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Theorem grpo2inv 20736
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 20723 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
4 eqid 2253 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
51, 4, 2grporinv 20726 . . . 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 20725 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G A )  =  (GId `  G
) )
86, 7eqtr4d 2288 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G ( N `
 ( N `  A ) ) )  =  ( ( N `
 A ) G A ) )
91, 2grpoinvcl 20723 . . . . 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 1137 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  ( N `  A )
)  e.  X  /\  A  e.  X  /\  ( N `  A )  e.  X ) )
131grpolcan 20730 . . 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 939    = wceq 1619    e. wcel 1621   ran crn 4581   ` cfv 4592  (class class class)co 5710   GrpOpcgr 20683  GIdcgi 20684   invcgn 20685
This theorem is referenced by:  grpoinvf  20737  grpodivinv  20741  grpoinvdiv  20742  gxneg  20763  gxneg2  20764  gxinv2  20768  gxsuc  20769  gxmul  20775  nvnegneg  21039  ghomf1olem  23172  mult2inv  24590  vec2inv  24627
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-iota 6143  df-riota 6190  df-grpo 20688  df-gid 20689  df-ginv 20690
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