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Theorem lsmcl 16157
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 15993 . . . 4  |-  ( W  e.  LMod  ->  W  e. 
Abel )
213ad2ant1 979 . . 3  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  W  e.  Abel )
3 lsmcl.s . . . . 5  |-  S  =  ( LSubSp `  W )
43lsssubg 16035 . . . 4  |-  ( ( W  e.  LMod  /\  T  e.  S )  ->  T  e.  (SubGrp `  W )
)
543adant3 978 . . 3  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  T  e.  (SubGrp `  W )
)
63lsssubg 16035 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
763adant2 977 . . 3  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
8 lsmcl.p . . . 4  |-  .(+)  =  (
LSSum `  W )
98lsmsubg2 15476 . . 3  |-  ( ( W  e.  Abel  /\  T  e.  (SubGrp `  W )  /\  U  e.  (SubGrp `  W ) )  -> 
( T  .(+)  U )  e.  (SubGrp `  W
) )
102, 5, 7, 9syl3anc 1185 . 2  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  (SubGrp `  W )
)
11 eqid 2438 . . . . . . . 8  |-  ( +g  `  W )  =  ( +g  `  W )
1211, 8lsmelval 15285 . . . . . . 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 644 . . . . . 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 453 . . . . 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 997 . . . . . . . . 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 733 . . . . . . . . 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 998 . . . . . . . . . 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 734 . . . . . . . . . 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 2438 . . . . . . . . . . 11  |-  ( Base `  W )  =  (
Base `  W )
2019, 3lssel 16016 . . . . . . . . . 10  |-  ( ( T  e.  S  /\  d  e.  T )  ->  d  e.  ( Base `  W ) )
2117, 18, 20syl2anc 644 . . . . . . . . 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 999 . . . . . . . . . 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 735 . . . . . . . . . 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 16016 . . . . . . . . . 10  |-  ( ( U  e.  S  /\  e  e.  U )  ->  e  e.  ( Base `  W ) )
2522, 23, 24syl2anc 644 . . . . . . . . 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 2438 . . . . . . . . . 10  |-  (Scalar `  W )  =  (Scalar `  W )
27 eqid 2438 . . . . . . . . . 10  |-  ( .s
`  W )  =  ( .s `  W
)
28 eqid 2438 . . . . . . . . . 10  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2919, 11, 26, 27, 28lmodvsdi 15975 . . . . . . . . 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 1187 . . . . . . . 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 644 . . . . . . . . 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 644 . . . . . . . . 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 16033 . . . . . . . . . 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 1186 . . . . . . . . 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 16033 . . . . . . . . . 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 1186 . . . . . . . . 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 15286 . . . . . . . . 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 1186 . . . . . . . 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 2512 . . . . . . 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 6091 . . . . . . . 8  |-  ( u  =  ( d ( +g  `  W ) e )  ->  (
a ( .s `  W ) u )  =  ( a ( .s `  W ) ( d ( +g  `  W ) e ) ) )
4140eleq1d 2504 . . . . . . 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 215 . . . . . 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 2839 . . . . 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 208 . . . 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 604 . . 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 2800 . 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 16040 . . 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 979 . 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 890 1  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531  Scalarcsca 13534   .scvsca 13535  SubGrpcsubg 14940   LSSumclsm 15270   Abelcabel 15415   LModclmod 15952   LSubSpclss 16010
This theorem is referenced by:  lsmelval2  16159  lsmsp  16160  lspprabs  16169  pj1lmhm  16174  lspabs3  16195  pjth  19342  kercvrlsm  27160  lshpnelb  29844  lsmsat  29868  lsmcv2  29889  lcvat  29890  lcvexchlem4  29897  lcvexchlem5  29898  lcv1  29901  lsatexch  29903  lsatcv0eq  29907  lsatcvatlem  29909  lsatcvat2  29911  lsatcvat3  29912  lkrlsp  29962  dia2dimlem7  31930  dihjustlem  32076  dihord1  32078  dihlsscpre  32094  dihjatcclem2  32279  dihjat1lem  32288  dochexmidlem5  32324  dochexmidlem6  32325  dochexmidlem8  32327  lcfrlem23  32425  mapdlsmcl  32523  mapdlsm  32524  mapdpglem1  32532  mapdpglem2a  32534  mapdindp0  32579  mapdheq4lem  32591  mapdh6lem1N  32593  mapdh6lem2N  32594  hdmap1l6lem1  32668  hdmap1l6lem2  32669  hdmaprnlem3eN  32721
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-0g 13729  df-mnd 14692  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-subg 14943  df-cntz 15118  df-lsm 15272  df-cmn 15416  df-abl 15417  df-mgp 15651  df-rng 15665  df-ur 15667  df-lmod 15954  df-lss 16011
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