Theorem cyggrp 19966

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 2762	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2762	. . 3 ⊢ (.g‘𝐺) = (.g‘𝐺)
3	1, 2	iscyg 19955	. 2 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = (Base‘𝐺)))
4	3	simplbi 501	1 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  ∃wrex 3088   ↦ cmpt 5191  ran crn 5661  ‘cfv 6536  (class class class)co 7412  ℤcz 12597  Basecbs 17275  Grpcgrp 19006  ._gcmg 19139  CycGrpccyg 19953

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-cnv 5668  df-dm 5670  df-rn 5671  df-iota 6492  df-fv 6544  df-ov 7415  df-cyg 19954

This theorem is used by:  fincygsubgodexd  20191  cygznlem1  21727  cygznlem2a  21728  cygznlem3  21730  prmsimpcyc  33557