Metamath Proof Explorer


Definition df-pridl

Description: Define the class of prime ideals of a ring R . A proper ideal I of R is prime if whenever A B C_ I for ideals A and B , either A C_ I or B C_ I . The more familiar definition using elements rather than ideals is equivalent provided R is commutative; see ispridl2 and ispridlc . (Contributed by Jeff Madsen, 10-Jun-2010)

Ref Expression
Assertion df-pridl PrIdl = r RingOps i Idl r | i ran 1 st r a Idl r b Idl r x a y b x 2 nd r y i a i b i

Detailed syntax breakdown

Step Hyp Ref Expression
0 cpridl class PrIdl
1 vr setvar r
2 crngo class RingOps
3 vi setvar i
4 cidl class Idl
5 1 cv setvar r
6 5 4 cfv class Idl r
7 3 cv setvar i
8 c1st class 1 st
9 5 8 cfv class 1 st r
10 9 crn class ran 1 st r
11 7 10 wne wff i ran 1 st r
12 va setvar a
13 vb setvar b
14 vx setvar x
15 12 cv setvar a
16 vy setvar y
17 13 cv setvar b
18 14 cv setvar x
19 c2nd class 2 nd
20 5 19 cfv class 2 nd r
21 16 cv setvar y
22 18 21 20 co class x 2 nd r y
23 22 7 wcel wff x 2 nd r y i
24 23 16 17 wral wff y b x 2 nd r y i
25 24 14 15 wral wff x a y b x 2 nd r y i
26 15 7 wss wff a i
27 17 7 wss wff b i
28 26 27 wo wff a i b i
29 25 28 wi wff x a y b x 2 nd r y i a i b i
30 29 13 6 wral wff b Idl r x a y b x 2 nd r y i a i b i
31 30 12 6 wral wff a Idl r b Idl r x a y b x 2 nd r y i a i b i
32 11 31 wa wff i ran 1 st r a Idl r b Idl r x a y b x 2 nd r y i a i b i
33 32 3 6 crab class i Idl r | i ran 1 st r a Idl r b Idl r x a y b x 2 nd r y i a i b i
34 1 2 33 cmpt class r RingOps i Idl r | i ran 1 st r a Idl r b Idl r x a y b x 2 nd r y i a i b i
35 0 34 wceq wff PrIdl = r RingOps i Idl r | i ran 1 st r a Idl r b Idl r x a y b x 2 nd r y i a i b i