df-prrngo - Mathbox for Jeff Madsen

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Definition df-prrngo 38055

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‘(1^st ‘𝑟))} ∈ (PrIdl‘𝑟)}

**Detailed syntax breakdown of Definition df-prrngo**
Step	Hyp	Ref	Expression
1		cprrng 38053	. 2 class PrRing
2		vr	. . . . . . . 8 setvar 𝑟
3	2	cv 1539	. . . . . . 7 class 𝑟
4		c1st 8012	. . . . . . 7 class 1^st
5	3, 4	cfv 6561	. . . . . 6 class (1^st ‘𝑟)
6		cgi 30509	. . . . . 6 class GId
7	5, 6	cfv 6561	. . . . 5 class (GId‘(1^st ‘𝑟))
8	7	csn 4626	. . . 4 class {(GId‘(1^st ‘𝑟))}
9		cpridl 38015	. . . . 5 class PrIdl
10	3, 9	cfv 6561	. . . 4 class (PrIdl‘𝑟)
11	8, 10	wcel 2108	. . 3 wff {(GId‘(1^st ‘𝑟))} ∈ (PrIdl‘𝑟)
12		crngo 37901	. . 3 class RingOps
13	11, 2, 12	crab 3436	. 2 class {𝑟 ∈ RingOps ∣ {(GId‘(1^st ‘𝑟))} ∈ (PrIdl‘𝑟)}
14	1, 13	wceq 1540	1 wff PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1^st ‘𝑟))} ∈ (PrIdl‘𝑟)}

Colors of variables: wff setvar class

This definition is referenced by: isprrngo 38057