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Theorem simpggrpd 20007
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 20006 . 2 (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)
31, 2syl 17 1 (𝜑𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  Grpcgrp 18856  SimpGrpcsimpg 20002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-simpg 20003
This theorem is referenced by:  simpgntrivd  20010  simpgnideld  20011  simpgnsgd  20012  ablsimpg1gend  20017  ablsimpgcygd  20018  ablsimpgfindlem1  20019  ablsimpgfindlem2  20020  ablsimpgfind  20022  ablsimpgprmd  20027  simpcntrab  45885
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