Step |
Hyp |
Ref |
Expression |
1 |
|
lsmelval.a |
|- .+ = ( +g ` G ) |
2 |
|
lsmelval.p |
|- .(+) = ( LSSum ` G ) |
3 |
|
subgrcl |
|- ( T e. ( SubGrp ` G ) -> G e. Grp ) |
4 |
3
|
adantr |
|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) -> G e. Grp ) |
5 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
6 |
5
|
subgss |
|- ( T e. ( SubGrp ` G ) -> T C_ ( Base ` G ) ) |
7 |
6
|
adantr |
|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) -> T C_ ( Base ` G ) ) |
8 |
5
|
subgss |
|- ( U e. ( SubGrp ` G ) -> U C_ ( Base ` G ) ) |
9 |
8
|
adantl |
|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) -> U C_ ( Base ` G ) ) |
10 |
4 7 9
|
3jca |
|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) -> ( G e. Grp /\ T C_ ( Base ` G ) /\ U C_ ( Base ` G ) ) ) |
11 |
5 1 2
|
lsmelvalix |
|- ( ( ( G e. Grp /\ T C_ ( Base ` G ) /\ U C_ ( Base ` G ) ) /\ ( X e. T /\ Y e. U ) ) -> ( X .+ Y ) e. ( T .(+) U ) ) |
12 |
10 11
|
sylan |
|- ( ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) /\ ( X e. T /\ Y e. U ) ) -> ( X .+ Y ) e. ( T .(+) U ) ) |