:: Representation Theorem for Heyting Lattices
:: by Jolanta Kamie\'nska
::
:: Received July 14, 1993
:: Copyright (c) 1993 Association of Mizar Users
theorem Th1: :: OPENLATT:1
theorem Th2: :: OPENLATT:2
:: deftheorem defines Topology_of OPENLATT:def 1 :
definition
let T be non
empty TopSpace;
func Top_Union T -> BinOp of
Topology_of T means :
Def2:
:: OPENLATT:def 2
for
P,
Q being
Element of
Topology_of T holds
it . P,
Q = P \/ Q;
existence
ex b1 being BinOp of Topology_of T st
for P, Q being Element of Topology_of T holds b1 . P,Q = P \/ Q
uniqueness
for b1, b2 being BinOp of Topology_of T st ( for P, Q being Element of Topology_of T holds b1 . P,Q = P \/ Q ) & ( for P, Q being Element of Topology_of T holds b2 . P,Q = P \/ Q ) holds
b1 = b2
func Top_Meet T -> BinOp of
Topology_of T means :
Def3:
:: OPENLATT:def 3
for
P,
Q being
Element of
Topology_of T holds
it . P,
Q = P /\ Q;
existence
ex b1 being BinOp of Topology_of T st
for P, Q being Element of Topology_of T holds b1 . P,Q = P /\ Q
uniqueness
for b1, b2 being BinOp of Topology_of T st ( for P, Q being Element of Topology_of T holds b1 . P,Q = P /\ Q ) & ( for P, Q being Element of Topology_of T holds b2 . P,Q = P /\ Q ) holds
b1 = b2
end;
:: deftheorem Def2 defines Top_Union OPENLATT:def 2 :
:: deftheorem Def3 defines Top_Meet OPENLATT:def 3 :
theorem :: OPENLATT:3
canceled;
theorem Th4: :: OPENLATT:4
:: deftheorem defines Open_setLatt OPENLATT:def 4 :
theorem :: OPENLATT:5
theorem :: OPENLATT:6
theorem Th7: :: OPENLATT:7
theorem Th8: :: OPENLATT:8
theorem Th9: :: OPENLATT:9
theorem Th10: :: OPENLATT:10
theorem Th11: :: OPENLATT:11
:: deftheorem defines F_primeSet OPENLATT:def 5 :
theorem Th12: :: OPENLATT:12
:: deftheorem Def6 defines StoneH OPENLATT:def 6 :
theorem Th13: :: OPENLATT:13
theorem Th14: :: OPENLATT:14
:: deftheorem defines StoneS OPENLATT:def 7 :
theorem Th15: :: OPENLATT:15
theorem Th16: :: OPENLATT:16
theorem Th17: :: OPENLATT:17
:: deftheorem defines SF_have OPENLATT:def 8 :
theorem Th18: :: OPENLATT:18
Lm1:
for L being D_Lattice
for F being Filter of L
for b, a being Element of L holds
( F in (SF_have b) \ (SF_have a) iff ( b in F & not a in F ) )
theorem Th19: :: OPENLATT:19
theorem Th20: :: OPENLATT:20
theorem Th21: :: OPENLATT:21
theorem Th22: :: OPENLATT:22
theorem Th23: :: OPENLATT:23
theorem Th24: :: OPENLATT:24
theorem Th25: :: OPENLATT:25
definition
let L be
D_Lattice;
func Set_Union L -> BinOp of
StoneS L means :
Def9:
:: OPENLATT:def 9
for
A,
B being
Element of
StoneS L holds
it . A,
B = A \/ B;
existence
ex b1 being BinOp of StoneS L st
for A, B being Element of StoneS L holds b1 . A,B = A \/ B
uniqueness
for b1, b2 being BinOp of StoneS L st ( for A, B being Element of StoneS L holds b1 . A,B = A \/ B ) & ( for A, B being Element of StoneS L holds b2 . A,B = A \/ B ) holds
b1 = b2
func Set_Meet L -> BinOp of
StoneS L means :
Def10:
:: OPENLATT:def 10
for
A,
B being
Element of
StoneS L holds
it . A,
B = A /\ B;
existence
ex b1 being BinOp of StoneS L st
for A, B being Element of StoneS L holds b1 . A,B = A /\ B
uniqueness
for b1, b2 being BinOp of StoneS L st ( for A, B being Element of StoneS L holds b1 . A,B = A /\ B ) & ( for A, B being Element of StoneS L holds b2 . A,B = A /\ B ) holds
b1 = b2
end;
:: deftheorem Def9 defines Set_Union OPENLATT:def 9 :
:: deftheorem Def10 defines Set_Meet OPENLATT:def 10 :
theorem Th26: :: OPENLATT:26
:: deftheorem defines StoneLatt OPENLATT:def 11 :
theorem :: OPENLATT:27
theorem :: OPENLATT:28
theorem :: OPENLATT:29
theorem Th30: :: OPENLATT:30
theorem :: OPENLATT:31
theorem :: OPENLATT:32
theorem Th33: :: OPENLATT:33
theorem Th34: :: OPENLATT:34
theorem Th35: :: OPENLATT:35
:: deftheorem Def12 defines HTopSpace OPENLATT:def 12 :
theorem :: OPENLATT:36
theorem Th37: :: OPENLATT:37
theorem :: OPENLATT:38
canceled;
theorem Th39: :: OPENLATT:39
theorem :: OPENLATT:40
theorem :: OPENLATT:41
theorem :: OPENLATT:42