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Theorem nmznsg 14947
Description: Any subgroup is a normal subgroup of its normalizer. (Contributed by Mario Carneiro, 19-Jan-2015.)
Hypotheses
Ref Expression
elnmz.1  |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S ) }
nmzsubg.2  |-  X  =  ( Base `  G
)
nmzsubg.3  |-  .+  =  ( +g  `  G )
nmznsg.4  |-  H  =  ( Gs  N )
Assertion
Ref Expression
nmznsg  |-  ( S  e.  (SubGrp `  G
)  ->  S  e.  (NrmSGrp `  H ) )
Distinct variable groups:    x, y, G    x, S, y    x,  .+ , y    x, X, y
Allowed substitution hints:    H( x, y)    N( x, y)

Proof of Theorem nmznsg
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 20 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
2 elnmz.1 . . . 4  |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S ) }
3 nmzsubg.2 . . . 4  |-  X  =  ( Base `  G
)
4 nmzsubg.3 . . . 4  |-  .+  =  ( +g  `  G )
52, 3, 4ssnmz 14945 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  N
)
6 subgrcl 14912 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
72, 3, 4nmzsubg 14944 . . . . 5  |-  ( G  e.  Grp  ->  N  e.  (SubGrp `  G )
)
86, 7syl 16 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  N  e.  (SubGrp `  G ) )
9 nmznsg.4 . . . . 5  |-  H  =  ( Gs  N )
109subsubg 14926 . . . 4  |-  ( N  e.  (SubGrp `  G
)  ->  ( S  e.  (SubGrp `  H )  <->  ( S  e.  (SubGrp `  G )  /\  S  C_  N ) ) )
118, 10syl 16 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  ( S  e.  (SubGrp `  H )  <->  ( S  e.  (SubGrp `  G )  /\  S  C_  N ) ) )
121, 5, 11mpbir2and 889 . 2  |-  ( S  e.  (SubGrp `  G
)  ->  S  e.  (SubGrp `  H ) )
13 ssrab2 3396 . . . . . . 7  |-  { x  e.  X  |  A. y  e.  X  (
( x  .+  y
)  e.  S  <->  ( y  .+  x )  e.  S
) }  C_  X
142, 13eqsstri 3346 . . . . . 6  |-  N  C_  X
1514sseli 3312 . . . . 5  |-  ( w  e.  N  ->  w  e.  X )
162nmzbi 14943 . . . . 5  |-  ( ( z  e.  N  /\  w  e.  X )  ->  ( ( z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S ) )
1715, 16sylan2 461 . . . 4  |-  ( ( z  e.  N  /\  w  e.  N )  ->  ( ( z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S ) )
1817rgen2a 2740 . . 3  |-  A. z  e.  N  A. w  e.  N  ( (
z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S
)
199subgbas 14911 . . . . 5  |-  ( N  e.  (SubGrp `  G
)  ->  N  =  ( Base `  H )
)
208, 19syl 16 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  N  =  ( Base `  H )
)
2120raleqdv 2878 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  ( A. w  e.  N  (
( z  .+  w
)  e.  S  <->  ( w  .+  z )  e.  S
)  <->  A. w  e.  (
Base `  H )
( ( z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S ) ) )
2220, 21raleqbidv 2884 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  ( A. z  e.  N  A. w  e.  N  (
( z  .+  w
)  e.  S  <->  ( w  .+  z )  e.  S
)  <->  A. z  e.  (
Base `  H ) A. w  e.  ( Base `  H ) ( ( z  .+  w
)  e.  S  <->  ( w  .+  z )  e.  S
) ) )
2318, 22mpbii 203 . 2  |-  ( S  e.  (SubGrp `  G
)  ->  A. z  e.  ( Base `  H
) A. w  e.  ( Base `  H
) ( ( z 
.+  w )  e.  S  <->  ( w  .+  z )  e.  S
) )
24 eqid 2412 . . 3  |-  ( Base `  H )  =  (
Base `  H )
25 fvex 5709 . . . . . 6  |-  ( Base `  G )  e.  _V
263, 25eqeltri 2482 . . . . 5  |-  X  e. 
_V
2726, 14ssexi 4316 . . . 4  |-  N  e. 
_V
289, 4ressplusg 13534 . . . 4  |-  ( N  e.  _V  ->  .+  =  ( +g  `  H ) )
2927, 28ax-mp 8 . . 3  |-  .+  =  ( +g  `  H )
3024, 29isnsg 14932 . 2  |-  ( S  e.  (NrmSGrp `  H
)  <->  ( S  e.  (SubGrp `  H )  /\  A. z  e.  (
Base `  H ) A. w  e.  ( Base `  H ) ( ( z  .+  w
)  e.  S  <->  ( w  .+  z )  e.  S
) ) )
3112, 23, 30sylanbrc 646 1  |-  ( S  e.  (SubGrp `  G
)  ->  S  e.  (NrmSGrp `  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   {crab 2678   _Vcvv 2924    C_ wss 3288   ` cfv 5421  (class class class)co 6048   Basecbs 13432   ↾s cress 13433   +g cplusg 13492   Grpcgrp 14648  SubGrpcsubg 14901  NrmSGrpcnsg 14902
This theorem is referenced by:  sylow3lem4  15227  sylow3lem6  15229
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-0g 13690  df-mnd 14653  df-grp 14775  df-minusg 14776  df-sbg 14777  df-subg 14904  df-nsg 14905
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