Theorem issimpgd 19611

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 19610	. 2 ⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))
4	1, 2, 3	sylanbrc 582	1 ⊢ (𝜑 → 𝐺 ∈ SimpGrp)

Syntax hints:   → wi 4   ∈ wcel 2108   class class class wbr 5070  ‘cfv 6418  2_oc2o 8261   ≈ cen 8688  Grpcgrp 18492  NrmSGrpcnsg 18665  SimpGrpcsimpg 19608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-simpg 19609

This theorem is referenced by:  2nsgsimpgd  19620