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Theorem simpggrpd 20115
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 20114 . 2 (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)
31, 2syl 17 1 (𝜑𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  Grpcgrp 18951  SimpGrpcsimpg 20110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-simpg 20111
This theorem is referenced by:  simpgntrivd  20118  simpgnideld  20119  simpgnsgd  20120  ablsimpg1gend  20125  ablsimpgcygd  20126  ablsimpgfindlem1  20127  ablsimpgfindlem2  20128  ablsimpgfind  20130  ablsimpgprmd  20135  simpcntrab  46885
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