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Theorem grpinvinv 14497
Description: Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 31-Mar-2014.)
Hypotheses
Ref Expression
grpinvinv.b  |-  B  =  ( Base `  G
)
grpinvinv.n  |-  N  =  ( inv g `  G )
Assertion
Ref Expression
grpinvinv  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  ( N `  X )
)  =  X )

Proof of Theorem grpinvinv
StepHypRef Expression
1 grpinvinv.b . . . . 5  |-  B  =  ( Base `  G
)
2 grpinvinv.n . . . . 5  |-  N  =  ( inv g `  G )
31, 2grpinvcl 14489 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
4 eqid 2258 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
5 eqid 2258 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
61, 4, 5, 2grprinv 14491 . . . 4  |-  ( ( G  e.  Grp  /\  ( N `  X )  e.  B )  -> 
( ( N `  X ) ( +g  `  G ) ( N `
 ( N `  X ) ) )  =  ( 0g `  G ) )
73, 6syldan 458 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X ) ( +g  `  G ) ( N `
 ( N `  X ) ) )  =  ( 0g `  G ) )
81, 4, 5, 2grplinv 14490 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X ) ( +g  `  G ) X )  =  ( 0g `  G ) )
97, 8eqtr4d 2293 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X ) ( +g  `  G ) ( N `
 ( N `  X ) ) )  =  ( ( N `
 X ) ( +g  `  G ) X ) )
10 simpl 445 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  G  e.  Grp )
111, 2grpinvcl 14489 . . . 4  |-  ( ( G  e.  Grp  /\  ( N `  X )  e.  B )  -> 
( N `  ( N `  X )
)  e.  B )
123, 11syldan 458 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  ( N `  X )
)  e.  B )
13 simpr 449 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  X  e.  B )
141, 4grplcan 14496 . . 3  |-  ( ( G  e.  Grp  /\  ( ( N `  ( N `  X ) )  e.  B  /\  X  e.  B  /\  ( N `  X )  e.  B ) )  ->  ( ( ( N `  X ) ( +g  `  G
) ( N `  ( N `  X ) ) )  =  ( ( N `  X
) ( +g  `  G
) X )  <->  ( N `  ( N `  X
) )  =  X ) )
1510, 12, 13, 3, 14syl13anc 1189 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( ( N `
 X ) ( +g  `  G ) ( N `  ( N `  X )
) )  =  ( ( N `  X
) ( +g  `  G
) X )  <->  ( N `  ( N `  X
) )  =  X ) )
169, 15mpbid 203 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  ( N `  X )
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   ` cfv 4673  (class class class)co 5792   Basecbs 13110   +g cplusg 13170   0gc0g 13362   Grpcgrp 14324   inv gcminusg 14325
This theorem is referenced by:  grpinv11  14499  grpinvnz  14501  grpsubinv  14503  grpinvsub  14510  grpsubeq0  14514  grpnpcan  14519  mulgneg  14547  mulgdir  14554  mulgass  14559  eqger  14629  frgpuptinv  15042  ablsub2inv  15074  mulgdi  15088  invghm  15092  rngm2neg  15346  unitinvinv  15419  unitnegcl  15425  irrednegb  15455  abvneg  15561  lspsnneg  15725  tgpconcomp  17757  islindf4  26675  baerlem5amN  31073  baerlem5bmN  31074  baerlem5abmN  31075
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 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-iota 6225  df-riota 6272  df-0g 13366  df-mnd 14329  df-grp 14451  df-minusg 14452
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