Article WELLORD1, MML version 4.99.1005
:: WELLORD1:funcnot 1 => WELLORD1:func 1
definition
let a1 be Relation-like set;
let a2 be set;
func A1 -Seg A2 -> set means
for b1 being set holds
b1 in it
iff
b1 <> a2 & [b1,a2] in a1;
end;
:: WELLORD1:def 1
theorem
for b1 being Relation-like set
for b2, b3 being set holds
b3 = b1 -Seg b2
iff
for b4 being set holds
b4 in b3
iff
b4 <> b2 & [b4,b2] in b1;
:: WELLORD1:th 2
theorem
for b1 being set
for b2 being Relation-like set
st not b1 in field b2
holds b2 -Seg b1 = {};
:: WELLORD1:attrnot 1 => WELLORD1:attr 1
definition
let a1 be Relation-like set;
attr a1 is well_founded means
for b1 being set
st b1 c= field a1 & b1 <> {}
holds ex b2 being set st
b2 in b1 & a1 -Seg b2 misses b1;
end;
:: WELLORD1:dfs 2
definiens
let a1 be Relation-like set;
To prove
a1 is well_founded
it is sufficient to prove
thus for b1 being set
st b1 c= field a1 & b1 <> {}
holds ex b2 being set st
b2 in b1 & a1 -Seg b2 misses b1;
:: WELLORD1:def 2
theorem
for b1 being Relation-like set holds
b1 is well_founded
iff
for b2 being set
st b2 c= field b1 & b2 <> {}
holds ex b3 being set st
b3 in b2 & b1 -Seg b3 misses b2;
:: WELLORD1:prednot 1 => WELLORD1:pred 1
definition
let a1 be Relation-like set;
let a2 be set;
pred A1 is_well_founded_in A2 means
for b1 being set
st b1 c= a2 & b1 <> {}
holds ex b2 being set st
b2 in b1 & a1 -Seg b2 misses b1;
end;
:: WELLORD1:dfs 3
definiens
let a1 be Relation-like set;
let a2 be set;
To prove
a1 is_well_founded_in a2
it is sufficient to prove
thus for b1 being set
st b1 c= a2 & b1 <> {}
holds ex b2 being set st
b2 in b1 & a1 -Seg b2 misses b1;
:: WELLORD1:def 3
theorem
for b1 being Relation-like set
for b2 being set holds
b1 is_well_founded_in b2
iff
for b3 being set
st b3 c= b2 & b3 <> {}
holds ex b4 being set st
b4 in b3 & b1 -Seg b4 misses b3;
:: WELLORD1:th 5
theorem
for b1 being Relation-like set holds
b1 is well_founded
iff
b1 is_well_founded_in field b1;
:: WELLORD1:attrnot 2 => WELLORD1:attr 2
definition
let a1 be Relation-like set;
attr a1 is well-ordering means
a1 is reflexive & a1 is transitive & a1 is antisymmetric & a1 is connected & a1 is well_founded;
end;
:: WELLORD1:dfs 4
definiens
let a1 be Relation-like set;
To prove
a1 is well-ordering
it is sufficient to prove
thus a1 is reflexive & a1 is transitive & a1 is antisymmetric & a1 is connected & a1 is well_founded;
:: WELLORD1:def 4
theorem
for b1 being Relation-like set holds
b1 is well-ordering
iff
b1 is reflexive & b1 is transitive & b1 is antisymmetric & b1 is connected & b1 is well_founded;
:: WELLORD1:prednot 2 => WELLORD1:pred 2
definition
let a1 be Relation-like set;
let a2 be set;
pred A1 well_orders A2 means
a1 is_reflexive_in a2 & a1 is_transitive_in a2 & a1 is_antisymmetric_in a2 & a1 is_connected_in a2 & a1 is_well_founded_in a2;
end;
:: WELLORD1:dfs 5
definiens
let a1 be Relation-like set;
let a2 be set;
To prove
a1 well_orders a2
it is sufficient to prove
thus a1 is_reflexive_in a2 & a1 is_transitive_in a2 & a1 is_antisymmetric_in a2 & a1 is_connected_in a2 & a1 is_well_founded_in a2;
:: WELLORD1:def 5
theorem
for b1 being Relation-like set
for b2 being set holds
b1 well_orders b2
iff
b1 is_reflexive_in b2 & b1 is_transitive_in b2 & b1 is_antisymmetric_in b2 & b1 is_connected_in b2 & b1 is_well_founded_in b2;
:: WELLORD1:th 8
theorem
for b1 being Relation-like set holds
b1 well_orders field b1
iff
b1 is well-ordering;
:: WELLORD1:th 9
theorem
for b1 being set
for b2 being Relation-like set
st b2 well_orders b1
for b3 being set
st b3 c= b1 & b3 <> {}
holds ex b4 being set st
b4 in b3 &
(for b5 being set
st b5 in b3
holds [b4,b5] in b2);
:: WELLORD1:th 10
theorem
for b1 being Relation-like set
st b1 is well-ordering
for b2 being set
st b2 c= field b1 & b2 <> {}
holds ex b3 being set st
b3 in b2 &
(for b4 being set
st b4 in b2
holds [b3,b4] in b1);
:: WELLORD1:th 11
theorem
for b1 being Relation-like set
st b1 is well-ordering & field b1 <> {}
holds ex b2 being set st
b2 in field b1 &
(for b3 being set
st b3 in field b1
holds [b2,b3] in b1);
:: WELLORD1:th 12
theorem
for b1 being Relation-like set
st b1 is well-ordering & field b1 <> {}
for b2 being set
st b2 in field b1 &
(ex b3 being set st
b3 in field b1 & not [b3,b2] in b1)
holds ex b3 being set st
b3 in field b1 &
[b2,b3] in b1 &
(for b4 being set
st b4 in field b1 & [b2,b4] in b1 & b4 <> b2
holds [b3,b4] in b1);
:: WELLORD1:th 13
theorem
for b1 being set
for b2 being Relation-like set holds
b2 -Seg b1 c= field b2;
:: WELLORD1:funcnot 2 => WELLORD1:func 2
definition
let a1 be Relation-like set;
let a2 be set;
func A1 |_2 A2 -> Relation-like set equals
a1 /\ [:a2,a2:];
end;
:: WELLORD1:def 6
theorem
for b1 being Relation-like set
for b2 being set holds
b1 |_2 b2 = b1 /\ [:b2,b2:];
:: WELLORD1:th 17
theorem
for b1 being set
for b2 being Relation-like set holds
b2 |_2 b1 = (b1 | b2) | b1;
:: WELLORD1:th 18
theorem
for b1 being set
for b2 being Relation-like set holds
b2 |_2 b1 = b1 | (b2 | b1);
:: WELLORD1:th 19
theorem
for b1, b2 being set
for b3 being Relation-like set
st b1 in field (b3 |_2 b2)
holds b1 in field b3 & b1 in b2;
:: WELLORD1:th 20
theorem
for b1 being set
for b2 being Relation-like set holds
field (b2 |_2 b1) c= field b2 & field (b2 |_2 b1) c= b1;
:: WELLORD1:th 21
theorem
for b1, b2 being set
for b3 being Relation-like set holds
(b3 |_2 b1) -Seg b2 c= b3 -Seg b2;
:: WELLORD1:th 22
theorem
for b1 being set
for b2 being Relation-like set
st b2 is reflexive
holds b2 |_2 b1 is reflexive;
:: WELLORD1:th 23
theorem
for b1 being set
for b2 being Relation-like set
st b2 is connected
holds b2 |_2 b1 is connected;
:: WELLORD1:th 24
theorem
for b1 being set
for b2 being Relation-like set
st b2 is transitive
holds b2 |_2 b1 is transitive;
:: WELLORD1:th 25
theorem
for b1 being set
for b2 being Relation-like set
st b2 is antisymmetric
holds b2 |_2 b1 is antisymmetric;
:: WELLORD1:th 26
theorem
for b1, b2 being set
for b3 being Relation-like set holds
(b3 |_2 b1) |_2 b2 = b3 |_2 (b1 /\ b2);
:: WELLORD1:th 27
theorem
for b1, b2 being set
for b3 being Relation-like set holds
(b3 |_2 b1) |_2 b2 = (b3 |_2 b2) |_2 b1;
:: WELLORD1:th 28
theorem
for b1 being set
for b2 being Relation-like set holds
(b2 |_2 b1) |_2 b1 = b2 |_2 b1;
:: WELLORD1:th 29
theorem
for b1, b2 being set
for b3 being Relation-like set
st b1 c= b2
holds (b3 |_2 b2) |_2 b1 = b3 |_2 b1;
:: WELLORD1:th 30
theorem
for b1 being Relation-like set holds
b1 |_2 field b1 = b1;
:: WELLORD1:th 31
theorem
for b1 being set
for b2 being Relation-like set
st b2 is well_founded
holds b2 |_2 b1 is well_founded;
:: WELLORD1:th 32
theorem
for b1 being set
for b2 being Relation-like set
st b2 is well-ordering
holds b2 |_2 b1 is well-ordering;
:: WELLORD1:th 33
theorem
for b1, b2 being set
for b3 being Relation-like set
st b3 is well-ordering
holds b3 -Seg b1,b3 -Seg b2 are_c=-comparable;
:: WELLORD1:th 35
theorem
for b1, b2 being set
for b3 being Relation-like set
st b3 is well-ordering & b1 in field b3 & b2 in b3 -Seg b1
holds (b3 |_2 (b3 -Seg b1)) -Seg b2 = b3 -Seg b2;
:: WELLORD1:th 36
theorem
for b1 being set
for b2 being Relation-like set
st b2 is well-ordering & b1 c= field b2
holds (b1 <> field b2 implies ex b3 being set st
b3 in field b2 & b1 = b2 -Seg b3)
iff
for b3 being set
st b3 in b1
for b4 being set
st [b4,b3] in b2
holds b4 in b1;
:: WELLORD1:th 37
theorem
for b1, b2 being set
for b3 being Relation-like set
st b3 is well-ordering & b1 in field b3 & b2 in field b3
holds [b1,b2] in b3
iff
b3 -Seg b1 c= b3 -Seg b2;
:: WELLORD1:th 38
theorem
for b1, b2 being set
for b3 being Relation-like set
st b3 is well-ordering & b1 in field b3 & b2 in field b3
holds b3 -Seg b1 c= b3 -Seg b2
iff
(b1 = b2 or b1 in b3 -Seg b2);
:: WELLORD1:th 39
theorem
for b1 being set
for b2 being Relation-like set
st b2 is well-ordering & b1 c= field b2
holds field (b2 |_2 b1) = b1;
:: WELLORD1:th 40
theorem
for b1 being set
for b2 being Relation-like set
st b2 is well-ordering
holds field (b2 |_2 (b2 -Seg b1)) = b2 -Seg b1;
:: WELLORD1:th 41
theorem
for b1 being Relation-like set
st b1 is well-ordering
for b2 being set
st for b3 being set
st b3 in field b1 & b1 -Seg b3 c= b2
holds b3 in b2
holds field b1 c= b2;
:: WELLORD1:th 42
theorem
for b1, b2 being set
for b3 being Relation-like set
st b3 is well-ordering &
b1 in field b3 &
b2 in field b3 &
(for b4 being set
st b4 in b3 -Seg b1
holds [b4,b2] in b3 & b4 <> b2)
holds [b1,b2] in b3;
:: WELLORD1:th 43
theorem
for b1 being Relation-like set
for b2 being Relation-like Function-like set
st b1 is well-ordering &
proj1 b2 = field b1 &
proj2 b2 c= field b1 &
(for b3, b4 being set
st [b3,b4] in b1 & b3 <> b4
holds [b2 . b3,b2 . b4] in b1 & b2 . b3 <> b2 . b4)
for b3 being set
st b3 in field b1
holds [b3,b2 . b3] in b1;
:: WELLORD1:prednot 3 => WELLORD1:pred 3
definition
let a1, a2 be Relation-like set;
let a3 be Relation-like Function-like set;
pred A3 is_isomorphism_of A1,A2 means
proj1 a3 = field a1 &
proj2 a3 = field a2 &
a3 is one-to-one &
(for b1, b2 being set holds
[b1,b2] in a1
iff
b1 in field a1 & b2 in field a1 & [a3 . b1,a3 . b2] in a2);
end;
:: WELLORD1:dfs 7
definiens
let a1, a2 be Relation-like set;
let a3 be Relation-like Function-like set;
To prove
a3 is_isomorphism_of a1,a2
it is sufficient to prove
thus proj1 a3 = field a1 &
proj2 a3 = field a2 &
a3 is one-to-one &
(for b1, b2 being set holds
[b1,b2] in a1
iff
b1 in field a1 & b2 in field a1 & [a3 . b1,a3 . b2] in a2);
:: WELLORD1:def 7
theorem
for b1, b2 being Relation-like set
for b3 being Relation-like Function-like set holds
b3 is_isomorphism_of b1,b2
iff
proj1 b3 = field b1 &
proj2 b3 = field b2 &
b3 is one-to-one &
(for b4, b5 being set holds
[b4,b5] in b1
iff
b4 in field b1 & b5 in field b1 & [b3 . b4,b3 . b5] in b2);
:: WELLORD1:th 45
theorem
for b1, b2 being Relation-like set
for b3 being Relation-like Function-like set
st b3 is_isomorphism_of b1,b2
for b4, b5 being set
st [b4,b5] in b1 & b4 <> b5
holds [b3 . b4,b3 . b5] in b2 & b3 . b4 <> b3 . b5;
:: WELLORD1:prednot 4 => WELLORD1:pred 4
definition
let a1, a2 be Relation-like set;
pred A1,A2 are_isomorphic means
ex b1 being Relation-like Function-like set st
b1 is_isomorphism_of a1,a2;
end;
:: WELLORD1:dfs 8
definiens
let a1, a2 be Relation-like set;
To prove
a1,a2 are_isomorphic
it is sufficient to prove
thus ex b1 being Relation-like Function-like set st
b1 is_isomorphism_of a1,a2;
:: WELLORD1:def 8
theorem
for b1, b2 being Relation-like set holds
b1,b2 are_isomorphic
iff
ex b3 being Relation-like Function-like set st
b3 is_isomorphism_of b1,b2;
:: WELLORD1:th 47
theorem
for b1 being Relation-like set holds
id field b1 is_isomorphism_of b1,b1;
:: WELLORD1:th 48
theorem
for b1 being Relation-like set holds
b1,b1 are_isomorphic;
:: WELLORD1:th 49
theorem
for b1, b2 being Relation-like set
for b3 being Relation-like Function-like set
st b3 is_isomorphism_of b1,b2
holds b3 " is_isomorphism_of b2,b1;
:: WELLORD1:th 50
theorem
for b1, b2 being Relation-like set
st b1,b2 are_isomorphic
holds b2,b1 are_isomorphic;
:: WELLORD1:th 51
theorem
for b1, b2, b3 being Relation-like set
for b4, b5 being Relation-like Function-like set
st b4 is_isomorphism_of b1,b2 & b5 is_isomorphism_of b2,b3
holds b4 * b5 is_isomorphism_of b1,b3;
:: WELLORD1:th 52
theorem
for b1, b2, b3 being Relation-like set
st b1,b2 are_isomorphic & b2,b3 are_isomorphic
holds b1,b3 are_isomorphic;
:: WELLORD1:th 53
theorem
for b1, b2 being Relation-like set
for b3 being Relation-like Function-like set
st b3 is_isomorphism_of b1,b2
holds (b1 is reflexive implies b2 is reflexive) & (b1 is transitive implies b2 is transitive) & (b1 is connected implies b2 is connected) & (b1 is antisymmetric implies b2 is antisymmetric) & (b1 is well_founded implies b2 is well_founded);
:: WELLORD1:th 54
theorem
for b1, b2 being Relation-like set
for b3 being Relation-like Function-like set
st b1 is well-ordering & b3 is_isomorphism_of b1,b2
holds b2 is well-ordering;
:: WELLORD1:th 55
theorem
for b1, b2 being Relation-like set
st b1 is well-ordering
for b3, b4 being Relation-like Function-like set
st b3 is_isomorphism_of b1,b2 & b4 is_isomorphism_of b1,b2
holds b3 = b4;
:: WELLORD1:funcnot 3 => WELLORD1:func 3
definition
let a1, a2 be Relation-like set;
assume a1 is well-ordering & a1,a2 are_isomorphic;
func canonical_isomorphism_of(A1,A2) -> Relation-like Function-like set means
it is_isomorphism_of a1,a2;
end;
:: WELLORD1:def 9
theorem
for b1, b2 being Relation-like set
st b1 is well-ordering & b1,b2 are_isomorphic
for b3 being Relation-like Function-like set holds
b3 = canonical_isomorphism_of(b1,b2)
iff
b3 is_isomorphism_of b1,b2;
:: WELLORD1:th 57
theorem
for b1 being Relation-like set
st b1 is well-ordering
for b2 being set
st b2 in field b1
holds not b1,b1 |_2 (b1 -Seg b2) are_isomorphic;
:: WELLORD1:th 58
theorem
for b1, b2 being set
for b3 being Relation-like set
st b3 is well-ordering & b1 in field b3 & b2 in field b3 & b1 <> b2
holds not b3 |_2 (b3 -Seg b1),b3 |_2 (b3 -Seg b2) are_isomorphic;
:: WELLORD1:th 59
theorem
for b1 being set
for b2, b3 being Relation-like set
for b4 being Relation-like Function-like set
st b2 is well-ordering & b1 c= field b2 & b4 is_isomorphism_of b2,b3
holds b4 | b1 is_isomorphism_of b2 |_2 b1,b3 |_2 (b4 .: b1) &
b2 |_2 b1,b3 |_2 (b4 .: b1) are_isomorphic;
:: WELLORD1:th 60
theorem
for b1, b2 being Relation-like set
for b3 being Relation-like Function-like set
st b1 is well-ordering & b3 is_isomorphism_of b1,b2
for b4 being set
st b4 in field b1
holds ex b5 being set st
b5 in field b2 & b3 .: (b1 -Seg b4) = b2 -Seg b5;
:: WELLORD1:th 61
theorem
for b1, b2 being Relation-like set
for b3 being Relation-like Function-like set
st b1 is well-ordering & b3 is_isomorphism_of b1,b2
for b4 being set
st b4 in field b1
holds ex b5 being set st
b5 in field b2 &
b1 |_2 (b1 -Seg b4),b2 |_2 (b2 -Seg b5) are_isomorphic;
:: WELLORD1:th 62
theorem
for b1, b2, b3 being set
for b4, b5 being Relation-like set
st b4 is well-ordering &
b5 is well-ordering &
b1 in field b4 &
b2 in field b5 &
b3 in field b5 &
b4,b5 |_2 (b5 -Seg b2) are_isomorphic &
b4 |_2 (b4 -Seg b1),b5 |_2 (b5 -Seg b3) are_isomorphic
holds b5 -Seg b3 c= b5 -Seg b2 & [b3,b2] in b5;
:: WELLORD1:th 63
theorem
for b1, b2 being Relation-like set
st b1 is well-ordering &
b2 is well-ordering &
not b1,b2 are_isomorphic &
(for b3 being set
st b3 in field b1
holds not b1 |_2 (b1 -Seg b3),b2 are_isomorphic)
holds ex b3 being set st
b3 in field b2 & b1,b2 |_2 (b2 -Seg b3) are_isomorphic;
:: WELLORD1:th 64
theorem
for b1 being set
for b2 being Relation-like set
st b1 c= field b2 & b2 is well-ordering & not b2,b2 |_2 b1 are_isomorphic
holds ex b3 being set st
b3 in field b2 & b2 |_2 (b2 -Seg b3),b2 |_2 b1 are_isomorphic;