Article YELLOW12, MML version 4.99.1005
:: YELLOW12:funcreg 1
registration
let a1 be empty set;
cluster union a1 -> empty;
end;
:: YELLOW12:th 1
theorem
for b1, b2 being set holds
(delta b1) .: b2 c= [:b2,b2:];
:: YELLOW12:th 2
theorem
for b1, b2 being set holds
(delta b1) " [:b2,b2:] c= b2;
:: YELLOW12:th 3
theorem
for b1 being set
for b2 being Element of bool b1 holds
(delta b1) " [:b2,b2:] = b2;
:: YELLOW12:th 4
theorem
for b1, b2 being set holds
proj1 <:pr2(b1,b2),pr1(b1,b2):> = [:b1,b2:] &
proj2 <:pr2(b1,b2),pr1(b1,b2):> = [:b2,b1:];
:: YELLOW12:th 5
theorem
for b1, b2, b3, b4 being set holds
<:pr2(b1,b2),pr1(b1,b2):> .: [:b3,b4:] c= [:b4,b3:];
:: YELLOW12:th 6
theorem
for b1, b2 being set
for b3 being Element of bool b1
for b4 being Element of bool b2 holds
<:pr2(b1,b2),pr1(b1,b2):> .: [:b3,b4:] = [:b4,b3:];
:: YELLOW12:th 7
theorem
for b1, b2 being set holds
<:pr2(b1,b2),pr1(b1,b2):> is one-to-one;
:: YELLOW12:funcreg 2
registration
let a1, a2 be set;
cluster <:pr2(a1,a2),pr1(a1,a2):> -> Relation-like Function-like one-to-one;
end;
:: YELLOW12:th 8
theorem
for b1, b2 being set holds
<:pr2(b1,b2),pr1(b1,b2):> " = <:pr2(b2,b1),pr1(b2,b1):>;
:: YELLOW12:th 9
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being non empty RelStr
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#) &
b3 = b5 &
b4 = b6
holds b3 "/\" b4 = b5 "/\" b6;
:: YELLOW12:th 10
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2 being non empty RelStr
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#) &
b3 = b5 &
b4 = b6
holds b3 "\/" b4 = b5 "\/" b6;
:: YELLOW12:th 11
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being non empty RelStr
for b3, b4 being Element of bool the carrier of b1
for b5, b6 being Element of bool the carrier of b2
st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#) &
b3 = b5 &
b4 = b6
holds b3 "/\" b4 = b5 "/\" b6;
:: YELLOW12:th 12
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2 being non empty RelStr
for b3, b4 being Element of bool the carrier of b1
for b5, b6 being Element of bool the carrier of b2
st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#) &
b3 = b5 &
b4 = b6
holds b3 "\/" b4 = b5 "\/" b6;
:: YELLOW12:th 13
theorem
for b1 being non empty reflexive antisymmetric up-complete RelStr
for b2 being non empty reflexive RelStr
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#) &
b3 = b4
holds waybelow b3 = waybelow b4 & wayabove b3 = wayabove b4;
:: YELLOW12:th 14
theorem
for b1 being reflexive transitive antisymmetric with_infima meet-continuous RelStr
for 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#)
holds b2 is meet-continuous;
:: YELLOW12:th 15
theorem
for b1 being non empty reflexive antisymmetric continuous RelStr
for 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#)
holds b2 is continuous;
:: YELLOW12:th 16
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
st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#) &
b3 = b4
holds subrelstr b3 = subrelstr b4;
:: YELLOW12:th 17
theorem
for b1, b2 being non empty RelStr
for b3 being SubRelStr of b1
for b4 being SubRelStr of b2
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#) &
b3 is meet-inheriting(b1)
holds b4 is meet-inheriting(b2);
:: YELLOW12:th 18
theorem
for b1, b2 being non empty RelStr
for b3 being SubRelStr of b1
for b4 being SubRelStr of b2
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#) &
b3 is join-inheriting(b1)
holds b4 is join-inheriting(b2);
:: YELLOW12:th 19
theorem
for b1 being non empty reflexive antisymmetric up-complete RelStr
for b2 being non empty reflexive RelStr
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#) &
b3 = b4 &
b3 is property(S)(b1)
holds b4 is property(S)(b2);
:: YELLOW12:th 20
theorem
for b1 being non empty reflexive antisymmetric up-complete RelStr
for b2 being non empty reflexive RelStr
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#) &
b3 = b4 &
b3 is closed_under_directed_sups(b1)
holds b4 is closed_under_directed_sups(b2);
:: YELLOW12:th 21
theorem
for b1 being antisymmetric with_infima RelStr
for b2, b3 being Element of bool the carrier of b1
for b4 being upper Element of bool the carrier of b1
st b2 misses b4
holds b2 "/\" b3 misses b4;
:: YELLOW12:th 22
theorem
for b1 being non empty reflexive RelStr holds
id the carrier of b1 c= (the InternalRel of b1) /\ the InternalRel of b1 ~;
:: YELLOW12:th 23
theorem
for b1 being antisymmetric RelStr holds
(the InternalRel of b1) /\ the InternalRel of b1 ~ c= id the carrier of b1;
:: YELLOW12:funcnot 1 => YELLOW12:func 1
definition
let a1 be non empty RelStr;
let a2 be Function-like quasi_total Relation of the carrier of [:a1,a1:],the carrier of a1;
let a3, a4 be Element of the carrier of a1;
redefine func a2 .(a3,a4) -> Element of the carrier of a1;
end;
:: YELLOW12:th 24
theorem
for b1 being reflexive transitive antisymmetric with_infima upper-bounded RelStr
for b2 being Element of bool the carrier of [:b1,b1:]
st ex_inf_of (inf_op b1) .: b2,b1
holds inf_op b1 preserves_inf_of b2;
:: YELLOW12:funcreg 3
registration
let a1 be reflexive transitive antisymmetric with_infima complete RelStr;
cluster inf_op a1 -> Function-like quasi_total infs-preserving;
end;
:: YELLOW12:th 25
theorem
for b1 being reflexive transitive antisymmetric with_suprema lower-bounded RelStr
for b2 being Element of bool the carrier of [:b1,b1:]
st ex_sup_of (sup_op b1) .: b2,b1
holds sup_op b1 preserves_sup_of b2;
:: YELLOW12:funcreg 4
registration
let a1 be reflexive transitive antisymmetric with_suprema complete RelStr;
cluster sup_op a1 -> Function-like quasi_total sups-preserving;
end;
:: YELLOW12:th 26
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being Element of bool the carrier of b1
st subrelstr b2 is meet-inheriting(b1)
holds b2 is filtered(b1);
:: YELLOW12:th 27
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2 being Element of bool the carrier of b1
st subrelstr b2 is join-inheriting(b1)
holds b2 is directed(b1);
:: YELLOW12:th 28
theorem
for b1 being transitive RelStr
for b2, b3 being Element of bool the carrier of b1
st b2 is_coarser_than uparrow b3
holds uparrow b2 c= uparrow b3;
:: YELLOW12:th 29
theorem
for b1 being transitive RelStr
for b2, b3 being Element of bool the carrier of b1
st b2 is_finer_than downarrow b3
holds downarrow b2 c= downarrow b3;
:: YELLOW12:th 30
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
st b2 in b3
holds uparrow b2 c= uparrow b3;
:: YELLOW12:th 31
theorem
for b1 being non empty reflexive RelStr
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
st b2 in b3
holds downarrow b2 c= downarrow b3;
:: YELLOW12:funcreg 5
registration
let a1 be non empty TopStruct;
cluster TopStruct(#the carrier of a1,the topology of a1#) -> non empty strict;
end;
:: YELLOW12:funcreg 6
registration
let a1 be TopSpace-like TopStruct;
cluster TopStruct(#the carrier of a1,the topology of a1#) -> strict TopSpace-like;
end;
:: YELLOW12:th 32
theorem
for b1, b2 being TopStruct
for b3 being Basis of b1
st TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#)
holds b3 is Basis of b2;
:: YELLOW12:th 33
theorem
for b1, b2 being TopStruct
for b3 being prebasis of b1
st TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#)
holds b3 is prebasis of b2;
:: YELLOW12:th 34
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Basis of b1 holds
b2 is not empty;
:: YELLOW12:condreg 1
registration
let a1 be non empty TopSpace-like TopStruct;
cluster -> non empty (Basis of a1);
end;
:: YELLOW12:th 35
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being Basis of b2 holds
b3 is not empty;
:: YELLOW12:condreg 2
registration
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of the carrier of a1;
cluster -> non empty (Basis of a2);
end;
:: YELLOW12:th 36
theorem
for b1, b2, b3, b4 being TopSpace-like TopStruct
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
for b6 being Function-like quasi_total Relation of the carrier of b2,the carrier of b4
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#) &
b5 = b6 &
b5 is continuous(b1, b3)
holds b6 is continuous(b2, b4);
:: YELLOW12:th 37
theorem
for b1 being non empty TopSpace-like TopStruct holds
id the carrier of b1 = {b2 where b2 is Element of the carrier of [:b1,b1:]: (pr1(the carrier of b1,the carrier of b1)) . b2 = (pr2(the carrier of b1,the carrier of b1)) . b2};
:: YELLOW12:th 38
theorem
for b1 being non empty TopSpace-like TopStruct holds
delta the carrier of b1 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of [:b1,b1:];
:: YELLOW12:th 39
theorem
for b1, b2 being non empty TopSpace-like TopStruct holds
pr1(the carrier of b1,the carrier of b2) is Function-like quasi_total continuous Relation of the carrier of [:b1,b2:],the carrier of b1;
:: YELLOW12:th 40
theorem
for b1, b2 being non empty TopSpace-like TopStruct holds
pr2(the carrier of b1,the carrier of b2) is Function-like quasi_total continuous Relation of the carrier of [:b1,b2:],the carrier of b2;
:: YELLOW12:th 41
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
for b4 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b3 holds
<:b4,b5:> is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of [:b2,b3:];
:: YELLOW12:th 42
theorem
for b1, b2 being non empty TopSpace-like TopStruct holds
<:pr2(the carrier of b1,the carrier of b2),pr1(the carrier of b1,the carrier of b2):> is Function-like quasi_total continuous Relation of the carrier of [:b1,b2:],the carrier of [:b2,b1:];
:: YELLOW12:th 43
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of [:b1,b2:],the carrier of [:b2,b1:]
st b3 = <:pr2(the carrier of b1,the carrier of b2),pr1(the carrier of b1,the carrier of b2):>
holds b3 is being_homeomorphism([:b1,b2:], [:b2,b1:]);
:: YELLOW12:th 44
theorem
for b1, b2 being non empty TopSpace-like TopStruct holds
[:b1,b2:],[:b2,b1:] are_homeomorphic;
:: YELLOW12:th 45
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like being_T2 TopStruct
for b3, b4 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2 holds
(for b5 being Element of bool the carrier of b1
st b5 = {b6 where b6 is Element of the carrier of b1: b3 . b6 <> b4 . b6}
holds b5 is open(b1)) &
(for b5 being Element of bool the carrier of b1
st b5 = {b6 where b6 is Element of the carrier of b1: b3 . b6 = b4 . b6}
holds b5 is closed(b1));
:: YELLOW12:th 46
theorem
for b1 being non empty TopSpace-like TopStruct holds
b1 is being_T2
iff
for b2 being Element of bool the carrier of [:b1,b1:]
st b2 = id the carrier of b1
holds b2 is closed([:b1,b1:]);
:: YELLOW12:exreg 1
registration
let a1, a2 be TopStruct;
cluster strict TopSpace-like Refinement of a1,a2;
end;
:: YELLOW12:exreg 2
registration
let a1 be non empty TopStruct;
let a2 be TopStruct;
cluster non empty strict TopSpace-like Refinement of a1,a2;
end;
:: YELLOW12:exreg 3
registration
let a1 be non empty TopStruct;
let a2 be TopStruct;
cluster non empty strict TopSpace-like Refinement of a2,a1;
end;
:: YELLOW12:th 47
theorem
for b1, b2, b3 being TopStruct holds
b1 is Refinement of b2,b3
iff
TopStruct(#the carrier of b1,the topology of b1#) is Refinement of b2,b3;
:: YELLOW12:th 48
theorem
for b1, b2, b3, b4 being non empty TopSpace-like TopStruct
for b5 being Refinement of [:b1,b3:],[:b2,b4:]
st the carrier of b1 = the carrier of b2 & the carrier of b3 = the carrier of b4
holds {[:b6,b8:] /\ [:b7,b9:] where b6 is Element of bool the carrier of b1, b7 is Element of bool the carrier of b2, b8 is Element of bool the carrier of b3, b9 is Element of bool the carrier of b4: b6 is open(b1) & b7 is open(b2) & b8 is open(b3) & b9 is open(b4)} is Basis of b5;
:: YELLOW12:th 49
theorem
for b1, b2, b3, b4 being non empty TopSpace-like TopStruct
for b5 being Refinement of [:b1,b3:],[:b2,b4:]
for b6 being Refinement of b1,b2
for b7 being Refinement of b3,b4
st the carrier of b1 = the carrier of b2 & the carrier of b3 = the carrier of b4
holds the carrier of [:b6,b7:] = the carrier of b5 & the topology of [:b6,b7:] = the topology of b5;
:: YELLOW12:th 50
theorem
for b1, b2, b3, b4 being non empty TopSpace-like TopStruct
for b5 being Refinement of b1,b2
for b6 being Refinement of b3,b4
st the carrier of b1 = the carrier of b2 & the carrier of b3 = the carrier of b4
holds [:b5,b6:] is Refinement of [:b1,b3:],[:b2,b4:];