Theorem cyggrp 20131

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 2763	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2763	. . 3 ⊢ (.g‘𝐺) = (.g‘𝐺)
3	1, 2	iscyg 20120	. 2 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = (Base‘𝐺)))
4	3	simplbi 502	1 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ↦ cmpt 5189  ran crn 5656  ‘cfv 6555  (class class class)co 7440  ℤcz 12717  Basecbs 17413  Grpcgrp 19170  ._gcmg 19303  CycGrpccyg 20118

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-cnv 5663  df-dm 5665  df-rn 5666  df-iota 6509  df-fv 6563  df-ov 7443  df-cyg 20119

This theorem is used by:  fincygsubgodexd  20356  cygznlem1  21899  cygznlem2a  21900  cygznlem3  21902  prmsimpcyc  33856