Theorem iscyg2 19905

Description: A cyclic group is a group which contains a generator. (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
iscyg2	⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ 𝐸 ≠ ∅))

Distinct variable groups:   𝑥,𝑛,𝐵   𝑛,𝐺,𝑥   · ,𝑛,𝑥

Allowed substitution hints:   𝐸(𝑥,𝑛)

Proof of Theorem iscyg2

Step	Hyp	Ref	Expression
1		iscyg.1	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		iscyg.2	. . 3 ⊢ · = (.g‘𝐺)
3	1, 2	iscyg 19902	. 2 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
4		iscyg3.e	. . . . 5 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}
5	4	neeq1i 3020	. . . 4 ⊢ (𝐸 ≠ ∅ ↔ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} ≠ ∅)
6		rabn0 4342	. . . 4 ⊢ ({𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} ≠ ∅ ↔ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵)
7	5, 6	bitri 277	. . 3 ⊢ (𝐸 ≠ ∅ ↔ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵)
8	7	anbi2i 632	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐸 ≠ ∅) ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
9	3, 8	bitr4i 280	1 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ 𝐸 ≠ ∅))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∃wrex 3085  {crab 3413  ∅c0 4285   ↦ cmpt 5180  ran crn 5646  ‘cfv 6517  (class class class)co 7392  ℤcz 12565  Basecbs 17228  Grpcgrp 18958  ._gcmg 19092  CycGrpccyg 19900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-cnv 5653  df-dm 5655  df-rn 5656  df-iota 6473  df-fv 6525  df-ov 7395  df-cyg 19901

This theorem is referenced by:  iscygd  19910  iscygodd  19911  cyggex2  19920  cyggexb  19922  cygzn  21602