| Step |
Hyp |
Ref |
Expression |
| 1 |
|
doch11.h |
|- H = ( LHyp ` K ) |
| 2 |
|
doch11.i |
|- I = ( ( DIsoH ` K ) ` W ) |
| 3 |
|
doch11.o |
|- ._|_ = ( ( ocH ` K ) ` W ) |
| 4 |
|
doch11.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
| 5 |
|
doch11.x |
|- ( ph -> X e. ran I ) |
| 6 |
|
doch11.y |
|- ( ph -> Y e. ran I ) |
| 7 |
|
eqid |
|- ( ( DVecH ` K ) ` W ) = ( ( DVecH ` K ) ` W ) |
| 8 |
|
eqid |
|- ( LSubSp ` ( ( DVecH ` K ) ` W ) ) = ( LSubSp ` ( ( DVecH ` K ) ` W ) ) |
| 9 |
1 7 2 8
|
dihrnlss |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> X e. ( LSubSp ` ( ( DVecH ` K ) ` W ) ) ) |
| 10 |
4 5 9
|
syl2anc |
|- ( ph -> X e. ( LSubSp ` ( ( DVecH ` K ) ` W ) ) ) |
| 11 |
|
eqid |
|- ( Base ` ( ( DVecH ` K ) ` W ) ) = ( Base ` ( ( DVecH ` K ) ` W ) ) |
| 12 |
11 8
|
lssss |
|- ( X e. ( LSubSp ` ( ( DVecH ` K ) ` W ) ) -> X C_ ( Base ` ( ( DVecH ` K ) ` W ) ) ) |
| 13 |
10 12
|
syl |
|- ( ph -> X C_ ( Base ` ( ( DVecH ` K ) ` W ) ) ) |
| 14 |
1 2 7 11 3
|
dochcl |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ ( Base ` ( ( DVecH ` K ) ` W ) ) ) -> ( ._|_ ` X ) e. ran I ) |
| 15 |
4 13 14
|
syl2anc |
|- ( ph -> ( ._|_ ` X ) e. ran I ) |
| 16 |
1 2 3 4 15 6
|
dochord |
|- ( ph -> ( ( ._|_ ` X ) C_ Y <-> ( ._|_ ` Y ) C_ ( ._|_ ` ( ._|_ ` X ) ) ) ) |
| 17 |
1 2 3
|
dochoc |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( ._|_ ` ( ._|_ ` X ) ) = X ) |
| 18 |
4 5 17
|
syl2anc |
|- ( ph -> ( ._|_ ` ( ._|_ ` X ) ) = X ) |
| 19 |
18
|
sseq2d |
|- ( ph -> ( ( ._|_ ` Y ) C_ ( ._|_ ` ( ._|_ ` X ) ) <-> ( ._|_ ` Y ) C_ X ) ) |
| 20 |
16 19
|
bitrd |
|- ( ph -> ( ( ._|_ ` X ) C_ Y <-> ( ._|_ ` Y ) C_ X ) ) |