MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscyg Structured version   Visualization version   GIF version

Theorem iscyg 19263
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 6717 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2 iscyg.1 . . . 4 𝐵 = (Base‘𝐺)
31, 2eqtr4di 2796 . . 3 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4 fveq2 6717 . . . . . . . 8 (𝑔 = 𝐺 → (.g𝑔) = (.g𝐺))
5 iscyg.2 . . . . . . . 8 · = (.g𝐺)
64, 5eqtr4di 2796 . . . . . . 7 (𝑔 = 𝐺 → (.g𝑔) = · )
76oveqd 7230 . . . . . 6 (𝑔 = 𝐺 → (𝑛(.g𝑔)𝑥) = (𝑛 · 𝑥))
87mpteq2dv 5151 . . . . 5 (𝑔 = 𝐺 → (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)))
98rneqd 5807 . . . 4 (𝑔 = 𝐺 → ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)))
109, 3eqeq12d 2753 . . 3 (𝑔 = 𝐺 → (ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (Base‘𝑔) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
113, 10rexeqbidv 3314 . 2 (𝑔 = 𝐺 → (∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (Base‘𝑔) ↔ ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
12 df-cyg 19262 . 2 CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (Base‘𝑔)}
1311, 12elrab2 3605 1 (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1543  wcel 2110  wrex 3062  cmpt 5135  ran crn 5552  cfv 6380  (class class class)co 7213  cz 12176  Basecbs 16760  Grpcgrp 18365  .gcmg 18488  CycGrpccyg 19261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-cnv 5559  df-dm 5561  df-rn 5562  df-iota 6338  df-fv 6388  df-ov 7216  df-cyg 19262
This theorem is referenced by:  iscyg2  19266  iscyg3  19270  cyggrp  19274  cygctb  19277  ghmcyg  19281  ablfac2  19476  fincygsubgodexd  19500  zncyg  20513
  Copyright terms: Public domain W3C validator