Article LAPLACE, MML version 4.99.1005
:: LAPLACE:th 1
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being natural set
st b2 in dom b1
holds len Del(b1,b2) = (len b1) -' 1;
:: LAPLACE:th 2
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being natural set
for b5 being Matrix of b4,b4,the carrier of b1
st b2 in dom b5
holds len Deleting(b5,b2,b3) = b4 -' 1;
:: LAPLACE:th 3
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being tabular FinSequence of (the carrier of b2) *
st b1 in Seg width b3
holds width DelCol(b3,b1) = (width b3) -' 1;
:: LAPLACE:th 4
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2 being tabular FinSequence of (the carrier of b1) *
for b3 being natural set
st 1 < len b2
holds width b2 = width DelLine(b2,b3);
:: LAPLACE:th 5
theorem
for b1, b2 being natural set
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b4 being Matrix of b2,b2,the carrier of b3
for b5 being natural set
st b1 in Seg width b4
holds width Deleting(b4,b5,b1) = b2 -' 1;
:: LAPLACE:funcnot 1 => LAPLACE:func 1
definition
let a1 be non empty multMagma;
let a2 be Function-like quasi_total Relation of [:the carrier of a1,NAT:],the carrier of a1;
let a3 be Element of the carrier of a1;
let a4 be natural set;
redefine func a2 .(a3,a4) -> Element of the carrier of a1;
end;
:: LAPLACE:th 6
theorem
for b1 being natural set holds
Card Permutations b1 = b1 !;
:: LAPLACE:th 7
theorem
for b1, b2, b3 being natural set
st b2 in Seg (b1 + 1) & b3 in Seg (b1 + 1)
holds Card {b4 where b4 is Element of Permutations (b1 + 1): b4 . b2 = b3} = b1 !;
:: LAPLACE:th 8
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive Fanoian doubleLoopStr
for b3 being Element of Permutations (b1 + 2)
for b4, b5 being Element of Fin TWOELEMENTSETS Seg (b1 + 2)
st b5 = {b6 where b6 is Element of TWOELEMENTSETS Seg (b1 + 2): b6 in b4 & (Part_sgn(b3,b2)) . b6 = - 1_ b2}
holds (the multF of b2) $$(b4,Part_sgn(b3,b2)) = (power b2) .(- 1_ b2,card b5);
:: LAPLACE:th 9
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive Fanoian doubleLoopStr
for b3 being Element of Permutations (b1 + 2)
for b4, b5 being natural set
st b4 in Seg (b1 + 2) & b3 . b4 = b5
holds ex b6 being Element of Fin TWOELEMENTSETS Seg (b1 + 2) st
b6 = {{b7,b4} where b7 is Element of NAT: {b7,b4} in TWOELEMENTSETS Seg (b1 + 2)} &
(the multF of b2) $$(b6,Part_sgn(b3,b2)) = (power b2) .(- 1_ b2,b4 + b5);
:: LAPLACE:th 10
theorem
for b1, b2, b3 being natural set
st b2 in Seg (b1 + 1) & b3 in Seg (b1 + 1) & 2 <= b1
holds ex b4 being Function-like quasi_total Relation of TWOELEMENTSETS Seg b1,TWOELEMENTSETS Seg (b1 + 1) st
proj2 b4 = (TWOELEMENTSETS Seg (b1 + 1)) \ {{b5,b2} where b5 is Element of NAT: {b5,b2} in TWOELEMENTSETS Seg (b1 + 1)} &
b4 is one-to-one &
(for b5, b6 being natural set
st b5 < b6 & {b5,b6} in TWOELEMENTSETS Seg b1
holds (b6 < b2 & b5 < b2 implies b4 . {b5,b6} = {b5,b6}) &
(b2 <= b6 & b5 < b2 implies b4 . {b5,b6} = {b5,b6 + 1}) &
(b2 <= b6 & b2 <= b5 implies b4 . {b5,b6} = {b5 + 1,b6 + 1}));
:: LAPLACE:th 11
theorem
for b1 being natural set
st b1 < 2
for b2 being Element of Permutations b1 holds
b2 is even(len Permutations b1) & b2 = idseq b1;
:: LAPLACE:th 12
theorem
for b1, b2, b3 being non empty set
for b4 being Function-like quasi_total Relation of b1,Fin b2
for b5 being Function-like quasi_total Relation of Fin b2,b3
for b6 being Function-like quasi_total Relation of [:b3,b3:],b3
st (for b7, b8 being Element of Fin b2
st b7 misses b8
holds b6 .(b5 . b7,b5 . b8) = b5 . (b7 \/ b8)) &
b6 is commutative(b3) &
b6 is associative(b3) &
b6 is having_a_unity(b3) &
b5 . {} = the_unity_wrt b6
for b7 being Element of Fin b1
st for b8, b9 being set
st b8 in b7 & b9 in b7 & b4 . b8 meets b4 . b9
holds b8 = b9
holds b6 $$(b7,b5 * b4) = b6 $$(b4 .: b7,b5) &
b6 $$(b4 .: b7,b5) = b5 . union (b4 .: b7) &
union (b4 .: b7) is Element of Fin b2;
:: LAPLACE:funcnot 2 => LAPLACE:func 2
definition
let a1, a2, a3 be natural set;
let a4 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
let a5 be Matrix of a3,a3,the carrier of a4;
assume a1 in Seg a3 & a2 in Seg a3;
func Delete(A5,A1,A2) -> Matrix of a3 -' 1,a3 -' 1,the carrier of a4 equals
Deleting(a5,a1,a2);
end;
:: LAPLACE:def 1
theorem
for b1, b2, b3 being natural set
for b4 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b5 being Matrix of b3,b3,the carrier of b4
st b1 in Seg b3 & b2 in Seg b3
holds Delete(b5,b1,b2) = Deleting(b5,b1,b2);
:: LAPLACE:th 13
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
for b4, b5 being natural set
st b4 in Seg b1 & b5 in Seg b1
for b6, b7 being natural set
st b6 in Seg (b1 -' 1) & b7 in Seg (b1 -' 1)
holds (b6 < b4 & b7 < b5 implies (Delete(b3,b4,b5)) *(b6,b7) = b3 *(b6,b7)) &
(b6 < b4 & b5 <= b7 implies (Delete(b3,b4,b5)) *(b6,b7) = b3 *(b6,b7 + 1)) &
(b4 <= b6 & b7 < b5 implies (Delete(b3,b4,b5)) *(b6,b7) = b3 *(b6 + 1,b7)) &
(b4 <= b6 & b5 <= b7 implies (Delete(b3,b4,b5)) *(b6,b7) = b3 *(b6 + 1,b7 + 1));
:: LAPLACE:th 14
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
for b4, b5 being natural set
st b4 in Seg b1 & b5 in Seg b1
holds (Delete(b3,b4,b5)) @ = Delete(b3 @,b5,b4);
:: LAPLACE:th 15
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
for b4 being FinSequence of the carrier of b2
for b5, b6 being natural set
st b5 in Seg b1 & b6 in Seg b1
holds Delete(b3,b5,b6) = Delete(ReplaceLine(b3,b5,b4),b5,b6);
:: LAPLACE:funcnot 3 => LAPLACE:func 3
definition
let a1, a2, a3 be natural set;
let a4 be non empty set;
let a5 be Matrix of a2,a3,a4;
let a6 be FinSequence of a4;
func ReplaceCol(A5,A1,A6) -> Matrix of a2,a3,a4 means
len it = len a5 &
width it = width a5 &
(for b1, b2 being natural set
st [b1,b2] in Indices a5
holds (b2 = a1 or it *(b1,b2) = a5 *(b1,b2)) & (b2 = a1 implies it *(b1,a1) = a6 . b1))
if len a6 = len a5
otherwise it = a5;
end;
:: LAPLACE:def 2
theorem
for b1, b2, b3 being natural set
for b4 being non empty set
for b5 being Matrix of b2,b3,b4
for b6 being FinSequence of b4
for b7 being Matrix of b2,b3,b4 holds
(len b6 = len b5 implies (b7 = ReplaceCol(b5,b1,b6)
iff
len b7 = len b5 &
width b7 = width b5 &
(for b8, b9 being natural set
st [b8,b9] in Indices b5
holds (b9 = b1 or b7 *(b8,b9) = b5 *(b8,b9)) &
(b9 = b1 implies b7 *(b8,b1) = b6 . b8)))) &
(len b6 = len b5 or (b7 = ReplaceCol(b5,b1,b6)
iff
b7 = b5));
:: LAPLACE:funcnot 4 => LAPLACE:func 3
notation
let a1, a2, a3 be natural set;
let a4 be non empty set;
let a5 be Matrix of a2,a3,a4;
let a6 be FinSequence of a4;
synonym RCol(a5,a1,a6) for ReplaceCol(a5,a1,a6);
end;
:: LAPLACE:th 16
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 FinSequence of b4
for b7 being natural set
st b7 in Seg width b5
holds (b7 = b3 & len b6 = len b5 implies Col(ReplaceCol(b5,b3,b6),b7) = b6) &
(b7 = b3 or Col(ReplaceCol(b5,b3,b6),b7) = Col(b5,b7));
:: LAPLACE:th 17
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 FinSequence of b4
st not b3 in Seg width b5
holds ReplaceCol(b5,b3,b6) = b5;
:: LAPLACE:th 18
theorem
for b1, b2, b3 being natural set
for b4 being non empty set
for b5 being Matrix of b1,b2,b4 holds
ReplaceCol(b5,b3,Col(b5,b3)) = b5;
:: LAPLACE:th 19
theorem
for b1, b2, b3 being natural set
for b4 being non empty set
for b5 being FinSequence of b4
for b6 being Matrix of b1,b2,b4
for b7 being Matrix of b2,b1,b4
st b7 = b6 @ & (b2 = {} implies b1 = {})
holds ReplaceCol(b6,b3,b5) = (ReplaceLine(b7,b3,b5)) @;
:: LAPLACE:funcnot 5 => LAPLACE:func 4
definition
let a1, a2 be natural set;
let a3 be Element of Permutations (a2 + 1);
assume a1 in Seg (a2 + 1);
func Rem(A3,A1) -> Element of Permutations a2 means
for b1 being natural set
st b1 in Seg a2
holds (a1 <= b1 or (a3 . a1 <= a3 . b1 or it . b1 = a3 . b1) &
(a3 . a1 <= a3 . b1 implies it . b1 = (a3 . b1) - 1)) &
(a1 <= b1 implies (a3 . a1 <= a3 . (b1 + 1) or it . b1 = a3 . (b1 + 1)) &
(a3 . a1 <= a3 . (b1 + 1) implies it . b1 = (a3 . (b1 + 1)) - 1));
end;
:: LAPLACE:def 3
theorem
for b1, b2 being natural set
for b3 being Element of Permutations (b2 + 1)
st b1 in Seg (b2 + 1)
for b4 being Element of Permutations b2 holds
b4 = Rem(b3,b1)
iff
for b5 being natural set
st b5 in Seg b2
holds (b1 <= b5 or (b3 . b1 <= b3 . b5 or b4 . b5 = b3 . b5) &
(b3 . b1 <= b3 . b5 implies b4 . b5 = (b3 . b5) - 1)) &
(b1 <= b5 implies (b3 . b1 <= b3 . (b5 + 1) or b4 . b5 = b3 . (b5 + 1)) &
(b3 . b1 <= b3 . (b5 + 1) implies b4 . b5 = (b3 . (b5 + 1)) - 1));
:: LAPLACE:th 20
theorem
for b1, b2, b3 being natural set
st b2 in Seg (b1 + 1) & b3 in Seg (b1 + 1)
for b4 being set
st b4 = {b5 where b5 is Element of Permutations (b1 + 1): b5 . b2 = b3}
holds ex b5 being Function-like quasi_total Relation of b4,Permutations b1 st
b5 is bijective(b4, Permutations b1) &
(for b6 being Element of Permutations (b1 + 1)
st b6 . b2 = b3
holds b5 . b6 = Rem(b6,b2));
:: LAPLACE:th 21
theorem
for b1 being natural set
for b2 being Element of Permutations (b1 + 1)
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b4 being Element of the carrier of b3
for b5, b6 being natural set
st b5 in Seg (b1 + 1) & b2 . b5 = b6
holds -(b4,b2) = ((power b3) .(- 1_ b3,b5 + b6)) * -(b4,Rem(b2,b5));
:: LAPLACE:th 22
theorem
for b1 being natural set
for b2 being Element of Permutations (b1 + 1)
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b4, b5 being natural set
st b4 in Seg (b1 + 1) & b2 . b4 = b5
for b6 being Matrix of b1 + 1,b1 + 1,the carrier of b3
for b7 being Matrix of b1,b1,the carrier of b3
st b7 = Delete(b6,b4,b5)
holds (Path_product b6) . b2 = (((power b3) .(- 1_ b3,b4 + b5)) * (b6 *(b4,b5))) * ((Path_product b7) . Rem(b2,b4));
:: LAPLACE:funcnot 6 => LAPLACE:func 5
definition
let a1, a2, a3 be natural set;
let a4 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
let a5 be Matrix of a3,a3,the carrier of a4;
func Minor(A5,A1,A2) -> Element of the carrier of a4 equals
Det Delete(a5,a1,a2);
end;
:: LAPLACE:def 4
theorem
for b1, b2, b3 being natural set
for b4 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b5 being Matrix of b3,b3,the carrier of b4 holds
Minor(b5,b1,b2) = Det Delete(b5,b1,b2);
:: LAPLACE:funcnot 7 => LAPLACE:func 6
definition
let a1, a2, a3 be natural set;
let a4 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
let a5 be Matrix of a3,a3,the carrier of a4;
func Cofactor(A5,A1,A2) -> Element of the carrier of a4 equals
((power a4) .(- 1_ a4,a1 + a2)) * Minor(a5,a1,a2);
end;
:: LAPLACE:def 5
theorem
for b1, b2, b3 being natural set
for b4 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b5 being Matrix of b3,b3,the carrier of b4 holds
Cofactor(b5,b1,b2) = ((power b4) .(- 1_ b4,b1 + b2)) * Minor(b5,b1,b2);
:: LAPLACE:th 23
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being natural set
st b3 in Seg b1 & b4 in Seg b1
for b5 being Element of Fin Permutations b1
st b5 = {b6 where b6 is Element of Permutations b1: b6 . b3 = b4}
for b6 being Matrix of b1,b1,the carrier of b2 holds
(the addF of b2) $$(b5,Path_product b6) = (b6 *(b3,b4)) * Cofactor(b6,b3,b4);
:: LAPLACE:th 24
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
for b4, b5 being natural set
st b4 in Seg b1 & b5 in Seg b1
holds Minor(b3,b4,b5) = Minor(b3 @,b5,b4);
:: LAPLACE:funcnot 8 => LAPLACE:func 7
definition
let a1 be natural set;
let a2 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
let a3 be Matrix of a1,a1,the carrier of a2;
func Matrix_of_Cofactor A3 -> Matrix of a1,a1,the carrier of a2 means
for b1, b2 being natural set
st [b1,b2] in Indices it
holds it *(b1,b2) = Cofactor(a3,b1,b2);
end;
:: LAPLACE:def 6
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2 holds
b4 = Matrix_of_Cofactor b3
iff
for b5, b6 being natural set
st [b5,b6] in Indices b4
holds b4 *(b5,b6) = Cofactor(b3,b5,b6);
:: LAPLACE:funcnot 9 => LAPLACE:func 8
definition
let a1, a2 be natural set;
let a3 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
let a4 be Matrix of a1,a1,the carrier of a3;
func LaplaceExpL(A4,A2) -> FinSequence of the carrier of a3 means
len it = a1 &
(for b1 being natural set
st b1 in dom it
holds it . b1 = (a4 *(a2,b1)) * Cofactor(a4,a2,b1));
end;
:: LAPLACE:def 7
theorem
for b1, b2 being natural set
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b4 being Matrix of b1,b1,the carrier of b3
for b5 being FinSequence of the carrier of b3 holds
b5 = LaplaceExpL(b4,b2)
iff
len b5 = b1 &
(for b6 being natural set
st b6 in dom b5
holds b5 . b6 = (b4 *(b2,b6)) * Cofactor(b4,b2,b6));
:: LAPLACE:funcnot 10 => LAPLACE:func 9
definition
let a1, a2 be natural set;
let a3 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
let a4 be Matrix of a1,a1,the carrier of a3;
func LaplaceExpC(A4,A2) -> FinSequence of the carrier of a3 means
len it = a1 &
(for b1 being natural set
st b1 in dom it
holds it . b1 = (a4 *(b1,a2)) * Cofactor(a4,b1,a2));
end;
:: LAPLACE:def 8
theorem
for b1, b2 being natural set
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b4 being Matrix of b1,b1,the carrier of b3
for b5 being FinSequence of the carrier of b3 holds
b5 = LaplaceExpC(b4,b2)
iff
len b5 = b1 &
(for b6 being natural set
st b6 in dom b5
holds b5 . b6 = (b4 *(b6,b2)) * Cofactor(b4,b6,b2));
:: LAPLACE:th 25
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being natural set
for b4 being Matrix of b1,b1,the carrier of b2
st b3 in Seg b1
holds Det b4 = Sum LaplaceExpL(b4,b3);
:: LAPLACE:th 26
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
for b4 being natural set
st b4 in Seg b1
holds LaplaceExpC(b3,b4) = LaplaceExpL(b3 @,b4);
:: LAPLACE:th 27
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being natural set
for b4 being Matrix of b1,b1,the carrier of b2
st b3 in Seg b1
holds Det b4 = Sum LaplaceExpC(b4,b3);
:: LAPLACE:th 28
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being FinSequence of the carrier of b2
for b4 being Matrix of b1,b1,the carrier of b2
for b5 being Element of Permutations b1
for b6 being natural set
st len b3 = b1 & b6 in Seg b1
holds mlt(Line(Matrix_of_Cofactor b4,b6),b3) = LaplaceExpL(ReplaceLine(b4,b6,b3),b6);
:: LAPLACE:th 29
theorem
for b1, b2, b3 being natural set
for b4 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b5 being Matrix of b2,b2,the carrier of b4
st b1 in Seg b2
holds (Line(b5,b3)) "*" Col((Matrix_of_Cofactor b5) @,b1) = Det ReplaceLine(b5,b1,Line(b5,b3));
:: LAPLACE:th 30
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
st Det b3 <> 0. b2
holds b3 * ((Det b3) " * ((Matrix_of_Cofactor b3) @)) = 1.(b2,b1);
:: LAPLACE:th 31
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
for b4 being FinSequence of the carrier of b2
for b5 being natural set
st len b4 = b1 & b5 in Seg b1
holds mlt(Col(Matrix_of_Cofactor b3,b5),b4) = LaplaceExpL(ReplaceLine(b3 @,b5,b4),b5);
:: LAPLACE:th 32
theorem
for b1, b2, b3 being natural set
for b4 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b5 being Matrix of b2,b2,the carrier of b4
st b1 in Seg b2 & b3 in Seg b2
holds (Line((Matrix_of_Cofactor b5) @,b1)) "*" Col(b5,b3) = Det ReplaceLine(b5 @,b1,Line(b5 @,b3));
:: LAPLACE:th 33
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
st Det b3 <> 0. b2
holds ((Det b3) " * ((Matrix_of_Cofactor b3) @)) * b3 = 1.(b2,b1);
:: LAPLACE:th 34
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2 holds
b3 is invertible(b1, b2)
iff
Det b3 <> 0. b2;
:: LAPLACE:th 35
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
st Det b3 <> 0. b2
holds b3 ~ = (Det b3) " * ((Matrix_of_Cofactor b3) @);
:: LAPLACE:th 36
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
st b3 is invertible(b1, b2)
for b4, b5 being natural set
st [b4,b5] in Indices (b3 ~)
holds b3 ~ *(b4,b5) = ((Det b3) " * ((power b2) .(- 1_ b2,b4 + b5))) * Minor(b3,b5,b4);
:: LAPLACE:th 37
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
st Det b3 <> 0. b2
for b4, b5 being tabular FinSequence of (the carrier of b2) *
st len b4 = b1 & b3 * b4 = b5
holds b4 = b3 ~ * b5 &
(for b6, b7 being natural set
st [b6,b7] in Indices b4
holds b4 *(b6,b7) = (Det b3) " * Det ReplaceCol(b3,b6,Col(b5,b7)));
:: LAPLACE:th 38
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
st Det b3 <> 0. b2
for b4, b5 being tabular FinSequence of (the carrier of b2) *
st width b4 = b1 & b4 * b3 = b5
holds b4 = b5 * (b3 ~) &
(for b6, b7 being natural set
st [b6,b7] in Indices b4
holds b4 *(b6,b7) = (Det b3) " * Det ReplaceLine(b3,b7,Line(b5,b6)));
:: LAPLACE:funcnot 11 => LAPLACE:func 10
definition
let a1 be non empty set;
let a2 be FinSequence of a1;
redefine func <*a2*> -> Matrix of 1,len a2,a1;
end;
:: LAPLACE:funcnot 12 => LAPLACE:func 11
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
let a2 be tabular FinSequence of (the carrier of a1) *;
let a3 be FinSequence of the carrier of a1;
func A2 * A3 -> tabular FinSequence of (the carrier of a1) * equals
a2 * (<*a3*> @);
end;
:: LAPLACE:def 9
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2 being tabular FinSequence of (the carrier of b1) *
for b3 being FinSequence of the carrier of b1 holds
b2 * b3 = b2 * (<*b3*> @);
:: LAPLACE:funcnot 13 => LAPLACE:func 12
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
let a2 be tabular FinSequence of (the carrier of a1) *;
let a3 be FinSequence of the carrier of a1;
func A3 * A2 -> tabular FinSequence of (the carrier of a1) * equals
<*a3*> * a2;
end;
:: LAPLACE:def 10
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2 being tabular FinSequence of (the carrier of b1) *
for b3 being FinSequence of the carrier of b1 holds
b3 * b2 = <*b3*> * b2;
:: LAPLACE:th 39
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
st Det b3 <> 0. b2
for b4, b5 being FinSequence of the carrier of b2
st len b4 = b1 & b3 * b4 = <*b5*> @
holds <*b4*> @ = b3 ~ * b5 &
(for b6 being natural set
st b6 in Seg b1
holds b4 . b6 = (Det b3) " * Det ReplaceCol(b3,b6,b5));
:: LAPLACE:th 40
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
st Det b3 <> 0. b2
for b4, b5 being FinSequence of the carrier of b2
st len b4 = b1 & b4 * b3 = <*b5*>
holds <*b4*> = b5 * (b3 ~) &
(for b6 being natural set
st b6 in Seg b1
holds b4 . b6 = (Det b3) " * Det ReplaceLine(b3,b6,b5));