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Theorem dsmmlss 27313
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 15650 . . . . 5  |-  ( a  e.  LMod  ->  a  e. 
Grp )
76ssriv 3197 . . . 4  |-  LMod  C_  Grp
8 fss 5413 . . . 4  |-  ( ( R : I --> LMod  /\  LMod  C_ 
Grp )  ->  R : I --> Grp )
95, 7, 8sylancl 643 . . 3  |-  ( ph  ->  R : I --> Grp )
101, 2, 3, 4, 9dsmmsubg 27312 . 2  |-  ( ph  ->  H  e.  (SubGrp `  P ) )
11 dsmmlss.k . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  (Scalar `  ( R `  x
) )  =  S )
121, 4, 3, 5, 11prdslmodd 15742 . . . . . 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 2296 . . . . . . . . 9  |-  ( S 
(+)m  R )  =  ( S  (+)m  R )
17 eqid 2296 . . . . . . . . 9  |-  ( Base `  P )  =  (
Base `  P )
18 ffn 5405 . . . . . . . . . 10  |-  ( R : I --> LMod  ->  R  Fn  I )
195, 18syl 15 . . . . . . . . 9  |-  ( ph  ->  R  Fn  I )
201, 16, 17, 2, 3, 19dsmmelbas 27308 . . . . . . . 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 2296 . . . . . 6  |-  (Scalar `  P )  =  (Scalar `  P )
25 eqid 2296 . . . . . 6  |-  ( .s
`  P )  =  ( .s `  P
)
26 eqid 2296 . . . . . 6  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
2717, 24, 25, 26lmodvscl 15660 . . . . 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 2296 . . . . . . . . . . 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 5765 . . . . . . . . . . . . . . . . . 18  |-  ( ( R : I --> LMod  /\  I  e.  W )  ->  R  e.  _V )
355, 3, 34syl2anc 642 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  R  e.  _V )
361, 4, 35prdssca 13372 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  S  =  (Scalar `  P ) )
3736fveq2d 5545 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( Base `  S
)  =  ( Base `  (Scalar `  P )
) )
3837eleq2d 2363 . . . . . . . . . . . . . 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 13395 . . . . . . . . . 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 5679 . . . . . . . . . . . . . 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 2328 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  I )  ->  (Scalar `  ( R `  x
) )  =  (Scalar `  P ) )
5251fveq2d 5545 . . . . . . . . . . . . . 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 2373 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
a  e.  ( Base `  (Scalar `  P )
)  /\  b  e.  H ) )  /\  x  e.  I )  ->  a  e.  ( Base `  (Scalar `  ( R `  x ) ) ) )
55 eqid 2296 . . . . . . . . . . . . 13  |-  (Scalar `  ( R `  x ) )  =  (Scalar `  ( R `  x ) )
56 eqid 2296 . . . . . . . . . . . . 13  |-  ( .s
`  ( R `  x ) )  =  ( .s `  ( R `  x )
)
57 eqid 2296 . . . . . . . . . . . . 13  |-  ( Base `  (Scalar `  ( R `  x ) ) )  =  ( Base `  (Scalar `  ( R `  x
) ) )
58 eqid 2296 . . . . . . . . . . . . 13  |-  ( 0g
`  ( R `  x ) )  =  ( 0g `  ( R `  x )
)
5955, 56, 57, 58lmodvs0 15680 . . . . . . . . . . . 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 5882 . . . . . . . . . . . 12  |-  ( ( b `  x )  =  ( 0g `  ( R `  x ) )  ->  ( a
( .s `  ( R `  x )
) ( b `  x ) )  =  ( a ( .s
`  ( R `  x ) ) ( 0g `  ( R `
 x ) ) ) )
6261eqeq1d 2304 . . . . . . . . . . 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 2328 . . . . . . . 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 2497 . . . . . 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 3267 . . . . 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 7099 . . . . 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 27308 . . . . 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 2648 . 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 15735 . . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   {crab 2560   _Vcvv 2801    C_ wss 3165    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   Fincfn 6879   Basecbs 13164  Scalarcsca 13227   .scvsca 13228   X_scprds 13362   0gc0g 13416   Grpcgrp 14378  SubGrpcsubg 14631   Ringcrg 15353   LModclmod 15643   LSubSpclss 15705    (+)m cdsmm 27300
This theorem is referenced by:  dsmmlmod  27314  frlmlss  27322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-prds 13364  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-mgp 15342  df-rng 15356  df-ur 15358  df-lmod 15645  df-lss 15706  df-dsmm 27301
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