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Theorem iscmn 16932
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 5872 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
2 iscmn.b . . . . 5  |-  B  =  ( Base `  G
)
31, 2syl6eqr 2516 . . . 4  |-  ( g  =  G  ->  ( Base `  g )  =  B )
4 raleq 3054 . . . . 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 3062 . . . 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 16 . . 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 5872 . . . . . . 7  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
8 iscmn.p . . . . . . 7  |-  .+  =  ( +g  `  G )
97, 8syl6eqr 2516 . . . . . 6  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
109oveqd 6313 . . . . 5  |-  ( g  =  G  ->  (
x ( +g  `  g
) y )  =  ( x  .+  y
) )
119oveqd 6313 . . . . 5  |-  ( g  =  G  ->  (
y ( +g  `  g
) x )  =  ( y  .+  x
) )
1210, 11eqeq12d 2479 . . . 4  |-  ( g  =  G  ->  (
( x ( +g  `  g ) y )  =  ( y ( +g  `  g ) x )  <->  ( x  .+  y )  =  ( y  .+  x ) ) )
13122ralbidv 2901 . . 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 253 . 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 16927 . 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 3259 1  |-  ( G  e. CMnd 
<->  ( G  e.  Mnd  /\ 
A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   ` cfv 5594  (class class class)co 6296   Basecbs 14644   +g cplusg 14712   Mndcmnd 16046  CMndccmn 16925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-cmn 16927
This theorem is referenced by:  isabl2  16933  cmnpropd  16934  iscmnd  16937  cmnmnd  16940  cmncom  16941  ghmcmn  16967  submcmn2  16974  iscrng2  17341  xrs1cmn  18585  abliso  27846  gicabl  31251  pgrpgt2nabl  33103
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