|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 19897 | . 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 3005 | . . . 4 ⊢ (𝐸 ≠ ∅ ↔ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} ≠ ∅) | 
| 6 | rabn0 4389 | . . . 4 ⊢ ({𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} ≠ ∅ ↔ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵) | |
| 7 | 5, 6 | bitri 275 | . . 3 ⊢ (𝐸 ≠ ∅ ↔ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵) | 
| 8 | 7 | anbi2i 623 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐸 ≠ ∅) ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵)) | 
| 9 | 3, 8 | bitr4i 278 | 1 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ 𝐸 ≠ ∅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 {crab 3436 ∅c0 4333 ↦ cmpt 5225 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ℤcz 12613 Basecbs 17247 Grpcgrp 18951 .gcmg 19085 CycGrpccyg 19895 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-cnv 5693 df-dm 5695 df-rn 5696 df-iota 6514 df-fv 6569 df-ov 7434 df-cyg 19896 | 
| This theorem is referenced by: iscygd 19905 iscygodd 19906 cyggex2 19915 cyggexb 19917 cygzn 21589 | 
| Copyright terms: Public domain | W3C validator |