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Metamath Proof Explorer

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
3 1 2 iscyg ⊢ G ∈ CycGrp ↔ G ∈ Grp ∧ ∃ x ∈ Base G ran ⁡ n ∈ ℤ ⟼ n ⋅ G x = Base G
4 3 simplbi ⊢ G ∈ CycGrp → G ∈ Grp