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Theorem ablcom 15106
Description: An Abelian group operation is commutative. (Contributed by NM, 26-Aug-2011.)
Hypotheses
Ref Expression
ablcom.b  |-  B  =  ( Base `  G
)
ablcom.p  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
ablcom  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )

Proof of Theorem ablcom
StepHypRef Expression
1 ablcmn 15095 . 2  |-  ( G  e.  Abel  ->  G  e. CMnd
)
2 ablcom.b . . 3  |-  B  =  ( Base `  G
)
3 ablcom.p . . 3  |-  .+  =  ( +g  `  G )
42, 3cmncom 15105 . 2  |-  ( ( G  e. CMnd  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
51, 4syl3an1 1215 1  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  CMndccmn 15089   Abelcabel 15090
This theorem is referenced by:  ablinvadd  15111  ablsub2inv  15112  ablsubadd  15113  abladdsub  15116  ablpncan3  15118  ablsub32  15123  eqgabl  15131  subgabl  15132  ablnsg  15139  lsmcomx  15148  divsabl  15157  frgpnabl  15163  ngplcan  18132  r1pid  19545  cnaddcom  29161  toycom  29162  lflsub  29257  lfladdcom  29262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-cmn 15091  df-abl 15092
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