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Theorem isgrpoi 21739
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 2763 . 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 6047 . . . . . . . 8  |-  ( y  =  N  ->  (
y G x )  =  ( N G x ) )
98eqeq1d 2412 . . . . . . 7  |-  ( y  =  N  ->  (
( y G x )  =  U  <->  ( N G x )  =  U ) )
109rspcev 3012 . . . . . 6  |-  ( ( N  e.  X  /\  ( N G x )  =  U )  ->  E. y  e.  X  ( y G x )  =  U )
116, 7, 10syl2anc 643 . . . . 5  |-  ( x  e.  X  ->  E. y  e.  X  ( y G x )  =  U )
125, 11jca 519 . . . 4  |-  ( x  e.  X  ->  (
( U G x )  =  x  /\  E. y  e.  X  ( y G x )  =  U ) )
1312rgen 2731 . . 3  |-  A. x  e.  X  ( ( U G x )  =  x  /\  E. y  e.  X  ( y G x )  =  U )
14 oveq1 6047 . . . . . . 7  |-  ( u  =  U  ->  (
u G x )  =  ( U G x ) )
1514eqeq1d 2412 . . . . . 6  |-  ( u  =  U  ->  (
( u G x )  =  x  <->  ( U G x )  =  x ) )
16 eqeq2 2413 . . . . . . 7  |-  ( u  =  U  ->  (
( y G x )  =  u  <->  ( y G x )  =  U ) )
1716rexbidv 2687 . . . . . 6  |-  ( u  =  U  ->  ( E. y  e.  X  ( y G x )  =  u  <->  E. y  e.  X  ( y G x )  =  U ) )
1815, 17anbi12d 692 . . . . 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 2686 . . . 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 3012 . . 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 654 . 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 4949 . . . 4  |-  ( X  X.  X )  e. 
_V
24 fex 5928 . . . 4  |-  ( ( G : ( X  X.  X ) --> X  /\  ( X  X.  X )  e.  _V )  ->  G  e.  _V )
251, 23, 24mp2an 654 . . 3  |-  G  e. 
_V
265eqcomd 2409 . . . . . . . . 9  |-  ( x  e.  X  ->  x  =  ( U G x ) )
27 rspceov 6075 . . . . . . . . . 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 1266 . . . . . . . . 9  |-  ( ( x  e.  X  /\  x  =  ( U G x ) )  ->  E. y  e.  X  E. z  e.  X  x  =  ( y G z ) )
2926, 28mpdan 650 . . . . . . . 8  |-  ( x  e.  X  ->  E. y  e.  X  E. z  e.  X  x  =  ( y G z ) )
3029rgen 2731 . . . . . . 7  |-  A. x  e.  X  E. y  e.  X  E. z  e.  X  x  =  ( y G z )
31 foov 6179 . . . . . . 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 887 . . . . . 6  |-  G :
( X  X.  X
) -onto-> X
33 forn 5615 . . . . . 6  |-  ( G : ( X  X.  X ) -onto-> X  ->  ran  G  =  X )
3432, 33ax-mp 8 . . . . 5  |-  ran  G  =  X
3534eqcomi 2408 . . . 4  |-  X  =  ran  G
3635isgrpo 21737 . . 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 1136 1  |-  G  e. 
GrpOp
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916    X. cxp 4835   ran crn 4838   -->wf 5409   -onto->wfo 5411  (class class class)co 6040   GrpOpcgr 21727
This theorem is referenced by:  grposn  21756  issubgoi  21851  cnaddablo  21891  ablomul  21896  hilablo  22615
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
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