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Theorem grpoinvcl 25799
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 2429 . . 3  |-  (GId `  G )  =  (GId
`  G )
3 grpinvcl.2 . . 3  |-  N  =  ( inv `  G
)
41, 2, 3grpoinvval 25798 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  =  ( iota_ y  e.  X  ( y G A )  =  (GId
`  G ) ) )
51, 2grpoinveu 25795 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  E! y  e.  X  (
y G A )  =  (GId `  G
) )
6 riotacl 6281 . . 3  |-  ( E! y  e.  X  ( y G A )  =  (GId `  G
)  ->  ( iota_ y  e.  X  ( y G A )  =  (GId `  G )
)  e.  X )
75, 6syl 17 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( iota_ y  e.  X  ( y G A )  =  (GId `  G
) )  e.  X
)
84, 7eqeltrd 2517 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   E!wreu 2784   ran crn 4855   ` cfv 5601   iota_crio 6266  (class class class)co 6305   GrpOpcgr 25759  GIdcgi 25760   invcgn 25761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-grpo 25764  df-gid 25765  df-ginv 25766
This theorem is referenced by:  grpoinvid1  25803  grpoinvid2  25804  grpolcan  25806  grpo2grp  25807  grpoasscan1  25810  grpoasscan2  25811  grpo2inv  25812  grpoinvf  25813  grpoinvop  25814  grpodivinv  25817  grpoinvdiv  25818  grpodivf  25819  grpomuldivass  25822  grponpcan  25825  grpopnpcan2  25826  grponnncan2  25827  gxcl  25838  gxcom  25842  gxinv  25843  gxinv2  25844  gxsuc  25845  ablodivdiv4  25864  subgoinv  25881  ghgrpOLD  25941  vcm  26035  ghomgrpilem2  30092  ghomf1olem  30100  rngonegcl  31888  isdrngo2  31901
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