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| Description: Define the class of left principal ideal rings, rings where every left ideal has a single generator. (Contributed by Stefan O'Rear, 3-Jan-2015.) | 
| Ref | Expression | 
|---|---|
| df-lpir | ⊢ LPIR = {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | clpir 21331 | . 2 class LPIR | |
| 2 | vw | . . . . . 6 setvar 𝑤 | |
| 3 | 2 | cv 1539 | . . . . 5 class 𝑤 | 
| 4 | clidl 21216 | . . . . 5 class LIdeal | |
| 5 | 3, 4 | cfv 6561 | . . . 4 class (LIdeal‘𝑤) | 
| 6 | clpidl 21330 | . . . . 5 class LPIdeal | |
| 7 | 3, 6 | cfv 6561 | . . . 4 class (LPIdeal‘𝑤) | 
| 8 | 5, 7 | wceq 1540 | . . 3 wff (LIdeal‘𝑤) = (LPIdeal‘𝑤) | 
| 9 | crg 20230 | . . 3 class Ring | |
| 10 | 8, 2, 9 | crab 3436 | . 2 class {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)} | 
| 11 | 1, 10 | wceq 1540 | 1 wff LPIR = {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)} | 
| Colors of variables: wff setvar class | 
| This definition is referenced by: islpir 21338 | 
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