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Theorem cyggrp 19758
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 2733 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2733 . . 3 (.g𝐺) = (.g𝐺)
31, 2iscyg 19747 . 2 (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)𝑥)) = (Base‘𝐺)))
43simplbi 499 1 (𝐺 ∈ CycGrp → 𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wrex 3071  cmpt 5232  ran crn 5678  cfv 6544  (class class class)co 7409  cz 12558  Basecbs 17144  Grpcgrp 18819  .gcmg 18950  CycGrpccyg 19745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-cnv 5685  df-dm 5687  df-rn 5688  df-iota 6496  df-fv 6552  df-ov 7412  df-cyg 19746
This theorem is referenced by:  fincygsubgodexd  19983  cygznlem1  21122  cygznlem2a  21123  cygznlem3  21125  prmsimpcyc  32373
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