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 |
2
|
adantr |
|- ( ( ph /\ ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) ) -> S e. ( SubGrp ` G ) ) |
7 |
3
|
adantr |
|- ( ( ph /\ ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) ) -> T e. ( SubGrp ` G ) ) |
8 |
4
|
adantr |
|- ( ( ph /\ ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) ) -> U e. ( SubGrp ` G ) ) |
9 |
|
simprl |
|- ( ( ph /\ ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) ) -> ( ( S .(+) T ) i^i U ) = { .0. } ) |
10 |
|
simprr |
|- ( ( ph /\ ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) ) -> ( S i^i T ) = { .0. } ) |
11 |
1 6 7 8 5 9 10
|
lsmdisj2 |
|- ( ( ph /\ ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) ) -> ( T i^i ( S .(+) U ) ) = { .0. } ) |
12 |
1 6 7 8 5 9
|
lsmdisj |
|- ( ( ph /\ ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) ) -> ( ( S i^i U ) = { .0. } /\ ( T i^i U ) = { .0. } ) ) |
13 |
12
|
simpld |
|- ( ( ph /\ ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) ) -> ( S i^i U ) = { .0. } ) |
14 |
11 13
|
jca |
|- ( ( ph /\ ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) ) -> ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) |
15 |
|
incom |
|- ( ( S .(+) T ) i^i U ) = ( U i^i ( S .(+) T ) ) |
16 |
2
|
adantr |
|- ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> S e. ( SubGrp ` G ) ) |
17 |
4
|
adantr |
|- ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> U e. ( SubGrp ` G ) ) |
18 |
3
|
adantr |
|- ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> T e. ( SubGrp ` G ) ) |
19 |
|
incom |
|- ( ( S .(+) U ) i^i T ) = ( T i^i ( S .(+) U ) ) |
20 |
|
simprl |
|- ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( T i^i ( S .(+) U ) ) = { .0. } ) |
21 |
19 20
|
syl5eq |
|- ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( ( S .(+) U ) i^i T ) = { .0. } ) |
22 |
|
simprr |
|- ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( S i^i U ) = { .0. } ) |
23 |
1 16 17 18 5 21 22
|
lsmdisj2 |
|- ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( U i^i ( S .(+) T ) ) = { .0. } ) |
24 |
15 23
|
syl5eq |
|- ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( ( S .(+) T ) i^i U ) = { .0. } ) |
25 |
|
incom |
|- ( S i^i T ) = ( T i^i S ) |
26 |
1 18 16 17 5 20
|
lsmdisjr |
|- ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( ( T i^i S ) = { .0. } /\ ( T i^i U ) = { .0. } ) ) |
27 |
26
|
simpld |
|- ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( T i^i S ) = { .0. } ) |
28 |
25 27
|
syl5eq |
|- ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( S i^i T ) = { .0. } ) |
29 |
24 28
|
jca |
|- ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) ) |
30 |
14 29
|
impbida |
|- ( ph -> ( ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) <-> ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) ) |