| Step |
Hyp |
Ref |
Expression |
| 1 |
|
dochkrsat.h |
|- H = ( LHyp ` K ) |
| 2 |
|
dochkrsat.o |
|- ._|_ = ( ( ocH ` K ) ` W ) |
| 3 |
|
dochkrsat.u |
|- U = ( ( DVecH ` K ) ` W ) |
| 4 |
|
dochkrsat.a |
|- A = ( LSAtoms ` U ) |
| 5 |
|
dochkrsat.f |
|- F = ( LFnl ` U ) |
| 6 |
|
dochkrsat.l |
|- L = ( LKer ` U ) |
| 7 |
|
dochkrsat.z |
|- .0. = ( 0g ` U ) |
| 8 |
|
dochkrsat.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
| 9 |
|
dochkrsat.g |
|- ( ph -> G e. F ) |
| 10 |
|
eqid |
|- ( Base ` U ) = ( Base ` U ) |
| 11 |
|
eqid |
|- ( LSHyp ` U ) = ( LSHyp ` U ) |
| 12 |
1 2 3 10 11 5 6 8 9
|
dochkrshp |
|- ( ph -> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) =/= ( Base ` U ) <-> ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) e. ( LSHyp ` U ) ) ) |
| 13 |
1 3 8
|
dvhlmod |
|- ( ph -> U e. LMod ) |
| 14 |
10 5 6 13 9
|
lkrssv |
|- ( ph -> ( L ` G ) C_ ( Base ` U ) ) |
| 15 |
1 2 3 10 7 8 14
|
dochn0nv |
|- ( ph -> ( ( ._|_ ` ( L ` G ) ) =/= { .0. } <-> ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) =/= ( Base ` U ) ) ) |
| 16 |
|
eqid |
|- ( LSubSp ` U ) = ( LSubSp ` U ) |
| 17 |
1 3 10 16 2
|
dochlss |
|- ( ( ( K e. HL /\ W e. H ) /\ ( L ` G ) C_ ( Base ` U ) ) -> ( ._|_ ` ( L ` G ) ) e. ( LSubSp ` U ) ) |
| 18 |
8 14 17
|
syl2anc |
|- ( ph -> ( ._|_ ` ( L ` G ) ) e. ( LSubSp ` U ) ) |
| 19 |
1 2 3 16 4 11 8 18
|
dochsatshpb |
|- ( ph -> ( ( ._|_ ` ( L ` G ) ) e. A <-> ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) e. ( LSHyp ` U ) ) ) |
| 20 |
12 15 19
|
3bitr4d |
|- ( ph -> ( ( ._|_ ` ( L ` G ) ) =/= { .0. } <-> ( ._|_ ` ( L ` G ) ) e. A ) ) |