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Theorem grpoidinv2 21798
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 21796 . . . . . 6  |-  ( G  e.  GrpOp  ->  U  =  ( iota_ u  e.  X A. x  e.  X  ( u G x )  =  x ) )
41grpoideu 21789 . . . . . . 7  |-  ( G  e.  GrpOp  ->  E! u  e.  X  A. x  e.  X  ( u G x )  =  x )
5 riotacl2 6555 . . . . . . 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 2509 . . . . 5  |-  ( G  e.  GrpOp  ->  U  e.  { u  e.  X  |  A. x  e.  X  ( u G x )  =  x }
)
8 simpll 731 . . . . . . . . . . 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 2773 . . . . . . . . . 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 2765 . . . . . . . . 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 21788 . . . . . . . 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 1134 . . . . . . 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 3619 . . . . . . 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 458 . . . . . 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 2939 . . . . 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 2511 . . . 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 6080 . . . . . . . . 9  |-  ( u  =  U  ->  (
u G x )  =  ( U G x ) )
1918eqeq1d 2443 . . . . . . . 8  |-  ( u  =  U  ->  (
( u G x )  =  x  <->  ( U G x )  =  x ) )
20 oveq2 6081 . . . . . . . . 9  |-  ( u  =  U  ->  (
x G u )  =  ( x G U ) )
2120eqeq1d 2443 . . . . . . . 8  |-  ( u  =  U  ->  (
( x G u )  =  x  <->  ( x G U )  =  x ) )
2219, 21anbi12d 692 . . . . . . 7  |-  ( u  =  U  ->  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  <->  ( ( U G x )  =  x  /\  ( x G U )  =  x ) ) )
23 eqeq2 2444 . . . . . . . . 9  |-  ( u  =  U  ->  (
( y G x )  =  u  <->  ( y G x )  =  U ) )
24 eqeq2 2444 . . . . . . . . 9  |-  ( u  =  U  ->  (
( x G y )  =  u  <->  ( x G y )  =  U ) )
2523, 24anbi12d 692 . . . . . . . 8  |-  ( u  =  U  ->  (
( ( y G x )  =  u  /\  ( x G y )  =  u )  <->  ( ( y G x )  =  U  /\  ( x G y )  =  U ) ) )
2625rexbidv 2718 . . . . . . 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 692 . . . . . 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 2717 . . . . 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 3084 . . . 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 189 . . 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 450 . 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 6081 . . . . . 6  |-  ( x  =  A  ->  ( U G x )  =  ( U G A ) )
33 id 20 . . . . . 6  |-  ( x  =  A  ->  x  =  A )
3432, 33eqeq12d 2449 . . . . 5  |-  ( x  =  A  ->  (
( U G x )  =  x  <->  ( U G A )  =  A ) )
35 oveq1 6080 . . . . . 6  |-  ( x  =  A  ->  (
x G U )  =  ( A G U ) )
3635, 33eqeq12d 2449 . . . . 5  |-  ( x  =  A  ->  (
( x G U )  =  x  <->  ( A G U )  =  A ) )
3734, 36anbi12d 692 . . . 4  |-  ( x  =  A  ->  (
( ( U G x )  =  x  /\  ( x G U )  =  x )  <->  ( ( U G A )  =  A  /\  ( A G U )  =  A ) ) )
38 oveq2 6081 . . . . . . 7  |-  ( x  =  A  ->  (
y G x )  =  ( y G A ) )
3938eqeq1d 2443 . . . . . 6  |-  ( x  =  A  ->  (
( y G x )  =  U  <->  ( y G A )  =  U ) )
40 oveq1 6080 . . . . . . 7  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
4140eqeq1d 2443 . . . . . 6  |-  ( x  =  A  ->  (
( x G y )  =  U  <->  ( A G y )  =  U ) )
4239, 41anbi12d 692 . . . . 5  |-  ( x  =  A  ->  (
( ( y G x )  =  U  /\  ( x G y )  =  U )  <->  ( ( y G A )  =  U  /\  ( A G y )  =  U ) ) )
4342rexbidv 2718 . . . 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 692 . . 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 3043 . 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 458 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   E!wreu 2699   {crab 2701   ran crn 4871   ` cfv 5446  (class class class)co 6073   iota_crio 6534   GrpOpcgr 21766  GIdcgi 21767
This theorem is referenced by:  grpolid  21799  grporid  21800  grporcan  21801  grpoinveu  21802  grpoinv  21807
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-riota 6541  df-grpo 21771  df-gid 21772
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