Theorem cyggrp

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

	Ref	Expression
Assertion	cyggrp	⊢ ( 𝐺 ∈ CycGrp → 𝐺 ∈ Grp )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
2		eqid	⊢ ( .g ‘ 𝐺 ) = ( .g ‘ 𝐺 )
3	1 2	iscyg	⊢ ( 𝐺 ∈ CycGrp ↔ ( 𝐺 ∈ Grp ∧ ∃ 𝑥 ∈ ( Base ‘ 𝐺 ) ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝐺 ) 𝑥 ) ) = ( Base ‘ 𝐺 ) ) )
4	3	simplbi	⊢ ( 𝐺 ∈ CycGrp → 𝐺 ∈ Grp )