Article WAYBEL28, MML version 4.99.1005
:: WAYBEL28:th 1
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty transitive directed NetStr over b1 holds
inf b2 <= lim_inf b2;
:: WAYBEL28:th 2
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty transitive directed NetStr over b1
for b3 being Element of the carrier of b1
st for b4 being subnet of b2 holds
b3 = lim_inf b4
holds b3 = lim_inf b2 &
(for b4 being subnet of b2 holds
inf b4 <= b3);
:: WAYBEL28:th 3
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty transitive directed NetStr over b1
for b3 being Element of the carrier of b1
st b2 in NetUniv b1 &
(for b4 being subnet of b2
st b4 in NetUniv b1
holds b3 = lim_inf b4)
holds b3 = lim_inf b2 &
(for b4 being subnet of b2
st b4 in NetUniv b1
holds inf b4 <= b3);
:: WAYBEL28:attrnot 1 => WAYBEL28:attr 1
definition
let a1 be non empty RelStr;
let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
attr a2 is greater_or_equal_to_id means
for b1 being Element of the carrier of a1 holds
b1 <= a2 . b1;
end;
:: WAYBEL28:dfs 1
definiens
let a1 be non empty RelStr;
let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
To prove
a2 is greater_or_equal_to_id
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
b1 <= a2 . b1;
:: WAYBEL28:def 1
theorem
for b1 being non empty RelStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
b2 is greater_or_equal_to_id(b1)
iff
for b3 being Element of the carrier of b1 holds
b3 <= b2 . b3;
:: WAYBEL28:th 4
theorem
for b1 being non empty reflexive RelStr holds
id b1 is greater_or_equal_to_id(b1);
:: WAYBEL28:th 5
theorem
for b1 being non empty directed RelStr
for b2, b3 being Element of the carrier of b1 holds
ex b4 being Element of the carrier of b1 st
b2 <= b4 & b3 <= b4;
:: WAYBEL28:th 6
theorem
for b1 being non empty directed RelStr holds
ex b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 st
b2 is greater_or_equal_to_id(b1);
:: WAYBEL28:exreg 1
registration
let a1 be non empty directed RelStr;
cluster non empty Relation-like Function-like total quasi_total greater_or_equal_to_id Relation of the carrier of a1,the carrier of a1;
end;
:: WAYBEL28:exreg 2
registration
let a1 be non empty reflexive RelStr;
cluster non empty Relation-like Function-like total quasi_total greater_or_equal_to_id Relation of the carrier of a1,the carrier of a1;
end;
:: WAYBEL28:funcnot 1 => WAYBEL28:func 1
definition
let a1 be non empty 1-sorted;
let a2 be non empty NetStr over a1;
let a3 be Function-like quasi_total Relation of the carrier of a2,the carrier of a2;
func A2 * A3 -> non empty strict NetStr over a1 means
RelStr(#the carrier of it,the InternalRel of it#) = RelStr(#the carrier of a2,the InternalRel of a2#) &
the mapping of it = (the mapping of a2) * a3;
end;
:: WAYBEL28:def 2
theorem
for b1 being non empty 1-sorted
for b2 being non empty NetStr over b1
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b2
for b4 being non empty strict NetStr over b1 holds
b4 = b2 * b3
iff
RelStr(#the carrier of b4,the InternalRel of b4#) = RelStr(#the carrier of b2,the InternalRel of b2#) &
the mapping of b4 = (the mapping of b2) * b3;
:: WAYBEL28:th 7
theorem
for b1 being non empty 1-sorted
for b2 being non empty NetStr over b1
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b2 holds
the carrier of b2 * b3 = the carrier of b2;
:: WAYBEL28:th 8
theorem
for b1 being non empty 1-sorted
for b2 being non empty NetStr over b1
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b2 holds
b2 * b3 = NetStr(#the carrier of b2,the InternalRel of b2,(the mapping of b2) * b3#);
:: WAYBEL28:th 9
theorem
for b1 being non empty 1-sorted
for b2 being non empty transitive directed RelStr
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1 holds
NetStr(#the carrier of b2,the InternalRel of b2,b3#) is non empty transitive directed NetStr over b1;
:: WAYBEL28:funcreg 1
registration
let a1 be non empty 1-sorted;
let a2 be non empty transitive directed RelStr;
let a3 be Function-like quasi_total Relation of the carrier of a2,the carrier of a1;
cluster NetStr(#the carrier of a2,the InternalRel of a2,a3#) -> non empty transitive strict directed;
end;
:: WAYBEL28:th 10
theorem
for b1 being non empty 1-sorted
for b2 being non empty transitive directed NetStr over b1
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b2 holds
b2 * b3 is non empty transitive directed NetStr over b1;
:: WAYBEL28:funcreg 2
registration
let a1 be non empty 1-sorted;
let a2 be non empty transitive directed NetStr over a1;
let a3 be Function-like quasi_total Relation of the carrier of a2,the carrier of a2;
cluster a2 * a3 -> non empty transitive strict directed;
end;
:: WAYBEL28:th 11
theorem
for b1 being non empty 1-sorted
for b2 being non empty transitive directed NetStr over b1
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b2
st b2 in NetUniv b1
holds b2 * b3 in NetUniv b1;
:: WAYBEL28:th 12
theorem
for b1 being non empty 1-sorted
for b2, b3 being non empty transitive directed NetStr over b1
st NetStr(#the carrier of b2,the InternalRel of b2,the mapping of b2#) = NetStr(#the carrier of b3,the InternalRel of b3,the mapping of b3#)
holds b3 is subnet of b2;
:: WAYBEL28:th 13
theorem
for b1 being non empty 1-sorted
for b2 being non empty transitive directed NetStr over b1
for b3 being Function-like quasi_total greater_or_equal_to_id Relation of the carrier of b2,the carrier of b2 holds
b2 * b3 is subnet of b2;
:: WAYBEL28:funcnot 2 => WAYBEL28:func 2
definition
let a1 be non empty 1-sorted;
let a2 be non empty transitive directed NetStr over a1;
let a3 be Function-like quasi_total greater_or_equal_to_id Relation of the carrier of a2,the carrier of a2;
redefine func a2 * a3 -> strict subnet of a2;
end;
:: WAYBEL28:th 14
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty transitive directed NetStr over b1
for b3 being Element of the carrier of b1
st b2 in NetUniv b1 &
b3 = lim_inf b2 &
(for b4 being subnet of b2
st b4 in NetUniv b1
holds inf b4 <= b3)
holds b3 = lim_inf b2 &
(for b4 being Function-like quasi_total greater_or_equal_to_id Relation of the carrier of b2,the carrier of b2 holds
inf (b2 * b4) <= b3);
:: WAYBEL28:th 15
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty transitive directed NetStr over b1
for b3 being Element of the carrier of b1
st b3 = lim_inf b2 &
(for b4 being Function-like quasi_total greater_or_equal_to_id Relation of the carrier of b2,the carrier of b2 holds
inf (b2 * b4) <= b3)
for b4 being subnet of b2 holds
b3 = lim_inf b4;
:: WAYBEL28:funcnot 3 => WAYBEL28:func 3
definition
let a1 be non empty RelStr;
func lim_inf-Convergence A1 -> Convergence-Class of a1 means
for b1 being non empty transitive directed NetStr over a1
st b1 in NetUniv a1
for b2 being Element of the carrier of a1 holds
[b1,b2] in it
iff
for b3 being subnet of b1 holds
b2 = lim_inf b3;
end;
:: WAYBEL28:def 3
theorem
for b1 being non empty RelStr
for b2 being Convergence-Class of b1 holds
b2 = lim_inf-Convergence b1
iff
for b3 being non empty transitive directed NetStr over b1
st b3 in NetUniv b1
for b4 being Element of the carrier of b1 holds
[b3,b4] in b2
iff
for b5 being subnet of b3 holds
b4 = lim_inf b5;
:: WAYBEL28:th 16
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty transitive directed NetStr over b1
for b3 being Element of the carrier of b1
st b2 in NetUniv b1
holds [b2,b3] in lim_inf-Convergence b1
iff
for b4 being subnet of b2
st b4 in NetUniv b1
holds b3 = lim_inf b4;
:: WAYBEL28:th 17
theorem
for b1 being non empty RelStr
for b2 being non empty transitive directed constant NetStr over b1
for b3 being subnet of b2 holds
b3 is constant(b1) & the_value_of b2 = the_value_of b3;
:: WAYBEL28:funcnot 4 => WAYBEL28:func 4
definition
let a1 be non empty RelStr;
func xi A1 -> Element of bool bool the carrier of a1 equals
the topology of ConvergenceSpace lim_inf-Convergence a1;
end;
:: WAYBEL28:def 4
theorem
for b1 being non empty RelStr holds
xi b1 = the topology of ConvergenceSpace lim_inf-Convergence b1;
:: WAYBEL28:th 18
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
lim_inf-Convergence b1 is (CONSTANTS)(b1);
:: WAYBEL28:th 19
theorem
for b1 being non empty RelStr holds
lim_inf-Convergence b1 is (SUBNETS)(b1);
:: WAYBEL28:th 20
theorem
for b1 being reflexive transitive antisymmetric continuous with_suprema with_infima complete RelStr holds
lim_inf-Convergence b1 is (DIVERGENCE)(b1);
:: WAYBEL28:th 21
theorem
for b1 being non empty RelStr
for b2, b3 being set
st [b2,b3] in lim_inf-Convergence b1
holds b2 in NetUniv b1;
:: WAYBEL28:th 22
theorem
for b1 being non empty 1-sorted
for b2, b3 being Convergence-Class of b1
st b2 c= b3
holds the topology of ConvergenceSpace b3 c= the topology of ConvergenceSpace b2;
:: WAYBEL28:th 23
theorem
for b1 being non empty reflexive RelStr holds
lim_inf-Convergence b1 c= Scott-Convergence b1;
:: WAYBEL28:th 24
theorem
for b1, b2 being set
st b1 c= b2
holds b1 in the_universe_of b2;
:: WAYBEL28:th 25
theorem
for b1 being non empty reflexive transitive RelStr
for b2 being non empty directed Element of bool the carrier of b1 holds
Net-Str b2 in NetUniv b1;
:: WAYBEL28:th 26
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty directed Element of bool the carrier of b1
for b3 being subnet of Net-Str b2 holds
lim_inf b3 = "\/"(b2,b1);
:: WAYBEL28:th 27
theorem
for b1 being non empty reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty directed Element of bool the carrier of b1 holds
[Net-Str b2,"\/"(b2,b1)] in lim_inf-Convergence b1;
:: WAYBEL28:th 28
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Element of bool the carrier of b1
st b2 in xi b1
holds b2 is property(S)(b1);
:: WAYBEL28:th 29
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of bool the carrier of b1
st b2 in sigma b1
holds b2 in xi b1;
:: WAYBEL28:th 30
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Element of bool the carrier of b1
st b2 is upper(b1) & b2 in xi b1
holds b2 in sigma b1;
:: WAYBEL28:th 31
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Element of bool the carrier of b1
st b2 is lower(b1)
holds b2 ` in xi b1
iff
b2 is closed_under_directed_sups(b1);