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Theorem nsgsubg 14859
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 2366 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2366 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
31, 2isnsg 14856 . 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 446 1  |-  ( S  e.  (NrmSGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1715   A.wral 2628   ` cfv 5358  (class class class)co 5981   Basecbs 13356   +g cplusg 13416  SubGrpcsubg 14825  NrmSGrpcnsg 14826
This theorem is referenced by:  nsgconj  14860  isnsg3  14861  eqgcpbl  14881  divsgrp  14882  divseccl  14883  divsadd  14884  divs0  14885  divsinv  14886  divssub  14887  ghmnsgima  14916  ghmnsgpreima  14917  conjnsg  14928  divsghm  14929  sylow3lem4  15151  clsnsg  18005  divstgpopn  18015  divstgphaus  18018
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fv 5366  df-ov 5984  df-subg 14828  df-nsg 14829
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