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Theorem lsmelval 16542
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 16078 . . 3  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
21adantr 465 . 2  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  G  e.  Grp )
3 eqid 2467 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
43subgss 16074 . . 3  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
54adantr 465 . 2  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  T  C_  ( Base `  G ) )
63subgss 16074 . . 3  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
76adantl 466 . 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 16533 . 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 1228 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2818    C_ wss 3481   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   Grpcgrp 15925  SubGrpcsubg 16067   LSSumclsm 16527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-subg 16070  df-lsm 16529
This theorem is referenced by:  lsmelvalm  16544  lsmsubg  16547  lsmcom2  16548  lsmmod  16566  lsmdisj2  16573  pj1eu  16587  lsmcl  17600  lsmspsn  17601  lsmelval2  17602  lsmcv  17658  lsmsat  34206  lshpsmreu  34307  dvhopellsm  36315  diblsmopel  36369  cdlemn11c  36407  dihord11c  36422  hdmapglem7a  37128
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