Theorem simpgnsgbid

Description: A nontrivial group is simple if and only if its normal subgroups are exactly the group itself and the trivial subgroup. (Contributed by Rohan Ridenour, 4-Aug-2023)

	Ref	Expression
Hypotheses	simpgnsgbid.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
	simpgnsgbid.2	⊢ 0 = ( 0g ‘ 𝐺 )
	simpgnsgbid.3	⊢ ( 𝜑 → 𝐺 ∈ Grp )
	simpgnsgbid.4	⊢ ( 𝜑 → ¬ { 0 } = 𝐵 )
Assertion	simpgnsgbid	⊢ ( 𝜑 → ( 𝐺 ∈ SimpGrp ↔ ( NrmSGrp ‘ 𝐺 ) = { { 0 } , 𝐵 } ) )

Proof

Step	Hyp	Ref	Expression
1		simpgnsgbid.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		simpgnsgbid.2	⊢ 0 = ( 0g ‘ 𝐺 )
3		simpgnsgbid.3	⊢ ( 𝜑 → 𝐺 ∈ Grp )
4		simpgnsgbid.4	⊢ ( 𝜑 → ¬ { 0 } = 𝐵 )
5		simpr	⊢ ( ( 𝜑 ∧ 𝐺 ∈ SimpGrp ) → 𝐺 ∈ SimpGrp )
6	1 2 5	simpgnsgd	⊢ ( ( 𝜑 ∧ 𝐺 ∈ SimpGrp ) → ( NrmSGrp ‘ 𝐺 ) = { { 0 } , 𝐵 } )
7	3	adantr	⊢ ( ( 𝜑 ∧ ( NrmSGrp ‘ 𝐺 ) = { { 0 } , 𝐵 } ) → 𝐺 ∈ Grp )
8	4	adantr	⊢ ( ( 𝜑 ∧ ( NrmSGrp ‘ 𝐺 ) = { { 0 } , 𝐵 } ) → ¬ { 0 } = 𝐵 )
9		simpr	⊢ ( ( ( 𝜑 ∧ ( NrmSGrp ‘ 𝐺 ) = { { 0 } , 𝐵 } ) ∧ 𝑥 ∈ ( NrmSGrp ‘ 𝐺 ) ) → 𝑥 ∈ ( NrmSGrp ‘ 𝐺 ) )
10		simplr	⊢ ( ( ( 𝜑 ∧ ( NrmSGrp ‘ 𝐺 ) = { { 0 } , 𝐵 } ) ∧ 𝑥 ∈ ( NrmSGrp ‘ 𝐺 ) ) → ( NrmSGrp ‘ 𝐺 ) = { { 0 } , 𝐵 } )
11	9 10	eleqtrd	⊢ ( ( ( 𝜑 ∧ ( NrmSGrp ‘ 𝐺 ) = { { 0 } , 𝐵 } ) ∧ 𝑥 ∈ ( NrmSGrp ‘ 𝐺 ) ) → 𝑥 ∈ { { 0 } , 𝐵 } )
12		elpri	⊢ ( 𝑥 ∈ { { 0 } , 𝐵 } → ( 𝑥 = { 0 } ∨ 𝑥 = 𝐵 ) )
13	11 12	syl	⊢ ( ( ( 𝜑 ∧ ( NrmSGrp ‘ 𝐺 ) = { { 0 } , 𝐵 } ) ∧ 𝑥 ∈ ( NrmSGrp ‘ 𝐺 ) ) → ( 𝑥 = { 0 } ∨ 𝑥 = 𝐵 ) )
14	1 2 7 8 13	2nsgsimpgd	⊢ ( ( 𝜑 ∧ ( NrmSGrp ‘ 𝐺 ) = { { 0 } , 𝐵 } ) → 𝐺 ∈ SimpGrp )
15	6 14	impbida	⊢ ( 𝜑 → ( 𝐺 ∈ SimpGrp ↔ ( NrmSGrp ‘ 𝐺 ) = { { 0 } , 𝐵 } ) )