Theorem issimpgd 20025

Description: Deduce a simple group from its properties. (Contributed by Rohan Ridenour, 3-Aug-2023.)

Hypotheses

Ref	Expression
issimpgd.1	⊢ (𝜑 → 𝐺 ∈ Grp)
issimpgd.2	⊢ (𝜑 → (NrmSGrp‘𝐺) ≈ 2o)

Assertion

Ref	Expression
issimpgd	⊢ (𝜑 → 𝐺 ∈ SimpGrp)

Proof of Theorem issimpgd

Step	Hyp	Ref	Expression
1		issimpgd.1	. 2 ⊢ (𝜑 → 𝐺 ∈ Grp)
2		issimpgd.2	. 2 ⊢ (𝜑 → (NrmSGrp‘𝐺) ≈ 2o)
3		issimpg 20024	. 2 ⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))
4	1, 2, 3	sylanbrc 583	1 ⊢ (𝜑 → 𝐺 ∈ SimpGrp)

Syntax hints:   → wi 4   ∈ wcel 2109   class class class wbr 5107  ‘cfv 6511  2_oc2o 8428   ≈ cen 8915  Grpcgrp 18865  NrmSGrpcnsg 19053  SimpGrpcsimpg 20022

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-simpg 20023

This theorem is referenced by:  2nsgsimpgd  20034