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Theorem nsgsubg 14972
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 2436 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2436 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
31, 2isnsg 14969 . 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 1725   A.wral 2705   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529  SubGrpcsubg 14938  NrmSGrpcnsg 14939
This theorem is referenced by:  nsgconj  14973  isnsg3  14974  eqgcpbl  14994  divsgrp  14995  divseccl  14996  divsadd  14997  divs0  14998  divsinv  14999  divssub  15000  ghmnsgima  15029  ghmnsgpreima  15030  conjnsg  15041  divsghm  15042  sylow3lem4  15264  clsnsg  18139  divstgpopn  18149  divstgphaus  18152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-subg 14941  df-nsg 14942
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