Article CLOPBAN3, MML version 4.99.1005
:: CLOPBAN3:th 1
theorem
for b1 being non empty right_complementable add-associative right_zeroed CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st for b3 being Element of NAT holds
b2 . b3 = 0. b1
for b3 being Element of NAT holds
(Partial_Sums b2) . b3 = 0. b1;
:: CLOPBAN3:attrnot 1 => CLOPBAN3:attr 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
attr a2 is summable means
Partial_Sums a2 is convergent(a1);
end;
:: CLOPBAN3:dfs 1
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
a2 is summable
it is sufficient to prove
thus Partial_Sums a2 is convergent(a1);
:: CLOPBAN3:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is summable(b1)
iff
Partial_Sums b2 is convergent(b1);
:: CLOPBAN3:exreg 1
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
cluster non empty Relation-like Function-like quasi_total total summable Relation of NAT,the carrier of a1;
end;
:: CLOPBAN3:funcnot 1 => CLOPBAN3:func 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
func Sum A2 -> Element of the carrier of a1 equals
lim Partial_Sums a2;
end;
:: CLOPBAN3:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
Sum b2 = lim Partial_Sums b2;
:: CLOPBAN3:attrnot 2 => CLOPBAN3:attr 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
attr a2 is norm_summable means
||.a2.|| is summable;
end;
:: CLOPBAN3:dfs 3
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
a2 is norm_summable
it is sufficient to prove
thus ||.a2.|| is summable;
:: CLOPBAN3:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is norm_summable(b1)
iff
||.b2.|| is summable;
:: CLOPBAN3:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of NAT holds
0 <= ||.b2.|| . b3;
:: CLOPBAN3:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3, b4 being Element of the carrier of b1 holds
||.b2 - b3.|| = ||.(b2 - b4) + (b4 - b3).||;
:: CLOPBAN3:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1)
for b3 being Element of REAL
st 0 < b3
holds ex b4 being Element of NAT st
for b5 being Element of NAT
st b4 <= b5
holds ||.(b2 . b5) - (b2 . b4).|| < b3;
:: CLOPBAN3:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is CCauchy(b1)
iff
for b3 being Element of REAL
st 0 < b3
holds ex b4 being Element of NAT st
for b5 being Element of NAT
st b4 <= b5
holds ||.(b2 . b5) - (b2 . b4).|| < b3;
:: CLOPBAN3:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st for b3 being Element of NAT holds
b2 . b3 = 0. b1
for b3 being Element of NAT holds
(Partial_Sums ||.b2.||) . b3 = 0;
:: CLOPBAN3:attrnot 3 => CLOPBAN3:attr 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
redefine attr a2 is constant means
ex b1 being Element of the carrier of a1 st
for b2 being Element of NAT holds
a2 . b2 = b1;
end;
:: CLOPBAN3:dfs 4
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
a1 is constant
it is sufficient to prove
thus ex b1 being Element of the carrier of a1 st
for b2 being Element of NAT holds
a2 . b2 = b1;
:: CLOPBAN3:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is constant
iff
ex b3 being Element of the carrier of b1 st
for b4 being Element of NAT holds
b2 . b4 = b3;
:: CLOPBAN3:funcnot 2 => CLOPBAN3:func 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
let a3 be Element of NAT;
func A2 ^\ A3 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
for b1 being Element of NAT holds
it . b1 = a2 . (b1 + a3);
end;
:: CLOPBAN3:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of NAT
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b4 = b2 ^\ b3
iff
for b5 being Element of NAT holds
b4 . b5 = b2 . (b5 + b3);
:: CLOPBAN3:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 ^\ 0 = b2;
:: CLOPBAN3:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3, b4 being Element of NAT holds
(b2 ^\ b3) ^\ b4 = (b2 ^\ b4) ^\ b3;
:: CLOPBAN3:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3, b4 being Element of NAT holds
(b2 ^\ b3) ^\ b4 = b2 ^\ (b3 + b4);
:: CLOPBAN3:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is subsequence of b2 & b2 is convergent(b1)
holds b3 is convergent(b1);
:: CLOPBAN3:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is subsequence of b2 & b2 is convergent(b1)
holds lim b3 = lim b2;
:: CLOPBAN3:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of NAT holds
b2 ^\ b3 is subsequence of b2;
:: CLOPBAN3:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
for b4 being Element of NAT
st b2 is convergent(b1)
holds b2 ^\ b4 is convergent(b1) & lim (b2 ^\ b4) = lim b2;
:: CLOPBAN3:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1) &
(ex b4 being Element of NAT st
b2 = b3 ^\ b4)
holds b3 is convergent(b1);
:: CLOPBAN3:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1) &
(ex b4 being Element of NAT st
b2 = b3 ^\ b4)
holds lim b3 = lim b2;
:: CLOPBAN3:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is constant
holds b2 is convergent(b1);
:: CLOPBAN3:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st for b3 being Element of NAT holds
b2 . b3 = 0. b1
holds b2 is norm_summable(b1);
:: CLOPBAN3:exreg 2
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
cluster non empty Relation-like Function-like quasi_total total norm_summable Relation of NAT,the carrier of a1;
end;
:: CLOPBAN3:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is summable(b1)
holds b2 is convergent(b1) & lim b2 = 0. b1;
:: CLOPBAN3:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
(Partial_Sums b2) + Partial_Sums b3 = Partial_Sums (b2 + b3);
:: CLOPBAN3:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
(Partial_Sums b2) - Partial_Sums b3 = Partial_Sums (b2 - b3);
:: CLOPBAN3:funcreg 1
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like quasi_total norm_summable Relation of NAT,the carrier of a1;
cluster ||.a2.|| -> Function-like quasi_total summable;
end;
:: CLOPBAN3:condreg 1
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
cluster Function-like quasi_total summable -> convergent (Relation of NAT,the carrier of a1);
end;
:: CLOPBAN3:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is summable(b1) & b3 is summable(b1)
holds b2 + b3 is summable(b1) &
Sum (b2 + b3) = (Sum b2) + Sum b3;
:: CLOPBAN3:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is summable(b1) & b3 is summable(b1)
holds b2 - b3 is summable(b1) &
Sum (b2 - b3) = (Sum b2) - Sum b3;
:: CLOPBAN3:funcreg 2
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2, a3 be Function-like quasi_total summable Relation of NAT,the carrier of a1;
cluster a2 + a3 -> Function-like quasi_total summable;
end;
:: CLOPBAN3:funcreg 3
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2, a3 be Function-like quasi_total summable Relation of NAT,the carrier of a1;
cluster a2 - a3 -> Function-like quasi_total summable;
end;
:: CLOPBAN3:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of COMPLEX holds
Partial_Sums (b3 * b2) = b3 * Partial_Sums b2;
:: CLOPBAN3:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total summable Relation of NAT,the carrier of b1
for b3 being Element of COMPLEX holds
b3 * b2 is summable(b1) & Sum (b3 * b2) = b3 * Sum b2;
:: CLOPBAN3:funcreg 4
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Element of COMPLEX;
let a3 be Function-like quasi_total summable Relation of NAT,the carrier of a1;
cluster a2 * a3 -> Function-like quasi_total summable;
end;
:: CLOPBAN3:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st for b4 being Element of NAT holds
b3 . b4 = b2 . 0
holds Partial_Sums (b2 ^\ 1) = ((Partial_Sums b2) ^\ 1) - b3;
:: CLOPBAN3:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is summable(b1)
for b3 being Element of NAT holds
b2 ^\ b3 is summable(b1);
:: CLOPBAN3:funcreg 5
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like quasi_total summable Relation of NAT,the carrier of a1;
let a3 be Element of NAT;
cluster a2 ^\ a3 -> Function-like quasi_total summable;
end;
:: CLOPBAN3:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
Partial_Sums ||.b2.|| is bounded_above
iff
b2 is norm_summable(b1);
:: CLOPBAN3:funcreg 6
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like quasi_total norm_summable Relation of NAT,the carrier of a1;
cluster Partial_Sums ||.a2.|| -> Function-like quasi_total bounded_above;
end;
:: CLOPBAN3:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like complete CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is summable(b1)
iff
for b3 being Element of REAL
st 0 < b3
holds ex b4 being Element of NAT st
for b5 being Element of NAT
st b4 <= b5
holds ||.((Partial_Sums b2) . b5) - ((Partial_Sums b2) . b4).|| < b3;
:: CLOPBAN3:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3, b4 being Element of NAT
st b3 <= b4
holds ||.((Partial_Sums b2) . b4) - ((Partial_Sums b2) . b3).|| <= abs (((Partial_Sums ||.b2.||) . b4) - ((Partial_Sums ||.b2.||) . b3));
:: CLOPBAN3:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like complete CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is norm_summable(b1)
holds b2 is summable(b1);
:: CLOPBAN3:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is summable &
(ex b4 being Element of NAT st
for b5 being Element of NAT
st b4 <= b5
holds ||.b3 . b5.|| <= b2 . b5)
holds b3 is norm_summable(b1);
:: CLOPBAN3:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st (for b4 being Element of NAT holds
0 <= ||.b2.|| . b4 &
||.b2.|| . b4 <= ||.b3.|| . b4) &
b3 is norm_summable(b1)
holds b2 is norm_summable(b1) &
Sum ||.b2.|| <= Sum ||.b3.||;
:: CLOPBAN3:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st (for b3 being Element of NAT holds
0 < ||.b2.|| . b3) &
(ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds 1 <= (||.b2.|| . (b4 + 1)) / (||.b2.|| . b4))
holds b2 is not norm_summable(b1);
:: CLOPBAN3:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,REAL
st (for b4 being Element of NAT holds
b3 . b4 = b4 -root (||.b2.|| . b4)) &
b3 is convergent &
lim b3 < 1
holds b2 is norm_summable(b1);
:: CLOPBAN3:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,REAL
st (for b4 being Element of NAT holds
b3 . b4 = b4 -root (||.b2.|| . b4)) &
(ex b4 being Element of NAT st
for b5 being Element of NAT
st b4 <= b5
holds 1 <= b3 . b5)
holds ||.b2.|| is not summable;
:: CLOPBAN3:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,REAL
st (for b4 being Element of NAT holds
b3 . b4 = b4 -root (||.b2.|| . b4)) &
b3 is convergent &
1 < lim b3
holds b2 is not norm_summable(b1);
:: CLOPBAN3:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,REAL
st ||.b2.|| is non-increasing &
(for b4 being Element of NAT holds
b3 . b4 = (2 to_power b4) * (||.b2.|| . (2 to_power b4)))
holds b2 is norm_summable(b1)
iff
b3 is summable;
:: CLOPBAN3:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of REAL
st 1 < b3 &
(for b4 being Element of NAT
st 1 <= b4
holds ||.b2.|| . b4 = 1 / (b4 to_power b3))
holds b2 is norm_summable(b1);
:: CLOPBAN3:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of REAL
st b3 <= 1 &
(for b4 being Element of NAT
st 1 <= b4
holds ||.b2.|| . b4 = 1 / (b4 to_power b3))
holds b2 is not norm_summable(b1);
:: CLOPBAN3:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,REAL
st (for b4 being Element of NAT holds
b2 . b4 <> 0. b1 &
b3 . b4 = (||.b2.|| . (b4 + 1)) / (||.b2.|| . b4)) &
b3 is convergent &
lim b3 < 1
holds b2 is norm_summable(b1);
:: CLOPBAN3:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st (for b3 being Element of NAT holds
b2 . b3 <> 0. b1) &
(ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds 1 <= (||.b2.|| . (b4 + 1)) / (||.b2.|| . b4))
holds b2 is not norm_summable(b1);
:: CLOPBAN3:condreg 2
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like complete CNORMSTR;
cluster Function-like quasi_total norm_summable -> summable (Relation of NAT,the carrier of a1);
end;
:: CLOPBAN3:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2, b3, b4 being Element of the carrier of b1
for b5, b6 being Element of COMPLEX holds
b2 + b3 = b3 + b2 &
(b2 + b3) + b4 = b2 + (b3 + b4) &
b2 + 0. b1 = b2 &
(ex b7 being Element of the carrier of b1 st
b2 + b7 = 0. b1) &
(b2 * b3) * b4 = b2 * (b3 * b4) &
1r * b2 = b2 &
0c * b2 = 0. b1 &
b5 * 0. b1 = 0. b1 &
(- 1r) * b2 = - b2 &
b2 * 1. b1 = b2 &
(1. b1) * b2 = b2 &
b2 * (b3 + b4) = (b2 * b3) + (b2 * b4) &
(b3 + b4) * b2 = (b3 * b2) + (b4 * b2) &
b5 * (b2 * b3) = (b5 * b2) * b3 &
b5 * (b2 + b3) = (b5 * b2) + (b5 * b3) &
(b5 + b6) * b2 = (b5 * b2) + (b6 * b2) &
(b5 * b6) * b2 = b5 * (b6 * b2) &
(b5 * b6) * (b2 * b3) = (b5 * b2) * (b6 * b3) &
b5 * (b2 * b3) = b2 * (b5 * b3) &
(0. b1) * b2 = 0. b1 &
b2 * 0. b1 = 0. b1 &
b2 * (b3 - b4) = (b2 * b3) - (b2 * b4) &
(b3 - b4) * b2 = (b3 * b2) - (b4 * b2) &
(b2 + b3) - b4 = b2 + (b3 - b4) &
(b2 - b3) + b4 = b2 - (b3 - b4) &
(b2 - b3) - b4 = b2 - (b3 + b4) &
b2 + b3 = (b2 - b4) + (b4 + b3) &
b2 - b3 = (b2 - b4) + (b4 - b3) &
b2 = (b2 - b3) + b3 &
b2 = b3 - (b3 - b2) &
(||.b2.|| = 0 implies b2 = 0. b1) &
(b2 = 0. b1 implies ||.b2.|| = 0) &
||.b5 * b2.|| = |.b5.| * ||.b2.|| &
||.b2 + b3.|| <= ||.b2.|| + ||.b3.|| &
||.b2 * b3.|| <= ||.b2.|| * ||.b3.|| &
||.1. b1.|| = 1 &
b1 is complete;
:: CLOPBAN3:condreg 3
registration
cluster non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like -> well-unital (Normed_Complex_AlgebraStr);
end;
:: CLOPBAN3:funcnot 3 => CLOPBAN3:func 3
definition
let a1 be non empty Normed_Complex_AlgebraStr;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
let a3 be Element of the carrier of a1;
func A3 * A2 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
for b1 being Element of NAT holds
it . b1 = a3 * (a2 . b1);
end;
:: CLOPBAN3:def 8
theorem
for b1 being non empty Normed_Complex_AlgebraStr
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b4 = b3 * b2
iff
for b5 being Element of NAT holds
b4 . b5 = b3 * (b2 . b5);
:: CLOPBAN3:funcnot 4 => CLOPBAN3:func 4
definition
let a1 be non empty Normed_Complex_AlgebraStr;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
let a3 be Element of the carrier of a1;
func A2 * A3 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
for b1 being Element of NAT holds
it . b1 = (a2 . b1) * a3;
end;
:: CLOPBAN3:def 9
theorem
for b1 being non empty Normed_Complex_AlgebraStr
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b4 = b2 * b3
iff
for b5 being Element of NAT holds
b4 . b5 = (b2 . b5) * b3;
:: CLOPBAN3:funcnot 5 => CLOPBAN3:func 5
definition
let a1 be non empty Normed_Complex_AlgebraStr;
let a2, a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
func A2 * A3 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
for b1 being Element of NAT holds
it . b1 = (a2 . b1) * (a3 . b1);
end;
:: CLOPBAN3:def 10
theorem
for b1 being non empty Normed_Complex_AlgebraStr
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b4 = b2 * b3
iff
for b5 being Element of NAT holds
b4 . b5 = (b2 . b5) * (b3 . b5);
:: CLOPBAN3:funcnot 6 => CLOPBAN3:func 6
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr;
let a2 be Element of the carrier of a1;
func A2 GeoSeq -> Function-like quasi_total Relation of NAT,the carrier of a1 means
it . 0 = 1. a1 &
(for b1 being Element of NAT holds
it . (b1 + 1) = (it . b1) * a2);
end;
:: CLOPBAN3:def 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b3 = b2 GeoSeq
iff
b3 . 0 = 1. b1 &
(for b4 being Element of NAT holds
b3 . (b4 + 1) = (b3 . b4) * b2);
:: CLOPBAN3:funcnot 7 => CLOPBAN3:func 7
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr;
let a2 be Element of the carrier of a1;
let a3 be Element of NAT;
func A2 #N A3 -> Element of the carrier of a1 equals
a2 GeoSeq . a3;
end;
:: CLOPBAN3:def 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of the carrier of b1
for b3 being Element of NAT holds
b2 #N b3 = b2 GeoSeq . b3;
:: CLOPBAN3:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of the carrier of b1 holds
b2 #N 0 = 1. b1;
:: CLOPBAN3:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of the carrier of b1
st ||.b2.|| < 1
holds b2 GeoSeq is summable(b1) & b2 GeoSeq is norm_summable(b1);
:: CLOPBAN3:th 45
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of the carrier of b1
st ||.(1. b1) - b2.|| < 1
holds ((1. b1) - b2) GeoSeq is summable(b1) & ((1. b1) - b2) GeoSeq is norm_summable(b1);
:: CLOPBAN3:th 46
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative right-distributive right_unital ComplexLinearSpace-like ComplexNormSpace-like ComplexAlgebra-like Banach_Algebra-like Normed_Complex_AlgebraStr
for b2 being Element of the carrier of b1
st ||.(1. b1) - b2.|| < 1
holds b2 is invertible(b1) &
/ b2 = Sum (((1. b1) - b2) GeoSeq);