| Step |
Hyp |
Ref |
Expression |
| 1 |
|
lsmcntz.p |
|- .(+) = ( LSSum ` G ) |
| 2 |
|
lsmcntz.s |
|- ( ph -> S e. ( SubGrp ` G ) ) |
| 3 |
|
lsmcntz.t |
|- ( ph -> T e. ( SubGrp ` G ) ) |
| 4 |
|
lsmcntz.u |
|- ( ph -> U e. ( SubGrp ` G ) ) |
| 5 |
|
lsmdisj.o |
|- .0. = ( 0g ` G ) |
| 6 |
|
lsmdisjr.i |
|- ( ph -> ( S i^i ( T .(+) U ) ) = { .0. } ) |
| 7 |
|
incom |
|- ( S i^i ( T .(+) U ) ) = ( ( T .(+) U ) i^i S ) |
| 8 |
7 6
|
eqtr3id |
|- ( ph -> ( ( T .(+) U ) i^i S ) = { .0. } ) |
| 9 |
1 3 4 2 5 8
|
lsmdisj |
|- ( ph -> ( ( T i^i S ) = { .0. } /\ ( U i^i S ) = { .0. } ) ) |
| 10 |
|
incom |
|- ( T i^i S ) = ( S i^i T ) |
| 11 |
10
|
eqeq1i |
|- ( ( T i^i S ) = { .0. } <-> ( S i^i T ) = { .0. } ) |
| 12 |
|
incom |
|- ( U i^i S ) = ( S i^i U ) |
| 13 |
12
|
eqeq1i |
|- ( ( U i^i S ) = { .0. } <-> ( S i^i U ) = { .0. } ) |
| 14 |
11 13
|
anbi12i |
|- ( ( ( T i^i S ) = { .0. } /\ ( U i^i S ) = { .0. } ) <-> ( ( S i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) |
| 15 |
9 14
|
sylib |
|- ( ph -> ( ( S i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) |