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Definition df-prrngo 32816
Description: Define the class of prime rings. A ring is prime if the zero ideal is a prime ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
Assertion
Ref Expression
df-prrngo PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1st𝑟))} ∈ (PrIdl‘𝑟)}

Detailed syntax breakdown of Definition df-prrngo
StepHypRef Expression
1 cprrng 32814 . 2 class PrRing
2 vr . . . . . . . 8 setvar 𝑟
32cv 1473 . . . . . . 7 class 𝑟
4 c1st 7030 . . . . . . 7 class 1st
53, 4cfv 5786 . . . . . 6 class (1st𝑟)
6 cgi 26490 . . . . . 6 class GId
75, 6cfv 5786 . . . . 5 class (GId‘(1st𝑟))
87csn 4120 . . . 4 class {(GId‘(1st𝑟))}
9 cpridl 32776 . . . . 5 class PrIdl
103, 9cfv 5786 . . . 4 class (PrIdl‘𝑟)
118, 10wcel 1975 . . 3 wff {(GId‘(1st𝑟))} ∈ (PrIdl‘𝑟)
12 crngo 32662 . . 3 class RingOps
1311, 2, 12crab 2895 . 2 class {𝑟 ∈ RingOps ∣ {(GId‘(1st𝑟))} ∈ (PrIdl‘𝑟)}
141, 13wceq 1474 1 wff PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1st𝑟))} ∈ (PrIdl‘𝑟)}
Colors of variables: wff setvar class
This definition is referenced by:  isprrngo  32818
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