Step |
Hyp |
Ref |
Expression |
1 |
|
trlset.b |
|- B = ( Base ` K ) |
2 |
|
trlset.l |
|- .<_ = ( le ` K ) |
3 |
|
trlset.j |
|- .\/ = ( join ` K ) |
4 |
|
trlset.m |
|- ./\ = ( meet ` K ) |
5 |
|
trlset.a |
|- A = ( Atoms ` K ) |
6 |
|
trlset.h |
|- H = ( LHyp ` K ) |
7 |
|
elex |
|- ( K e. C -> K e. _V ) |
8 |
|
fveq2 |
|- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) ) |
9 |
8 6
|
eqtr4di |
|- ( k = K -> ( LHyp ` k ) = H ) |
10 |
|
fveq2 |
|- ( k = K -> ( LTrn ` k ) = ( LTrn ` K ) ) |
11 |
10
|
fveq1d |
|- ( k = K -> ( ( LTrn ` k ) ` w ) = ( ( LTrn ` K ) ` w ) ) |
12 |
|
fveq2 |
|- ( k = K -> ( Base ` k ) = ( Base ` K ) ) |
13 |
12 1
|
eqtr4di |
|- ( k = K -> ( Base ` k ) = B ) |
14 |
|
fveq2 |
|- ( k = K -> ( Atoms ` k ) = ( Atoms ` K ) ) |
15 |
14 5
|
eqtr4di |
|- ( k = K -> ( Atoms ` k ) = A ) |
16 |
|
fveq2 |
|- ( k = K -> ( le ` k ) = ( le ` K ) ) |
17 |
16 2
|
eqtr4di |
|- ( k = K -> ( le ` k ) = .<_ ) |
18 |
17
|
breqd |
|- ( k = K -> ( p ( le ` k ) w <-> p .<_ w ) ) |
19 |
18
|
notbid |
|- ( k = K -> ( -. p ( le ` k ) w <-> -. p .<_ w ) ) |
20 |
|
fveq2 |
|- ( k = K -> ( meet ` k ) = ( meet ` K ) ) |
21 |
20 4
|
eqtr4di |
|- ( k = K -> ( meet ` k ) = ./\ ) |
22 |
|
fveq2 |
|- ( k = K -> ( join ` k ) = ( join ` K ) ) |
23 |
22 3
|
eqtr4di |
|- ( k = K -> ( join ` k ) = .\/ ) |
24 |
23
|
oveqd |
|- ( k = K -> ( p ( join ` k ) ( f ` p ) ) = ( p .\/ ( f ` p ) ) ) |
25 |
|
eqidd |
|- ( k = K -> w = w ) |
26 |
21 24 25
|
oveq123d |
|- ( k = K -> ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) = ( ( p .\/ ( f ` p ) ) ./\ w ) ) |
27 |
26
|
eqeq2d |
|- ( k = K -> ( x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) <-> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) |
28 |
19 27
|
imbi12d |
|- ( k = K -> ( ( -. p ( le ` k ) w -> x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) ) <-> ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) |
29 |
15 28
|
raleqbidv |
|- ( k = K -> ( A. p e. ( Atoms ` k ) ( -. p ( le ` k ) w -> x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) ) <-> A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) |
30 |
13 29
|
riotaeqbidv |
|- ( k = K -> ( iota_ x e. ( Base ` k ) A. p e. ( Atoms ` k ) ( -. p ( le ` k ) w -> x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) ) ) = ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) |
31 |
11 30
|
mpteq12dv |
|- ( k = K -> ( f e. ( ( LTrn ` k ) ` w ) |-> ( iota_ x e. ( Base ` k ) A. p e. ( Atoms ` k ) ( -. p ( le ` k ) w -> x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) ) ) ) = ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) ) |
32 |
9 31
|
mpteq12dv |
|- ( k = K -> ( w e. ( LHyp ` k ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( iota_ x e. ( Base ` k ) A. p e. ( Atoms ` k ) ( -. p ( le ` k ) w -> x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) ) ) ) ) = ( w e. H |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) ) ) |
33 |
|
df-trl |
|- trL = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( iota_ x e. ( Base ` k ) A. p e. ( Atoms ` k ) ( -. p ( le ` k ) w -> x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) ) ) ) ) ) |
34 |
32 33 6
|
mptfvmpt |
|- ( K e. _V -> ( trL ` K ) = ( w e. H |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) ) ) |
35 |
7 34
|
syl |
|- ( K e. C -> ( trL ` K ) = ( w e. H |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) ) ) |