Theorem issimpgd 19696

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 19695	. 2 ⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))
4	1, 2, 3	sylanbrc 583	1 ⊢ (𝜑 → 𝐺 ∈ SimpGrp)

Syntax hints:   → wi 4   ∈ wcel 2106   class class class wbr 5074  ‘cfv 6433  2_oc2o 8291   ≈ cen 8730  Grpcgrp 18577  NrmSGrpcnsg 18750  SimpGrpcsimpg 19693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-simpg 19694

This theorem is referenced by:  2nsgsimpgd  19705