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Theorem cyggrp 18272
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 2620 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2620 . . 3 (.g𝐺) = (.g𝐺)
31, 2iscyg 18262 . 2 (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = (Base‘𝐺)))
43simplbi 476 1 (𝐺 ∈ CycGrp → 𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  wrex 2910  cmpt 4720  ran crn 5105  cfv 5876  (class class class)co 6635  cz 11362  Basecbs 15838  Grpcgrp 17403  .gcmg 17521  CycGrpccyg 18260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-cnv 5112  df-dm 5114  df-rn 5115  df-iota 5839  df-fv 5884  df-ov 6638  df-cyg 18261
This theorem is referenced by:  cygznlem1  19896  cygznlem2a  19897  cygznlem3  19899
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