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Theorem iscyggen 19898
Description: The property of being a cyclic generator for a group. (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
iscyggen (𝑋𝐸 ↔ (𝑋𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
Distinct variable groups:   𝑥,𝑛,𝐵   𝑛,𝑋,𝑥   𝑛,𝐺,𝑥   · ,𝑛,𝑥
Allowed substitution hints:   𝐸(𝑥,𝑛)

Proof of Theorem iscyggen
StepHypRef Expression
1 simpl 482 . . . . . 6 ((𝑥 = 𝑋𝑛 ∈ ℤ) → 𝑥 = 𝑋)
21oveq2d 7447 . . . . 5 ((𝑥 = 𝑋𝑛 ∈ ℤ) → (𝑛 · 𝑥) = (𝑛 · 𝑋))
32mpteq2dva 5242 . . . 4 (𝑥 = 𝑋 → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)))
43rneqd 5949 . . 3 (𝑥 = 𝑋 → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)))
54eqeq1d 2739 . 2 (𝑥 = 𝑋 → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵 ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
6 iscyg3.e . 2 𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}
75, 6elrab2 3695 1 (𝑋𝐸 ↔ (𝑋𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  {crab 3436  cmpt 5225  ran crn 5686  cfv 6561  (class class class)co 7431  cz 12613  Basecbs 17247  .gcmg 19085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-cnv 5693  df-dm 5695  df-rn 5696  df-iota 6514  df-fv 6569  df-ov 7434
This theorem is referenced by:  iscyggen2  19899  cyggenod  19902  cyggenod2  19903  cygznlem1  21585  cygznlem3  21588
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