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Theorem cyggrp 19908
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 2737 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2737 . . 3 (.g𝐺) = (.g𝐺)
31, 2iscyg 19897 . 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 1540  wcel 2108  wrex 3070  cmpt 5225  ran crn 5686  cfv 6561  (class class class)co 7431  cz 12613  Basecbs 17247  Grpcgrp 18951  .gcmg 19085  CycGrpccyg 19895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-cnv 5693  df-dm 5695  df-rn 5696  df-iota 6514  df-fv 6569  df-ov 7434  df-cyg 19896
This theorem is referenced by:  fincygsubgodexd  20133  cygznlem1  21585  cygznlem2a  21586  cygznlem3  21588  prmsimpcyc  33234
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