Theorem issimpg

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

	Ref	Expression
Assertion	issimpg	$$⊢G\in \mathrm{SimpGrp}\leftrightarrow \left(G\in \mathrm{Grp}\wedge \mathrm{NrmSGrp}\left(G\right)\approx 2_{𝑜}\right)$$

Proof

Step	Hyp	Ref	Expression
1		fveq2	$$⊢g=G\to \mathrm{NrmSGrp}\left(g\right)=\mathrm{NrmSGrp}\left(G\right)$$
2	1	breq1d	$$⊢g=G\to \left(\mathrm{NrmSGrp}\left(g\right)\approx 2_{𝑜}\leftrightarrow \mathrm{NrmSGrp}\left(G\right)\approx 2_{𝑜}\right)$$
3		df-simpg	$$⊢\mathrm{SimpGrp}=\left\{g\in \mathrm{Grp}|\mathrm{NrmSGrp}\left(g\right)\approx 2_{𝑜}\right\}$$
4	2 3	elrab2	$$⊢G\in \mathrm{SimpGrp}\leftrightarrow \left(G\in \mathrm{Grp}\wedge \mathrm{NrmSGrp}\left(G\right)\approx 2_{𝑜}\right)$$