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Theorem issmgrp 21770
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 21767 . . 3  |-  SemiGrp  =  (
Magma  i^i  Ass )
21eleq2i 2451 . 2  |-  ( G  e.  SemiGrp 
<->  G  e.  ( Magma  i^i 
Ass ) )
3 elin 3473 . . 3  |-  ( G  e.  ( Magma  i^i  Ass ) 
<->  ( G  e.  Magma  /\  G  e.  Ass )
)
4 issmgrp.1 . . . . 5  |-  X  =  dom  dom  G
54ismgm 21756 . . . 4  |-  ( G  e.  A  ->  ( G  e.  Magma  <->  G :
( X  X.  X
) --> X ) )
64isass 21752 . . . 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 692 . . 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 249 . 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 249 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649    i^i cin 3262    X. cxp 4816   dom cdm 4818   -->wf 5390  (class class class)co 6020   Asscass 21748   Magmacmagm 21754   SemiGrpcsem 21766
This theorem is referenced by:  smgrpmgm  21771  smgrpass  21772  ismndo1  21780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402  df-ov 6023  df-ass 21749  df-mgm 21755  df-sgr 21767
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