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Theorem grpoinv 21815
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 21813 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  =  ( iota_ y  e.  X ( y G A )  =  U ) )
51, 2grpoinveu 21810 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  E! y  e.  X  (
y G A )  =  U )
6 riotacl2 6563 . . . . . 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 16 . . . . 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 2510 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  { y  e.  X  |  ( y G A )  =  U } )
9 simpl 444 . . . . . . . . 9  |-  ( ( ( y G A )  =  U  /\  ( A G y )  =  U )  -> 
( y G A )  =  U )
109rgenw 2773 . . . . . . . 8  |-  A. y  e.  X  ( (
( y G A )  =  U  /\  ( A G y )  =  U )  -> 
( y G A )  =  U )
1110a1i 11 . . . . . . 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 21806 . . . . . . . 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 450 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  E. y  e.  X  ( (
y G A )  =  U  /\  ( A G y )  =  U ) )
1411, 13, 53jca 1134 . . . . . 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 3627 . . . . . 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 458 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  y  e.  X
)  ->  ( (
y G A )  =  U  <->  ( (
y G A )  =  U  /\  ( A G y )  =  U ) ) )
1716rabbidva 2947 . . . 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 2512 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  { y  e.  X  |  ( ( y G A )  =  U  /\  ( A G y )  =  U ) } )
19 oveq1 6088 . . . . . 6  |-  ( y  =  ( N `  A )  ->  (
y G A )  =  ( ( N `
 A ) G A ) )
2019eqeq1d 2444 . . . . 5  |-  ( y  =  ( N `  A )  ->  (
( y G A )  =  U  <->  ( ( N `  A ) G A )  =  U ) )
21 oveq2 6089 . . . . . 6  |-  ( y  =  ( N `  A )  ->  ( A G y )  =  ( A G ( N `  A ) ) )
2221eqeq1d 2444 . . . . 5  |-  ( y  =  ( N `  A )  ->  (
( A G y )  =  U  <->  ( A G ( N `  A ) )  =  U ) )
2320, 22anbi12d 692 . . . 4  |-  ( y  =  ( N `  A )  ->  (
( ( y G A )  =  U  /\  ( A G y )  =  U )  <->  ( ( ( N `  A ) G A )  =  U  /\  ( A G ( N `  A ) )  =  U ) ) )
2423elrab 3092 . . 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 189 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
)  e.  X  /\  ( ( ( N `
 A ) G A )  =  U  /\  ( A G ( N `  A
) )  =  U ) ) )
2625simprd 450 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   E!wreu 2707   {crab 2709   ran crn 4879   ` cfv 5454  (class class class)co 6081   iota_crio 6542   GrpOpcgr 21774  GIdcgi 21775   invcgn 21776
This theorem is referenced by:  grpolinv  21816  grporinv  21817
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-riota 6549  df-grpo 21779  df-gid 21780  df-ginv 21781
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