| Step |
Hyp |
Ref |
Expression |
| 1 |
|
lsatcv1.o |
|- .0. = ( 0g ` W ) |
| 2 |
|
lsatcv1.p |
|- .(+) = ( LSSum ` W ) |
| 3 |
|
lsatcv1.s |
|- S = ( LSubSp ` W ) |
| 4 |
|
lsatcv1.a |
|- A = ( LSAtoms ` W ) |
| 5 |
|
lsatcv1.c |
|- C = (
|
| 6 |
|
lsatcv1.w |
|- ( ph -> W e. LVec ) |
| 7 |
|
lsatcv1.u |
|- ( ph -> U e. S ) |
| 8 |
|
lsatcv1.q |
|- ( ph -> Q e. A ) |
| 9 |
|
lsatcv1.r |
|- ( ph -> R e. A ) |
| 10 |
|
lsatcv1.l |
|- ( ph -> U C ( Q .(+) R ) ) |
| 11 |
|
breq1 |
|- ( U = { .0. } -> ( U C ( Q .(+) R ) <-> { .0. } C ( Q .(+) R ) ) ) |
| 12 |
10 11
|
syl5ibcom |
|- ( ph -> ( U = { .0. } -> { .0. } C ( Q .(+) R ) ) ) |
| 13 |
1 2 4 5 6 8 9
|
lsatcv0eq |
|- ( ph -> ( { .0. } C ( Q .(+) R ) <-> Q = R ) ) |
| 14 |
12 13
|
sylibd |
|- ( ph -> ( U = { .0. } -> Q = R ) ) |
| 15 |
10
|
adantr |
|- ( ( ph /\ Q = R ) -> U C ( Q .(+) R ) ) |
| 16 |
6
|
adantr |
|- ( ( ph /\ Q = R ) -> W e. LVec ) |
| 17 |
7
|
adantr |
|- ( ( ph /\ Q = R ) -> U e. S ) |
| 18 |
|
oveq1 |
|- ( Q = R -> ( Q .(+) R ) = ( R .(+) R ) ) |
| 19 |
|
lveclmod |
|- ( W e. LVec -> W e. LMod ) |
| 20 |
6 19
|
syl |
|- ( ph -> W e. LMod ) |
| 21 |
3 4 20 9
|
lsatlssel |
|- ( ph -> R e. S ) |
| 22 |
3
|
lsssubg |
|- ( ( W e. LMod /\ R e. S ) -> R e. ( SubGrp ` W ) ) |
| 23 |
20 21 22
|
syl2anc |
|- ( ph -> R e. ( SubGrp ` W ) ) |
| 24 |
2
|
lsmidm |
|- ( R e. ( SubGrp ` W ) -> ( R .(+) R ) = R ) |
| 25 |
23 24
|
syl |
|- ( ph -> ( R .(+) R ) = R ) |
| 26 |
18 25
|
sylan9eqr |
|- ( ( ph /\ Q = R ) -> ( Q .(+) R ) = R ) |
| 27 |
9
|
adantr |
|- ( ( ph /\ Q = R ) -> R e. A ) |
| 28 |
26 27
|
eqeltrd |
|- ( ( ph /\ Q = R ) -> ( Q .(+) R ) e. A ) |
| 29 |
1 3 4 5 16 17 28
|
lsatcveq0 |
|- ( ( ph /\ Q = R ) -> ( U C ( Q .(+) R ) <-> U = { .0. } ) ) |
| 30 |
15 29
|
mpbid |
|- ( ( ph /\ Q = R ) -> U = { .0. } ) |
| 31 |
30
|
ex |
|- ( ph -> ( Q = R -> U = { .0. } ) ) |
| 32 |
14 31
|
impbid |
|- ( ph -> ( U = { .0. } <-> Q = R ) ) |