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Theorem cyggrp 18114
 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 2610 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2610 . . 3 (.g𝐺) = (.g𝐺)
31, 2iscyg 18104 . 2 (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = (Base‘𝐺)))
43simplbi 475 1 (𝐺 ∈ CycGrp → 𝐺 ∈ Grp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ↦ cmpt 4643  ran crn 5039  ‘cfv 5804  (class class class)co 6549  ℤcz 11254  Basecbs 15695  Grpcgrp 17245  .gcmg 17363  CycGrpccyg 18102 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-cnv 5046  df-dm 5048  df-rn 5049  df-iota 5768  df-fv 5812  df-ov 6552  df-cyg 18103 This theorem is referenced by:  cygznlem1  19734  cygznlem2a  19735  cygznlem3  19737
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