Article BHSP_6, MML version 4.99.1005
:: BHSP_6:funcnot 1 => BHSP_6:func 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be finite Element of bool the carrier of a1;
assume the addF of a1 is commutative(the carrier of a1) & the addF of a1 is associative(the carrier of a1) & the addF of a1 is having_a_unity(the carrier of a1);
func setsum A2 -> Element of the carrier of a1 means
ex b1 being FinSequence of the carrier of a1 st
b1 is one-to-one & rng b1 = a2 & it = (the addF of a1) "**" b1;
end;
:: BHSP_6:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
st the addF of b1 is commutative(the carrier of b1) & the addF of b1 is associative(the carrier of b1) & the addF of b1 is having_a_unity(the carrier of b1)
for b2 being finite Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
b3 = setsum b2
iff
ex b4 being FinSequence of the carrier of b1 st
b4 is one-to-one & rng b4 = b2 & b3 = (the addF of b1) "**" b4;
:: BHSP_6:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
st the addF of b1 is commutative(the carrier of b1) & the addF of b1 is associative(the carrier of b1) & the addF of b1 is having_a_unity(the carrier of b1)
for b2 being finite Element of bool the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
st b2 c= dom b3 &
(for b4 being set
st b4 in dom b3
holds b3 . b4 = b4)
holds setsum b2 = setopfunc(b2,the carrier of b1,the carrier of b1,b3,the addF of b1);
:: BHSP_6:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
st the addF of b1 is commutative(the carrier of b1) & the addF of b1 is associative(the carrier of b1) & the addF of b1 is having_a_unity(the carrier of b1)
for b2, b3 being finite Element of bool the carrier of b1
st b2 misses b3
for b4 being finite Element of bool the carrier of b1
st b4 = b2 \/ b3
holds setsum b4 = (setsum b2) + setsum b3;
:: BHSP_6:attrnot 1 => BHSP_6:attr 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Element of bool the carrier of a1;
attr a2 is summable_set means
ex b1 being Element of the carrier of a1 st
for b2 being Element of REAL
st 0 < b2
holds ex b3 being finite Element of bool the carrier of a1 st
b3 is not empty &
b3 c= a2 &
(for b4 being finite Element of bool the carrier of a1
st b3 c= b4 & b4 c= a2
holds ||.b1 - setsum b4.|| < b2);
end;
:: BHSP_6:dfs 2
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Element of bool the carrier of a1;
To prove
a2 is summable_set
it is sufficient to prove
thus ex b1 being Element of the carrier of a1 st
for b2 being Element of REAL
st 0 < b2
holds ex b3 being finite Element of bool the carrier of a1 st
b3 is not empty &
b3 c= a2 &
(for b4 being finite Element of bool the carrier of a1
st b3 c= b4 & b4 c= a2
holds ||.b1 - setsum b4.|| < b2);
:: BHSP_6:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1 holds
b2 is summable_set(b1)
iff
ex b3 being Element of the carrier of b1 st
for b4 being Element of REAL
st 0 < b4
holds ex b5 being finite Element of bool the carrier of b1 st
b5 is not empty &
b5 c= b2 &
(for b6 being finite Element of bool the carrier of b1
st b5 c= b6 & b6 c= b2
holds ||.b3 - setsum b6.|| < b4);
:: BHSP_6:funcnot 2 => BHSP_6:func 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Element of bool the carrier of a1;
assume a2 is summable_set(a1);
func sum A2 -> Element of the carrier of a1 means
for b1 being Element of REAL
st 0 < b1
holds ex b2 being finite Element of bool the carrier of a1 st
b2 is not empty &
b2 c= a2 &
(for b3 being finite Element of bool the carrier of a1
st b2 c= b3 & b3 c= a2
holds ||.it - setsum b3.|| < b1);
end;
:: BHSP_6:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
st b2 is summable_set(b1)
for b3 being Element of the carrier of b1 holds
b3 = sum b2
iff
for b4 being Element of REAL
st 0 < b4
holds ex b5 being finite Element of bool the carrier of b1 st
b5 is not empty &
b5 c= b2 &
(for b6 being finite Element of bool the carrier of b1
st b5 c= b6 & b6 c= b2
holds ||.b3 - setsum b6.|| < b4);
:: BHSP_6:attrnot 2 => BHSP_6:attr 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Function-like quasi_total additive homogeneous Relation of the carrier of a1,REAL;
attr a2 is Bounded means
ex b1 being Element of REAL st
0 < b1 &
(for b2 being Element of the carrier of a1 holds
abs (a2 . b2) <= b1 * ||.b2.||);
end;
:: BHSP_6:dfs 4
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Function-like quasi_total additive homogeneous Relation of the carrier of a1,REAL;
To prove
a2 is Bounded
it is sufficient to prove
thus ex b1 being Element of REAL st
0 < b1 &
(for b2 being Element of the carrier of a1 holds
abs (a2 . b2) <= b1 * ||.b2.||);
:: BHSP_6:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,REAL holds
b2 is Bounded(b1)
iff
ex b3 being Element of REAL st
0 < b3 &
(for b4 being Element of the carrier of b1 holds
abs (b2 . b4) <= b3 * ||.b4.||);
:: BHSP_6:attrnot 3 => BHSP_6:attr 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Element of bool the carrier of a1;
attr a2 is weakly_summable_set means
ex b1 being Element of the carrier of a1 st
for b2 being Function-like quasi_total additive homogeneous Relation of the carrier of a1,REAL
st b2 is Bounded(a1)
for b3 being Element of REAL
st 0 < b3
holds ex b4 being finite Element of bool the carrier of a1 st
b4 is not empty &
b4 c= a2 &
(for b5 being finite Element of bool the carrier of a1
st b4 c= b5 & b5 c= a2
holds abs (b2 . (b1 - setsum b5)) < b3);
end;
:: BHSP_6:dfs 5
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Element of bool the carrier of a1;
To prove
a2 is weakly_summable_set
it is sufficient to prove
thus ex b1 being Element of the carrier of a1 st
for b2 being Function-like quasi_total additive homogeneous Relation of the carrier of a1,REAL
st b2 is Bounded(a1)
for b3 being Element of REAL
st 0 < b3
holds ex b4 being finite Element of bool the carrier of a1 st
b4 is not empty &
b4 c= a2 &
(for b5 being finite Element of bool the carrier of a1
st b4 c= b5 & b5 c= a2
holds abs (b2 . (b1 - setsum b5)) < b3);
:: BHSP_6:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1 holds
b2 is weakly_summable_set(b1)
iff
ex b3 being Element of the carrier of b1 st
for b4 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,REAL
st b4 is Bounded(b1)
for b5 being Element of REAL
st 0 < b5
holds ex b6 being finite Element of bool the carrier of b1 st
b6 is not empty &
b6 c= b2 &
(for b7 being finite Element of bool the carrier of b1
st b6 c= b7 & b7 c= b2
holds abs (b4 . (b3 - setsum b7)) < b5);
:: BHSP_6:prednot 1 => BHSP_6:pred 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Element of bool the carrier of a1;
let a3 be Function-like quasi_total Relation of the carrier of a1,REAL;
pred A2 is_summable_set_by A3 means
ex b1 being Element of REAL st
for b2 being Element of REAL
st 0 < b2
holds ex b3 being finite Element of bool the carrier of a1 st
b3 is not empty &
b3 c= a2 &
(for b4 being finite Element of bool the carrier of a1
st b3 c= b4 & b4 c= a2
holds abs (b1 - setopfunc(b4,the carrier of a1,REAL,a3,addreal)) < b2);
end;
:: BHSP_6:dfs 6
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Element of bool the carrier of a1;
let a3 be Function-like quasi_total Relation of the carrier of a1,REAL;
To prove
a2 is_summable_set_by a3
it is sufficient to prove
thus ex b1 being Element of REAL st
for b2 being Element of REAL
st 0 < b2
holds ex b3 being finite Element of bool the carrier of a1 st
b3 is not empty &
b3 c= a2 &
(for b4 being finite Element of bool the carrier of a1
st b3 c= b4 & b4 c= a2
holds abs (b1 - setopfunc(b4,the carrier of a1,REAL,a3,addreal)) < b2);
:: BHSP_6:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,REAL holds
b2 is_summable_set_by b3
iff
ex b4 being Element of REAL st
for b5 being Element of REAL
st 0 < b5
holds ex b6 being finite Element of bool the carrier of b1 st
b6 is not empty &
b6 c= b2 &
(for b7 being finite Element of bool the carrier of b1
st b6 c= b7 & b7 c= b2
holds abs (b4 - setopfunc(b7,the carrier of b1,REAL,b3,addreal)) < b5);
:: BHSP_6:funcnot 3 => BHSP_6:func 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Element of bool the carrier of a1;
let a3 be Function-like quasi_total Relation of the carrier of a1,REAL;
assume a2 is_summable_set_by a3;
func sum_byfunc(A2,A3) -> Element of REAL means
for b1 being Element of REAL
st 0 < b1
holds ex b2 being finite Element of bool the carrier of a1 st
b2 is not empty &
b2 c= a2 &
(for b3 being finite Element of bool the carrier of a1
st b2 c= b3 & b3 c= a2
holds abs (it - setopfunc(b3,the carrier of a1,REAL,a3,addreal)) < b1);
end;
:: BHSP_6:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,REAL
st b2 is_summable_set_by b3
for b4 being Element of REAL holds
b4 = sum_byfunc(b2,b3)
iff
for b5 being Element of REAL
st 0 < b5
holds ex b6 being finite Element of bool the carrier of b1 st
b6 is not empty &
b6 c= b2 &
(for b7 being finite Element of bool the carrier of b1
st b6 c= b7 & b7 c= b2
holds abs (b4 - setopfunc(b7,the carrier of b1,REAL,b3,addreal)) < b5);
:: BHSP_6:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
st b2 is summable_set(b1)
holds b2 is weakly_summable_set(b1);
:: BHSP_6:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,REAL
for b3 being FinSequence of the carrier of b1
st 1 <= len b3
for b4 being FinSequence of REAL
st dom b3 = dom b4 &
(for b5 being Element of NAT
st b5 in dom b4
holds b4 . b5 = b2 . (b3 . b5))
holds b2 . ((the addF of b1) "**" b3) = addreal "**" b4;
:: BHSP_6:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
st the addF of b1 is commutative(the carrier of b1) & the addF of b1 is associative(the carrier of b1) & the addF of b1 is having_a_unity(the carrier of b1)
for b2 being finite Element of bool the carrier of b1
st b2 is not empty
for b3 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,REAL holds
b3 . setsum b2 = setopfunc(b2,the carrier of b1,REAL,b3,addreal);
:: BHSP_6:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
st the addF of b1 is commutative(the carrier of b1) & the addF of b1 is associative(the carrier of b1) & the addF of b1 is having_a_unity(the carrier of b1)
for b2 being Element of bool the carrier of b1
st b2 is weakly_summable_set(b1)
holds ex b3 being Element of the carrier of b1 st
for b4 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,REAL
st b4 is Bounded(b1)
for b5 being Element of REAL
st 0 < b5
holds ex b6 being finite Element of bool the carrier of b1 st
b6 is not empty &
b6 c= b2 &
(for b7 being finite Element of bool the carrier of b1
st b6 c= b7 & b7 c= b2
holds abs ((b4 . b3) - setopfunc(b7,the carrier of b1,REAL,b4,addreal)) < b5);
:: BHSP_6:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
st the addF of b1 is commutative(the carrier of b1) & the addF of b1 is associative(the carrier of b1) & the addF of b1 is having_a_unity(the carrier of b1)
for b2 being Element of bool the carrier of b1
st b2 is weakly_summable_set(b1)
for b3 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,REAL
st b3 is Bounded(b1)
holds b2 is_summable_set_by b3;
:: BHSP_6:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
st the addF of b1 is commutative(the carrier of b1) & the addF of b1 is associative(the carrier of b1) & the addF of b1 is having_a_unity(the carrier of b1)
for b2 being Element of bool the carrier of b1
st b2 is summable_set(b1)
for b3 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,REAL
st b3 is Bounded(b1)
holds b2 is_summable_set_by b3;
:: BHSP_6:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being finite Element of bool the carrier of b1
st b2 is not empty
holds b2 is summable_set(b1);
:: BHSP_6:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
st the addF of b1 is commutative(the carrier of b1) & the addF of b1 is associative(the carrier of b1) & the addF of b1 is having_a_unity(the carrier of b1) & b1 is Hilbert
for b2 being Element of bool the carrier of b1 holds
b2 is summable_set(b1)
iff
for b3 being Element of REAL
st 0 < b3
holds ex b4 being finite Element of bool the carrier of b1 st
b4 is not empty &
b4 c= b2 &
(for b5 being finite Element of bool the carrier of b1
st b5 is not empty & b5 c= b2 & b4 misses b5
holds ||.setsum b5.|| < b3);