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Theorem conjnsg 16442
Description: A normal subgroup is unchanged under conjugation. (Contributed by Mario Carneiro, 18-Jan-2015.)
Hypotheses
Ref Expression
conjghm.x  |-  X  =  ( Base `  G
)
conjghm.p  |-  .+  =  ( +g  `  G )
conjghm.m  |-  .-  =  ( -g `  G )
conjsubg.f  |-  F  =  ( x  e.  S  |->  ( ( A  .+  x )  .-  A
) )
Assertion
Ref Expression
conjnsg  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X )  ->  S  =  ran  F )
Distinct variable groups:    x,  .-    x,  .+    x, A    x, G    x, S    x, X
Allowed substitution hint:    F( x)

Proof of Theorem conjnsg
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2396 . . . . . 6  |-  { y  e.  X  |  A. z  e.  X  (
( y  .+  z
)  e.  S  <->  ( z  .+  y )  e.  S
) }  =  {
y  e.  X  |  A. z  e.  X  ( ( y  .+  z )  e.  S  <->  ( z  .+  y )  e.  S ) }
2 conjghm.x . . . . . 6  |-  X  =  ( Base `  G
)
3 conjghm.p . . . . . 6  |-  .+  =  ( +g  `  G )
41, 2, 3isnsg4 16384 . . . . 5  |-  ( S  e.  (NrmSGrp `  G
)  <->  ( S  e.  (SubGrp `  G )  /\  { y  e.  X  |  A. z  e.  X  ( ( y  .+  z )  e.  S  <->  ( z  .+  y )  e.  S ) }  =  X ) )
54simprbi 462 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  { y  e.  X  |  A. z  e.  X  (
( y  .+  z
)  e.  S  <->  ( z  .+  y )  e.  S
) }  =  X )
65eleq2d 2466 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  ( A  e.  { y  e.  X  |  A. z  e.  X  ( ( y  .+  z )  e.  S  <->  ( z  .+  y )  e.  S ) }  <-> 
A  e.  X ) )
76biimpar 483 . 2  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X )  ->  A  e.  { y  e.  X  |  A. z  e.  X  ( ( y  .+  z )  e.  S  <->  ( z  .+  y )  e.  S ) } )
8 nsgsubg 16373 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
9 conjghm.m . . . 4  |-  .-  =  ( -g `  G )
10 conjsubg.f . . . 4  |-  F  =  ( x  e.  S  |->  ( ( A  .+  x )  .-  A
) )
112, 3, 9, 10, 1conjnmz 16440 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  { y  e.  X  |  A. z  e.  X  ( ( y  .+  z )  e.  S  <->  ( z  .+  y )  e.  S ) } )  ->  S  =  ran  F )
128, 11sylan 469 . 2  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  { y  e.  X  |  A. z  e.  X  ( ( y  .+  z )  e.  S  <->  ( z  .+  y )  e.  S ) } )  ->  S  =  ran  F )
137, 12syldan 468 1  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X )  ->  S  =  ran  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1836   A.wral 2746   {crab 2750    |-> cmpt 4442   ran crn 4931   ` cfv 5513  (class class class)co 6218   Basecbs 14657   +g cplusg 14725   -gcsg 16195  SubGrpcsubg 16335  NrmSGrpcnsg 16336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-1st 6721  df-2nd 6722  df-0g 14872  df-mgm 16012  df-sgrp 16051  df-mnd 16061  df-grp 16197  df-minusg 16198  df-sbg 16199  df-subg 16338  df-nsg 16339
This theorem is referenced by:  sylow3lem6  16792
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