Theorem iscyg

Description: Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016)

	Ref	Expression
Hypotheses	iscyg.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
	iscyg.2	⊢ · = ( .g ‘ 𝐺 )
Assertion	iscyg	⊢ ( 𝐺 ∈ CycGrp ↔ ( 𝐺 ∈ Grp ∧ ∃ 𝑥 ∈ 𝐵 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		iscyg.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		iscyg.2	⊢ · = ( .g ‘ 𝐺 )
3		fveq2	⊢ ( 𝑔 = 𝐺 → ( Base ‘ 𝑔 ) = ( Base ‘ 𝐺 ) )
4	3 1	eqtr4di	⊢ ( 𝑔 = 𝐺 → ( Base ‘ 𝑔 ) = 𝐵 )
5		fveq2	⊢ ( 𝑔 = 𝐺 → ( .g ‘ 𝑔 ) = ( .g ‘ 𝐺 ) )
6	5 2	eqtr4di	⊢ ( 𝑔 = 𝐺 → ( .g ‘ 𝑔 ) = · )
7	6	oveqd	⊢ ( 𝑔 = 𝐺 → ( 𝑛 ( .g ‘ 𝑔 ) 𝑥 ) = ( 𝑛 · 𝑥 ) )
8	7	mpteq2dv	⊢ ( 𝑔 = 𝐺 → ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝑔 ) 𝑥 ) ) = ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) )
9	8	rneqd	⊢ ( 𝑔 = 𝐺 → ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝑔 ) 𝑥 ) ) = ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) )
10	9 4	eqeq12d	⊢ ( 𝑔 = 𝐺 → ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝑔 ) 𝑥 ) ) = ( Base ‘ 𝑔 ) ↔ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = 𝐵 ) )
11	4 10	rexeqbidv	⊢ ( 𝑔 = 𝐺 → ( ∃ 𝑥 ∈ ( Base ‘ 𝑔 ) ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝑔 ) 𝑥 ) ) = ( Base ‘ 𝑔 ) ↔ ∃ 𝑥 ∈ 𝐵 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = 𝐵 ) )
12		df-cyg	⊢ CycGrp = { 𝑔 ∈ Grp ∣ ∃ 𝑥 ∈ ( Base ‘ 𝑔 ) ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝑔 ) 𝑥 ) ) = ( Base ‘ 𝑔 ) }
13	11 12	elrab2	⊢ ( 𝐺 ∈ CycGrp ↔ ( 𝐺 ∈ Grp ∧ ∃ 𝑥 ∈ 𝐵 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = 𝐵 ) )