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Theorem simpggrp 19348
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 19346 . 2 (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))
21simplbi 501 1 (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114   class class class wbr 5040  cfv 6350  2oc2o 8138  cen 8565  Grpcgrp 18232  NrmSGrpcnsg 18405  SimpGrpcsimpg 19344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-rab 3063  df-v 3402  df-un 3858  df-in 3860  df-ss 3870  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-iota 6308  df-fv 6358  df-simpg 19345
This theorem is referenced by:  simpggrpd  19349  prmsimpcyc  31071
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