Step |
Hyp |
Ref |
Expression |
1 |
|
docaval.j |
|- .\/ = ( join ` K ) |
2 |
|
docaval.m |
|- ./\ = ( meet ` K ) |
3 |
|
docaval.o |
|- ._|_ = ( oc ` K ) |
4 |
|
docaval.h |
|- H = ( LHyp ` K ) |
5 |
|
docaval.t |
|- T = ( ( LTrn ` K ) ` W ) |
6 |
|
docaval.i |
|- I = ( ( DIsoA ` K ) ` W ) |
7 |
|
docaval.n |
|- N = ( ( ocA ` K ) ` W ) |
8 |
1 2 3 4
|
docaffvalN |
|- ( K e. V -> ( ocA ` K ) = ( w e. H |-> ( x e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( DIsoA ` K ) ` w ) ` ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) ) ) ) ) |
9 |
8
|
fveq1d |
|- ( K e. V -> ( ( ocA ` K ) ` W ) = ( ( w e. H |-> ( x e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( DIsoA ` K ) ` w ) ` ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) ) ) ) ` W ) ) |
10 |
7 9
|
syl5eq |
|- ( K e. V -> N = ( ( w e. H |-> ( x e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( DIsoA ` K ) ` w ) ` ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) ) ) ) ` W ) ) |
11 |
|
fveq2 |
|- ( w = W -> ( ( LTrn ` K ) ` w ) = ( ( LTrn ` K ) ` W ) ) |
12 |
11 5
|
eqtr4di |
|- ( w = W -> ( ( LTrn ` K ) ` w ) = T ) |
13 |
12
|
pweqd |
|- ( w = W -> ~P ( ( LTrn ` K ) ` w ) = ~P T ) |
14 |
|
fveq2 |
|- ( w = W -> ( ( DIsoA ` K ) ` w ) = ( ( DIsoA ` K ) ` W ) ) |
15 |
14 6
|
eqtr4di |
|- ( w = W -> ( ( DIsoA ` K ) ` w ) = I ) |
16 |
15
|
cnveqd |
|- ( w = W -> `' ( ( DIsoA ` K ) ` w ) = `' I ) |
17 |
15
|
rneqd |
|- ( w = W -> ran ( ( DIsoA ` K ) ` w ) = ran I ) |
18 |
17
|
rabeqdv |
|- ( w = W -> { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } = { z e. ran I | x C_ z } ) |
19 |
18
|
inteqd |
|- ( w = W -> |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } = |^| { z e. ran I | x C_ z } ) |
20 |
16 19
|
fveq12d |
|- ( w = W -> ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) = ( `' I ` |^| { z e. ran I | x C_ z } ) ) |
21 |
20
|
fveq2d |
|- ( w = W -> ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) = ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) ) |
22 |
|
fveq2 |
|- ( w = W -> ( ._|_ ` w ) = ( ._|_ ` W ) ) |
23 |
21 22
|
oveq12d |
|- ( w = W -> ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) = ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ) |
24 |
|
id |
|- ( w = W -> w = W ) |
25 |
23 24
|
oveq12d |
|- ( w = W -> ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) = ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) |
26 |
15 25
|
fveq12d |
|- ( w = W -> ( ( ( DIsoA ` K ) ` w ) ` ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) ) = ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) |
27 |
13 26
|
mpteq12dv |
|- ( w = W -> ( x e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( DIsoA ` K ) ` w ) ` ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) ) ) = ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) ) |
28 |
|
eqid |
|- ( w e. H |-> ( x e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( DIsoA ` K ) ` w ) ` ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) ) ) ) = ( w e. H |-> ( x e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( DIsoA ` K ) ` w ) ` ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) ) ) ) |
29 |
5
|
fvexi |
|- T e. _V |
30 |
29
|
pwex |
|- ~P T e. _V |
31 |
30
|
mptex |
|- ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) e. _V |
32 |
27 28 31
|
fvmpt |
|- ( W e. H -> ( ( w e. H |-> ( x e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( DIsoA ` K ) ` w ) ` ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) ) ) ) ` W ) = ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) ) |
33 |
10 32
|
sylan9eq |
|- ( ( K e. V /\ W e. H ) -> N = ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) ) |