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Theorem simpggrpd 20130
Description: A simple group is a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
Hypothesis
Ref Expression
simpggrpd.1 (𝜑𝐺 ∈ SimpGrp)
Assertion
Ref Expression
simpggrpd (𝜑𝐺 ∈ Grp)

Proof of Theorem simpggrpd
StepHypRef Expression
1 simpggrpd.1 . 2 (𝜑𝐺 ∈ SimpGrp)
2 simpggrp 20129 . 2 (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)
31, 2syl 17 1 (𝜑𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Grpcgrp 18964  SimpGrpcsimpg 20125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-simpg 20126
This theorem is referenced by:  simpgntrivd  20133  simpgnideld  20134  simpgnsgd  20135  ablsimpg1gend  20140  ablsimpgcygd  20141  ablsimpgfindlem1  20142  ablsimpgfindlem2  20143  ablsimpgfind  20145  ablsimpgprmd  20150  simpcntrab  46826
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