| Step |
Hyp |
Ref |
Expression |
| 1 |
|
dochsnkr.h |
|- H = ( LHyp ` K ) |
| 2 |
|
dochsnkr.o |
|- ._|_ = ( ( ocH ` K ) ` W ) |
| 3 |
|
dochsnkr.u |
|- U = ( ( DVecH ` K ) ` W ) |
| 4 |
|
dochsnkr.v |
|- V = ( Base ` U ) |
| 5 |
|
dochsnkr.z |
|- .0. = ( 0g ` U ) |
| 6 |
|
dochsnkr.f |
|- F = ( LFnl ` U ) |
| 7 |
|
dochsnkr.l |
|- L = ( LKer ` U ) |
| 8 |
|
dochsnkr.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
| 9 |
|
dochsnkr.g |
|- ( ph -> G e. F ) |
| 10 |
|
dochsnkr.x |
|- ( ph -> X e. ( ( ._|_ ` ( L ` G ) ) \ { .0. } ) ) |
| 11 |
|
eldif |
|- ( X e. ( ( ._|_ ` ( L ` G ) ) \ { .0. } ) <-> ( X e. ( ._|_ ` ( L ` G ) ) /\ -. X e. { .0. } ) ) |
| 12 |
|
nelne1 |
|- ( ( X e. ( ._|_ ` ( L ` G ) ) /\ -. X e. { .0. } ) -> ( ._|_ ` ( L ` G ) ) =/= { .0. } ) |
| 13 |
11 12
|
sylbi |
|- ( X e. ( ( ._|_ ` ( L ` G ) ) \ { .0. } ) -> ( ._|_ ` ( L ` G ) ) =/= { .0. } ) |
| 14 |
10 13
|
syl |
|- ( ph -> ( ._|_ ` ( L ` G ) ) =/= { .0. } ) |
| 15 |
1 3 8
|
dvhlmod |
|- ( ph -> U e. LMod ) |
| 16 |
4 6 7 15 9
|
lkrssv |
|- ( ph -> ( L ` G ) C_ V ) |
| 17 |
1 2 3 4 5 8 16
|
dochn0nv |
|- ( ph -> ( ( ._|_ ` ( L ` G ) ) =/= { .0. } <-> ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) =/= V ) ) |
| 18 |
14 17
|
mpbid |
|- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) =/= V ) |