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Theorem issmgrp 21001
Description: The predicate "is a semi-group". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
Hypothesis
Ref Expression
issmgrp.1  |-  X  =  dom  dom  G
Assertion
Ref Expression
issmgrp  |-  ( G  e.  A  ->  ( G  e.  SemiGrp  <->  ( G : ( X  X.  X ) --> X  /\  A. x  e.  X  A. y  e.  X  A. z  e.  X  (
( x G y ) G z )  =  ( x G ( y G z ) ) ) ) )
Distinct variable groups:    x, G, y, z    x, X, y, z
Allowed substitution hints:    A( x, y, z)

Proof of Theorem issmgrp
StepHypRef Expression
1 df-sgr 20998 . . 3  |-  SemiGrp  =  (
Magma  i^i  Ass )
21eleq2i 2347 . 2  |-  ( G  e.  SemiGrp 
<->  G  e.  ( Magma  i^i 
Ass ) )
3 elin 3358 . . 3  |-  ( G  e.  ( Magma  i^i  Ass ) 
<->  ( G  e.  Magma  /\  G  e.  Ass )
)
4 issmgrp.1 . . . . 5  |-  X  =  dom  dom  G
54ismgm 20987 . . . 4  |-  ( G  e.  A  ->  ( G  e.  Magma  <->  G :
( X  X.  X
) --> X ) )
64isass 20983 . . . 4  |-  ( G  e.  A  ->  ( G  e.  Ass  <->  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( (
x G y ) G z )  =  ( x G ( y G z ) ) ) )
75, 6anbi12d 691 . . 3  |-  ( G  e.  A  ->  (
( G  e.  Magma  /\  G  e.  Ass )  <->  ( G : ( X  X.  X ) --> X  /\  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( x G y ) G z )  =  ( x G ( y G z ) ) ) ) )
83, 7syl5bb 248 . 2  |-  ( G  e.  A  ->  ( G  e.  ( Magma  i^i 
Ass )  <->  ( G : ( X  X.  X ) --> X  /\  A. x  e.  X  A. y  e.  X  A. z  e.  X  (
( x G y ) G z )  =  ( x G ( y G z ) ) ) ) )
92, 8syl5bb 248 1  |-  ( G  e.  A  ->  ( G  e.  SemiGrp  <->  ( G : ( X  X.  X ) --> X  /\  A. x  e.  X  A. y  e.  X  A. z  e.  X  (
( x G y ) G z )  =  ( x G ( y G z ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    X. cxp 4687   dom cdm 4689   -->wf 5251  (class class class)co 5858   Asscass 20979   Magmacmagm 20985   SemiGrpcsem 20997
This theorem is referenced by:  smgrpmgm  21002  smgrpass  21003  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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  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-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-ass 20980  df-mgm 20986  df-sgr 20998
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