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Theorem dsmmlss 27078
Description: The finite hull of a product of modules is additionally closed under scalar multiplication and thus is a linear subspace of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
Hypotheses
Ref Expression
dsmmlss.i  |-  ( ph  ->  I  e.  W )
dsmmlss.s  |-  ( ph  ->  S  e.  Ring )
dsmmlss.r  |-  ( ph  ->  R : I --> LMod )
dsmmlss.k  |-  ( (
ph  /\  x  e.  I )  ->  (Scalar `  ( R `  x
) )  =  S )
dsmmlss.p  |-  P  =  ( S X_s R )
dsmmlss.u  |-  U  =  ( LSubSp `  P )
dsmmlss.h  |-  H  =  ( Base `  ( S  (+)m  R ) )
Assertion
Ref Expression
dsmmlss  |-  ( ph  ->  H  e.  U )
Distinct variable groups:    ph, x    x, S    x, R    x, I    x, P    x, H
Allowed substitution hints:    U( x)    W( x)

Proof of Theorem dsmmlss
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dsmmlss.p . . 3  |-  P  =  ( S X_s R )
2 dsmmlss.h . . 3  |-  H  =  ( Base `  ( S  (+)m  R ) )
3 dsmmlss.i . . 3  |-  ( ph  ->  I  e.  W )
4 dsmmlss.s . . 3  |-  ( ph  ->  S  e.  Ring )
5 dsmmlss.r . . . 4  |-  ( ph  ->  R : I --> LMod )
6 lmodgrp 15912 . . . . 5  |-  ( a  e.  LMod  ->  a  e. 
Grp )
76ssriv 3312 . . . 4  |-  LMod  C_  Grp
8 fss 5558 . . . 4  |-  ( ( R : I --> LMod  /\  LMod  C_ 
Grp )  ->  R : I --> Grp )
95, 7, 8sylancl 644 . . 3  |-  ( ph  ->  R : I --> Grp )
101, 2, 3, 4, 9dsmmsubg 27077 . 2  |-  ( ph  ->  H  e.  (SubGrp `  P ) )
11 dsmmlss.k . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  (Scalar `  ( R `  x
) )  =  S )
121, 4, 3, 5, 11prdslmodd 16000 . . . . . 6  |-  ( ph  ->  P  e.  LMod )
1312adantr 452 . . . . 5  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  P  e.  LMod )
14 simprl 733 . . . . 5  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  a  e.  ( Base `  (Scalar `  P
) ) )
15 simprr 734 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  b  e.  H )
16 eqid 2404 . . . . . . . . 9  |-  ( S 
(+)m  R )  =  ( S  (+)m  R )
17 eqid 2404 . . . . . . . . 9  |-  ( Base `  P )  =  (
Base `  P )
18 ffn 5550 . . . . . . . . . 10  |-  ( R : I --> LMod  ->  R  Fn  I )
195, 18syl 16 . . . . . . . . 9  |-  ( ph  ->  R  Fn  I )
201, 16, 17, 2, 3, 19dsmmelbas 27073 . . . . . . . 8  |-  ( ph  ->  ( b  e.  H  <->  ( b  e.  ( Base `  P )  /\  {
x  e.  I  |  ( b `  x
)  =/=  ( 0g
`  ( R `  x ) ) }  e.  Fin ) ) )
2120adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  ( b  e.  H  <->  ( b  e.  ( Base `  P
)  /\  { x  e.  I  |  (
b `  x )  =/=  ( 0g `  ( R `  x )
) }  e.  Fin ) ) )
2215, 21mpbid 202 . . . . . 6  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  ( b  e.  ( Base `  P
)  /\  { x  e.  I  |  (
b `  x )  =/=  ( 0g `  ( R `  x )
) }  e.  Fin ) )
2322simpld 446 . . . . 5  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  b  e.  ( Base `  P )
)
24 eqid 2404 . . . . . 6  |-  (Scalar `  P )  =  (Scalar `  P )
25 eqid 2404 . . . . . 6  |-  ( .s
`  P )  =  ( .s `  P
)
26 eqid 2404 . . . . . 6  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
2717, 24, 25, 26lmodvscl 15922 . . . . 5  |-  ( ( P  e.  LMod  /\  a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  ( Base `  P ) )  -> 
( a ( .s
`  P ) b )  e.  ( Base `  P ) )
2813, 14, 23, 27syl3anc 1184 . . . 4  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  ( a
( .s `  P
) b )  e.  ( Base `  P
) )
2922simprd 450 . . . . 5  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  { x  e.  I  |  (
b `  x )  =/=  ( 0g `  ( R `  x )
) }  e.  Fin )
30 eqid 2404 . . . . . . . . . . 11  |-  ( Base `  S )  =  (
Base `  S )
314ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  S  e.  Ring )
323ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  I  e.  W )
3319ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  R  Fn  I )
34 fex 5928 . . . . . . . . . . . . . . . . . 18  |-  ( ( R : I --> LMod  /\  I  e.  W )  ->  R  e.  _V )
355, 3, 34syl2anc 643 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  R  e.  _V )
361, 4, 35prdssca 13634 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  S  =  (Scalar `  P ) )
3736fveq2d 5691 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( Base `  S
)  =  ( Base `  (Scalar `  P )
) )
3837eleq2d 2471 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( a  e.  (
Base `  S )  <->  a  e.  ( Base `  (Scalar `  P ) ) ) )
3938biimpar 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  ( Base `  (Scalar `  P
) ) )  -> 
a  e.  ( Base `  S ) )
4039adantrr 698 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  a  e.  ( Base `  S )
)
4140adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  a  e.  ( Base `  S ) )
4223adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  b  e.  ( Base `  P ) )
43 simpr 448 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  x  e.  I )
441, 17, 25, 30, 31, 32, 33, 41, 42, 43prdsvscafval 13657 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  ( ( a ( .s `  P ) b ) `  x
)  =  ( a ( .s `  ( R `  x )
) ( b `  x ) ) )
4544adantrr 698 . . . . . . . . 9  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  ( x  e.  I  /\  ( b `  x
)  =  ( 0g
`  ( R `  x ) ) ) )  ->  ( (
a ( .s `  P ) b ) `
 x )  =  ( a ( .s
`  ( R `  x ) ) ( b `  x ) ) )
465ffvelrnda 5829 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  I )  ->  ( R `  x )  e.  LMod )
4746adantlr 696 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  ( R `  x
)  e.  LMod )
48 simplrl 737 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  a  e.  ( Base `  (Scalar `  P )
) )
4936adantr 452 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  I )  ->  S  =  (Scalar `  P )
)
5011, 49eqtrd 2436 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  I )  ->  (Scalar `  ( R `  x
) )  =  (Scalar `  P ) )
5150fveq2d 5691 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  I )  ->  ( Base `  (Scalar `  ( R `  x )
) )  =  (
Base `  (Scalar `  P
) ) )
5251adantlr 696 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  ( Base `  (Scalar `  ( R `  x
) ) )  =  ( Base `  (Scalar `  P ) ) )
5348, 52eleqtrrd 2481 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  a  e.  ( Base `  (Scalar `  ( R `  x ) ) ) )
54 eqid 2404 . . . . . . . . . . . . 13  |-  (Scalar `  ( R `  x ) )  =  (Scalar `  ( R `  x ) )
55 eqid 2404 . . . . . . . . . . . . 13  |-  ( .s
`  ( R `  x ) )  =  ( .s `  ( R `  x )
)
56 eqid 2404 . . . . . . . . . . . . 13  |-  ( Base `  (Scalar `  ( R `  x ) ) )  =  ( Base `  (Scalar `  ( R `  x
) ) )
57 eqid 2404 . . . . . . . . . . . . 13  |-  ( 0g
`  ( R `  x ) )  =  ( 0g `  ( R `  x )
)
5854, 55, 56, 57lmodvs0 15939 . . . . . . . . . . . 12  |-  ( ( ( R `  x
)  e.  LMod  /\  a  e.  ( Base `  (Scalar `  ( R `  x
) ) ) )  ->  ( a ( .s `  ( R `
 x ) ) ( 0g `  ( R `  x )
) )  =  ( 0g `  ( R `
 x ) ) )
5947, 53, 58syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  ( a ( .s
`  ( R `  x ) ) ( 0g `  ( R `
 x ) ) )  =  ( 0g
`  ( R `  x ) ) )
60 oveq2 6048 . . . . . . . . . . . 12  |-  ( ( b `  x )  =  ( 0g `  ( R `  x ) )  ->  ( a
( .s `  ( R `  x )
) ( b `  x ) )  =  ( a ( .s
`  ( R `  x ) ) ( 0g `  ( R `
 x ) ) ) )
6160eqeq1d 2412 . . . . . . . . . . 11  |-  ( ( b `  x )  =  ( 0g `  ( R `  x ) )  ->  ( (
a ( .s `  ( R `  x ) ) ( b `  x ) )  =  ( 0g `  ( R `  x )
)  <->  ( a ( .s `  ( R `
 x ) ) ( 0g `  ( R `  x )
) )  =  ( 0g `  ( R `
 x ) ) ) )
6259, 61syl5ibrcom 214 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  ( ( b `  x )  =  ( 0g `  ( R `
 x ) )  ->  ( a ( .s `  ( R `
 x ) ) ( b `  x
) )  =  ( 0g `  ( R `
 x ) ) ) )
6362impr 603 . . . . . . . . 9  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  ( x  e.  I  /\  ( b `  x
)  =  ( 0g
`  ( R `  x ) ) ) )  ->  ( a
( .s `  ( R `  x )
) ( b `  x ) )  =  ( 0g `  ( R `  x )
) )
6445, 63eqtrd 2436 . . . . . . . 8  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  ( x  e.  I  /\  ( b `  x
)  =  ( 0g
`  ( R `  x ) ) ) )  ->  ( (
a ( .s `  P ) b ) `
 x )  =  ( 0g `  ( R `  x )
) )
6564expr 599 . . . . . . 7  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  ( ( b `  x )  =  ( 0g `  ( R `
 x ) )  ->  ( ( a ( .s `  P
) b ) `  x )  =  ( 0g `  ( R `
 x ) ) ) )
6665necon3d 2605 . . . . . 6  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  ( ( ( a ( .s `  P
) b ) `  x )  =/=  ( 0g `  ( R `  x ) )  -> 
( b `  x
)  =/=  ( 0g
`  ( R `  x ) ) ) )
6766ss2rabdv 3384 . . . . 5  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  { x  e.  I  |  (
( a ( .s
`  P ) b ) `  x )  =/=  ( 0g `  ( R `  x ) ) }  C_  { x  e.  I  |  (
b `  x )  =/=  ( 0g `  ( R `  x )
) } )
68 ssfi 7288 . . . . 5  |-  ( ( { x  e.  I  |  ( b `  x )  =/=  ( 0g `  ( R `  x ) ) }  e.  Fin  /\  {
x  e.  I  |  ( ( a ( .s `  P ) b ) `  x
)  =/=  ( 0g
`  ( R `  x ) ) } 
C_  { x  e.  I  |  ( b `
 x )  =/=  ( 0g `  ( R `  x )
) } )  ->  { x  e.  I  |  ( ( a ( .s `  P
) b ) `  x )  =/=  ( 0g `  ( R `  x ) ) }  e.  Fin )
6929, 67, 68syl2anc 643 . . . 4  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  { x  e.  I  |  (
( a ( .s
`  P ) b ) `  x )  =/=  ( 0g `  ( R `  x ) ) }  e.  Fin )
701, 16, 17, 2, 3, 19dsmmelbas 27073 . . . . 5  |-  ( ph  ->  ( ( a ( .s `  P ) b )  e.  H  <->  ( ( a ( .s
`  P ) b )  e.  ( Base `  P )  /\  {
x  e.  I  |  ( ( a ( .s `  P ) b ) `  x
)  =/=  ( 0g
`  ( R `  x ) ) }  e.  Fin ) ) )
7170adantr 452 . . . 4  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  ( (
a ( .s `  P ) b )  e.  H  <->  ( (
a ( .s `  P ) b )  e.  ( Base `  P
)  /\  { x  e.  I  |  (
( a ( .s
`  P ) b ) `  x )  =/=  ( 0g `  ( R `  x ) ) }  e.  Fin ) ) )
7228, 69, 71mpbir2and 889 . . 3  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  ( a
( .s `  P
) b )  e.  H )
7372ralrimivva 2758 . 2  |-  ( ph  ->  A. a  e.  (
Base `  (Scalar `  P
) ) A. b  e.  H  ( a
( .s `  P
) b )  e.  H )
74 dsmmlss.u . . . 4  |-  U  =  ( LSubSp `  P )
7524, 26, 17, 25, 74islss4 15993 . . 3  |-  ( P  e.  LMod  ->  ( H  e.  U  <->  ( H  e.  (SubGrp `  P )  /\  A. a  e.  (
Base `  (Scalar `  P
) ) A. b  e.  H  ( a
( .s `  P
) b )  e.  H ) ) )
7612, 75syl 16 . 2  |-  ( ph  ->  ( H  e.  U  <->  ( H  e.  (SubGrp `  P )  /\  A. a  e.  ( Base `  (Scalar `  P )
) A. b  e.  H  ( a ( .s `  P ) b )  e.  H
) ) )
7710, 73, 76mpbir2and 889 1  |-  ( ph  ->  H  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   {crab 2670   _Vcvv 2916    C_ wss 3280    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   Fincfn 7068   Basecbs 13424  Scalarcsca 13487   .scvsca 13488   X_scprds 13624   0gc0g 13678   Grpcgrp 14640  SubGrpcsubg 14893   Ringcrg 15615   LModclmod 15905   LSubSpclss 15963    (+)m cdsmm 27065
This theorem is referenced by:  dsmmlmod  27079  frlmlss  27087
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-hom 13508  df-cco 13509  df-prds 13626  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-mgp 15604  df-rng 15618  df-ur 15620  df-lmod 15907  df-lss 15964  df-dsmm 27066
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