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Metamath Proof Explorer

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 ) ) ) ) )