D --> ( Base `  R ) )
54 eqid 2443 . . . . . . . . 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 3483 . . . . . . . . . 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 17468 . . . . . . . 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 17455 . . . . . . . . . 10  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
v : D --> ( Base `  R ) )
61 ssid 3380 . . . . . . . . . . 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 6121 . . . . . . . . . . . 12  |-  ( NN0 
^m  I )  e. 
_V
6412, 63rabex2 4450 . . . . . . . . . . 11  |-  D  e. 
_V
6564a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  D  e.  _V )
66 fvex 5706 . . . . . . . . . . . 12  |-  ( 0g
`  R )  e. 
_V
6711, 66eqeltri 2513 . . . . . . . . . . 11  |-  .0.  e.  _V
6867a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  .0.  e.  _V )
6960, 62, 65, 68suppssr 6725 . . . . . . . . 9  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( v supp  .0.  )
) )  ->  (
v `  k )  =  .0.  )
7069oveq2d 6112 . . . . . . . 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 16687 . . . . . . . . . 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 2479 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( v supp  .0.  )
) )  ->  (
( u ( .s
`  S ) v ) `  k )  =  .0.  )
7553, 74suppss 6724 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( ( u ( .s `  S ) v ) supp  .0.  )  C_  ( v supp  .0.  )
)
76 sseq1 3382 . . . . . . . 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 2503 . . . . . . . 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 3062 . . . . . 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 2510 . . . . . 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 6103 . . . . . . . 8  |-  ( g  =  ( u ( .s `  S ) v )  ->  (
g supp  .0.  )  =  ( ( u ( .s `  S ) v ) supp  .0.  )
)
8382eleq1d 2509 . . . . . . 7  |-  ( g  =  ( u ( .s `  S ) v )  ->  (
( g supp  .0.  )  e.  A  <->  ( ( u ( .s `  S
) v ) supp  .0.  )  e.  A )
)
8483elrab 3122 . . . . . 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 913 . . . 4  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .s
`  S ) v )  e.  U )
87863adantr3 1149 . . 3  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U  /\  w  e.  U
) )  ->  (
u ( .s `  S ) v )  e.  U )
88 simpr3 996 . . 3  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U  /\  w  e.  U
) )  ->  w  e.  U )
89 eqid 2443 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
9089subgcl 15696 . . 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 1218 . 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 17022 1  |-  ( ph  ->  U  e.  ( LSubSp `  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965   A.wal 1367    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   {crab 2724   _Vcvv 2977    \ cdif 3330    u. cun 3331    C_ wss 3333   (/)c0 3642   `'ccnv 4844   "cima 4848   ` cfv 5423  (class class class)co 6096   supp csupp 6695    ^m cmap 7219   Fincfn 7315   NNcn 10327   NN0cn0 10584   Basecbs 14179   +g cplusg 14243   .rcmulr 14244   .scvsca 14247   0gc0g 14383   Grpcgrp 15415  SubGrpcsubg 15680   Ringcrg 16650   LSubSpclss 17018   mPwSer cmps 17423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-tset 14262  df-0g 14385  df-mnd 15420  df-grp 15550  df-minusg 15551  df-subg 15683  df-mgp 16597  df-rng 16652  df-lss 17019  df-psr 17428
This theorem is referenced by:  mpllss  17521
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