Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dsmmlss Unicode version

Theorem dsmmlss 27210
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 15634 . . . . 5  |-  ( a  e.  LMod  ->  a  e. 
Grp )
76ssriv 3184 . . . 4  |-  LMod  C_  Grp
8 fss 5397 . . . 4  |-  ( ( R : I --> LMod  /\  LMod  C_ 
Grp )  ->  R : I --> Grp )
95, 7, 8sylancl 643 . . 3  |-  ( ph  ->  R : I --> Grp )
101, 2, 3, 4, 9dsmmsubg 27209 . 2  |-  ( ph  ->  H  e.  (SubGrp `  P ) )
11 dsmmlss.k . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  (Scalar `  ( R `  x
) )  =  S )
121, 4, 3, 5, 11prdslmodd 15726 . . . . . 6  |-  ( ph  ->  P  e.  LMod )
1312adantr 451 . . . . 5  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  P  e.  LMod )
14 simprl 732 . . . . 5  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  a  e.  ( Base `  (Scalar `  P
) ) )
15 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  b  e.  H )
16 eqid 2283 . . . . . . . . 9  |-  ( S 
(+)m  R )  =  ( S  (+)m  R )
17 eqid 2283 . . . . . . . . 9  |-  ( Base `  P )  =  (
Base `  P )
18 ffn 5389 . . . . . . . . . 10  |-  ( R : I --> LMod  ->  R  Fn  I )
195, 18syl 15 . . . . . . . . 9  |-  ( ph  ->  R  Fn  I )
201, 16, 17, 2, 3, 19dsmmelbas 27205 . . . . . . . 8  |-  ( ph  ->  ( b  e.  H  <->  ( b  e.  ( Base `  P )  /\  {
x  e.  I  |  ( b `  x
)  =/=  ( 0g
`  ( R `  x ) ) }  e.  Fin ) ) )
2120adantr 451 . . . . . . 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 201 . . . . . 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 445 . . . . 5  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  b  e.  ( Base `  P )
)
24 eqid 2283 . . . . . 6  |-  (Scalar `  P )  =  (Scalar `  P )
25 eqid 2283 . . . . . 6  |-  ( .s
`  P )  =  ( .s `  P
)
26 eqid 2283 . . . . . 6  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
2717, 24, 25, 26lmodvscl 15644 . . . . 5  |-  ( ( P  e.  LMod  /\  a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  ( Base `  P ) )  -> 
( a ( .s
`  P ) b )  e.  ( Base `  P ) )
2813, 14, 23, 27syl3anc 1182 . . . 4  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  ( a
( .s `  P
) b )  e.  ( Base `  P
) )
2922simprd 449 . . . . 5  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  { x  e.  I  |  (
b `  x )  =/=  ( 0g `  ( R `  x )
) }  e.  Fin )
30 eqid 2283 . . . . . . . . . . 11  |-  ( Base `  S )  =  (
Base `  S )
314ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  S  e.  Ring )
323ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  I  e.  W )
3319ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  R  Fn  I )
34 fex 5749 . . . . . . . . . . . . . . . . . 18  |-  ( ( R : I --> LMod  /\  I  e.  W )  ->  R  e.  _V )
355, 3, 34syl2anc 642 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  R  e.  _V )
361, 4, 35prdssca 13356 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  S  =  (Scalar `  P ) )
3736fveq2d 5529 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( Base `  S
)  =  ( Base `  (Scalar `  P )
) )
3837eleq2d 2350 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( a  e.  (
Base `  S )  <->  a  e.  ( Base `  (Scalar `  P ) ) ) )
3938biimpar 471 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  ( Base `  (Scalar `  P
) ) )  -> 
a  e.  ( Base `  S ) )
4039adantrr 697 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  a  e.  ( Base `  S )
)
4140adantr 451 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  a  e.  ( Base `  S ) )
4223adantr 451 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  b  e.  ( Base `  P ) )
43 simpr 447 . . . . . . . . . . 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 13379 . . . . . . . . . 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 697 . . . . . . . . 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 ) ) )
46 ffvelrn 5663 . . . . . . . . . . . . . 14  |-  ( ( R : I --> LMod  /\  x  e.  I )  ->  ( R `  x )  e.  LMod )
475, 46sylan 457 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  I )  ->  ( R `  x )  e.  LMod )
4847adantlr 695 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  ( R `  x
)  e.  LMod )
49 simplrl 736 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  a  e.  ( Base `  (Scalar `  P )
) )
5036adantr 451 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  I )  ->  S  =  (Scalar `  P )
)
5111, 50eqtrd 2315 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  I )  ->  (Scalar `  ( R `  x
) )  =  (Scalar `  P ) )
5251fveq2d 5529 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  I )  ->  ( Base `  (Scalar `  ( R `  x )
) )  =  (
Base `  (Scalar `  P
) ) )
5352adantlr 695 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  ( Base `  (Scalar `  ( R `  x
) ) )  =  ( Base `  (Scalar `  P ) ) )
5449, 53eleqtrrd 2360 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  a  e.  ( Base `  (Scalar `  ( R `  x ) ) ) )
55 eqid 2283 . . . . . . . . . . . . 13  |-  (Scalar `  ( R `  x ) )  =  (Scalar `  ( R `  x ) )
56 eqid 2283 . . . . . . . . . . . . 13  |-  ( .s
`  ( R `  x ) )  =  ( .s `  ( R `  x )
)
57 eqid 2283 . . . . . . . . . . . . 13  |-  ( Base `  (Scalar `  ( R `  x ) ) )  =  ( Base `  (Scalar `  ( R `  x
) ) )
58 eqid 2283 . . . . . . . . . . . . 13  |-  ( 0g
`  ( R `  x ) )  =  ( 0g `  ( R `  x )
)
5955, 56, 57, 58lmodvs0 15664 . . . . . . . . . . . 12  |-  ( ( ( R `  x
)  e.  LMod  /\  a  e.  ( Base `  (Scalar `  ( R `  x
) ) ) )  ->  ( a ( .s `  ( R `
 x ) ) ( 0g `  ( R `  x )
) )  =  ( 0g `  ( R `
 x ) ) )
6048, 54, 59syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  ( a ( .s
`  ( R `  x ) ) ( 0g `  ( R `
 x ) ) )  =  ( 0g
`  ( R `  x ) ) )
61 oveq2 5866 . . . . . . . . . . . 12  |-  ( ( b `  x )  =  ( 0g `  ( R `  x ) )  ->  ( a
( .s `  ( R `  x )
) ( b `  x ) )  =  ( a ( .s
`  ( R `  x ) ) ( 0g `  ( R `
 x ) ) ) )
6261eqeq1d 2291 . . . . . . . . . . 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 ) ) ) )
6360, 62syl5ibrcom 213 . . . . . . . . . 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 ) ) ) )
6463impr 602 . . . . . . . . 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 )
) )
6545, 64eqtrd 2315 . . . . . . . 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 )
) )
6665expr 598 . . . . . . 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 ) ) ) )
6766necon3d 2484 . . . . . 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 ) ) ) )
6867ss2rabdv 3254 . . . . 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 )
) } )
69 ssfi 7083 . . . . 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 )
7029, 68, 69syl2anc 642 . . . 4  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  { x  e.  I  |  (
( a ( .s
`  P ) b ) `  x )  =/=  ( 0g `  ( R `  x ) ) }  e.  Fin )
711, 16, 17, 2, 3, 19dsmmelbas 27205 . . . . 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 ) ) )
7271adantr 451 . . . 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 ) ) )
7328, 70, 72mpbir2and 888 . . 3  |-  ( (
ph  /\  ( a  e.  ( Base `  (Scalar `  P ) )  /\  b  e.  H )
)  ->  ( a
( .s `  P
) b )  e.  H )
7473ralrimivva 2635 . 2  |-  ( ph  ->  A. a  e.  (
Base `  (Scalar `  P
) ) A. b  e.  H  ( a
( .s `  P
) b )  e.  H )
75 dsmmlss.u . . . 4  |-  U  =  ( LSubSp `  P )
7624, 26, 17, 25, 75islss4 15719 . . 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 ) ) )
7712, 76syl 15 . 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
) ) )
7810, 74, 77mpbir2and 888 1  |-  ( ph  ->  H  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547   _Vcvv 2788    C_ wss 3152    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   X_scprds 13346   0gc0g 13400   Grpcgrp 14362  SubGrpcsubg 14615   Ringcrg 15337   LModclmod 15627   LSubSpclss 15689    (+)m cdsmm 27197
This theorem is referenced by:  dsmmlmod  27211  frlmlss  27219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lss 15690  df-dsmm 27198
  Copyright terms: Public domain W3C validator