Theorem ismxidl

Description: The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011) (Revised by Thierry Arnoux, 19-Jan-2024)

	Ref	Expression
Hypothesis	mxidlval.1	|- B = ( Base ` R )
Assertion	ismxidl	|- ( R e. Ring -> ( M e. ( MaxIdeal ` R ) <-> ( M e. ( LIdeal ` R ) /\ M =/= B /\ A. j e. ( LIdeal ` R ) ( M C_ j -> ( j = M \/ j = B ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		mxidlval.1	|- B = ( Base ` R )
2	1	mxidlval	|- ( R e. Ring -> ( MaxIdeal ` R ) = { i e. ( LIdeal ` R ) | ( i =/= B /\ A. j e. ( LIdeal ` R ) ( i C_ j -> ( j = i \/ j = B ) ) ) } )
3	2	eleq2d	|- ( R e. Ring -> ( M e. ( MaxIdeal ` R ) <-> M e. { i e. ( LIdeal ` R ) | ( i =/= B /\ A. j e. ( LIdeal ` R ) ( i C_ j -> ( j = i \/ j = B ) ) ) } ) )
4		neeq1	|- ( i = M -> ( i =/= B <-> M =/= B ) )
5		sseq1	|- ( i = M -> ( i C_ j <-> M C_ j ) )
6		eqeq2	|- ( i = M -> ( j = i <-> j = M ) )
7	6	orbi1d	|- ( i = M -> ( ( j = i \/ j = B ) <-> ( j = M \/ j = B ) ) )
8	5 7	imbi12d	|- ( i = M -> ( ( i C_ j -> ( j = i \/ j = B ) ) <-> ( M C_ j -> ( j = M \/ j = B ) ) ) )
9	8	ralbidv	|- ( i = M -> ( A. j e. ( LIdeal ` R ) ( i C_ j -> ( j = i \/ j = B ) ) <-> A. j e. ( LIdeal ` R ) ( M C_ j -> ( j = M \/ j = B ) ) ) )
10	4 9	anbi12d	|- ( i = M -> ( ( i =/= B /\ A. j e. ( LIdeal ` R ) ( i C_ j -> ( j = i \/ j = B ) ) ) <-> ( M =/= B /\ A. j e. ( LIdeal ` R ) ( M C_ j -> ( j = M \/ j = B ) ) ) ) )
11	10	elrab	|- ( M e. { i e. ( LIdeal ` R ) | ( i =/= B /\ A. j e. ( LIdeal ` R ) ( i C_ j -> ( j = i \/ j = B ) ) ) } <-> ( M e. ( LIdeal ` R ) /\ ( M =/= B /\ A. j e. ( LIdeal ` R ) ( M C_ j -> ( j = M \/ j = B ) ) ) ) )
12		3anass	|- ( ( M e. ( LIdeal ` R ) /\ M =/= B /\ A. j e. ( LIdeal ` R ) ( M C_ j -> ( j = M \/ j = B ) ) ) <-> ( M e. ( LIdeal ` R ) /\ ( M =/= B /\ A. j e. ( LIdeal ` R ) ( M C_ j -> ( j = M \/ j = B ) ) ) ) )
13	11 12	bitr4i	|- ( M e. { i e. ( LIdeal ` R ) | ( i =/= B /\ A. j e. ( LIdeal ` R ) ( i C_ j -> ( j = i \/ j = B ) ) ) } <-> ( M e. ( LIdeal ` R ) /\ M =/= B /\ A. j e. ( LIdeal ` R ) ( M C_ j -> ( j = M \/ j = B ) ) ) )
14	3 13	bitrdi	|- ( R e. Ring -> ( M e. ( MaxIdeal ` R ) <-> ( M e. ( LIdeal ` R ) /\ M =/= B /\ A. j e. ( LIdeal ` R ) ( M C_ j -> ( j = M \/ j = B ) ) ) ) )