Article YELLOW_2, MML version 4.99.1005
:: YELLOW_2:th 1
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
b3 c= downarrow b2
iff
b3 is_<=_than b2;
:: YELLOW_2:th 2
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
b3 c= uparrow b2
iff
b2 is_<=_than b3;
:: YELLOW_2:th 3
theorem
for b1 being transitive antisymmetric with_suprema RelStr
for b2, b3 being set
st ex_sup_of b2,b1 & ex_sup_of b3,b1
holds ex_sup_of b2 \/ b3,b1 &
"\/"(b2 \/ b3,b1) = ("\/"(b2,b1)) "\/" "\/"(b3,b1);
:: YELLOW_2:th 4
theorem
for b1 being transitive antisymmetric with_infima RelStr
for b2, b3 being set
st ex_inf_of b2,b1 & ex_inf_of b3,b1
holds ex_inf_of b2 \/ b3,b1 &
"/\"(b2 \/ b3,b1) = ("/\"(b2,b1)) "/\" "/\"(b3,b1);
:: YELLOW_2:th 5
theorem
for b1 being Relation-like set
for b2, b3 being set
st b2 c= b3
holds b1 |_2 b2 c= b1 |_2 b3;
:: YELLOW_2:th 6
theorem
for b1 being RelStr
for b2, b3 being full SubRelStr of b1
st the carrier of b2 c= the carrier of b3
holds the InternalRel of b2 c= the InternalRel of b3;
:: YELLOW_2:th 7
theorem
for b1 being set
for b2 being non empty RelStr
for b3 being non empty SubRelStr of b2 holds
(b1 is directed Element of bool the carrier of b3 implies b1 is directed Element of bool the carrier of b2) &
(b1 is filtered Element of bool the carrier of b3 implies b1 is filtered Element of bool the carrier of b2);
:: YELLOW_2:th 8
theorem
for b1 being non empty RelStr
for b2, b3 being non empty full SubRelStr of b1
st the carrier of b2 c= the carrier of b3
for b4 being Element of bool the carrier of b2 holds
b4 is Element of bool the carrier of b3 &
(for b5 being Element of bool the carrier of b3
st b4 = b5
holds (b4 is filtered(b2) implies b5 is filtered(b3)) & (b4 is directed(b2) implies b5 is directed(b3)));
:: YELLOW_2:prednot 1 => YELLOW_2:pred 1
definition
let a1 be set;
let a2 be RelStr;
let a3, a4 be Function-like quasi_total Relation of a1,the carrier of a2;
pred A3 <= A4 means
for b1 being set
st b1 in a1
holds ex b2, b3 being Element of the carrier of a2 st
b2 = a3 . b1 & b3 = a4 . b1 & b2 <= b3;
end;
:: YELLOW_2:dfs 1
definiens
let a1 be set;
let a2 be RelStr;
let a3, a4 be Function-like quasi_total Relation of a1,the carrier of a2;
To prove
a3 <= a4
it is sufficient to prove
thus for b1 being set
st b1 in a1
holds ex b2, b3 being Element of the carrier of a2 st
b2 = a3 . b1 & b3 = a4 . b1 & b2 <= b3;
:: YELLOW_2:def 1
theorem
for b1 being set
for b2 being RelStr
for b3, b4 being Function-like quasi_total Relation of b1,the carrier of b2 holds
b3 <= b4
iff
for b5 being set
st b5 in b1
holds ex b6, b7 being Element of the carrier of b2 st
b6 = b3 . b5 & b7 = b4 . b5 & b6 <= b7;
:: YELLOW_2:prednot 2 => YELLOW_2:pred 1
notation
let a1 be set;
let a2 be RelStr;
let a3, a4 be Function-like quasi_total Relation of a1,the carrier of a2;
synonym a4 >= a3 for a3 <= a4;
end;
:: YELLOW_2:th 10
theorem
for b1, b2 being non empty RelStr
for b3, b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 <= b4
iff
for b5 being Element of the carrier of b1 holds
b3 . b5 <= b4 . b5;
:: YELLOW_2:funcnot 1 => YELLOW_2:func 1
definition
let a1, a2 be non empty RelStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
func Image A3 -> strict full SubRelStr of a2 equals
subrelstr rng a3;
end;
:: YELLOW_2:def 2
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
Image b3 = subrelstr rng b3;
:: YELLOW_2:th 12
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of Image b3 holds
ex b5 being Element of the carrier of b1 st
b3 . b5 = b4;
:: YELLOW_2:funcreg 1
registration
let a1 be non empty RelStr;
let a2 be non empty Element of bool the carrier of a1;
cluster subrelstr a2 -> non empty strict full;
end;
:: YELLOW_2:funcreg 2
registration
let a1, a2 be non empty RelStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
cluster Image a3 -> non empty strict full;
end;
:: YELLOW_2:th 13
theorem
for b1 being non empty RelStr holds
id b1 is monotone(b1, b1);
:: YELLOW_2:th 14
theorem
for b1, b2, b3 being non empty RelStr
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 b2,the carrier of b3
st b4 is monotone(b1, b2) & b5 is monotone(b2, b3)
holds b5 * b4 is monotone(b1, b3);
:: YELLOW_2:th 15
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1
for b5 being Element of the carrier of b1
st b3 is monotone(b1, b2) & b5 is_<=_than b4
holds b3 . b5 is_<=_than b3 .: b4;
:: YELLOW_2:th 16
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1
for b5 being Element of the carrier of b1
st b3 is monotone(b1, b2) & b4 is_<=_than b5
holds b3 .: b4 is_<=_than b3 . b5;
:: YELLOW_2:th 17
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being directed Element of bool the carrier of b1
st b3 is monotone(b1, b2)
holds b3 .: b4 is directed(b2);
:: YELLOW_2:th 18
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
st b2 is directed-sups-preserving(b1, b1)
holds b2 is monotone(b1, b1);
:: YELLOW_2:th 19
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
st b2 is filtered-infs-preserving(b1, b1)
holds b2 is monotone(b1, b1);
:: YELLOW_2:th 20
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
st b2 is idempotent
for b3 being Element of the carrier of b1 holds
b2 . (b2 . b3) = b2 . b3;
:: YELLOW_2:th 21
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
st b2 is idempotent
holds rng b2 = {b3 where b3 is Element of the carrier of b1: b3 = b2 . b3};
:: YELLOW_2:th 22
theorem
for b1 being set
for b2 being non empty 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b2
st b3 is idempotent & b1 c= rng b3
holds b3 .: b1 = b1;
:: YELLOW_2:th 23
theorem
for b1 being non empty RelStr holds
id b1 is idempotent;
:: YELLOW_2:th 24
theorem
for b1 being set
for b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Element of the carrier of b2
st b3 in b1
holds b3 <= "\/"(b1,b2) & "/\"(b1,b2) <= b3;
:: YELLOW_2:th 25
theorem
for b1 being non empty RelStr holds
for b2 being set holds
ex_sup_of b2,b1
iff
for b2 being set holds
ex_inf_of b2,b1;
:: YELLOW_2:th 26
theorem
for b1 being non empty RelStr
st for b2 being set holds
ex_sup_of b2,b1
holds b1 is complete;
:: YELLOW_2:th 27
theorem
for b1 being non empty RelStr
st for b2 being set holds
ex_inf_of b2,b1
holds b1 is complete;
:: YELLOW_2:th 28
theorem
for b1 being non empty RelStr
st for b2 being Element of bool the carrier of b1 holds
ex_inf_of b2,b1
for b2 being set holds
ex_inf_of b2,b1 &
"/\"(b2,b1) = "/\"(b2 /\ the carrier of b1,b1);
:: YELLOW_2:th 29
theorem
for b1 being non empty RelStr
st for b2 being Element of bool the carrier of b1 holds
ex_sup_of b2,b1
for b2 being set holds
ex_sup_of b2,b1 &
"\/"(b2,b1) = "\/"(b2 /\ the carrier of b1,b1);
:: YELLOW_2:th 30
theorem
for b1 being non empty RelStr
st for b2 being Element of bool the carrier of b1 holds
ex_inf_of b2,b1
holds b1 is complete;
:: YELLOW_2:condreg 1
registration
cluster non empty reflexive transitive antisymmetric upper-bounded up-complete /\-complete -> complete (RelStr);
end;
:: YELLOW_2:th 31
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
st b2 is monotone(b1, b1)
for b3 being Element of bool the carrier of b1
st b3 = {b4 where b4 is Element of the carrier of b1: b4 = b2 . b4}
holds subrelstr b3 is reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
:: YELLOW_2:th 32
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty full infs-inheriting SubRelStr of b1 holds
b2 is reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
:: YELLOW_2:th 33
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty full sups-inheriting SubRelStr of b1 holds
b2 is reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
:: YELLOW_2:th 34
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b3 is sups-preserving(b1, b2)
holds Image b3 is sups-inheriting(b2);
:: YELLOW_2:th 35
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b3 is infs-preserving(b1, b2)
holds Image b3 is infs-inheriting(b2);
:: YELLOW_2:th 36
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st (b3 is sups-preserving(b1, b2) or b3 is infs-preserving(b1, b2))
holds Image b3 is reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
:: YELLOW_2:th 37
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
st b2 is idempotent & b2 is directed-sups-preserving(b1, b1)
holds Image b2 is directed-sups-inheriting(b1) &
Image b2 is reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
:: YELLOW_2:th 38
theorem
for b1 being RelStr
for b2 being Element of bool bool the carrier of b1
st for b3 being Element of bool the carrier of b1
st b3 in b2
holds b3 is lower(b1)
holds meet b2 is lower Element of bool the carrier of b1;
:: YELLOW_2:th 39
theorem
for b1 being RelStr
for b2 being Element of bool bool the carrier of b1
st for b3 being Element of bool the carrier of b1
st b3 in b2
holds b3 is upper(b1)
holds meet b2 is upper Element of bool the carrier of b1;
:: YELLOW_2:th 40
theorem
for b1 being antisymmetric with_suprema RelStr
for b2 being Element of bool bool the carrier of b1
st for b3 being Element of bool the carrier of b1
st b3 in b2
holds b3 is lower(b1) & b3 is directed(b1)
holds meet b2 is directed Element of bool the carrier of b1;
:: YELLOW_2:th 41
theorem
for b1 being antisymmetric with_infima RelStr
for b2 being Element of bool bool the carrier of b1
st for b3 being Element of bool the carrier of b1
st b3 in b2
holds b3 is upper(b1) & b3 is filtered(b1)
holds meet b2 is filtered Element of bool the carrier of b1;
:: YELLOW_2:th 42
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2, b3 being non empty directed lower Element of bool the carrier of b1 holds
b2 /\ b3 is non empty directed lower Element of bool the carrier of b1;
:: YELLOW_2:funcreg 3
registration
let a1 be non empty reflexive transitive RelStr;
cluster Ids a1 -> non empty;
end;
:: YELLOW_2:th 43
theorem
for b1 being set
for b2 being non empty reflexive transitive RelStr holds
b1 is Element of the carrier of InclPoset Ids b2
iff
b1 is non empty directed lower Element of bool the carrier of b2;
:: YELLOW_2:th 44
theorem
for b1 being set
for b2 being non empty reflexive transitive RelStr
for b3 being Element of the carrier of InclPoset Ids b2
st b1 in b3
holds b1 is Element of the carrier of b2;
:: YELLOW_2:th 45
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2, b3 being Element of the carrier of InclPoset Ids b1 holds
b2 "/\" b3 = b2 /\ b3;
:: YELLOW_2:funcreg 4
registration
let a1 be reflexive transitive antisymmetric with_infima RelStr;
cluster InclPoset Ids a1 -> strict with_infima;
end;
:: YELLOW_2:th 46
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2, b3 being Element of the carrier of InclPoset Ids b1 holds
ex b4 being Element of bool the carrier of b1 st
b4 = {b5 where b5 is Element of the carrier of b1: (not b5 in b2 & not b5 in b3 implies ex b6, b7 being Element of the carrier of b1 st
b6 in b2 & b7 in b3 & b5 = b6 "\/" b7)} &
ex_sup_of {b2,b3},InclPoset Ids b1 &
b2 "\/" b3 = downarrow b4;
:: YELLOW_2:funcreg 5
registration
let a1 be reflexive transitive antisymmetric with_suprema RelStr;
cluster InclPoset Ids a1 -> strict with_suprema;
end;
:: YELLOW_2:th 47
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded RelStr
for b2 being non empty Element of bool Ids b1 holds
meet b2 is non empty directed lower Element of bool the carrier of b1;
:: YELLOW_2:th 48
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded RelStr
for b2 being non empty Element of bool the carrier of InclPoset Ids b1 holds
ex_inf_of b2,InclPoset Ids b1 & "/\"(b2,InclPoset Ids b1) = meet b2;
:: YELLOW_2:th 49
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr holds
ex_inf_of {},InclPoset Ids b1 &
"/\"({},InclPoset Ids b1) = [#] b1;
:: YELLOW_2:th 50
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded RelStr holds
InclPoset Ids b1 is complete;
:: YELLOW_2:funcreg 6
registration
let a1 be reflexive transitive antisymmetric with_suprema lower-bounded RelStr;
cluster InclPoset Ids a1 -> strict complete;
end;
:: YELLOW_2:funcnot 2 => YELLOW_2:func 2
definition
let a1 be non empty reflexive transitive antisymmetric RelStr;
func SupMap A1 -> Function-like quasi_total Relation of the carrier of InclPoset Ids a1,the carrier of a1 means
for b1 being non empty directed lower Element of bool the carrier of a1 holds
it . b1 = "\/"(b1,a1);
end;
:: YELLOW_2:def 3
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being Function-like quasi_total Relation of the carrier of InclPoset Ids b1,the carrier of b1 holds
b2 = SupMap b1
iff
for b3 being non empty directed lower Element of bool the carrier of b1 holds
b2 . b3 = "\/"(b3,b1);
:: YELLOW_2:th 51
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr holds
dom SupMap b1 = Ids b1 & rng SupMap b1 is Element of bool the carrier of b1;
:: YELLOW_2:th 52
theorem
for b1 being set
for b2 being non empty reflexive transitive antisymmetric RelStr holds
b1 in dom SupMap b2
iff
b1 is non empty directed lower Element of bool the carrier of b2;
:: YELLOW_2:th 53
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr holds
SupMap b1 is monotone(InclPoset Ids b1, b1);
:: YELLOW_2:funcreg 7
registration
let a1 be non empty reflexive transitive antisymmetric up-complete RelStr;
cluster SupMap a1 -> Function-like quasi_total monotone;
end;
:: YELLOW_2:funcnot 3 => YELLOW_2:func 3
definition
let a1 be non empty reflexive transitive antisymmetric RelStr;
func IdsMap A1 -> Function-like quasi_total Relation of the carrier of a1,the carrier of InclPoset Ids a1 means
for b1 being Element of the carrier of a1 holds
it . b1 = downarrow b1;
end;
:: YELLOW_2:def 4
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of InclPoset Ids b1 holds
b2 = IdsMap b1
iff
for b3 being Element of the carrier of b1 holds
b2 . b3 = downarrow b3;
:: YELLOW_2:th 54
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr holds
IdsMap b1 is monotone(b1, InclPoset Ids b1);
:: YELLOW_2:funcreg 8
registration
let a1 be non empty reflexive transitive antisymmetric RelStr;
cluster IdsMap a1 -> Function-like quasi_total monotone;
end;
:: YELLOW_2:funcnot 4 => YELLOW_2:func 4
definition
let a1 be non empty RelStr;
let a2 be Relation-like set;
func \\/(A2,A1) -> Element of the carrier of a1 equals
"\/"(proj2 a2,a1);
end;
:: YELLOW_2:def 5
theorem
for b1 being non empty RelStr
for b2 being Relation-like set holds
\\/(b2,b1) = "\/"(proj2 b2,b1);
:: YELLOW_2:funcnot 5 => YELLOW_2:func 5
definition
let a1 be non empty RelStr;
let a2 be Relation-like set;
func //\(A2,A1) -> Element of the carrier of a1 equals
"/\"(proj2 a2,a1);
end;
:: YELLOW_2:def 6
theorem
for b1 being non empty RelStr
for b2 being Relation-like set holds
//\(b2,b1) = "/\"(proj2 b2,b1);
:: YELLOW_2:funcnot 6 => YELLOW_2:func 4
notation
let a1 be set;
let a2 be non empty RelStr;
let a3 be Function-like quasi_total Relation of a1,the carrier of a2;
synonym Sup a3 for \\/(a2,a1);
end;
:: YELLOW_2:funcnot 7 => YELLOW_2:func 5
notation
let a1 be set;
let a2 be non empty RelStr;
let a3 be Function-like quasi_total Relation of a1,the carrier of a2;
synonym Inf a3 for //\(a2,a1);
end;
:: YELLOW_2:th 55
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty set
for b3 being Element of b2
for b4 being Function-like quasi_total Relation of b2,the carrier of b1 holds
b4 . b3 <= \\/(b4,b1) & //\(b4,b1) <= b4 . b3;
:: YELLOW_2:th 56
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 non empty set
for b4 being Function-like quasi_total Relation of b3,the carrier of b1
st for b5 being Element of b3 holds
b4 . b5 <= b2
holds \\/(b4,b1) <= b2;
:: YELLOW_2:th 57
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 non empty set
for b4 being Function-like quasi_total Relation of b3,the carrier of b1
st for b5 being Element of b3 holds
b2 <= b4 . b5
holds b2 <= //\(b4,b1);