Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  simpggrpd Structured version   Visualization version   GIF version

Theorem simpggrpd 19285
 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 19284 . 2 (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)
31, 2syl 17 1 (𝜑𝐺 ∈ Grp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111  Grpcgrp 18169  SimpGrpcsimpg 19280 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-rab 3079  df-v 3411  df-un 3863  df-in 3865  df-ss 3875  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-iota 6294  df-fv 6343  df-simpg 19281 This theorem is referenced by:  simpgntrivd  19288  simpgnideld  19289  simpgnsgd  19290  ablsimpg1gend  19295  ablsimpgcygd  19296  ablsimpgfindlem1  19297  ablsimpgfindlem2  19298  ablsimpgfind  19300  ablsimpgprmd  19305  simpcntrab  43872
 Copyright terms: Public domain W3C validator