| Step |
Hyp |
Ref |
Expression |
| 1 |
|
lsatcvat2.s |
|- S = ( LSubSp ` W ) |
| 2 |
|
lsatcvat2.p |
|- .(+) = ( LSSum ` W ) |
| 3 |
|
lsatcvat2.a |
|- A = ( LSAtoms ` W ) |
| 4 |
|
lsatcvat2.c |
|- C = (
|
| 5 |
|
lsatcvat2.w |
|- ( ph -> W e. LVec ) |
| 6 |
|
lsatcvat2.u |
|- ( ph -> U e. S ) |
| 7 |
|
lsatcvat2.q |
|- ( ph -> Q e. A ) |
| 8 |
|
lsatcvat2.r |
|- ( ph -> R e. A ) |
| 9 |
|
lsatcvat2.n |
|- ( ph -> Q =/= R ) |
| 10 |
|
lsatcvat2.l |
|- ( ph -> U C ( Q .(+) R ) ) |
| 11 |
|
eqid |
|- ( 0g ` W ) = ( 0g ` W ) |
| 12 |
11 2 1 3 4 5 6 7 8 10
|
lsatcv1 |
|- ( ph -> ( U = { ( 0g ` W ) } <-> Q = R ) ) |
| 13 |
12
|
necon3bid |
|- ( ph -> ( U =/= { ( 0g ` W ) } <-> Q =/= R ) ) |
| 14 |
9 13
|
mpbird |
|- ( ph -> U =/= { ( 0g ` W ) } ) |
| 15 |
|
lveclmod |
|- ( W e. LVec -> W e. LMod ) |
| 16 |
5 15
|
syl |
|- ( ph -> W e. LMod ) |
| 17 |
1 3 16 7
|
lsatlssel |
|- ( ph -> Q e. S ) |
| 18 |
1 3 16 8
|
lsatlssel |
|- ( ph -> R e. S ) |
| 19 |
1 2
|
lsmcl |
|- ( ( W e. LMod /\ Q e. S /\ R e. S ) -> ( Q .(+) R ) e. S ) |
| 20 |
16 17 18 19
|
syl3anc |
|- ( ph -> ( Q .(+) R ) e. S ) |
| 21 |
1 4 5 6 20 10
|
lcvpss |
|- ( ph -> U C. ( Q .(+) R ) ) |
| 22 |
11 1 2 3 5 6 7 8 14 21
|
lsatcvat |
|- ( ph -> U e. A ) |