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Theorem lsmcom 16667
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 22 . 2  |-  ( G  e.  Abel  ->  G  e. 
Abel )
2 eqid 2467 . . 3  |-  ( Base `  G )  =  (
Base `  G )
32subgss 16007 . 2  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
42subgss 16007 . 2  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
5 lsmcom.s . . 3  |-  .(+)  =  (
LSSum `  G )
62, 5lsmcomx 16665 . 2  |-  ( ( G  e.  Abel  /\  T  C_  ( Base `  G
)  /\  U  C_  ( Base `  G ) )  ->  ( T  .(+)  U )  =  ( U 
.(+)  T ) )
71, 3, 4, 6syl3an 1270 1  |-  ( ( G  e.  Abel  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  -> 
( T  .(+)  U )  =  ( U  .(+)  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3476   ` cfv 5588  (class class class)co 6284   Basecbs 14490  SubGrpcsubg 16000   LSSumclsm 16460   Abelcabl 16605
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-subg 16003  df-lsm 16462  df-cmn 16606  df-abl 16607
This theorem is referenced by:  lsm4  16669  pgpfac1lem4  16931  pgpfaclem1  16934  lspprabs  17541  ocvpj  18543  lcvexchlem3  33851  lcvexchlem4  33852  lcvexchlem5  33853  lsatcvatlem  33864  lsatcvat  33865  lsatcvat3  33867  l1cvat  33870  dia2dimlem5  35883  dihjatc3  36128  dihmeetlem9N  36130  dihjat  36238  lclkrlem2b  36323
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