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Theorem iscyg 18327
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 6229 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2 iscyg.1 . . . 4 𝐵 = (Base‘𝐺)
31, 2syl6eqr 2703 . . 3 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4 fveq2 6229 . . . . . . . 8 (𝑔 = 𝐺 → (.g𝑔) = (.g𝐺))
5 iscyg.2 . . . . . . . 8 · = (.g𝐺)
64, 5syl6eqr 2703 . . . . . . 7 (𝑔 = 𝐺 → (.g𝑔) = · )
76oveqd 6707 . . . . . 6 (𝑔 = 𝐺 → (𝑛(.g𝑔)𝑥) = (𝑛 · 𝑥))
87mpteq2dv 4778 . . . . 5 (𝑔 = 𝐺 → (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)))
98rneqd 5385 . . . 4 (𝑔 = 𝐺 → ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)))
109, 3eqeq12d 2666 . . 3 (𝑔 = 𝐺 → (ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (Base‘𝑔) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
113, 10rexeqbidv 3183 . 2 (𝑔 = 𝐺 → (∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (Base‘𝑔) ↔ ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
12 df-cyg 18326 . 2 CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (Base‘𝑔)}
1311, 12elrab2 3399 1 (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wcel 2030  wrex 2942  cmpt 4762  ran crn 5144  cfv 5926  (class class class)co 6690  cz 11415  Basecbs 15904  Grpcgrp 17469  .gcmg 17587  CycGrpccyg 18325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-cnv 5151  df-dm 5153  df-rn 5154  df-iota 5889  df-fv 5934  df-ov 6693  df-cyg 18326
This theorem is referenced by:  iscyg2  18330  iscyg3  18334  cyggrp  18337  cygctb  18339  ghmcyg  18343  ablfac2  18534  zncyg  19945
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