Article YELLOW10, MML version 4.99.1005
:: YELLOW10:funcreg 1
registration
let a1, a2 be non empty upper-bounded RelStr;
cluster [:a1,a2:] -> strict upper-bounded;
end;
:: YELLOW10:funcreg 2
registration
let a1, a2 be non empty lower-bounded RelStr;
cluster [:a1,a2:] -> strict lower-bounded;
end;
:: YELLOW10:th 1
theorem
for b1, b2 being non empty RelStr
st [:b1,b2:] is upper-bounded
holds b1 is upper-bounded & b2 is upper-bounded;
:: YELLOW10:th 2
theorem
for b1, b2 being non empty RelStr
st [:b1,b2:] is lower-bounded
holds b1 is lower-bounded & b2 is lower-bounded;
:: YELLOW10:th 3
theorem
for b1, b2 being non empty antisymmetric upper-bounded RelStr holds
Top [:b1,b2:] = [Top b1,Top b2];
:: YELLOW10:th 4
theorem
for b1, b2 being non empty antisymmetric lower-bounded RelStr holds
Bottom [:b1,b2:] = [Bottom b1,Bottom b2];
:: YELLOW10:th 5
theorem
for b1, b2 being non empty antisymmetric lower-bounded RelStr
for b3 being Element of bool the carrier of [:b1,b2:]
st ([:b1,b2:] is complete or ex_sup_of b3,[:b1,b2:])
holds "\/"(b3,[:b1,b2:]) = ["\/"(proj1 b3,b1),"\/"(proj2 b3,b2)];
:: YELLOW10:th 6
theorem
for b1, b2 being non empty antisymmetric upper-bounded RelStr
for b3 being Element of bool the carrier of [:b1,b2:]
st ([:b1,b2:] is complete or ex_inf_of b3,[:b1,b2:])
holds "/\"(b3,[:b1,b2:]) = ["/\"(proj1 b3,b1),"/\"(proj2 b3,b2)];
:: YELLOW10:th 7
theorem
for b1, b2 being non empty RelStr
for b3, b4 being Element of the carrier of [:b1,b2:] holds
b3 is_<=_than {b4}
iff
b3 `1 is_<=_than {b4 `1} & b3 `2 is_<=_than {b4 `2};
:: YELLOW10:th 8
theorem
for b1, b2 being non empty RelStr
for b3, b4, b5 being Element of the carrier of [:b1,b2:] holds
b3 is_<=_than {b4,b5}
iff
b3 `1 is_<=_than {b4 `1,b5 `1} &
b3 `2 is_<=_than {b4 `2,b5 `2};
:: YELLOW10:th 9
theorem
for b1, b2 being non empty RelStr
for b3, b4 being Element of the carrier of [:b1,b2:] holds
{b4} is_<=_than b3
iff
{b4 `1} is_<=_than b3 `1 & {b4 `2} is_<=_than b3 `2;
:: YELLOW10:th 10
theorem
for b1, b2 being non empty RelStr
for b3, b4, b5 being Element of the carrier of [:b1,b2:] holds
{b4,b5} is_<=_than b3
iff
{b4 `1,b5 `1} is_<=_than b3 `1 &
{b4 `2,b5 `2} is_<=_than b3 `2;
:: YELLOW10:th 11
theorem
for b1, b2 being non empty antisymmetric RelStr
for b3, b4 being Element of the carrier of [:b1,b2:] holds
ex_inf_of {b3,b4},[:b1,b2:]
iff
ex_inf_of {b3 `1,b4 `1},b1 & ex_inf_of {b3 `2,b4 `2},b2;
:: YELLOW10:th 12
theorem
for b1, b2 being non empty antisymmetric RelStr
for b3, b4 being Element of the carrier of [:b1,b2:] holds
ex_sup_of {b3,b4},[:b1,b2:]
iff
ex_sup_of {b3 `1,b4 `1},b1 & ex_sup_of {b3 `2,b4 `2},b2;
:: YELLOW10:th 13
theorem
for b1, b2 being antisymmetric with_infima RelStr
for b3, b4 being Element of the carrier of [:b1,b2:] holds
(b3 "/\" b4) `1 = b3 `1 "/\" (b4 `1) &
(b3 "/\" b4) `2 = b3 `2 "/\" (b4 `2);
:: YELLOW10:th 14
theorem
for b1, b2 being antisymmetric with_suprema RelStr
for b3, b4 being Element of the carrier of [:b1,b2:] holds
(b3 "\/" b4) `1 = b3 `1 "\/" (b4 `1) &
(b3 "\/" b4) `2 = b3 `2 "\/" (b4 `2);
:: YELLOW10:th 15
theorem
for b1, b2 being antisymmetric with_infima RelStr
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2 holds
[b3 "/\" b4,b5 "/\" b6] = [b3,b5] "/\" [b4,b6];
:: YELLOW10:th 16
theorem
for b1, b2 being antisymmetric with_suprema RelStr
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2 holds
[b3 "\/" b4,b5 "\/" b6] = [b3,b5] "\/" [b4,b6];
:: YELLOW10:prednot 1 => YELLOW10:pred 1
definition
let a1 be antisymmetric with_suprema with_infima RelStr;
let a2, a3 be Element of the carrier of a1;
redefine pred a3 is_a_complement_of a2;
symmetry;
:: for a1 being antisymmetric with_suprema with_infima RelStr
:: for a2, a3 being Element of the carrier of a1
:: st a3 is_a_complement_of a2
:: holds a2 is_a_complement_of a3;
end;
:: YELLOW10:th 17
theorem
for b1, b2 being antisymmetric bounded with_suprema with_infima RelStr
for b3, b4 being Element of the carrier of [:b1,b2:] holds
b3 is_a_complement_of b4
iff
b3 `1 is_a_complement_of b4 `1 & b3 `2 is_a_complement_of b4 `2;
:: YELLOW10:th 18
theorem
for b1, b2 being non empty reflexive antisymmetric up-complete RelStr
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
st [b3,b5] is_way_below [b4,b6]
holds b3 is_way_below b4 & b5 is_way_below b6;
:: YELLOW10:th 19
theorem
for b1, b2 being non empty reflexive transitive antisymmetric up-complete RelStr
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2 holds
[b3,b5] is_way_below [b4,b6]
iff
b3 is_way_below b4 & b5 is_way_below b6;
:: YELLOW10:th 20
theorem
for b1, b2 being non empty reflexive antisymmetric up-complete RelStr
for b3, b4 being Element of the carrier of [:b1,b2:]
st b3 is_way_below b4
holds b3 `1 is_way_below b4 `1 & b3 `2 is_way_below b4 `2;
:: YELLOW10:th 21
theorem
for b1, b2 being non empty reflexive transitive antisymmetric up-complete RelStr
for b3, b4 being Element of the carrier of [:b1,b2:] holds
b3 is_way_below b4
iff
b3 `1 is_way_below b4 `1 & b3 `2 is_way_below b4 `2;
:: YELLOW10:th 22
theorem
for b1, b2 being non empty reflexive antisymmetric up-complete RelStr
for b3 being Element of the carrier of [:b1,b2:]
st b3 is compact([:b1,b2:])
holds b3 `1 is compact(b1) & b3 `2 is compact(b2);
:: YELLOW10:th 23
theorem
for b1, b2 being non empty reflexive transitive antisymmetric up-complete RelStr
for b3 being Element of the carrier of [:b1,b2:]
st b3 `1 is compact(b1) & b3 `2 is compact(b2)
holds b3 is compact([:b1,b2:]);
:: YELLOW10:th 24
theorem
for b1, b2 being antisymmetric with_infima RelStr
for b3, b4 being Element of bool the carrier of [:b1,b2:] holds
proj1 (b3 "/\" b4) = (proj1 b3) "/\" proj1 b4 &
proj2 (b3 "/\" b4) = (proj2 b3) "/\" proj2 b4;
:: YELLOW10:th 25
theorem
for b1, b2 being antisymmetric with_suprema RelStr
for b3, b4 being Element of bool the carrier of [:b1,b2:] holds
proj1 (b3 "\/" b4) = (proj1 b3) "\/" proj1 b4 &
proj2 (b3 "\/" b4) = (proj2 b3) "\/" proj2 b4;
:: YELLOW10:th 26
theorem
for b1, b2 being RelStr
for b3 being Element of bool the carrier of [:b1,b2:] holds
downarrow b3 c= [:downarrow proj1 b3,downarrow proj2 b3:];
:: YELLOW10:th 27
theorem
for b1, b2 being RelStr
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2 holds
[:downarrow b3,downarrow b4:] = downarrow [:b3,b4:];
:: YELLOW10:th 28
theorem
for b1, b2 being RelStr
for b3 being Element of bool the carrier of [:b1,b2:] holds
proj1 downarrow b3 c= downarrow proj1 b3 & proj2 downarrow b3 c= downarrow proj2 b3;
:: YELLOW10:th 29
theorem
for b1 being RelStr
for b2 being reflexive RelStr
for b3 being Element of bool the carrier of [:b1,b2:] holds
proj1 downarrow b3 = downarrow proj1 b3;
:: YELLOW10:th 30
theorem
for b1 being reflexive RelStr
for b2 being RelStr
for b3 being Element of bool the carrier of [:b1,b2:] holds
proj2 downarrow b3 = downarrow proj2 b3;
:: YELLOW10:th 31
theorem
for b1, b2 being RelStr
for b3 being Element of bool the carrier of [:b1,b2:] holds
uparrow b3 c= [:uparrow proj1 b3,uparrow proj2 b3:];
:: YELLOW10:th 32
theorem
for b1, b2 being RelStr
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2 holds
[:uparrow b3,uparrow b4:] = uparrow [:b3,b4:];
:: YELLOW10:th 33
theorem
for b1, b2 being RelStr
for b3 being Element of bool the carrier of [:b1,b2:] holds
proj1 uparrow b3 c= uparrow proj1 b3 & proj2 uparrow b3 c= uparrow proj2 b3;
:: YELLOW10:th 34
theorem
for b1 being RelStr
for b2 being reflexive RelStr
for b3 being Element of bool the carrier of [:b1,b2:] holds
proj1 uparrow b3 = uparrow proj1 b3;
:: YELLOW10:th 35
theorem
for b1 being reflexive RelStr
for b2 being RelStr
for b3 being Element of bool the carrier of [:b1,b2:] holds
proj2 uparrow b3 = uparrow proj2 b3;
:: YELLOW10:th 36
theorem
for b1, b2 being non empty RelStr
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2 holds
[:downarrow b3,downarrow b4:] = downarrow [b3,b4];
:: YELLOW10:th 37
theorem
for b1, b2 being non empty RelStr
for b3 being Element of the carrier of [:b1,b2:] holds
proj1 downarrow b3 c= downarrow (b3 `1) & proj2 downarrow b3 c= downarrow (b3 `2);
:: YELLOW10:th 38
theorem
for b1 being non empty RelStr
for b2 being non empty reflexive RelStr
for b3 being Element of the carrier of [:b1,b2:] holds
proj1 downarrow b3 = downarrow (b3 `1);
:: YELLOW10:th 39
theorem
for b1 being non empty reflexive RelStr
for b2 being non empty RelStr
for b3 being Element of the carrier of [:b1,b2:] holds
proj2 downarrow b3 = downarrow (b3 `2);
:: YELLOW10:th 40
theorem
for b1, b2 being non empty RelStr
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2 holds
[:uparrow b3,uparrow b4:] = uparrow [b3,b4];
:: YELLOW10:th 41
theorem
for b1, b2 being non empty RelStr
for b3 being Element of the carrier of [:b1,b2:] holds
proj1 uparrow b3 c= uparrow (b3 `1) & proj2 uparrow b3 c= uparrow (b3 `2);
:: YELLOW10:th 42
theorem
for b1 being non empty RelStr
for b2 being non empty reflexive RelStr
for b3 being Element of the carrier of [:b1,b2:] holds
proj1 uparrow b3 = uparrow (b3 `1);
:: YELLOW10:th 43
theorem
for b1 being non empty reflexive RelStr
for b2 being non empty RelStr
for b3 being Element of the carrier of [:b1,b2:] holds
proj2 uparrow b3 = uparrow (b3 `2);
:: YELLOW10:th 44
theorem
for b1, b2 being non empty reflexive transitive antisymmetric up-complete RelStr
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2 holds
[:waybelow b3,waybelow b4:] = waybelow [b3,b4];
:: YELLOW10:th 45
theorem
for b1, b2 being non empty reflexive antisymmetric up-complete RelStr
for b3 being Element of the carrier of [:b1,b2:] holds
proj1 waybelow b3 c= waybelow (b3 `1) & proj2 waybelow b3 c= waybelow (b3 `2);
:: YELLOW10:th 46
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr
for b2 being non empty reflexive transitive antisymmetric lower-bounded up-complete RelStr
for b3 being Element of the carrier of [:b1,b2:] holds
proj1 waybelow b3 = waybelow (b3 `1);
:: YELLOW10:th 47
theorem
for b1 being non empty reflexive transitive antisymmetric lower-bounded up-complete RelStr
for b2 being non empty reflexive transitive antisymmetric up-complete RelStr
for b3 being Element of the carrier of [:b1,b2:] holds
proj2 waybelow b3 = waybelow (b3 `2);
:: YELLOW10:th 48
theorem
for b1, b2 being non empty reflexive transitive antisymmetric up-complete RelStr
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2 holds
[:wayabove b3,wayabove b4:] = wayabove [b3,b4];
:: YELLOW10:th 49
theorem
for b1, b2 being non empty reflexive antisymmetric up-complete RelStr
for b3 being Element of the carrier of [:b1,b2:] holds
proj1 wayabove b3 c= wayabove (b3 `1) & proj2 wayabove b3 c= wayabove (b3 `2);
:: YELLOW10:th 50
theorem
for b1, b2 being non empty reflexive transitive antisymmetric up-complete RelStr
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2 holds
[:compactbelow b3,compactbelow b4:] = compactbelow [b3,b4];
:: YELLOW10:th 51
theorem
for b1, b2 being non empty reflexive antisymmetric up-complete RelStr
for b3 being Element of the carrier of [:b1,b2:] holds
proj1 compactbelow b3 c= compactbelow (b3 `1) & proj2 compactbelow b3 c= compactbelow (b3 `2);
:: YELLOW10:th 52
theorem
for b1 being non empty reflexive transitive antisymmetric up-complete RelStr
for b2 being non empty reflexive transitive antisymmetric lower-bounded up-complete RelStr
for b3 being Element of the carrier of [:b1,b2:] holds
proj1 compactbelow b3 = compactbelow (b3 `1);
:: YELLOW10:th 53
theorem
for b1 being non empty reflexive transitive antisymmetric lower-bounded up-complete RelStr
for b2 being non empty reflexive transitive antisymmetric up-complete RelStr
for b3 being Element of the carrier of [:b1,b2:] holds
proj2 compactbelow b3 = compactbelow (b3 `2);
:: YELLOW10:condreg 1
registration
let a1 be non empty reflexive RelStr;
cluster empty -> Open (Element of bool the carrier of a1);
end;
:: YELLOW10:th 54
theorem
for b1, b2 being non empty reflexive antisymmetric up-complete RelStr
for b3 being Element of bool the carrier of [:b1,b2:]
st b3 is Open([:b1,b2:])
holds proj1 b3 is Open(b1) & proj2 b3 is Open(b2);
:: YELLOW10:th 55
theorem
for b1, b2 being non empty reflexive transitive antisymmetric up-complete RelStr
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
st b3 is Open(b1) & b4 is Open(b2)
holds [:b3,b4:] is Open([:b1,b2:]);
:: YELLOW10:th 56
theorem
for b1, b2 being non empty reflexive antisymmetric up-complete RelStr
for b3 being Element of bool the carrier of [:b1,b2:]
st b3 is inaccessible_by_directed_joins([:b1,b2:])
holds proj1 b3 is inaccessible_by_directed_joins(b1) & proj2 b3 is inaccessible_by_directed_joins(b2);
:: YELLOW10:th 57
theorem
for b1, b2 being non empty reflexive antisymmetric up-complete RelStr
for b3 being upper Element of bool the carrier of b1
for b4 being upper Element of bool the carrier of b2
st b3 is inaccessible_by_directed_joins(b1) & b4 is inaccessible_by_directed_joins(b2)
holds [:b3,b4:] is inaccessible_by_directed_joins([:b1,b2:]);
:: YELLOW10:th 58
theorem
for b1, b2 being non empty reflexive antisymmetric up-complete RelStr
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
st [:b3,b4:] is closed_under_directed_sups([:b1,b2:])
holds (b4 = {} or b3 is closed_under_directed_sups(b1)) & (b3 = {} or b4 is closed_under_directed_sups(b2));
:: YELLOW10:th 59
theorem
for b1, b2 being non empty reflexive antisymmetric up-complete RelStr
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
st b3 is closed_under_directed_sups(b1) & b4 is closed_under_directed_sups(b2)
holds [:b3,b4:] is closed_under_directed_sups([:b1,b2:]);
:: YELLOW10:th 60
theorem
for b1, b2 being non empty reflexive antisymmetric up-complete RelStr
for b3 being Element of bool the carrier of [:b1,b2:]
st b3 is property(S)([:b1,b2:])
holds proj1 b3 is property(S)(b1) & proj2 b3 is property(S)(b2);
:: YELLOW10:th 61
theorem
for b1, b2 being non empty reflexive transitive antisymmetric up-complete RelStr
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
st b3 is property(S)(b1) & b4 is property(S)(b2)
holds [:b3,b4:] is property(S)([:b1,b2:]);
:: YELLOW10:th 62
theorem
for b1, b2 being non empty reflexive RelStr
st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#) &
b1 is /\-complete
holds b2 is /\-complete;
:: YELLOW10:funcreg 3
registration
let a1 be non empty reflexive /\-complete RelStr;
cluster RelStr(#the carrier of a1,the InternalRel of a1#) -> strict /\-complete;
end;
:: YELLOW10:funcreg 4
registration
let a1, a2 be non empty reflexive /\-complete RelStr;
cluster [:a1,a2:] -> strict /\-complete;
end;
:: YELLOW10:th 63
theorem
for b1, b2 being non empty reflexive RelStr
st [:b1,b2:] is /\-complete
holds b1 is /\-complete & b2 is /\-complete;
:: YELLOW10:funcreg 5
registration
let a1, a2 be non empty antisymmetric bounded complemented with_suprema with_infima RelStr;
cluster [:a1,a2:] -> strict complemented;
end;
:: YELLOW10:th 64
theorem
for b1, b2 being antisymmetric bounded with_suprema with_infima RelStr
st [:b1,b2:] is complemented
holds b1 is complemented & b2 is complemented;
:: YELLOW10:funcreg 6
registration
let a1, a2 be non empty antisymmetric distributive with_suprema with_infima RelStr;
cluster [:a1,a2:] -> strict distributive;
end;
:: YELLOW10:th 65
theorem
for b1 being antisymmetric with_suprema with_infima RelStr
for b2 being reflexive antisymmetric with_suprema with_infima RelStr
st [:b1,b2:] is distributive
holds b1 is distributive;
:: YELLOW10:th 66
theorem
for b1 being reflexive antisymmetric with_suprema with_infima RelStr
for b2 being antisymmetric with_suprema with_infima RelStr
st [:b1,b2:] is distributive
holds b2 is distributive;
:: YELLOW10:funcreg 7
registration
let a1, a2 be reflexive transitive antisymmetric with_infima meet-continuous RelStr;
cluster [:a1,a2:] -> strict satisfying_MC;
end;
:: YELLOW10:th 67
theorem
for b1, b2 being reflexive transitive antisymmetric with_infima RelStr
st [:b1,b2:] is meet-continuous
holds b1 is meet-continuous & b2 is meet-continuous;
:: YELLOW10:funcreg 8
registration
let a1, a2 be non empty reflexive transitive antisymmetric up-complete /\-complete satisfying_axiom_of_approximation RelStr;
cluster [:a1,a2:] -> strict satisfying_axiom_of_approximation;
end;
:: YELLOW10:funcreg 9
registration
let a1, a2 be non empty reflexive transitive antisymmetric /\-complete continuous RelStr;
cluster [:a1,a2:] -> strict continuous;
end;
:: YELLOW10:th 68
theorem
for b1, b2 being non empty reflexive transitive antisymmetric lower-bounded up-complete RelStr
st [:b1,b2:] is continuous
holds b1 is continuous & b2 is continuous;
:: YELLOW10:funcreg 10
registration
let a1, a2 be reflexive transitive antisymmetric lower-bounded up-complete with_suprema satisfying_axiom_K RelStr;
cluster [:a1,a2:] -> strict satisfying_axiom_K;
end;
:: YELLOW10:funcreg 11
registration
let a1, a2 be reflexive transitive antisymmetric lower-bounded with_suprema complete algebraic RelStr;
cluster [:a1,a2:] -> strict algebraic;
end;
:: YELLOW10:th 69
theorem
for b1, b2 being non empty reflexive transitive antisymmetric lower-bounded RelStr
st [:b1,b2:] is algebraic
holds b1 is algebraic & b2 is algebraic;
:: YELLOW10:funcreg 12
registration
let a1, a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima arithmetic RelStr;
cluster [:a1,a2:] -> strict arithmetic;
end;
:: YELLOW10:th 70
theorem
for b1, b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
st [:b1,b2:] is arithmetic
holds b1 is arithmetic & b2 is arithmetic;