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Theorem isnsg4 13003
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 12993 . 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 2179 . . . 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 2188 . . . 4  |-  ( X  =  N  <->  X  =  { x  e.  X  |  A. y  e.  X  ( ( x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S ) } )
7 rabid2 2653 . . . 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 206 . . 3  |-  ( N  =  X  <->  A. x  e.  X  A. y  e.  X  ( (
x  .+  y )  e.  S  <->  ( y  .+  x )  e.  S
) )
98anbi2i 457 . 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 187 1  |-  ( S  e.  (NrmSGrp `  G
)  <->  ( S  e.  (SubGrp `  G )  /\  N  =  X
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455   {crab 2459   ` cfv 5215  (class class class)co 5872   Basecbs 12454   +g cplusg 12528  SubGrpcsubg 12958  NrmSGrpcnsg 12959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-cnex 7899  ax-resscn 7900  ax-1re 7902  ax-addrcl 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223  df-ov 5875  df-inn 8916  df-2 8974  df-ndx 12457  df-slot 12458  df-base 12460  df-plusg 12541  df-subg 12961  df-nsg 12962
This theorem is referenced by: (None)
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