Article HAHNBAN, MML version 4.99.1005
:: HAHNBAN:th 3
theorem
for b1 being non empty set
for b2 being set
st b1 <> {b2}
holds ex b3 being Element of b1 st
b3 <> b2;
:: HAHNBAN:th 4
theorem
for b1, b2 being set
for b3 being non empty Element of bool PFuncs(b1,b2) holds
b3 is non empty functional set;
:: HAHNBAN:th 5
theorem
for b1 being non empty functional set
for b2 being Relation-like Function-like set
st b2 = union b1
holds proj1 b2 = union {proj1 b3 where b3 is Element of b1: TRUE} &
proj2 b2 = union {proj2 b3 where b3 is Element of b1: TRUE};
:: HAHNBAN:th 6
theorem
for b1 being non empty Element of bool ExtREAL
st for b2 being Element of ExtREAL
st b2 in b1
holds b2 <= -infty
holds b1 = {-infty};
:: HAHNBAN:th 7
theorem
for b1 being non empty Element of bool ExtREAL
st for b2 being Element of ExtREAL
st b2 in b1
holds +infty <= b2
holds b1 = {+infty};
:: HAHNBAN:th 8
theorem
for b1 being non empty Element of bool ExtREAL
for b2 being ext-real set
st b2 < sup b1
holds ex b3 being Element of ExtREAL st
b3 in b1 & b2 < b3;
:: HAHNBAN:th 9
theorem
for b1 being non empty Element of bool ExtREAL
for b2 being Element of ExtREAL
st inf b1 < b2
holds ex b3 being Element of ExtREAL st
b3 in b1 & b3 < b2;
:: HAHNBAN:th 10
theorem
for b1, b2 being non empty Element of bool ExtREAL
st for b3, b4 being Element of ExtREAL
st b3 in b1 & b4 in b2
holds b3 <= b4
holds sup b1 <= inf b2;
:: HAHNBAN:exreg 1
registration
let a1 be non empty set;
cluster non empty c=-linear Element of bool a1;
end;
:: HAHNBAN:th 13
theorem
for b1, b2 being set
for b3 being c=-linear Element of bool PFuncs(b1,b2) holds
union b3 in PFuncs(b1,b2);
:: HAHNBAN:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1 holds
the carrier of b2 c= the carrier of b2 + b3;
:: HAHNBAN:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1
st b1 is_the_direct_sum_of b2,b3
for b4, b5, b6 being Element of the carrier of b1
st b5 in b2 & b6 in b3 & b4 = b5 + b6
holds b4 |--(b2,b3) = [b5,b6];
:: HAHNBAN:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1
st b1 is_the_direct_sum_of b2,b3
for b4, b5, b6 being Element of the carrier of b1
st b4 |--(b2,b3) = [b5,b6]
holds b4 = b5 + b6;
:: HAHNBAN:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1
st b1 is_the_direct_sum_of b2,b3
for b4, b5, b6 being Element of the carrier of b1
st b4 |--(b2,b3) = [b5,b6]
holds b5 in b2 & b6 in b3;
:: HAHNBAN:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1
st b1 is_the_direct_sum_of b2,b3
for b4, b5, b6 being Element of the carrier of b1
st b4 |--(b2,b3) = [b5,b6]
holds b4 |--(b3,b2) = [b6,b5];
:: HAHNBAN:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1
st b1 is_the_direct_sum_of b2,b3
for b4 being Element of the carrier of b1
st b4 in b2
holds b4 |--(b2,b3) = [b4,0. b1];
:: HAHNBAN:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1
st b1 is_the_direct_sum_of b2,b3
for b4 being Element of the carrier of b1
st b4 in b3
holds b4 |--(b2,b3) = [0. b1,b4];
:: HAHNBAN:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Subspace of b2
for b4 being Element of the carrier of b1
st b4 in b3
holds b4 is Element of the carrier of b2;
:: HAHNBAN:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Subspace of b1
for b5, b6 being Subspace of b4
st b5 = b2 & b6 = b3
holds b5 + b6 = b2 + b3;
:: HAHNBAN:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
st b3 = b4
holds Lin {b4} = Lin {b3};
:: HAHNBAN:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1
st not b2 in b3
for b4 being Element of the carrier of b3 + Lin {b2}
for b5 being Subspace of b3 + Lin {b2}
st b2 = b4 & b5 = b3
holds b3 + Lin {b2} is_the_direct_sum_of b5,Lin {b4};
:: HAHNBAN:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1
for b4 being Element of the carrier of b3 + Lin {b2}
for b5 being Subspace of b3 + Lin {b2}
st b2 = b4 & b3 = b5 & not b2 in b3
holds b4 |--(b5,Lin {b4}) = [0. b5,b4];
:: HAHNBAN:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1
for b4 being Element of the carrier of b3 + Lin {b2}
for b5 being Subspace of b3 + Lin {b2}
st b2 = b4 & b3 = b5 & not b2 in b3
for b6 being Element of the carrier of b3 + Lin {b2}
st b6 in b3
holds b6 |--(b5,Lin {b4}) = [b6,0. b1];
:: HAHNBAN:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1
for b3, b4 being Subspace of b1 holds
ex b5, b6 being Element of the carrier of b1 st
b2 |--(b3,b4) = [b5,b6];
:: HAHNBAN:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1
for b4 being Element of the carrier of b3 + Lin {b2}
for b5 being Subspace of b3 + Lin {b2}
st b2 = b4 & b3 = b5 & not b2 in b3
for b6 being Element of the carrier of b3 + Lin {b2} holds
ex b7 being Element of the carrier of b3 st
ex b8 being Element of REAL st
b6 |--(b5,Lin {b4}) = [b7,b8 * b2];
:: HAHNBAN:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1
for b4 being Element of the carrier of b3 + Lin {b2}
for b5 being Subspace of b3 + Lin {b2}
st b2 = b4 & b3 = b5 & not b2 in b3
for b6, b7 being Element of the carrier of b3 + Lin {b2}
for b8, b9 being Element of the carrier of b3
for b10, b11 being Element of REAL
st b6 |--(b5,Lin {b4}) = [b8,b10 * b2] &
b7 |--(b5,Lin {b4}) = [b9,b11 * b2]
holds (b6 + b7) |--(b5,Lin {b4}) = [b8 + b9,(b10 + b11) * b2];
:: HAHNBAN:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1
for b4 being Element of the carrier of b3 + Lin {b2}
for b5 being Subspace of b3 + Lin {b2}
st b2 = b4 & b3 = b5 & not b2 in b3
for b6 being Element of the carrier of b3 + Lin {b2}
for b7 being Element of the carrier of b3
for b8, b9 being Element of REAL
st b6 |--(b5,Lin {b4}) = [b7,b9 * b2]
holds (b8 * b6) |--(b5,Lin {b4}) = [b8 * b7,(b8 * b9) * b2];
:: HAHNBAN:modenot 1
definition
let a1 be RLSStruct;
mode Functional of a1 is Function-like quasi_total Relation of the carrier of a1,REAL;
end;
:: HAHNBAN:attrnot 1 => HAHNBAN:attr 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Function-like quasi_total Relation of the carrier of a1,REAL;
attr a2 is subadditive means
for b1, b2 being Element of the carrier of a1 holds
a2 . (b1 + b2) <= (a2 . b1) + (a2 . b2);
end;
:: HAHNBAN:dfs 1
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Function-like quasi_total Relation of the carrier of a1,REAL;
To prove
a2 is subadditive
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
a2 . (b1 + b2) <= (a2 . b1) + (a2 . b2);
:: HAHNBAN:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,REAL holds
b2 is subadditive(b1)
iff
for b3, b4 being Element of the carrier of b1 holds
b2 . (b3 + b4) <= (b2 . b3) + (b2 . b4);
:: HAHNBAN:attrnot 2 => HAHNBAN:attr 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Function-like quasi_total Relation of the carrier of a1,REAL;
attr a2 is additive means
for b1, b2 being Element of the carrier of a1 holds
a2 . (b1 + b2) = (a2 . b1) + (a2 . b2);
end;
:: HAHNBAN:dfs 2
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Function-like quasi_total Relation of the carrier of a1,REAL;
To prove
a2 is additive
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
a2 . (b1 + b2) = (a2 . b1) + (a2 . b2);
:: HAHNBAN:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,REAL holds
b2 is additive(b1)
iff
for b3, b4 being Element of the carrier of b1 holds
b2 . (b3 + b4) = (b2 . b3) + (b2 . b4);
:: HAHNBAN:attrnot 3 => HAHNBAN:attr 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Function-like quasi_total Relation of the carrier of a1,REAL;
attr a2 is homogeneous means
for b1 being Element of the carrier of a1
for b2 being Element of REAL holds
a2 . (b2 * b1) = b2 * (a2 . b1);
end;
:: HAHNBAN:dfs 3
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Function-like quasi_total Relation of the carrier of a1,REAL;
To prove
a2 is homogeneous
it is sufficient to prove
thus for b1 being Element of the carrier of a1
for b2 being Element of REAL holds
a2 . (b2 * b1) = b2 * (a2 . b1);
:: HAHNBAN:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,REAL holds
b2 is homogeneous(b1)
iff
for b3 being Element of the carrier of b1
for b4 being Element of REAL holds
b2 . (b4 * b3) = b4 * (b2 . b3);
:: HAHNBAN:attrnot 4 => HAHNBAN:attr 4
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Function-like quasi_total Relation of the carrier of a1,REAL;
attr a2 is positively_homogeneous means
for b1 being Element of the carrier of a1
for b2 being Element of REAL
st 0 < b2
holds a2 . (b2 * b1) = b2 * (a2 . b1);
end;
:: HAHNBAN:dfs 4
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Function-like quasi_total Relation of the carrier of a1,REAL;
To prove
a2 is positively_homogeneous
it is sufficient to prove
thus for b1 being Element of the carrier of a1
for b2 being Element of REAL
st 0 < b2
holds a2 . (b2 * b1) = b2 * (a2 . b1);
:: HAHNBAN:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,REAL holds
b2 is positively_homogeneous(b1)
iff
for b3 being Element of the carrier of b1
for b4 being Element of REAL
st 0 < b4
holds b2 . (b4 * b3) = b4 * (b2 . b3);
:: HAHNBAN:attrnot 5 => HAHNBAN:attr 5
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Function-like quasi_total Relation of the carrier of a1,REAL;
attr a2 is semi-homogeneous means
for b1 being Element of the carrier of a1
for b2 being Element of REAL
st 0 <= b2
holds a2 . (b2 * b1) = b2 * (a2 . b1);
end;
:: HAHNBAN:dfs 5
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Function-like quasi_total Relation of the carrier of a1,REAL;
To prove
a2 is semi-homogeneous
it is sufficient to prove
thus for b1 being Element of the carrier of a1
for b2 being Element of REAL
st 0 <= b2
holds a2 . (b2 * b1) = b2 * (a2 . b1);
:: HAHNBAN:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,REAL holds
b2 is semi-homogeneous(b1)
iff
for b3 being Element of the carrier of b1
for b4 being Element of REAL
st 0 <= b4
holds b2 . (b4 * b3) = b4 * (b2 . b3);
:: HAHNBAN:attrnot 6 => HAHNBAN:attr 6
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Function-like quasi_total Relation of the carrier of a1,REAL;
attr a2 is absolutely_homogeneous means
for b1 being Element of the carrier of a1
for b2 being Element of REAL holds
a2 . (b2 * b1) = (abs b2) * (a2 . b1);
end;
:: HAHNBAN:dfs 6
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Function-like quasi_total Relation of the carrier of a1,REAL;
To prove
a2 is absolutely_homogeneous
it is sufficient to prove
thus for b1 being Element of the carrier of a1
for b2 being Element of REAL holds
a2 . (b2 * b1) = (abs b2) * (a2 . b1);
:: HAHNBAN:def 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,REAL holds
b2 is absolutely_homogeneous(b1)
iff
for b3 being Element of the carrier of b1
for b4 being Element of REAL holds
b2 . (b4 * b3) = (abs b4) * (b2 . b3);
:: HAHNBAN:attrnot 7 => HAHNBAN:attr 7
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Function-like quasi_total Relation of the carrier of a1,REAL;
attr a2 is 0-preserving means
a2 . 0. a1 = 0;
end;
:: HAHNBAN:dfs 7
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Function-like quasi_total Relation of the carrier of a1,REAL;
To prove
a2 is 0-preserving
it is sufficient to prove
thus a2 . 0. a1 = 0;
:: HAHNBAN:def 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,REAL holds
b2 is 0-preserving(b1)
iff
b2 . 0. b1 = 0;
:: HAHNBAN:condreg 1
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
cluster Function-like quasi_total additive -> subadditive (Relation of the carrier of a1,REAL);
end;
:: HAHNBAN:condreg 2
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
cluster Function-like quasi_total homogeneous -> positively_homogeneous (Relation of the carrier of a1,REAL);
end;
:: HAHNBAN:condreg 3
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
cluster Function-like quasi_total semi-homogeneous -> positively_homogeneous (Relation of the carrier of a1,REAL);
end;
:: HAHNBAN:condreg 4
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
cluster Function-like quasi_total semi-homogeneous -> 0-preserving (Relation of the carrier of a1,REAL);
end;
:: HAHNBAN:condreg 5
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
cluster Function-like quasi_total absolutely_homogeneous -> semi-homogeneous (Relation of the carrier of a1,REAL);
end;
:: HAHNBAN:condreg 6
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
cluster Function-like quasi_total positively_homogeneous 0-preserving -> semi-homogeneous (Relation of the carrier of a1,REAL);
end;
:: HAHNBAN:exreg 2
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
cluster Relation-like Function-like non empty quasi_total total additive homogeneous absolutely_homogeneous Relation of the carrier of a1,REAL;
end;
:: HAHNBAN:modenot 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
mode Banach-Functional of a1 is Function-like quasi_total subadditive positively_homogeneous Relation of the carrier of a1,REAL;
end;
:: HAHNBAN:modenot 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
mode linear-Functional of a1 is Function-like quasi_total additive homogeneous Relation of the carrier of a1,REAL;
end;
:: HAHNBAN:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Function-like quasi_total homogeneous Relation of the carrier of b1,REAL
for b3 being Element of the carrier of b1 holds
b2 . - b3 = - (b2 . b3);
:: HAHNBAN:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,REAL
for b3, b4 being Element of the carrier of b1 holds
b2 . (b3 - b4) = (b2 . b3) - (b2 . b4);
:: HAHNBAN:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Function-like quasi_total additive Relation of the carrier of b1,REAL holds
b2 . 0. b1 = 0;
:: HAHNBAN:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Function-like quasi_total additive homogeneous Relation of the carrier of b2,REAL
for b4 being Element of the carrier of b1
for b5 being Element of the carrier of b2 + Lin {b4}
st b4 = b5 & not b4 in b2
for b6 being Element of REAL holds
ex b7 being Function-like quasi_total additive homogeneous Relation of the carrier of b2 + Lin {b4},REAL st
b7 | the carrier of b2 = b3 & b7 . b5 = b6;
:: HAHNBAN:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Function-like quasi_total subadditive positively_homogeneous Relation of the carrier of b1,REAL
for b4 being Function-like quasi_total additive homogeneous Relation of the carrier of b2,REAL
st for b5 being Element of the carrier of b2
for b6 being Element of the carrier of b1
st b5 = b6
holds b4 . b5 <= b3 . b6
holds ex b5 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,REAL st
b5 | the carrier of b2 = b4 &
(for b6 being Element of the carrier of b1 holds
b5 . b6 <= b3 . b6);
:: HAHNBAN:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
the norm of b1 is Function-like quasi_total subadditive absolutely_homogeneous Relation of the carrier of b1,REAL;
:: HAHNBAN:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Subspace of b1
for b3 being Function-like quasi_total additive homogeneous Relation of the carrier of b2,REAL
st for b4 being Element of the carrier of b2
for b5 being Element of the carrier of b1
st b4 = b5
holds b3 . b4 <= ||.b5.||
holds ex b4 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,REAL st
b4 | the carrier of b2 = b3 &
(for b5 being Element of the carrier of b1 holds
b4 . b5 <= ||.b5.||);