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Theorem grpoinv 20910
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 20908 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  =  ( iota_ y  e.  X ( y G A )  =  U ) )
51, 2grpoinveu 20905 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  E! y  e.  X  (
y G A )  =  U )
6 riotacl2 6334 . . . . . 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 2370 . . . 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 2623 . . . . . . . 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 20901 . . . . . . . 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 3467 . . . . . 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 2792 . . . 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 2372 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  { y  e.  X  |  ( ( y G A )  =  U  /\  ( A G y )  =  U ) } )
19 oveq1 5881 . . . . . 6  |-  ( y  =  ( N `  A )  ->  (
y G A )  =  ( ( N `
 A ) G A ) )
2019eqeq1d 2304 . . . . 5  |-  ( y  =  ( N `  A )  ->  (
( y G A )  =  U  <->  ( ( N `  A ) G A )  =  U ) )
21 oveq2 5882 . . . . . 6  |-  ( y  =  ( N `  A )  ->  ( A G y )  =  ( A G ( N `  A ) ) )
2221eqeq1d 2304 . . . . 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 2936 . . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   E!wreu 2558   {crab 2560   ran crn 4706   ` cfv 5271  (class class class)co 5874   iota_crio 6313   GrpOpcgr 20869  GIdcgi 20870   invcgn 20871
This theorem is referenced by:  grpolinv  20911  grporinv  20912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-riota 6320  df-grpo 20874  df-gid 20875  df-ginv 20876
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