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Theorem lsmelvali 16994
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 16530 . . . 4  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
21adantr 463 . . 3  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  G  e.  Grp )
3 eqid 2402 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
43subgss 16526 . . . 4  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
54adantr 463 . . 3  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  T  C_  ( Base `  G ) )
63subgss 16526 . . . 4  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
76adantl 464 . . 3  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  U  C_  ( Base `  G ) )
82, 5, 73jca 1177 . 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 16985 . 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 469 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    C_ wss 3414   ` cfv 5569  (class class class)co 6278   Basecbs 14841   +g cplusg 14909   Grpcgrp 16377  SubGrpcsubg 16519   LSSumclsm 16978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-subg 16522  df-lsm 16980
This theorem is referenced by:  lsmsubg  16998  lsmmod  17017  lsmdisj2  17024  lsmhash  17047  ablfacrp  17437  lsmcl  18049  lsmelval2  18051  lsppreli  18056  lspprabs  18061  lspabs3  18087  pjthlem2  22145  lkrlsp  32120  dia2dimlem5  34088  mapdindp0  34739  hdmaprnlem3eN  34881
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