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

Theorem ismndo 21010
Description: The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
ismndo.1  |-  X  =  dom  dom  G
Assertion
Ref Expression
ismndo  |-  ( G  e.  A  ->  ( G  e. MndOp  <->  ( G  e.  SemiGrp 
/\  E. x  e.  X  A. y  e.  X  ( ( x G y )  =  y  /\  ( y G x )  =  y ) ) ) )
Distinct variable groups:    x, G, y    x, X, y
Allowed substitution hints:    A( x, y)

Proof of Theorem ismndo
StepHypRef Expression
1 df-mndo 21005 . . 3  |- MndOp  =  (
SemiGrp  i^i  ExId  )
21eleq2i 2347 . 2  |-  ( G  e. MndOp 
<->  G  e.  ( SemiGrp  i^i 
ExId  ) )
3 elin 3358 . . 3  |-  ( G  e.  ( SemiGrp  i^i  ExId  )  <-> 
( G  e.  SemiGrp  /\  G  e.  ExId  ) )
4 ismndo.1 . . . . 5  |-  X  =  dom  dom  G
54isexid 20984 . . . 4  |-  ( G  e.  A  ->  ( G  e.  ExId  <->  E. x  e.  X  A. y  e.  X  ( (
x G y )  =  y  /\  (
y G x )  =  y ) ) )
65anbi2d 684 . . 3  |-  ( G  e.  A  ->  (
( G  e.  SemiGrp  /\  G  e.  ExId  )  <->  ( G  e.  SemiGrp  /\  E. x  e.  X  A. y  e.  X  ( (
x G y )  =  y  /\  (
y G x )  =  y ) ) ) )
73, 6syl5bb 248 . 2  |-  ( G  e.  A  ->  ( G  e.  ( SemiGrp  i^i 
ExId  )  <->  ( G  e.  SemiGrp 
/\  E. x  e.  X  A. y  e.  X  ( ( x G y )  =  y  /\  ( y G x )  =  y ) ) ) )
82, 7syl5bb 248 1  |-  ( G  e.  A  ->  ( G  e. MndOp  <->  ( G  e.  SemiGrp 
/\  E. x  e.  X  A. y  e.  X  ( ( x G y )  =  y  /\  ( y G x )  =  y ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151   dom cdm 4689  (class class class)co 5858    ExId cexid 20981   SemiGrpcsem 20997  MndOpcmndo 21004
This theorem is referenced by:  ismndo1  21011
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-dm 4699  df-iota 5219  df-fv 5263  df-ov 5861  df-exid 20982  df-mndo 21005
  Copyright terms: Public domain W3C validator