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Theorem grpoinvcl 21845
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 2442 . . 3  |-  (GId `  G )  =  (GId
`  G )
3 grpinvcl.2 . . 3  |-  N  =  ( inv `  G
)
41, 2, 3grpoinvval 21844 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  =  ( iota_ y  e.  X ( y G A )  =  (GId
`  G ) ) )
51, 2grpoinveu 21841 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  E! y  e.  X  (
y G A )  =  (GId `  G
) )
6 riotacl 6593 . . 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 2516 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727   E!wreu 2713   ran crn 4908   ` cfv 5483  (class class class)co 6110   iota_crio 6571   GrpOpcgr 21805  GIdcgi 21806   invcgn 21807
This theorem is referenced by:  grpoinvid1  21849  grpoinvid2  21850  grpolcan  21852  grpo2grp  21853  grpoasscan1  21856  grpoasscan2  21857  grpo2inv  21858  grpoinvf  21859  grpoinvop  21860  grpodivinv  21863  grpoinvdiv  21864  grpodivf  21865  grpomuldivass  21868  grponpcan  21871  grpopnpcan2  21872  grponnncan2  21873  gxcl  21884  gxcom  21888  gxinv  21889  gxinv2  21890  gxsuc  21891  ablodivdiv4  21910  subgoinv  21927  ghgrp  21987  vcm  22081  ghomgrpilem2  25128  ghomf1olem  25136  rngonegcl  26599  isdrngo2  26612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-riota 6578  df-grpo 21810  df-gid 21811  df-ginv 21812
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