Theorem simpggrpd 20083

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 20082	. 2 ⊢ (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2109  Grpcgrp 18921  SimpGrpcsimpg 20078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-simpg 20079

This theorem is referenced by:  simpgntrivd  20086  simpgnideld  20087  simpgnsgd  20088  ablsimpg1gend  20093  ablsimpgcygd  20094  ablsimpgfindlem1  20095  ablsimpgfindlem2  20096  ablsimpgfind  20098  ablsimpgprmd  20103  simpcntrab  46866