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Theorem cmncom 15204
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 15195 . . . . 5  |-  ( G  e. CMnd 
<->  ( G  e.  Mnd  /\ 
A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
43simprbi 450 . . . 4  |-  ( G  e. CMnd  ->  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) )
5 rsp2 2681 . . . . 5  |-  ( A. x  e.  B  A. y  e.  B  (
x  .+  y )  =  ( y  .+  x )  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( x  .+  y )  =  ( y  .+  x ) ) )
65imp 418 . . . 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 457 . . 3  |-  ( ( G  e. CMnd  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x  .+  y )  =  ( y  .+  x ) )
87caovcomg 6102 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )
)  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
983impb 1147 1  |-  ( ( G  e. CMnd  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   A.wral 2619   ` cfv 5337  (class class class)co 5945   Basecbs 13245   +g cplusg 13305   Mndcmnd 14460  CMndccmn 15188
This theorem is referenced by:  ablcom  15205  cmn32  15206  cmn4  15207  cmn12  15208  mulgnn0di  15224  subcmn  15232  cntzcmn  15235  prdscmnd  15252  crngcom  15454  ip2di  16651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-iota 5301  df-fv 5345  df-ov 5948  df-cmn 15190
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