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Theorem iscmn 15112
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 5541 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
2 iscmn.b . . . . 5  |-  B  =  ( Base `  G
)
31, 2syl6eqr 2346 . . . 4  |-  ( g  =  G  ->  ( Base `  g )  =  B )
4 raleq 2749 . . . . 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 2757 . . . 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 5541 . . . . . . 7  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
8 iscmn.p . . . . . . 7  |-  .+  =  ( +g  `  G )
97, 8syl6eqr 2346 . . . . . 6  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
109oveqd 5891 . . . . 5  |-  ( g  =  G  ->  (
x ( +g  `  g
) y )  =  ( x  .+  y
) )
119oveqd 5891 . . . . 5  |-  ( g  =  G  ->  (
y ( +g  `  g
) x )  =  ( y  .+  x
) )
1210, 11eqeq12d 2310 . . . 4  |-  ( g  =  G  ->  (
( x ( +g  `  g ) y )  =  ( y ( +g  `  g ) x )  <->  ( x  .+  y )  =  ( y  .+  x ) ) )
13122ralbidv 2598 . . 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 15107 . 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 2938 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 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   Mndcmnd 14377  CMndccmn 15105
This theorem is referenced by:  isabl2  15113  cmnpropd  15114  iscmnd  15117  cmnmnd  15120  cmncom  15121  submcmn2  15151  iscrng2  15372  xrs1cmn  16427  gicabl  27366
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-cmn 15107
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