Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
|- ( 1st ` R ) = ( 1st ` R ) |
2 |
|
eqid |
|- ran ( 1st ` R ) = ran ( 1st ` R ) |
3 |
1 2
|
ismaxidl |
|- ( R e. RingOps -> ( M e. ( MaxIdl ` R ) <-> ( M e. ( Idl ` R ) /\ M =/= ran ( 1st ` R ) /\ A. j e. ( Idl ` R ) ( M C_ j -> ( j = M \/ j = ran ( 1st ` R ) ) ) ) ) ) |
4 |
|
3anass |
|- ( ( M e. ( Idl ` R ) /\ M =/= ran ( 1st ` R ) /\ A. j e. ( Idl ` R ) ( M C_ j -> ( j = M \/ j = ran ( 1st ` R ) ) ) ) <-> ( M e. ( Idl ` R ) /\ ( M =/= ran ( 1st ` R ) /\ A. j e. ( Idl ` R ) ( M C_ j -> ( j = M \/ j = ran ( 1st ` R ) ) ) ) ) ) |
5 |
3 4
|
bitrdi |
|- ( R e. RingOps -> ( M e. ( MaxIdl ` R ) <-> ( M e. ( Idl ` R ) /\ ( M =/= ran ( 1st ` R ) /\ A. j e. ( Idl ` R ) ( M C_ j -> ( j = M \/ j = ran ( 1st ` R ) ) ) ) ) ) ) |
6 |
5
|
simprbda |
|- ( ( R e. RingOps /\ M e. ( MaxIdl ` R ) ) -> M e. ( Idl ` R ) ) |