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Theorem nsgsubg 19038
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 2733 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2733 . . 3 (+g𝐺) = (+g𝐺)
31, 2isnsg 19035 . 2 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦) ∈ 𝑆 ↔ (𝑦(+g𝐺)𝑥) ∈ 𝑆)))
43simplbi 499 1 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2107  wral 3062  cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  SubGrpcsubg 19000  NrmSGrpcnsg 19001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-subg 19003  df-nsg 19004
This theorem is referenced by:  nsgconj  19039  isnsg3  19040  trivnsgd  19052  eqgcpbl  19062  qusgrp  19065  quseccl  19066  qusadd  19067  qus0  19068  qusinv  19069  qussub  19070  ghmnsgima  19116  ghmnsgpreima  19117  conjnsg  19128  qusghm  19129  sylow3lem4  19498  prmgrpsimpgd  19984  clsnsg  23614  qustgpopn  23624  qustgphaus  23627  cyc3genpm  32311  qusker  32464  qus0g  32518  qusima  32519  qusrn  32520  nsgqus0  32521  nsgmgclem  32522  nsgmgc  32523  nsgqusf1olem1  32524  nsgqusf1olem2  32525  nsgqusf1olem3  32526  ghmquskerlem1  32528  ghmquskerlem2  32530  ghmquskerlem3  32531  ghmqusker  32532  lmhmqusker  32534  rhmquskerlem  32543  qsnzr  32574  opprqusplusg  32603  opprqus0g  32604  qsdrngilem  32608  qsdrngi  32609  ecqusaddcl  46769  rngqiprngimf1lem  46779  rngqiprngimf1  46785  rngqiprngimfo  46786  rngqiprngfulem4  46799  rngqipring1  46801
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