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Theorem cmncom 15428
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 15419 . . . . 5  |-  ( G  e. CMnd 
<->  ( G  e.  Mnd  /\ 
A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
43simprbi 451 . . . 4  |-  ( G  e. CMnd  ->  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) )
5 rsp2 2768 . . . . 5  |-  ( A. x  e.  B  A. y  e.  B  (
x  .+  y )  =  ( y  .+  x )  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( x  .+  y )  =  ( y  .+  x ) ) )
65imp 419 . . . 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 458 . . 3  |-  ( ( G  e. CMnd  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x  .+  y )  =  ( y  .+  x ) )
87caovcomg 6242 . 2  |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )
)  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
983impb 1149 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   Mndcmnd 14684  CMndccmn 15412
This theorem is referenced by:  ablcom  15429  cmn32  15430  cmn4  15431  cmn12  15432  mulgnn0di  15448  subcmn  15456  cntzcmn  15459  prdscmnd  15476  crngcom  15678  ip2di  16872  ofldchr  24244
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-cmn 15414
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