Theorem iscyg 19776

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 6826	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2		iscyg.1	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3	1, 2	eqtr4di 2782	. . 3 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4		fveq2 6826	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = (.g‘𝐺))
5		iscyg.2	. . . . . . . 8 ⊢ · = (.g‘𝐺)
6	4, 5	eqtr4di 2782	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (.g‘𝑔) = · )
7	6	oveqd 7370	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑛(.g‘𝑔)𝑥) = (𝑛 · 𝑥))
8	7	mpteq2dv 5189	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)))
9	8	rneqd 5884	. . . 4 ⊢ (𝑔 = 𝐺 → ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)))
10	9, 3	eqeq12d 2745	. . 3 ⊢ (𝑔 = 𝐺 → (ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔) ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
11	3, 10	rexeqbidv 3311	. 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 19775	. 2 ⊢ CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔)}
13	11, 12	elrab2 3653	1 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ↦ cmpt 5176  ran crn 5624  ‘cfv 6486  (class class class)co 7353  ℤcz 12489  Basecbs 17138  Grpcgrp 18830  ._gcmg 18964  CycGrpccyg 19774

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-cnv 5631  df-dm 5633  df-rn 5634  df-iota 6442  df-fv 6494  df-ov 7356  df-cyg 19775

This theorem is referenced by:  iscyg2  19779  iscyg3  19783  cyggrp  19787  cygctb  19789  ghmcyg  19793  ablfac2  19988  fincygsubgodexd  20012  zncyg  21473