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Theorem cyggrp 19923
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.
StepHypRef Expression
1 eqid 2735 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2735 . . 3 (.g𝐺) = (.g𝐺)
31, 2iscyg 19912 . 2 (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = (Base‘𝐺)))
43simplbi 497 1 (𝐺 ∈ CycGrp → 𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wrex 3068  cmpt 5231  ran crn 5690  cfv 6563  (class class class)co 7431  cz 12611  Basecbs 17245  Grpcgrp 18964  .gcmg 19098  CycGrpccyg 19910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-cnv 5697  df-dm 5699  df-rn 5700  df-iota 6516  df-fv 6571  df-ov 7434  df-cyg 19911
This theorem is referenced by:  fincygsubgodexd  20148  cygznlem1  21603  cygznlem2a  21604  cygznlem3  21606  prmsimpcyc  33217
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