Theorem cyggrp 19932

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 2740	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2740	. . 3 ⊢ (.g‘𝐺) = (.g‘𝐺)
3	1, 2	iscyg 19921	. 2 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = (Base‘𝐺)))
4	3	simplbi 497	1 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ∃wrex 3076   ↦ cmpt 5249  ran crn 5701  ‘cfv 6573  (class class class)co 7448  ℤcz 12639  Basecbs 17258  Grpcgrp 18973  ._gcmg 19107  CycGrpccyg 19919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-cnv 5708  df-dm 5710  df-rn 5711  df-iota 6525  df-fv 6581  df-ov 7451  df-cyg 19920

This theorem is referenced by:  fincygsubgodexd  20157  cygznlem1  21608  cygznlem2a  21609  cygznlem3  21611  prmsimpcyc  33207