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Theorem nsgsubg 19176
Description: A normal subgroup is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
Assertion
Ref Expression
nsgsubg (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))

Proof of Theorem nsgsubg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2737 . . 3 (+g𝐺) = (+g𝐺)
31, 2isnsg 19173 . 2 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦) ∈ 𝑆 ↔ (𝑦(+g𝐺)𝑥) ∈ 𝑆)))
43simplbi 497 1 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2108  wral 3061  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  SubGrpcsubg 19138  NrmSGrpcnsg 19139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-subg 19141  df-nsg 19142
This theorem is referenced by:  nsgconj  19177  isnsg3  19178  trivnsgd  19190  eqgcpbl  19200  qusgrp  19204  quseccl  19205  qusadd  19206  qus0  19207  qusinv  19208  qussub  19209  ecqusaddcl  19211  ghmnsgima  19258  ghmnsgpreima  19259  conjnsg  19272  qusghm  19273  ghmqusnsglem1  19298  ghmqusnsglem2  19299  ghmqusnsg  19300  ghmquskerlem1  19301  ghmquskerlem2  19303  ghmquskerlem3  19304  ghmqusker  19305  sylow3lem4  19648  prmgrpsimpgd  20134  rhmqusnsg  21295  rngqiprngimf1lem  21304  rngqiprngimf1  21310  rngqiprngimfo  21311  rngqiprngfulem4  21324  rngqipring1  21326  clsnsg  24118  qustgpopn  24128  qustgphaus  24131  cyc3genpm  33172  qusker  33377  qus0g  33435  qusima  33436  qusrn  33437  nsgqus0  33438  nsgmgclem  33439  nsgmgc  33440  nsgqusf1olem1  33441  nsgqusf1olem2  33442  nsgqusf1olem3  33443  lmhmqusker  33445  rhmquskerlem  33453  qsnzr  33483  opprqusplusg  33517  opprqus0g  33518  qsdrngilem  33522  qsdrngi  33523
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