Theorem simpggrpd 20326

Description: A simple group is a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)

Hypothesis

Ref	Expression
simpggrpd.1	⊢ (𝜑 → 𝐺 ∈ SimpGrp)

Assertion

Ref	Expression
simpggrpd	⊢ (𝜑 → 𝐺 ∈ Grp)

Proof of Theorem simpggrpd

Step	Hyp	Ref	Expression
1		simpggrpd.1	. 2 ⊢ (𝜑 → 𝐺 ∈ SimpGrp)
2		simpggrp 20325	. 2 ⊢ (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐺 ∈ Grp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  Grpcgrp 19159  SimpGrpcsimpg 20321

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

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 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6501  df-fv 6555  df-simpg 20322

This theorem is used by:  simpgntrivd  20329  simpgnideld  20330  simpgnsgd  20331  ablsimpg1gend  20336  ablsimpgcygd  20337  ablsimpgfindlem1  20338  ablsimpgfindlem2  20339  ablsimpgfind  20341  ablsimpgprmd  20346  simpcntrab  48022