Theorem cyggrp 20104

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 2760	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2760	. . 3 ⊢ (.g‘𝐺) = (.g‘𝐺)
3	1, 2	iscyg 20093	. 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 1569   ∈ wcel 2144  ∃wrex 3086   ↦ cmpt 5185  ran crn 5651  ‘cfv 6538  (class class class)co 7420  ℤcz 12693  Basecbs 17387  Grpcgrp 19144  ._gcmg 19277  CycGrpccyg 20091

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-cnv 5658  df-dm 5660  df-rn 5661  df-iota 6494  df-fv 6546  df-ov 7423  df-cyg 20092

This theorem is used by:  fincygsubgodexd  20329  cygznlem1  21872  cygznlem2a  21873  cygznlem3  21875  prmsimpcyc  33789