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Theorem grpoinvcl 21325
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 2366 . . 3  |-  (GId `  G )  =  (GId
`  G )
3 grpinvcl.2 . . 3  |-  N  =  ( inv `  G
)
41, 2, 3grpoinvval 21324 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  =  ( iota_ y  e.  X ( y G A )  =  (GId
`  G ) ) )
51, 2grpoinveu 21321 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  E! y  e.  X  (
y G A )  =  (GId `  G
) )
6 riotacl 6461 . . 3  |-  ( E! y  e.  X  ( y G A )  =  (GId `  G
)  ->  ( iota_ y  e.  X ( y G A )  =  (GId `  G )
)  e.  X )
75, 6syl 15 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( iota_ y  e.  X ( y G A )  =  (GId `  G
) )  e.  X
)
84, 7eqeltrd 2440 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   E!wreu 2630   ran crn 4793   ` cfv 5358  (class class class)co 5981   iota_crio 6439   GrpOpcgr 21285  GIdcgi 21286   invcgn 21287
This theorem is referenced by:  grpoinvid1  21329  grpoinvid2  21330  grpolcan  21332  grpo2grp  21333  grpoasscan1  21336  grpoasscan2  21337  grpo2inv  21338  grpoinvf  21339  grpoinvop  21340  grpodivinv  21343  grpoinvdiv  21344  grpodivf  21345  grpomuldivass  21348  grponpcan  21351  grpopnpcan2  21352  grponnncan2  21353  gxcl  21364  gxcom  21368  gxinv  21369  gxinv2  21370  gxsuc  21371  ablodivdiv4  21390  subgoinv  21407  ghgrp  21467  vcm  21561  ghomgrpilem2  24665  ghomf1olem  24673  rngonegcl  26167  isdrngo2  26180
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-riota 6446  df-grpo 21290  df-gid 21291  df-ginv 21292
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