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Theorem cmncom 16417
Description: A commutative monoid is commutative. (Contributed by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
ablcom.b  |-  B  =  ( Base `  G
)
ablcom.p  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
cmncom  |-  ( ( G  e. CMnd  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )

Proof of Theorem cmncom
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ablcom.b . . . . . 6  |-  B  =  ( Base `  G
)
2 ablcom.p . . . . . 6  |-  .+  =  ( +g  `  G )
31, 2iscmn 16408 . . . . 5  |-  ( G  e. CMnd 
<->  ( G  e.  Mnd  /\ 
A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
43simprbi 464 . . . 4  |-  ( G  e. CMnd  ->  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) )
5 rsp2 2896 . . . . 5  |-  ( A. x  e.  B  A. y  e.  B  (
x  .+  y )  =  ( y  .+  x )  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( x  .+  y )  =  ( y  .+  x ) ) )
65imp 429 . . . 4  |-  ( ( A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x  .+  y )  =  ( y  .+  x ) )
74, 6sylan 471 . . 3  |-  ( ( G  e. CMnd  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x  .+  y )  =  ( y  .+  x ) )
87caovcomg 6371 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )
)  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
983impb 1184 1  |-  ( ( G  e. CMnd  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   ` cfv 5529  (class class class)co 6203   Basecbs 14295   +g cplusg 14360   Mndcmnd 15531  CMndccmn 16401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-iota 5492  df-fv 5537  df-ov 6206  df-cmn 16403
This theorem is referenced by:  ablcom  16418  cmn32  16419  cmn4  16420  cmn12  16421  mulgnn0di  16437  subcmn  16445  cntzcmn  16448  prdscmnd  16467  srgcom  16751  srgbinomlem4  16767  csrgbinom  16770  crngcom  16785  ip2di  18198  omndadd2d  26336  ofldchr  26447  lply1binom  31029  chfacfscmulgsum  31366  chfacfpmmulgsum  31370  cpmadugsumlemF  31382
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