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Theorem nsgsubg 16435
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 2454 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2454 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
31, 2isnsg 16432 . 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 458 1  |-  ( S  e.  (NrmSGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1823   A.wral 2804   ` cfv 5570  (class class class)co 6270   Basecbs 14719   +g cplusg 14787  SubGrpcsubg 16397  NrmSGrpcnsg 16398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-subg 16400  df-nsg 16401
This theorem is referenced by:  nsgconj  16436  isnsg3  16437  eqgcpbl  16457  qusgrp  16458  quseccl  16459  qusadd  16460  qus0  16461  qusinv  16462  qussub  16463  ghmnsgima  16492  ghmnsgpreima  16493  conjnsg  16504  qusghm  16505  sylow3lem4  16852  clsnsg  20777  qustgpopn  20787  qustgphaus  20790
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