Theorem iscyg 19479

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 6774	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2		iscyg.1	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3	1, 2	eqtr4di 2796	. . 3 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4		fveq2 6774	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = (.g‘𝐺))
5		iscyg.2	. . . . . . . 8 ⊢ · = (.g‘𝐺)
6	4, 5	eqtr4di 2796	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = · )
7	6	oveqd 7292	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑛(.g‘𝑔)𝑥) = (𝑛 · 𝑥))
8	7	mpteq2dv 5176	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)))
9	8	rneqd 5847	. . . 4 ⊢ (𝑔 = 𝐺 → ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)))
10	9, 3	eqeq12d 2754	. . 3 ⊢ (𝑔 = 𝐺 → (ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
11	3, 10	rexeqbidv 3337	. 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 19478	. 2 ⊢ CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔)}
13	11, 12	elrab2 3627	1 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065   ↦ cmpt 5157  ran crn 5590  ‘cfv 6433  (class class class)co 7275  ℤcz 12319  Basecbs 16912  Grpcgrp 18577  ._gcmg 18700  CycGrpccyg 19477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-cnv 5597  df-dm 5599  df-rn 5600  df-iota 6391  df-fv 6441  df-ov 7278  df-cyg 19478

This theorem is referenced by:  iscyg2  19482  iscyg3  19486  cyggrp  19490  cygctb  19493  ghmcyg  19497  ablfac2  19692  fincygsubgodexd  19716  zncyg  20756