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 |
|
lsmcntz.z |
|- Z = ( Cntz ` G ) |
6 |
|
subgrcl |
|- ( U e. ( SubGrp ` G ) -> G e. Grp ) |
7 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
8 |
7
|
subgss |
|- ( U e. ( SubGrp ` G ) -> U C_ ( Base ` G ) ) |
9 |
7 5
|
cntzsubg |
|- ( ( G e. Grp /\ U C_ ( Base ` G ) ) -> ( Z ` U ) e. ( SubGrp ` G ) ) |
10 |
6 8 9
|
syl2anc |
|- ( U e. ( SubGrp ` G ) -> ( Z ` U ) e. ( SubGrp ` G ) ) |
11 |
4 10
|
syl |
|- ( ph -> ( Z ` U ) e. ( SubGrp ` G ) ) |
12 |
1
|
lsmlub |
|- ( ( S e. ( SubGrp ` G ) /\ T e. ( SubGrp ` G ) /\ ( Z ` U ) e. ( SubGrp ` G ) ) -> ( ( S C_ ( Z ` U ) /\ T C_ ( Z ` U ) ) <-> ( S .(+) T ) C_ ( Z ` U ) ) ) |
13 |
2 3 11 12
|
syl3anc |
|- ( ph -> ( ( S C_ ( Z ` U ) /\ T C_ ( Z ` U ) ) <-> ( S .(+) T ) C_ ( Z ` U ) ) ) |
14 |
13
|
bicomd |
|- ( ph -> ( ( S .(+) T ) C_ ( Z ` U ) <-> ( S C_ ( Z ` U ) /\ T C_ ( Z ` U ) ) ) ) |