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Definition df-lpir 19176
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.)
Assertion
Ref Expression
df-lpir LPIR = {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)}

Detailed syntax breakdown of Definition df-lpir
StepHypRef Expression
1 clpir 19174 . 2 class LPIR
2 vw . . . . . 6 setvar 𝑤
32cv 1479 . . . . 5 class 𝑤
4 clidl 19102 . . . . 5 class LIdeal
53, 4cfv 5852 . . . 4 class (LIdeal‘𝑤)
6 clpidl 19173 . . . . 5 class LPIdeal
73, 6cfv 5852 . . . 4 class (LPIdeal‘𝑤)
85, 7wceq 1480 . . 3 wff (LIdeal‘𝑤) = (LPIdeal‘𝑤)
9 crg 18479 . . 3 class Ring
108, 2, 9crab 2911 . 2 class {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)}
111, 10wceq 1480 1 wff LPIR = {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)}
Colors of variables: wff setvar class
This definition is referenced by:  islpir  19181
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