Article WAYBEL_5, MML version 4.99.1005
:: WAYBEL_5:th 1
theorem
for b1 being reflexive transitive antisymmetric with_infima up-complete RelStr holds
b1 is continuous
iff
for b2 being Element of the carrier of b1 holds
waybelow b2 is non empty directed lower Element of bool the carrier of b1 &
b2 <= "\/"(waybelow b2,b1) &
(for b3 being non empty directed lower Element of bool the carrier of b1
st b2 <= "\/"(b3,b1)
holds waybelow b2 c= b3);
:: WAYBEL_5:th 2
theorem
for b1 being reflexive transitive antisymmetric with_infima up-complete RelStr holds
b1 is continuous
iff
for b2 being Element of the carrier of b1 holds
ex b3 being non empty directed lower Element of bool the carrier of b1 st
b2 <= "\/"(b3,b1) &
(for b4 being non empty directed lower Element of bool the carrier of b1
st b2 <= "\/"(b4,b1)
holds b3 c= b4);
:: WAYBEL_5:th 3
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded continuous RelStr holds
SupMap b1 is upper_adjoint(InclPoset Ids b1, b1);
:: WAYBEL_5:th 4
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded up-complete RelStr
st SupMap b1 is upper_adjoint(InclPoset Ids b1, b1)
holds b1 is continuous;
:: WAYBEL_5:th 5
theorem
for b1 being reflexive transitive antisymmetric with_infima complete RelStr
st SupMap b1 is infs-preserving(InclPoset Ids b1, b1) & SupMap b1 is sups-preserving(InclPoset Ids b1, b1)
holds SupMap b1 is upper_adjoint(InclPoset Ids b1, b1);
:: WAYBEL_5:modenot 1
definition
let a1, a2 be set;
let a3 be ManySortedSet of a1;
mode DoubleIndexedSet of a3,a2 is ManySortedFunction of a3,a1 --> a2;
end;
:: WAYBEL_5:modenot 2
definition
let a1 be set;
let a2 be ManySortedSet of a1;
let a3 be 1-sorted;
mode DoubleIndexedSet of a2,a3 is ManySortedFunction of a2,a1 --> the carrier of a3;
end;
:: WAYBEL_5:th 6
theorem
for b1, b2 being set
for b3 being ManySortedSet of b1
for b4 being ManySortedFunction of b3,b1 --> b2
for b5 being set
st b5 in b1
holds b4 . b5 is Function-like quasi_total Relation of b3 . b5,b2;
:: WAYBEL_5:funcnot 1 => WAYBEL_5:func 1
definition
let a1, a2 be non empty set;
let a3 be ManySortedSet of a1;
let a4 be ManySortedFunction of a3,a1 --> a2;
let a5 be Element of a1;
redefine func a4 . a5 -> Function-like quasi_total Relation of a3 . a5,a2;
end;
:: WAYBEL_5:funcreg 1
registration
let a1, a2 be non empty set;
let a3 be non-empty ManySortedSet of a1;
let a4 be ManySortedFunction of a3,a1 --> a2;
let a5 be Element of a1;
cluster proj2 (a4 . a5) -> non empty;
end;
:: WAYBEL_5:condreg 1
registration
let a1 be set;
let a2 be non empty set;
let a3 be non-empty ManySortedSet of a1;
cluster -> non-empty (ManySortedFunction of a3,a1 --> a2);
end;
:: WAYBEL_5:th 7
theorem
for b1 being Relation-like Function-like Function-yielding set
for b2 being set
st b2 in proj1 Frege b1
holds b2 is Relation-like Function-like set;
:: WAYBEL_5:th 8
theorem
for b1 being Relation-like Function-like Function-yielding set
for b2 being Relation-like Function-like set
st b2 in proj1 Frege b1
holds proj1 b2 = proj1 b1 & proj1 b1 = proj1 ((Frege b1) . b2);
:: WAYBEL_5:th 9
theorem
for b1 being Relation-like Function-like Function-yielding set
for b2 being Relation-like Function-like set
st b2 in proj1 Frege b1
for b3 being set
st b3 in proj1 b1
holds b2 . b3 in proj1 (b1 . b3) &
((Frege b1) . b2) . b3 = (b1 . b3) . (b2 . b3) &
(b1 . b3) . (b2 . b3) in proj2 ((Frege b1) . b2);
:: WAYBEL_5:th 10
theorem
for b1, b2 being set
for b3 being ManySortedSet of b1
for b4 being ManySortedFunction of b3,b1 --> b2
for b5 being Relation-like Function-like set
st b5 in proj1 Frege b4
holds (Frege b4) . b5 is Function-like quasi_total Relation of b1,b2;
:: WAYBEL_5:funcreg 2
registration
let a1 be Relation-like non-empty Function-like set;
cluster doms a1 -> Relation-like non-empty Function-like;
end;
:: WAYBEL_5:funcnot 2 => WAYBEL_5:func 2
definition
let a1, a2 be set;
let a3 be ManySortedSet of a1;
let a4 be ManySortedFunction of a3,a1 --> a2;
redefine func Frege a4 -> ManySortedFunction of (product doms a4) --> a1,(product doms a4) --> a2;
end;
:: WAYBEL_5:funcnot 3 => WAYBEL_5:func 3
definition
let a1, a2 be non empty set;
let a3 be non-empty ManySortedSet of a1;
let a4 be ManySortedFunction of a3,a1 --> a2;
let a5 be ManySortedFunction of (product doms a4) --> a1,(product doms a4) --> a2;
let a6 be Element of product doms a4;
redefine func a5 . a6 -> Function-like quasi_total Relation of a1,a2;
end;
:: WAYBEL_5:funcnot 4 => WAYBEL_5:func 4
definition
let a1 be non empty RelStr;
let a2 be Relation-like Function-like Function-yielding set;
func \//(A2,A1) -> Function-like quasi_total Relation of proj1 a2,the carrier of a1 means
for b1 being set
st b1 in proj1 a2
holds it . b1 = \\/(a2 . b1,a1);
end;
:: WAYBEL_5:def 1
theorem
for b1 being non empty RelStr
for b2 being Relation-like Function-like Function-yielding set
for b3 being Function-like quasi_total Relation of proj1 b2,the carrier of b1 holds
b3 = \//(b2,b1)
iff
for b4 being set
st b4 in proj1 b2
holds b3 . b4 = \\/(b2 . b4,b1);
:: WAYBEL_5:funcnot 5 => WAYBEL_5:func 5
definition
let a1 be non empty RelStr;
let a2 be Relation-like Function-like Function-yielding set;
func /\\(A2,A1) -> Function-like quasi_total Relation of proj1 a2,the carrier of a1 means
for b1 being set
st b1 in proj1 a2
holds it . b1 = //\(a2 . b1,a1);
end;
:: WAYBEL_5:def 2
theorem
for b1 being non empty RelStr
for b2 being Relation-like Function-like Function-yielding set
for b3 being Function-like quasi_total Relation of proj1 b2,the carrier of b1 holds
b3 = /\\(b2,b1)
iff
for b4 being set
st b4 in proj1 b2
holds b3 . b4 = //\(b2 . b4,b1);
:: WAYBEL_5:funcnot 6 => WAYBEL_5:func 4
notation
let a1 be set;
let a2 be ManySortedSet of a1;
let a3 be non empty RelStr;
let a4 be ManySortedFunction of a2,a1 --> the carrier of a3;
synonym Sups a4 for \//(a2,a1);
end;
:: WAYBEL_5:funcnot 7 => WAYBEL_5:func 5
notation
let a1 be set;
let a2 be ManySortedSet of a1;
let a3 be non empty RelStr;
let a4 be ManySortedFunction of a2,a1 --> the carrier of a3;
synonym Infs a4 for /\\(a2,a1);
end;
:: WAYBEL_5:funcnot 8 => WAYBEL_5:func 4
notation
let a1, a2 be set;
let a3 be non empty RelStr;
let a4 be ManySortedFunction of a1 --> a2,a1 --> the carrier of a3;
synonym Sups a4 for \//(a2,a1);
end;
:: WAYBEL_5:funcnot 9 => WAYBEL_5:func 5
notation
let a1, a2 be set;
let a3 be non empty RelStr;
let a4 be ManySortedFunction of a1 --> a2,a1 --> the carrier of a3;
synonym Infs a4 for /\\(a2,a1);
end;
:: WAYBEL_5:th 11
theorem
for b1 being non empty RelStr
for b2, b3 being Relation-like Function-like Function-yielding set
st proj1 b2 = proj1 b3 &
(for b4 being set
st b4 in proj1 b2
holds \\/(b2 . b4,b1) = \\/(b3 . b4,b1))
holds \//(b2,b1) = \//(b3,b1);
:: WAYBEL_5:th 12
theorem
for b1 being non empty RelStr
for b2, b3 being Relation-like Function-like Function-yielding set
st proj1 b2 = proj1 b3 &
(for b4 being set
st b4 in proj1 b2
holds //\(b2 . b4,b1) = //\(b3 . b4,b1))
holds /\\(b2,b1) = /\\(b3,b1);
:: WAYBEL_5:th 13
theorem
for b1 being set
for b2 being non empty RelStr
for b3 being Relation-like Function-like Function-yielding set holds
(b1 in rng \//(b3,b2) implies ex b4 being set st
b4 in proj1 b3 & b1 = \\/(b3 . b4,b2)) &
(for b4 being set
st b4 in proj1 b3
holds b1 <> \\/(b3 . b4,b2) or b1 in rng \//(b3,b2)) &
(b1 in rng /\\(b3,b2) implies ex b4 being set st
b4 in proj1 b3 & b1 = //\(b3 . b4,b2)) &
(for b4 being set
st b4 in proj1 b3
holds b1 <> //\(b3 . b4,b2) or b1 in rng /\\(b3,b2));
:: WAYBEL_5:th 14
theorem
for b1 being set
for b2 being non empty RelStr
for b3 being non empty set
for b4 being ManySortedSet of b3
for b5 being ManySortedFunction of b4,b3 --> the carrier of b2 holds
(b1 in rng \//(b5,b2) implies ex b6 being Element of b3 st
b1 = \\/(b5 . b6,b2)) &
(for b6 being Element of b3 holds
b1 <> \\/(b5 . b6,b2) or b1 in rng \//(b5,b2)) &
(b1 in rng /\\(b5,b2) implies ex b6 being Element of b3 st
b1 = //\(b5 . b6,b2)) &
(for b6 being Element of b3 holds
b1 <> //\(b5 . b6,b2) or b1 in rng /\\(b5,b2));
:: WAYBEL_5:funcreg 3
registration
let a1 be non empty set;
let a2 be ManySortedSet of a1;
let a3 be non empty RelStr;
let a4 be ManySortedFunction of a2,a1 --> the carrier of a3;
cluster proj2 \//(a4,a3) -> non empty;
end;
:: WAYBEL_5:funcreg 4
registration
let a1 be non empty set;
let a2 be ManySortedSet of a1;
let a3 be non empty RelStr;
let a4 be ManySortedFunction of a2,a1 --> the carrier of a3;
cluster proj2 /\\(a4,a3) -> non empty;
end;
:: WAYBEL_5:th 15
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Element of the carrier of b1
for b3 being Relation-like Function-like Function-yielding set
st for b4 being Relation-like Function-like set
st b4 in proj1 Frege b3
holds //\((Frege b3) . b4,b1) <= b2
holds \\/(/\\(Frege b3,b1),b1) <= b2;
:: WAYBEL_5:th 16
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty set
for b3 being non-empty ManySortedSet of b2
for b4 being ManySortedFunction of b3,b2 --> the carrier of b1 holds
\\/(/\\(Frege b4,b1),b1) <= //\(\//(b4,b1),b1);
:: WAYBEL_5:th 17
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2, b3 being Element of the carrier of b1
st b1 is continuous &
(for b4 being Element of the carrier of b1
st b4 is_way_below b2
holds b4 <= b3)
holds b2 <= b3;
:: WAYBEL_5:th 18
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
st for b2 being non empty set
st b2 in the_universe_of the carrier of b1
for b3 being non-empty ManySortedSet of b2
st for b4 being Element of b2 holds
b3 . b4 in the_universe_of the carrier of b1
for b4 being ManySortedFunction of b3,b2 --> the carrier of b1
st for b5 being Element of b2 holds
rng (b4 . b5) is directed(b1)
holds //\(\//(b4,b1),b1) = \\/(/\\(Frege b4,b1),b1)
holds b1 is continuous;
:: WAYBEL_5:th 19
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
b1 is continuous
iff
for b2 being non empty set
for b3 being non-empty ManySortedSet of b2
for b4 being ManySortedFunction of b3,b2 --> the carrier of b1
st for b5 being Element of b2 holds
rng (b4 . b5) is directed(b1)
holds //\(\//(b4,b1),b1) = \\/(/\\(Frege b4,b1),b1);
:: WAYBEL_5:funcnot 10 => WAYBEL_5:func 6
definition
let a1, a2, a3 be non empty set;
let a4 be Function-like quasi_total Relation of [:a1,a2:],a3;
redefine func curry a4 -> ManySortedFunction of a1 --> a2,a1 --> a3;
end;
:: WAYBEL_5:th 20
theorem
for b1, b2, b3 being non empty set
for b4 being Element of b1
for b5 being Element of b2
for b6 being Function-like quasi_total Relation of [:b1,b2:],b3 holds
proj1 curry b6 = b1 &
dom ((curry b6) . b4) = b2 &
b6 .(b4,b5) = ((curry b6) . b4) . b5;
:: WAYBEL_5:th 21
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
b1 is continuous
iff
for b2, b3 being non empty set
for b4 being Function-like quasi_total Relation of [:b2,b3:],the carrier of b1
st for b5 being Element of b2 holds
rng ((curry b4) . b5) is directed(b1)
holds //\(\//(curry b4,b1),b1) = \\/(/\\(Frege curry b4,b1),b1);
:: WAYBEL_5:th 22
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2, b3 being non empty set
for b4 being Function-like quasi_total Relation of [:b2,b3:],the carrier of b1
for b5 being Element of bool the carrier of b1
st b5 = {b6 where b6 is Element of the carrier of b1: ex b7 being non-empty ManySortedSet of b2 st
b7 in Funcs(b2,Fin b3) &
(ex b8 being ManySortedFunction of b7,b2 --> the carrier of b1 st
(for b9 being Element of b2
for b10 being set
st b10 in b7 . b9
holds (b8 . b9) . b10 = b4 .(b9,b10)) &
b6 = //\(\//(b8,b1),b1))}
holds "\/"(b5,b1) <= //\(\//(curry b4,b1),b1);
:: WAYBEL_5:th 23
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
b1 is continuous
iff
for b2, b3 being non empty set
for b4 being Function-like quasi_total Relation of [:b2,b3:],the carrier of b1
for b5 being Element of bool the carrier of b1
st b5 = {b6 where b6 is Element of the carrier of b1: ex b7 being non-empty ManySortedSet of b2 st
b7 in Funcs(b2,Fin b3) &
(ex b8 being ManySortedFunction of b7,b2 --> the carrier of b1 st
(for b9 being Element of b2
for b10 being set
st b10 in b7 . b9
holds (b8 . b9) . b10 = b4 .(b9,b10)) &
b6 = //\(\//(b8,b1),b1))}
holds //\(\//(curry b4,b1),b1) = "\/"(b5,b1);
:: WAYBEL_5:attrnot 1 => WAYBEL_5:attr 1
definition
let a1 be non empty RelStr;
attr a1 is completely-distributive means
a1 is complete &
(for b1 being non empty set
for b2 being non-empty ManySortedSet of b1
for b3 being ManySortedFunction of b2,b1 --> the carrier of a1 holds
//\(\//(b3,a1),a1) = \\/(/\\(Frege b3,a1),a1));
end;
:: WAYBEL_5:dfs 3
definiens
let a1 be non empty RelStr;
To prove
a1 is completely-distributive
it is sufficient to prove
thus a1 is complete &
(for b1 being non empty set
for b2 being non-empty ManySortedSet of b1
for b3 being ManySortedFunction of b2,b1 --> the carrier of a1 holds
//\(\//(b3,a1),a1) = \\/(/\\(Frege b3,a1),a1));
:: WAYBEL_5:def 3
theorem
for b1 being non empty RelStr holds
b1 is completely-distributive
iff
b1 is complete &
(for b2 being non empty set
for b3 being non-empty ManySortedSet of b2
for b4 being ManySortedFunction of b3,b2 --> the carrier of b1 holds
//\(\//(b4,b1),b1) = \\/(/\\(Frege b4,b1),b1));
:: WAYBEL_5:condreg 2
registration
cluster non empty trivial reflexive transitive antisymmetric -> completely-distributive (RelStr);
end;
:: WAYBEL_5:exreg 1
registration
cluster non empty total reflexive transitive antisymmetric with_suprema with_infima completely-distributive RelStr;
end;
:: WAYBEL_5:th 24
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima completely-distributive RelStr holds
b1 is continuous;
:: WAYBEL_5:condreg 3
registration
cluster reflexive transitive antisymmetric with_suprema with_infima completely-distributive -> complete continuous (RelStr);
end;
:: WAYBEL_5:th 25
theorem
for b1 being non empty transitive antisymmetric with_infima RelStr
for b2 being Element of the carrier of b1
for b3, b4 being Element of bool the carrier of b1
st ex_sup_of b3,b1 &
ex_sup_of b4,b1 &
b4 = {b2 "/\" b5 where b5 is Element of the carrier of b1: b5 in b3}
holds "\/"(b4,b1) <= b2 "/\" "\/"(b3,b1);
:: WAYBEL_5:th 26
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima completely-distributive RelStr
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
b3 "/\" "\/"(b2,b1) = "\/"({b3 "/\" b4 where b4 is Element of the carrier of b1: b4 in b2},b1);
:: WAYBEL_5:condreg 4
registration
cluster reflexive transitive antisymmetric with_suprema with_infima completely-distributive -> Heyting (RelStr);
end;
:: WAYBEL_5:modenot 3
definition
let a1 be non empty RelStr;
mode CLSubFrame of a1 is non empty full infs-inheriting directed-sups-inheriting SubRelStr of a1;
end;
:: WAYBEL_5:th 27
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty set
for b3 being non-empty ManySortedSet of b2
for b4 being ManySortedFunction of b3,b2 --> the carrier of b1
st for b5 being Element of b2 holds
rng (b4 . b5) is directed(b1)
holds rng /\\(Frege b4,b1) is directed(b1);
:: WAYBEL_5:th 28
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
st b1 is continuous
for b2 being non empty full infs-inheriting directed-sups-inheriting SubRelStr of b1 holds
b2 is reflexive transitive antisymmetric with_suprema with_infima continuous RelStr;
:: WAYBEL_5:th 29
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty reflexive transitive antisymmetric RelStr
st ex b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 st
b3 is infs-preserving(b1, b2) & b3 is onto(the carrier of b1, the carrier of b2)
holds b2 is reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
:: WAYBEL_5:funcnot 11 => FUNCOP_1:func 2
notation
let a1, a2 be set;
synonym a1 => a2 for a1 --> a2;
end;
:: WAYBEL_5:funcnot 12 => WAYBEL_5:func 7
definition
let a1, a2 be set;
redefine func a1 => a2 -> ManySortedSet of a1;
end;
:: WAYBEL_5:funcnot 13 => WAYBEL_5:func 8
definition
let a1, a2, a3 be set;
let a4 be Function-like quasi_total Relation of a1,a2;
redefine func a3 => a4 -> ManySortedFunction of a3 --> a1,a3 --> a2;
end;
:: WAYBEL_5:th 30
theorem
for b1 being non empty set
for b2, b3 being set
for b4 being Function-like quasi_total Relation of b2,b3
for b5 being Function-like quasi_total Relation of b3,b2
st b5 * b4 = id b2
holds (b1 => b5) ** (b1 => b4) = id (b1 --> b2);
:: WAYBEL_5:th 31
theorem
for b1, b2 being non empty set
for b3 being set
for b4 being ManySortedSet of b1
for b5 being ManySortedFunction of b4,b1 --> b2
for b6 being Function-like quasi_total Relation of b2,b3 holds
(b1 => b6) ** b5 is ManySortedFunction of b4,b1 --> b3;
:: WAYBEL_5:th 32
theorem
for b1, b2, b3 being non empty set
for b4 being ManySortedSet of b1
for b5 being ManySortedFunction of b4,b1 --> b2
for b6 being Function-like quasi_total Relation of b2,b3 holds
doms ((b1 => b6) ** b5) = doms b5;
:: WAYBEL_5:th 33
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
st b1 is continuous
for b2 being non empty reflexive transitive antisymmetric RelStr
st ex b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 st
b3 is infs-preserving(b1, b2) & b3 is directed-sups-preserving(b1, b2) & b3 is onto(the carrier of b1, the carrier of b2)
holds b2 is reflexive transitive antisymmetric with_suprema with_infima continuous RelStr;