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Theorem isgrp2i 20849
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 20848 . 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 2419   E.wrex 2517   _Vcvv 2757   (/)c0 3416    X. cxp 4645   -->wf 4655  (class class class)co 5778   GrpOpcgr 20799
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 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-grpo 20804
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