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Theorem lsmcom 15473
Description: Subgroup sum commutes. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
Hypothesis
Ref Expression
lsmcom.s  |-  .(+)  =  (
LSSum `  G )
Assertion
Ref Expression
lsmcom  |-  ( ( G  e.  Abel  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  -> 
( T  .(+)  U )  =  ( U  .(+)  T ) )

Proof of Theorem lsmcom
StepHypRef Expression
1 id 20 . 2  |-  ( G  e.  Abel  ->  G  e. 
Abel )
2 eqid 2436 . . 3  |-  ( Base `  G )  =  (
Base `  G )
32subgss 14945 . 2  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
42subgss 14945 . 2  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
5 lsmcom.s . . 3  |-  .(+)  =  (
LSSum `  G )
62, 5lsmcomx 15471 . 2  |-  ( ( G  e.  Abel  /\  T  C_  ( Base `  G
)  /\  U  C_  ( Base `  G ) )  ->  ( T  .(+)  U )  =  ( U 
.(+)  T ) )
71, 3, 4, 6syl3an 1226 1  |-  ( ( G  e.  Abel  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  -> 
( T  .(+)  U )  =  ( U  .(+)  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3320   ` cfv 5454  (class class class)co 6081   Basecbs 13469  SubGrpcsubg 14938   LSSumclsm 15268   Abelcabel 15413
This theorem is referenced by:  lsm4  15475  pgpfac1lem4  15636  pgpfaclem1  15639  lspprabs  16167  ocvpj  16944  lcvexchlem3  29834  lcvexchlem4  29835  lcvexchlem5  29836  lsatcvatlem  29847  lsatcvat  29848  lsatcvat3  29850  l1cvat  29853  dia2dimlem5  31866  dihjatc3  32111  dihmeetlem9N  32113  dihjat  32221  lclkrlem2b  32306
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 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-subg 14941  df-lsm 15270  df-cmn 15414  df-abl 15415
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