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Theorem iscmn 15096
Description: The predicate "is a commutative monoid." (Contributed by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
iscmn.b  |-  B  =  ( Base `  G
)
iscmn.p  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
iscmn  |-  ( G  e. CMnd 
<->  ( G  e.  Mnd  /\ 
A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
Distinct variable groups:    x, y, B    x, G, y
Allowed substitution hints:    .+ ( x, y)

Proof of Theorem iscmn
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
2 iscmn.b . . . . 5  |-  B  =  ( Base `  G
)
31, 2syl6eqr 2333 . . . 4  |-  ( g  =  G  ->  ( Base `  g )  =  B )
4 raleq 2736 . . . . 5  |-  ( (
Base `  g )  =  B  ->  ( A. y  e.  ( Base `  g ) ( x ( +g  `  g
) y )  =  ( y ( +g  `  g ) x )  <->  A. y  e.  B  ( x ( +g  `  g ) y )  =  ( y ( +g  `  g ) x ) ) )
54raleqbi1dv 2744 . . . 4  |-  ( (
Base `  g )  =  B  ->  ( A. x  e.  ( Base `  g ) A. y  e.  ( Base `  g
) ( x ( +g  `  g ) y )  =  ( y ( +g  `  g
) x )  <->  A. x  e.  B  A. y  e.  B  ( x
( +g  `  g ) y )  =  ( y ( +g  `  g
) x ) ) )
63, 5syl 15 . . 3  |-  ( g  =  G  ->  ( A. x  e.  ( Base `  g ) A. y  e.  ( Base `  g ) ( x ( +g  `  g
) y )  =  ( y ( +g  `  g ) x )  <->  A. x  e.  B  A. y  e.  B  ( x ( +g  `  g ) y )  =  ( y ( +g  `  g ) x ) ) )
7 fveq2 5525 . . . . . . 7  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
8 iscmn.p . . . . . . 7  |-  .+  =  ( +g  `  G )
97, 8syl6eqr 2333 . . . . . 6  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
109oveqd 5875 . . . . 5  |-  ( g  =  G  ->  (
x ( +g  `  g
) y )  =  ( x  .+  y
) )
119oveqd 5875 . . . . 5  |-  ( g  =  G  ->  (
y ( +g  `  g
) x )  =  ( y  .+  x
) )
1210, 11eqeq12d 2297 . . . 4  |-  ( g  =  G  ->  (
( x ( +g  `  g ) y )  =  ( y ( +g  `  g ) x )  <->  ( x  .+  y )  =  ( y  .+  x ) ) )
13122ralbidv 2585 . . 3  |-  ( g  =  G  ->  ( A. x  e.  B  A. y  e.  B  ( x ( +g  `  g ) y )  =  ( y ( +g  `  g ) x )  <->  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
146, 13bitrd 244 . 2  |-  ( g  =  G  ->  ( A. x  e.  ( Base `  g ) A. y  e.  ( Base `  g ) ( x ( +g  `  g
) y )  =  ( y ( +g  `  g ) x )  <->  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
15 df-cmn 15091 . 2  |- CMnd  =  {
g  e.  Mnd  |  A. x  e.  ( Base `  g ) A. y  e.  ( Base `  g ) ( x ( +g  `  g
) y )  =  ( y ( +g  `  g ) x ) }
1614, 15elrab2 2925 1  |-  ( G  e. CMnd 
<->  ( G  e.  Mnd  /\ 
A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Mndcmnd 14361  CMndccmn 15089
This theorem is referenced by:  isabl2  15097  cmnpropd  15098  iscmnd  15101  cmnmnd  15104  cmncom  15105  submcmn2  15135  iscrng2  15356  xrs1cmn  16411  gicabl  27263
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
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