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Theorem grpoinvop 21786
 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 957 . . . 4
2 simp2 958 . . . 4
3 simp3 959 . . . 4
4 grpasscan1.1 . . . . . . 7
5 grpasscan1.2 . . . . . . 7
64, 5grpoinvcl 21771 . . . . . 6
763adant2 976 . . . . 5
84, 5grpoinvcl 21771 . . . . . 6
983adant3 977 . . . . 5
104grpocl 21745 . . . . 5
111, 7, 9, 10syl3anc 1184 . . . 4
124grpoass 21748 . . . 4
131, 2, 3, 11, 12syl13anc 1186 . . 3
14 eqid 2408 . . . . . . . 8 GId GId
154, 14, 5grporinv 21774 . . . . . . 7 GId
16153adant2 976 . . . . . 6 GId
1716oveq1d 6059 . . . . 5 GId
184grpoass 21748 . . . . . 6
191, 3, 7, 9, 18syl13anc 1186 . . . . 5
204, 14grpolid 21764 . . . . . . 7 GId
218, 20syldan 457 . . . . . 6 GId
22213adant3 977 . . . . 5 GId
2317, 19, 223eqtr3d 2448 . . . 4
2423oveq2d 6060 . . 3
254, 14, 5grporinv 21774 . . . 4 GId
26253adant3 977 . . 3 GId
2713, 24, 263eqtrd 2444 . 2 GId
284grpocl 21745 . . 3
294, 14, 5grpoinvid1 21775 . . 3 GId
301, 28, 11, 29syl3anc 1184 . 2 GId
3127, 30mpbird 224 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   w3a 936   wceq 1649   wcel 1721   crn 4842  cfv 5417  (class class class)co 6044  cgr 21731  GIdcgi 21732  cgn 21733 This theorem is referenced by:  grpoinvdiv  21790  grpopnpcan2  21798  gxcom  21814  gxinv  21815  gxsuc  21817  gxdi  21841 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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-un 4664 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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-riota 6512  df-grpo 21736  df-gid 21737  df-ginv 21738
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