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Theorem nmznsg 14661
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 19 . . 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 14659 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  N
)
6 subgrcl 14626 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
72, 3, 4nmzsubg 14658 . . . . 5  |-  ( G  e.  Grp  ->  N  e.  (SubGrp `  G )
)
86, 7syl 15 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  N  e.  (SubGrp `  G ) )
9 nmznsg.4 . . . . 5  |-  H  =  ( Gs  N )
109subsubg 14640 . . . 4  |-  ( N  e.  (SubGrp `  G
)  ->  ( S  e.  (SubGrp `  H )  <->  ( S  e.  (SubGrp `  G )  /\  S  C_  N ) ) )
118, 10syl 15 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  ( S  e.  (SubGrp `  H )  <->  ( S  e.  (SubGrp `  G )  /\  S  C_  N ) ) )
121, 5, 11mpbir2and 888 . 2  |-  ( S  e.  (SubGrp `  G
)  ->  S  e.  (SubGrp `  H ) )
13 ssrab2 3258 . . . . . . 7  |-  { x  e.  X  |  A. y  e.  X  (
( x  .+  y
)  e.  S  <->  ( y  .+  x )  e.  S
) }  C_  X
142, 13eqsstri 3208 . . . . . 6  |-  N  C_  X
1514sseli 3176 . . . . 5  |-  ( w  e.  N  ->  w  e.  X )
162nmzbi 14657 . . . . 5  |-  ( ( z  e.  N  /\  w  e.  X )  ->  ( ( z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S ) )
1715, 16sylan2 460 . . . 4  |-  ( ( z  e.  N  /\  w  e.  N )  ->  ( ( z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S ) )
1817rgen2a 2609 . . 3  |-  A. z  e.  N  A. w  e.  N  ( (
z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S
)
199subgbas 14625 . . . . 5  |-  ( N  e.  (SubGrp `  G
)  ->  N  =  ( Base `  H )
)
208, 19syl 15 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  N  =  ( Base `  H )
)
2120raleqdv 2742 . . . 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 2748 . . 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 202 . 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 2283 . . 3  |-  ( Base `  H )  =  (
Base `  H )
25 fvex 5539 . . . . . 6  |-  ( Base `  G )  e.  _V
263, 25eqeltri 2353 . . . . 5  |-  X  e. 
_V
2726, 14ssexi 4159 . . . 4  |-  N  e. 
_V
289, 4ressplusg 13250 . . . 4  |-  ( N  e.  _V  ->  .+  =  ( +g  `  H ) )
2927, 28ax-mp 8 . . 3  |-  .+  =  ( +g  `  H )
3024, 29isnsg 14646 . 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 645 1  |-  ( S  e.  (SubGrp `  G
)  ->  S  e.  (NrmSGrp `  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   Grpcgrp 14362  SubGrpcsubg 14615  NrmSGrpcnsg 14616
This theorem is referenced by:  sylow3lem4  14941  sylow3lem6  14943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-nsg 14619
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