Step |
Hyp |
Ref |
Expression |
1 |
|
0ringprmidl.1 |
|- B = ( Base ` R ) |
2 |
|
prmidlssidl |
|- ( R e. Ring -> ( PrmIdeal ` R ) C_ ( LIdeal ` R ) ) |
3 |
2
|
adantr |
|- ( ( R e. Ring /\ ( # ` B ) = 1 ) -> ( PrmIdeal ` R ) C_ ( LIdeal ` R ) ) |
4 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
5 |
1 4
|
0ringidl |
|- ( ( R e. Ring /\ ( # ` B ) = 1 ) -> ( LIdeal ` R ) = { { ( 0g ` R ) } } ) |
6 |
3 5
|
sseqtrd |
|- ( ( R e. Ring /\ ( # ` B ) = 1 ) -> ( PrmIdeal ` R ) C_ { { ( 0g ` R ) } } ) |
7 |
6
|
sselda |
|- ( ( ( R e. Ring /\ ( # ` B ) = 1 ) /\ i e. ( PrmIdeal ` R ) ) -> i e. { { ( 0g ` R ) } } ) |
8 |
|
elsni |
|- ( i e. { { ( 0g ` R ) } } -> i = { ( 0g ` R ) } ) |
9 |
7 8
|
syl |
|- ( ( ( R e. Ring /\ ( # ` B ) = 1 ) /\ i e. ( PrmIdeal ` R ) ) -> i = { ( 0g ` R ) } ) |
10 |
|
eqid |
|- ( .r ` R ) = ( .r ` R ) |
11 |
1 10
|
prmidlnr |
|- ( ( R e. Ring /\ i e. ( PrmIdeal ` R ) ) -> i =/= B ) |
12 |
11
|
adantlr |
|- ( ( ( R e. Ring /\ ( # ` B ) = 1 ) /\ i e. ( PrmIdeal ` R ) ) -> i =/= B ) |
13 |
1 4
|
0ring |
|- ( ( R e. Ring /\ ( # ` B ) = 1 ) -> B = { ( 0g ` R ) } ) |
14 |
13
|
adantr |
|- ( ( ( R e. Ring /\ ( # ` B ) = 1 ) /\ i e. ( PrmIdeal ` R ) ) -> B = { ( 0g ` R ) } ) |
15 |
12 14
|
neeqtrd |
|- ( ( ( R e. Ring /\ ( # ` B ) = 1 ) /\ i e. ( PrmIdeal ` R ) ) -> i =/= { ( 0g ` R ) } ) |
16 |
15
|
neneqd |
|- ( ( ( R e. Ring /\ ( # ` B ) = 1 ) /\ i e. ( PrmIdeal ` R ) ) -> -. i = { ( 0g ` R ) } ) |
17 |
9 16
|
pm2.65da |
|- ( ( R e. Ring /\ ( # ` B ) = 1 ) -> -. i e. ( PrmIdeal ` R ) ) |
18 |
17
|
eq0rdv |
|- ( ( R e. Ring /\ ( # ` B ) = 1 ) -> ( PrmIdeal ` R ) = (/) ) |