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Theorem nsgsubg 19074
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 2732 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2732 . . 3 (+g𝐺) = (+g𝐺)
31, 2isnsg 19071 . 2 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦) ∈ 𝑆 ↔ (𝑦(+g𝐺)𝑥) ∈ 𝑆)))
43simplbi 498 1 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2106  wral 3061  cfv 6543  (class class class)co 7411  Basecbs 17148  +gcplusg 17201  SubGrpcsubg 19036  NrmSGrpcnsg 19037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-subg 19039  df-nsg 19040
This theorem is referenced by:  nsgconj  19075  isnsg3  19076  trivnsgd  19088  eqgcpbl  19098  qusgrp  19101  quseccl  19102  qusadd  19103  qus0  19104  qusinv  19105  qussub  19106  ecqusaddcl  19108  ghmnsgima  19154  ghmnsgpreima  19155  conjnsg  19168  qusghm  19169  sylow3lem4  19539  prmgrpsimpgd  20025  rngqiprngimf1lem  21053  rngqiprngimf1  21059  rngqiprngimfo  21060  rngqiprngfulem4  21073  rngqipring1  21075  clsnsg  23834  qustgpopn  23844  qustgphaus  23847  cyc3genpm  32569  qusker  32722  qus0g  32780  qusima  32781  qusrn  32782  nsgqus0  32783  nsgmgclem  32784  nsgmgc  32785  nsgqusf1olem1  32786  nsgqusf1olem2  32787  nsgqusf1olem3  32788  ghmquskerlem1  32790  ghmquskerlem2  32792  ghmquskerlem3  32793  ghmqusker  32794  lmhmqusker  32796  rhmquskerlem  32805  qsnzr  32836  opprqusplusg  32865  opprqus0g  32866  qsdrngilem  32870  qsdrngi  32871
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