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Theorem iscyg 18197
Description: Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
iscyg.1 𝐵 = (Base‘𝐺)
iscyg.2 · = (.g𝐺)
Assertion
Ref Expression
iscyg (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
Distinct variable groups:   𝑥,𝑛,𝐵   𝑛,𝐺,𝑥   · ,𝑛,𝑥

Proof of Theorem iscyg
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6150 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2 iscyg.1 . . . 4 𝐵 = (Base‘𝐺)
31, 2syl6eqr 2678 . . 3 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4 fveq2 6150 . . . . . . . 8 (𝑔 = 𝐺 → (.g𝑔) = (.g𝐺))
5 iscyg.2 . . . . . . . 8 · = (.g𝐺)
64, 5syl6eqr 2678 . . . . . . 7 (𝑔 = 𝐺 → (.g𝑔) = · )
76oveqd 6622 . . . . . 6 (𝑔 = 𝐺 → (𝑛(.g𝑔)𝑥) = (𝑛 · 𝑥))
87mpteq2dv 4710 . . . . 5 (𝑔 = 𝐺 → (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)))
98rneqd 5317 . . . 4 (𝑔 = 𝐺 → ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)))
109, 3eqeq12d 2641 . . 3 (𝑔 = 𝐺 → (ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (Base‘𝑔) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
113, 10rexeqbidv 3147 . 2 (𝑔 = 𝐺 → (∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (Base‘𝑔) ↔ ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
12 df-cyg 18196 . 2 CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (Base‘𝑔)}
1311, 12elrab2 3353 1 (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1992  wrex 2913  cmpt 4678  ran crn 5080  cfv 5850  (class class class)co 6605  cz 11322  Basecbs 15776  Grpcgrp 17338  .gcmg 17456  CycGrpccyg 18195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-cnv 5087  df-dm 5089  df-rn 5090  df-iota 5813  df-fv 5858  df-ov 6608  df-cyg 18196
This theorem is referenced by:  iscyg2  18200  iscyg3  18204  cyggrp  18207  cygctb  18209  ghmcyg  18213  ablfac2  18404  zncyg  19811
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