| 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 |
|
lsmdisj3b.z |
|- Z = ( Cntz ` G ) |
| 7 |
|
lsmdisj3a.2 |
|- ( ph -> S C_ ( Z ` T ) ) |
| 8 |
1 6
|
lsmcom2 |
|- ( ( S e. ( SubGrp ` G ) /\ T e. ( SubGrp ` G ) /\ S C_ ( Z ` T ) ) -> ( S .(+) T ) = ( T .(+) S ) ) |
| 9 |
2 3 7 8
|
syl3anc |
|- ( ph -> ( S .(+) T ) = ( T .(+) S ) ) |
| 10 |
9
|
ineq1d |
|- ( ph -> ( ( S .(+) T ) i^i U ) = ( ( T .(+) S ) i^i U ) ) |
| 11 |
10
|
eqeq1d |
|- ( ph -> ( ( ( S .(+) T ) i^i U ) = { .0. } <-> ( ( T .(+) S ) i^i U ) = { .0. } ) ) |
| 12 |
|
incom |
|- ( S i^i T ) = ( T i^i S ) |
| 13 |
12
|
a1i |
|- ( ph -> ( S i^i T ) = ( T i^i S ) ) |
| 14 |
13
|
eqeq1d |
|- ( ph -> ( ( S i^i T ) = { .0. } <-> ( T i^i S ) = { .0. } ) ) |
| 15 |
11 14
|
anbi12d |
|- ( ph -> ( ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) <-> ( ( ( T .(+) S ) i^i U ) = { .0. } /\ ( T i^i S ) = { .0. } ) ) ) |
| 16 |
1 3 2 4 5
|
lsmdisj2a |
|- ( ph -> ( ( ( ( T .(+) S ) i^i U ) = { .0. } /\ ( T i^i S ) = { .0. } ) <-> ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( T i^i U ) = { .0. } ) ) ) |
| 17 |
15 16
|
bitrd |
|- ( ph -> ( ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) <-> ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( T i^i U ) = { .0. } ) ) ) |