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Theorem conjnsg 14970
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 2389 . . . . . 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 14912 . . . . 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 451 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  { y  e.  X  |  A. z  e.  X  (
( y  .+  z
)  e.  S  <->  ( z  .+  y )  e.  S
) }  =  X )
65eleq2d 2456 . . 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 472 . 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 14901 . . 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 14968 . . 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 458 . 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 457 1  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X )  ->  S  =  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   {crab 2655    e. cmpt 4209   ran crn 4821   ` cfv 5396  (class class class)co 6022   Basecbs 13398   +g cplusg 13458   -gcsg 14617  SubGrpcsubg 14867  NrmSGrpcnsg 14868
This theorem is referenced by:  sylow3lem6  15195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-0g 13656  df-mnd 14619  df-grp 14741  df-minusg 14742  df-sbg 14743  df-subg 14870  df-nsg 14871
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