Article ORDINAL2, MML version 4.99.1005
:: ORDINAL2:sch 1
scheme ORDINAL2:sch 1
for b1 being ordinal set holds
P1[b1]
provided
P1[{}]
and
for b1 being ordinal set
st P1[b1]
holds P1[succ b1]
and
for b1 being ordinal set
st b1 <> {} &
b1 is being_limit_ordinal &
(for b2 being ordinal set
st b2 in b1
holds P1[b2])
holds P1[b1];
:: ORDINAL2:th 1
theorem
for b1, b2 being ordinal set holds
b1 c= b2
iff
succ b1 c= succ b2;
:: ORDINAL2:th 2
theorem
for b1 being ordinal set holds
union succ b1 = b1;
:: ORDINAL2:th 3
theorem
for b1 being ordinal set holds
succ b1 c= bool b1;
:: ORDINAL2:th 4
theorem
{} is being_limit_ordinal;
:: ORDINAL2:th 5
theorem
for b1 being ordinal set holds
union b1 c= b1;
:: ORDINAL2:funcnot 1 => ORDINAL2:func 1
definition
let a1 be Relation-like Function-like T-Sequence-like set;
func last A1 -> set equals
a1 . union proj1 a1;
end;
:: ORDINAL2:def 1
theorem
for b1 being Relation-like Function-like T-Sequence-like set holds
last b1 = b1 . union proj1 b1;
:: ORDINAL2:th 7
theorem
for b1 being ordinal set
for b2 being Relation-like Function-like T-Sequence-like set
st proj1 b2 = succ b1
holds last b2 = b2 . b1;
:: ORDINAL2:th 9
theorem
for b1 being set holds
On b1 c= b1;
:: ORDINAL2:th 10
theorem
for b1 being ordinal set holds
On b1 = b1;
:: ORDINAL2:th 11
theorem
for b1, b2 being set
st b1 c= b2
holds On b1 c= On b2;
:: ORDINAL2:th 13
theorem
for b1 being set holds
Lim b1 c= b1;
:: ORDINAL2:th 14
theorem
for b1, b2 being set
st b1 c= b2
holds Lim b1 c= Lim b2;
:: ORDINAL2:th 15
theorem
for b1 being set holds
Lim b1 c= On b1;
:: ORDINAL2:th 17
theorem
for b1 being set
st for b2 being set
st b2 in b1
holds b2 is ordinal set
holds meet b1 is ordinal set;
:: ORDINAL2:exreg 1
registration
cluster epsilon-transitive epsilon-connected ordinal being_limit_ordinal set;
end;
:: ORDINAL2:funcnot 2 => ORDINAL2:func 2
definition
let a1 be set;
func inf A1 -> ordinal set equals
meet On a1;
end;
:: ORDINAL2:def 6
theorem
for b1 being set holds
inf b1 = meet On b1;
:: ORDINAL2:funcnot 3 => ORDINAL2:func 3
definition
let a1 be set;
func sup A1 -> ordinal set means
On a1 c= it &
(for b1 being ordinal set
st On a1 c= b1
holds it c= b1);
end;
:: ORDINAL2:def 7
theorem
for b1 being set
for b2 being ordinal set holds
b2 = sup b1
iff
On b1 c= b2 &
(for b3 being ordinal set
st On b1 c= b3
holds b2 c= b3);
:: ORDINAL2:th 22
theorem
for b1 being ordinal set
for b2 being set
st b1 in b2
holds inf b2 c= b1;
:: ORDINAL2:th 23
theorem
for b1 being ordinal set
for b2 being set
st On b2 <> {} &
(for b3 being ordinal set
st b3 in b2
holds b1 c= b3)
holds b1 c= inf b2;
:: ORDINAL2:th 24
theorem
for b1 being ordinal set
for b2, b3 being set
st b1 in b2 & b2 c= b3
holds inf b3 c= inf b2;
:: ORDINAL2:th 25
theorem
for b1 being ordinal set
for b2 being set
st b1 in b2
holds inf b2 in b2;
:: ORDINAL2:th 26
theorem
for b1 being ordinal set holds
sup b1 = b1;
:: ORDINAL2:th 27
theorem
for b1 being ordinal set
for b2 being set
st b1 in b2
holds b1 in sup b2;
:: ORDINAL2:th 28
theorem
for b1 being ordinal set
for b2 being set
st for b3 being ordinal set
st b3 in b2
holds b3 in b1
holds sup b2 c= b1;
:: ORDINAL2:th 29
theorem
for b1 being ordinal set
for b2 being set
st b1 in sup b2
holds ex b3 being ordinal set st
b3 in b2 & b1 c= b3;
:: ORDINAL2:th 30
theorem
for b1, b2 being set
st b1 c= b2
holds sup b1 c= sup b2;
:: ORDINAL2:th 31
theorem
for b1 being ordinal set holds
sup {b1} = succ b1;
:: ORDINAL2:th 32
theorem
for b1 being set holds
inf b1 c= sup b1;
:: ORDINAL2:sch 2
scheme ORDINAL2:sch 2
{F1 -> ordinal set,
F2 -> set}:
ex b1 being Relation-like Function-like T-Sequence-like set st
proj1 b1 = F1() &
(for b2 being ordinal set
st b2 in F1()
holds b1 . b2 = F2(b2))
:: ORDINAL2:attrnot 1 => ORDINAL2:attr 1
definition
let a1 be Relation-like Function-like set;
attr a1 is Ordinal-yielding means
ex b1 being ordinal set st
proj2 a1 c= b1;
end;
:: ORDINAL2:dfs 4
definiens
let a1 be Relation-like Function-like set;
To prove
a1 is Ordinal-yielding
it is sufficient to prove
thus ex b1 being ordinal set st
proj2 a1 c= b1;
:: ORDINAL2:def 8
theorem
for b1 being Relation-like Function-like set holds
b1 is Ordinal-yielding
iff
ex b2 being ordinal set st
proj2 b1 c= b2;
:: ORDINAL2:exreg 2
registration
cluster Relation-like Function-like T-Sequence-like Ordinal-yielding set;
end;
:: ORDINAL2:modenot 1
definition
mode Ordinal-Sequence is Relation-like Function-like T-Sequence-like Ordinal-yielding set;
end;
:: ORDINAL2:condreg 1
registration
let a1 be ordinal set;
cluster -> Ordinal-yielding (T-Sequence of a1);
end;
:: ORDINAL2:funcreg 1
registration
let a1 be Relation-like Function-like T-Sequence-like Ordinal-yielding set;
let a2 be ordinal set;
cluster a1 | a2 -> Relation-like Ordinal-yielding;
end;
:: ORDINAL2:th 34
theorem
for b1 being ordinal set
for b2 being Relation-like Function-like T-Sequence-like Ordinal-yielding set
st b1 in proj1 b2
holds b2 . b1 is ordinal set;
:: ORDINAL2:funcreg 2
registration
let a1 be Relation-like Function-like T-Sequence-like Ordinal-yielding set;
let a2 be ordinal set;
cluster a1 . a2 -> ordinal;
end;
:: ORDINAL2:sch 3
scheme ORDINAL2:sch 3
{F1 -> ordinal set,
F2 -> ordinal set}:
ex b1 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
proj1 b1 = F1() &
(for b2 being ordinal set
st b2 in F1()
holds b1 . b2 = F2(b2))
:: ORDINAL2:sch 4
scheme ORDINAL2:sch 4
{F1 -> ordinal set,
F2 -> set,
F3 -> set,
F4 -> set,
F5 -> Relation-like Function-like T-Sequence-like set,
F6 -> Relation-like Function-like T-Sequence-like set}:
F5() = F6()
provided
proj1 F5() = F1()
and
({} in F1() implies F5() . {} = F2())
and
for b1 being ordinal set
st succ b1 in F1()
holds F5() . succ b1 = F3(b1, F5() . b1)
and
for b1 being ordinal set
st b1 in F1() & b1 <> {} & b1 is being_limit_ordinal
holds F5() . b1 = F4(b1, F5() | b1)
and
proj1 F6() = F1()
and
({} in F1() implies F6() . {} = F2())
and
for b1 being ordinal set
st succ b1 in F1()
holds F6() . succ b1 = F3(b1, F6() . b1)
and
for b1 being ordinal set
st b1 in F1() & b1 <> {} & b1 is being_limit_ordinal
holds F6() . b1 = F4(b1, F6() | b1);
:: ORDINAL2:sch 5
scheme ORDINAL2:sch 5
{F1 -> ordinal set,
F2 -> set,
F3 -> set,
F4 -> set}:
ex b1 being Relation-like Function-like T-Sequence-like set st
proj1 b1 = F1() &
({} in F1() implies b1 . {} = F2()) &
(for b2 being ordinal set
st succ b2 in F1()
holds b1 . succ b2 = F3(b2, b1 . b2)) &
(for b2 being ordinal set
st b2 in F1() & b2 <> {} & b2 is being_limit_ordinal
holds b1 . b2 = F4(b2, b1 | b2))
:: ORDINAL2:sch 6
scheme ORDINAL2:sch 6
{F1 -> Relation-like Function-like T-Sequence-like set,
F2 -> set,
F3 -> ordinal set,
F4 -> set,
F5 -> set,
F6 -> set}:
for b1 being ordinal set
st b1 in proj1 F1()
holds F1() . b1 = F2(b1)
provided
for b1 being ordinal set
for b2 being set holds
b2 = F2(b1)
iff
ex b3 being Relation-like Function-like T-Sequence-like set st
b2 = last b3 &
proj1 b3 = succ b1 &
b3 . {} = F4() &
(for b4 being ordinal set
st succ b4 in succ b1
holds b3 . succ b4 = F5(b4, b3 . b4)) &
(for b4 being ordinal set
st b4 in succ b1 & b4 <> {} & b4 is being_limit_ordinal
holds b3 . b4 = F6(b4, b3 | b4))
and
proj1 F1() = F3()
and
({} in F3() implies F1() . {} = F4())
and
for b1 being ordinal set
st succ b1 in F3()
holds F1() . succ b1 = F5(b1, F1() . b1)
and
for b1 being ordinal set
st b1 in F3() & b1 <> {} & b1 is being_limit_ordinal
holds F1() . b1 = F6(b1, F1() | b1);
:: ORDINAL2:sch 7
scheme ORDINAL2:sch 7
{F1 -> ordinal set,
F2 -> set,
F3 -> set,
F4 -> set}:
(ex b1 being set st
ex b2 being Relation-like Function-like T-Sequence-like set st
b1 = last b2 &
proj1 b2 = succ F1() &
b2 . {} = F2() &
(for b3 being ordinal set
st succ b3 in succ F1()
holds b2 . succ b3 = F3(b3, b2 . b3)) &
(for b3 being ordinal set
st b3 in succ F1() & b3 <> {} & b3 is being_limit_ordinal
holds b2 . b3 = F4(b3, b2 | b3))) &
(for b1, b2 being set
st (ex b3 being Relation-like Function-like T-Sequence-like set st
b1 = last b3 &
proj1 b3 = succ F1() &
b3 . {} = F2() &
(for b4 being ordinal set
st succ b4 in succ F1()
holds b3 . succ b4 = F3(b4, b3 . b4)) &
(for b4 being ordinal set
st b4 in succ F1() & b4 <> {} & b4 is being_limit_ordinal
holds b3 . b4 = F4(b4, b3 | b4))) &
(ex b3 being Relation-like Function-like T-Sequence-like set st
b2 = last b3 &
proj1 b3 = succ F1() &
b3 . {} = F2() &
(for b4 being ordinal set
st succ b4 in succ F1()
holds b3 . succ b4 = F3(b4, b3 . b4)) &
(for b4 being ordinal set
st b4 in succ F1() & b4 <> {} & b4 is being_limit_ordinal
holds b3 . b4 = F4(b4, b3 | b4)))
holds b1 = b2)
:: ORDINAL2:sch 8
scheme ORDINAL2:sch 8
{F1 -> set,
F2 -> set,
F3 -> set,
F4 -> set}:
F1({}) = F2()
provided
for b1 being ordinal set
for b2 being set holds
b2 = F1(b1)
iff
ex b3 being Relation-like Function-like T-Sequence-like set st
b2 = last b3 &
proj1 b3 = succ b1 &
b3 . {} = F2() &
(for b4 being ordinal set
st succ b4 in succ b1
holds b3 . succ b4 = F3(b4, b3 . b4)) &
(for b4 being ordinal set
st b4 in succ b1 & b4 <> {} & b4 is being_limit_ordinal
holds b3 . b4 = F4(b4, b3 | b4));
:: ORDINAL2:sch 9
scheme ORDINAL2:sch 9
{F1 -> set,
F2 -> set,
F3 -> set,
F4 -> set}:
for b1 being ordinal set holds
F4(succ b1) = F2(b1, F4(b1))
provided
for b1 being ordinal set
for b2 being set holds
b2 = F4(b1)
iff
ex b3 being Relation-like Function-like T-Sequence-like set st
b2 = last b3 &
proj1 b3 = succ b1 &
b3 . {} = F1() &
(for b4 being ordinal set
st succ b4 in succ b1
holds b3 . succ b4 = F2(b4, b3 . b4)) &
(for b4 being ordinal set
st b4 in succ b1 & b4 <> {} & b4 is being_limit_ordinal
holds b3 . b4 = F3(b4, b3 | b4));
:: ORDINAL2:sch 10
scheme ORDINAL2:sch 10
{F1 -> Relation-like Function-like T-Sequence-like set,
F2 -> ordinal set,
F3 -> set,
F4 -> set,
F5 -> set,
F6 -> set}:
F3(F2()) = F6(F2(), F1())
provided
for b1 being ordinal set
for b2 being set holds
b2 = F3(b1)
iff
ex b3 being Relation-like Function-like T-Sequence-like set st
b2 = last b3 &
proj1 b3 = succ b1 &
b3 . {} = F4() &
(for b4 being ordinal set
st succ b4 in succ b1
holds b3 . succ b4 = F5(b4, b3 . b4)) &
(for b4 being ordinal set
st b4 in succ b1 & b4 <> {} & b4 is being_limit_ordinal
holds b3 . b4 = F6(b4, b3 | b4))
and
F2() <> {} & F2() is being_limit_ordinal
and
proj1 F1() = F2()
and
for b1 being ordinal set
st b1 in F2()
holds F1() . b1 = F3(b1);
:: ORDINAL2:sch 11
scheme ORDINAL2:sch 11
{F1 -> ordinal set,
F2 -> ordinal set,
F3 -> ordinal set,
F4 -> ordinal set}:
ex b1 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
proj1 b1 = F1() &
({} in F1() implies b1 . {} = F2()) &
(for b2 being ordinal set
st succ b2 in F1()
holds b1 . succ b2 = F3(b2, b1 . b2)) &
(for b2 being ordinal set
st b2 in F1() & b2 <> {} & b2 is being_limit_ordinal
holds b1 . b2 = F4(b2, b1 | b2))
:: ORDINAL2:sch 12
scheme ORDINAL2:sch 12
{F1 -> Relation-like Function-like T-Sequence-like Ordinal-yielding set,
F2 -> ordinal set,
F3 -> ordinal set,
F4 -> ordinal set,
F5 -> ordinal set,
F6 -> ordinal set}:
for b1 being ordinal set
st b1 in proj1 F1()
holds F1() . b1 = F2(b1)
provided
for b1, b2 being ordinal set holds
b2 = F2(b1)
iff
ex b3 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
b2 = last b3 &
proj1 b3 = succ b1 &
b3 . {} = F4() &
(for b4 being ordinal set
st succ b4 in succ b1
holds b3 . succ b4 = F5(b4, b3 . b4)) &
(for b4 being ordinal set
st b4 in succ b1 & b4 <> {} & b4 is being_limit_ordinal
holds b3 . b4 = F6(b4, b3 | b4))
and
proj1 F1() = F3()
and
({} in F3() implies F1() . {} = F4())
and
for b1 being ordinal set
st succ b1 in F3()
holds F1() . succ b1 = F5(b1, F1() . b1)
and
for b1 being ordinal set
st b1 in F3() & b1 <> {} & b1 is being_limit_ordinal
holds F1() . b1 = F6(b1, F1() | b1);
:: ORDINAL2:sch 13
scheme ORDINAL2:sch 13
{F1 -> ordinal set,
F2 -> ordinal set,
F3 -> ordinal set,
F4 -> ordinal set}:
(ex b1 being ordinal set st
ex b2 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
b1 = last b2 &
proj1 b2 = succ F1() &
b2 . {} = F2() &
(for b3 being ordinal set
st succ b3 in succ F1()
holds b2 . succ b3 = F3(b3, b2 . b3)) &
(for b3 being ordinal set
st b3 in succ F1() & b3 <> {} & b3 is being_limit_ordinal
holds b2 . b3 = F4(b3, b2 | b3))) &
(for b1, b2 being ordinal set
st (ex b3 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
b1 = last b3 &
proj1 b3 = succ F1() &
b3 . {} = F2() &
(for b4 being ordinal set
st succ b4 in succ F1()
holds b3 . succ b4 = F3(b4, b3 . b4)) &
(for b4 being ordinal set
st b4 in succ F1() & b4 <> {} & b4 is being_limit_ordinal
holds b3 . b4 = F4(b4, b3 | b4))) &
(ex b3 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
b2 = last b3 &
proj1 b3 = succ F1() &
b3 . {} = F2() &
(for b4 being ordinal set
st succ b4 in succ F1()
holds b3 . succ b4 = F3(b4, b3 . b4)) &
(for b4 being ordinal set
st b4 in succ F1() & b4 <> {} & b4 is being_limit_ordinal
holds b3 . b4 = F4(b4, b3 | b4)))
holds b1 = b2)
:: ORDINAL2:sch 14
scheme ORDINAL2:sch 14
{F1 -> ordinal set,
F2 -> ordinal set,
F3 -> ordinal set,
F4 -> ordinal set}:
F1({}) = F2()
provided
for b1, b2 being ordinal set holds
b2 = F1(b1)
iff
ex b3 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
b2 = last b3 &
proj1 b3 = succ b1 &
b3 . {} = F2() &
(for b4 being ordinal set
st succ b4 in succ b1
holds b3 . succ b4 = F3(b4, b3 . b4)) &
(for b4 being ordinal set
st b4 in succ b1 & b4 <> {} & b4 is being_limit_ordinal
holds b3 . b4 = F4(b4, b3 | b4));
:: ORDINAL2:sch 15
scheme ORDINAL2:sch 15
{F1 -> ordinal set,
F2 -> ordinal set,
F3 -> ordinal set,
F4 -> ordinal set}:
for b1 being ordinal set holds
F4(succ b1) = F2(b1, F4(b1))
provided
for b1, b2 being ordinal set holds
b2 = F4(b1)
iff
ex b3 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
b2 = last b3 &
proj1 b3 = succ b1 &
b3 . {} = F1() &
(for b4 being ordinal set
st succ b4 in succ b1
holds b3 . succ b4 = F2(b4, b3 . b4)) &
(for b4 being ordinal set
st b4 in succ b1 & b4 <> {} & b4 is being_limit_ordinal
holds b3 . b4 = F3(b4, b3 | b4));
:: ORDINAL2:sch 16
scheme ORDINAL2:sch 16
{F1 -> Relation-like Function-like T-Sequence-like Ordinal-yielding set,
F2 -> ordinal set,
F3 -> ordinal set,
F4 -> ordinal set,
F5 -> ordinal set,
F6 -> ordinal set}:
F3(F2()) = F6(F2(), F1())
provided
for b1, b2 being ordinal set holds
b2 = F3(b1)
iff
ex b3 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
b2 = last b3 &
proj1 b3 = succ b1 &
b3 . {} = F4() &
(for b4 being ordinal set
st succ b4 in succ b1
holds b3 . succ b4 = F5(b4, b3 . b4)) &
(for b4 being ordinal set
st b4 in succ b1 & b4 <> {} & b4 is being_limit_ordinal
holds b3 . b4 = F6(b4, b3 | b4))
and
F2() <> {} & F2() is being_limit_ordinal
and
proj1 F1() = F2()
and
for b1 being ordinal set
st b1 in F2()
holds F1() . b1 = F3(b1);
:: ORDINAL2:funcnot 4 => ORDINAL2:func 4
definition
let a1 be Relation-like Function-like T-Sequence-like set;
func sup A1 -> ordinal set equals
sup proj2 a1;
end;
:: ORDINAL2:def 9
theorem
for b1 being Relation-like Function-like T-Sequence-like set holds
sup b1 = sup proj2 b1;
:: ORDINAL2:funcnot 5 => ORDINAL2:func 5
definition
let a1 be Relation-like Function-like T-Sequence-like set;
func inf A1 -> ordinal set equals
inf proj2 a1;
end;
:: ORDINAL2:def 10
theorem
for b1 being Relation-like Function-like T-Sequence-like set holds
inf b1 = inf proj2 b1;
:: ORDINAL2:th 35
theorem
for b1 being Relation-like Function-like T-Sequence-like set holds
sup b1 = sup proj2 b1 & inf b1 = inf proj2 b1;
:: ORDINAL2:funcnot 6 => ORDINAL2:func 6
definition
let a1 be Relation-like Function-like T-Sequence-like set;
func lim_sup A1 -> ordinal set means
ex b1 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
it = inf b1 &
proj1 b1 = proj1 a1 &
(for b2 being ordinal set
st b2 in proj1 a1
holds b1 . b2 = sup proj2 (a1 | ((proj1 a1) \ b2)));
end;
:: ORDINAL2:def 11
theorem
for b1 being Relation-like Function-like T-Sequence-like set
for b2 being ordinal set holds
b2 = lim_sup b1
iff
ex b3 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
b2 = inf b3 &
proj1 b3 = proj1 b1 &
(for b4 being ordinal set
st b4 in proj1 b1
holds b3 . b4 = sup proj2 (b1 | ((proj1 b1) \ b4)));
:: ORDINAL2:funcnot 7 => ORDINAL2:func 7
definition
let a1 be Relation-like Function-like T-Sequence-like set;
func lim_inf A1 -> ordinal set means
ex b1 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
it = sup b1 &
proj1 b1 = proj1 a1 &
(for b2 being ordinal set
st b2 in proj1 a1
holds b1 . b2 = inf proj2 (a1 | ((proj1 a1) \ b2)));
end;
:: ORDINAL2:def 12
theorem
for b1 being Relation-like Function-like T-Sequence-like set
for b2 being ordinal set holds
b2 = lim_inf b1
iff
ex b3 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
b2 = sup b3 &
proj1 b3 = proj1 b1 &
(for b4 being ordinal set
st b4 in proj1 b1
holds b3 . b4 = inf proj2 (b1 | ((proj1 b1) \ b4)));
:: ORDINAL2:prednot 1 => ORDINAL2:pred 1
definition
let a1 be ordinal set;
let a2 be Relation-like Function-like T-Sequence-like Ordinal-yielding set;
pred A1 is_limes_of A2 means
ex b1 being ordinal set st
b1 in proj1 a2 &
(for b2 being ordinal set
st b1 c= b2 & b2 in proj1 a2
holds a2 . b2 = {})
if a1 = {}
otherwise for b1, b2 being ordinal set
st b1 in a1 & a1 in b2
holds ex b3 being ordinal set st
b3 in proj1 a2 &
(for b4 being ordinal set
st b3 c= b4 & b4 in proj1 a2
holds b1 in a2 . b4 & a2 . b4 in b2);
end;
:: ORDINAL2:dfs 9
definiens
let a1 be ordinal set;
let a2 be Relation-like Function-like T-Sequence-like Ordinal-yielding set;
To prove
a1 is_limes_of a2
it is sufficient to prove
per cases;
case a1 = {};
thus ex b1 being ordinal set st
b1 in proj1 a2 &
(for b2 being ordinal set
st b1 c= b2 & b2 in proj1 a2
holds a2 . b2 = {});
end;
case a1 <> {};
thus for b1, b2 being ordinal set
st b1 in a1 & a1 in b2
holds ex b3 being ordinal set st
b3 in proj1 a2 &
(for b4 being ordinal set
st b3 c= b4 & b4 in proj1 a2
holds b1 in a2 . b4 & a2 . b4 in b2);
end;
:: ORDINAL2:def 13
theorem
for b1 being ordinal set
for b2 being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
(b1 = {} implies (b1 is_limes_of b2
iff
ex b3 being ordinal set st
b3 in proj1 b2 &
(for b4 being ordinal set
st b3 c= b4 & b4 in proj1 b2
holds b2 . b4 = {}))) &
(b1 = {} or (b1 is_limes_of b2
iff
for b3, b4 being ordinal set
st b3 in b1 & b1 in b4
holds ex b5 being ordinal set st
b5 in proj1 b2 &
(for b6 being ordinal set
st b5 c= b6 & b6 in proj1 b2
holds b3 in b2 . b6 & b2 . b6 in b4)));
:: ORDINAL2:funcnot 8 => ORDINAL2:func 8
definition
let a1 be Relation-like Function-like T-Sequence-like Ordinal-yielding set;
assume ex b1 being ordinal set st
b1 is_limes_of a1;
func lim A1 -> ordinal set means
it is_limes_of a1;
end;
:: ORDINAL2:def 14
theorem
for b1 being Relation-like Function-like T-Sequence-like Ordinal-yielding set
st ex b2 being ordinal set st
b2 is_limes_of b1
for b2 being ordinal set holds
b2 = lim b1
iff
b2 is_limes_of b1;
:: ORDINAL2:funcnot 9 => ORDINAL2:func 9
definition
let a1 be ordinal set;
let a2 be Relation-like Function-like T-Sequence-like Ordinal-yielding set;
func lim(A1,A2) -> ordinal set equals
lim (a2 | a1);
end;
:: ORDINAL2:def 15
theorem
for b1 being ordinal set
for b2 being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
lim(b1,b2) = lim (b2 | b1);
:: ORDINAL2:attrnot 2 => ORDINAL2:attr 2
definition
let a1 be Relation-like Function-like T-Sequence-like Ordinal-yielding set;
attr a1 is increasing means
for b1, b2 being ordinal set
st b1 in b2 & b2 in proj1 a1
holds a1 . b1 in a1 . b2;
end;
:: ORDINAL2:dfs 12
definiens
let a1 be Relation-like Function-like T-Sequence-like Ordinal-yielding set;
To prove
a1 is increasing
it is sufficient to prove
thus for b1, b2 being ordinal set
st b1 in b2 & b2 in proj1 a1
holds a1 . b1 in a1 . b2;
:: ORDINAL2:def 16
theorem
for b1 being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
b1 is increasing
iff
for b2, b3 being ordinal set
st b2 in b3 & b3 in proj1 b1
holds b1 . b2 in b1 . b3;
:: ORDINAL2:attrnot 3 => ORDINAL2:attr 3
definition
let a1 be Relation-like Function-like T-Sequence-like Ordinal-yielding set;
attr a1 is continuous means
for b1, b2 being ordinal set
st b1 in proj1 a1 & b1 <> {} & b1 is being_limit_ordinal & b2 = a1 . b1
holds b2 is_limes_of a1 | b1;
end;
:: ORDINAL2:dfs 13
definiens
let a1 be Relation-like Function-like T-Sequence-like Ordinal-yielding set;
To prove
a1 is continuous
it is sufficient to prove
thus for b1, b2 being ordinal set
st b1 in proj1 a1 & b1 <> {} & b1 is being_limit_ordinal & b2 = a1 . b1
holds b2 is_limes_of a1 | b1;
:: ORDINAL2:def 17
theorem
for b1 being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
b1 is continuous
iff
for b2, b3 being ordinal set
st b2 in proj1 b1 & b2 <> {} & b2 is being_limit_ordinal & b3 = b1 . b2
holds b3 is_limes_of b1 | b2;
:: ORDINAL2:funcnot 10 => ORDINAL2:func 10
definition
let a1, a2 be ordinal set;
func A1 +^ A2 -> ordinal set means
ex b1 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
it = last b1 &
proj1 b1 = succ a2 &
b1 . {} = a1 &
(for b2 being ordinal set
st succ b2 in succ a2
holds b1 . succ b2 = succ (b1 . b2)) &
(for b2 being ordinal set
st b2 in succ a2 & b2 <> {} & b2 is being_limit_ordinal
holds b1 . b2 = sup (b1 | b2));
end;
:: ORDINAL2:def 18
theorem
for b1, b2, b3 being ordinal set holds
b3 = b1 +^ b2
iff
ex b4 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
b3 = last b4 &
proj1 b4 = succ b2 &
b4 . {} = b1 &
(for b5 being ordinal set
st succ b5 in succ b2
holds b4 . succ b5 = succ (b4 . b5)) &
(for b5 being ordinal set
st b5 in succ b2 & b5 <> {} & b5 is being_limit_ordinal
holds b4 . b5 = sup (b4 | b5));
:: ORDINAL2:funcnot 11 => ORDINAL2:func 11
definition
let a1, a2 be ordinal set;
func A1 *^ A2 -> ordinal set means
ex b1 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
it = last b1 &
proj1 b1 = succ a1 &
b1 . {} = {} &
(for b2 being ordinal set
st succ b2 in succ a1
holds b1 . succ b2 = (b1 . b2) +^ a2) &
(for b2 being ordinal set
st b2 in succ a1 & b2 <> {} & b2 is being_limit_ordinal
holds b1 . b2 = union sup (b1 | b2));
end;
:: ORDINAL2:def 19
theorem
for b1, b2, b3 being ordinal set holds
b3 = b1 *^ b2
iff
ex b4 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
b3 = last b4 &
proj1 b4 = succ b1 &
b4 . {} = {} &
(for b5 being ordinal set
st succ b5 in succ b1
holds b4 . succ b5 = (b4 . b5) +^ b2) &
(for b5 being ordinal set
st b5 in succ b1 & b5 <> {} & b5 is being_limit_ordinal
holds b4 . b5 = union sup (b4 | b5));
:: ORDINAL2:condreg 2
registration
let a1 be ordinal set;
cluster -> ordinal (Element of a1);
end;
:: ORDINAL2:funcnot 12 => ORDINAL2:func 12
definition
let a1, a2 be ordinal set;
func exp(A1,A2) -> ordinal set means
ex b1 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
it = last b1 &
proj1 b1 = succ a2 &
b1 . {} = 1 &
(for b2 being ordinal set
st succ b2 in succ a2
holds b1 . succ b2 = a1 *^ (b1 . b2)) &
(for b2 being ordinal set
st b2 in succ a2 & b2 <> {} & b2 is being_limit_ordinal
holds b1 . b2 = lim (b1 | b2));
end;
:: ORDINAL2:def 20
theorem
for b1, b2, b3 being ordinal set holds
b3 = exp(b1,b2)
iff
ex b4 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
b3 = last b4 &
proj1 b4 = succ b2 &
b4 . {} = 1 &
(for b5 being ordinal set
st succ b5 in succ b2
holds b4 . succ b5 = b1 *^ (b4 . b5)) &
(for b5 being ordinal set
st b5 in succ b2 & b5 <> {} & b5 is being_limit_ordinal
holds b4 . b5 = lim (b4 | b5));
:: ORDINAL2:th 44
theorem
for b1 being ordinal set holds
b1 +^ {} = b1;
:: ORDINAL2:th 45
theorem
for b1, b2 being ordinal set holds
b1 +^ succ b2 = succ (b1 +^ b2);
:: ORDINAL2:th 46
theorem
for b1, b2 being ordinal set
st b1 <> {} & b1 is being_limit_ordinal
for b3 being Relation-like Function-like T-Sequence-like Ordinal-yielding set
st proj1 b3 = b1 &
(for b4 being ordinal set
st b4 in b1
holds b3 . b4 = b2 +^ b4)
holds b2 +^ b1 = sup b3;
:: ORDINAL2:th 47
theorem
for b1 being ordinal set holds
{} +^ b1 = b1;
:: ORDINAL2:th 48
theorem
for b1 being ordinal set holds
b1 +^ 1 = succ b1;
:: ORDINAL2:th 49
theorem
for b1, b2, b3 being ordinal set
st b1 in b2
holds b3 +^ b1 in b3 +^ b2;
:: ORDINAL2:th 50
theorem
for b1, b2, b3 being ordinal set
st b1 c= b2
holds b3 +^ b1 c= b3 +^ b2;
:: ORDINAL2:th 51
theorem
for b1, b2, b3 being ordinal set
st b1 c= b2
holds b1 +^ b3 c= b2 +^ b3;
:: ORDINAL2:th 52
theorem
for b1 being ordinal set holds
{} *^ b1 = {};
:: ORDINAL2:th 53
theorem
for b1, b2 being ordinal set holds
(succ b1) *^ b2 = (b1 *^ b2) +^ b2;
:: ORDINAL2:th 54
theorem
for b1, b2 being ordinal set
st b1 <> {} & b1 is being_limit_ordinal
for b3 being Relation-like Function-like T-Sequence-like Ordinal-yielding set
st proj1 b3 = b1 &
(for b4 being ordinal set
st b4 in b1
holds b3 . b4 = b4 *^ b2)
holds b1 *^ b2 = union sup b3;
:: ORDINAL2:th 55
theorem
for b1 being ordinal set holds
b1 *^ {} = {};
:: ORDINAL2:th 56
theorem
for b1 being ordinal set holds
1 *^ b1 = b1 & b1 *^ 1 = b1;
:: ORDINAL2:th 57
theorem
for b1, b2, b3 being ordinal set
st b1 <> {} & b2 in b3
holds b2 *^ b1 in b3 *^ b1;
:: ORDINAL2:th 58
theorem
for b1, b2, b3 being ordinal set
st b1 c= b2
holds b1 *^ b3 c= b2 *^ b3;
:: ORDINAL2:th 59
theorem
for b1, b2, b3 being ordinal set
st b1 c= b2
holds b3 *^ b1 c= b3 *^ b2;
:: ORDINAL2:th 60
theorem
for b1 being ordinal set holds
exp(b1,{}) = 1;
:: ORDINAL2:th 61
theorem
for b1, b2 being ordinal set holds
exp(b1,succ b2) = b1 *^ exp(b1,b2);
:: ORDINAL2:th 62
theorem
for b1, b2 being ordinal set
st b1 <> {} & b1 is being_limit_ordinal
for b3 being Relation-like Function-like T-Sequence-like Ordinal-yielding set
st proj1 b3 = b1 &
(for b4 being ordinal set
st b4 in b1
holds b3 . b4 = exp(b2,b4))
holds exp(b2,b1) = lim b3;
:: ORDINAL2:th 63
theorem
for b1 being ordinal set holds
exp(b1,1) = b1 & exp(1,b1) = 1;
:: ORDINAL2:th 65
theorem
for b1 being ordinal set holds
ex b2, b3 being ordinal set st
b2 is being_limit_ordinal & b3 is natural & b1 = b2 +^ b3;
:: ORDINAL2:funcreg 3
registration
let a1 be set;
let a2 be ordinal set;
cluster a1 --> a2 -> Ordinal-yielding;
end;
:: ORDINAL2:prednot 2 => ORDINAL2:pred 2
definition
let a1, a2 be ordinal set;
pred A1 is_cofinal_with A2 means
ex b1 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
proj1 b1 = a2 & proj2 b1 c= a1 & b1 is increasing & a1 = sup b1;
reflexivity;
:: for a1 being ordinal set holds
:: a1 is_cofinal_with a1;
end;
:: ORDINAL2:dfs 17
definiens
let a1, a2 be ordinal set;
To prove
a1 is_cofinal_with a2
it is sufficient to prove
thus ex b1 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
proj1 b1 = a2 & proj2 b1 c= a1 & b1 is increasing & a1 = sup b1;
:: ORDINAL2:def 21
theorem
for b1, b2 being ordinal set holds
b1 is_cofinal_with b2
iff
ex b3 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
proj1 b3 = b2 & proj2 b3 c= b1 & b3 is increasing & b1 = sup b3;
:: ORDINAL2:th 66
theorem
for b1 being Relation-like Function-like T-Sequence-like Ordinal-yielding set
for b2 being set
st b2 in proj2 b1
holds b2 is ordinal set;
:: ORDINAL2:th 67
theorem
for b1 being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
proj2 b1 c= sup b1;
:: ORDINAL2:th 68
theorem
for b1 being ordinal set
st b1 is_cofinal_with {}
holds b1 = {};
:: ORDINAL2:sch 17
scheme ORDINAL2:sch 17
{F1 -> natural set}:
P1[F1()]
provided
P1[{}]
and
for b1 being natural set
st P1[b1]
holds P1[succ b1];
:: ORDINAL2:funcreg 4
registration
let a1, a2 be natural set;
cluster a1 +^ a2 -> ordinal natural;
end;
:: ORDINAL2:funcreg 5
registration
let a1, a2 be set;
let a3, a4 be natural set;
cluster IFEQ(a1,a2,a3,a4) -> natural;
end;