Theorem simpggrpd 19976

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 19975	. 2 ⊢ (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2109  Grpcgrp 18812  SimpGrpcsimpg 19971

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 2701

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 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-iota 6438  df-fv 6490  df-simpg 19972

This theorem is referenced by:  simpgntrivd  19979  simpgnideld  19980  simpgnsgd  19981  ablsimpg1gend  19986  ablsimpgcygd  19987  ablsimpgfindlem1  19988  ablsimpgfindlem2  19989  ablsimpgfind  19991  ablsimpgprmd  19996  simpcntrab  46861