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Theorem grpoinvval 21663
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 21662 . . 3  |-  ( G  e.  GrpOp  ->  N  =  ( x  e.  X  |->  ( iota_ y  e.  X
( y G x )  =  U ) ) )
54fveq1d 5672 . 2  |-  ( G  e.  GrpOp  ->  ( N `  A )  =  ( ( x  e.  X  |->  ( iota_ y  e.  X
( y G x )  =  U ) ) `  A ) )
6 oveq2 6030 . . . . 5  |-  ( x  =  A  ->  (
y G x )  =  ( y G A ) )
76eqeq1d 2397 . . . 4  |-  ( x  =  A  ->  (
( y G x )  =  U  <->  ( y G A )  =  U ) )
87riotabidv 6489 . . 3  |-  ( x  =  A  ->  ( iota_ y  e.  X ( y G x )  =  U )  =  ( iota_ y  e.  X
( y G A )  =  U ) )
9 eqid 2389 . . 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 6491 . . 3  |-  ( iota_ y  e.  X ( y G A )  =  U )  e.  _V
118, 9, 10fvmpt 5747 . 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 2441 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 359    = wceq 1649    e. wcel 1717    e. cmpt 4209   ran crn 4821   ` cfv 5396  (class class class)co 6022   iota_crio 6480   GrpOpcgr 21624  GIdcgi 21625   invcgn 21626
This theorem is referenced by:  grpoinvcl  21664  grpoinv  21665  addinv  21790
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-riota 6487  df-ginv 21631
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