Metamath Proof Explorer

Theorem ablsimpgcygd

Description: An abelian simple group is cyclic. (Contributed by Rohan Ridenour, 3-Aug-2023) (Proof shortened by Rohan Ridenour, 31-Oct-2023)

		Ref	Expression
	Hypotheses	ablsimpgcygd.1	⊢ ( 𝜑 → 𝐺 ∈ Abel )
		ablsimpgcygd.2	⊢ ( 𝜑 → 𝐺 ∈ SimpGrp )
	Assertion	ablsimpgcygd	⊢ ( 𝜑 → 𝐺 ∈ CycGrp )

Proof

Step	Hyp	Ref	Expression
1		ablsimpgcygd.1	⊢ ( 𝜑 → 𝐺 ∈ Abel )
2		ablsimpgcygd.2	⊢ ( 𝜑 → 𝐺 ∈ SimpGrp )
3		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
4		eqid	⊢ ( 0_g ‘ 𝐺 ) = ( 0_g ‘ 𝐺 )
5	3 4 2	simpgnideld	⊢ ( 𝜑 → ∃ 𝑥 ∈ ( Base ‘ 𝐺 ) ¬ 𝑥 = ( 0_g ‘ 𝐺 ) )
6		eqid	⊢ ( ._g ‘ 𝐺 ) = ( ._g ‘ 𝐺 )
7	2	simpggrpd	⊢ ( 𝜑 → 𝐺 ∈ Grp )
8	7	adantr	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ ¬ 𝑥 = ( 0_g ‘ 𝐺 ) ) ) → 𝐺 ∈ Grp )
9		simprl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ ¬ 𝑥 = ( 0_g ‘ 𝐺 ) ) ) → 𝑥 ∈ ( Base ‘ 𝐺 ) )
10	1	ad2antrr	⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ ¬ 𝑥 = ( 0_g ‘ 𝐺 ) ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) → 𝐺 ∈ Abel )
11	2	ad2antrr	⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ ¬ 𝑥 = ( 0_g ‘ 𝐺 ) ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) → 𝐺 ∈ SimpGrp )
12		simplrl	⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ ¬ 𝑥 = ( 0_g ‘ 𝐺 ) ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) → 𝑥 ∈ ( Base ‘ 𝐺 ) )
13		simplrr	⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ ¬ 𝑥 = ( 0_g ‘ 𝐺 ) ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) → ¬ 𝑥 = ( 0_g ‘ 𝐺 ) )
14		simpr	⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ ¬ 𝑥 = ( 0_g ‘ 𝐺 ) ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) → 𝑦 ∈ ( Base ‘ 𝐺 ) )
15	3 4 6 10 11 12 13 14	ablsimpg1gend	⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ ¬ 𝑥 = ( 0_g ‘ 𝐺 ) ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) → ∃ 𝑧 ∈ ℤ 𝑦 = ( 𝑧 ( ._g ‘ 𝐺 ) 𝑥 ) )
16	3 6 8 9 15	iscygd	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ ¬ 𝑥 = ( 0_g ‘ 𝐺 ) ) ) → 𝐺 ∈ CycGrp )
17	5 16	rexlimddv	⊢ ( 𝜑 → 𝐺 ∈ CycGrp )