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Theorem nmznsg 15836
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 22 . . 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 15834 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  N
)
6 subgrcl 15797 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
72, 3, 4nmzsubg 15833 . . . . 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 15815 . . . 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 913 . 2  |-  ( S  e.  (SubGrp `  G
)  ->  S  e.  (SubGrp `  H ) )
13 ssrab2 3538 . . . . . . 7  |-  { x  e.  X  |  A. y  e.  X  (
( x  .+  y
)  e.  S  <->  ( y  .+  x )  e.  S
) }  C_  X
142, 13eqsstri 3487 . . . . . 6  |-  N  C_  X
1514sseli 3453 . . . . 5  |-  ( w  e.  N  ->  w  e.  X )
162nmzbi 15832 . . . . 5  |-  ( ( z  e.  N  /\  w  e.  X )  ->  ( ( z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S ) )
1715, 16sylan2 474 . . . 4  |-  ( ( z  e.  N  /\  w  e.  N )  ->  ( ( z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S ) )
1817rgen2a 2893 . . 3  |-  A. z  e.  N  A. w  e.  N  ( (
z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S
)
199subgbas 15796 . . . . 5  |-  ( N  e.  (SubGrp `  G
)  ->  N  =  ( Base `  H )
)
208, 19syl 16 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  N  =  ( Base `  H )
)
2120raleqdv 3022 . . . 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 3030 . . 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 211 . 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 2451 . . 3  |-  ( Base `  H )  =  (
Base `  H )
25 fvex 5802 . . . . . 6  |-  ( Base `  G )  e.  _V
263, 25eqeltri 2535 . . . . 5  |-  X  e. 
_V
2726, 14ssexi 4538 . . . 4  |-  N  e. 
_V
289, 4ressplusg 14391 . . . 4  |-  ( N  e.  _V  ->  .+  =  ( +g  `  H ) )
2927, 28ax-mp 5 . . 3  |-  .+  =  ( +g  `  H )
3024, 29isnsg 15821 . 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 664 1  |-  ( S  e.  (SubGrp `  G
)  ->  S  e.  (NrmSGrp `  H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   {crab 2799   _Vcvv 3071    C_ wss 3429   ` cfv 5519  (class class class)co 6193   Basecbs 14285   ↾s cress 14286   +g cplusg 14349   Grpcgrp 15521  SubGrpcsubg 15786  NrmSGrpcnsg 15787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-0g 14491  df-mnd 15526  df-grp 15656  df-minusg 15657  df-sbg 15658  df-subg 15789  df-nsg 15790
This theorem is referenced by:  sylow3lem4  16242  sylow3lem6  16244
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