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Theorem nmznsg 14871
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 14869 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  N
)
6 subgrcl 14836 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
72, 3, 4nmzsubg 14868 . . . . 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 14850 . . . 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 3344 . . . . . . 7  |-  { x  e.  X  |  A. y  e.  X  (
( x  .+  y
)  e.  S  <->  ( y  .+  x )  e.  S
) }  C_  X
142, 13eqsstri 3294 . . . . . 6  |-  N  C_  X
1514sseli 3262 . . . . 5  |-  ( w  e.  N  ->  w  e.  X )
162nmzbi 14867 . . . . 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 2694 . . 3  |-  A. z  e.  N  A. w  e.  N  ( (
z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S
)
199subgbas 14835 . . . . 5  |-  ( N  e.  (SubGrp `  G
)  ->  N  =  ( Base `  H )
)
208, 19syl 15 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  N  =  ( Base `  H )
)
2120raleqdv 2827 . . . 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 2833 . . 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 2366 . . 3  |-  ( Base `  H )  =  (
Base `  H )
25 fvex 5646 . . . . . 6  |-  ( Base `  G )  e.  _V
263, 25eqeltri 2436 . . . . 5  |-  X  e. 
_V
2726, 14ssexi 4261 . . . 4  |-  N  e. 
_V
289, 4ressplusg 13458 . . . 4  |-  ( N  e.  _V  ->  .+  =  ( +g  `  H ) )
2927, 28ax-mp 8 . . 3  |-  .+  =  ( +g  `  H )
3024, 29isnsg 14856 . 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 1647    e. wcel 1715   A.wral 2628   {crab 2632   _Vcvv 2873    C_ wss 3238   ` cfv 5358  (class class class)co 5981   Basecbs 13356   ↾s cress 13357   +g cplusg 13416   Grpcgrp 14572  SubGrpcsubg 14825  NrmSGrpcnsg 14826
This theorem is referenced by:  sylow3lem4  15151  sylow3lem6  15153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-0g 13614  df-mnd 14577  df-grp 14699  df-minusg 14700  df-sbg 14701  df-subg 14828  df-nsg 14829
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