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Theorem dsmmlss 27187
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 15957 . . . . 5  |-  ( a  e.  LMod  ->  a  e. 
Grp )
76ssriv 3352 . . . 4  |-  LMod  C_  Grp
8 fss 5599 . . . 4  |-  ( ( R : I --> LMod  /\  LMod  C_ 
Grp )  ->  R : I --> Grp )
95, 7, 8sylancl 644 . . 3  |-  ( ph  ->  R : I --> Grp )
101, 2, 3, 4, 9dsmmsubg 27186 . 2  |-  ( ph  ->  H  e.  (SubGrp `  P ) )
11 dsmmlss.k . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  (Scalar `  ( R `  x
) )  =  S )
121, 4, 3, 5, 11prdslmodd 16045 . . . . . 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 2436 . . . . . . . . 9  |-  ( S 
(+)m  R )  =  ( S  (+)m  R )
17 eqid 2436 . . . . . . . . 9  |-  ( Base `  P )  =  (
Base `  P )
18 ffn 5591 . . . . . . . . . 10  |-  ( R : I --> LMod  ->  R  Fn  I )
195, 18syl 16 . . . . . . . . 9  |-  ( ph  ->  R  Fn  I )
201, 16, 17, 2, 3, 19dsmmelbas 27182 . . . . . . . 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 2436 . . . . . 6  |-  (Scalar `  P )  =  (Scalar `  P )
25 eqid 2436 . . . . . 6  |-  ( .s
`  P )  =  ( .s `  P
)
26 eqid 2436 . . . . . 6  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
2717, 24, 25, 26lmodvscl 15967 . . . . 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 2436 . . . . . . . . . . 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 5969 . . . . . . . . . . . . . . . . . 18  |-  ( ( R : I --> LMod  /\  I  e.  W )  ->  R  e.  _V )
355, 3, 34syl2anc 643 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  R  e.  _V )
361, 4, 35prdssca 13679 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  S  =  (Scalar `  P ) )
3736fveq2d 5732 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( Base `  S
)  =  ( Base `  (Scalar `  P )
) )
3837eleq2d 2503 . . . . . . . . . . . . . 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 13702 . . . . . . . . . 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 5870 . . . . . . . . . . . . 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 2468 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  I )  ->  (Scalar `  ( R `  x
) )  =  (Scalar `  P ) )
5150fveq2d 5732 . . . . . . . . . . . . . 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 2513 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  a  e.  ( Base `  (Scalar `  ( R `  x ) ) ) )
54 eqid 2436 . . . . . . . . . . . . 13  |-  (Scalar `  ( R `  x ) )  =  (Scalar `  ( R `  x ) )
55 eqid 2436 . . . . . . . . . . . . 13  |-  ( .s
`  ( R `  x ) )  =  ( .s `  ( R `  x )
)
56 eqid 2436 . . . . . . . . . . . . 13  |-  ( Base `  (Scalar `  ( R `  x ) ) )  =  ( Base `  (Scalar `  ( R `  x
) ) )
57 eqid 2436 . . . . . . . . . . . . 13  |-  ( 0g
`  ( R `  x ) )  =  ( 0g `  ( R `  x )
)
5854, 55, 56, 57lmodvs0 15984 . . . . . . . . . . . 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 6089 . . . . . . . . . . . 12  |-  ( ( b `  x )  =  ( 0g `  ( R `  x ) )  ->  ( a
( .s `  ( R `  x )
) ( b `  x ) )  =  ( a ( .s
`  ( R `  x ) ) ( 0g `  ( R `
 x ) ) ) )
6160eqeq1d 2444 . . . . . . . . . . 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 2468 . . . . . . . 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 2639 . . . . . 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 3424 . . . . 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 7329 . . . . 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 27182 . . . . 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 2798 . 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 16038 . . 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 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   {crab 2709   _Vcvv 2956    C_ wss 3320    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   Fincfn 7109   Basecbs 13469  Scalarcsca 13532   .scvsca 13533   X_scprds 13669   0gc0g 13723   Grpcgrp 14685  SubGrpcsubg 14938   Ringcrg 15660   LModclmod 15950   LSubSpclss 16008    (+)m cdsmm 27174
This theorem is referenced by:  dsmmlmod  27188  frlmlss  27196
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-hom 13553  df-cco 13554  df-prds 13671  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-mgp 15649  df-rng 15663  df-ur 15665  df-lmod 15952  df-lss 16009  df-dsmm 27175
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