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Theorem grpoinvop 25034
 Description: The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpasscan1.1
grpasscan1.2
Assertion
Ref Expression
grpoinvop

Proof of Theorem grpoinvop
StepHypRef Expression
1 simp1 996 . . . 4
2 simp2 997 . . . 4
3 simp3 998 . . . 4
4 grpasscan1.1 . . . . . . 7
5 grpasscan1.2 . . . . . . 7
64, 5grpoinvcl 25019 . . . . . 6
763adant2 1015 . . . . 5
84, 5grpoinvcl 25019 . . . . . 6
983adant3 1016 . . . . 5
104grpocl 24993 . . . . 5
111, 7, 9, 10syl3anc 1228 . . . 4
124grpoass 24996 . . . 4
131, 2, 3, 11, 12syl13anc 1230 . . 3
14 eqid 2467 . . . . . . . 8 GId GId
154, 14, 5grporinv 25022 . . . . . . 7 GId
16153adant2 1015 . . . . . 6 GId
1716oveq1d 6309 . . . . 5 GId
184grpoass 24996 . . . . . 6
191, 3, 7, 9, 18syl13anc 1230 . . . . 5
204, 14grpolid 25012 . . . . . . 7 GId
218, 20syldan 470 . . . . . 6 GId
22213adant3 1016 . . . . 5 GId
2317, 19, 223eqtr3d 2516 . . . 4
2423oveq2d 6310 . . 3
254, 14, 5grporinv 25022 . . . 4 GId
26253adant3 1016 . . 3 GId
2713, 24, 263eqtrd 2512 . 2 GId
284grpocl 24993 . . 3
294, 14, 5grpoinvid1 25023 . . 3 GId
301, 28, 11, 29syl3anc 1228 . 2 GId
3127, 30mpbird 232 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   w3a 973   wceq 1379   wcel 1767   crn 5005  cfv 5593  (class class class)co 6294  cgr 24979  GIdcgi 24980  cgn 24981 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 4563  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-un 6586 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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-grpo 24984  df-gid 24985  df-ginv 24986 This theorem is referenced by:  grpoinvdiv  25038  grpopnpcan2  25046  gxcom  25062  gxinv  25063  gxsuc  25065  gxdi  25089
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