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Theorem isgrpoi 20865
Description: Properties that determine a group operation. Read  N as  N ( x ). (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
isgrpi.1  |-  X  e. 
_V
isgrpi.2  |-  G :
( X  X.  X
) --> X
isgrpi.3  |-  ( ( x  e.  X  /\  y  e.  X  /\  z  e.  X )  ->  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )
isgrpi.4  |-  U  e.  X
isgrpi.5  |-  ( x  e.  X  ->  ( U G x )  =  x )
isgrpi.6  |-  ( x  e.  X  ->  N  e.  X )
isgrpi.7  |-  ( x  e.  X  ->  ( N G x )  =  U )
Assertion
Ref Expression
isgrpoi  |-  G  e. 
GrpOp
Distinct variable groups:    x, y,
z, G    x, U, y, z    x, X, y, z    y, N
Allowed substitution hints:    N( x, z)

Proof of Theorem isgrpoi
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 isgrpi.2 . 2  |-  G :
( X  X.  X
) --> X
2 isgrpi.3 . . 3  |-  ( ( x  e.  X  /\  y  e.  X  /\  z  e.  X )  ->  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )
32rgen3 2640 . 2  |-  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( (
x G y ) G z )  =  ( x G ( y G z ) )
4 isgrpi.4 . . 3  |-  U  e.  X
5 isgrpi.5 . . . . 5  |-  ( x  e.  X  ->  ( U G x )  =  x )
6 isgrpi.6 . . . . . 6  |-  ( x  e.  X  ->  N  e.  X )
7 isgrpi.7 . . . . . 6  |-  ( x  e.  X  ->  ( N G x )  =  U )
8 oveq1 5865 . . . . . . . 8  |-  ( y  =  N  ->  (
y G x )  =  ( N G x ) )
98eqeq1d 2291 . . . . . . 7  |-  ( y  =  N  ->  (
( y G x )  =  U  <->  ( N G x )  =  U ) )
109rspcev 2884 . . . . . 6  |-  ( ( N  e.  X  /\  ( N G x )  =  U )  ->  E. y  e.  X  ( y G x )  =  U )
116, 7, 10syl2anc 642 . . . . 5  |-  ( x  e.  X  ->  E. y  e.  X  ( y G x )  =  U )
125, 11jca 518 . . . 4  |-  ( x  e.  X  ->  (
( U G x )  =  x  /\  E. y  e.  X  ( y G x )  =  U ) )
1312rgen 2608 . . 3  |-  A. x  e.  X  ( ( U G x )  =  x  /\  E. y  e.  X  ( y G x )  =  U )
14 oveq1 5865 . . . . . . 7  |-  ( u  =  U  ->  (
u G x )  =  ( U G x ) )
1514eqeq1d 2291 . . . . . 6  |-  ( u  =  U  ->  (
( u G x )  =  x  <->  ( U G x )  =  x ) )
16 eqeq2 2292 . . . . . . 7  |-  ( u  =  U  ->  (
( y G x )  =  u  <->  ( y G x )  =  U ) )
1716rexbidv 2564 . . . . . 6  |-  ( u  =  U  ->  ( E. y  e.  X  ( y G x )  =  u  <->  E. y  e.  X  ( y G x )  =  U ) )
1815, 17anbi12d 691 . . . . 5  |-  ( u  =  U  ->  (
( ( u G x )  =  x  /\  E. y  e.  X  ( y G x )  =  u )  <->  ( ( U G x )  =  x  /\  E. y  e.  X  ( y G x )  =  U ) ) )
1918ralbidv 2563 . . . 4  |-  ( u  =  U  ->  ( A. x  e.  X  ( ( u G x )  =  x  /\  E. y  e.  X  ( y G x )  =  u )  <->  A. x  e.  X  ( ( U G x )  =  x  /\  E. y  e.  X  ( y G x )  =  U ) ) )
2019rspcev 2884 . . 3  |-  ( ( U  e.  X  /\  A. x  e.  X  ( ( U G x )  =  x  /\  E. y  e.  X  ( y G x )  =  U ) )  ->  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  E. y  e.  X  ( y G x )  =  u ) )
214, 13, 20mp2an 653 . 2  |-  E. u  e.  X  A. x  e.  X  ( (
u G x )  =  x  /\  E. y  e.  X  (
y G x )  =  u )
22 isgrpi.1 . . . . 5  |-  X  e. 
_V
2322, 22xpex 4801 . . . 4  |-  ( X  X.  X )  e. 
_V
24 fex 5749 . . . 4  |-  ( ( G : ( X  X.  X ) --> X  /\  ( X  X.  X )  e.  _V )  ->  G  e.  _V )
251, 23, 24mp2an 653 . . 3  |-  G  e. 
_V
265eqcomd 2288 . . . . . . . . 9  |-  ( x  e.  X  ->  x  =  ( U G x ) )
27 rspceov 5893 . . . . . . . . . 10  |-  ( ( U  e.  X  /\  x  e.  X  /\  x  =  ( U G x ) )  ->  E. y  e.  X  E. z  e.  X  x  =  ( y G z ) )
284, 27mp3an1 1264 . . . . . . . . 9  |-  ( ( x  e.  X  /\  x  =  ( U G x ) )  ->  E. y  e.  X  E. z  e.  X  x  =  ( y G z ) )
2926, 28mpdan 649 . . . . . . . 8  |-  ( x  e.  X  ->  E. y  e.  X  E. z  e.  X  x  =  ( y G z ) )
3029rgen 2608 . . . . . . 7  |-  A. x  e.  X  E. y  e.  X  E. z  e.  X  x  =  ( y G z )
31 foov 5994 . . . . . . 7  |-  ( G : ( X  X.  X ) -onto-> X  <->  ( G : ( X  X.  X ) --> X  /\  A. x  e.  X  E. y  e.  X  E. z  e.  X  x  =  ( y G z ) ) )
321, 30, 31mpbir2an 886 . . . . . 6  |-  G :
( X  X.  X
) -onto-> X
33 forn 5454 . . . . . 6  |-  ( G : ( X  X.  X ) -onto-> X  ->  ran  G  =  X )
3432, 33ax-mp 8 . . . . 5  |-  ran  G  =  X
3534eqcomi 2287 . . . 4  |-  X  =  ran  G
3635isgrpo 20863 . . 3  |-  ( G  e.  _V  ->  ( G  e.  GrpOp  <->  ( 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 ) )  /\  E. u  e.  X  A. x  e.  X  (
( u G x )  =  x  /\  E. y  e.  X  ( y G x )  =  u ) ) ) )
3725, 36ax-mp 8 . 2  |-  ( G  e.  GrpOp 
<->  ( 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 ) )  /\  E. u  e.  X  A. x  e.  X  ( (
u G x )  =  x  /\  E. y  e.  X  (
y G x )  =  u ) ) )
381, 3, 21, 37mpbir3an 1134 1  |-  G  e. 
GrpOp
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    X. cxp 4687   ran crn 4690   -->wf 5251   -onto->wfo 5253  (class class class)co 5858   GrpOpcgr 20853
This theorem is referenced by:  grposn  20882  issubgoi  20977  cnaddablo  21017  ablomul  21022  hilablo  21739  hmeogrpi  25536
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-grpo 20858
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