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Theorem lsmelval 16151
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 15689 . . 3  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
21adantr 465 . 2  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  G  e.  Grp )
3 eqid 2443 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
43subgss 15685 . . 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 15685 . . 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 16142 . 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 1218 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 1369    e. wcel 1756   E.wrex 2719    C_ wss 3331   ` cfv 5421  (class class class)co 6094   Basecbs 14177   +g cplusg 14241   Grpcgrp 15413  SubGrpcsubg 15678   LSSumclsm 16136
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 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-reu 2725  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-1st 6580  df-2nd 6581  df-subg 15681  df-lsm 16138
This theorem is referenced by:  lsmelvalm  16153  lsmsubg  16156  lsmcom2  16157  lsmmod  16175  lsmdisj2  16182  pj1eu  16196  lsmcl  17167  lsmspsn  17168  lsmelval2  17169  lsmcv  17225  lsmsat  32656  lshpsmreu  32757  dvhopellsm  34765  diblsmopel  34819  cdlemn11c  34857  dihord11c  34872  hdmapglem7a  35578
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