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

Theorem isgrpoi 21791
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 2805 . 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 6091 . . . . . . . 8  |-  ( y  =  N  ->  (
y G x )  =  ( N G x ) )
98eqeq1d 2446 . . . . . . 7  |-  ( y  =  N  ->  (
( y G x )  =  U  <->  ( N G x )  =  U ) )
109rspcev 3054 . . . . . 6  |-  ( ( N  e.  X  /\  ( N G x )  =  U )  ->  E. y  e.  X  ( y G x )  =  U )
116, 7, 10syl2anc 644 . . . . 5  |-  ( x  e.  X  ->  E. y  e.  X  ( y G x )  =  U )
125, 11jca 520 . . . 4  |-  ( x  e.  X  ->  (
( U G x )  =  x  /\  E. y  e.  X  ( y G x )  =  U ) )
1312rgen 2773 . . 3  |-  A. x  e.  X  ( ( U G x )  =  x  /\  E. y  e.  X  ( y G x )  =  U )
14 oveq1 6091 . . . . . . 7  |-  ( u  =  U  ->  (
u G x )  =  ( U G x ) )
1514eqeq1d 2446 . . . . . 6  |-  ( u  =  U  ->  (
( u G x )  =  x  <->  ( U G x )  =  x ) )
16 eqeq2 2447 . . . . . . 7  |-  ( u  =  U  ->  (
( y G x )  =  u  <->  ( y G x )  =  U ) )
1716rexbidv 2728 . . . . . 6  |-  ( u  =  U  ->  ( E. y  e.  X  ( y G x )  =  u  <->  E. y  e.  X  ( y G x )  =  U ) )
1815, 17anbi12d 693 . . . . 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 2727 . . . 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 3054 . . 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 655 . 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 4993 . . . 4  |-  ( X  X.  X )  e. 
_V
24 fex 5972 . . . 4  |-  ( ( G : ( X  X.  X ) --> X  /\  ( X  X.  X )  e.  _V )  ->  G  e.  _V )
251, 23, 24mp2an 655 . . 3  |-  G  e. 
_V
265eqcomd 2443 . . . . . . . . 9  |-  ( x  e.  X  ->  x  =  ( U G x ) )
27 rspceov 6119 . . . . . . . . . 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 1267 . . . . . . . . 9  |-  ( ( x  e.  X  /\  x  =  ( U G x ) )  ->  E. y  e.  X  E. z  e.  X  x  =  ( y G z ) )
2926, 28mpdan 651 . . . . . . . 8  |-  ( x  e.  X  ->  E. y  e.  X  E. z  e.  X  x  =  ( y G z ) )
3029rgen 2773 . . . . . . 7  |-  A. x  e.  X  E. y  e.  X  E. z  e.  X  x  =  ( y G z )
31 foov 6223 . . . . . . 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 888 . . . . . 6  |-  G :
( X  X.  X
) -onto-> X
33 forn 5659 . . . . . 6  |-  ( G : ( X  X.  X ) -onto-> X  ->  ran  G  =  X )
3432, 33ax-mp 5 . . . . 5  |-  ran  G  =  X
3534eqcomi 2442 . . . 4  |-  X  =  ran  G
3635isgrpo 21789 . . 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 5 . 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 1137 1  |-  G  e. 
GrpOp
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958    X. cxp 4879   ran crn 4882   -->wf 5453   -onto->wfo 5455  (class class class)co 6084   GrpOpcgr 21779
This theorem is referenced by:  grposn  21808  issubgoi  21903  cnaddablo  21943  ablomul  21948  hilablo  22667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-grpo 21784
  Copyright terms: Public domain W3C validator