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Theorem cmncom 13110
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 13101 . . . . 5  |-  ( G  e. CMnd 
<->  ( G  e.  Mnd  /\ 
A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
43simprbi 275 . . . 4  |-  ( G  e. CMnd  ->  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) )
5 rsp2 2527 . . . . 5  |-  ( A. x  e.  B  A. y  e.  B  (
x  .+  y )  =  ( y  .+  x )  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( x  .+  y )  =  ( y  .+  x ) ) )
65imp 124 . . . 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 283 . . 3  |-  ( ( G  e. CMnd  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x  .+  y )  =  ( y  .+  x ) )
87caovcomg 6032 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )
)  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
983impb 1199 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 104    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   ` cfv 5218  (class class class)co 5877   Basecbs 12464   +g cplusg 12538   Mndcmnd 12822  CMndccmn 13093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880  df-cmn 13095
This theorem is referenced by:  ablcom  13111  cmn32  13112  cmn4  13113  cmn12  13114  rinvmod  13117  subcmnd  13134  srgcom  13171  crngcom  13202
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