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Theorem grpo2inv 21832
Description: Double inverse law for groups. Lemma 2.2.1(c) 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
grpo2inv  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  ( N `  A ) )  =  A )

Proof of Theorem grpo2inv
StepHypRef Expression
1 grpasscan1.1 . . . . 5  |-  X  =  ran  G
2 grpasscan1.2 . . . . 5  |-  N  =  ( inv `  G
)
31, 2grpoinvcl 21819 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
4 eqid 2438 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
51, 4, 2grporinv 21822 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X )  ->  (
( N `  A
) G ( N `
 ( N `  A ) ) )  =  (GId `  G
) )
63, 5syldan 458 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G ( N `
 ( N `  A ) ) )  =  (GId `  G
) )
71, 4, 2grpolinv 21821 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G A )  =  (GId `  G
) )
86, 7eqtr4d 2473 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G ( N `
 ( N `  A ) ) )  =  ( ( N `
 A ) G A ) )
91, 2grpoinvcl 21819 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X )  ->  ( N `  ( N `  A ) )  e.  X )
103, 9syldan 458 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  ( N `  A ) )  e.  X )
11 simpr 449 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  A  e.  X )
1210, 11, 33jca 1135 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  ( N `  A )
)  e.  X  /\  A  e.  X  /\  ( N `  A )  e.  X ) )
131grpolcan 21826 . . 3  |-  ( ( G  e.  GrpOp  /\  (
( N `  ( N `  A )
)  e.  X  /\  A  e.  X  /\  ( N `  A )  e.  X ) )  ->  ( ( ( N `  A ) G ( N `  ( N `  A ) ) )  =  ( ( N `  A
) G A )  <-> 
( N `  ( N `  A )
)  =  A ) )
1412, 13syldan 458 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( ( N `  A ) G ( N `  ( N `
 A ) ) )  =  ( ( N `  A ) G A )  <->  ( N `  ( N `  A
) )  =  A ) )
158, 14mpbid 203 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  ( N `  A ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ran crn 4882   ` cfv 5457  (class class class)co 6084   GrpOpcgr 21779  GIdcgi 21780   invcgn 21781
This theorem is referenced by:  grpoinvf  21833  grpodivinv  21837  grpoinvdiv  21838  gxneg  21859  gxneg2  21860  gxinv2  21864  gxsuc  21865  gxmul  21871  nvnegneg  22137  ghomf1olem  25110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4323  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-un 4704
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 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-riota 6552  df-grpo 21784  df-gid 21785  df-ginv 21786
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