Theorem iscyggen 18999

Description: The property of being a cyclic generator for a group. (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
iscyggen	⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))

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

Allowed substitution hints:   𝐸(𝑥,𝑛)

Proof of Theorem iscyggen

Step	Hyp	Ref	Expression
1		simpl 485	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑛 ∈ ℤ) → 𝑥 = 𝑋)
2	1	oveq2d 7172	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑛 ∈ ℤ) → (𝑛 · 𝑥) = (𝑛 · 𝑋))
3	2	mpteq2dva 5161	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)))
4	3	rneqd 5808	. . 3 ⊢ (𝑥 = 𝑋 → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)))
5	4	eqeq1d 2823	. 2 ⊢ (𝑥 = 𝑋 → (ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵 ↔ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
6		iscyg3.e	. 2 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}
7	5, 6	elrab2 3683	1 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {crab 3142   ↦ cmpt 5146  ran crn 5556  ‘cfv 6355  (class class class)co 7156  ℤcz 11982  Basecbs 16483  ._gcmg 18224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-cnv 5563  df-dm 5565  df-rn 5566  df-iota 6314  df-fv 6363  df-ov 7159

This theorem is referenced by:  iscyggen2  19000  cyggenod  19003  cyggenod2  19004  cygznlem1  20713  cygznlem3  20716