| 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 |
|
lsmdisj.i |
|- ( ph -> ( ( S .(+) T ) i^i U ) = { .0. } ) |
| 7 |
1
|
lsmub1 |
|- ( ( S e. ( SubGrp ` G ) /\ T e. ( SubGrp ` G ) ) -> S C_ ( S .(+) T ) ) |
| 8 |
2 3 7
|
syl2anc |
|- ( ph -> S C_ ( S .(+) T ) ) |
| 9 |
8
|
ssrind |
|- ( ph -> ( S i^i U ) C_ ( ( S .(+) T ) i^i U ) ) |
| 10 |
9 6
|
sseqtrd |
|- ( ph -> ( S i^i U ) C_ { .0. } ) |
| 11 |
5
|
subg0cl |
|- ( S e. ( SubGrp ` G ) -> .0. e. S ) |
| 12 |
2 11
|
syl |
|- ( ph -> .0. e. S ) |
| 13 |
5
|
subg0cl |
|- ( U e. ( SubGrp ` G ) -> .0. e. U ) |
| 14 |
4 13
|
syl |
|- ( ph -> .0. e. U ) |
| 15 |
12 14
|
elind |
|- ( ph -> .0. e. ( S i^i U ) ) |
| 16 |
15
|
snssd |
|- ( ph -> { .0. } C_ ( S i^i U ) ) |
| 17 |
10 16
|
eqssd |
|- ( ph -> ( S i^i U ) = { .0. } ) |
| 18 |
1
|
lsmub2 |
|- ( ( S e. ( SubGrp ` G ) /\ T e. ( SubGrp ` G ) ) -> T C_ ( S .(+) T ) ) |
| 19 |
2 3 18
|
syl2anc |
|- ( ph -> T C_ ( S .(+) T ) ) |
| 20 |
19
|
ssrind |
|- ( ph -> ( T i^i U ) C_ ( ( S .(+) T ) i^i U ) ) |
| 21 |
20 6
|
sseqtrd |
|- ( ph -> ( T i^i U ) C_ { .0. } ) |
| 22 |
5
|
subg0cl |
|- ( T e. ( SubGrp ` G ) -> .0. e. T ) |
| 23 |
3 22
|
syl |
|- ( ph -> .0. e. T ) |
| 24 |
23 14
|
elind |
|- ( ph -> .0. e. ( T i^i U ) ) |
| 25 |
24
|
snssd |
|- ( ph -> { .0. } C_ ( T i^i U ) ) |
| 26 |
21 25
|
eqssd |
|- ( ph -> ( T i^i U ) = { .0. } ) |
| 27 |
17 26
|
jca |
|- ( ph -> ( ( S i^i U ) = { .0. } /\ ( T i^i U ) = { .0. } ) ) |