Theorem issimpgd 20309

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 20308	. 2 ⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))
4	1, 2, 3	sylanbrc 595	1 ⊢ (𝜑 → 𝐺 ∈ SimpGrp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144   class class class wbr 5102  ‘cfv 6538  2_oc2o 8470   ≈ cen 8970  Grpcgrp 19144  NrmSGrpcnsg 19331  SimpGrpcsimpg 20306

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-simpg 20307

This theorem is used by:  2nsgsimpgd  20318