| 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 |
|
lsmdisj3b.2 |
|- ( ph -> T C_ ( Z ` U ) ) |
| 8 |
1 2 4 3 5
|
lsmdisj2b |
|- ( ph -> ( ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) <-> ( ( S i^i ( U .(+) T ) ) = { .0. } /\ ( U i^i T ) = { .0. } ) ) ) |
| 9 |
1 6
|
lsmcom2 |
|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) /\ T C_ ( Z ` U ) ) -> ( T .(+) U ) = ( U .(+) T ) ) |
| 10 |
3 4 7 9
|
syl3anc |
|- ( ph -> ( T .(+) U ) = ( U .(+) T ) ) |
| 11 |
10
|
ineq2d |
|- ( ph -> ( S i^i ( T .(+) U ) ) = ( S i^i ( U .(+) T ) ) ) |
| 12 |
11
|
eqeq1d |
|- ( ph -> ( ( S i^i ( T .(+) U ) ) = { .0. } <-> ( S i^i ( U .(+) T ) ) = { .0. } ) ) |
| 13 |
|
incom |
|- ( T i^i U ) = ( U i^i T ) |
| 14 |
13
|
a1i |
|- ( ph -> ( T i^i U ) = ( U i^i T ) ) |
| 15 |
14
|
eqeq1d |
|- ( ph -> ( ( T i^i U ) = { .0. } <-> ( U i^i T ) = { .0. } ) ) |
| 16 |
12 15
|
anbi12d |
|- ( ph -> ( ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( T i^i U ) = { .0. } ) <-> ( ( S i^i ( U .(+) T ) ) = { .0. } /\ ( U i^i T ) = { .0. } ) ) ) |
| 17 |
8 16
|
bitr4d |
|- ( ph -> ( ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) <-> ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( T i^i U ) = { .0. } ) ) ) |