Theorem iscyg2 19004

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 19001	. 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 3083	. . . 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 624	. 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 398   = wceq 1536   ∈ wcel 2113   ≠ wne 3019  ∃wrex 3142  {crab 3145  ∅c0 4294   ↦ cmpt 5149  ran crn 5559  ‘cfv 6358  (class class class)co 7159  ℤcz 11984  Basecbs 16486  Grpcgrp 18106  ._gcmg 18227  CycGrpccyg 18999

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-cnv 5566  df-dm 5568  df-rn 5569  df-iota 6317  df-fv 6366  df-ov 7162  df-cyg 19000

This theorem is referenced by:  iscygd  19009  iscygodd  19010  cyggex2  19020  cyggexb  19022  cygzn  20720