Article NORMSP_1, MML version 4.99.1005
:: NORMSP_1:structnot 1 => NORMSP_1:struct 1
definition
struct(RLSStruct) NORMSTR(#
carrier -> set,
ZeroF -> Element of the carrier of it,
addF -> Function-like quasi_total Relation of [:the carrier of it,the carrier of it:],the carrier of it,
Mult -> Function-like quasi_total Relation of [:REAL,the carrier of it:],the carrier of it,
norm -> Function-like quasi_total Relation of the carrier of it,REAL
#);
end;
:: NORMSP_1:attrnot 1 => NORMSP_1:attr 1
definition
let a1 be NORMSTR;
attr a1 is strict;
end;
:: NORMSP_1:exreg 1
registration
cluster strict NORMSTR;
end;
:: NORMSP_1:aggrnot 1 => NORMSP_1:aggr 1
definition
let a1 be set;
let a2 be Element of a1;
let a3 be Function-like quasi_total Relation of [:a1,a1:],a1;
let a4 be Function-like quasi_total Relation of [:REAL,a1:],a1;
let a5 be Function-like quasi_total Relation of a1,REAL;
aggr NORMSTR(#a1,a2,a3,a4,a5#) -> strict NORMSTR;
end;
:: NORMSP_1:selnot 1 => NORMSP_1:sel 1
definition
let a1 be NORMSTR;
sel the norm of a1 -> Function-like quasi_total Relation of the carrier of a1,REAL;
end;
:: NORMSP_1:exreg 2
registration
cluster non empty strict NORMSTR;
end;
:: NORMSP_1:funcnot 1 => NORMSP_1:func 1
definition
let a1 be non empty NORMSTR;
let a2 be Element of the carrier of a1;
func ||.A2.|| -> Element of REAL equals
(the norm of a1) . a2;
end;
:: NORMSP_1:def 1
theorem
for b1 being non empty NORMSTR
for b2 being Element of the carrier of b1 holds
||.b2.|| = (the norm of b1) . b2;
:: NORMSP_1:attrnot 2 => NORMSP_1:attr 2
definition
let a1 be non empty NORMSTR;
attr a1 is RealNormSpace-like means
for b1, b2 being Element of the carrier of a1
for b3 being Element of REAL holds
(||.b1.|| = 0 implies b1 = 0. a1) &
(b1 = 0. a1 implies ||.b1.|| = 0) &
||.b3 * b1.|| = (abs b3) * ||.b1.|| &
||.b1 + b2.|| <= ||.b1.|| + ||.b2.||;
end;
:: NORMSP_1:dfs 2
definiens
let a1 be non empty NORMSTR;
To prove
a1 is RealNormSpace-like
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
for b3 being Element of REAL holds
(||.b1.|| = 0 implies b1 = 0. a1) &
(b1 = 0. a1 implies ||.b1.|| = 0) &
||.b3 * b1.|| = (abs b3) * ||.b1.|| &
||.b1 + b2.|| <= ||.b1.|| + ||.b2.||;
:: NORMSP_1:def 2
theorem
for b1 being non empty NORMSTR holds
b1 is RealNormSpace-like
iff
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL holds
(||.b2.|| = 0 implies b2 = 0. b1) &
(b2 = 0. b1 implies ||.b2.|| = 0) &
||.b4 * b2.|| = (abs b4) * ||.b2.|| &
||.b2 + b3.|| <= ||.b2.|| + ||.b3.||;
:: NORMSP_1:exreg 3
registration
cluster non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like strict RealNormSpace-like NORMSTR;
end;
:: NORMSP_1:modenot 1
definition
mode RealNormSpace is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
end;
:: NORMSP_1:funcreg 1
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
cluster ||.0. a1.|| -> empty;
end;
:: NORMSP_1:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
||.0. b1.|| = 0;
:: NORMSP_1:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of the carrier of b1 holds
||.- b2.|| = ||.b2.||;
:: NORMSP_1:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2 - b3.|| <= ||.b2.|| + ||.b3.||;
:: NORMSP_1:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of the carrier of b1 holds
0 <= ||.b2.||;
:: NORMSP_1:th 9
theorem
for b1, b2 being Element of REAL
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4, b5 being Element of the carrier of b3 holds
||.(b1 * b4) + (b2 * b5).|| <= ((abs b1) * ||.b4.||) + ((abs b2) * ||.b5.||);
:: NORMSP_1:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2 - b3.|| = 0
iff
b2 = b3;
:: NORMSP_1:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2 - b3.|| = ||.b3 - b2.||;
:: NORMSP_1:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2.|| - ||.b3.|| <= ||.b2 - b3.||;
:: NORMSP_1:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Element of the carrier of b1 holds
abs (||.b2.|| - ||.b3.||) <= ||.b2 - b3.||;
:: NORMSP_1:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3, b4 being Element of the carrier of b1 holds
||.b2 - b3.|| <= ||.b2 - b4.|| + ||.b4 - b3.||;
:: NORMSP_1:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Element of the carrier of b1
st b2 <> b3
holds ||.b2 - b3.|| <> 0;
:: NORMSP_1:th 17
theorem
for b1 being Relation-like Function-like set
for b2 being non empty 1-sorted
for b3 being Element of the carrier of b2 holds
b1 is Function-like quasi_total Relation of NAT,the carrier of b2
iff
proj1 b1 = NAT &
(for b4 being set
st b4 in NAT
holds b1 . b4 is Element of the carrier of b2);
:: NORMSP_1:th 19
theorem
for b1 being non empty 1-sorted
for b2 being Element of the carrier of b1 holds
ex b3 being Function-like quasi_total Relation of NAT,the carrier of b1 st
proj2 b3 = {b2};
:: NORMSP_1:th 20
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st ex b3 being Element of the carrier of b1 st
for b4 being Element of NAT holds
b2 . b4 = b3
holds ex b3 being Element of the carrier of b1 st
proj2 b2 = {b3};
:: NORMSP_1:th 21
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st ex b3 being Element of the carrier of b1 st
proj2 b2 = {b3}
for b3 being Element of NAT holds
b2 . b3 = b2 . (b3 + 1);
:: NORMSP_1:th 22
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st for b3 being Element of NAT holds
b2 . b3 = b2 . (b3 + 1)
for b3, b4 being Element of NAT holds
b2 . b3 = b2 . (b3 + b4);
:: NORMSP_1:th 23
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st for b3, b4 being Element of NAT holds
b2 . b3 = b2 . (b3 + b4)
for b3, b4 being Element of NAT holds
b2 . b3 = b2 . b4;
:: NORMSP_1:th 24
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st for b3, b4 being Element of NAT holds
b2 . b3 = b2 . b4
holds ex b3 being Element of the carrier of b1 st
for b4 being Element of NAT holds
b2 . b4 = b3;
:: NORMSP_1:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
ex b2 being Function-like quasi_total Relation of NAT,the carrier of b1 st
proj2 b2 = {0. b1};
:: NORMSP_1:attrnot 3 => NORMSP_1:attr 3
definition
let a1 be non empty 1-sorted;
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;
:: NORMSP_1:dfs 3
definiens
let a1 be non empty 1-sorted;
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;
:: NORMSP_1:def 4
theorem
for b1 being non empty 1-sorted
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;
:: NORMSP_1:th 27
theorem
for b1 being non empty 1-sorted
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
proj2 b2 = {b3};
:: NORMSP_1:funcnot 2 => NORMSP_1:func 2
definition
let a1 be non empty 1-sorted;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
let a3 be Element of NAT;
redefine func a2 . a3 -> Element of the carrier of a1;
end;
:: NORMSP_1:funcnot 3 => NORMSP_1:func 3
definition
let a1 be non empty addLoopStr;
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;
:: NORMSP_1:def 5
theorem
for b1 being non empty addLoopStr
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);
:: NORMSP_1:funcnot 4 => NORMSP_1:func 4
definition
let a1 be non empty addLoopStr;
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;
:: NORMSP_1:def 6
theorem
for b1 being non empty addLoopStr
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);
:: NORMSP_1:funcnot 5 => NORMSP_1:func 5
definition
let a1 be non empty addLoopStr;
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;
:: NORMSP_1:def 7
theorem
for b1 being non empty addLoopStr
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;
:: NORMSP_1:funcnot 6 => NORMSP_1:func 6
definition
let a1 be non empty RLSStruct;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
let a3 be Element of REAL;
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;
:: NORMSP_1:def 8
theorem
for b1 being non empty RLSStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of REAL
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);
:: NORMSP_1:attrnot 4 => NORMSP_1:attr 4
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
attr a2 is convergent means
ex b1 being Element of the carrier of a1 st
for b2 being Element of REAL
st 0 < b2
holds ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds ||.(a2 . b4) - b1.|| < b2;
end;
:: NORMSP_1:dfs 8
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
a2 is convergent
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 Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds ||.(a2 . b4) - b1.|| < b2;
:: NORMSP_1:def 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is convergent(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 Element of NAT st
for b6 being Element of NAT
st b5 <= b6
holds ||.(b2 . b6) - b3.|| < b4;
:: NORMSP_1:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1) & b3 is convergent(b1)
holds b2 + b3 is convergent(b1);
:: NORMSP_1:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1) & b3 is convergent(b1)
holds b2 - b3 is convergent(b1);
:: NORMSP_1:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is convergent(b1)
holds b3 - b2 is convergent(b1);
:: NORMSP_1:th 37
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
st b3 is convergent(b2)
holds b1 * b3 is convergent(b2);
:: NORMSP_1:funcnot 7 => NORMSP_1:func 7
definition
let a1 be non empty NORMSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
func ||.A2.|| -> Function-like quasi_total Relation of NAT,REAL means
for b1 being Element of NAT holds
it . b1 = ||.a2 . b1.||;
end;
:: NORMSP_1:def 10
theorem
for b1 being non empty NORMSTR
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 holds
b3 = ||.b2.||
iff
for b4 being Element of NAT holds
b3 . b4 = ||.b2 . b4.||;
:: NORMSP_1:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1)
holds ||.b2.|| is convergent;
:: NORMSP_1:funcnot 8 => NORMSP_1:func 8
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
assume a2 is convergent(a1);
func lim A2 -> Element of the carrier of a1 means
for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of NAT st
for b3 being Element of NAT
st b2 <= b3
holds ||.(a2 . b3) - it.|| < b1;
end;
:: NORMSP_1:def 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1)
for b3 being Element of the carrier of b1 holds
b3 = lim b2
iff
for b4 being Element of REAL
st 0 < b4
holds ex b5 being Element of NAT st
for b6 being Element of NAT
st b5 <= b6
holds ||.(b2 . b6) - b3.|| < b4;
:: NORMSP_1:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is convergent(b1) & lim b3 = b2
holds ||.b3 - b2.|| is convergent & lim ||.b3 - b2.|| = 0;
:: NORMSP_1:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1) & b3 is convergent(b1)
holds lim (b2 + b3) = (lim b2) + lim b3;
:: NORMSP_1:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1) & b3 is convergent(b1)
holds lim (b2 - b3) = (lim b2) - lim b3;
:: NORMSP_1:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is convergent(b1)
holds lim (b3 - b2) = (lim b3) - b2;
:: NORMSP_1:th 45
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
st b3 is convergent(b2)
holds lim (b1 * b3) = b1 * lim b3;