MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isgrp2i Unicode version

Theorem isgrp2i 21777
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 11 . . 3  |-  (  T. 
->  X  e.  _V )
3 isgrp2i.2 . . . 4  |-  X  =/=  (/)
43a1i 11 . . 3  |-  (  T. 
->  X  =/=  (/) )
5 isgrp2i.3 . . . 4  |-  G :
( X  X.  X
) --> X
65a1i 11 . . 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 453 . . 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 453 . . 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 453 . . 3  |-  ( (  T.  /\  ( x  e.  X  /\  y  e.  X ) )  ->  E. z  e.  X  ( x G z )  =  y )
132, 4, 6, 8, 10, 12isgrp2d 21776 . 2  |-  (  T. 
->  G  e.  GrpOp )
1413trud 1329 1  |-  G  e. 
GrpOp
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    T. wtru 1322    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   _Vcvv 2916   (/)c0 3588    X. cxp 4835   -->wf 5409  (class class class)co 6040   GrpOpcgr 21727
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-grpo 21732
  Copyright terms: Public domain W3C validator