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Theorem iscmn 15421
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 5730 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
2 iscmn.b . . . . 5  |-  B  =  ( Base `  G
)
31, 2syl6eqr 2488 . . . 4  |-  ( g  =  G  ->  ( Base `  g )  =  B )
4 raleq 2906 . . . . 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 2914 . . . 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 5730 . . . . . . 7  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
8 iscmn.p . . . . . . 7  |-  .+  =  ( +g  `  G )
97, 8syl6eqr 2488 . . . . . 6  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
109oveqd 6100 . . . . 5  |-  ( g  =  G  ->  (
x ( +g  `  g
) y )  =  ( x  .+  y
) )
119oveqd 6100 . . . . 5  |-  ( g  =  G  ->  (
y ( +g  `  g
) x )  =  ( y  .+  x
) )
1210, 11eqeq12d 2452 . . . 4  |-  ( g  =  G  ->  (
( x ( +g  `  g ) y )  =  ( y ( +g  `  g ) x )  <->  ( x  .+  y )  =  ( y  .+  x ) ) )
13122ralbidv 2749 . . 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 246 . 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 15416 . 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 3096 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   Mndcmnd 14686  CMndccmn 15414
This theorem is referenced by:  isabl2  15422  cmnpropd  15423  iscmnd  15426  cmnmnd  15429  cmncom  15430  submcmn2  15460  iscrng2  15681  xrs1cmn  16740  gicabl  27242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-cmn 15416
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