Article BHSP_5, MML version 4.99.1005
:: BHSP_5:th 1
theorem
for b1 being set
for b2, b3 being FinSequence of b1
st b2 is one-to-one & b3 is one-to-one & proj2 b2 = proj2 b3
holds dom b2 = dom b3 &
(ex b4 being Function-like quasi_total bijective Relation of dom b2,dom b2 st
b3 = b4 * b2 & proj1 b4 = dom b2 & proj2 b4 = dom b2);
:: BHSP_5:funcnot 1 => BHSP_5:func 1
definition
let a1 be non empty set;
let a2 be Function-like quasi_total Relation of [:a1,a1:],a1;
let a3 be finite Element of bool a1;
assume a2 is commutative(a1) & a2 is associative(a1) & a2 is having_a_unity(a1);
func A2 ++ A3 -> Element of a1 means
ex b1 being FinSequence of a1 st
b1 is one-to-one & proj2 b1 = a3 & it = a2 "**" b1;
end;
:: BHSP_5:def 1
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],b1
st b2 is commutative(b1) & b2 is associative(b1) & b2 is having_a_unity(b1)
for b3 being finite Element of bool b1
for b4 being Element of b1 holds
b4 = b2 ++ b3
iff
ex b5 being FinSequence of b1 st
b5 is one-to-one & proj2 b5 = b3 & b4 = b2 "**" b5;
:: BHSP_5:funcnot 2 => BHSP_5:func 2
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;
func setop_SUM(A2,A1) -> set equals
(the addF of a1) ++ a2
if a2 <> {}
otherwise 0. a1;
end;
:: BHSP_5:def 2
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 holds
(b2 = {} or setop_SUM(b2,b1) = (the addF of b1) ++ b2) &
(b2 = {} implies setop_SUM(b2,b1) = 0. b1);
:: BHSP_5:funcnot 3 => BHSP_5: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 the carrier of a1;
let a3 be Relation-like Function-like FinSequence-like set;
let a4 be natural set;
func PO(A4,A3,A2) -> set equals
(the scalar of a1) . [a2,a3 . a4];
end;
:: BHSP_5: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 the carrier of b1
for b3 being Relation-like Function-like FinSequence-like set
for b4 being natural set holds
PO(b4,b3,b2) = (the scalar of b1) . [b2,b3 . b4];
:: BHSP_5:funcnot 4 => BHSP_5:func 4
definition
let a1, a2 be non empty set;
let a3 be Function-like quasi_total Relation of a2,a1;
let a4 be FinSequence of a2;
func Func_Seq(A3,A4) -> FinSequence of a1 equals
a4 * a3;
end;
:: BHSP_5:def 4
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total Relation of b2,b1
for b4 being FinSequence of b2 holds
Func_Seq(b3,b4) = b4 * b3;
:: BHSP_5:funcnot 5 => BHSP_5:func 5
definition
let a1, a2 be non empty set;
let a3 be Function-like quasi_total Relation of [:a1,a1:],a1;
let a4 be finite Element of bool a2;
let a5 be Function-like quasi_total Relation of a2,a1;
assume a3 is commutative(a1) & a3 is associative(a1) & a3 is having_a_unity(a1) & a4 c= proj1 a5;
func setopfunc(A4,A2,A1,A5,A3) -> Element of a1 means
ex b1 being FinSequence of a2 st
b1 is one-to-one & proj2 b1 = a4 & it = a3 "**" Func_Seq(a5,b1);
end;
:: BHSP_5:def 5
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total Relation of [:b1,b1:],b1
st b3 is commutative(b1) & b3 is associative(b1) & b3 is having_a_unity(b1)
for b4 being finite Element of bool b2
for b5 being Function-like quasi_total Relation of b2,b1
st b4 c= proj1 b5
for b6 being Element of b1 holds
b6 = setopfunc(b4,b2,b1,b5,b3)
iff
ex b7 being FinSequence of b2 st
b7 is one-to-one & proj2 b7 = b4 & b6 = b3 "**" Func_Seq(b5,b7);
:: BHSP_5:funcnot 6 => BHSP_5:func 6
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Element of the carrier of a1;
let a3 be finite Element of bool the carrier of a1;
func setop_xPre_PROD(A2,A3,A1) -> Element of REAL means
ex b1 being FinSequence of the carrier of a1 st
b1 is one-to-one &
proj2 b1 = a3 &
(ex b2 being FinSequence of REAL st
dom b2 = dom b1 &
(for b3 being Element of NAT
st b3 in dom b2
holds b2 . b3 = PO(b3,b1,a2)) &
it = addreal "**" b2);
end;
:: BHSP_5: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 the carrier of b1
for b3 being finite Element of bool the carrier of b1
for b4 being Element of REAL holds
b4 = setop_xPre_PROD(b2,b3,b1)
iff
ex b5 being FinSequence of the carrier of b1 st
b5 is one-to-one &
proj2 b5 = b3 &
(ex b6 being FinSequence of REAL st
dom b6 = dom b5 &
(for b7 being Element of NAT
st b7 in dom b6
holds b6 . b7 = PO(b7,b5,b2)) &
b4 = addreal "**" b6);
:: BHSP_5:funcnot 7 => BHSP_5:func 7
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Element of the carrier of a1;
let a3 be finite Element of bool the carrier of a1;
func setop_xPROD(A2,A3,A1) -> Element of REAL equals
setop_xPre_PROD(a2,a3,a1)
if a3 <> {}
otherwise 0;
end;
:: BHSP_5: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 the carrier of b1
for b3 being finite Element of bool the carrier of b1 holds
(b3 = {} or setop_xPROD(b2,b3,b1) = setop_xPre_PROD(b2,b3,b1)) &
(b3 = {} implies setop_xPROD(b2,b3,b1) = 0);
:: BHSP_5:modenot 1 => BHSP_5:mode 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
mode OrthogonalFamily of A1 -> Element of bool the carrier of a1 means
for b1, b2 being Element of the carrier of a1
st b1 in it & b2 in it & b1 <> b2
holds b1 .|. b2 = 0;
end;
:: BHSP_5:dfs 8
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 OrthogonalFamily of a1
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
st b1 in a2 & b2 in a2 & b1 <> b2
holds b1 .|. b2 = 0;
:: BHSP_5:def 8
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 OrthogonalFamily of b1
iff
for b3, b4 being Element of the carrier of b1
st b3 in b2 & b4 in b2 & b3 <> b4
holds b3 .|. b4 = 0;
:: BHSP_5:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
{} is OrthogonalFamily of b1;
:: BHSP_5:exreg 1
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
cluster finite OrthogonalFamily of a1;
end;
:: BHSP_5:modenot 2 => BHSP_5:mode 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
mode OrthonormalFamily of A1 -> Element of bool the carrier of a1 means
it is OrthogonalFamily of a1 &
(for b1 being Element of the carrier of a1
st b1 in it
holds b1 .|. b1 = 1);
end;
:: BHSP_5:dfs 9
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 OrthonormalFamily of a1
it is sufficient to prove
thus a2 is OrthogonalFamily of a1 &
(for b1 being Element of the carrier of a1
st b1 in a2
holds b1 .|. b1 = 1);
:: BHSP_5:def 9
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 OrthonormalFamily of b1
iff
b2 is OrthogonalFamily of b1 &
(for b3 being Element of the carrier of b1
st b3 in b2
holds b3 .|. b3 = 1);
:: BHSP_5:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
{} is OrthonormalFamily of b1;
:: BHSP_5:exreg 2
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
cluster finite OrthonormalFamily of a1;
end;
:: BHSP_5:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1 holds
b2 = 0. b1
iff
for b3 being Element of the carrier of b1 holds
b2 .|. b3 = 0;
:: BHSP_5:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2 + b3.|| ^2 + (||.b2 - b3.|| ^2) = (2 * (||.b2.|| ^2)) + (2 * (||.b3.|| ^2));
:: BHSP_5:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
st b2,b3 are_orthogonal
holds ||.b2 + b3.|| ^2 = ||.b2.|| ^2 + (||.b3.|| ^2);
:: BHSP_5:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being FinSequence of the carrier of b1
st 1 <= len b2 &
(for b3, b4 being Element of NAT
st b3 in dom b2 & b4 in dom b2 & b3 <> b4
holds (the scalar of b1) . [b2 . b3,b2 . b4] = 0)
for b3 being FinSequence of REAL
st dom b2 = dom b3 &
(for b4 being Element of NAT
st b4 in dom b3
holds b3 . b4 = (the scalar of b1) . [b2 . b4,b2 . b4])
holds ((the addF of b1) "**" b2) .|. ((the addF of b1) "**" b2) = addreal "**" b3;
:: BHSP_5:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1
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 = (the scalar of b1) . [b2,b3 . b5])
holds b2 .|. ((the addF of b1) "**" b3) = addreal "**" b4;
:: BHSP_5:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being non empty 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= proj1 b3 &
(for b4, b5 being Element of the carrier of b1
st b4 in b2 & b5 in b2 & b4 <> b5
holds (the scalar of b1) . [b3 . b4,b3 . b5] = 0)
for b4 being Function-like quasi_total Relation of the carrier of b1,REAL
st b2 c= proj1 b4 &
(for b5 being Element of the carrier of b1
st b5 in b2
holds b4 . b5 = (the scalar of b1) . [b3 . b5,b3 . b5])
for b5 being FinSequence of the carrier of b1
st b5 is one-to-one & proj2 b5 = b2
holds (the scalar of b1) . [(the addF of b1) "**" Func_Seq(b3,b5),(the addF of b1) "**" Func_Seq(b3,b5)] = addreal "**" Func_Seq(b4,b5);
:: BHSP_5:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1
for b3 being non empty finite Element of bool the carrier of b1
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
st b3 c= proj1 b4
for b5 being Function-like quasi_total Relation of the carrier of b1,REAL
st b3 c= proj1 b5 &
(for b6 being Element of the carrier of b1
st b6 in b3
holds b5 . b6 = (the scalar of b1) . [b2,b4 . b6])
for b6 being FinSequence of the carrier of b1
st b6 is one-to-one & proj2 b6 = b3
holds (the scalar of b1) . [b2,(the addF of b1) "**" Func_Seq(b4,b6)] = addreal "**" Func_Seq(b5,b6);
:: BHSP_5:th 11
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 the carrier of b1
for b3 being finite OrthonormalFamily of b1
st b3 is not empty
for b4 being Function-like quasi_total Relation of the carrier of b1,REAL
st b3 c= proj1 b4 &
(for b5 being Element of the carrier of b1
st b5 in b3
holds b4 . b5 = (b2 .|. b5) ^2)
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
st b3 c= proj1 b5 &
(for b6 being Element of the carrier of b1
st b6 in b3
holds b5 . b6 = (b2 .|. b6) * b6)
holds b2 .|. setopfunc(b3,the carrier of b1,the carrier of b1,b5,the addF of b1) = setopfunc(b3,the carrier of b1,REAL,b4,addreal);
:: BHSP_5:th 12
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 the carrier of b1
for b3 being finite OrthonormalFamily of b1
st b3 is not empty
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
st b3 c= proj1 b4 &
(for b5 being Element of the carrier of b1
st b5 in b3
holds b4 . b5 = (b2 .|. b5) * b5)
for b5 being Function-like quasi_total Relation of the carrier of b1,REAL
st b3 c= proj1 b5 &
(for b6 being Element of the carrier of b1
st b6 in b3
holds b5 . b6 = (b2 .|. b6) ^2)
holds (setopfunc(b3,the carrier of b1,the carrier of b1,b4,the addF of b1)) .|. setopfunc(b3,the carrier of b1,the carrier of b1,b4,the addF of b1) = setopfunc(b3,the carrier of b1,REAL,b5,addreal);
:: BHSP_5:th 13
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 the carrier of b1
for b3 being finite OrthonormalFamily of b1
st b3 is not empty
for b4 being Function-like quasi_total Relation of the carrier of b1,REAL
st b3 c= proj1 b4 &
(for b5 being Element of the carrier of b1
st b5 in b3
holds b4 . b5 = (b2 .|. b5) ^2)
holds setopfunc(b3,the carrier of b1,REAL,b4,addreal) <= ||.b2.|| ^2;
:: BHSP_5:th 14
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total Relation of [:b1,b1:],b1
st b3 is commutative(b1) & b3 is associative(b1) & b3 is having_a_unity(b1)
for b4, b5 being finite Element of bool b2
st b4 misses b5
for b6 being Function-like quasi_total Relation of b2,b1
st b4 c= proj1 b6 & b5 c= proj1 b6
for b7 being finite Element of bool b2
st b7 = b4 \/ b5
holds setopfunc(b7,b2,b1,b6,b3) = b3 .(setopfunc(b4,b2,b1,b6,b3),setopfunc(b5,b2,b1,b6,b3));