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Theorem lsmcl 18299
Description: The sum of two subspaces is a subspace. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmcl.s  |-  S  =  ( LSubSp `  W )
lsmcl.p  |-  .(+)  =  (
LSSum `  W )
Assertion
Ref Expression
lsmcl  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  S )

Proof of Theorem lsmcl
Dummy variables  a 
d  e  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmodabl 18128 . . . 4  |-  ( W  e.  LMod  ->  W  e. 
Abel )
213ad2ant1 1027 . . 3  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  W  e.  Abel )
3 lsmcl.s . . . . 5  |-  S  =  ( LSubSp `  W )
43lsssubg 18173 . . . 4  |-  ( ( W  e.  LMod  /\  T  e.  S )  ->  T  e.  (SubGrp `  W )
)
543adant3 1026 . . 3  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  T  e.  (SubGrp `  W )
)
63lsssubg 18173 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
763adant2 1025 . . 3  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
8 lsmcl.p . . . 4  |-  .(+)  =  (
LSSum `  W )
98lsmsubg2 17490 . . 3  |-  ( ( W  e.  Abel  /\  T  e.  (SubGrp `  W )  /\  U  e.  (SubGrp `  W ) )  -> 
( T  .(+)  U )  e.  (SubGrp `  W
) )
102, 5, 7, 9syl3anc 1265 . 2  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  (SubGrp `  W )
)
11 eqid 2423 . . . . . . . 8  |-  ( +g  `  W )  =  ( +g  `  W )
1211, 8lsmelval 17294 . . . . . . 7  |-  ( ( T  e.  (SubGrp `  W )  /\  U  e.  (SubGrp `  W )
)  ->  ( u  e.  ( T  .(+)  U )  <->  E. d  e.  T  E. e  e.  U  u  =  ( d
( +g  `  W ) e ) ) )
135, 7, 12syl2anc 666 . . . . . 6  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  (
u  e.  ( T 
.(+)  U )  <->  E. d  e.  T  E. e  e.  U  u  =  ( d ( +g  `  W ) e ) ) )
1413adantr 467 . . . . 5  |-  ( ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  /\  a  e.  ( Base `  (Scalar `  W
) ) )  -> 
( u  e.  ( T  .(+)  U )  <->  E. d  e.  T  E. e  e.  U  u  =  ( d ( +g  `  W ) e ) ) )
15 simpll1 1045 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LMod  /\  T  e.  S  /\  U  e.  S
)  /\  a  e.  ( Base `  (Scalar `  W
) ) )  /\  ( d  e.  T  /\  e  e.  U
) )  ->  W  e.  LMod )
16 simplr 761 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LMod  /\  T  e.  S  /\  U  e.  S
)  /\  a  e.  ( Base `  (Scalar `  W
) ) )  /\  ( d  e.  T  /\  e  e.  U
) )  ->  a  e.  ( Base `  (Scalar `  W ) ) )
17 simpll2 1046 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LMod  /\  T  e.  S  /\  U  e.  S
)  /\  a  e.  ( Base `  (Scalar `  W
) ) )  /\  ( d  e.  T  /\  e  e.  U
) )  ->  T  e.  S )
18 simprl 763 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LMod  /\  T  e.  S  /\  U  e.  S
)  /\  a  e.  ( Base `  (Scalar `  W
) ) )  /\  ( d  e.  T  /\  e  e.  U
) )  ->  d  e.  T )
19 eqid 2423 . . . . . . . . . . 11  |-  ( Base `  W )  =  (
Base `  W )
2019, 3lssel 18154 . . . . . . . . . 10  |-  ( ( T  e.  S  /\  d  e.  T )  ->  d  e.  ( Base `  W ) )
2117, 18, 20syl2anc 666 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LMod  /\  T  e.  S  /\  U  e.  S
)  /\  a  e.  ( Base `  (Scalar `  W
) ) )  /\  ( d  e.  T  /\  e  e.  U
) )  ->  d  e.  ( Base `  W
) )
22 simpll3 1047 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LMod  /\  T  e.  S  /\  U  e.  S
)  /\  a  e.  ( Base `  (Scalar `  W
) ) )  /\  ( d  e.  T  /\  e  e.  U
) )  ->  U  e.  S )
23 simprr 765 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LMod  /\  T  e.  S  /\  U  e.  S
)  /\  a  e.  ( Base `  (Scalar `  W
) ) )  /\  ( d  e.  T  /\  e  e.  U
) )  ->  e  e.  U )
2419, 3lssel 18154 . . . . . . . . . 10  |-  ( ( U  e.  S  /\  e  e.  U )  ->  e  e.  ( Base `  W ) )
2522, 23, 24syl2anc 666 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LMod  /\  T  e.  S  /\  U  e.  S
)  /\  a  e.  ( Base `  (Scalar `  W
) ) )  /\  ( d  e.  T  /\  e  e.  U
) )  ->  e  e.  ( Base `  W
) )
26 eqid 2423 . . . . . . . . . 10  |-  (Scalar `  W )  =  (Scalar `  W )
27 eqid 2423 . . . . . . . . . 10  |-  ( .s
`  W )  =  ( .s `  W
)
28 eqid 2423 . . . . . . . . . 10  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2919, 11, 26, 27, 28lmodvsdi 18107 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  (
a  e.  ( Base `  (Scalar `  W )
)  /\  d  e.  ( Base `  W )  /\  e  e.  ( Base `  W ) ) )  ->  ( a
( .s `  W
) ( d ( +g  `  W ) e ) )  =  ( ( a ( .s `  W ) d ) ( +g  `  W ) ( a ( .s `  W
) e ) ) )
3015, 16, 21, 25, 29syl13anc 1267 . . . . . . . 8  |-  ( ( ( ( W  e. 
LMod  /\  T  e.  S  /\  U  e.  S
)  /\  a  e.  ( Base `  (Scalar `  W
) ) )  /\  ( d  e.  T  /\  e  e.  U
) )  ->  (
a ( .s `  W ) ( d ( +g  `  W
) e ) )  =  ( ( a ( .s `  W
) d ) ( +g  `  W ) ( a ( .s
`  W ) e ) ) )
3115, 17, 4syl2anc 666 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LMod  /\  T  e.  S  /\  U  e.  S
)  /\  a  e.  ( Base `  (Scalar `  W
) ) )  /\  ( d  e.  T  /\  e  e.  U
) )  ->  T  e.  (SubGrp `  W )
)
3215, 22, 6syl2anc 666 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LMod  /\  T  e.  S  /\  U  e.  S
)  /\  a  e.  ( Base `  (Scalar `  W
) ) )  /\  ( d  e.  T  /\  e  e.  U
) )  ->  U  e.  (SubGrp `  W )
)
3326, 27, 28, 3lssvscl 18171 . . . . . . . . . 10  |-  ( ( ( W  e.  LMod  /\  T  e.  S )  /\  ( a  e.  ( Base `  (Scalar `  W ) )  /\  d  e.  T )
)  ->  ( a
( .s `  W
) d )  e.  T )
3415, 17, 16, 18, 33syl22anc 1266 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LMod  /\  T  e.  S  /\  U  e.  S
)  /\  a  e.  ( Base `  (Scalar `  W
) ) )  /\  ( d  e.  T  /\  e  e.  U
) )  ->  (
a ( .s `  W ) d )  e.  T )
3526, 27, 28, 3lssvscl 18171 . . . . . . . . . 10  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( a  e.  ( Base `  (Scalar `  W ) )  /\  e  e.  U )
)  ->  ( a
( .s `  W
) e )  e.  U )
3615, 22, 16, 23, 35syl22anc 1266 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LMod  /\  T  e.  S  /\  U  e.  S
)  /\  a  e.  ( Base `  (Scalar `  W
) ) )  /\  ( d  e.  T  /\  e  e.  U
) )  ->  (
a ( .s `  W ) e )  e.  U )
3711, 8lsmelvali 17295 . . . . . . . . 9  |-  ( ( ( T  e.  (SubGrp `  W )  /\  U  e.  (SubGrp `  W )
)  /\  ( (
a ( .s `  W ) d )  e.  T  /\  (
a ( .s `  W ) e )  e.  U ) )  ->  ( ( a ( .s `  W
) d ) ( +g  `  W ) ( a ( .s
`  W ) e ) )  e.  ( T  .(+)  U )
)
3831, 32, 34, 36, 37syl22anc 1266 . . . . . . . 8  |-  ( ( ( ( W  e. 
LMod  /\  T  e.  S  /\  U  e.  S
)  /\  a  e.  ( Base `  (Scalar `  W
) ) )  /\  ( d  e.  T  /\  e  e.  U
) )  ->  (
( a ( .s
`  W ) d ) ( +g  `  W
) ( a ( .s `  W ) e ) )  e.  ( T  .(+)  U ) )
3930, 38eqeltrd 2511 . . . . . . 7  |-  ( ( ( ( W  e. 
LMod  /\  T  e.  S  /\  U  e.  S
)  /\  a  e.  ( Base `  (Scalar `  W
) ) )  /\  ( d  e.  T  /\  e  e.  U
) )  ->  (
a ( .s `  W ) ( d ( +g  `  W
) e ) )  e.  ( T  .(+)  U ) )
40 oveq2 6311 . . . . . . . 8  |-  ( u  =  ( d ( +g  `  W ) e )  ->  (
a ( .s `  W ) u )  =  ( a ( .s `  W ) ( d ( +g  `  W ) e ) ) )
4140eleq1d 2492 . . . . . . 7  |-  ( u  =  ( d ( +g  `  W ) e )  ->  (
( a ( .s
`  W ) u )  e.  ( T 
.(+)  U )  <->  ( a
( .s `  W
) ( d ( +g  `  W ) e ) )  e.  ( T  .(+)  U ) ) )
4239, 41syl5ibrcom 226 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  T  e.  S  /\  U  e.  S
)  /\  a  e.  ( Base `  (Scalar `  W
) ) )  /\  ( d  e.  T  /\  e  e.  U
) )  ->  (
u  =  ( d ( +g  `  W
) e )  -> 
( a ( .s
`  W ) u )  e.  ( T 
.(+)  U ) ) )
4342rexlimdvva 2925 . . . . 5  |-  ( ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  /\  a  e.  ( Base `  (Scalar `  W
) ) )  -> 
( E. d  e.  T  E. e  e.  U  u  =  ( d ( +g  `  W
) e )  -> 
( a ( .s
`  W ) u )  e.  ( T 
.(+)  U ) ) )
4414, 43sylbid 219 . . . 4  |-  ( ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  /\  a  e.  ( Base `  (Scalar `  W
) ) )  -> 
( u  e.  ( T  .(+)  U )  ->  ( a ( .s
`  W ) u )  e.  ( T 
.(+)  U ) ) )
4544impr 624 . . 3  |-  ( ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  /\  ( a  e.  (
Base `  (Scalar `  W
) )  /\  u  e.  ( T  .(+)  U ) ) )  ->  (
a ( .s `  W ) u )  e.  ( T  .(+)  U ) )
4645ralrimivva 2847 . 2  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  A. a  e.  ( Base `  (Scalar `  W ) ) A. u  e.  ( T  .(+) 
U ) ( a ( .s `  W
) u )  e.  ( T  .(+)  U ) )
4726, 28, 19, 27, 3islss4 18178 . . 3  |-  ( W  e.  LMod  ->  ( ( T  .(+)  U )  e.  S  <->  ( ( T 
.(+)  U )  e.  (SubGrp `  W )  /\  A. a  e.  ( Base `  (Scalar `  W )
) A. u  e.  ( T  .(+)  U ) ( a ( .s
`  W ) u )  e.  ( T 
.(+)  U ) ) ) )
48473ad2ant1 1027 . 2  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  (
( T  .(+)  U )  e.  S  <->  ( ( T  .(+)  U )  e.  (SubGrp `  W )  /\  A. a  e.  (
Base `  (Scalar `  W
) ) A. u  e.  ( T  .(+)  U ) ( a ( .s
`  W ) u )  e.  ( T 
.(+)  U ) ) ) )
4910, 46, 48mpbir2and 931 1  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   A.wral 2776   E.wrex 2777   ` cfv 5599  (class class class)co 6303   Basecbs 15114   +g cplusg 15183  Scalarcsca 15186   .scvsca 15187  SubGrpcsubg 16804   LSSumclsm 17279   Abelcabl 17424   LModclmod 18084   LSubSpclss 18148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-2 10670  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-0g 15333  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-grp 16666  df-minusg 16667  df-sbg 16668  df-subg 16807  df-cntz 16964  df-lsm 17281  df-cmn 17425  df-abl 17426  df-mgp 17717  df-ur 17729  df-ring 17775  df-lmod 18086  df-lss 18149
This theorem is referenced by:  lsmelval2  18301  lsmsp  18302  lspprabs  18311  pj1lmhm  18316  lspabs3  18337  pjth  22385  lshpnelb  32513  lsmsat  32537  lsmcv2  32558  lcvat  32559  lcvexchlem4  32566  lcvexchlem5  32567  lcv1  32570  lsatexch  32572  lsatcv0eq  32576  lsatcvatlem  32578  lsatcvat2  32580  lsatcvat3  32581  lkrlsp  32631  dia2dimlem7  34601  dihjustlem  34747  dihord1  34749  dihlsscpre  34765  dihjatcclem2  34950  dihjat1lem  34959  dochexmidlem5  34995  dochexmidlem6  34996  dochexmidlem8  34998  lcfrlem23  35096  mapdlsmcl  35194  mapdlsm  35195  mapdpglem1  35203  mapdpglem2a  35205  mapdindp0  35250  mapdheq4lem  35262  mapdh6lem1N  35264  mapdh6lem2N  35265  hdmap1l6lem1  35339  hdmap1l6lem2  35340  hdmaprnlem3eN  35392  kercvrlsm  35905
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