Theorem cyggrp 20065

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 2760	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2760	. . 3 ⊢ (.g‘𝐺) = (.g‘𝐺)
3	1, 2	iscyg 20054	. 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 2145  ∃wrex 3086   ↦ cmpt 5185  ran crn 5648  ‘cfv 6527  (class class class)co 7408  ℤcz 12662  Basecbs 17348  Grpcgrp 19105  ._gcmg 19238  CycGrpccyg 20052

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 2147  ax-9 2155  ax-ext 2732

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 2739  df-cleq 2752  df-clel 2835  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-cnv 5655  df-dm 5657  df-rn 5658  df-iota 6483  df-fv 6535  df-ov 7411  df-cyg 20053

This theorem is used by:  fincygsubgodexd  20290  cygznlem1  21833  cygznlem2a  21834  cygznlem3  21836  prmsimpcyc  33722