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Theorem lsmelval 15276
Description: Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmelval.a  |-  .+  =  ( +g  `  G )
lsmelval.p  |-  .(+)  =  (
LSSum `  G )
Assertion
Ref Expression
lsmelval  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( X  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. z  e.  U  X  =  ( y  .+  z ) ) )
Distinct variable groups:    y, z,  .+    y, T, z    y, U, z    y, G, z   
y, X, z
Allowed substitution hints:    .(+) ( y, z)

Proof of Theorem lsmelval
StepHypRef Expression
1 subgrcl 14942 . . 3  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
21adantr 452 . 2  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  G  e.  Grp )
3 eqid 2436 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
43subgss 14938 . . 3  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
54adantr 452 . 2  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  T  C_  ( Base `  G ) )
63subgss 14938 . . 3  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
76adantl 453 . 2  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  U  C_  ( Base `  G ) )
8 lsmelval.a . . 3  |-  .+  =  ( +g  `  G )
9 lsmelval.p . . 3  |-  .(+)  =  (
LSSum `  G )
103, 8, 9lsmelvalx 15267 . 2  |-  ( ( G  e.  Grp  /\  T  C_  ( Base `  G
)  /\  U  C_  ( Base `  G ) )  ->  ( X  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. z  e.  U  X  =  ( y  .+  z ) ) )
112, 5, 7, 10syl3anc 1184 1  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( X  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. z  e.  U  X  =  ( y  .+  z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2699    C_ wss 3313   ` cfv 5447  (class class class)co 6074   Basecbs 13462   +g cplusg 13522   Grpcgrp 14678  SubGrpcsubg 14931   LSSumclsm 15261
This theorem is referenced by:  lsmelvalm  15278  lsmsubg  15281  lsmcom2  15282  lsmmod  15300  lsmdisj2  15307  pj1eu  15321  lsmcl  16148  lsmspsn  16149  lsmelval2  16150  lsmcv  16206  lsmsat  29744  lshpsmreu  29845  dvhopellsm  31853  diblsmopel  31907  cdlemn11c  31945  dihord11c  31960  hdmapglem7a  32666
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-subg 14934  df-lsm 15263
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