Definition df-maxidl 33811

Description: Define the class of maximal ideals of a ring R. A proper ideal is called maximal if it is maximal with respect to inclusion among proper ideals. (Contributed by Jeff Madsen, 5-Jan-2011.)

Assertion

Ref	Expression
df-maxidl	|- MaxIdl = ( r e. RingOps |-> { i e. ( Idl ` r ) | ( i =/= ran ( 1st ` r ) /\ A. j e. ( Idl ` r ) ( i C_ j -> ( j = i \/ j = ran ( 1st ` r ) ) ) ) } )

Distinct variable group:   i, r, j

Detailed syntax breakdown of Definition df-maxidl

Step	Hyp	Ref	Expression
1		cmaxidl 33808	. 2 class MaxIdl
2		vr	. . 3 setvar r
3		crngo 33693	. . 3 class RingOps
4		vi	. . . . . . 7 setvar i
5	4	cv 1482	. . . . . 6 class i
6	2	cv 1482	. . . . . . . 8 class r
7		c1st 7166	. . . . . . . 8 class 1st
8	6, 7	cfv 5888	. . . . . . 7 class ( 1st ` r )
9	8	crn 5115	. . . . . 6 class ran ( 1st ` r )
10	5, 9	wne 2794	. . . . 5 wff i =/= ran ( 1st ` r )
11		vj	. . . . . . . . 9 setvar j
12	11	cv 1482	. . . . . . . 8 class j
13	5, 12	wss 3574	. . . . . . 7 wff i C_ j
14	11, 4	weq 1874	. . . . . . . 8 wff j = i
15	12, 9	wceq 1483	. . . . . . . 8 wff j = ran ( 1st ` r )
16	14, 15	wo 383	. . . . . . 7 wff ( j = i \/ j = ran ( 1st ` r ) )
17	13, 16	wi 4	. . . . . 6 wff ( i C_ j -> ( j = i \/ j = ran ( 1st ` r ) ) )
18		cidl 33806	. . . . . . 7 class Idl
19	6, 18	cfv 5888	. . . . . 6 class ( Idl ` r )
20	17, 11, 19	wral 2912	. . . . 5 wff A. j e. ( Idl ` r ) ( i C_ j -> ( j = i \/ j = ran ( 1st ` r ) ) )
21	10, 20	wa 384	. . . 4 wff ( i =/= ran ( 1st ` r ) /\ A. j e. ( Idl ` r ) ( i C_ j -> ( j = i \/ j = ran ( 1st ` r ) ) ) )
22	21, 4, 19	crab 2916	. . 3 class { i e. ( Idl ` r ) | ( i =/= ran ( 1st ` r ) /\ A. j e. ( Idl ` r ) ( i C_ j -> ( j = i \/ j = ran ( 1st ` r ) ) ) ) }
23	2, 3, 22	cmpt 4729	. 2 class ( r e. RingOps |-> { i e. ( Idl ` r ) | ( i =/= ran ( 1st ` r ) /\ A. j e. ( Idl ` r ) ( i C_ j -> ( j = i \/ j = ran ( 1st ` r ) ) ) ) } )
24	1, 23	wceq 1483	1 wff MaxIdl = ( r e. RingOps |-> { i e. ( Idl ` r ) | ( i =/= ran ( 1st ` r ) /\ A. j e. ( Idl ` r ) ( i C_ j -> ( j = i \/ j = ran ( 1st ` r ) ) ) ) } )

This definition is referenced by:  maxidlval  33838