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Theorem grpoinvcl 21775
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 2412 . . 3  |-  (GId `  G )  =  (GId
`  G )
3 grpinvcl.2 . . 3  |-  N  =  ( inv `  G
)
41, 2, 3grpoinvval 21774 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  =  ( iota_ y  e.  X ( y G A )  =  (GId
`  G ) ) )
51, 2grpoinveu 21771 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  E! y  e.  X  (
y G A )  =  (GId `  G
) )
6 riotacl 6531 . . 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 2486 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   E!wreu 2676   ran crn 4846   ` cfv 5421  (class class class)co 6048   iota_crio 6509   GrpOpcgr 21735  GIdcgi 21736   invcgn 21737
This theorem is referenced by:  grpoinvid1  21779  grpoinvid2  21780  grpolcan  21782  grpo2grp  21783  grpoasscan1  21786  grpoasscan2  21787  grpo2inv  21788  grpoinvf  21789  grpoinvop  21790  grpodivinv  21793  grpoinvdiv  21794  grpodivf  21795  grpomuldivass  21798  grponpcan  21801  grpopnpcan2  21802  grponnncan2  21803  gxcl  21814  gxcom  21818  gxinv  21819  gxinv2  21820  gxsuc  21821  ablodivdiv4  21840  subgoinv  21857  ghgrp  21917  vcm  22011  ghomgrpilem2  25058  ghomf1olem  25066  rngonegcl  26459  isdrngo2  26472
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-riota 6516  df-grpo 21740  df-gid 21741  df-ginv 21742
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