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Theorem nmznsg 16444
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 16442 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  N
)
6 subgrcl 16405 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
72, 3, 4nmzsubg 16441 . . . . 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 16423 . . . 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 920 . 2  |-  ( S  e.  (SubGrp `  G
)  ->  S  e.  (SubGrp `  H ) )
13 ssrab2 3571 . . . . . . 7  |-  { x  e.  X  |  A. y  e.  X  (
( x  .+  y
)  e.  S  <->  ( y  .+  x )  e.  S
) }  C_  X
142, 13eqsstri 3519 . . . . . 6  |-  N  C_  X
1514sseli 3485 . . . . 5  |-  ( w  e.  N  ->  w  e.  X )
162nmzbi 16440 . . . . 5  |-  ( ( z  e.  N  /\  w  e.  X )  ->  ( ( z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S ) )
1715, 16sylan2 472 . . . 4  |-  ( ( z  e.  N  /\  w  e.  N )  ->  ( ( z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S ) )
1817rgen2a 2881 . . 3  |-  A. z  e.  N  A. w  e.  N  ( (
z  .+  w )  e.  S  <->  ( w  .+  z )  e.  S
)
199subgbas 16404 . . . . 5  |-  ( N  e.  (SubGrp `  G
)  ->  N  =  ( Base `  H )
)
208, 19syl 16 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  N  =  ( Base `  H )
)
2120raleqdv 3057 . . . 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 3065 . . 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 2454 . . 3  |-  ( Base `  H )  =  (
Base `  H )
25 fvex 5858 . . . . . 6  |-  ( Base `  G )  e.  _V
263, 25eqeltri 2538 . . . . 5  |-  X  e. 
_V
2726, 14ssexi 4582 . . . 4  |-  N  e. 
_V
289, 4ressplusg 14830 . . . 4  |-  ( N  e.  _V  ->  .+  =  ( +g  `  H ) )
2927, 28ax-mp 5 . . 3  |-  .+  =  ( +g  `  H )
3024, 29isnsg 16429 . 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 662 1  |-  ( S  e.  (SubGrp `  G
)  ->  S  e.  (NrmSGrp `  H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808   _Vcvv 3106    C_ wss 3461   ` cfv 5570  (class class class)co 6270   Basecbs 14716   ↾s cress 14717   +g cplusg 14784   Grpcgrp 16252  SubGrpcsubg 16394  NrmSGrpcnsg 16395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257  df-sbg 16258  df-subg 16397  df-nsg 16398
This theorem is referenced by:  sylow3lem4  16849  sylow3lem6  16851
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