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Theorem grpoinv 20894
Description: The properties of a group element's inverse. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinv.1  |-  X  =  ran  G
grpinv.2  |-  U  =  (GId `  G )
grpinv.3  |-  N  =  ( inv `  G
)
Assertion
Ref Expression
grpoinv  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( ( N `  A ) G A )  =  U  /\  ( A G ( N `
 A ) )  =  U ) )

Proof of Theorem grpoinv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 grpinv.1 . . . . . 6  |-  X  =  ran  G
2 grpinv.2 . . . . . 6  |-  U  =  (GId `  G )
3 grpinv.3 . . . . . 6  |-  N  =  ( inv `  G
)
41, 2, 3grpoinvval 20892 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  =  ( iota_ y  e.  X ( y G A )  =  U ) )
51, 2grpoinveu 20889 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  E! y  e.  X  (
y G A )  =  U )
6 riotacl2 6318 . . . . . 6  |-  ( E! y  e.  X  ( y G A )  =  U  ->  ( iota_ y  e.  X ( y G A )  =  U )  e. 
{ y  e.  X  |  ( y G A )  =  U } )
75, 6syl 15 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( iota_ y  e.  X ( y G A )  =  U )  e. 
{ y  e.  X  |  ( y G A )  =  U } )
84, 7eqeltrd 2357 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  { y  e.  X  |  ( y G A )  =  U } )
9 simpl 443 . . . . . . . . 9  |-  ( ( ( y G A )  =  U  /\  ( A G y )  =  U )  -> 
( y G A )  =  U )
109rgenw 2610 . . . . . . . 8  |-  A. y  e.  X  ( (
( y G A )  =  U  /\  ( A G y )  =  U )  -> 
( y G A )  =  U )
1110a1i 10 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  A. y  e.  X  ( (
( y G A )  =  U  /\  ( A G y )  =  U )  -> 
( y G A )  =  U ) )
121, 2grpoidinv2 20885 . . . . . . . 8  |-  ( ( 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 ) ) )
1312simprd 449 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  E. y  e.  X  ( (
y G A )  =  U  /\  ( A G y )  =  U ) )
1411, 13, 53jca 1132 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A. y  e.  X  ( ( ( y G A )  =  U  /\  ( A G y )  =  U )  ->  (
y G A )  =  U )  /\  E. y  e.  X  ( ( y G A )  =  U  /\  ( A G y )  =  U )  /\  E! y  e.  X  ( y G A )  =  U ) )
15 reupick2 3454 . . . . . 6  |-  ( ( ( A. y  e.  X  ( ( ( y G A )  =  U  /\  ( A G y )  =  U )  ->  (
y G A )  =  U )  /\  E. y  e.  X  ( ( y G A )  =  U  /\  ( A G y )  =  U )  /\  E! y  e.  X  ( y G A )  =  U )  /\  y  e.  X
)  ->  ( (
y G A )  =  U  <->  ( (
y G A )  =  U  /\  ( A G y )  =  U ) ) )
1614, 15sylan 457 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  ( (
y G A )  =  U  <->  ( (
y G A )  =  U  /\  ( A G y )  =  U ) ) )
1716rabbidva 2779 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  { y  e.  X  |  ( y G A )  =  U }  =  { y  e.  X  |  ( ( y G A )  =  U  /\  ( A G y )  =  U ) } )
188, 17eleqtrd 2359 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  { y  e.  X  |  ( ( y G A )  =  U  /\  ( A G y )  =  U ) } )
19 oveq1 5865 . . . . . 6  |-  ( y  =  ( N `  A )  ->  (
y G A )  =  ( ( N `
 A ) G A ) )
2019eqeq1d 2291 . . . . 5  |-  ( y  =  ( N `  A )  ->  (
( y G A )  =  U  <->  ( ( N `  A ) G A )  =  U ) )
21 oveq2 5866 . . . . . 6  |-  ( y  =  ( N `  A )  ->  ( A G y )  =  ( A G ( N `  A ) ) )
2221eqeq1d 2291 . . . . 5  |-  ( y  =  ( N `  A )  ->  (
( A G y )  =  U  <->  ( A G ( N `  A ) )  =  U ) )
2320, 22anbi12d 691 . . . 4  |-  ( y  =  ( N `  A )  ->  (
( ( y G A )  =  U  /\  ( A G y )  =  U )  <->  ( ( ( N `  A ) G A )  =  U  /\  ( A G ( N `  A ) )  =  U ) ) )
2423elrab 2923 . . 3  |-  ( ( N `  A )  e.  { y  e.  X  |  ( ( y G A )  =  U  /\  ( A G y )  =  U ) }  <->  ( ( N `  A )  e.  X  /\  (
( ( N `  A ) G A )  =  U  /\  ( A G ( N `
 A ) )  =  U ) ) )
2518, 24sylib 188 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
)  e.  X  /\  ( ( ( N `
 A ) G A )  =  U  /\  ( A G ( N `  A
) )  =  U ) ) )
2625simprd 449 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( ( N `  A ) G A )  =  U  /\  ( A G ( N `
 A ) )  =  U ) )
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   E!wreu 2545   {crab 2547   ran crn 4690   ` cfv 5255  (class class class)co 5858   iota_crio 6297   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855
This theorem is referenced by:  grpolinv  20895  grporinv  20896
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-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-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-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860
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