Step |
Hyp |
Ref |
Expression |
1 |
|
lsatnem0.o |
|- .0. = ( 0g ` W ) |
2 |
|
lsatnem0.a |
|- A = ( LSAtoms ` W ) |
3 |
|
lsatnem0.w |
|- ( ph -> W e. LVec ) |
4 |
|
lsatnem0.q |
|- ( ph -> Q e. A ) |
5 |
|
lsatnem0.r |
|- ( ph -> R e. A ) |
6 |
2 3 5 4
|
lsatcmp |
|- ( ph -> ( R C_ Q <-> R = Q ) ) |
7 |
|
eqcom |
|- ( R = Q <-> Q = R ) |
8 |
6 7
|
bitrdi |
|- ( ph -> ( R C_ Q <-> Q = R ) ) |
9 |
8
|
necon3bbid |
|- ( ph -> ( -. R C_ Q <-> Q =/= R ) ) |
10 |
|
eqid |
|- ( LSubSp ` W ) = ( LSubSp ` W ) |
11 |
|
lveclmod |
|- ( W e. LVec -> W e. LMod ) |
12 |
3 11
|
syl |
|- ( ph -> W e. LMod ) |
13 |
10 2 12 4
|
lsatlssel |
|- ( ph -> Q e. ( LSubSp ` W ) ) |
14 |
1 10 2 3 13 5
|
lsatnle |
|- ( ph -> ( -. R C_ Q <-> ( Q i^i R ) = { .0. } ) ) |
15 |
9 14
|
bitr3d |
|- ( ph -> ( Q =/= R <-> ( Q i^i R ) = { .0. } ) ) |