Step |
Hyp |
Ref |
Expression |
1 |
|
ltrnset.l |
|- .<_ = ( le ` K ) |
2 |
|
ltrnset.j |
|- .\/ = ( join ` K ) |
3 |
|
ltrnset.m |
|- ./\ = ( meet ` K ) |
4 |
|
ltrnset.a |
|- A = ( Atoms ` K ) |
5 |
|
ltrnset.h |
|- H = ( LHyp ` K ) |
6 |
|
elex |
|- ( K e. C -> K e. _V ) |
7 |
|
fveq2 |
|- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) ) |
8 |
7 5
|
eqtr4di |
|- ( k = K -> ( LHyp ` k ) = H ) |
9 |
|
fveq2 |
|- ( k = K -> ( LDil ` k ) = ( LDil ` K ) ) |
10 |
9
|
fveq1d |
|- ( k = K -> ( ( LDil ` k ) ` w ) = ( ( LDil ` K ) ` w ) ) |
11 |
|
fveq2 |
|- ( k = K -> ( Atoms ` k ) = ( Atoms ` K ) ) |
12 |
11 4
|
eqtr4di |
|- ( k = K -> ( Atoms ` k ) = A ) |
13 |
|
fveq2 |
|- ( k = K -> ( le ` k ) = ( le ` K ) ) |
14 |
13 1
|
eqtr4di |
|- ( k = K -> ( le ` k ) = .<_ ) |
15 |
14
|
breqd |
|- ( k = K -> ( p ( le ` k ) w <-> p .<_ w ) ) |
16 |
15
|
notbid |
|- ( k = K -> ( -. p ( le ` k ) w <-> -. p .<_ w ) ) |
17 |
14
|
breqd |
|- ( k = K -> ( q ( le ` k ) w <-> q .<_ w ) ) |
18 |
17
|
notbid |
|- ( k = K -> ( -. q ( le ` k ) w <-> -. q .<_ w ) ) |
19 |
16 18
|
anbi12d |
|- ( k = K -> ( ( -. p ( le ` k ) w /\ -. q ( le ` k ) w ) <-> ( -. p .<_ w /\ -. q .<_ w ) ) ) |
20 |
|
fveq2 |
|- ( k = K -> ( meet ` k ) = ( meet ` K ) ) |
21 |
20 3
|
eqtr4di |
|- ( k = K -> ( meet ` k ) = ./\ ) |
22 |
|
fveq2 |
|- ( k = K -> ( join ` k ) = ( join ` K ) ) |
23 |
22 2
|
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 |
23
|
oveqd |
|- ( k = K -> ( q ( join ` k ) ( f ` q ) ) = ( q .\/ ( f ` q ) ) ) |
28 |
21 27 25
|
oveq123d |
|- ( k = K -> ( ( q ( join ` k ) ( f ` q ) ) ( meet ` k ) w ) = ( ( q .\/ ( f ` q ) ) ./\ w ) ) |
29 |
26 28
|
eqeq12d |
|- ( k = K -> ( ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) = ( ( q ( join ` k ) ( f ` q ) ) ( meet ` k ) w ) <-> ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( q .\/ ( f ` q ) ) ./\ w ) ) ) |
30 |
19 29
|
imbi12d |
|- ( k = K -> ( ( ( -. p ( le ` k ) w /\ -. q ( le ` k ) w ) -> ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) = ( ( q ( join ` k ) ( f ` q ) ) ( meet ` k ) w ) ) <-> ( ( -. p .<_ w /\ -. q .<_ w ) -> ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( q .\/ ( f ` q ) ) ./\ w ) ) ) ) |
31 |
12 30
|
raleqbidv |
|- ( k = K -> ( A. q e. ( Atoms ` k ) ( ( -. p ( le ` k ) w /\ -. q ( le ` k ) w ) -> ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) = ( ( q ( join ` k ) ( f ` q ) ) ( meet ` k ) w ) ) <-> A. q e. A ( ( -. p .<_ w /\ -. q .<_ w ) -> ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( q .\/ ( f ` q ) ) ./\ w ) ) ) ) |
32 |
12 31
|
raleqbidv |
|- ( k = K -> ( A. p e. ( Atoms ` k ) A. q e. ( Atoms ` k ) ( ( -. p ( le ` k ) w /\ -. q ( le ` k ) w ) -> ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) = ( ( q ( join ` k ) ( f ` q ) ) ( meet ` k ) w ) ) <-> A. p e. A A. q e. A ( ( -. p .<_ w /\ -. q .<_ w ) -> ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( q .\/ ( f ` q ) ) ./\ w ) ) ) ) |
33 |
10 32
|
rabeqbidv |
|- ( k = K -> { f e. ( ( LDil ` k ) ` w ) | A. p e. ( Atoms ` k ) A. q e. ( Atoms ` k ) ( ( -. p ( le ` k ) w /\ -. q ( le ` k ) w ) -> ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) = ( ( q ( join ` k ) ( f ` q ) ) ( meet ` k ) w ) ) } = { f e. ( ( LDil ` K ) ` w ) | A. p e. A A. q e. A ( ( -. p .<_ w /\ -. q .<_ w ) -> ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( q .\/ ( f ` q ) ) ./\ w ) ) } ) |
34 |
8 33
|
mpteq12dv |
|- ( k = K -> ( w e. ( LHyp ` k ) |-> { f e. ( ( LDil ` k ) ` w ) | A. p e. ( Atoms ` k ) A. q e. ( Atoms ` k ) ( ( -. p ( le ` k ) w /\ -. q ( le ` k ) w ) -> ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) = ( ( q ( join ` k ) ( f ` q ) ) ( meet ` k ) w ) ) } ) = ( w e. H |-> { f e. ( ( LDil ` K ) ` w ) | A. p e. A A. q e. A ( ( -. p .<_ w /\ -. q .<_ w ) -> ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( q .\/ ( f ` q ) ) ./\ w ) ) } ) ) |
35 |
|
df-ltrn |
|- LTrn = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { f e. ( ( LDil ` k ) ` w ) | A. p e. ( Atoms ` k ) A. q e. ( Atoms ` k ) ( ( -. p ( le ` k ) w /\ -. q ( le ` k ) w ) -> ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) = ( ( q ( join ` k ) ( f ` q ) ) ( meet ` k ) w ) ) } ) ) |
36 |
34 35 5
|
mptfvmpt |
|- ( K e. _V -> ( LTrn ` K ) = ( w e. H |-> { f e. ( ( LDil ` K ) ` w ) | A. p e. A A. q e. A ( ( -. p .<_ w /\ -. q .<_ w ) -> ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( q .\/ ( f ` q ) ) ./\ w ) ) } ) ) |
37 |
6 36
|
syl |
|- ( K e. C -> ( LTrn ` K ) = ( w e. H |-> { f e. ( ( LDil ` K ) ` w ) | A. p e. A A. q e. A ( ( -. p .<_ w /\ -. q .<_ w ) -> ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( q .\/ ( f ` q ) ) ./\ w ) ) } ) ) |