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Theorem grpo2inv 20906
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 20893 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
4 eqid 2283 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
51, 4, 2grporinv 20896 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X )  ->  (
( N `  A
) G ( N `
 ( N `  A ) ) )  =  (GId `  G
) )
63, 5syldan 456 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G ( N `
 ( N `  A ) ) )  =  (GId `  G
) )
71, 4, 2grpolinv 20895 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G A )  =  (GId `  G
) )
86, 7eqtr4d 2318 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G ( N `
 ( N `  A ) ) )  =  ( ( N `
 A ) G A ) )
91, 2grpoinvcl 20893 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X )  ->  ( N `  ( N `  A ) )  e.  X )
103, 9syldan 456 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  ( N `  A ) )  e.  X )
11 simpr 447 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  A  e.  X )
1210, 11, 33jca 1132 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  ( N `  A )
)  e.  X  /\  A  e.  X  /\  ( N `  A )  e.  X ) )
131grpolcan 20900 . . 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 456 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( ( N `  A ) G ( N `  ( N `
 A ) ) )  =  ( ( N `  A ) G A )  <->  ( N `  ( N `  A
) )  =  A ) )
158, 14mpbid 201 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  ( N `  A ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855
This theorem is referenced by:  grpoinvf  20907  grpodivinv  20911  grpoinvdiv  20912  gxneg  20933  gxneg2  20934  gxinv2  20938  gxsuc  20939  gxmul  20945  nvnegneg  21209  ghomf1olem  24001  mult2inv  25424  vec2inv  25461
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860
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