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Theorem grpoinvval 24903
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 24902 . . 3  |-  ( G  e.  GrpOp  ->  N  =  ( x  e.  X  |->  ( iota_ y  e.  X  ( y G x )  =  U ) ) )
54fveq1d 5866 . 2  |-  ( G  e.  GrpOp  ->  ( N `  A )  =  ( ( x  e.  X  |->  ( iota_ y  e.  X  ( y G x )  =  U ) ) `  A ) )
6 oveq2 6290 . . . . 5  |-  ( x  =  A  ->  (
y G x )  =  ( y G A ) )
76eqeq1d 2469 . . . 4  |-  ( x  =  A  ->  (
( y G x )  =  U  <->  ( y G A )  =  U ) )
87riotabidv 6245 . . 3  |-  ( x  =  A  ->  ( iota_ y  e.  X  ( y G x )  =  U )  =  ( iota_ y  e.  X  ( y G A )  =  U ) )
9 eqid 2467 . . 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 6247 . . 3  |-  ( iota_ y  e.  X  ( y G A )  =  U )  e.  _V
118, 9, 10fvmpt 5948 . 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 2528 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  =  ( iota_ y  e.  X  ( y G A )  =  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    |-> cmpt 4505   ran crn 5000   ` cfv 5586   iota_crio 6242  (class class class)co 6282   GrpOpcgr 24864  GIdcgi 24865   invcgn 24866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-ginv 24871
This theorem is referenced by:  grpoinvcl  24904  grpoinv  24905  addinv  25030
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