Step |
Hyp |
Ref |
Expression |
1 |
|
lidl0.u |
|- U = ( LIdeal ` R ) |
2 |
|
lidl0.z |
|- .0. = ( 0g ` R ) |
3 |
|
rlmlmod |
|- ( R e. Ring -> ( ringLMod ` R ) e. LMod ) |
4 |
|
rlm0 |
|- ( 0g ` R ) = ( 0g ` ( ringLMod ` R ) ) |
5 |
2 4
|
eqtri |
|- .0. = ( 0g ` ( ringLMod ` R ) ) |
6 |
|
eqid |
|- ( LSubSp ` ( ringLMod ` R ) ) = ( LSubSp ` ( ringLMod ` R ) ) |
7 |
5 6
|
lsssn0 |
|- ( ( ringLMod ` R ) e. LMod -> { .0. } e. ( LSubSp ` ( ringLMod ` R ) ) ) |
8 |
3 7
|
syl |
|- ( R e. Ring -> { .0. } e. ( LSubSp ` ( ringLMod ` R ) ) ) |
9 |
|
lidlval |
|- ( LIdeal ` R ) = ( LSubSp ` ( ringLMod ` R ) ) |
10 |
1 9
|
eqtri |
|- U = ( LSubSp ` ( ringLMod ` R ) ) |
11 |
8 10
|
eleqtrrdi |
|- ( R e. Ring -> { .0. } e. U ) |