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Theorem grpoinvcl 24932
Description: A group element's inverse is a group element. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinvcl.1  |-  X  =  ran  G
grpinvcl.2  |-  N  =  ( inv `  G
)
Assertion
Ref Expression
grpoinvcl  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )

Proof of Theorem grpoinvcl
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 grpinvcl.1 . . 3  |-  X  =  ran  G
2 eqid 2467 . . 3  |-  (GId `  G )  =  (GId
`  G )
3 grpinvcl.2 . . 3  |-  N  =  ( inv `  G
)
41, 2, 3grpoinvval 24931 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  =  ( iota_ y  e.  X  ( y G A )  =  (GId
`  G ) ) )
51, 2grpoinveu 24928 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  E! y  e.  X  (
y G A )  =  (GId `  G
) )
6 riotacl 6260 . . 3  |-  ( E! y  e.  X  ( y G A )  =  (GId `  G
)  ->  ( iota_ y  e.  X  ( y G A )  =  (GId `  G )
)  e.  X )
75, 6syl 16 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( iota_ y  e.  X  ( y G A )  =  (GId `  G
) )  e.  X
)
84, 7eqeltrd 2555 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   E!wreu 2816   ran crn 5000   ` cfv 5588   iota_crio 6244  (class class class)co 6284   GrpOpcgr 24892  GIdcgi 24893   invcgn 24894
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 6576
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-grpo 24897  df-gid 24898  df-ginv 24899
This theorem is referenced by:  grpoinvid1  24936  grpoinvid2  24937  grpolcan  24939  grpo2grp  24940  grpoasscan1  24943  grpoasscan2  24944  grpo2inv  24945  grpoinvf  24946  grpoinvop  24947  grpodivinv  24950  grpoinvdiv  24951  grpodivf  24952  grpomuldivass  24955  grponpcan  24958  grpopnpcan2  24959  grponnncan2  24960  gxcl  24971  gxcom  24975  gxinv  24976  gxinv2  24977  gxsuc  24978  ablodivdiv4  24997  subgoinv  25014  ghgrp  25074  vcm  25168  ghomgrpilem2  28529  ghomf1olem  28537  rngonegcl  29979  isdrngo2  29992
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