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Theorem grpoinvval 20892
Description: The inverse of a group element. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinvfval.1  |-  X  =  ran  G
grpinvfval.2  |-  U  =  (GId `  G )
grpinvfval.3  |-  N  =  ( inv `  G
)
Assertion
Ref Expression
grpoinvval  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  =  ( iota_ y  e.  X ( y G A )  =  U ) )
Distinct variable groups:    y, A    y, G    y, X
Allowed substitution hints:    U( y)    N( y)

Proof of Theorem grpoinvval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 grpinvfval.1 . . . 4  |-  X  =  ran  G
2 grpinvfval.2 . . . 4  |-  U  =  (GId `  G )
3 grpinvfval.3 . . . 4  |-  N  =  ( inv `  G
)
41, 2, 3grpoinvfval 20891 . . 3  |-  ( G  e.  GrpOp  ->  N  =  ( x  e.  X  |->  ( iota_ y  e.  X
( y G x )  =  U ) ) )
54fveq1d 5527 . 2  |-  ( G  e.  GrpOp  ->  ( N `  A )  =  ( ( x  e.  X  |->  ( iota_ y  e.  X
( y G x )  =  U ) ) `  A ) )
6 oveq2 5866 . . . . 5  |-  ( x  =  A  ->  (
y G x )  =  ( y G A ) )
76eqeq1d 2291 . . . 4  |-  ( x  =  A  ->  (
( y G x )  =  U  <->  ( y G A )  =  U ) )
87riotabidv 6306 . . 3  |-  ( x  =  A  ->  ( iota_ y  e.  X ( y G x )  =  U )  =  ( iota_ y  e.  X
( y G A )  =  U ) )
9 eqid 2283 . . 3  |-  ( x  e.  X  |->  ( iota_ y  e.  X ( y G x )  =  U ) )  =  ( x  e.  X  |->  ( iota_ y  e.  X
( y G x )  =  U ) )
10 riotaex 6308 . . 3  |-  ( iota_ y  e.  X ( y G A )  =  U )  e.  _V
118, 9, 10fvmpt 5602 . 2  |-  ( A  e.  X  ->  (
( x  e.  X  |->  ( iota_ y  e.  X
( y G x )  =  U ) ) `  A )  =  ( iota_ y  e.  X ( y G A )  =  U ) )
125, 11sylan9eq 2335 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  =  ( iota_ y  e.  X ( y G A )  =  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858   iota_crio 6297   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855
This theorem is referenced by:  grpoinvcl  20893  grpoinv  20894  addinv  21019
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-ginv 20860
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