Theorem issimpgd 20114

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 20113	. 2 ⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))
4	1, 2, 3	sylanbrc 583	1 ⊢ (𝜑 → 𝐺 ∈ SimpGrp)

Syntax hints:   → wi 4   ∈ wcel 2107   class class class wbr 5142  ‘cfv 6560  2_oc2o 8501   ≈ cen 8983  Grpcgrp 18952  NrmSGrpcnsg 19140  SimpGrpcsimpg 20111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-simpg 20112

This theorem is referenced by:  2nsgsimpgd  20123