Theorem iscyggen 19949

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 487	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑛 ∈ ℤ) → 𝑥 = 𝑋)
2	1	oveq2d 7427	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑛 ∈ ℤ) → (𝑛 · 𝑥) = (𝑛 · 𝑋))
3	2	mpteq2dva 5208	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)))
4	3	rneqd 5929	. . 3 ⊢ (𝑥 = 𝑋 → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)))
5	4	eqeq1d 2771	. 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 3663	1 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {crab 3423   ↦ cmpt 5196  ran crn 5663  ‘cfv 6537  (class class class)co 7411  ℤcz 12590  Basecbs 17268  ._gcmg 19132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-cnv 5670  df-dm 5672  df-rn 5673  df-iota 6493  df-fv 6545  df-ov 7414

This theorem is referenced by:  iscyggen2  19950  cyggenod  19953  cyggenod2  19954  cygznlem1  21684  cygznlem3  21687