Theorem simpggrpd 20213

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 20212	. 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 19046  SimpGrpcsimpg 20208

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-simpg 20209

This theorem is used by:  simpgntrivd  20216  simpgnideld  20217  simpgnsgd  20218  ablsimpg1gend  20223  ablsimpgcygd  20224  ablsimpgfindlem1  20225  ablsimpgfindlem2  20226  ablsimpgfind  20228  ablsimpgprmd  20233  simpcntrab  47644