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Theorem lsmcom 16738
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 2443 . . 3  |-  ( Base `  G )  =  (
Base `  G )
32subgss 16076 . 2  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
42subgss 16076 . 2  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
5 lsmcom.s . . 3  |-  .(+)  =  (
LSSum `  G )
62, 5lsmcomx 16736 . 2  |-  ( ( G  e.  Abel  /\  T  C_  ( Base `  G
)  /\  U  C_  ( Base `  G ) )  ->  ( T  .(+)  U )  =  ( U 
.(+)  T ) )
71, 3, 4, 6syl3an 1271 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 974    = wceq 1383    e. wcel 1804    C_ wss 3461   ` cfv 5578  (class class class)co 6281   Basecbs 14509  SubGrpcsubg 16069   LSSumclsm 16528   Abelcabl 16673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-subg 16072  df-lsm 16530  df-cmn 16674  df-abl 16675
This theorem is referenced by:  lsm4  16740  pgpfac1lem4  17003  pgpfaclem1  17006  lspprabs  17615  ocvpj  18621  lcvexchlem3  34501  lcvexchlem4  34502  lcvexchlem5  34503  lsatcvatlem  34514  lsatcvat  34515  lsatcvat3  34517  l1cvat  34520  dia2dimlem5  36535  dihjatc3  36780  dihmeetlem9N  36782  dihjat  36890  lclkrlem2b  36975
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