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Mirrors > Home > MPE Home > Th. List > iscyg2 | Structured version Visualization version GIF version |
Description: A cyclic group is a group which contains a generator. (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 |
---|---|
iscyg2 | ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ 𝐸 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscyg.1 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | iscyg.2 | . . 3 ⊢ · = (.g‘𝐺) | |
3 | 1, 2 | iscyg 18991 | . 2 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵)) |
4 | iscyg3.e | . . . . 5 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} | |
5 | 4 | neeq1i 3051 | . . . 4 ⊢ (𝐸 ≠ ∅ ↔ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} ≠ ∅) |
6 | rabn0 4293 | . . . 4 ⊢ ({𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} ≠ ∅ ↔ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵) | |
7 | 5, 6 | bitri 278 | . . 3 ⊢ (𝐸 ≠ ∅ ↔ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵) |
8 | 7 | anbi2i 625 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐸 ≠ ∅) ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵)) |
9 | 3, 8 | bitr4i 281 | 1 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ 𝐸 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 {crab 3110 ∅c0 4243 ↦ cmpt 5110 ran crn 5520 ‘cfv 6324 (class class class)co 7135 ℤcz 11969 Basecbs 16475 Grpcgrp 18095 .gcmg 18216 CycGrpccyg 18989 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-cnv 5527 df-dm 5529 df-rn 5530 df-iota 6283 df-fv 6332 df-ov 7138 df-cyg 18990 |
This theorem is referenced by: iscygd 18999 iscygodd 19000 cyggex2 19010 cyggexb 19012 cygzn 20262 |
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