Article TDLAT_1, MML version 4.99.1005
:: TDLAT_1:th 1
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
b2 \/ b3 = [#] b1
iff
b2 ` c= b3;
:: TDLAT_1:th 2
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
b2 misses b3
iff
b3 c= b2 `;
:: TDLAT_1:th 3
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
Cl Int Cl b2 c= Cl b2;
:: TDLAT_1:th 4
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
Int b2 c= Int Cl Int b2;
:: TDLAT_1:th 5
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
Int Cl b2 = Int Cl Int Cl b2;
:: TDLAT_1:th 6
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st (b2 is closed(b1) or b3 is closed(b1))
holds (Cl Int b2) \/ Cl Int b3 = Cl Int (b2 \/ b3);
:: TDLAT_1:th 7
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st (b2 is open(b1) or b3 is open(b1))
holds (Int Cl b2) /\ Int Cl b3 = Int Cl (b2 /\ b3);
:: TDLAT_1:th 8
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
Int (b2 /\ Cl (b2 `)) = {} b1;
:: TDLAT_1:th 9
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
Cl (b2 \/ Int (b2 `)) = [#] b1;
:: TDLAT_1:th 10
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
(Int Cl (b2 \/ ((Int Cl b3) \/ b3))) \/ (b2 \/ ((Int Cl b3) \/ b3)) = (Int Cl (b2 \/ b3)) \/ (b2 \/ b3);
:: TDLAT_1:th 11
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
(Int Cl (((Int Cl b2) \/ b2) \/ b3)) \/ (((Int Cl b2) \/ b2) \/ b3) = (Int Cl (b2 \/ b3)) \/ (b2 \/ b3);
:: TDLAT_1:th 12
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
(Cl Int (b2 /\ ((Cl Int b3) /\ b3))) /\ (b2 /\ ((Cl Int b3) /\ b3)) = (Cl Int (b2 /\ b3)) /\ (b2 /\ b3);
:: TDLAT_1:th 13
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
(Cl Int (((Cl Int b2) /\ b2) /\ b3)) /\ (((Cl Int b2) /\ b2) /\ b3) = (Cl Int (b2 /\ b3)) /\ (b2 /\ b3);
:: TDLAT_1:th 14
theorem
for b1 being TopSpace-like TopStruct holds
{} b1 is condensed(b1);
:: TDLAT_1:th 15
theorem
for b1 being TopSpace-like TopStruct holds
[#] b1 is condensed(b1);
:: TDLAT_1:th 16
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is condensed(b1)
holds b2 ` is condensed(b1);
:: TDLAT_1:th 17
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is condensed(b1) & b3 is condensed(b1)
holds (Int Cl (b2 \/ b3)) \/ (b2 \/ b3) is condensed(b1) &
(Cl Int (b2 /\ b3)) /\ (b2 /\ b3) is condensed(b1);
:: TDLAT_1:th 18
theorem
for b1 being TopSpace-like TopStruct holds
{} b1 is closed_condensed(b1);
:: TDLAT_1:th 19
theorem
for b1 being TopSpace-like TopStruct holds
[#] b1 is closed_condensed(b1);
:: TDLAT_1:th 20
theorem
for b1 being TopSpace-like TopStruct holds
{} b1 is open_condensed(b1);
:: TDLAT_1:th 21
theorem
for b1 being TopSpace-like TopStruct holds
[#] b1 is open_condensed(b1);
:: TDLAT_1:th 22
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
Cl Int b2 is closed_condensed(b1);
:: TDLAT_1:th 23
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
Int Cl b2 is open_condensed(b1);
:: TDLAT_1:th 24
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is condensed(b1)
holds Cl b2 is closed_condensed(b1);
:: TDLAT_1:th 25
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is condensed(b1)
holds Int b2 is open_condensed(b1);
:: TDLAT_1:th 26
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is condensed(b1)
holds Cl (b2 `) is closed_condensed(b1);
:: TDLAT_1:th 27
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is condensed(b1)
holds Int (b2 `) is open_condensed(b1);
:: TDLAT_1:th 28
theorem
for b1 being TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
st b2 is closed_condensed(b1) & b3 is closed_condensed(b1) & b4 is closed_condensed(b1)
holds Cl Int (b2 /\ Cl Int (b3 /\ b4)) = Cl Int ((Cl Int (b2 /\ b3)) /\ b4);
:: TDLAT_1:th 29
theorem
for b1 being TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
st b2 is open_condensed(b1) & b3 is open_condensed(b1) & b4 is open_condensed(b1)
holds Int Cl (b2 \/ Int Cl (b3 \/ b4)) = Int Cl ((Int Cl (b2 \/ b3)) \/ b4);
:: TDLAT_1:funcnot 1 => TDLAT_1:func 1
definition
let a1 be TopStruct;
func Domains_of A1 -> Element of bool bool the carrier of a1 equals
{b1 where b1 is Element of bool the carrier of a1: b1 is condensed(a1)};
end;
:: TDLAT_1:def 1
theorem
for b1 being TopStruct holds
Domains_of b1 = {b2 where b2 is Element of bool the carrier of b1: b2 is condensed(b1)};
:: TDLAT_1:funcreg 1
registration
let a1 be TopSpace-like TopStruct;
cluster Domains_of a1 -> non empty;
end;
:: TDLAT_1:funcnot 2 => TDLAT_1:func 2
definition
let a1 be TopSpace-like TopStruct;
func Domains_Union A1 -> Function-like quasi_total Relation of [:Domains_of a1,Domains_of a1:],Domains_of a1 means
for b1, b2 being Element of Domains_of a1 holds
it .(b1,b2) = (Int Cl (b1 \/ b2)) \/ (b1 \/ b2);
end;
:: TDLAT_1:def 2
theorem
for b1 being TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of [:Domains_of b1,Domains_of b1:],Domains_of b1 holds
b2 = Domains_Union b1
iff
for b3, b4 being Element of Domains_of b1 holds
b2 .(b3,b4) = (Int Cl (b3 \/ b4)) \/ (b3 \/ b4);
:: TDLAT_1:funcnot 3 => TDLAT_1:func 2
notation
let a1 be TopSpace-like TopStruct;
synonym D-Union a1 for Domains_Union a1;
end;
:: TDLAT_1:funcnot 4 => TDLAT_1:func 3
definition
let a1 be TopSpace-like TopStruct;
func Domains_Meet A1 -> Function-like quasi_total Relation of [:Domains_of a1,Domains_of a1:],Domains_of a1 means
for b1, b2 being Element of Domains_of a1 holds
it .(b1,b2) = (Cl Int (b1 /\ b2)) /\ (b1 /\ b2);
end;
:: TDLAT_1:def 3
theorem
for b1 being TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of [:Domains_of b1,Domains_of b1:],Domains_of b1 holds
b2 = Domains_Meet b1
iff
for b3, b4 being Element of Domains_of b1 holds
b2 .(b3,b4) = (Cl Int (b3 /\ b4)) /\ (b3 /\ b4);
:: TDLAT_1:funcnot 5 => TDLAT_1:func 3
notation
let a1 be TopSpace-like TopStruct;
synonym D-Meet a1 for Domains_Meet a1;
end;
:: TDLAT_1:th 30
theorem
for b1 being TopSpace-like TopStruct holds
LattStr(#Domains_of b1,Domains_Union b1,Domains_Meet b1#) is non empty Lattice-like bounded complemented LattStr;
:: TDLAT_1:funcnot 6 => TDLAT_1:func 4
definition
let a1 be TopSpace-like TopStruct;
func Domains_Lattice A1 -> non empty Lattice-like bounded complemented LattStr equals
LattStr(#Domains_of a1,Domains_Union a1,Domains_Meet a1#);
end;
:: TDLAT_1:def 4
theorem
for b1 being TopSpace-like TopStruct holds
Domains_Lattice b1 = LattStr(#Domains_of b1,Domains_Union b1,Domains_Meet b1#);
:: TDLAT_1:funcnot 7 => TDLAT_1:func 5
definition
let a1 be TopStruct;
func Closed_Domains_of A1 -> Element of bool bool the carrier of a1 equals
{b1 where b1 is Element of bool the carrier of a1: b1 is closed_condensed(a1)};
end;
:: TDLAT_1:def 5
theorem
for b1 being TopStruct holds
Closed_Domains_of b1 = {b2 where b2 is Element of bool the carrier of b1: b2 is closed_condensed(b1)};
:: TDLAT_1:funcreg 2
registration
let a1 be TopSpace-like TopStruct;
cluster Closed_Domains_of a1 -> non empty;
end;
:: TDLAT_1:th 31
theorem
for b1 being TopSpace-like TopStruct holds
Closed_Domains_of b1 c= Domains_of b1;
:: TDLAT_1:funcnot 8 => TDLAT_1:func 6
definition
let a1 be TopSpace-like TopStruct;
func Closed_Domains_Union A1 -> Function-like quasi_total Relation of [:Closed_Domains_of a1,Closed_Domains_of a1:],Closed_Domains_of a1 means
for b1, b2 being Element of Closed_Domains_of a1 holds
it .(b1,b2) = b1 \/ b2;
end;
:: TDLAT_1:def 6
theorem
for b1 being TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of [:Closed_Domains_of b1,Closed_Domains_of b1:],Closed_Domains_of b1 holds
b2 = Closed_Domains_Union b1
iff
for b3, b4 being Element of Closed_Domains_of b1 holds
b2 .(b3,b4) = b3 \/ b4;
:: TDLAT_1:funcnot 9 => TDLAT_1:func 6
notation
let a1 be TopSpace-like TopStruct;
synonym CLD-Union a1 for Closed_Domains_Union a1;
end;
:: TDLAT_1:th 32
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of Closed_Domains_of b1 holds
(Closed_Domains_Union b1) .(b2,b3) = (Domains_Union b1) .(b2,b3);
:: TDLAT_1:funcnot 10 => TDLAT_1:func 7
definition
let a1 be TopSpace-like TopStruct;
func Closed_Domains_Meet A1 -> Function-like quasi_total Relation of [:Closed_Domains_of a1,Closed_Domains_of a1:],Closed_Domains_of a1 means
for b1, b2 being Element of Closed_Domains_of a1 holds
it .(b1,b2) = Cl Int (b1 /\ b2);
end;
:: TDLAT_1:def 7
theorem
for b1 being TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of [:Closed_Domains_of b1,Closed_Domains_of b1:],Closed_Domains_of b1 holds
b2 = Closed_Domains_Meet b1
iff
for b3, b4 being Element of Closed_Domains_of b1 holds
b2 .(b3,b4) = Cl Int (b3 /\ b4);
:: TDLAT_1:funcnot 11 => TDLAT_1:func 7
notation
let a1 be TopSpace-like TopStruct;
synonym CLD-Meet a1 for Closed_Domains_Meet a1;
end;
:: TDLAT_1:th 33
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of Closed_Domains_of b1 holds
(Closed_Domains_Meet b1) .(b2,b3) = (Domains_Meet b1) .(b2,b3);
:: TDLAT_1:th 34
theorem
for b1 being TopSpace-like TopStruct holds
LattStr(#Closed_Domains_of b1,Closed_Domains_Union b1,Closed_Domains_Meet b1#) is non empty Lattice-like Boolean LattStr;
:: TDLAT_1:funcnot 12 => TDLAT_1:func 8
definition
let a1 be TopSpace-like TopStruct;
func Closed_Domains_Lattice A1 -> non empty Lattice-like Boolean LattStr equals
LattStr(#Closed_Domains_of a1,Closed_Domains_Union a1,Closed_Domains_Meet a1#);
end;
:: TDLAT_1:def 8
theorem
for b1 being TopSpace-like TopStruct holds
Closed_Domains_Lattice b1 = LattStr(#Closed_Domains_of b1,Closed_Domains_Union b1,Closed_Domains_Meet b1#);
:: TDLAT_1:funcnot 13 => TDLAT_1:func 9
definition
let a1 be TopStruct;
func Open_Domains_of A1 -> Element of bool bool the carrier of a1 equals
{b1 where b1 is Element of bool the carrier of a1: b1 is open_condensed(a1)};
end;
:: TDLAT_1:def 9
theorem
for b1 being TopStruct holds
Open_Domains_of b1 = {b2 where b2 is Element of bool the carrier of b1: b2 is open_condensed(b1)};
:: TDLAT_1:funcreg 3
registration
let a1 be TopSpace-like TopStruct;
cluster Open_Domains_of a1 -> non empty;
end;
:: TDLAT_1:th 35
theorem
for b1 being TopSpace-like TopStruct holds
Open_Domains_of b1 c= Domains_of b1;
:: TDLAT_1:funcnot 14 => TDLAT_1:func 10
definition
let a1 be TopSpace-like TopStruct;
func Open_Domains_Union A1 -> Function-like quasi_total Relation of [:Open_Domains_of a1,Open_Domains_of a1:],Open_Domains_of a1 means
for b1, b2 being Element of Open_Domains_of a1 holds
it .(b1,b2) = Int Cl (b1 \/ b2);
end;
:: TDLAT_1:def 10
theorem
for b1 being TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of [:Open_Domains_of b1,Open_Domains_of b1:],Open_Domains_of b1 holds
b2 = Open_Domains_Union b1
iff
for b3, b4 being Element of Open_Domains_of b1 holds
b2 .(b3,b4) = Int Cl (b3 \/ b4);
:: TDLAT_1:funcnot 15 => TDLAT_1:func 10
notation
let a1 be TopSpace-like TopStruct;
synonym OPD-Union a1 for Open_Domains_Union a1;
end;
:: TDLAT_1:th 36
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of Open_Domains_of b1 holds
(Open_Domains_Union b1) .(b2,b3) = (Domains_Union b1) .(b2,b3);
:: TDLAT_1:funcnot 16 => TDLAT_1:func 11
definition
let a1 be TopSpace-like TopStruct;
func Open_Domains_Meet A1 -> Function-like quasi_total Relation of [:Open_Domains_of a1,Open_Domains_of a1:],Open_Domains_of a1 means
for b1, b2 being Element of Open_Domains_of a1 holds
it .(b1,b2) = b1 /\ b2;
end;
:: TDLAT_1:def 11
theorem
for b1 being TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of [:Open_Domains_of b1,Open_Domains_of b1:],Open_Domains_of b1 holds
b2 = Open_Domains_Meet b1
iff
for b3, b4 being Element of Open_Domains_of b1 holds
b2 .(b3,b4) = b3 /\ b4;
:: TDLAT_1:funcnot 17 => TDLAT_1:func 11
notation
let a1 be TopSpace-like TopStruct;
synonym OPD-Meet a1 for Open_Domains_Meet a1;
end;
:: TDLAT_1:th 37
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of Open_Domains_of b1 holds
(Open_Domains_Meet b1) .(b2,b3) = (Domains_Meet b1) .(b2,b3);
:: TDLAT_1:th 38
theorem
for b1 being TopSpace-like TopStruct holds
LattStr(#Open_Domains_of b1,Open_Domains_Union b1,Open_Domains_Meet b1#) is non empty Lattice-like Boolean LattStr;
:: TDLAT_1:funcnot 18 => TDLAT_1:func 12
definition
let a1 be TopSpace-like TopStruct;
func Open_Domains_Lattice A1 -> non empty Lattice-like Boolean LattStr equals
LattStr(#Open_Domains_of a1,Open_Domains_Union a1,Open_Domains_Meet a1#);
end;
:: TDLAT_1:def 12
theorem
for b1 being TopSpace-like TopStruct holds
Open_Domains_Lattice b1 = LattStr(#Open_Domains_of b1,Open_Domains_Union b1,Open_Domains_Meet b1#);
:: TDLAT_1:th 39
theorem
for b1 being TopSpace-like TopStruct holds
Closed_Domains_Union b1 = (Domains_Union b1) || Closed_Domains_of b1;
:: TDLAT_1:th 40
theorem
for b1 being TopSpace-like TopStruct holds
Closed_Domains_Meet b1 = (Domains_Meet b1) || Closed_Domains_of b1;
:: TDLAT_1:th 41
theorem
for b1 being TopSpace-like TopStruct holds
Closed_Domains_Lattice b1 is SubLattice of Domains_Lattice b1;
:: TDLAT_1:th 42
theorem
for b1 being TopSpace-like TopStruct holds
Open_Domains_Union b1 = (Domains_Union b1) || Open_Domains_of b1;
:: TDLAT_1:th 43
theorem
for b1 being TopSpace-like TopStruct holds
Open_Domains_Meet b1 = (Domains_Meet b1) || Open_Domains_of b1;
:: TDLAT_1:th 44
theorem
for b1 being TopSpace-like TopStruct holds
Open_Domains_Lattice b1 is SubLattice of Domains_Lattice b1;