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 |
1 3 4 2 5
|
lsmcntz |
|- ( ph -> ( ( T .(+) U ) C_ ( Z ` S ) <-> ( T C_ ( Z ` S ) /\ U C_ ( Z ` S ) ) ) ) |
7 |
|
subgrcl |
|- ( S e. ( SubGrp ` G ) -> G e. Grp ) |
8 |
|
grpmnd |
|- ( G e. Grp -> G e. Mnd ) |
9 |
2 7 8
|
3syl |
|- ( ph -> G e. Mnd ) |
10 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
11 |
10
|
subgss |
|- ( T e. ( SubGrp ` G ) -> T C_ ( Base ` G ) ) |
12 |
3 11
|
syl |
|- ( ph -> T C_ ( Base ` G ) ) |
13 |
10
|
subgss |
|- ( U e. ( SubGrp ` G ) -> U C_ ( Base ` G ) ) |
14 |
4 13
|
syl |
|- ( ph -> U C_ ( Base ` G ) ) |
15 |
10 1
|
lsmssv |
|- ( ( G e. Mnd /\ T C_ ( Base ` G ) /\ U C_ ( Base ` G ) ) -> ( T .(+) U ) C_ ( Base ` G ) ) |
16 |
9 12 14 15
|
syl3anc |
|- ( ph -> ( T .(+) U ) C_ ( Base ` G ) ) |
17 |
10
|
subgss |
|- ( S e. ( SubGrp ` G ) -> S C_ ( Base ` G ) ) |
18 |
2 17
|
syl |
|- ( ph -> S C_ ( Base ` G ) ) |
19 |
10 5
|
cntzrec |
|- ( ( ( T .(+) U ) C_ ( Base ` G ) /\ S C_ ( Base ` G ) ) -> ( ( T .(+) U ) C_ ( Z ` S ) <-> S C_ ( Z ` ( T .(+) U ) ) ) ) |
20 |
16 18 19
|
syl2anc |
|- ( ph -> ( ( T .(+) U ) C_ ( Z ` S ) <-> S C_ ( Z ` ( T .(+) U ) ) ) ) |
21 |
10 5
|
cntzrec |
|- ( ( T C_ ( Base ` G ) /\ S C_ ( Base ` G ) ) -> ( T C_ ( Z ` S ) <-> S C_ ( Z ` T ) ) ) |
22 |
12 18 21
|
syl2anc |
|- ( ph -> ( T C_ ( Z ` S ) <-> S C_ ( Z ` T ) ) ) |
23 |
10 5
|
cntzrec |
|- ( ( U C_ ( Base ` G ) /\ S C_ ( Base ` G ) ) -> ( U C_ ( Z ` S ) <-> S C_ ( Z ` U ) ) ) |
24 |
14 18 23
|
syl2anc |
|- ( ph -> ( U C_ ( Z ` S ) <-> S C_ ( Z ` U ) ) ) |
25 |
22 24
|
anbi12d |
|- ( ph -> ( ( T C_ ( Z ` S ) /\ U C_ ( Z ` S ) ) <-> ( S C_ ( Z ` T ) /\ S C_ ( Z ` U ) ) ) ) |
26 |
6 20 25
|
3bitr3d |
|- ( ph -> ( S C_ ( Z ` ( T .(+) U ) ) <-> ( S C_ ( Z ` T ) /\ S C_ ( Z ` U ) ) ) ) |