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Theorem iscyg 18501
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 6417 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2 iscyg.1 . . . 4 𝐵 = (Base‘𝐺)
31, 2syl6eqr 2869 . . 3 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4 fveq2 6417 . . . . . . . 8 (𝑔 = 𝐺 → (.g𝑔) = (.g𝐺))
5 iscyg.2 . . . . . . . 8 · = (.g𝐺)
64, 5syl6eqr 2869 . . . . . . 7 (𝑔 = 𝐺 → (.g𝑔) = · )
76oveqd 6900 . . . . . 6 (𝑔 = 𝐺 → (𝑛(.g𝑔)𝑥) = (𝑛 · 𝑥))
87mpteq2dv 4950 . . . . 5 (𝑔 = 𝐺 → (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)))
98rneqd 5567 . . . 4 (𝑔 = 𝐺 → ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)))
109, 3eqeq12d 2832 . . 3 (𝑔 = 𝐺 → (ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (Base‘𝑔) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
113, 10rexeqbidv 3353 . 2 (𝑔 = 𝐺 → (∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (Base‘𝑔) ↔ ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
12 df-cyg 18500 . 2 CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (Base‘𝑔)}
1311, 12elrab2 3573 1 (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1637  wcel 2157  wrex 3108  cmpt 4934  ran crn 5325  cfv 6110  (class class class)co 6883  cz 11662  Basecbs 16087  Grpcgrp 17646  .gcmg 17764  CycGrpccyg 18499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-br 4856  df-opab 4918  df-mpt 4935  df-cnv 5332  df-dm 5334  df-rn 5335  df-iota 6073  df-fv 6118  df-ov 6886  df-cyg 18500
This theorem is referenced by:  iscyg2  18504  iscyg3  18508  cyggrp  18511  cygctb  18513  ghmcyg  18517  ablfac2  18709  zncyg  20123
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