Theorem iscyg 19833

Description: Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
iscyg.1	⊢ 𝐵 = (Base‘𝐺)
iscyg.2	⊢ · = (.g‘𝐺)

Assertion

Ref	Expression
iscyg	⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))

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

Proof of Theorem iscyg

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6897	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2		iscyg.1	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3	1, 2	eqtr4di 2786	. . 3 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4		fveq2 6897	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = (.g‘𝐺))
5		iscyg.2	. . . . . . . 8 ⊢ · = (.g‘𝐺)
6	4, 5	eqtr4di 2786	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = · )
7	6	oveqd 7437	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑛(.g‘𝑔)𝑥) = (𝑛 · 𝑥))
8	7	mpteq2dv 5250	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)))
9	8	rneqd 5940	. . . 4 ⊢ (𝑔 = 𝐺 → ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)))
10	9, 3	eqeq12d 2744	. . 3 ⊢ (𝑔 = 𝐺 → (ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
11	3, 10	rexeqbidv 3340	. 2 ⊢ (𝑔 = 𝐺 → (∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔) ↔ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
12		df-cyg 19832	. 2 ⊢ CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔)}
13	11, 12	elrab2 3685	1 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∃wrex 3067   ↦ cmpt 5231  ran crn 5679  ‘cfv 6548  (class class class)co 7420  ℤcz 12588  Basecbs 17179  Grpcgrp 18889  ._gcmg 19022  CycGrpccyg 19831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-cnv 5686  df-dm 5688  df-rn 5689  df-iota 6500  df-fv 6556  df-ov 7423  df-cyg 19832

This theorem is referenced by:  iscyg2  19836  iscyg3  19840  cyggrp  19844  cygctb  19846  ghmcyg  19850  ablfac2  20045  fincygsubgodexd  20069  zncyg  21481