MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  conjnsg Unicode version

Theorem conjnsg 14734
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 2296 . . . . . 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 14676 . . . . 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 450 . . . 4  |-  ( S  e.  (NrmSGrp `  G
)  ->  { y  e.  X  |  A. z  e.  X  (
( y  .+  z
)  e.  S  <->  ( z  .+  y )  e.  S
) }  =  X )
65eleq2d 2363 . . 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 471 . 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 14665 . . 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 14732 . . 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 457 . 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 456 1  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X )  ->  S  =  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    e. cmpt 4093   ran crn 4706   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   -gcsg 14381  SubGrpcsubg 14631  NrmSGrpcnsg 14632
This theorem is referenced by:  sylow3lem6  14959
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-nsg 14635
  Copyright terms: Public domain W3C validator