Theorem iscyg 19854

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 6840	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2		iscyg.1	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3	1, 2	eqtr4di 2789	. . 3 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4		fveq2 6840	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = (.g‘𝐺))
5		iscyg.2	. . . . . . . 8 ⊢ · = (.g‘𝐺)
6	4, 5	eqtr4di 2789	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = · )
7	6	oveqd 7384	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑛(.g‘𝑔)𝑥) = (𝑛 · 𝑥))
8	7	mpteq2dv 5179	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)))
9	8	rneqd 5893	. . . 4 ⊢ (𝑔 = 𝐺 → ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)))
10	9, 3	eqeq12d 2752	. . 3 ⊢ (𝑔 = 𝐺 → (ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
11	3, 10	rexeqbidv 3312	. 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 19853	. 2 ⊢ CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔)}
13	11, 12	elrab2 3637	1 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3061   ↦ cmpt 5166  ran crn 5632  ‘cfv 6498  (class class class)co 7367  ℤcz 12524  Basecbs 17179  Grpcgrp 18909  ._gcmg 19043  CycGrpccyg 19852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-cnv 5639  df-dm 5641  df-rn 5642  df-iota 6454  df-fv 6506  df-ov 7370  df-cyg 19853

This theorem is referenced by:  iscyg2  19857  iscyg3  19861  cyggrp  19865  cygctb  19867  ghmcyg  19871  ablfac2  20066  fincygsubgodexd  20090  zncyg  21528