Article MATRLIN, MML version 4.99.1005
:: MATRLIN:funcnot 1 => MATRLIN:func 1
definition
let a1, a2 be set;
let a3 be non empty FinSequenceSet of a1;
let a4 be Function-like Relation of a2,a3;
let a5 be set;
redefine func a4 /. a5 -> Element of a3;
end;
:: MATRLIN:funcnot 2 => MATRLIN:func 2
definition
let a1 be non empty set;
let a2 be natural set;
let a3 be tabular FinSequence of a1 *;
redefine func Del(a3,a2) -> tabular FinSequence of a1 *;
end;
:: MATRLIN:th 3
theorem
for b1 being natural set
for b2 being Relation-like Function-like FinSequence-like set
st len b2 = b1 + 1
holds len Del(b2,b1 + 1) = b1;
:: MATRLIN:th 4
theorem
for b1, b2 being natural set
for b3 being non empty set
for b4 being Matrix of b1 + 1,b2,b3
for b5 being tabular FinSequence of b3 * holds
(b1 <= {} or width b4 = width Del(b4,b1 + 1)) &
(b5 = <*b4 . (b1 + 1)*> implies width b4 = width b5);
:: MATRLIN:th 5
theorem
for b1, b2 being natural set
for b3 being non empty set
for b4 being Matrix of b1 + 1,b2,b3 holds
Del(b4,b1 + 1) is Matrix of b1,b2,b3;
:: MATRLIN:th 6
theorem
for b1 being natural set
for b2 being Relation-like Function-like FinSequence-like set
st len b2 = b1 + 1
holds b2 = (Del(b2,len b2)) ^ <*b2 . len b2*>;
:: MATRLIN:funcnot 3 => MATRLIN:func 3
definition
let a1 be non empty set;
let a2 be FinSequence of a1;
redefine func <*a2*> -> Matrix of 1,len a2,a1;
end;
:: MATRLIN:th 7
theorem
for b1 being set
for b2 being Relation-like Function-like FinSequence-like set holds
(Sgm (b2 " b1)) ^ Sgm (b2 " ((proj2 b2) \ b1)) is Function-like quasi_total bijective Relation of proj1 b2,proj1 b2;
:: MATRLIN:attrnot 1 => MATRLIN:attr 1
definition
let a1 be Relation-like Function-like set;
attr a1 is FinSequence-yielding means
for b1 being set
st b1 in proj1 a1
holds a1 . b1 is Relation-like Function-like FinSequence-like set;
end;
:: MATRLIN:dfs 1
definiens
let a1 be Relation-like Function-like set;
To prove
a1 is FinSequence-yielding
it is sufficient to prove
thus for b1 being set
st b1 in proj1 a1
holds a1 . b1 is Relation-like Function-like FinSequence-like set;
:: MATRLIN:def 1
theorem
for b1 being Relation-like Function-like set holds
b1 is FinSequence-yielding
iff
for b2 being set
st b2 in proj1 b1
holds b1 . b2 is Relation-like Function-like FinSequence-like set;
:: MATRLIN:exreg 1
registration
cluster Relation-like Function-like FinSequence-yielding set;
end;
:: MATRLIN:funcnot 4 => MATRLIN:func 4
definition
let a1, a2 be Relation-like Function-like FinSequence-yielding set;
func A1 ^^ A2 -> Relation-like Function-like FinSequence-yielding set means
proj1 it = (proj1 a1) /\ proj1 a2 &
(for b1 being set
st b1 in proj1 it
for b2, b3 being Relation-like Function-like FinSequence-like set
st b2 = a1 . b1 & b3 = a2 . b1
holds it . b1 = b2 ^ b3);
end;
:: MATRLIN:def 2
theorem
for b1, b2, b3 being Relation-like Function-like FinSequence-yielding set holds
b3 = b1 ^^ b2
iff
proj1 b3 = (proj1 b1) /\ proj1 b2 &
(for b4 being set
st b4 in proj1 b3
for b5, b6 being Relation-like Function-like FinSequence-like set
st b5 = b1 . b4 & b6 = b2 . b4
holds b3 . b4 = b5 ^ b6);
:: MATRLIN:th 9
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Linear_Combination of b2
for b5 being Element of bool the carrier of b2
st b5 is linearly-independent(b1, b2) & Carrier b3 c= b5 & Carrier b4 c= b5 & Sum b3 = Sum b4
holds b3 = b4;
:: MATRLIN:th 10
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4, b5 being Linear_Combination of b2
for b6 being Element of bool the carrier of b2
st b6 is linearly-independent(b1, b2) & Carrier b3 c= b6 & Carrier b4 c= b6 & Carrier b5 c= b6 & Sum b3 = (Sum b4) + Sum b5
holds b3 = b4 + b5;
:: MATRLIN:th 11
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b1
for b4, b5 being Linear_Combination of b2
for b6 being Element of bool the carrier of b2
st b6 is linearly-independent(b1, b2) & Carrier b4 c= b6 & Carrier b5 c= b6 & b3 <> 0. b1 & Sum b4 = b3 * Sum b5
holds b4 = b3 * b5;
:: MATRLIN:th 12
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2
for b4 being Basis of b2 holds
ex b5 being Linear_Combination of b2 st
b3 = Sum b5 & Carrier b5 c= b4;
:: MATRLIN:attrnot 2 => MATRLIN:attr 2
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
attr a2 is finite-dimensional means
ex b1 being finite Element of bool the carrier of a2 st
b1 is Basis of a2;
end;
:: MATRLIN:dfs 3
definiens
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
To prove
a2 is finite-dimensional
it is sufficient to prove
thus ex b1 being finite Element of bool the carrier of a2 st
b1 is Basis of a2;
:: MATRLIN:def 3
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
b2 is finite-dimensional(b1)
iff
ex b3 being finite Element of bool the carrier of b2 st
b3 is Basis of b2;
:: MATRLIN:exreg 2
registration
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
cluster non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over a1;
end;
:: MATRLIN:modenot 1 => MATRLIN:mode 1
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over a1;
mode OrdBasis of A2 -> FinSequence of the carrier of a2 means
it is one-to-one & proj2 it is Basis of a2;
end;
:: MATRLIN:dfs 4
definiens
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over a1;
let a3 be FinSequence of the carrier of a2;
To prove
a3 is OrdBasis of a2
it is sufficient to prove
thus a3 is one-to-one & proj2 a3 is Basis of a2;
:: MATRLIN:def 4
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being FinSequence of the carrier of b2 holds
b3 is OrdBasis of b2
iff
b3 is one-to-one & proj2 b3 is Basis of b2;
:: MATRLIN:funcnot 5 => MATRLIN:func 5
definition
let a1 be non empty doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4, a5 be Function-like quasi_total Relation of the carrier of a2,the carrier of a3;
func A4 + A5 -> Function-like quasi_total Relation of the carrier of a2,the carrier of a3 means
for b1 being Element of the carrier of a2 holds
it . b1 = (a4 . b1) + (a5 . b1);
end;
:: MATRLIN:def 5
theorem
for b1 being non empty doubleLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4, b5, b6 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3 holds
b6 = b4 + b5
iff
for b7 being Element of the carrier of b2 holds
b6 . b7 = (b4 . b7) + (b5 . b7);
:: MATRLIN:funcnot 6 => MATRLIN:func 6
definition
let a1 be non empty doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of the carrier of a2,the carrier of a3;
let a5 be Element of the carrier of a1;
func A5 * A4 -> Function-like quasi_total Relation of the carrier of a2,the carrier of a3 means
for b1 being Element of the carrier of a2 holds
it . b1 = a5 * (a4 . b1);
end;
:: MATRLIN:def 6
theorem
for b1 being non empty doubleLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
for b5 being Element of the carrier of b1
for b6 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3 holds
b6 = b5 * b4
iff
for b7 being Element of the carrier of b2 holds
b6 . b7 = b5 * (b4 . b7);
:: MATRLIN:th 13
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being Element of the carrier of b2
for b4 being FinSequence of the carrier of b2
for b5 being FinSequence of the carrier of b1
st len b4 = len b5 &
(for b6 being natural set
for b7 being Element of the carrier of b1
st b6 in proj1 b4 & b7 = b5 . b6
holds b4 . b6 = b7 * b3)
holds Sum b4 = (Sum b5) * b3;
:: MATRLIN:th 14
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being Element of the carrier of b2
for b4 being FinSequence of the carrier of b1
for b5 being FinSequence of the carrier of b2
st len b4 = len b5 &
(for b6 being natural set
st b6 in proj1 b4
holds b5 . b6 = (b4 /. b6) * b3)
holds Sum b5 = (Sum b4) * b3;
:: MATRLIN:th 15
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being FinSequence of the carrier of b1
st for b3 being natural set
st b3 in proj1 b2
holds b2 /. b3 = 0. b1
holds Sum b2 = 0. b1;
:: MATRLIN:funcnot 7 => MATRLIN:func 7
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over a1;
let a3 be FinSequence of the carrier of a1;
let a4 be FinSequence of the carrier of a2;
func lmlt(A3,A4) -> FinSequence of the carrier of a2 equals
(the lmult of a2) .:(a3,a4);
end;
:: MATRLIN:def 7
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being FinSequence of the carrier of b1
for b4 being FinSequence of the carrier of b2 holds
lmlt(b3,b4) = (the lmult of b2) .:(b3,b4);
:: MATRLIN:th 16
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being FinSequence of the carrier of b2
for b4 being FinSequence of the carrier of b1
st proj1 b4 = proj1 b3
holds proj1 lmlt(b4,b3) = proj1 b4;
:: MATRLIN:funcnot 8 => MATRLIN:func 8
definition
let a1 be non empty addLoopStr;
let a2 be FinSequence of (the carrier of a1) *;
func Sum A2 -> FinSequence of the carrier of a1 means
len it = len a2 &
(for b1 being natural set
st b1 in proj1 it
holds it /. b1 = Sum (a2 /. b1));
end;
:: MATRLIN:def 8
theorem
for b1 being non empty addLoopStr
for b2 being FinSequence of (the carrier of b1) *
for b3 being FinSequence of the carrier of b1 holds
b3 = Sum b2
iff
len b3 = len b2 &
(for b4 being natural set
st b4 in proj1 b3
holds b3 /. b4 = Sum (b2 /. b4));
:: MATRLIN:th 17
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being tabular FinSequence of (the carrier of b2) *
st len b3 = {}
holds Sum Sum b3 = 0. b2;
:: MATRLIN:th 18
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b2
for b4 being Matrix of b1 + 1,{},the carrier of b3 holds
Sum Sum b4 = 0. b3;
:: MATRLIN:th 19
theorem
for b1 being non empty set
for b2 being Element of b1 holds
<*<*b2*>*> = <*<*b2*>*> @;
:: MATRLIN:th 20
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
for b5 being FinSequence of the carrier of b2
st b4 is linear(b1, b2, b3)
holds b4 . Sum b5 = Sum (b4 * b5);
:: MATRLIN:th 21
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
for b5 being FinSequence of the carrier of b1
for b6 being FinSequence of the carrier of b3
st len b6 = len b5 & b4 is linear(b1, b3, b2)
holds b4 * lmlt(b5,b6) = lmlt(b5,b4 * b6);
:: MATRLIN:th 22
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
for b5 being OrdBasis of b3
for b6 being FinSequence of the carrier of b1
st len b6 = len b5 & b4 is linear(b1, b3, b2)
holds b4 . Sum lmlt(b6,b5) = Sum lmlt(b6,b4 * b5);
:: MATRLIN:th 23
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3, b4 being FinSequence of the carrier of b2
for b5 being Linear_Combination of b2
for b6 being Function-like quasi_total bijective Relation of proj1 b3,proj1 b3
st b4 = b3 * b6
holds b5 (#) b4 = (b5 (#) b3) * b6;
:: MATRLIN:th 24
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being FinSequence of the carrier of b2
for b4 being Linear_Combination of b2
st b3 is one-to-one & Carrier b4 c= proj2 b3
holds Sum (b4 (#) b3) = Sum b4;
:: MATRLIN:th 25
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b4, b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
for b6 being set
for b7 being FinSequence of the carrier of b3
st proj2 b7 c= b6 &
b4 is linear(b1, b3, b2) &
b5 is linear(b1, b3, b2) &
(for b8 being Element of the carrier of b3
st b8 in b6
holds b4 . b8 = b5 . b8)
holds b4 . Sum b7 = b5 . Sum b7;
:: MATRLIN:th 26
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b4, b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
st b4 is linear(b1, b3, b2) & b5 is linear(b1, b3, b2)
for b6 being OrdBasis of b3
st {} < len b6 & b4 * b6 = b5 * b6
holds b4 = b5;
:: MATRLIN:condreg 1
registration
let a1 be non empty set;
cluster tabular -> FinSequence-yielding (FinSequence of a1 *);
end;
:: MATRLIN:funcnot 9 => MATRLIN:func 9
definition
let a1 be non empty set;
let a2, a3 be tabular FinSequence of a1 *;
redefine func a2 ^^ a3 -> tabular FinSequence of a1 *;
end;
:: MATRLIN:funcnot 10 => MATRLIN:func 10
definition
let a1 be non empty set;
let a2, a3, a4 be natural set;
let a5 be Matrix of a2,a4,a1;
let a6 be Matrix of a3,a4,a1;
redefine func a5 ^ a6 -> Matrix of a2 + a3,a4,a1;
end;
:: MATRLIN:th 27
theorem
for b1, b2, b3, b4 being natural set
for b5 being non empty set
for b6 being Matrix of b1,b2,b5
for b7 being Matrix of b3,b2,b5
st b4 in proj1 b6
holds Line(b6 ^ b7,b4) = Line(b6,b4);
:: MATRLIN:th 28
theorem
for b1, b2, b3 being natural set
for b4 being non empty set
for b5 being Matrix of b1,b2,b4
for b6 being Matrix of b3,b2,b4
st width b5 = width b6
holds width (b5 ^ b6) = width b5;
:: MATRLIN:th 29
theorem
for b1, b2, b3, b4, b5 being natural set
for b6 being non empty set
for b7 being Matrix of b1,b2,b6
for b8 being Matrix of b3,b2,b6
st b4 in proj1 b8 & b5 = (len b7) + b4
holds Line(b7 ^ b8,b5) = Line(b8,b4);
:: MATRLIN:th 30
theorem
for b1, b2, b3 being natural set
for b4 being non empty set
for b5 being Matrix of b1,b2,b4
for b6 being Matrix of b3,b2,b4
st width b5 = width b6
for b7 being natural set
st b7 in Seg width b5
holds Col(b5 ^ b6,b7) = (Col(b5,b7)) ^ Col(b6,b7);
:: MATRLIN:th 31
theorem
for b1, b2, b3 being natural set
for b4 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b5 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b4
for b6 being Matrix of b1,b2,the carrier of b5
for b7 being Matrix of b3,b2,the carrier of b5 holds
Sum (b6 ^ b7) = (Sum b6) ^ Sum b7;
:: MATRLIN:th 32
theorem
for b1, b2, b3 being natural set
for b4 being non empty set
for b5 being Matrix of b1,b2,b4
for b6 being Matrix of b3,b2,b4
st width b5 = width b6
holds (b5 ^ b6) @ = b5 @ ^^ (b6 @);
:: MATRLIN:th 33
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3, b4 being tabular FinSequence of (the carrier of b2) * holds
(Sum b3) + Sum b4 = Sum (b3 ^^ b4);
:: MATRLIN:th 34
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3, b4 being FinSequence of the carrier of b2
st len b3 = len b4
holds Sum (b3 + b4) = (Sum b3) + Sum b4;
:: MATRLIN:th 35
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3, b4 being tabular FinSequence of (the carrier of b2) *
st len b3 = len b4
holds (Sum Sum b3) + Sum Sum b4 = Sum Sum (b3 ^^ b4);
:: MATRLIN:th 37
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being tabular FinSequence of (the carrier of b2) * holds
Sum Sum b3 = Sum Sum (b3 @);
:: MATRLIN:th 38
theorem
for b1, b2 being natural set
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b4 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b3
for b5 being Matrix of b1,b2,the carrier of b3
st {} < b1 & {} < b2
for b6, b7 being FinSequence of the carrier of b3
st len b6 = b1 &
len b7 = b2 &
(for b8 being natural set
st b8 in proj1 b7
holds b7 /. b8 = Sum mlt(b6,Col(b5,b8)))
for b8, b9 being FinSequence of the carrier of b4
st len b8 = b2 &
len b9 = b1 &
(for b10 being natural set
st b10 in proj1 b9
holds b9 /. b10 = Sum lmlt(Line(b5,b10),b8))
holds Sum lmlt(b6,b9) = Sum lmlt(b7,b8);
:: MATRLIN:funcnot 11 => MATRLIN:func 11
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over a1;
let a3 be OrdBasis of a2;
let a4 be Element of the carrier of a2;
func A4 |-- A3 -> FinSequence of the carrier of a1 means
len it = len a3 &
(ex b1 being Linear_Combination of a2 st
a4 = Sum b1 &
Carrier b1 c= proj2 a3 &
(for b2 being natural set
st 1 <= b2 & b2 <= len it
holds it /. b2 = b1 . (a3 /. b2)));
end;
:: MATRLIN:def 9
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being OrdBasis of b2
for b4 being Element of the carrier of b2
for b5 being FinSequence of the carrier of b1 holds
b5 = b4 |-- b3
iff
len b5 = len b3 &
(ex b6 being Linear_Combination of b2 st
b4 = Sum b6 &
Carrier b6 c= proj2 b3 &
(for b7 being natural set
st 1 <= b7 & b7 <= len b5
holds b5 /. b7 = b6 . (b3 /. b7)));
:: MATRLIN:th 39
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being OrdBasis of b2
for b4, b5 being Element of the carrier of b2
st b4 |-- b3 = b5 |-- b3
holds b4 = b5;
:: MATRLIN:th 40
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being OrdBasis of b2
for b4 being Element of the carrier of b2 holds
b4 = Sum lmlt(b4 |-- b3,b3);
:: MATRLIN:th 41
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being OrdBasis of b2
for b4 being FinSequence of the carrier of b1
st len b4 = len b3
holds b4 = (Sum lmlt(b4,b3)) |-- b3;
:: MATRLIN:th 42
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
for b5 being OrdBasis of b2
for b6 being OrdBasis of b3
for b7, b8 being FinSequence of the carrier of b1
st len b7 = len b5
for b9 being natural set
st b9 in proj1 b6 &
len b8 = len b5 &
(for b10 being natural set
st b10 in proj1 b5
holds b8 . b10 = ((b4 . (b5 /. b10)) |-- b6) /. b9) &
{} < len b5
holds ((Sum lmlt(b7,b4 * b5)) |-- b6) /. b9 = Sum mlt(b7,b8);
:: MATRLIN:funcnot 12 => MATRLIN:func 12
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2, a3 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over a1;
let a4 be Function-like quasi_total Relation of the carrier of a2,the carrier of a3;
let a5 be FinSequence of the carrier of a2;
let a6 be OrdBasis of a3;
func AutMt(A4,A5,A6) -> tabular FinSequence of (the carrier of a1) * means
len it = len a5 &
(for b1 being natural set
st b1 in proj1 a5
holds it /. b1 = (a4 . (a5 /. b1)) |-- a6);
end;
:: MATRLIN:def 10
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
for b5 being FinSequence of the carrier of b2
for b6 being OrdBasis of b3
for b7 being tabular FinSequence of (the carrier of b1) * holds
b7 = AutMt(b4,b5,b6)
iff
len b7 = len b5 &
(for b8 being natural set
st b8 in proj1 b5
holds b7 /. b8 = (b4 . (b5 /. b8)) |-- b6);
:: MATRLIN:th 43
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
for b5 being OrdBasis of b2
for b6 being OrdBasis of b3
st len b5 = {}
holds AutMt(b4,b5,b6) = {};
:: MATRLIN:th 44
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
for b5 being OrdBasis of b2
for b6 being OrdBasis of b3
st {} < len b5
holds width AutMt(b4,b5,b6) = len b6;
:: MATRLIN:th 45
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b4, b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
for b6 being OrdBasis of b2
for b7 being OrdBasis of b3
st b4 is linear(b1, b2, b3) & b5 is linear(b1, b2, b3) & AutMt(b4,b6,b7) = AutMt(b5,b6,b7) & {} < len b6
holds b4 = b5;
:: MATRLIN:th 46
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3, b4 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
for b6 being Function-like quasi_total Relation of the carrier of b3,the carrier of b4
for b7 being OrdBasis of b2
for b8 being OrdBasis of b3
for b9 being OrdBasis of b4
st b5 is linear(b1, b2, b3) & b6 is linear(b1, b3, b4) & {} < len b7 & {} < len b8 & {} < len b9
holds AutMt(b6 * b5,b7,b9) = (AutMt(b5,b7,b8)) * AutMt(b6,b8,b9);
:: MATRLIN:th 47
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b4, b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
for b6 being OrdBasis of b2
for b7 being OrdBasis of b3 holds
AutMt(b4 + b5,b6,b7) = (AutMt(b4,b6,b7)) + AutMt(b5,b6,b7);
:: MATRLIN:th 48
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being Element of the carrier of b1
for b3, b4 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b4
for b6 being OrdBasis of b3
for b7 being OrdBasis of b4
st b2 <> 0. b1
holds AutMt(b2 * b5,b6,b7) = b2 * AutMt(b5,b6,b7);