Theorem cyggrp

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

	Ref	Expression
Assertion	cyggrp	$$⊢G\in \mathrm{CycGrp}\to G\in \mathrm{Grp}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢{\mathrm{Base}}_G={\mathrm{Base}}_G$$
2		eqid	$$⊢{\cdot }_G={\cdot }_G$$
3	1 2	iscyg	$$⊢G\in \mathrm{CycGrp}\leftrightarrow \left(G\in \mathrm{Grp}\wedge \exists x\in {\mathrm{Base}}_G\mathrm{ran}\left(n\in \mathbb{Z}⟼n{\cdot }_{G}x\right)={\mathrm{Base}}_G\right)$$
4	3	simplbi	$$⊢G\in \mathrm{CycGrp}\to G\in \mathrm{Grp}$$