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Theorem grpo2inv 21815
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 21802 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
4 eqid 2435 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
51, 4, 2grporinv 21805 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X )  ->  (
( N `  A
) G ( N `
 ( N `  A ) ) )  =  (GId `  G
) )
63, 5syldan 457 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G ( N `
 ( N `  A ) ) )  =  (GId `  G
) )
71, 4, 2grpolinv 21804 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G A )  =  (GId `  G
) )
86, 7eqtr4d 2470 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G ( N `
 ( N `  A ) ) )  =  ( ( N `
 A ) G A ) )
91, 2grpoinvcl 21802 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X )  ->  ( N `  ( N `  A ) )  e.  X )
103, 9syldan 457 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  ( N `  A ) )  e.  X )
11 simpr 448 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  A  e.  X )
1210, 11, 33jca 1134 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  ( N `  A )
)  e.  X  /\  A  e.  X  /\  ( N `  A )  e.  X ) )
131grpolcan 21809 . . 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 457 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( ( N `  A ) G ( N `  ( N `
 A ) ) )  =  ( ( N `  A ) G A )  <->  ( N `  ( N `  A
) )  =  A ) )
158, 14mpbid 202 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  ( N `  A ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ran crn 4870   ` cfv 5445  (class class class)co 6072   GrpOpcgr 21762  GIdcgi 21763   invcgn 21764
This theorem is referenced by:  grpoinvf  21816  grpodivinv  21820  grpoinvdiv  21821  gxneg  21842  gxneg2  21843  gxinv2  21847  gxsuc  21848  gxmul  21854  nvnegneg  22120  ghomf1olem  25093
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-riota 6540  df-grpo 21767  df-gid 21768  df-ginv 21769
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