MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscmn Structured version   Visualization version   Unicode version

Theorem iscmn 17437
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 5865 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
2 iscmn.b . . . . 5  |-  B  =  ( Base `  G
)
31, 2syl6eqr 2503 . . . 4  |-  ( g  =  G  ->  ( Base `  g )  =  B )
4 raleq 2987 . . . . 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 2995 . . . 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 17 . . 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 5865 . . . . . . 7  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
8 iscmn.p . . . . . . 7  |-  .+  =  ( +g  `  G )
97, 8syl6eqr 2503 . . . . . 6  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
109oveqd 6307 . . . . 5  |-  ( g  =  G  ->  (
x ( +g  `  g
) y )  =  ( x  .+  y
) )
119oveqd 6307 . . . . 5  |-  ( g  =  G  ->  (
y ( +g  `  g
) x )  =  ( y  .+  x
) )
1210, 11eqeq12d 2466 . . . 4  |-  ( g  =  G  ->  (
( x ( +g  `  g ) y )  =  ( y ( +g  `  g ) x )  <->  ( x  .+  y )  =  ( y  .+  x ) ) )
13122ralbidv 2832 . . 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 257 . 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 17432 . 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 3198 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 188    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   ` cfv 5582  (class class class)co 6290   Basecbs 15121   +g cplusg 15190   Mndcmnd 16535  CMndccmn 17430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-iota 5546  df-fv 5590  df-ov 6293  df-cmn 17432
This theorem is referenced by:  isabl2  17438  cmnpropd  17439  iscmnd  17442  cmnmnd  17445  cmncom  17446  ghmcmn  17472  submcmn2  17479  iscrng2  17796  xrs1cmn  19008  abliso  28459  gicabl  35957  pgrpgt2nabl  40204
  Copyright terms: Public domain W3C validator