Step |
Hyp |
Ref |
Expression |
1 |
|
lsmidl.1 |
|- B = ( Base ` R ) |
2 |
|
lsmidl.3 |
|- .(+) = ( LSSum ` R ) |
3 |
|
lsmidl.4 |
|- K = ( RSpan ` R ) |
4 |
|
lsmidl.5 |
|- ( ph -> R e. Ring ) |
5 |
|
lsmidl.6 |
|- ( ph -> I e. ( LIdeal ` R ) ) |
6 |
|
lsmidl.7 |
|- ( ph -> J e. ( LIdeal ` R ) ) |
7 |
|
rlmlsm |
|- ( R e. Ring -> ( LSSum ` R ) = ( LSSum ` ( ringLMod ` R ) ) ) |
8 |
4 7
|
syl |
|- ( ph -> ( LSSum ` R ) = ( LSSum ` ( ringLMod ` R ) ) ) |
9 |
2 8
|
syl5eq |
|- ( ph -> .(+) = ( LSSum ` ( ringLMod ` R ) ) ) |
10 |
9
|
oveqd |
|- ( ph -> ( I .(+) J ) = ( I ( LSSum ` ( ringLMod ` R ) ) J ) ) |
11 |
|
rlmlmod |
|- ( R e. Ring -> ( ringLMod ` R ) e. LMod ) |
12 |
4 11
|
syl |
|- ( ph -> ( ringLMod ` R ) e. LMod ) |
13 |
|
lidlval |
|- ( LIdeal ` R ) = ( LSubSp ` ( ringLMod ` R ) ) |
14 |
|
rspval |
|- ( RSpan ` R ) = ( LSpan ` ( ringLMod ` R ) ) |
15 |
3 14
|
eqtri |
|- K = ( LSpan ` ( ringLMod ` R ) ) |
16 |
|
eqid |
|- ( LSSum ` ( ringLMod ` R ) ) = ( LSSum ` ( ringLMod ` R ) ) |
17 |
13 15 16
|
lsmsp |
|- ( ( ( ringLMod ` R ) e. LMod /\ I e. ( LIdeal ` R ) /\ J e. ( LIdeal ` R ) ) -> ( I ( LSSum ` ( ringLMod ` R ) ) J ) = ( K ` ( I u. J ) ) ) |
18 |
12 5 6 17
|
syl3anc |
|- ( ph -> ( I ( LSSum ` ( ringLMod ` R ) ) J ) = ( K ` ( I u. J ) ) ) |
19 |
10 18
|
eqtrd |
|- ( ph -> ( I .(+) J ) = ( K ` ( I u. J ) ) ) |