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Theorem isgrp2i 20895
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 20894 . 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 1628    e. wcel 1688    =/= wne 2447   E.wrex 2545   _Vcvv 2789   (/)c0 3456    X. cxp 4686   -->wf 5217  (class class class)co 5819   GrpOpcgr 20845
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-grpo 20850
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