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Definition df-prrngo 38034
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 38032 . 2 class PrRing
2 vr . . . . . . . 8 setvar 𝑟
32cv 1535 . . . . . . 7 class 𝑟
4 c1st 8010 . . . . . . 7 class 1st
53, 4cfv 6562 . . . . . 6 class (1st𝑟)
6 cgi 30518 . . . . . 6 class GId
75, 6cfv 6562 . . . . 5 class (GId‘(1st𝑟))
87csn 4630 . . . 4 class {(GId‘(1st𝑟))}
9 cpridl 37994 . . . . 5 class PrIdl
103, 9cfv 6562 . . . 4 class (PrIdl‘𝑟)
118, 10wcel 2105 . . 3 wff {(GId‘(1st𝑟))} ∈ (PrIdl‘𝑟)
12 crngo 37880 . . 3 class RingOps
1311, 2, 12crab 3432 . 2 class {𝑟 ∈ RingOps ∣ {(GId‘(1st𝑟))} ∈ (PrIdl‘𝑟)}
141, 13wceq 1536 1 wff PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1st𝑟))} ∈ (PrIdl‘𝑟)}
Colors of variables: wff setvar class
This definition is referenced by:  isprrngo  38036
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