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Theorem isnsg4 14660
Description: A subgroup is normal iff its normalizer is the entire group. (Contributed by Mario Carneiro, 18-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 )
Assertion
Ref Expression
isnsg4  |-  ( S  e.  (NrmSGrp `  G
)  <->  ( S  e.  (SubGrp `  G )  /\  N  =  X
) )
Distinct variable groups:    x, y, G    x, S, y    x,  .+ , y    x, X, y
Allowed substitution hints:    N( x, y)

Proof of Theorem isnsg4
StepHypRef Expression
1 nmzsubg.2 . . 3  |-  X  =  ( Base `  G
)
2 nmzsubg.3 . . 3  |-  .+  =  ( +g  `  G )
31, 2isnsg 14646 . 2  |-  ( S  e.  (NrmSGrp `  G
)  <->  ( S  e.  (SubGrp `  G )  /\  A. x  e.  X  A. y  e.  X  ( ( x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S ) ) )
4 eqcom 2285 . . . 4  |-  ( N  =  X  <->  X  =  N )
5 elnmz.1 . . . . 5  |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S ) }
65eqeq2i 2293 . . . 4  |-  ( X  =  N  <->  X  =  { x  e.  X  |  A. y  e.  X  ( ( x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S ) } )
7 rabid2 2717 . . . 4  |-  ( X  =  { x  e.  X  |  A. y  e.  X  ( (
x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S
) }  <->  A. x  e.  X  A. y  e.  X  ( (
x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S
) )
84, 6, 73bitri 262 . . 3  |-  ( N  =  X  <->  A. x  e.  X  A. y  e.  X  ( (
x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S
) )
98anbi2i 675 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  N  =  X )  <->  ( S  e.  (SubGrp `  G )  /\  A. x  e.  X  A. y  e.  X  ( ( x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S ) ) )
103, 9bitr4i 243 1  |-  ( S  e.  (NrmSGrp `  G
)  <->  ( S  e.  (SubGrp `  G )  /\  N  =  X
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  SubGrpcsubg 14615  NrmSGrpcnsg 14616
This theorem is referenced by:  conjnsg  14718
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-subg 14618  df-nsg 14619
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