Theorem cyggrp 19012

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 2824	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2824	. . 3 ⊢ (.g‘𝐺) = (.g‘𝐺)
3	1, 2	iscyg 19001	. 2 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = (Base‘𝐺)))
4	3	simplbi 500	1 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2113  ∃wrex 3142   ↦ cmpt 5149  ran crn 5559  ‘cfv 6358  (class class class)co 7159  ℤcz 11984  Basecbs 16486  Grpcgrp 18106  ._gcmg 18227  CycGrpccyg 18999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-cnv 5566  df-dm 5568  df-rn 5569  df-iota 6317  df-fv 6366  df-ov 7162  df-cyg 19000

This theorem is referenced by:  fincygsubgodexd  19238  cygznlem1  20716  cygznlem2a  20717  cygznlem3  20719  prmsimpcyc  30860