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

Theorem lsmcl 17186
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 17014 . . . 4  |-  ( W  e.  LMod  ->  W  e. 
Abel )
213ad2ant1 1009 . . 3  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  W  e.  Abel )
3 lsmcl.s . . . . 5  |-  S  =  ( LSubSp `  W )
43lsssubg 17060 . . . 4  |-  ( ( W  e.  LMod  /\  T  e.  S )  ->  T  e.  (SubGrp `  W )
)
543adant3 1008 . . 3  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  T  e.  (SubGrp `  W )
)
63lsssubg 17060 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
763adant2 1007 . . 3  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
8 lsmcl.p . . . 4  |-  .(+)  =  (
LSSum `  W )
98lsmsubg2 16362 . . 3  |-  ( ( W  e.  Abel  /\  T  e.  (SubGrp `  W )  /\  U  e.  (SubGrp `  W ) )  -> 
( T  .(+)  U )  e.  (SubGrp `  W
) )
102, 5, 7, 9syl3anc 1218 . 2  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  (SubGrp `  W )
)
11 eqid 2443 . . . . . . . 8  |-  ( +g  `  W )  =  ( +g  `  W )
1211, 8lsmelval 16169 . . . . . . 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 1027 . . . . . . . . 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 1028 . . . . . . . . . 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 2443 . . . . . . . . . . 11  |-  ( Base `  W )  =  (
Base `  W )
2019, 3lssel 17041 . . . . . . . . . 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 1029 . . . . . . . . . 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 17041 . . . . . . . . . 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 2443 . . . . . . . . . 10  |-  (Scalar `  W )  =  (Scalar `  W )
27 eqid 2443 . . . . . . . . . 10  |-  ( .s
`  W )  =  ( .s `  W
)
28 eqid 2443 . . . . . . . . . 10  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2919, 11, 26, 27, 28lmodvsdi 16993 . . . . . . . . 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 1220 . . . . . . . 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 17058 . . . . . . . . . 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 1219 . . . . . . . . 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 17058 . . . . . . . . . 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 1219 . . . . . . . . 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 16170 . . . . . . . . 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 1219 . . . . . . . 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 2517 . . . . . . 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 6120 . . . . . . . 8  |-  ( u  =  ( d ( +g  `  W ) e )  ->  (
a ( .s `  W ) u )  =  ( a ( .s `  W ) ( d ( +g  `  W ) e ) ) )
4140eleq1d 2509 . . . . . . 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 2869 . . . . 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 2829 . 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 17065 . . 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 1009 . 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 913 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 965    = wceq 1369    e. wcel 1756   A.wral 2736   E.wrex 2737   ` cfv 5439  (class class class)co 6112   Basecbs 14195   +g cplusg 14259  Scalarcsca 14262   .scvsca 14263  SubGrpcsubg 15696   LSSumclsm 16154   Abelcabel 16299   LModclmod 16970   LSubSpclss 17035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-0g 14401  df-mnd 15436  df-submnd 15486  df-grp 15566  df-minusg 15567  df-sbg 15568  df-subg 15699  df-cntz 15856  df-lsm 16156  df-cmn 16300  df-abl 16301  df-mgp 16614  df-ur 16626  df-rng 16669  df-lmod 16972  df-lss 17036
This theorem is referenced by:  lsmelval2  17188  lsmsp  17189  lspprabs  17198  pj1lmhm  17203  lspabs3  17224  pjth  20948  kercvrlsm  29462  lshpnelb  32725  lsmsat  32749  lsmcv2  32770  lcvat  32771  lcvexchlem4  32778  lcvexchlem5  32779  lcv1  32782  lsatexch  32784  lsatcv0eq  32788  lsatcvatlem  32790  lsatcvat2  32792  lsatcvat3  32793  lkrlsp  32843  dia2dimlem7  34811  dihjustlem  34957  dihord1  34959  dihlsscpre  34975  dihjatcclem2  35160  dihjat1lem  35169  dochexmidlem5  35205  dochexmidlem6  35206  dochexmidlem8  35208  lcfrlem23  35306  mapdlsmcl  35404  mapdlsm  35405  mapdpglem1  35413  mapdpglem2a  35415  mapdindp0  35460  mapdheq4lem  35472  mapdh6lem1N  35474  mapdh6lem2N  35475  hdmap1l6lem1  35549  hdmap1l6lem2  35550  hdmaprnlem3eN  35602
  Copyright terms: Public domain W3C validator