MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lsmcl Structured version   Unicode version

Theorem lsmcl 18049
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 17877 . . . 4  |-  ( W  e.  LMod  ->  W  e. 
Abel )
213ad2ant1 1018 . . 3  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  W  e.  Abel )
3 lsmcl.s . . . . 5  |-  S  =  ( LSubSp `  W )
43lsssubg 17923 . . . 4  |-  ( ( W  e.  LMod  /\  T  e.  S )  ->  T  e.  (SubGrp `  W )
)
543adant3 1017 . . 3  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  T  e.  (SubGrp `  W )
)
63lsssubg 17923 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
763adant2 1016 . . 3  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
8 lsmcl.p . . . 4  |-  .(+)  =  (
LSSum `  W )
98lsmsubg2 17189 . . 3  |-  ( ( W  e.  Abel  /\  T  e.  (SubGrp `  W )  /\  U  e.  (SubGrp `  W ) )  -> 
( T  .(+)  U )  e.  (SubGrp `  W
) )
102, 5, 7, 9syl3anc 1230 . 2  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  (SubGrp `  W )
)
11 eqid 2402 . . . . . . . 8  |-  ( +g  `  W )  =  ( +g  `  W )
1211, 8lsmelval 16993 . . . . . . 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 659 . . . . . 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 463 . . . . 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 1036 . . . . . . . . 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 754 . . . . . . . . 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 1037 . . . . . . . . . 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 756 . . . . . . . . . 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 2402 . . . . . . . . . . 11  |-  ( Base `  W )  =  (
Base `  W )
2019, 3lssel 17904 . . . . . . . . . 10  |-  ( ( T  e.  S  /\  d  e.  T )  ->  d  e.  ( Base `  W ) )
2117, 18, 20syl2anc 659 . . . . . . . . 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 1038 . . . . . . . . . 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 758 . . . . . . . . . 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 17904 . . . . . . . . . 10  |-  ( ( U  e.  S  /\  e  e.  U )  ->  e  e.  ( Base `  W ) )
2522, 23, 24syl2anc 659 . . . . . . . . 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 2402 . . . . . . . . . 10  |-  (Scalar `  W )  =  (Scalar `  W )
27 eqid 2402 . . . . . . . . . 10  |-  ( .s
`  W )  =  ( .s `  W
)
28 eqid 2402 . . . . . . . . . 10  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2919, 11, 26, 27, 28lmodvsdi 17855 . . . . . . . . 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 1232 . . . . . . . 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 659 . . . . . . . . 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 659 . . . . . . . . 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 17921 . . . . . . . . . 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 1231 . . . . . . . . 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 17921 . . . . . . . . . 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 1231 . . . . . . . . 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 16994 . . . . . . . . 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 1231 . . . . . . . 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 2490 . . . . . . 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 6286 . . . . . . . 8  |-  ( u  =  ( d ( +g  `  W ) e )  ->  (
a ( .s `  W ) u )  =  ( a ( .s `  W ) ( d ( +g  `  W ) e ) ) )
4140eleq1d 2471 . . . . . . 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 222 . . . . . 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 2903 . . . . 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 215 . . . 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 617 . . 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 2825 . 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 17928 . . 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 1018 . 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 923 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 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   E.wrex 2755   ` cfv 5569  (class class class)co 6278   Basecbs 14841   +g cplusg 14909  Scalarcsca 14912   .scvsca 14913  SubGrpcsubg 16519   LSSumclsm 16978   Abelcabl 17123   LModclmod 17832   LSubSpclss 17898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-0g 15056  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-grp 16381  df-minusg 16382  df-sbg 16383  df-subg 16522  df-cntz 16679  df-lsm 16980  df-cmn 17124  df-abl 17125  df-mgp 17462  df-ur 17474  df-ring 17520  df-lmod 17834  df-lss 17899
This theorem is referenced by:  lsmelval2  18051  lsmsp  18052  lspprabs  18061  pj1lmhm  18066  lspabs3  18087  pjth  22146  lshpnelb  32002  lsmsat  32026  lsmcv2  32047  lcvat  32048  lcvexchlem4  32055  lcvexchlem5  32056  lcv1  32059  lsatexch  32061  lsatcv0eq  32065  lsatcvatlem  32067  lsatcvat2  32069  lsatcvat3  32070  lkrlsp  32120  dia2dimlem7  34090  dihjustlem  34236  dihord1  34238  dihlsscpre  34254  dihjatcclem2  34439  dihjat1lem  34448  dochexmidlem5  34484  dochexmidlem6  34485  dochexmidlem8  34487  lcfrlem23  34585  mapdlsmcl  34683  mapdlsm  34684  mapdpglem1  34692  mapdpglem2a  34694  mapdindp0  34739  mapdheq4lem  34751  mapdh6lem1N  34753  mapdh6lem2N  34754  hdmap1l6lem1  34828  hdmap1l6lem2  34829  hdmaprnlem3eN  34881  kercvrlsm  35391
  Copyright terms: Public domain W3C validator