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Mirrors > Home > MPE Home > Th. List > df-lpir | Structured version Visualization version GIF version |
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 19063 | . 2 class LPIR | |
2 | vw | . . . . . 6 setvar 𝑤 | |
3 | 2 | cv 1474 | . . . . 5 class 𝑤 |
4 | clidl 18991 | . . . . 5 class LIdeal | |
5 | 3, 4 | cfv 5804 | . . . 4 class (LIdeal‘𝑤) |
6 | clpidl 19062 | . . . . 5 class LPIdeal | |
7 | 3, 6 | cfv 5804 | . . . 4 class (LPIdeal‘𝑤) |
8 | 5, 7 | wceq 1475 | . . 3 wff (LIdeal‘𝑤) = (LPIdeal‘𝑤) |
9 | crg 18370 | . . 3 class Ring | |
10 | 8, 2, 9 | crab 2900 | . 2 class {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)} |
11 | 1, 10 | wceq 1475 | 1 wff LPIR = {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)} |
Colors of variables: wff setvar class |
This definition is referenced by: islpir 19070 |
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