Step |
Hyp |
Ref |
Expression |
1 |
|
dochf.h |
|- H = ( LHyp ` K ) |
2 |
|
dochf.i |
|- I = ( ( DIsoH ` K ) ` W ) |
3 |
|
dochf.u |
|- U = ( ( DVecH ` K ) ` W ) |
4 |
|
dochf.v |
|- V = ( Base ` U ) |
5 |
|
dochf.o |
|- ._|_ = ( ( ocH ` K ) ` W ) |
6 |
|
dochf.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
7 |
|
fvexd |
|- ( ( ph /\ x e. ~P V ) -> ( I ` ( ( oc ` K ) ` ( ( glb ` K ) ` { y e. ( Base ` K ) | x C_ ( I ` y ) } ) ) ) e. _V ) |
8 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
9 |
|
eqid |
|- ( glb ` K ) = ( glb ` K ) |
10 |
|
eqid |
|- ( oc ` K ) = ( oc ` K ) |
11 |
8 9 10 1 2 3 4 5
|
dochfval |
|- ( ( K e. HL /\ W e. H ) -> ._|_ = ( x e. ~P V |-> ( I ` ( ( oc ` K ) ` ( ( glb ` K ) ` { y e. ( Base ` K ) | x C_ ( I ` y ) } ) ) ) ) ) |
12 |
6 11
|
syl |
|- ( ph -> ._|_ = ( x e. ~P V |-> ( I ` ( ( oc ` K ) ` ( ( glb ` K ) ` { y e. ( Base ` K ) | x C_ ( I ` y ) } ) ) ) ) ) |
13 |
|
elpwi |
|- ( y e. ~P V -> y C_ V ) |
14 |
1 2 3 4 5
|
dochcl |
|- ( ( ( K e. HL /\ W e. H ) /\ y C_ V ) -> ( ._|_ ` y ) e. ran I ) |
15 |
6 13 14
|
syl2an |
|- ( ( ph /\ y e. ~P V ) -> ( ._|_ ` y ) e. ran I ) |
16 |
7 12 15
|
fmpt2d |
|- ( ph -> ._|_ : ~P V --> ran I ) |