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Theorem nsgsubg 15815
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 2451 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2451 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
31, 2isnsg 15812 . 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 460 1  |-  ( S  e.  (NrmSGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1758   A.wral 2795   ` cfv 5516  (class class class)co 6190   Basecbs 14276   +g cplusg 14340  SubGrpcsubg 15777  NrmSGrpcnsg 15778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fv 5524  df-ov 6193  df-subg 15780  df-nsg 15781
This theorem is referenced by:  nsgconj  15816  isnsg3  15817  eqgcpbl  15837  divsgrp  15838  divseccl  15839  divsadd  15840  divs0  15841  divsinv  15842  divssub  15843  ghmnsgima  15872  ghmnsgpreima  15873  conjnsg  15884  divsghm  15885  sylow3lem4  16233  clsnsg  19796  divstgpopn  19806  divstgphaus  19809
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