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Theorem iscyg 18754
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 6499 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2 iscyg.1 . . . 4 𝐵 = (Base‘𝐺)
31, 2syl6eqr 2833 . . 3 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4 fveq2 6499 . . . . . . . 8 (𝑔 = 𝐺 → (.g𝑔) = (.g𝐺))
5 iscyg.2 . . . . . . . 8 · = (.g𝐺)
64, 5syl6eqr 2833 . . . . . . 7 (𝑔 = 𝐺 → (.g𝑔) = · )
76oveqd 6993 . . . . . 6 (𝑔 = 𝐺 → (𝑛(.g𝑔)𝑥) = (𝑛 · 𝑥))
87mpteq2dv 5023 . . . . 5 (𝑔 = 𝐺 → (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)))
98rneqd 5651 . . . 4 (𝑔 = 𝐺 → ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)))
109, 3eqeq12d 2794 . . 3 (𝑔 = 𝐺 → (ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (Base‘𝑔) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
113, 10rexeqbidv 3343 . 2 (𝑔 = 𝐺 → (∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (Base‘𝑔) ↔ ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
12 df-cyg 18753 . 2 CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (Base‘𝑔)}
1311, 12elrab2 3600 1 (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387   = wceq 1507  wcel 2050  wrex 3090  cmpt 5008  ran crn 5408  cfv 6188  (class class class)co 6976  cz 11793  Basecbs 16339  Grpcgrp 17891  .gcmg 18011  CycGrpccyg 18752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2751
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-cnv 5415  df-dm 5417  df-rn 5418  df-iota 6152  df-fv 6196  df-ov 6979  df-cyg 18753
This theorem is referenced by:  iscyg2  18757  iscyg3  18761  cyggrp  18764  cygctb  18766  ghmcyg  18770  ablfac2  18961  zncyg  20397  fincygsubgodexd  40045
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