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Mirrors > Home > MPE Home > Th. List > iscyggen | Structured version Visualization version GIF version |
Description: The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
iscyg.1 | ⊢ 𝐵 = (Base‘𝐺) |
iscyg.2 | ⊢ · = (.g‘𝐺) |
iscyg3.e | ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} |
Ref | Expression |
---|---|
iscyggen | ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 483 | . . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑛 ∈ ℤ) → 𝑥 = 𝑋) | |
2 | 1 | oveq2d 7367 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑛 ∈ ℤ) → (𝑛 · 𝑥) = (𝑛 · 𝑋)) |
3 | 2 | mpteq2dva 5203 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))) |
4 | 3 | rneqd 5891 | . . 3 ⊢ (𝑥 = 𝑋 → ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋))) |
5 | 4 | eqeq1d 2739 | . 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 3646 | 1 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3405 ↦ cmpt 5186 ran crn 5632 ‘cfv 6493 (class class class)co 7351 ℤcz 12457 Basecbs 17043 .gcmg 18831 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-cnv 5639 df-dm 5641 df-rn 5642 df-iota 6445 df-fv 6501 df-ov 7354 |
This theorem is referenced by: iscyggen2 19617 cyggenod 19620 cyggenod2 19621 cygznlem1 20926 cygznlem3 20929 |
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