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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  |-  X  =  ran  G
grpasscan1.2  |-  N  =  ( inv `  G
)
Assertion
Ref Expression
grpoinvop  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G B ) )  =  ( ( N `  B ) G ( N `  A ) ) )

Proof of Theorem grpoinvop
StepHypRef Expression
1 simp1 996 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  G  e.  GrpOp )
2 simp2 997 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
3 simp3 998 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  B  e.  X )
4 grpasscan1.1 . . . . . . 7  |-  X  =  ran  G
5 grpasscan1.2 . . . . . . 7  |-  N  =  ( inv `  G
)
64, 5grpoinvcl 25019 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( N `  B )  e.  X )
763adant2 1015 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  B )  e.  X )
84, 5grpoinvcl 25019 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
983adant3 1016 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  A )  e.  X )
104grpocl 24993 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( N `  B )  e.  X  /\  ( N `  A )  e.  X )  ->  (
( N `  B
) G ( N `
 A ) )  e.  X )
111, 7, 9, 10syl3anc 1228 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  B
) G ( N `
 A ) )  e.  X )
124grpoass 24996 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  ( ( N `  B ) G ( N `  A ) )  e.  X ) )  ->  ( ( A G B ) G ( ( N `  B ) G ( N `  A ) ) )  =  ( A G ( B G ( ( N `
 B ) G ( N `  A
) ) ) ) )
131, 2, 3, 11, 12syl13anc 1230 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G B ) G ( ( N `  B ) G ( N `  A ) ) )  =  ( A G ( B G ( ( N `  B
) G ( N `
 A ) ) ) ) )
14 eqid 2467 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
154, 14, 5grporinv 25022 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( B G ( N `  B ) )  =  (GId `  G )
)
16153adant2 1015 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( B G ( N `  B ) )  =  (GId `  G )
)
1716oveq1d 6309 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( B G ( N `  B ) ) G ( N `
 A ) )  =  ( (GId `  G ) G ( N `  A ) ) )
184grpoass 24996 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  ( N `  B )  e.  X  /\  ( N `  A )  e.  X ) )  -> 
( ( B G ( N `  B
) ) G ( N `  A ) )  =  ( B G ( ( N `
 B ) G ( N `  A
) ) ) )
191, 3, 7, 9, 18syl13anc 1230 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( B G ( N `  B ) ) G ( N `
 A ) )  =  ( B G ( ( N `  B ) G ( N `  A ) ) ) )
204, 14grpolid 25012 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X )  ->  (
(GId `  G ) G ( N `  A ) )  =  ( N `  A
) )
218, 20syldan 470 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
(GId `  G ) G ( N `  A ) )  =  ( N `  A
) )
22213adant3 1016 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
(GId `  G ) G ( N `  A ) )  =  ( N `  A
) )
2317, 19, 223eqtr3d 2516 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( B G ( ( N `
 B ) G ( N `  A
) ) )  =  ( N `  A
) )
2423oveq2d 6310 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( B G ( ( N `  B ) G ( N `  A ) ) ) )  =  ( A G ( N `  A ) ) )
254, 14, 5grporinv 25022 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( N `  A ) )  =  (GId `  G )
)
26253adant3 1016 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( N `  A ) )  =  (GId `  G )
)
2713, 24, 263eqtrd 2512 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G B ) G ( ( N `  B ) G ( N `  A ) ) )  =  (GId `  G
) )
284grpocl 24993 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
294, 14, 5grpoinvid1 25023 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A G B )  e.  X  /\  ( ( N `  B ) G ( N `  A ) )  e.  X )  ->  (
( N `  ( A G B ) )  =  ( ( N `
 B ) G ( N `  A
) )  <->  ( ( A G B ) G ( ( N `  B ) G ( N `  A ) ) )  =  (GId
`  G ) ) )
301, 28, 11, 29syl3anc 1228 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  ( A G B ) )  =  ( ( N `
 B ) G ( N `  A
) )  <->  ( ( A G B ) G ( ( N `  B ) G ( N `  A ) ) )  =  (GId
`  G ) ) )
3127, 30mpbird 232 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G B ) )  =  ( ( N `  B ) G ( N `  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1379    e. wcel 1767   ran crn 5005   ` cfv 5593  (class class class)co 6294   GrpOpcgr 24979  GIdcgi 24980   invcgn 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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