Article WAYBEL27, MML version 4.99.1005
:: WAYBEL27:attrnot 1 => WAYBEL27:attr 1
definition
let a1 be Relation-like Function-like set;
attr a1 is uncurrying means
(for b1 being set
st b1 in proj1 a1
holds b1 is Relation-like Function-like Function-yielding set) &
(for b1 being Relation-like Function-like set
st b1 in proj1 a1
holds a1 . b1 = uncurry b1);
end;
:: WAYBEL27:dfs 1
definiens
let a1 be Relation-like Function-like set;
To prove
a1 is uncurrying
it is sufficient to prove
thus (for b1 being set
st b1 in proj1 a1
holds b1 is Relation-like Function-like Function-yielding set) &
(for b1 being Relation-like Function-like set
st b1 in proj1 a1
holds a1 . b1 = uncurry b1);
:: WAYBEL27:def 1
theorem
for b1 being Relation-like Function-like set holds
b1 is uncurrying
iff
(for b2 being set
st b2 in proj1 b1
holds b2 is Relation-like Function-like Function-yielding set) &
(for b2 being Relation-like Function-like set
st b2 in proj1 b1
holds b1 . b2 = uncurry b2);
:: WAYBEL27:attrnot 2 => WAYBEL27:attr 2
definition
let a1 be Relation-like Function-like set;
attr a1 is currying means
(for b1 being set
st b1 in proj1 a1
holds b1 is Relation-like Function-like set & proj1 b1 is Relation-like set) &
(for b1 being Relation-like Function-like set
st b1 in proj1 a1
holds a1 . b1 = curry b1);
end;
:: WAYBEL27:dfs 2
definiens
let a1 be Relation-like Function-like set;
To prove
a1 is currying
it is sufficient to prove
thus (for b1 being set
st b1 in proj1 a1
holds b1 is Relation-like Function-like set & proj1 b1 is Relation-like set) &
(for b1 being Relation-like Function-like set
st b1 in proj1 a1
holds a1 . b1 = curry b1);
:: WAYBEL27:def 2
theorem
for b1 being Relation-like Function-like set holds
b1 is currying
iff
(for b2 being set
st b2 in proj1 b1
holds b2 is Relation-like Function-like set & proj1 b2 is Relation-like set) &
(for b2 being Relation-like Function-like set
st b2 in proj1 b1
holds b1 . b2 = curry b2);
:: WAYBEL27:attrnot 3 => WAYBEL27:attr 3
definition
let a1 be Relation-like Function-like set;
attr a1 is commuting means
(for b1 being set
st b1 in proj1 a1
holds b1 is Relation-like Function-like Function-yielding set) &
(for b1 being Relation-like Function-like set
st b1 in proj1 a1
holds a1 . b1 = commute b1);
end;
:: WAYBEL27:dfs 3
definiens
let a1 be Relation-like Function-like set;
To prove
a1 is commuting
it is sufficient to prove
thus (for b1 being set
st b1 in proj1 a1
holds b1 is Relation-like Function-like Function-yielding set) &
(for b1 being Relation-like Function-like set
st b1 in proj1 a1
holds a1 . b1 = commute b1);
:: WAYBEL27:def 3
theorem
for b1 being Relation-like Function-like set holds
b1 is commuting
iff
(for b2 being set
st b2 in proj1 b1
holds b2 is Relation-like Function-like Function-yielding set) &
(for b2 being Relation-like Function-like set
st b2 in proj1 b1
holds b1 . b2 = commute b2);
:: WAYBEL27:condreg 1
registration
cluster empty Relation-like Function-like -> uncurrying currying commuting (set);
end;
:: WAYBEL27:exreg 1
registration
cluster Relation-like Function-like uncurrying currying commuting set;
end;
:: WAYBEL27:funcreg 1
registration
let a1 be Relation-like Function-like uncurrying set;
let a2 be set;
cluster a1 | a2 -> Relation-like uncurrying;
end;
:: WAYBEL27:funcreg 2
registration
let a1 be Relation-like Function-like currying set;
let a2 be set;
cluster a1 | a2 -> Relation-like currying;
end;
:: WAYBEL27:th 1
theorem
for b1, b2, b3, b4 being set
st b4 c= Funcs(b1,Funcs(b2,b3))
holds ex b5 being ManySortedSet of b4 st
b5 is uncurrying & proj2 b5 c= Funcs([:b1,b2:],b3);
:: WAYBEL27:th 2
theorem
for b1, b2, b3, b4 being set
st b4 c= Funcs([:b1,b2:],b3)
holds ex b5 being ManySortedSet of b4 st
b5 is currying &
(b2 = {} & b1 <> {} or proj2 b5 c= Funcs(b1,Funcs(b2,b3)));
:: WAYBEL27:exreg 2
registration
let a1, a2, a3 be set;
cluster Relation-like Function-like uncurrying ManySortedSet of Funcs(a1,Funcs(a2,a3));
end;
:: WAYBEL27:exreg 3
registration
let a1, a2, a3 be set;
cluster Relation-like Function-like currying ManySortedSet of Funcs([:a1,a2:],a3);
end;
:: WAYBEL27:th 3
theorem
for b1, b2 being non empty set
for b3 being set
for b4, b5 being Relation-like Function-like commuting set
st proj1 b4 c= Funcs(b1,Funcs(b2,b3)) & proj2 b4 c= proj1 b5
holds b4 * b5 = id proj1 b4;
:: WAYBEL27:th 4
theorem
for b1 being non empty set
for b2, b3 being set
for b4 being Relation-like Function-like uncurrying set
for b5 being Relation-like Function-like currying set
st proj1 b4 c= Funcs(b2,Funcs(b1,b3)) & proj2 b4 c= proj1 b5
holds b4 * b5 = id proj1 b4;
:: WAYBEL27:th 5
theorem
for b1, b2, b3 being set
for b4 being Relation-like Function-like currying set
for b5 being Relation-like Function-like uncurrying set
st proj1 b4 c= Funcs([:b1,b2:],b3) & proj2 b4 c= proj1 b5
holds b4 * b5 = id proj1 b4;
:: WAYBEL27:th 6
theorem
for b1 being Relation-like Function-like Function-yielding set
for b2, b3 being set
st b2 in proj1 commute b1
holds ((commute b1) . b2) .: b3 c= pi(b1 .: b3,b2);
:: WAYBEL27:th 7
theorem
for b1 being Relation-like Function-like Function-yielding set
for b2, b3 being set
st for b4 being Relation-like Function-like set
st b4 in b1 .: b3
holds b2 in proj1 b4
holds pi(b1 .: b3,b2) c= ((commute b1) . b2) .: b3;
:: WAYBEL27:th 8
theorem
for b1, b2 being set
for b3 being Relation-like Function-like set
st proj2 b3 c= Funcs(b1,b2)
for b4, b5 being set
st b4 in b1
holds ((commute b3) . b4) .: b5 = pi(b3 .: b5,b4);
:: WAYBEL27:th 9
theorem
for b1 being Relation-like Function-like set
for b2, b3 being set
st [:b3,{b2}:] c= proj1 b1
holds pi((curry b1) .: b3,b2) = b1 .: [:b3,{b2}:];
:: WAYBEL27:condreg 2
registration
let a1 be set;
let a2 be non empty functional set;
cluster Function-like quasi_total -> Function-yielding (Relation of a1,a2);
end;
:: WAYBEL27:funcreg 3
registration
let a1 be constituted-Functions 1-sorted;
cluster the carrier of a1 -> functional;
end;
:: WAYBEL27:funcreg 4
registration
let a1 be set;
let a2 be non empty RelStr;
cluster a2 |^ a1 -> constituted-Functions strict;
end;
:: WAYBEL27:exreg 4
registration
cluster non empty constituted-Functions strict reflexive transitive antisymmetric with_suprema with_infima complete non void RelStr;
end;
:: WAYBEL27:exreg 5
registration
cluster non empty constituted-Functions 1-sorted;
end;
:: WAYBEL27:condreg 3
registration
let a1 be non empty constituted-Functions RelStr;
cluster non empty -> constituted-Functions (SubRelStr of a1);
end;
:: WAYBEL27:th 10
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total idempotent Relation of the carrier of b2,the carrier of b2
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of Image b3 holds
b3 * b4 = b4;
:: WAYBEL27:th 11
theorem
for b1, b2, b3 being non empty RelStr
st b2 is SubRelStr of b3
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
st b4 is monotone(b1, b2) & b4 = b5
holds b5 is monotone(b1, b3);
:: WAYBEL27:th 12
theorem
for b1, b2, b3 being non empty RelStr
st b2 is full SubRelStr of b3
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
st b5 is monotone(b1, b3) & b4 = b5
holds b4 is monotone(b1, b2);
:: WAYBEL27:th 13
theorem
for b1 being set
for b2 being Element of bool b1 holds
(chi(b2,b1)) " {1} = b2 &
(chi(b2,b1)) " {0} = b1 \ b2;
:: WAYBEL27:funcnot 1 => WAYBEL27:func 1
definition
let a1 be non empty set;
let a2 be non empty RelStr;
let a3 be Element of the carrier of a2 |^ a1;
let a4 be Element of a1;
redefine func a3 . a4 -> Element of the carrier of a2;
end;
:: WAYBEL27:th 14
theorem
for b1 being non empty set
for b2 being non empty RelStr
for b3, b4 being Element of the carrier of b2 |^ b1 holds
b3 <= b4
iff
for b5 being Element of b1 holds
b3 . b5 <= b4 . b5;
:: WAYBEL27:th 15
theorem
for b1 being set
for b2, b3 being non empty RelStr
st RelStr(#the carrier of b2,the InternalRel of b2#) = RelStr(#the carrier of b3,the InternalRel of b3#)
holds b2 |^ b1 = b3 |^ b1;
:: WAYBEL27:th 16
theorem
for b1, b2, b3, b4 being non empty TopSpace-like TopStruct
st TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#) &
TopStruct(#the carrier of b3,the topology of b3#) = TopStruct(#the carrier of b4,the topology of b4#)
holds oContMaps(b1,b3) = oContMaps(b2,b4);
:: WAYBEL27:th 17
theorem
for b1 being set holds
ex b2 being Function-like quasi_total Relation of the carrier of BoolePoset b1,the carrier of (BoolePoset 1) |^ b1 st
b2 is isomorphic(BoolePoset b1, (BoolePoset 1) |^ b1) &
(for b3 being Element of bool b1 holds
b2 . b3 = chi(b3,b1));
:: WAYBEL27:th 18
theorem
for b1 being set holds
BoolePoset b1,(BoolePoset 1) |^ b1 are_isomorphic;
:: WAYBEL27:th 19
theorem
for b1, b2 being non empty set
for b3 being non empty reflexive transitive antisymmetric RelStr
for b4 being non empty full SubRelStr of (b3 |^ b1) |^ b2
for b5 being non empty full SubRelStr of (b3 |^ b2) |^ b1
for b6 being Function-like quasi_total Relation of the carrier of b4,the carrier of b5
st b6 is commuting
holds b6 is monotone(b4, b5);
:: WAYBEL27:th 20
theorem
for b1, b2 being non empty set
for b3 being non empty reflexive transitive antisymmetric RelStr
for b4 being non empty full SubRelStr of (b3 |^ b2) |^ b1
for b5 being non empty full SubRelStr of b3 |^ [:b1,b2:]
for b6 being Function-like quasi_total Relation of the carrier of b4,the carrier of b5
st b6 is uncurrying
holds b6 is monotone(b4, b5);
:: WAYBEL27:th 21
theorem
for b1, b2 being non empty set
for b3 being non empty reflexive transitive antisymmetric RelStr
for b4 being non empty full SubRelStr of (b3 |^ b2) |^ b1
for b5 being non empty full SubRelStr of b3 |^ [:b1,b2:]
for b6 being Function-like quasi_total Relation of the carrier of b5,the carrier of b4
st b6 is currying
holds b6 is monotone(b5, b4);
:: WAYBEL27:funcnot 2 => WAYBEL27:func 2
definition
let a1 be non empty RelStr;
let a2 be non empty reflexive antisymmetric RelStr;
func UPS(A1,A2) -> strict RelStr means
it is full SubRelStr of a2 |^ the carrier of a1 &
(for b1 being set holds
b1 in the carrier of it
iff
b1 is Function-like quasi_total directed-sups-preserving Relation of the carrier of a1,the carrier of a2);
end;
:: WAYBEL27:def 4
theorem
for b1 being non empty RelStr
for b2 being non empty reflexive antisymmetric RelStr
for b3 being strict RelStr holds
b3 = UPS(b1,b2)
iff
b3 is full SubRelStr of b2 |^ the carrier of b1 &
(for b4 being set holds
b4 in the carrier of b3
iff
b4 is Function-like quasi_total directed-sups-preserving Relation of the carrier of b1,the carrier of b2);
:: WAYBEL27:funcreg 5
registration
let a1 be non empty RelStr;
let a2 be non empty reflexive antisymmetric RelStr;
cluster UPS(a1,a2) -> non empty constituted-Functions strict reflexive antisymmetric;
end;
:: WAYBEL27:funcreg 6
registration
let a1 be non empty RelStr;
let a2 be non empty reflexive transitive antisymmetric RelStr;
cluster UPS(a1,a2) -> strict transitive;
end;
:: WAYBEL27:th 22
theorem
for b1 being non empty RelStr
for b2 being non empty reflexive antisymmetric RelStr holds
the carrier of UPS(b1,b2) c= Funcs(the carrier of b1,the carrier of b2);
:: WAYBEL27:funcnot 3 => WAYBEL27:func 3
definition
let a1 be non empty RelStr;
let a2 be non empty reflexive antisymmetric RelStr;
let a3 be Element of the carrier of UPS(a1,a2);
let a4 be Element of the carrier of a1;
redefine func a3 . a4 -> Element of the carrier of a2;
end;
:: WAYBEL27:th 23
theorem
for b1 being non empty RelStr
for b2 being non empty reflexive antisymmetric RelStr
for b3, b4 being Element of the carrier of UPS(b1,b2) holds
b3 <= b4
iff
for b5 being Element of the carrier of b1 holds
b3 . b5 <= b4 . b5;
:: WAYBEL27:th 24
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr holds
UPS(b1,b2) = SCMaps(b1,b2);
:: WAYBEL27:th 25
theorem
for b1, b2 being non empty RelStr
for b3, b4 being non empty reflexive antisymmetric RelStr
st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#) &
RelStr(#the carrier of b3,the InternalRel of b3#) = RelStr(#the carrier of b4,the InternalRel of b4#)
holds UPS(b1,b3) = UPS(b2,b4);
:: WAYBEL27:funcreg 7
registration
let a1, a2 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
cluster UPS(a1,a2) -> strict complete;
end;
:: WAYBEL27:th 26
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
UPS(b1,b2) is sups-inheriting SubRelStr of b2 |^ the carrier of b1;
:: WAYBEL27:th 27
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Element of bool the carrier of UPS(b1,b2) holds
"\/"(b3,UPS(b1,b2)) = "\/"(b3,b2 |^ the carrier of b1);
:: WAYBEL27:funcnot 4 => WAYBEL27:func 4
definition
let a1, a2, a3, a4 be non empty reflexive antisymmetric RelStr;
let a5 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a6 be Function-like quasi_total Relation of the carrier of a3,the carrier of a4;
assume a5 is directed-sups-preserving(a1, a2) & a6 is directed-sups-preserving(a3, a4);
func UPS(A5,A6) -> Function-like quasi_total Relation of the carrier of UPS(a2,a3),the carrier of UPS(a1,a4) means
for b1 being Function-like quasi_total directed-sups-preserving Relation of the carrier of a2,the carrier of a3 holds
it . b1 = (a6 * b1) * a5;
end;
:: WAYBEL27:def 5
theorem
for b1, b2, b3, b4 being non empty reflexive antisymmetric RelStr
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b5 is directed-sups-preserving(b1, b2)
for b6 being Function-like quasi_total Relation of the carrier of b3,the carrier of b4
st b6 is directed-sups-preserving(b3, b4)
for b7 being Function-like quasi_total Relation of the carrier of UPS(b2,b3),the carrier of UPS(b1,b4) holds
b7 = UPS(b5,b6)
iff
for b8 being Function-like quasi_total directed-sups-preserving Relation of the carrier of b2,the carrier of b3 holds
b7 . b8 = (b6 * b8) * b5;
:: WAYBEL27:th 28
theorem
for b1, b2, b3, b4, b5, b6 being non empty reflexive transitive antisymmetric RelStr
for b7 being Function-like quasi_total directed-sups-preserving Relation of the carrier of b2,the carrier of b3
for b8 being Function-like quasi_total directed-sups-preserving Relation of the carrier of b1,the carrier of b2
for b9 being Function-like quasi_total directed-sups-preserving Relation of the carrier of b4,the carrier of b5
for b10 being Function-like quasi_total directed-sups-preserving Relation of the carrier of b5,the carrier of b6 holds
(UPS(b8,b10)) * UPS(b7,b9) = UPS(b7 * b8,b10 * b9);
:: WAYBEL27:th 29
theorem
for b1, b2 being non empty reflexive antisymmetric RelStr holds
UPS(id b1,id b2) = id UPS(b1,b2);
:: WAYBEL27:th 30
theorem
for b1, b2, b3, b4 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b5 being Function-like quasi_total directed-sups-preserving Relation of the carrier of b1,the carrier of b2
for b6 being Function-like quasi_total directed-sups-preserving Relation of the carrier of b3,the carrier of b4 holds
UPS(b5,b6) is directed-sups-preserving(UPS(b2,b3), UPS(b1,b4));
:: WAYBEL27:th 31
theorem
Omega Sierpinski_Space is Scott;
:: WAYBEL27:th 32
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr holds
oContMaps(b1,Sierpinski_Space) = UPS(b1,BoolePoset 1);
:: WAYBEL27:th 33
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
ex b2 being Function-like quasi_total Relation of the carrier of UPS(b1,BoolePoset 1),the carrier of InclPoset sigma b1 st
b2 is isomorphic(UPS(b1,BoolePoset 1), InclPoset sigma b1) &
(for b3 being Function-like quasi_total directed-sups-preserving Relation of the carrier of b1,the carrier of BoolePoset 1 holds
b2 . b3 = b3 " {1});
:: WAYBEL27:th 34
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
UPS(b1,BoolePoset 1),InclPoset sigma b1 are_isomorphic;
:: WAYBEL27:th 35
theorem
for b1, b2, b3, b4 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b3,the carrier of b4
st b5 is isomorphic(b1, b2) & b6 is isomorphic(b3, b4)
holds UPS(b5,b6) is isomorphic(UPS(b2,b3), UPS(b1,b4));
:: WAYBEL27:th 36
theorem
for b1, b2, b3, b4 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
st b1,b2 are_isomorphic & b3,b4 are_isomorphic
holds UPS(b2,b3),UPS(b1,b4) are_isomorphic;
:: WAYBEL27:th 37
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total directed-sups-preserving projection Relation of the carrier of b2,the carrier of b2 holds
Image UPS(id b1,b3) = UPS(b1,Image b3);
:: WAYBEL27:th 38
theorem
for b1 being non empty set
for b2, b3 being non empty reflexive transitive antisymmetric RelStr
for b4 being Function-like quasi_total directed-sups-preserving Relation of the carrier of b2,the carrier of b3 |^ b1
for b5 being Element of b1 holds
(commute b4) . b5 is Function-like quasi_total directed-sups-preserving Relation of the carrier of b2,the carrier of b3;
:: WAYBEL27:th 39
theorem
for b1 being non empty set
for b2, b3 being non empty reflexive transitive antisymmetric RelStr
for b4 being Function-like quasi_total directed-sups-preserving Relation of the carrier of b2,the carrier of b3 |^ b1 holds
commute b4 is Function-like quasi_total Relation of b1,the carrier of UPS(b2,b3);
:: WAYBEL27:th 40
theorem
for b1 being non empty set
for b2, b3 being non empty reflexive transitive antisymmetric RelStr
for b4 being Function-like quasi_total Relation of b1,the carrier of UPS(b2,b3) holds
commute b4 is Function-like quasi_total directed-sups-preserving Relation of the carrier of b2,the carrier of b3 |^ b1;
:: WAYBEL27:th 41
theorem
for b1 being non empty set
for b2, b3 being non empty reflexive transitive antisymmetric RelStr holds
ex b4 being Function-like quasi_total Relation of the carrier of UPS(b2,b3 |^ b1),the carrier of (UPS(b2,b3)) |^ b1 st
b4 is commuting &
b4 is isomorphic(UPS(b2,b3 |^ b1), (UPS(b2,b3)) |^ b1);
:: WAYBEL27:th 42
theorem
for b1 being non empty set
for b2, b3 being non empty reflexive transitive antisymmetric RelStr holds
UPS(b2,b3 |^ b1),(UPS(b2,b3)) |^ b1 are_isomorphic;
:: WAYBEL27:th 43
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete continuous RelStr holds
UPS(b1,b2) is continuous;
:: WAYBEL27:th 44
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete algebraic RelStr holds
UPS(b1,b2) is algebraic;
:: WAYBEL27:th 45
theorem
for b1, b2, b3 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b4 being Function-like quasi_total directed-sups-preserving Relation of the carrier of b1,the carrier of UPS(b2,b3) holds
uncurry b4 is Function-like quasi_total directed-sups-preserving Relation of the carrier of [:b1,b2:],the carrier of b3;
:: WAYBEL27:th 46
theorem
for b1, b2, b3 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b4 being Function-like quasi_total directed-sups-preserving Relation of the carrier of [:b1,b2:],the carrier of b3 holds
curry b4 is Function-like quasi_total directed-sups-preserving Relation of the carrier of b1,the carrier of UPS(b2,b3);
:: WAYBEL27:th 47
theorem
for b1, b2, b3 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
ex b4 being Function-like quasi_total Relation of the carrier of UPS(b1,UPS(b2,b3)),the carrier of UPS([:b1,b2:],b3) st
b4 is uncurrying &
b4 is isomorphic(UPS(b1,UPS(b2,b3)), UPS([:b1,b2:],b3));