Step |
Hyp |
Ref |
Expression |
1 |
|
dochval.b |
|- B = ( Base ` K ) |
2 |
|
dochval.g |
|- G = ( glb ` K ) |
3 |
|
dochval.o |
|- ._|_ = ( oc ` K ) |
4 |
|
dochval.h |
|- H = ( LHyp ` K ) |
5 |
|
dochval.i |
|- I = ( ( DIsoH ` K ) ` W ) |
6 |
|
dochval.u |
|- U = ( ( DVecH ` K ) ` W ) |
7 |
|
dochval.v |
|- V = ( Base ` U ) |
8 |
|
dochval.n |
|- N = ( ( ocH ` K ) ` W ) |
9 |
1 2 3 4
|
dochffval |
|- ( K e. X -> ( ocH ` K ) = ( w e. H |-> ( x e. ~P ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( ( ( DIsoH ` K ) ` w ) ` ( ._|_ ` ( G ` { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) ) |
10 |
9
|
fveq1d |
|- ( K e. X -> ( ( ocH ` K ) ` W ) = ( ( w e. H |-> ( x e. ~P ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( ( ( DIsoH ` K ) ` w ) ` ( ._|_ ` ( G ` { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) ` W ) ) |
11 |
8 10
|
syl5eq |
|- ( K e. X -> N = ( ( w e. H |-> ( x e. ~P ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( ( ( DIsoH ` K ) ` w ) ` ( ._|_ ` ( G ` { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) ` W ) ) |
12 |
|
fveq2 |
|- ( w = W -> ( ( DVecH ` K ) ` w ) = ( ( DVecH ` K ) ` W ) ) |
13 |
12 6
|
eqtr4di |
|- ( w = W -> ( ( DVecH ` K ) ` w ) = U ) |
14 |
13
|
fveq2d |
|- ( w = W -> ( Base ` ( ( DVecH ` K ) ` w ) ) = ( Base ` U ) ) |
15 |
14 7
|
eqtr4di |
|- ( w = W -> ( Base ` ( ( DVecH ` K ) ` w ) ) = V ) |
16 |
15
|
pweqd |
|- ( w = W -> ~P ( Base ` ( ( DVecH ` K ) ` w ) ) = ~P V ) |
17 |
|
fveq2 |
|- ( w = W -> ( ( DIsoH ` K ) ` w ) = ( ( DIsoH ` K ) ` W ) ) |
18 |
17 5
|
eqtr4di |
|- ( w = W -> ( ( DIsoH ` K ) ` w ) = I ) |
19 |
18
|
fveq1d |
|- ( w = W -> ( ( ( DIsoH ` K ) ` w ) ` y ) = ( I ` y ) ) |
20 |
19
|
sseq2d |
|- ( w = W -> ( x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) <-> x C_ ( I ` y ) ) ) |
21 |
20
|
rabbidv |
|- ( w = W -> { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } = { y e. B | x C_ ( I ` y ) } ) |
22 |
21
|
fveq2d |
|- ( w = W -> ( G ` { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) = ( G ` { y e. B | x C_ ( I ` y ) } ) ) |
23 |
22
|
fveq2d |
|- ( w = W -> ( ._|_ ` ( G ` { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) = ( ._|_ ` ( G ` { y e. B | x C_ ( I ` y ) } ) ) ) |
24 |
18 23
|
fveq12d |
|- ( w = W -> ( ( ( DIsoH ` K ) ` w ) ` ( ._|_ ` ( G ` { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) = ( I ` ( ._|_ ` ( G ` { y e. B | x C_ ( I ` y ) } ) ) ) ) |
25 |
16 24
|
mpteq12dv |
|- ( w = W -> ( x e. ~P ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( ( ( DIsoH ` K ) ` w ) ` ( ._|_ ` ( G ` { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) = ( x e. ~P V |-> ( I ` ( ._|_ ` ( G ` { y e. B | x C_ ( I ` y ) } ) ) ) ) ) |
26 |
|
eqid |
|- ( w e. H |-> ( x e. ~P ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( ( ( DIsoH ` K ) ` w ) ` ( ._|_ ` ( G ` { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) = ( w e. H |-> ( x e. ~P ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( ( ( DIsoH ` K ) ` w ) ` ( ._|_ ` ( G ` { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) |
27 |
7
|
fvexi |
|- V e. _V |
28 |
27
|
pwex |
|- ~P V e. _V |
29 |
28
|
mptex |
|- ( x e. ~P V |-> ( I ` ( ._|_ ` ( G ` { y e. B | x C_ ( I ` y ) } ) ) ) ) e. _V |
30 |
25 26 29
|
fvmpt |
|- ( W e. H -> ( ( w e. H |-> ( x e. ~P ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( ( ( DIsoH ` K ) ` w ) ` ( ._|_ ` ( G ` { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) ` W ) = ( x e. ~P V |-> ( I ` ( ._|_ ` ( G ` { y e. B | x C_ ( I ` y ) } ) ) ) ) ) |
31 |
11 30
|
sylan9eq |
|- ( ( K e. X /\ W e. H ) -> N = ( x e. ~P V |-> ( I ` ( ._|_ ` ( G ` { y e. B | x C_ ( I ` y ) } ) ) ) ) ) |