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Theorem simpggrp 20016
Description: A simple group is a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
Assertion
Ref Expression
simpggrp (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)

Proof of Theorem simpggrp
StepHypRef Expression
1 issimpg 20014 . 2 (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))
21simplbi 497 1 (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098   class class class wbr 5141  cfv 6537  2oc2o 8461  cen 8938  Grpcgrp 18863  NrmSGrpcnsg 19048  SimpGrpcsimpg 20012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6489  df-fv 6545  df-simpg 20013
This theorem is referenced by:  simpggrpd  20017  prmsimpcyc  32879
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