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Theorem lsmcl 17505
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 17333 . . . 4  |-  ( W  e.  LMod  ->  W  e. 
Abel )
213ad2ant1 1012 . . 3  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  W  e.  Abel )
3 lsmcl.s . . . . 5  |-  S  =  ( LSubSp `  W )
43lsssubg 17379 . . . 4  |-  ( ( W  e.  LMod  /\  T  e.  S )  ->  T  e.  (SubGrp `  W )
)
543adant3 1011 . . 3  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  T  e.  (SubGrp `  W )
)
63lsssubg 17379 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
763adant2 1010 . . 3  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
8 lsmcl.p . . . 4  |-  .(+)  =  (
LSSum `  W )
98lsmsubg2 16651 . . 3  |-  ( ( W  e.  Abel  /\  T  e.  (SubGrp `  W )  /\  U  e.  (SubGrp `  W ) )  -> 
( T  .(+)  U )  e.  (SubGrp `  W
) )
102, 5, 7, 9syl3anc 1223 . 2  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  (SubGrp `  W )
)
11 eqid 2460 . . . . . . . 8  |-  ( +g  `  W )  =  ( +g  `  W )
1211, 8lsmelval 16458 . . . . . . 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 661 . . . . . 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 465 . . . . 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 1030 . . . . . . . . 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 1031 . . . . . . . . . 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 755 . . . . . . . . . 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 2460 . . . . . . . . . . 11  |-  ( Base `  W )  =  (
Base `  W )
2019, 3lssel 17360 . . . . . . . . . 10  |-  ( ( T  e.  S  /\  d  e.  T )  ->  d  e.  ( Base `  W ) )
2117, 18, 20syl2anc 661 . . . . . . . . 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 1032 . . . . . . . . . 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 756 . . . . . . . . . 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 17360 . . . . . . . . . 10  |-  ( ( U  e.  S  /\  e  e.  U )  ->  e  e.  ( Base `  W ) )
2522, 23, 24syl2anc 661 . . . . . . . . 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 2460 . . . . . . . . . 10  |-  (Scalar `  W )  =  (Scalar `  W )
27 eqid 2460 . . . . . . . . . 10  |-  ( .s
`  W )  =  ( .s `  W
)
28 eqid 2460 . . . . . . . . . 10  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2919, 11, 26, 27, 28lmodvsdi 17311 . . . . . . . . 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 1225 . . . . . . . 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 661 . . . . . . . . 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 661 . . . . . . . . 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 17377 . . . . . . . . . 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 1224 . . . . . . . . 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 17377 . . . . . . . . . 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 1224 . . . . . . . . 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 16459 . . . . . . . . 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 1224 . . . . . . . 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 2548 . . . . . . 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 6283 . . . . . . . 8  |-  ( u  =  ( d ( +g  `  W ) e )  ->  (
a ( .s `  W ) u )  =  ( a ( .s `  W ) ( d ( +g  `  W ) e ) ) )
4140eleq1d 2529 . . . . . . 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 2955 . . . . 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 619 . . 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 2878 . 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 17384 . . 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 1012 . 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 915 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 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544  Scalarcsca 14547   .scvsca 14548  SubGrpcsubg 15983   LSSumclsm 16443   Abelcabel 16588   LModclmod 17288   LSubSpclss 17354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-0g 14686  df-mnd 15721  df-submnd 15771  df-grp 15851  df-minusg 15852  df-sbg 15853  df-subg 15986  df-cntz 16143  df-lsm 16445  df-cmn 16589  df-abl 16590  df-mgp 16925  df-ur 16937  df-rng 16981  df-lmod 17290  df-lss 17355
This theorem is referenced by:  lsmelval2  17507  lsmsp  17508  lspprabs  17517  pj1lmhm  17522  lspabs3  17543  pjth  21582  kercvrlsm  30622  lshpnelb  33656  lsmsat  33680  lsmcv2  33701  lcvat  33702  lcvexchlem4  33709  lcvexchlem5  33710  lcv1  33713  lsatexch  33715  lsatcv0eq  33719  lsatcvatlem  33721  lsatcvat2  33723  lsatcvat3  33724  lkrlsp  33774  dia2dimlem7  35742  dihjustlem  35888  dihord1  35890  dihlsscpre  35906  dihjatcclem2  36091  dihjat1lem  36100  dochexmidlem5  36136  dochexmidlem6  36137  dochexmidlem8  36139  lcfrlem23  36237  mapdlsmcl  36335  mapdlsm  36336  mapdpglem1  36344  mapdpglem2a  36346  mapdindp0  36391  mapdheq4lem  36403  mapdh6lem1N  36405  mapdh6lem2N  36406  hdmap1l6lem1  36480  hdmap1l6lem2  36481  hdmaprnlem3eN  36533
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