Step |
Hyp |
Ref |
Expression |
1 |
|
dochsncom.h |
|- H = ( LHyp ` K ) |
2 |
|
dochsncom.o |
|- ._|_ = ( ( ocH ` K ) ` W ) |
3 |
|
dochsncom.u |
|- U = ( ( DVecH ` K ) ` W ) |
4 |
|
dochsncom.v |
|- V = ( Base ` U ) |
5 |
|
dochsncom.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
6 |
|
dochsncom.x |
|- ( ph -> X e. V ) |
7 |
|
dochsncom.y |
|- ( ph -> Y e. V ) |
8 |
|
eqid |
|- ( ( DIsoH ` K ) ` W ) = ( ( DIsoH ` K ) ` W ) |
9 |
|
eqid |
|- ( LSpan ` U ) = ( LSpan ` U ) |
10 |
1 3 4 9 8
|
dihlsprn |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. V ) -> ( ( LSpan ` U ) ` { X } ) e. ran ( ( DIsoH ` K ) ` W ) ) |
11 |
5 6 10
|
syl2anc |
|- ( ph -> ( ( LSpan ` U ) ` { X } ) e. ran ( ( DIsoH ` K ) ` W ) ) |
12 |
1 3 4 9 8
|
dihlsprn |
|- ( ( ( K e. HL /\ W e. H ) /\ Y e. V ) -> ( ( LSpan ` U ) ` { Y } ) e. ran ( ( DIsoH ` K ) ` W ) ) |
13 |
5 7 12
|
syl2anc |
|- ( ph -> ( ( LSpan ` U ) ` { Y } ) e. ran ( ( DIsoH ` K ) ` W ) ) |
14 |
1 8 2 5 11 13
|
dochord3 |
|- ( ph -> ( ( ( LSpan ` U ) ` { X } ) C_ ( ._|_ ` ( ( LSpan ` U ) ` { Y } ) ) <-> ( ( LSpan ` U ) ` { Y } ) C_ ( ._|_ ` ( ( LSpan ` U ) ` { X } ) ) ) ) |
15 |
7
|
snssd |
|- ( ph -> { Y } C_ V ) |
16 |
1 3 2 4 9 5 15
|
dochocsp |
|- ( ph -> ( ._|_ ` ( ( LSpan ` U ) ` { Y } ) ) = ( ._|_ ` { Y } ) ) |
17 |
16
|
sseq2d |
|- ( ph -> ( ( ( LSpan ` U ) ` { X } ) C_ ( ._|_ ` ( ( LSpan ` U ) ` { Y } ) ) <-> ( ( LSpan ` U ) ` { X } ) C_ ( ._|_ ` { Y } ) ) ) |
18 |
6
|
snssd |
|- ( ph -> { X } C_ V ) |
19 |
1 3 2 4 9 5 18
|
dochocsp |
|- ( ph -> ( ._|_ ` ( ( LSpan ` U ) ` { X } ) ) = ( ._|_ ` { X } ) ) |
20 |
19
|
sseq2d |
|- ( ph -> ( ( ( LSpan ` U ) ` { Y } ) C_ ( ._|_ ` ( ( LSpan ` U ) ` { X } ) ) <-> ( ( LSpan ` U ) ` { Y } ) C_ ( ._|_ ` { X } ) ) ) |
21 |
14 17 20
|
3bitr3d |
|- ( ph -> ( ( ( LSpan ` U ) ` { X } ) C_ ( ._|_ ` { Y } ) <-> ( ( LSpan ` U ) ` { Y } ) C_ ( ._|_ ` { X } ) ) ) |
22 |
|
eqid |
|- ( LSubSp ` U ) = ( LSubSp ` U ) |
23 |
1 3 5
|
dvhlmod |
|- ( ph -> U e. LMod ) |
24 |
1 3 4 22 2
|
dochlss |
|- ( ( ( K e. HL /\ W e. H ) /\ { Y } C_ V ) -> ( ._|_ ` { Y } ) e. ( LSubSp ` U ) ) |
25 |
5 15 24
|
syl2anc |
|- ( ph -> ( ._|_ ` { Y } ) e. ( LSubSp ` U ) ) |
26 |
4 22 9 23 25 6
|
lspsnel5 |
|- ( ph -> ( X e. ( ._|_ ` { Y } ) <-> ( ( LSpan ` U ) ` { X } ) C_ ( ._|_ ` { Y } ) ) ) |
27 |
1 3 4 22 2
|
dochlss |
|- ( ( ( K e. HL /\ W e. H ) /\ { X } C_ V ) -> ( ._|_ ` { X } ) e. ( LSubSp ` U ) ) |
28 |
5 18 27
|
syl2anc |
|- ( ph -> ( ._|_ ` { X } ) e. ( LSubSp ` U ) ) |
29 |
4 22 9 23 28 7
|
lspsnel5 |
|- ( ph -> ( Y e. ( ._|_ ` { X } ) <-> ( ( LSpan ` U ) ` { Y } ) C_ ( ._|_ ` { X } ) ) ) |
30 |
21 26 29
|
3bitr4d |
|- ( ph -> ( X e. ( ._|_ ` { Y } ) <-> Y e. ( ._|_ ` { X } ) ) ) |