Article HILBERT3, MML version 4.99.1005
:: HILBERT3:th 1
theorem
for b1 being integer set holds
b1 is even
iff
b1 - 1 is odd;
:: HILBERT3:th 2
theorem
for b1 being integer set holds
b1 is odd
iff
b1 - 1 is even;
:: HILBERT3:th 3
theorem
for b1 being trivial set
for b2 being set
st b2 in b1
for b3 being Function-like quasi_total Relation of b1,b1 holds
b2 is_a_fixpoint_of b3;
:: HILBERT3:condreg 1
registration
let a1, a2, a3 be set;
cluster Function-like quasi_total -> Function-yielding (Relation of a1,Funcs(a2,a3));
end;
:: HILBERT3:th 4
theorem
for b1 being Relation-like Function-like Function-yielding set holds
SubFuncs proj2 b1 = proj2 b1;
:: HILBERT3:th 5
theorem
for b1, b2, b3 being set
for b4 being Relation-like Function-like set
st b3 in b1 & b4 in Funcs(b1,b2)
holds b4 . b3 in b2;
:: HILBERT3:th 6
theorem
for b1, b2, b3 being set
st (b3 = {} & b2 <> {} implies b1 = {})
for b4 being Function-like quasi_total Relation of b1,Funcs(b2,b3) holds
doms b4 = b1 --> b2;
:: HILBERT3:funcreg 1
registration
cluster {} -> Function-yielding;
end;
:: HILBERT3:th 7
theorem
for b1 being set holds
{} . b1 = {};
:: HILBERT3:condreg 2
registration
let a1 be set;
let a2 be functional set;
cluster Function-like quasi_total -> Function-yielding (Relation of a1,a2);
end;
:: HILBERT3:th 8
theorem
for b1 being set
for b2 being Element of bool b1 holds
((0,1)-->(1,0)) * chi(b2,b1) = chi(b2 `,b1);
:: HILBERT3:th 9
theorem
for b1 being set
for b2 being Element of bool b1 holds
((0,1)-->(1,0)) * chi(b2 `,b1) = chi(b2,b1);
:: HILBERT3:th 10
theorem
for b1, b2, b3, b4, b5, b6 being set
st b1 <> b2 & (b1,b2)-->(b3,b4) = (b1,b2)-->(b5,b6)
holds b3 = b5 & b4 = b6;
:: HILBERT3:th 11
theorem
for b1, b2, b3, b4, b5, b6 being set
st b1 <> b2 & b3 in b5 & b4 in b6
holds (b1,b2)-->(b3,b4) in product ((b1,b2)-->(b5,b6));
:: HILBERT3:th 12
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of 2,b1 holds
ex b3, b4 being Element of b1 st
b2 = (0,1)-->(b3,b4);
:: HILBERT3:th 13
theorem
for b1, b2, b3, b4 being set
st b1 <> b2
holds ((b1,b2)-->(b2,b1)) * ((b1,b2)-->(b3,b4)) = (b1,b2)-->(b4,b3);
:: HILBERT3:th 14
theorem
for b1, b2, b3, b4 being set
for b5 being Relation-like Function-like set
st b1 <> b2 & b3 in proj1 b5 & b4 in proj1 b5
holds ((b1,b2)-->(b3,b4)) * b5 = (b1,b2)-->(b5 . b3,b5 . b4);
:: HILBERT3:funcreg 2
registration
let a1, a2 be Relation-like Function-like one-to-one set;
cluster [:a1,a2:] -> Relation-like Function-like one-to-one;
end;
:: HILBERT3:th 15
theorem
for b1, b2 being non empty set
for b3, b4 being set
for b5 being Function-like quasi_total Relation of b3,b1
for b6 being Function-like quasi_total Relation of b4,b2 holds
(pr1(b1,b2)) * [:b5,b6:] = b5 * pr1(b3,b4);
:: HILBERT3:th 16
theorem
for b1, b2 being non empty set
for b3, b4 being set
for b5 being Function-like quasi_total Relation of b3,b1
for b6 being Function-like quasi_total Relation of b4,b2 holds
(pr2(b1,b2)) * [:b5,b6:] = b6 * pr2(b3,b4);
:: HILBERT3:th 17
theorem
for b1 being Relation-like Function-like set holds
{} .. b1 = {};
:: HILBERT3:th 18
theorem
for b1 being Relation-like Function-like Function-yielding set
for b2, b3 being Relation-like Function-like set holds
b3 * (b1 .. b2) = (b3 * b1) .. (b3 * b2);
:: HILBERT3:th 19
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like quasi_total Relation of b2,Funcs({},b1)
for b4 being Function-like quasi_total Relation of b2,{} holds
proj2 (b3 .. b4) = {{}};
:: HILBERT3:th 20
theorem
for b1, b2, b3 being set
st (b2 = {} implies b1 = {})
for b4 being Function-like quasi_total Relation of b1,Funcs(b2,b3)
for b5 being Function-like quasi_total Relation of b1,b2 holds
proj2 (b4 .. b5) c= b3;
:: HILBERT3:th 21
theorem
for b1, b2, b3 being set
st (b3 = {} & b2 <> {} implies b1 = {})
for b4 being Function-like quasi_total Relation of b1,Funcs(b2,b3) holds
proj1 Frege b4 = Funcs(b1,b2);
:: HILBERT3:th 23
theorem
for b1, b2, b3 being set
st (b3 = {} & b2 <> {} implies b1 = {})
for b4 being Function-like quasi_total Relation of b1,Funcs(b2,b3) holds
proj2 Frege b4 c= Funcs(b1,b3);
:: HILBERT3:th 24
theorem
for b1, b2, b3 being set
st (b3 = {} & b2 <> {} implies b1 = {})
for b4 being Function-like quasi_total Relation of b1,Funcs(b2,b3) holds
Frege b4 is Function-like quasi_total Relation of Funcs(b1,b2),Funcs(b1,b3);
:: HILBERT3:th 25
theorem
for b1, b2 being set
for b3 being Function-like quasi_total bijective Relation of b1,b1
for b4 being Function-like quasi_total bijective Relation of b2,b2 holds
[:b3,b4:] is bijective([:b1,b2:], [:b1,b2:]);
:: HILBERT3:funcnot 1 => HILBERT3:func 1
definition
let a1, a2 be non empty set;
let a3 be Function-like quasi_total bijective Relation of a1,a1;
let a4 be Function-like quasi_total Relation of a2,a2;
func A3 => A4 -> Function-like quasi_total Relation of Funcs(a1,a2),Funcs(a1,a2) means
for b1 being Function-like quasi_total Relation of a1,a2 holds
it . b1 = (a4 * b1) * (a3 ");
end;
:: HILBERT3:def 1
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total bijective Relation of b1,b1
for b4 being Function-like quasi_total Relation of b2,b2
for b5 being Function-like quasi_total Relation of Funcs(b1,b2),Funcs(b1,b2) holds
b5 = b3 => b4
iff
for b6 being Function-like quasi_total Relation of b1,b2 holds
b5 . b6 = (b4 * b6) * (b3 ");
:: HILBERT3:funcreg 3
registration
let a1, a2 be non empty set;
let a3 be Function-like quasi_total bijective Relation of a1,a1;
let a4 be Function-like quasi_total bijective Relation of a2,a2;
cluster a3 => a4 -> Function-like quasi_total bijective;
end;
:: HILBERT3:th 26
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total bijective Relation of b1,b1
for b4 being Function-like quasi_total bijective Relation of b2,b2
for b5 being Function-like quasi_total Relation of b1,b2 holds
(b3 => b4) " . b5 = (b4 " * b5) * b3;
:: HILBERT3:th 27
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total bijective Relation of b1,b1
for b4 being Function-like quasi_total bijective Relation of b2,b2 holds
(b3 => b4) " = b3 " => (b4 ");
:: HILBERT3:th 28
theorem
for b1, b2, b3 being non empty set
for b4 being Function-like quasi_total Relation of b1,Funcs(b2,b3)
for b5 being Function-like quasi_total Relation of b1,b2
for b6 being Function-like quasi_total bijective Relation of b2,b2
for b7 being Function-like quasi_total bijective Relation of b3,b3 holds
((b6 => b7) * b4) .. (b6 * b5) = (b4 .. b5) * b7;
:: HILBERT3:modenot 1
definition
mode SetValuation is non-empty ManySortedSet of NAT;
end;
:: HILBERT3:funcnot 2 => HILBERT3:func 2
definition
let a1 be non-empty ManySortedSet of NAT;
func SetVal A1 -> ManySortedSet of HP-WFF means
it . VERUM = 1 &
(for b1 being Element of NAT holds
it . prop b1 = a1 . b1) &
(for b1, b2 being Element of HP-WFF holds
it . (b1 '&' b2) = [:it . b1,it . b2:] &
it . (b1 => b2) = Funcs(it . b1,it . b2));
end;
:: HILBERT3:def 2
theorem
for b1 being non-empty ManySortedSet of NAT
for b2 being ManySortedSet of HP-WFF holds
b2 = SetVal b1
iff
b2 . VERUM = 1 &
(for b3 being Element of NAT holds
b2 . prop b3 = b1 . b3) &
(for b3, b4 being Element of HP-WFF holds
b2 . (b3 '&' b4) = [:b2 . b3,b2 . b4:] &
b2 . (b3 => b4) = Funcs(b2 . b3,b2 . b4));
:: HILBERT3:funcnot 3 => HILBERT3:func 3
definition
let a1 be non-empty ManySortedSet of NAT;
let a2 be Element of HP-WFF;
func SetVal(A1,A2) -> set equals
(SetVal a1) . a2;
end;
:: HILBERT3:def 3
theorem
for b1 being non-empty ManySortedSet of NAT
for b2 being Element of HP-WFF holds
SetVal(b1,b2) = (SetVal b1) . b2;
:: HILBERT3:funcreg 4
registration
let a1 be non-empty ManySortedSet of NAT;
let a2 be Element of HP-WFF;
cluster SetVal(a1,a2) -> non empty;
end;
:: HILBERT3:th 29
theorem
for b1 being non-empty ManySortedSet of NAT holds
SetVal(b1,VERUM) = 1;
:: HILBERT3:th 30
theorem
for b1 being Element of NAT
for b2 being non-empty ManySortedSet of NAT holds
SetVal(b2,prop b1) = b2 . b1;
:: HILBERT3:th 31
theorem
for b1, b2 being Element of HP-WFF
for b3 being non-empty ManySortedSet of NAT holds
SetVal(b3,b1 '&' b2) = [:SetVal(b3,b1),SetVal(b3,b2):];
:: HILBERT3:th 32
theorem
for b1, b2 being Element of HP-WFF
for b3 being non-empty ManySortedSet of NAT holds
SetVal(b3,b1 => b2) = Funcs(SetVal(b3,b1),SetVal(b3,b2));
:: HILBERT3:funcreg 5
registration
let a1 be non-empty ManySortedSet of NAT;
let a2, a3 be Element of HP-WFF;
cluster SetVal(a1,a2 => a3) -> functional;
end;
:: HILBERT3:condreg 3
registration
let a1 be non-empty ManySortedSet of NAT;
let a2, a3, a4 be Element of HP-WFF;
cluster -> Function-yielding (Element of SetVal(a1,a2 => (a3 => a4)));
end;
:: HILBERT3:exreg 1
registration
let a1 be non-empty ManySortedSet of NAT;
let a2, a3, a4 be Element of HP-WFF;
cluster Relation-like Function-like non empty quasi_total Function-yielding total Relation of SetVal(a1,a2 => a3),SetVal(a1,a2 => a4);
end;
:: HILBERT3:exreg 2
registration
let a1 be non-empty ManySortedSet of NAT;
let a2, a3, a4 be Element of HP-WFF;
cluster Relation-like Function-like Function-yielding Element of SetVal(a1,a2 => (a3 => a4));
end;
:: HILBERT3:modenot 2 => HILBERT3:mode 1
definition
let a1 be non-empty ManySortedSet of NAT;
mode Permutation of A1 -> Relation-like Function-like set means
proj1 it = NAT &
(for b1 being Element of NAT holds
it . b1 is Function-like quasi_total bijective Relation of a1 . b1,a1 . b1);
end;
:: HILBERT3:dfs 4
definiens
let a1 be non-empty ManySortedSet of NAT;
let a2 be Relation-like Function-like set;
To prove
a2 is Permutation of a1
it is sufficient to prove
thus proj1 a2 = NAT &
(for b1 being Element of NAT holds
a2 . b1 is Function-like quasi_total bijective Relation of a1 . b1,a1 . b1);
:: HILBERT3:def 4
theorem
for b1 being non-empty ManySortedSet of NAT
for b2 being Relation-like Function-like set holds
b2 is Permutation of b1
iff
proj1 b2 = NAT &
(for b3 being Element of NAT holds
b2 . b3 is Function-like quasi_total bijective Relation of b1 . b3,b1 . b3);
:: HILBERT3:funcnot 4 => HILBERT3:func 4
definition
let a1 be non-empty ManySortedSet of NAT;
let a2 be Permutation of a1;
func Perm A2 -> ManySortedFunction of SetVal a1,SetVal a1 means
it . VERUM = id 1 &
(for b1 being Element of NAT holds
it . prop b1 = a2 . b1) &
(for b1, b2 being Element of HP-WFF holds
ex b3 being Function-like quasi_total bijective Relation of SetVal(a1,b1),SetVal(a1,b1) st
ex b4 being Function-like quasi_total bijective Relation of SetVal(a1,b2),SetVal(a1,b2) st
b3 = it . b1 & b4 = it . b2 & it . (b1 '&' b2) = [:b3,b4:] & it . (b1 => b2) = b3 => b4);
end;
:: HILBERT3:def 5
theorem
for b1 being non-empty ManySortedSet of NAT
for b2 being Permutation of b1
for b3 being ManySortedFunction of SetVal b1,SetVal b1 holds
b3 = Perm b2
iff
b3 . VERUM = id 1 &
(for b4 being Element of NAT holds
b3 . prop b4 = b2 . b4) &
(for b4, b5 being Element of HP-WFF holds
ex b6 being Function-like quasi_total bijective Relation of SetVal(b1,b4),SetVal(b1,b4) st
ex b7 being Function-like quasi_total bijective Relation of SetVal(b1,b5),SetVal(b1,b5) st
b6 = b3 . b4 & b7 = b3 . b5 & b3 . (b4 '&' b5) = [:b6,b7:] & b3 . (b4 => b5) = b6 => b7);
:: HILBERT3:funcnot 5 => HILBERT3:func 5
definition
let a1 be non-empty ManySortedSet of NAT;
let a2 be Permutation of a1;
let a3 be Element of HP-WFF;
func Perm(A2,A3) -> Function-like quasi_total Relation of SetVal(a1,a3),SetVal(a1,a3) equals
(Perm a2) . a3;
end;
:: HILBERT3:def 6
theorem
for b1 being non-empty ManySortedSet of NAT
for b2 being Permutation of b1
for b3 being Element of HP-WFF holds
Perm(b2,b3) = (Perm b2) . b3;
:: HILBERT3:th 33
theorem
for b1 being non-empty ManySortedSet of NAT
for b2 being Permutation of b1 holds
Perm(b2,VERUM) = id SetVal(b1,VERUM);
:: HILBERT3:th 34
theorem
for b1 being Element of NAT
for b2 being non-empty ManySortedSet of NAT
for b3 being Permutation of b2 holds
Perm(b3,prop b1) = b3 . b1;
:: HILBERT3:th 35
theorem
for b1, b2 being Element of HP-WFF
for b3 being non-empty ManySortedSet of NAT
for b4 being Permutation of b3 holds
Perm(b4,b1 '&' b2) = [:Perm(b4,b1),Perm(b4,b2):];
:: HILBERT3:th 36
theorem
for b1, b2 being Element of HP-WFF
for b3 being non-empty ManySortedSet of NAT
for b4 being Permutation of b3
for b5 being Function-like quasi_total bijective Relation of SetVal(b3,b1),SetVal(b3,b1)
for b6 being Function-like quasi_total bijective Relation of SetVal(b3,b2),SetVal(b3,b2)
st b5 = Perm(b4,b1) & b6 = Perm(b4,b2)
holds Perm(b4,b1 => b2) = b5 => b6;
:: HILBERT3:funcreg 6
registration
let a1 be non-empty ManySortedSet of NAT;
let a2 be Permutation of a1;
let a3 be Element of HP-WFF;
cluster Perm(a2,a3) -> Function-like quasi_total bijective;
end;
:: HILBERT3:th 37
theorem
for b1, b2 being Element of HP-WFF
for b3 being non-empty ManySortedSet of NAT
for b4 being Permutation of b3
for b5 being Function-like quasi_total Relation of SetVal(b3,b1),SetVal(b3,b2) holds
(Perm(b4,b1 => b2)) . b5 = ((Perm(b4,b2)) * b5) * ((Perm(b4,b1)) ");
:: HILBERT3:th 38
theorem
for b1, b2 being Element of HP-WFF
for b3 being non-empty ManySortedSet of NAT
for b4 being Permutation of b3
for b5 being Function-like quasi_total Relation of SetVal(b3,b1),SetVal(b3,b2) holds
(Perm(b4,b1 => b2)) " . b5 = ((Perm(b4,b2)) " * b5) * Perm(b4,b1);
:: HILBERT3:th 39
theorem
for b1, b2 being Element of HP-WFF
for b3 being non-empty ManySortedSet of NAT
for b4 being Permutation of b3
for b5, b6 being Function-like quasi_total Relation of SetVal(b3,b1),SetVal(b3,b2)
st b5 = (Perm(b4,b1 => b2)) . b6
holds (Perm(b4,b2)) * b6 = b5 * Perm(b4,b1);
:: HILBERT3:th 40
theorem
for b1, b2 being Element of HP-WFF
for b3 being non-empty ManySortedSet of NAT
for b4 being Permutation of b3
for b5 being set
st b5 is_a_fixpoint_of Perm(b4,b1)
for b6 being Relation-like Function-like set
st b6 is_a_fixpoint_of Perm(b4,b1 => b2)
holds b6 . b5 is_a_fixpoint_of Perm(b4,b2);
:: HILBERT3:attrnot 1 => HILBERT3:attr 1
definition
let a1 be Element of HP-WFF;
attr a1 is canonical means
for b1 being non-empty ManySortedSet of NAT holds
ex b2 being set st
for b3 being Permutation of b1 holds
b2 is_a_fixpoint_of Perm(b3,a1);
end;
:: HILBERT3:dfs 7
definiens
let a1 be Element of HP-WFF;
To prove
a1 is canonical
it is sufficient to prove
thus for b1 being non-empty ManySortedSet of NAT holds
ex b2 being set st
for b3 being Permutation of b1 holds
b2 is_a_fixpoint_of Perm(b3,a1);
:: HILBERT3:def 7
theorem
for b1 being Element of HP-WFF holds
b1 is canonical
iff
for b2 being non-empty ManySortedSet of NAT holds
ex b3 being set st
for b4 being Permutation of b2 holds
b3 is_a_fixpoint_of Perm(b4,b1);
:: HILBERT3:funcreg 7
registration
cluster VERUM -> canonical;
end;
:: HILBERT3:th 41
theorem
for b1, b2 being Element of HP-WFF holds
b1 => (b2 => b1) is canonical;
:: HILBERT3:th 42
theorem
for b1, b2, b3 being Element of HP-WFF holds
(b1 => (b2 => b3)) => ((b1 => b2) => (b1 => b3)) is canonical;
:: HILBERT3:th 43
theorem
for b1, b2 being Element of HP-WFF holds
(b1 '&' b2) => b1 is canonical;
:: HILBERT3:th 44
theorem
for b1, b2 being Element of HP-WFF holds
(b1 '&' b2) => b2 is canonical;
:: HILBERT3:th 45
theorem
for b1, b2 being Element of HP-WFF holds
b1 => (b2 => (b1 '&' b2)) is canonical;
:: HILBERT3:th 46
theorem
for b1, b2 being Element of HP-WFF
st b1 is canonical & b1 => b2 is canonical
holds b2 is canonical;
:: HILBERT3:th 47
theorem
for b1 being Element of HP-WFF
st b1 in HP_TAUT
holds b1 is canonical;
:: HILBERT3:exreg 3
registration
cluster Relation-like Function-like FinSequence-like canonical Element of HP-WFF;
end;
:: HILBERT3:attrnot 2 => HILBERT3:attr 2
definition
let a1 be Element of HP-WFF;
attr a1 is pseudo-canonical means
for b1 being non-empty ManySortedSet of NAT
for b2 being Permutation of b1 holds
ex b3 being set st
b3 is_a_fixpoint_of Perm(b2,a1);
end;
:: HILBERT3:dfs 8
definiens
let a1 be Element of HP-WFF;
To prove
a1 is pseudo-canonical
it is sufficient to prove
thus for b1 being non-empty ManySortedSet of NAT
for b2 being Permutation of b1 holds
ex b3 being set st
b3 is_a_fixpoint_of Perm(b2,a1);
:: HILBERT3:def 8
theorem
for b1 being Element of HP-WFF holds
b1 is pseudo-canonical
iff
for b2 being non-empty ManySortedSet of NAT
for b3 being Permutation of b2 holds
ex b4 being set st
b4 is_a_fixpoint_of Perm(b3,b1);
:: HILBERT3:condreg 4
registration
cluster canonical -> pseudo-canonical (Element of HP-WFF);
end;
:: HILBERT3:th 48
theorem
for b1, b2 being Element of HP-WFF holds
b1 => (b2 => b1) is pseudo-canonical;
:: HILBERT3:th 49
theorem
for b1, b2, b3 being Element of HP-WFF holds
(b1 => (b2 => b3)) => ((b1 => b2) => (b1 => b3)) is pseudo-canonical;
:: HILBERT3:th 50
theorem
for b1, b2 being Element of HP-WFF holds
(b1 '&' b2) => b1 is pseudo-canonical;
:: HILBERT3:th 51
theorem
for b1, b2 being Element of HP-WFF holds
(b1 '&' b2) => b2 is pseudo-canonical;
:: HILBERT3:th 52
theorem
for b1, b2 being Element of HP-WFF holds
b1 => (b2 => (b1 '&' b2)) is pseudo-canonical;
:: HILBERT3:th 53
theorem
for b1, b2 being Element of HP-WFF
st b1 is pseudo-canonical & b1 => b2 is pseudo-canonical
holds b2 is pseudo-canonical;
:: HILBERT3:th 54
theorem
for b1, b2 being Element of HP-WFF
for b3 being non-empty ManySortedSet of NAT
for b4 being Permutation of b3
st (ex b5 being set st
b5 is_a_fixpoint_of Perm(b4,b1)) &
(for b5 being set holds
not b5 is_a_fixpoint_of Perm(b4,b2))
holds b1 => b2 is not pseudo-canonical;
:: HILBERT3:th 55
theorem
(((prop 0) => prop 1) => prop 0) => prop 0 is not pseudo-canonical;