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Theorem grpo2inv 20898
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 20885 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
4 eqid 2284 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
51, 4, 2grporinv 20888 . . . 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 20887 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G A )  =  (GId `  G
) )
86, 7eqtr4d 2319 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G ( N `
 ( N `  A ) ) )  =  ( ( N `
 A ) G A ) )
91, 2grpoinvcl 20885 . . . . 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 1134 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  ( N `  A )
)  e.  X  /\  A  e.  X  /\  ( N `  A )  e.  X ) )
131grpolcan 20892 . . 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 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685   ran crn 4689   ` cfv 5221  (class class class)co 5819   GrpOpcgr 20845  GIdcgi 20846   invcgn 20847
This theorem is referenced by:  grpoinvf  20899  grpodivinv  20903  grpoinvdiv  20904  gxneg  20925  gxneg2  20926  gxinv2  20930  gxsuc  20931  gxmul  20937  nvnegneg  21201  ghomf1olem  23405  mult2inv  24823  vec2inv  24860
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-iota 6252  df-riota 6299  df-grpo 20850  df-gid 20851  df-ginv 20852
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