Step |
Hyp |
Ref |
Expression |
1 |
|
lsmub1.p |
|- .(+) = ( LSSum ` G ) |
2 |
|
ssid |
|- U C_ U |
3 |
1
|
lsmlub |
|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) -> ( ( T C_ U /\ U C_ U ) <-> ( T .(+) U ) C_ U ) ) |
4 |
3
|
3anidm23 |
|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) -> ( ( T C_ U /\ U C_ U ) <-> ( T .(+) U ) C_ U ) ) |
5 |
4
|
biimpd |
|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) -> ( ( T C_ U /\ U C_ U ) -> ( T .(+) U ) C_ U ) ) |
6 |
2 5
|
mpan2i |
|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) -> ( T C_ U -> ( T .(+) U ) C_ U ) ) |
7 |
6
|
3impia |
|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) /\ T C_ U ) -> ( T .(+) U ) C_ U ) |
8 |
1
|
lsmub2 |
|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) -> U C_ ( T .(+) U ) ) |
9 |
8
|
3adant3 |
|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) /\ T C_ U ) -> U C_ ( T .(+) U ) ) |
10 |
7 9
|
eqssd |
|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) /\ T C_ U ) -> ( T .(+) U ) = U ) |