MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpoidinv2 Structured version   Unicode version

Theorem grpoidinv2 25418
Description: A group's properties using the explicit identity element. (Contributed by NM, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpoidval.1  |-  X  =  ran  G
grpoidval.2  |-  U  =  (GId `  G )
Assertion
Ref Expression
grpoidinv2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( ( U G A )  =  A  /\  ( A G U )  =  A )  /\  E. y  e.  X  ( (
y G A )  =  U  /\  ( A G y )  =  U ) ) )
Distinct variable groups:    y, A    y, G    y, U    y, X

Proof of Theorem grpoidinv2
Dummy variables  x  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpoidval.1 . . . . . . 7  |-  X  =  ran  G
2 grpoidval.2 . . . . . . 7  |-  U  =  (GId `  G )
31, 2grpoidval 25416 . . . . . 6  |-  ( G  e.  GrpOp  ->  U  =  ( iota_ u  e.  X  A. x  e.  X  ( u G x )  =  x ) )
41grpoideu 25409 . . . . . . 7  |-  ( G  e.  GrpOp  ->  E! u  e.  X  A. x  e.  X  ( u G x )  =  x )
5 riotacl2 6245 . . . . . . 7  |-  ( E! u  e.  X  A. x  e.  X  (
u G x )  =  x  ->  ( iota_ u  e.  X  A. x  e.  X  (
u G x )  =  x )  e. 
{ u  e.  X  |  A. x  e.  X  ( u G x )  =  x }
)
64, 5syl 16 . . . . . 6  |-  ( G  e.  GrpOp  ->  ( iota_ u  e.  X  A. x  e.  X  ( u G x )  =  x )  e.  {
u  e.  X  |  A. x  e.  X  ( u G x )  =  x }
)
73, 6eqeltrd 2542 . . . . 5  |-  ( G  e.  GrpOp  ->  U  e.  { u  e.  X  |  A. x  e.  X  ( u G x )  =  x }
)
8 simpll 751 . . . . . . . . . . 11  |-  ( ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( (
y G x )  =  u  /\  (
x G y )  =  u ) )  ->  ( u G x )  =  x )
98ralimi 2847 . . . . . . . . . 10  |-  ( A. x  e.  X  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( (
y G x )  =  u  /\  (
x G y )  =  u ) )  ->  A. x  e.  X  ( u G x )  =  x )
109rgenw 2815 . . . . . . . . 9  |-  A. u  e.  X  ( A. x  e.  X  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( (
y G x )  =  u  /\  (
x G y )  =  u ) )  ->  A. x  e.  X  ( u G x )  =  x )
1110a1i 11 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  A. u  e.  X  ( A. x  e.  X  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( (
y G x )  =  u  /\  (
x G y )  =  u ) )  ->  A. x  e.  X  ( u G x )  =  x ) )
121grpoidinv 25408 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  E. u  e.  X  A. x  e.  X  ( (
( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( ( y G x )  =  u  /\  ( x G y )  =  u ) ) )
1311, 12, 43jca 1174 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ( A. u  e.  X  ( A. x  e.  X  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  (
( y G x )  =  u  /\  ( x G y )  =  u ) )  ->  A. x  e.  X  ( u G x )  =  x )  /\  E. u  e.  X  A. x  e.  X  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( (
y G x )  =  u  /\  (
x G y )  =  u ) )  /\  E! u  e.  X  A. x  e.  X  ( u G x )  =  x ) )
14 reupick2 3781 . . . . . . 7  |-  ( ( ( A. u  e.  X  ( A. x  e.  X  ( (
( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( ( y G x )  =  u  /\  ( x G y )  =  u ) )  ->  A. x  e.  X  ( u G x )  =  x )  /\  E. u  e.  X  A. x  e.  X  ( ( ( u G x )  =  x  /\  (
x G u )  =  x )  /\  E. y  e.  X  ( ( y G x )  =  u  /\  ( x G y )  =  u ) )  /\  E! u  e.  X  A. x  e.  X  ( u G x )  =  x )  /\  u  e.  X )  ->  ( A. x  e.  X  ( u G x )  =  x  <->  A. x  e.  X  ( (
( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( ( y G x )  =  u  /\  ( x G y )  =  u ) ) ) )
1513, 14sylan 469 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  u  e.  X )  ->  ( A. x  e.  X  ( u G x )  =  x  <->  A. x  e.  X  ( (
( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( ( y G x )  =  u  /\  ( x G y )  =  u ) ) ) )
1615rabbidva 3097 . . . . 5  |-  ( G  e.  GrpOp  ->  { u  e.  X  |  A. x  e.  X  (
u G x )  =  x }  =  { u  e.  X  |  A. x  e.  X  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  (
( y G x )  =  u  /\  ( x G y )  =  u ) ) } )
177, 16eleqtrd 2544 . . . 4  |-  ( G  e.  GrpOp  ->  U  e.  { u  e.  X  |  A. x  e.  X  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  (
( y G x )  =  u  /\  ( x G y )  =  u ) ) } )
18 oveq1 6277 . . . . . . . . 9  |-  ( u  =  U  ->  (
u G x )  =  ( U G x ) )
1918eqeq1d 2456 . . . . . . . 8  |-  ( u  =  U  ->  (
( u G x )  =  x  <->  ( U G x )  =  x ) )
20 oveq2 6278 . . . . . . . . 9  |-  ( u  =  U  ->  (
x G u )  =  ( x G U ) )
2120eqeq1d 2456 . . . . . . . 8  |-  ( u  =  U  ->  (
( x G u )  =  x  <->  ( x G U )  =  x ) )
2219, 21anbi12d 708 . . . . . . 7  |-  ( u  =  U  ->  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  <->  ( ( U G x )  =  x  /\  ( x G U )  =  x ) ) )
23 eqeq2 2469 . . . . . . . . 9  |-  ( u  =  U  ->  (
( y G x )  =  u  <->  ( y G x )  =  U ) )
24 eqeq2 2469 . . . . . . . . 9  |-  ( u  =  U  ->  (
( x G y )  =  u  <->  ( x G y )  =  U ) )
2523, 24anbi12d 708 . . . . . . . 8  |-  ( u  =  U  ->  (
( ( y G x )  =  u  /\  ( x G y )  =  u )  <->  ( ( y G x )  =  U  /\  ( x G y )  =  U ) ) )
2625rexbidv 2965 . . . . . . 7  |-  ( u  =  U  ->  ( E. y  e.  X  ( ( y G x )  =  u  /\  ( x G y )  =  u )  <->  E. y  e.  X  ( ( y G x )  =  U  /\  ( x G y )  =  U ) ) )
2722, 26anbi12d 708 . . . . . 6  |-  ( u  =  U  ->  (
( ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  (
( y G x )  =  u  /\  ( x G y )  =  u ) )  <->  ( ( ( U G x )  =  x  /\  (
x G U )  =  x )  /\  E. y  e.  X  ( ( y G x )  =  U  /\  ( x G y )  =  U ) ) ) )
2827ralbidv 2893 . . . . 5  |-  ( u  =  U  ->  ( A. x  e.  X  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  (
( y G x )  =  u  /\  ( x G y )  =  u ) )  <->  A. x  e.  X  ( ( ( U G x )  =  x  /\  ( x G U )  =  x )  /\  E. y  e.  X  (
( y G x )  =  U  /\  ( x G y )  =  U ) ) ) )
2928elrab 3254 . . . 4  |-  ( U  e.  { u  e.  X  |  A. x  e.  X  ( (
( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( ( y G x )  =  u  /\  ( x G y )  =  u ) ) }  <-> 
( U  e.  X  /\  A. x  e.  X  ( ( ( U G x )  =  x  /\  ( x G U )  =  x )  /\  E. y  e.  X  (
( y G x )  =  U  /\  ( x G y )  =  U ) ) ) )
3017, 29sylib 196 . . 3  |-  ( G  e.  GrpOp  ->  ( U  e.  X  /\  A. x  e.  X  ( (
( U G x )  =  x  /\  ( x G U )  =  x )  /\  E. y  e.  X  ( ( y G x )  =  U  /\  ( x G y )  =  U ) ) ) )
3130simprd 461 . 2  |-  ( G  e.  GrpOp  ->  A. x  e.  X  ( (
( U G x )  =  x  /\  ( x G U )  =  x )  /\  E. y  e.  X  ( ( y G x )  =  U  /\  ( x G y )  =  U ) ) )
32 oveq2 6278 . . . . . 6  |-  ( x  =  A  ->  ( U G x )  =  ( U G A ) )
33 id 22 . . . . . 6  |-  ( x  =  A  ->  x  =  A )
3432, 33eqeq12d 2476 . . . . 5  |-  ( x  =  A  ->  (
( U G x )  =  x  <->  ( U G A )  =  A ) )
35 oveq1 6277 . . . . . 6  |-  ( x  =  A  ->  (
x G U )  =  ( A G U ) )
3635, 33eqeq12d 2476 . . . . 5  |-  ( x  =  A  ->  (
( x G U )  =  x  <->  ( A G U )  =  A ) )
3734, 36anbi12d 708 . . . 4  |-  ( x  =  A  ->  (
( ( U G x )  =  x  /\  ( x G U )  =  x )  <->  ( ( U G A )  =  A  /\  ( A G U )  =  A ) ) )
38 oveq2 6278 . . . . . . 7  |-  ( x  =  A  ->  (
y G x )  =  ( y G A ) )
3938eqeq1d 2456 . . . . . 6  |-  ( x  =  A  ->  (
( y G x )  =  U  <->  ( y G A )  =  U ) )
40 oveq1 6277 . . . . . . 7  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
4140eqeq1d 2456 . . . . . 6  |-  ( x  =  A  ->  (
( x G y )  =  U  <->  ( A G y )  =  U ) )
4239, 41anbi12d 708 . . . . 5  |-  ( x  =  A  ->  (
( ( y G x )  =  U  /\  ( x G y )  =  U )  <->  ( ( y G A )  =  U  /\  ( A G y )  =  U ) ) )
4342rexbidv 2965 . . . 4  |-  ( x  =  A  ->  ( E. y  e.  X  ( ( y G x )  =  U  /\  ( x G y )  =  U )  <->  E. y  e.  X  ( ( y G A )  =  U  /\  ( A G y )  =  U ) ) )
4437, 43anbi12d 708 . . 3  |-  ( x  =  A  ->  (
( ( ( U G x )  =  x  /\  ( x G U )  =  x )  /\  E. y  e.  X  (
( y G x )  =  U  /\  ( x G y )  =  U ) )  <->  ( ( ( U G A )  =  A  /\  ( A G U )  =  A )  /\  E. y  e.  X  (
( y G A )  =  U  /\  ( A G y )  =  U ) ) ) )
4544rspccva 3206 . 2  |-  ( ( A. x  e.  X  ( ( ( U G x )  =  x  /\  ( x G U )  =  x )  /\  E. y  e.  X  (
( y G x )  =  U  /\  ( x G y )  =  U ) )  /\  A  e.  X )  ->  (
( ( U G A )  =  A  /\  ( A G U )  =  A )  /\  E. y  e.  X  ( (
y G A )  =  U  /\  ( A G y )  =  U ) ) )
4631, 45sylan 469 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( ( U G A )  =  A  /\  ( A G U )  =  A )  /\  E. y  e.  X  ( (
y G A )  =  U  /\  ( A G y )  =  U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   E!wreu 2806   {crab 2808   ran crn 4989   ` cfv 5570   iota_crio 6231  (class class class)co 6270   GrpOpcgr 25386  GIdcgi 25387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-riota 6232  df-ov 6273  df-grpo 25391  df-gid 25392
This theorem is referenced by:  grpolid  25419  grporid  25420  grporcan  25421  grpoinveu  25422  grpoinv  25427
  Copyright terms: Public domain W3C validator