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Theorem lsmelvali 16170
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
lsmelvali  |-  ( ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( X  e.  T  /\  Y  e.  U ) )  -> 
( X  .+  Y
)  e.  ( T 
.(+)  U ) )

Proof of Theorem lsmelvali
StepHypRef Expression
1 subgrcl 15707 . . . 4  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
21adantr 465 . . 3  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  G  e.  Grp )
3 eqid 2443 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
43subgss 15703 . . . 4  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
54adantr 465 . . 3  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  T  C_  ( Base `  G ) )
63subgss 15703 . . . 4  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
76adantl 466 . . 3  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  U  C_  ( Base `  G ) )
82, 5, 73jca 1168 . 2  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( G  e.  Grp  /\  T  C_  ( Base `  G )  /\  U  C_  ( Base `  G ) ) )
9 lsmelval.a . . 3  |-  .+  =  ( +g  `  G )
10 lsmelval.p . . 3  |-  .(+)  =  (
LSSum `  G )
113, 9, 10lsmelvalix 16161 . 2  |-  ( ( ( G  e.  Grp  /\  T  C_  ( Base `  G )  /\  U  C_  ( Base `  G
) )  /\  ( X  e.  T  /\  Y  e.  U )
)  ->  ( X  .+  Y )  e.  ( T  .(+)  U )
)
128, 11sylan 471 1  |-  ( ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  /\  ( X  e.  T  /\  Y  e.  U ) )  -> 
( X  .+  Y
)  e.  ( T 
.(+)  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    C_ wss 3349   ` cfv 5439  (class class class)co 6112   Basecbs 14195   +g cplusg 14259   Grpcgrp 15431  SubGrpcsubg 15696   LSSumclsm 16154
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
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 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  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-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  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-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-subg 15699  df-lsm 16156
This theorem is referenced by:  lsmsubg  16174  lsmmod  16193  lsmdisj2  16200  lsmhash  16223  ablfacrp  16589  lsmcl  17186  lsmelval2  17188  lsppreli  17193  lspprabs  17198  lspabs3  17224  pjthlem2  20947  lkrlsp  32843  dia2dimlem5  34809  mapdindp0  35460  hdmaprnlem3eN  35602
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