Theorem iscyggen 19922

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	⊢ · = (.g‘𝐺)
iscyg3.e	⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}

Assertion

Ref	Expression
iscyggen	⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ 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

Step	Hyp	Ref	Expression
1		simpl 482	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑛 ∈ ℤ) → 𝑥 = 𝑋)
2	1	oveq2d 7464	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑛 ∈ ℤ) → (𝑛 · 𝑥) = (𝑛 · 𝑋))
3	2	mpteq2dva 5266	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)))
4	3	rneqd 5963	. . 3 ⊢ (𝑥 = 𝑋 → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)))
5	4	eqeq1d 2742	. 2 ⊢ (𝑥 = 𝑋 → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵 ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
6		iscyg3.e	. 2 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}
7	5, 6	elrab2 3711	1 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {crab 3443   ↦ cmpt 5249  ran crn 5701  ‘cfv 6573  (class class class)co 7448  ℤcz 12639  Basecbs 17258  ._gcmg 19107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-cnv 5708  df-dm 5710  df-rn 5711  df-iota 6525  df-fv 6581  df-ov 7451

This theorem is referenced by:  iscyggen2  19923  cyggenod  19926  cyggenod2  19927  cygznlem1  21608  cygznlem3  21611