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Theorem isgrp2i 20733
Description: An alternate way to show a group operation. Exercise 1 of [Herstein] p. 57. (Contributed by NM, 5-Dec-2006.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
isgrp2i.1  |-  X  e. 
_V
isgrp2i.2  |-  X  =/=  (/)
isgrp2i.3  |-  G :
( X  X.  X
) --> X
isgrp2i.4  |-  ( ( x  e.  X  /\  y  e.  X  /\  z  e.  X )  ->  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )
isgrp2i.5  |-  ( ( x  e.  X  /\  y  e.  X )  ->  E. z  e.  X  ( z G x )  =  y )
isgrp2i.6  |-  ( ( x  e.  X  /\  y  e.  X )  ->  E. z  e.  X  ( x G z )  =  y )
Assertion
Ref Expression
isgrp2i  |-  G  e. 
GrpOp
Distinct variable groups:    x, y,
z, G    x, X, y, z

Proof of Theorem isgrp2i
StepHypRef Expression
1 isgrp2i.1 . . . 4  |-  X  e. 
_V
21a1i 12 . . 3  |-  (  T. 
->  X  e.  _V )
3 isgrp2i.2 . . . 4  |-  X  =/=  (/)
43a1i 12 . . 3  |-  (  T. 
->  X  =/=  (/) )
5 isgrp2i.3 . . . 4  |-  G :
( X  X.  X
) --> X
65a1i 12 . . 3  |-  (  T. 
->  G : ( X  X.  X ) --> X )
7 isgrp2i.4 . . . 4  |-  ( ( x  e.  X  /\  y  e.  X  /\  z  e.  X )  ->  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )
87adantl 454 . . 3  |-  ( (  T.  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  -> 
( ( x G y ) G z )  =  ( x G ( y G z ) ) )
9 isgrp2i.5 . . . 4  |-  ( ( x  e.  X  /\  y  e.  X )  ->  E. z  e.  X  ( z G x )  =  y )
109adantl 454 . . 3  |-  ( (  T.  /\  ( x  e.  X  /\  y  e.  X ) )  ->  E. z  e.  X  ( z G x )  =  y )
11 isgrp2i.6 . . . 4  |-  ( ( x  e.  X  /\  y  e.  X )  ->  E. z  e.  X  ( x G z )  =  y )
1211adantl 454 . . 3  |-  ( (  T.  /\  ( x  e.  X  /\  y  e.  X ) )  ->  E. z  e.  X  ( x G z )  =  y )
132, 4, 6, 8, 10, 12isgrp2d 20732 . 2  |-  (  T. 
->  G  e.  GrpOp )
1413trud 1320 1  |-  G  e. 
GrpOp
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    T. wtru 1312    = wceq 1619    e. wcel 1621    =/= wne 2412   E.wrex 2510   _Vcvv 2727   (/)c0 3362    X. cxp 4578   -->wf 4588  (class class class)co 5710   GrpOpcgr 20683
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-grpo 20688
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