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Theorem iscyggen 19913
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 5248 . . . 4 (𝑥 = 𝑋 → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)))
43rneqd 5952 . . 3 (𝑥 = 𝑋 → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)))
54eqeq1d 2737 . 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 3698 1 (𝑋𝐸 ↔ (𝑋𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  {crab 3433  cmpt 5231  ran crn 5690  cfv 6563  (class class class)co 7431  cz 12611  Basecbs 17245  .gcmg 19098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-cnv 5697  df-dm 5699  df-rn 5700  df-iota 6516  df-fv 6571  df-ov 7434
This theorem is referenced by:  iscyggen2  19914  cyggenod  19917  cyggenod2  19918  cygznlem1  21603  cygznlem3  21606
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