| Step |
Hyp |
Ref |
Expression |
| 1 |
|
dochoc0.h |
|- H = ( LHyp ` K ) |
| 2 |
|
dochoc0.u |
|- U = ( ( DVecH ` K ) ` W ) |
| 3 |
|
dochoc0.o |
|- ._|_ = ( ( ocH ` K ) ` W ) |
| 4 |
|
dochoc0.z |
|- .0. = ( 0g ` U ) |
| 5 |
|
dochoc0.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
| 6 |
|
eqid |
|- ( Base ` U ) = ( Base ` U ) |
| 7 |
1 2 3 6 4
|
doch0 |
|- ( ( K e. HL /\ W e. H ) -> ( ._|_ ` { .0. } ) = ( Base ` U ) ) |
| 8 |
7
|
fveq2d |
|- ( ( K e. HL /\ W e. H ) -> ( ._|_ ` ( ._|_ ` { .0. } ) ) = ( ._|_ ` ( Base ` U ) ) ) |
| 9 |
1 2 3 6 4
|
doch1 |
|- ( ( K e. HL /\ W e. H ) -> ( ._|_ ` ( Base ` U ) ) = { .0. } ) |
| 10 |
8 9
|
eqtrd |
|- ( ( K e. HL /\ W e. H ) -> ( ._|_ ` ( ._|_ ` { .0. } ) ) = { .0. } ) |
| 11 |
5 10
|
syl |
|- ( ph -> ( ._|_ ` ( ._|_ ` { .0. } ) ) = { .0. } ) |