Theorem cyggrp 19970

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 2766	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2766	. . 3 ⊢ (.g‘𝐺) = (.g‘𝐺)
3	1, 2	iscyg 19959	. 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 1570   ∈ wcel 2146  ∃wrex 3092   ↦ cmpt 5195  ran crn 5665  ‘cfv 6540  (class class class)co 7416  ℤcz 12601  Basecbs 17279  Grpcgrp 19010  ._gcmg 19143  CycGrpccyg 19957

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4876  df-br 5113  df-opab 5177  df-mpt 5196  df-cnv 5672  df-dm 5674  df-rn 5675  df-iota 6496  df-fv 6548  df-ov 7419  df-cyg 19958

This theorem is used by:  fincygsubgodexd  20195  cygznlem1  21731  cygznlem2a  21732  cygznlem3  21734  prmsimpcyc  33561