Theorem cyggrp 19801

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

Assertion

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

Proof of Theorem cyggrp

Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2730	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2730	. . 3 ⊢ (.g‘𝐺) = (.g‘𝐺)
3	1, 2	iscyg 19790	. 2 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = (Base‘𝐺)))
4	3	simplbi 496	1 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2104  ∃wrex 3068   ↦ cmpt 5232  ran crn 5678  ‘cfv 6544  (class class class)co 7413  ℤcz 12564  Basecbs 17150  Grpcgrp 18857  ._gcmg 18988  CycGrpccyg 19788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-cnv 5685  df-dm 5687  df-rn 5688  df-iota 6496  df-fv 6552  df-ov 7416  df-cyg 19789

This theorem is referenced by:  fincygsubgodexd  20026  cygznlem1  21343  cygznlem2a  21344  cygznlem3  21346  prmsimpcyc  32641