Theorem cyggrp 19859

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 2741	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2741	. . 3 ⊢ (.g‘𝐺) = (.g‘𝐺)
3	1, 2	iscyg 19848	. 2 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = (Base‘𝐺)))
4	3	simplbi 498	1 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   = wceq 1548   ∈ wcel 2121  ∃wrex 3065   ↦ cmpt 5155  ran crn 5621  ‘cfv 6488  (class class class)co 7359  ℤcz 12519  Basecbs 17174  Grpcgrp 18904  ._gcmg 19038  CycGrpccyg 19846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-mpt 5156  df-cnv 5628  df-dm 5630  df-rn 5631  df-iota 6444  df-fv 6496  df-ov 7362  df-cyg 19847

This theorem is referenced by:  fincygsubgodexd  20084  cygznlem1  21544  cygznlem2a  21545  cygznlem3  21547  prmsimpcyc  33311