MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscyg2 Structured version   Visualization version   GIF version

Theorem iscyg2 18980
Description: A cyclic group is a group which contains a generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
iscyg.1 𝐵 = (Base‘𝐺)
iscyg.2 · = (.g𝐺)
iscyg3.e 𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}
Assertion
Ref Expression
iscyg2 (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ 𝐸 ≠ ∅))
Distinct variable groups:   𝑥,𝑛,𝐵   𝑛,𝐺,𝑥   · ,𝑛,𝑥
Allowed substitution hints:   𝐸(𝑥,𝑛)

Proof of Theorem iscyg2
StepHypRef Expression
1 iscyg.1 . . 3 𝐵 = (Base‘𝐺)
2 iscyg.2 . . 3 · = (.g𝐺)
31, 2iscyg 18977 . 2 (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
4 iscyg3.e . . . . 5 𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}
54neeq1i 3071 . . . 4 (𝐸 ≠ ∅ ↔ {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} ≠ ∅)
6 rabn0 4312 . . . 4 ({𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} ≠ ∅ ↔ ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵)
75, 6bitri 278 . . 3 (𝐸 ≠ ∅ ↔ ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵)
87anbi2i 625 . 2 ((𝐺 ∈ Grp ∧ 𝐸 ≠ ∅) ↔ (𝐺 ∈ Grp ∧ ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
93, 8bitr4i 281 1 (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ 𝐸 ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2115  wne 3007  wrex 3127  {crab 3130  c0 4266  cmpt 5119  ran crn 5529  cfv 6328  (class class class)co 7130  cz 11959  Basecbs 16462  Grpcgrp 18082  .gcmg 18203  CycGrpccyg 18975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-cnv 5536  df-dm 5538  df-rn 5539  df-iota 6287  df-fv 6336  df-ov 7133  df-cyg 18976
This theorem is referenced by:  iscygd  18985  iscygodd  18986  cyggex2  18996  cyggexb  18998  cygzn  20693
  Copyright terms: Public domain W3C validator