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Theorem grpoinvop 21831
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 958 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  G  e.  GrpOp )
2 simp2 959 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
3 simp3 960 . . . 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 21816 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( N `  B )  e.  X )
763adant2 977 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  B )  e.  X )
84, 5grpoinvcl 21816 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
983adant3 978 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  A )  e.  X )
104grpocl 21790 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( N `  B )  e.  X  /\  ( N `  A )  e.  X )  ->  (
( N `  B
) G ( N `
 A ) )  e.  X )
111, 7, 9, 10syl3anc 1185 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  B
) G ( N `
 A ) )  e.  X )
124grpoass 21793 . . . 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 1187 . . 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 2438 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
154, 14, 5grporinv 21819 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( B G ( N `  B ) )  =  (GId `  G )
)
16153adant2 977 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( B G ( N `  B ) )  =  (GId `  G )
)
1716oveq1d 6098 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( B G ( N `  B ) ) G ( N `
 A ) )  =  ( (GId `  G ) G ( N `  A ) ) )
184grpoass 21793 . . . . . 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 1187 . . . . 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 21809 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X )  ->  (
(GId `  G ) G ( N `  A ) )  =  ( N `  A
) )
218, 20syldan 458 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
(GId `  G ) G ( N `  A ) )  =  ( N `  A
) )
22213adant3 978 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
(GId `  G ) G ( N `  A ) )  =  ( N `  A
) )
2317, 19, 223eqtr3d 2478 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( B G ( ( N `
 B ) G ( N `  A
) ) )  =  ( N `  A
) )
2423oveq2d 6099 . . 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 21819 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( N `  A ) )  =  (GId `  G )
)
26253adant3 978 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( N `  A ) )  =  (GId `  G )
)
2713, 24, 263eqtrd 2474 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G B ) G ( ( N `  B ) G ( N `  A ) ) )  =  (GId `  G
) )
284grpocl 21790 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
294, 14, 5grpoinvid1 21820 . . 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 1185 . 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 225 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G B ) )  =  ( ( N `  B ) G ( N `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937    = wceq 1653    e. wcel 1726   ran crn 4881   ` cfv 5456  (class class class)co 6083   GrpOpcgr 21776  GIdcgi 21777   invcgn 21778
This theorem is referenced by:  grpoinvdiv  21835  grpopnpcan2  21843  gxcom  21859  gxinv  21860  gxsuc  21862  gxdi  21886
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-riota 6551  df-grpo 21781  df-gid 21782  df-ginv 21783
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