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Theorem grpoinvop 20910
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 955 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  G  e.  GrpOp )
2 simp2 956 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
3 simp3 957 . . . 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 20895 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( N `  B )  e.  X )
763adant2 974 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  B )  e.  X )
84, 5grpoinvcl 20895 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
983adant3 975 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  A )  e.  X )
104grpocl 20869 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( N `  B )  e.  X  /\  ( N `  A )  e.  X )  ->  (
( N `  B
) G ( N `
 A ) )  e.  X )
111, 7, 9, 10syl3anc 1182 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  B
) G ( N `
 A ) )  e.  X )
124grpoass 20872 . . . 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 1184 . . 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 2285 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
154, 14, 5grporinv 20898 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( B G ( N `  B ) )  =  (GId `  G )
)
16153adant2 974 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( B G ( N `  B ) )  =  (GId `  G )
)
1716oveq1d 5875 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( B G ( N `  B ) ) G ( N `
 A ) )  =  ( (GId `  G ) G ( N `  A ) ) )
184grpoass 20872 . . . . . 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 1184 . . . . 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 20888 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X )  ->  (
(GId `  G ) G ( N `  A ) )  =  ( N `  A
) )
218, 20syldan 456 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
(GId `  G ) G ( N `  A ) )  =  ( N `  A
) )
22213adant3 975 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
(GId `  G ) G ( N `  A ) )  =  ( N `  A
) )
2317, 19, 223eqtr3d 2325 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( B G ( ( N `
 B ) G ( N `  A
) ) )  =  ( N `  A
) )
2423oveq2d 5876 . . 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 20898 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( N `  A ) )  =  (GId `  G )
)
26253adant3 975 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( N `  A ) )  =  (GId `  G )
)
2713, 24, 263eqtrd 2321 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G B ) G ( ( N `  B ) G ( N `  A ) ) )  =  (GId `  G
) )
284grpocl 20869 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
294, 14, 5grpoinvid1 20899 . . 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 1182 . 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 223 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 176    /\ w3a 934    = wceq 1625    e. wcel 1686   ran crn 4692   ` cfv 5257  (class class class)co 5860   GrpOpcgr 20855  GIdcgi 20856   invcgn 20857
This theorem is referenced by:  grpoinvdiv  20914  grpopnpcan2  20922  gxcom  20938  gxinv  20939  gxsuc  20941  gxdi  20965  abloinvop  25364  fprodneg  25389  invaddvec  25478
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-riota 6306  df-grpo 20860  df-gid 20861  df-ginv 20862
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