Theorem 2nsgsimpgd

Description: If any normal subgroup of a nontrivial group is either the trivial subgroup or the whole group, the group is simple. (Contributed by Rohan Ridenour, 3-Aug-2023)

	Ref	Expression
Hypotheses	2nsgsimpgd.1	$$⊢B={\mathrm{Base}}_G$$
	2nsgsimpgd.2	$$⊢\underset{\dot{}}{0}=0_G$$
	2nsgsimpgd.3	$$⊢\phi \to G\in \mathrm{Grp}$$
	2nsgsimpgd.4	$$⊢\phi \to \neg \left\{\underset{\dot{}}{0}\right\}=B$$
	2nsgsimpgd.5	$$⊢\left(\phi \wedge x\in \mathrm{NrmSGrp}\left(G\right)\right)\to \left(x=\left\{\underset{\dot{}}{0}\right\}\vee x=B\right)$$
Assertion	2nsgsimpgd	$$⊢\phi \to G\in \mathrm{SimpGrp}$$

Proof

Step	Hyp	Ref	Expression
1		2nsgsimpgd.1	$$⊢B={\mathrm{Base}}_G$$
2		2nsgsimpgd.2	$$⊢\underset{\dot{}}{0}=0_G$$
3		2nsgsimpgd.3	$$⊢\phi \to G\in \mathrm{Grp}$$
4		2nsgsimpgd.4	$$⊢\phi \to \neg \left\{\underset{\dot{}}{0}\right\}=B$$
5		2nsgsimpgd.5	$$⊢\left(\phi \wedge x\in \mathrm{NrmSGrp}\left(G\right)\right)\to \left(x=\left\{\underset{\dot{}}{0}\right\}\vee x=B\right)$$
6		elprg	$$⊢x\in \mathrm{NrmSGrp}\left(G\right)\to \left(x\in \left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\leftrightarrow \left(x=\left\{\underset{\dot{}}{0}\right\}\vee x=B\right)\right)$$
7	6	adantl	$$⊢\left(\phi \wedge x\in \mathrm{NrmSGrp}\left(G\right)\right)\to \left(x\in \left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\leftrightarrow \left(x=\left\{\underset{\dot{}}{0}\right\}\vee x=B\right)\right)$$
8	5 7	mpbird	$$⊢\left(\phi \wedge x\in \mathrm{NrmSGrp}\left(G\right)\right)\to x\in \left\{\left\{\underset{\dot{}}{0}\right\},B\right\}$$
9		simpr	$$⊢\left(\phi \wedge x=\left\{\underset{\dot{}}{0}\right\}\right)\to x=\left\{\underset{\dot{}}{0}\right\}$$
10	2	0nsg	$$⊢G\in \mathrm{Grp}\to \left\{\underset{\dot{}}{0}\right\}\in \mathrm{NrmSGrp}\left(G\right)$$
11	3 10	syl	$$⊢\phi \to \left\{\underset{\dot{}}{0}\right\}\in \mathrm{NrmSGrp}\left(G\right)$$
12	11	adantr	$$⊢\left(\phi \wedge x=\left\{\underset{\dot{}}{0}\right\}\right)\to \left\{\underset{\dot{}}{0}\right\}\in \mathrm{NrmSGrp}\left(G\right)$$
13	9 12	eqeltrd	$$⊢\left(\phi \wedge x=\left\{\underset{\dot{}}{0}\right\}\right)\to x\in \mathrm{NrmSGrp}\left(G\right)$$
14	13	adantlr	$$⊢\left(\left(\phi \wedge x\in \left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)\wedge x=\left\{\underset{\dot{}}{0}\right\}\right)\to x\in \mathrm{NrmSGrp}\left(G\right)$$
15		simpr	$$⊢\left(\phi \wedge x=B\right)\to x=B$$
16	1	nsgid	$$⊢G\in \mathrm{Grp}\to B\in \mathrm{NrmSGrp}\left(G\right)$$
17	3 16	syl	$$⊢\phi \to B\in \mathrm{NrmSGrp}\left(G\right)$$
18	17	adantr	$$⊢\left(\phi \wedge x=B\right)\to B\in \mathrm{NrmSGrp}\left(G\right)$$
19	15 18	eqeltrd	$$⊢\left(\phi \wedge x=B\right)\to x\in \mathrm{NrmSGrp}\left(G\right)$$
20	19	adantlr	$$⊢\left(\left(\phi \wedge x\in \left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)\wedge x=B\right)\to x\in \mathrm{NrmSGrp}\left(G\right)$$
21		elpri	$$⊢x\in \left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\to \left(x=\left\{\underset{\dot{}}{0}\right\}\vee x=B\right)$$
22	21	adantl	$$⊢\left(\phi \wedge x\in \left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)\to \left(x=\left\{\underset{\dot{}}{0}\right\}\vee x=B\right)$$
23	14 20 22	mpjaodan	$$⊢\left(\phi \wedge x\in \left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)\to x\in \mathrm{NrmSGrp}\left(G\right)$$
24	8 23	impbida	$$⊢\phi \to \left(x\in \mathrm{NrmSGrp}\left(G\right)\leftrightarrow x\in \left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\right)$$
25	24	eqrdv	$$⊢\phi \to \mathrm{NrmSGrp}\left(G\right)=\left\{\left\{\underset{\dot{}}{0}\right\},B\right\}$$
26		snex	$$⊢\left\{\underset{\dot{}}{0}\right\}\in \mathrm{V}$$
27	26	a1i	$$⊢\phi \to \left\{\underset{\dot{}}{0}\right\}\in \mathrm{V}$$
28	1	fvexi	$$⊢B\in \mathrm{V}$$
29	28	a1i	$$⊢\phi \to B\in \mathrm{V}$$
30	27 29 4	enpr2d	$$⊢\phi \to \left\{\left\{\underset{\dot{}}{0}\right\},B\right\}\approx 2_{𝑜}$$
31	25 30	eqbrtrd	$$⊢\phi \to \mathrm{NrmSGrp}\left(G\right)\approx 2_{𝑜}$$
32	3 31	issimpgd	$$⊢\phi \to G\in \mathrm{SimpGrp}$$