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Theorem nsgsubg 14927
Description: A normal subgroup is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
Assertion
Ref Expression
nsgsubg  |-  ( S  e.  (NrmSGrp `  G
)  ->  S  e.  (SubGrp `  G ) )

Proof of Theorem nsgsubg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2404 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
31, 2isnsg 14924 . 2  |-  ( S  e.  (NrmSGrp `  G
)  <->  ( S  e.  (SubGrp `  G )  /\  A. x  e.  (
Base `  G ) A. y  e.  ( Base `  G ) ( ( x ( +g  `  G ) y )  e.  S  <->  ( y
( +g  `  G ) x )  e.  S
) ) )
43simplbi 447 1  |-  ( S  e.  (NrmSGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1721   A.wral 2666   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484  SubGrpcsubg 14893  NrmSGrpcnsg 14894
This theorem is referenced by:  nsgconj  14928  isnsg3  14929  eqgcpbl  14949  divsgrp  14950  divseccl  14951  divsadd  14952  divs0  14953  divsinv  14954  divssub  14955  ghmnsgima  14984  ghmnsgpreima  14985  conjnsg  14996  divsghm  14997  sylow3lem4  15219  clsnsg  18092  divstgpopn  18102  divstgphaus  18105
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-subg 14896  df-nsg 14897
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