Article MATRIX11, MML version 4.99.1005
:: MATRIX11:funcnot 1 => SGRAPH1:func 2
notation
let a1 be set;
synonym 2Set a1 for TWOELEMENTSETS a1;
end;
:: MATRIX11:th 1
theorem
for b1 being set
for b2 being natural set holds
b1 in TWOELEMENTSETS Seg b2
iff
ex b3, b4 being natural set st
b3 in Seg b2 & b4 in Seg b2 & b3 < b4 & b1 = {b3,b4};
:: MATRIX11:th 2
theorem
TWOELEMENTSETS Seg {} = {} & TWOELEMENTSETS Seg 1 = {};
:: MATRIX11:th 3
theorem
for b1 being natural set
st 2 <= b1
holds {1,2} in TWOELEMENTSETS Seg b1;
:: MATRIX11:funcreg 1
registration
let a1 be natural set;
cluster TWOELEMENTSETS Seg (a1 + 2) -> non empty finite;
end;
:: MATRIX11:funcreg 2
registration
let a1 be natural set;
let a2 be set;
let a3 be Element of Permutations a1;
cluster a3 . a2 -> natural;
end;
:: MATRIX11:funcreg 3
registration
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
cluster the multF of a1 -> Function-like quasi_total having_a_unity;
end;
:: MATRIX11:funcreg 4
registration
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
cluster the multF of a1 -> Function-like quasi_total associative;
end;
:: MATRIX11:funcnot 2 => MATRIX11:func 1
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 Element of Permutations (a1 + 2);
func Part_sgn(A3,A2) -> Function-like quasi_total Relation of TWOELEMENTSETS Seg (a1 + 2),the carrier of a2 means
for b1, b2 being Element of NAT
st b1 in Seg (a1 + 2) & b2 in Seg (a1 + 2) & b1 < b2
holds (a3 . b2 <= a3 . b1 or it . {b1,b2} = 1_ a2) &
(a3 . b1 <= a3 . b2 or it . {b1,b2} = - 1_ a2);
end;
:: MATRIX11:def 1
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 Element of Permutations (b1 + 2)
for b4 being Function-like quasi_total Relation of TWOELEMENTSETS Seg (b1 + 2),the carrier of b2 holds
b4 = Part_sgn(b3,b2)
iff
for b5, b6 being Element of NAT
st b5 in Seg (b1 + 2) & b6 in Seg (b1 + 2) & b5 < b6
holds (b3 . b6 <= b3 . b5 or b4 . {b5,b6} = 1_ b2) &
(b3 . b5 <= b3 . b6 or b4 . {b5,b6} = - 1_ b2);
:: MATRIX11:th 4
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 Element of Permutations (b1 + 2)
for b4 being Element of Fin TWOELEMENTSETS Seg (b1 + 2)
st for b5 being set
st b5 in b4
holds (Part_sgn(b3,b2)) . b5 = 1_ b2
holds (the multF of b2) $$(b4,Part_sgn(b3,b2)) = 1_ b2;
:: MATRIX11:th 5
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Element of Permutations (b1 + 2)
for b4 being Element of TWOELEMENTSETS Seg (b1 + 2)
st (Part_sgn(b3,b2)) . b4 <> 1_ b2
holds (Part_sgn(b3,b2)) . b4 = - 1_ b2;
:: MATRIX11:th 6
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Element of Permutations (b1 + 2)
for b5, b6 being natural set
st b5 in Seg (b1 + 2) & b6 in Seg (b1 + 2) & b5 < b6 & b3 . b5 = b4 . b5 & b3 . b6 = b4 . b6
holds (Part_sgn(b3,b2)) . {b5,b6} = (Part_sgn(b4,b2)) . {b5,b6};
:: MATRIX11:th 7
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Element of Fin TWOELEMENTSETS Seg (b1 + 2)
for b4, b5 being Element of Permutations (b1 + 2)
for b6 being finite set
st b6 = {b7 where b7 is Element of TWOELEMENTSETS Seg (b1 + 2): b7 in b3 &
(Part_sgn(b4,b2)) . b7 <> (Part_sgn(b5,b2)) . b7}
holds ((card b6) mod 2 = {} implies (the multF of b2) $$(b3,Part_sgn(b4,b2)) = (the multF of b2) $$(b3,Part_sgn(b5,b2))) &
((card b6) mod 2 = 1 implies (the multF of b2) $$(b3,Part_sgn(b4,b2)) = - ((the multF of b2) $$(b3,Part_sgn(b5,b2))));
:: MATRIX11:th 8
theorem
for b1 being natural set
for b2 being Function-like quasi_total bijective Relation of Seg b1,Seg b1
st b2 is being_transposition(b1)
for b3, b4 being natural set
st b3 < b4
holds b2 . b3 = b4
iff
b3 in proj1 b2 &
b4 in proj1 b2 &
b2 . b3 = b4 &
b2 . b4 = b3 &
(for b5 being natural set
st b5 <> b3 & b5 <> b4 & b5 in proj1 b2
holds b2 . b5 = b5);
:: MATRIX11: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 doubleLoopStr
for b3, b4, b5 being Element of Permutations (b1 + 2)
for b6, b7 being natural set
st b5 = b3 * b4 & b4 is being_transposition(len Permutations (b1 + 2)) & b4 . b6 = b7 & b6 < b7
for b8 being Element of TWOELEMENTSETS Seg (b1 + 2)
st (Part_sgn(b3,b2)) . b8 <> (Part_sgn(b5,b2)) . b8 &
not b6 in b8
holds b7 in b8;
:: MATRIX11:th 10
theorem
for b1 being natural set
for b2, b3, b4 being Element of Permutations (b1 + 2)
for b5, b6 being natural set
for b7 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
st b4 = b2 * b3 & b3 is being_transposition(len Permutations (b1 + 2)) & b3 . b5 = b6 & b5 < b6 & 1_ b7 <> - 1_ b7
holds (Part_sgn(b2,b7)) . {b5,b6} <> (Part_sgn(b4,b7)) . {b5,b6} &
(for b8 being natural set
st b8 in Seg (b1 + 2) & b5 <> b8 & b6 <> b8
holds (Part_sgn(b2,b7)) . {b5,b8} <> (Part_sgn(b4,b7)) . {b5,b8}
iff
(Part_sgn(b2,b7)) . {b6,b8} <> (Part_sgn(b4,b7)) . {b6,b8});
:: MATRIX11:funcnot 3 => MATRIX11:func 2
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 Element of Permutations (a1 + 2);
func sgn(A3,A2) -> Element of the carrier of a2 equals
(the multF of a2) $$(FinOmega TWOELEMENTSETS Seg (a1 + 2),Part_sgn(a3,a2));
end;
:: MATRIX11:def 2
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 Element of Permutations (b1 + 2) holds
sgn(b3,b2) = (the multF of b2) $$(FinOmega TWOELEMENTSETS Seg (b1 + 2),Part_sgn(b3,b2));
:: MATRIX11:th 11
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 Element of Permutations (b1 + 2)
st sgn(b3,b2) <> 1_ b2
holds sgn(b3,b2) = - 1_ b2;
:: MATRIX11:th 12
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 Element of Permutations (b1 + 2)
st b3 = idseq (b1 + 2)
holds sgn(b3,b2) = 1_ b2;
:: MATRIX11: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, b4, b5 being Element of Permutations (b1 + 2)
st b5 = b3 * b4 & b4 is being_transposition(len Permutations (b1 + 2))
holds sgn(b5,b2) = - sgn(b3,b2);
:: MATRIX11: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 Element of Permutations (b1 + 2)
st b3 is being_transposition(len Permutations (b1 + 2))
holds sgn(b3,b2) = - 1_ b2;
:: MATRIX11: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 FinSequence of the carrier of Group_of_Perm (b1 + 2)
for b4 being Element of Permutations (b1 + 2)
st b4 = Product b3 &
(for b5 being natural set
st b5 in dom b3
holds ex b6 being Element of Permutations (b1 + 2) st
b3 . b5 = b6 & b6 is being_transposition(len Permutations (b1 + 2)))
holds ((len b3) mod 2 = {} implies sgn(b4,b2) = 1_ b2) &
((len b3) mod 2 = 1 implies sgn(b4,b2) = - 1_ b2);
:: MATRIX11:th 16
theorem
for b1, b2, b3 being natural set
st b1 < b2 & b1 in Seg b3 & b2 in Seg b3
holds ex b4 being Element of Permutations b3 st
b4 is being_transposition(len Permutations b3) & b4 . b1 = b2;
:: MATRIX11:th 17
theorem
for b1 being natural set
for b2 being Element of Permutations (b1 + 1)
st b2 . (b1 + 1) <> b1 + 1
holds ex b3 being Element of Permutations (b1 + 1) st
b3 is being_transposition(len Permutations (b1 + 1)) &
b3 . (b2 . (b1 + 1)) = b1 + 1 &
(b3 * b2) . (b1 + 1) = b1 + 1;
:: MATRIX11:th 18
theorem
for b1, b2 being set
st not b2 in b1
for b3 being Function-like quasi_total bijective Relation of b1 \/ {b2},b1 \/ {b2}
st b3 . b2 = b2
holds ex b4 being Function-like quasi_total bijective Relation of b1,b1 st
b3 | b1 = b4;
:: MATRIX11:th 19
theorem
for b1, b2 being set
for b3, b4 being Function-like quasi_total bijective Relation of b1,b1
for b5, b6 being Function-like quasi_total bijective Relation of b1 \/ {b2},b1 \/ {b2}
st b5 | b1 = b3 & b6 | b1 = b4 & b5 . b2 = b2 & b6 . b2 = b2
holds (b5 * b6) | b1 = b3 * b4 & (b5 * b6) . b2 = b2;
:: MATRIX11:th 20
theorem
for b1 being natural set
for b2 being Element of Permutations b1
st b2 is being_transposition(len Permutations b1)
holds b2 * b2 = idseq b1 & b2 = b2 ";
:: MATRIX11:th 21
theorem
for b1 being natural set
for b2 being Element of Permutations b1 holds
ex b3 being FinSequence of the carrier of Group_of_Perm b1 st
b2 = Product b3 &
(for b4 being natural set
st b4 in dom b3
holds ex b5 being Element of Permutations b1 st
b3 . b4 = b5 & b5 is being_transposition(len Permutations b1));
:: MATRIX11:th 22
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr holds
b1 is Fanoian
iff
1_ b1 <> - 1_ b1;
:: MATRIX11:th 23
theorem
for b1 being natural set
for b2 being Element of Permutations (b1 + 2)
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive Fanoian doubleLoopStr holds
(b2 is even(len Permutations (b1 + 2)) implies sgn(b2,b3) = 1_ b3) &
(sgn(b2,b3) = 1_ b3 implies b2 is even(len Permutations (b1 + 2))) &
(b2 is even(len Permutations (b1 + 2)) or sgn(b2,b3) = - 1_ b3) &
(sgn(b2,b3) = - 1_ b3 implies b2 is even(not len Permutations (b1 + 2)));
:: MATRIX11: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, b4, b5 being Element of Permutations (b1 + 2)
st b5 = b3 * b4
holds sgn(b5,b2) = (sgn(b3,b2)) * sgn(b4,b2);
:: MATRIX11:th 25
theorem
for b1 being natural set
for b2, b3 being Element of Permutations b1 holds
(b2 is even(len Permutations b1) & b3 is even(len Permutations b1) or b2 is even(not len Permutations b1) & b3 is even(not len Permutations b1))
iff
b2 * b3 is even(len Permutations b1);
:: MATRIX11: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 Element of the carrier of b2
for b4 being Element of Permutations (b1 + 2) holds
-(b3,b4) = (sgn(b4,b2)) * b3;
:: MATRIX11:th 27
theorem
for b1 being natural set
for b2 being Element of Permutations (b1 + 2)
st b2 is being_transposition(len Permutations (b1 + 2))
holds b2 is even(not len Permutations (b1 + 2));
:: MATRIX11:exreg 1
registration
let a1 be natural set;
cluster Relation-like Function-like one-to-one quasi_total onto bijective finite odd Relation of Seg (a1 + 2),Seg (a1 + 2);
end;
:: MATRIX11:funcnot 4 => MATRIX11: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 ReplaceLine(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 (b1 = a1 or it *(b1,b2) = a5 *(b1,b2)) & (b1 = a1 implies it *(a1,b2) = a6 . b2))
if len a6 = width a5
otherwise it = a5;
end;
:: MATRIX11:def 3
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 = width b5 implies (b7 = ReplaceLine(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 (b8 = b1 or b7 *(b8,b9) = b5 *(b8,b9)) &
(b8 = b1 implies b7 *(b1,b9) = b6 . b9)))) &
(len b6 = width b5 or (b7 = ReplaceLine(b5,b1,b6)
iff
b7 = b5));
:: MATRIX11:funcnot 5 => MATRIX11: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 RLine(a5,a1,a6) for ReplaceLine(a5,a1,a6);
end;
:: MATRIX11:th 28
theorem
for b1, b2 being natural set
for b3 being non empty set
for b4 being natural set
for b5 being Matrix of b2,b1,b3
for b6 being FinSequence of b3
for b7 being natural set
st b7 in Seg b2
holds (b7 = b4 & len b6 = width b5 implies Line(ReplaceLine(b5,b4,b6),b7) = b6) &
(b7 = b4 or Line(ReplaceLine(b5,b4,b6),b7) = Line(b5,b7));
:: MATRIX11:th 29
theorem
for b1, b2, b3 being natural set
for b4 being non empty set
for b5 being Matrix of b2,b1,b4
for b6 being FinSequence of b4
st len b6 = width b5
for b7 being Element of b4 *
st b6 = b7
holds ReplaceLine(b5,b3,b6) = Replace(b5,b3,b7);
:: MATRIX11:th 30
theorem
for b1, b2, b3 being natural set
for b4 being non empty set
for b5 being Matrix of b1,b2,b4 holds
b5 = ReplaceLine(b5,b3,Line(b5,b3));
:: MATRIX11: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, b4 being Element of the carrier of b2
for b5 being natural set
for b6, b7 being FinSequence of the carrier of b2
for b8 being Element of Permutations b1
st b5 in Seg b1 & len b6 = b1 & len b7 = b1
for b9 being Matrix of b1,b1,the carrier of b2 holds
(the multF of b2) "**" Path_matrix(b8,ReplaceLine(b9,b5,(b3 * b6) + (b4 * b7))) = (b3 * ((the multF of b2) "**" Path_matrix(b8,ReplaceLine(b9,b5,b6)))) + (b4 * ((the multF of b2) "**" Path_matrix(b8,ReplaceLine(b9,b5,b7))));
:: MATRIX11:th 32
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 Element of the carrier of b2
for b5 being natural set
for b6, b7 being FinSequence of the carrier of b2
for b8 being Element of Permutations b1
st b5 in Seg b1 & len b6 = b1 & len b7 = b1
for b9 being Matrix of b1,b1,the carrier of b2 holds
(Path_product ReplaceLine(b9,b5,(b3 * b6) + (b4 * b7))) . b8 = (b3 * ((Path_product ReplaceLine(b9,b5,b6)) . b8)) + (b4 * ((Path_product ReplaceLine(b9,b5,b7)) . b8));
:: MATRIX11: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, b4 being Element of the carrier of b2
for b5 being natural set
for b6, b7 being FinSequence of the carrier of b2
st b5 in Seg b1 & len b6 = b1 & len b7 = b1
for b8 being Matrix of b1,b1,the carrier of b2 holds
Det ReplaceLine(b8,b5,(b3 * b6) + (b4 * b7)) = (b3 * Det ReplaceLine(b8,b5,b6)) + (b4 * Det ReplaceLine(b8,b5,b7));
:: MATRIX11:th 34
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 Element of the carrier of b3
for b5 being FinSequence of the carrier of b3
for b6 being Matrix of b2,b2,the carrier of b3
st b1 in Seg b2 & len b5 = b2
holds Det ReplaceLine(b6,b1,b4 * b5) = b4 * Det ReplaceLine(b6,b1,b5);
:: MATRIX11:th 35
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 Element of the carrier of b3
for b5 being Matrix of b2,b2,the carrier of b3
st b1 in Seg b2
holds Det ReplaceLine(b5,b1,b4 * Line(b5,b1)) = b4 * Det b5;
:: MATRIX11:th 36
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, b5 being FinSequence of the carrier of b3
for b6 being Matrix of b2,b2,the carrier of b3
st b1 in Seg b2 & len b4 = b2 & len b5 = b2
holds Det ReplaceLine(b6,b1,b4 + b5) = (Det ReplaceLine(b6,b1,b4)) + Det ReplaceLine(b6,b1,b5);
:: MATRIX11:funcnot 6 => MATRIX11:func 4
definition
let a1, a2 be natural set;
let a3 be non empty set;
let a4 be Function-like quasi_total Relation of Seg a1,Seg a1;
let a5 be Matrix of a1,a2,a3;
redefine func A5 * A4 -> Matrix of a1,a2,a3 means
len it = len a5 &
width it = width a5 &
(for b1, b2, b3 being natural set
st [b1,b2] in Indices a5 & a4 . b1 = b3
holds it *(b1,b2) = a5 *(b3,b2));
end;
:: MATRIX11:def 4
theorem
for b1, b2 being natural set
for b3 being non empty set
for b4 being Function-like quasi_total Relation of Seg b1,Seg b1
for b5, b6 being Matrix of b1,b2,b3 holds
b6 = b5 * b4
iff
len b6 = len b5 &
width b6 = width b5 &
(for b7, b8, b9 being natural set
st [b7,b8] in Indices b5 & b4 . b7 = b9
holds b6 *(b7,b8) = b5 *(b9,b8));
:: MATRIX11:th 37
theorem
for b1, b2 being natural set
for b3 being non empty set
for b4 being Function-like quasi_total Relation of Seg b1,Seg b1
for b5 being Matrix of b1,b2,b3 holds
Indices b5 = Indices (b5 * b4) &
(for b6, b7 being natural set
st [b6,b7] in Indices b5
holds ex b8 being natural set st
b4 . b6 = b8 & [b8,b7] in Indices b5 & (b5 * b4) *(b6,b7) = b5 *(b8,b7));
:: MATRIX11:th 38
theorem
for b1, b2 being natural set
for b3 being non empty set
for b4 being Matrix of b1,b2,b3
for b5 being Function-like quasi_total Relation of Seg b1,Seg b1
for b6 being natural set
st b6 in Seg b1
holds Line(b4 * b5,b6) = b4 . (b5 . b6);
:: MATRIX11:th 39
theorem
for b1, b2 being natural set
for b3 being non empty set
for b4 being Matrix of b2,b1,b3 holds
(idseq b2) * b4 = b4;
:: MATRIX11: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
for b4 being Element of Permutations b1
for b5 being Function-like quasi_total bijective Relation of Seg b1,Seg b1
for b6 being Element of Permutations b1
st b6 = b4 * (b5 ")
holds Path_matrix(b4,b3 * b5) = (Path_matrix(b6,b3)) * b5;
:: MATRIX11:th 41
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 Element of Permutations b1
for b5 being Function-like quasi_total bijective Relation of Seg b1,Seg b1
for b6 being Element of Permutations b1
st b6 = b4 * (b5 ")
holds (the multF of b2) "**" Path_matrix(b4,b3 * b5) = (the multF of b2) "**" Path_matrix(b6,b3);
:: MATRIX11:th 42
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 Element of Permutations (b1 + 2)
st b4 = b3 "
holds sgn(b3,b2) = sgn(b4,b2);
:: MATRIX11:th 43
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 + 2,b1 + 2,the carrier of b2
for b4 being Element of Permutations (b1 + 2)
for b5 being Function-like quasi_total bijective Relation of Seg (b1 + 2),Seg (b1 + 2)
st b4 = b5
for b6, b7 being Element of Permutations (b1 + 2)
st b7 = b6 * (b5 ")
holds (Path_product b3) . b7 = (sgn(b4,b2)) * ((Path_product (b3 * b5)) . b6);
:: MATRIX11:th 44
theorem
for b1 being natural set
for b2 being Element of Permutations b1 holds
ex b3 being Function-like quasi_total bijective Relation of Permutations b1,Permutations b1 st
for b4 being Element of Permutations b1 holds
b3 . b4 = b4 * b2;
:: MATRIX11:th 45
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 + 2,b1 + 2,the carrier of b2
for b4 being Element of Permutations (b1 + 2)
for b5 being Function-like quasi_total bijective Relation of Seg (b1 + 2),Seg (b1 + 2)
st b4 = b5
holds Det (b3 * b5) = (sgn(b4,b2)) * Det b3;
:: MATRIX11:th 46
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 Element of Permutations b1
for b5 being Function-like quasi_total bijective Relation of Seg b1,Seg b1
st b4 = b5
holds Det (b3 * b5) = -(Det b3,b4);
:: MATRIX11:th 47
theorem
for b1 being natural set
for b2 being Function-like quasi_total bijective Relation of Permutations b1,Permutations b1
for b3 being Element of Permutations b1
st b3 is even(not len Permutations b1) &
(for b4 being Element of Permutations b1 holds
b2 . b4 = b4 * b3)
holds b2 .: {b4 where b4 is Element of Permutations b1: b4 is even(len Permutations b1)} = {b4 where b4 is Element of Permutations b1: b4 is even(not len Permutations b1)};
:: MATRIX11:th 48
theorem
for b1 being natural set
st 2 <= b1
holds ex b2, b3 being finite set st
b3 = {b4 where b4 is Element of Permutations b1: b4 is even(len Permutations b1)} &
b2 = {b4 where b4 is Element of Permutations b1: b4 is even(not len Permutations b1)} &
b3 /\ b2 = {} &
b3 \/ b2 = Permutations b1 &
card b3 = card b2;
:: MATRIX11:th 49
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 & b3 < b4
for b5 being Matrix of b1,b1,the carrier of b2
st Line(b5,b3) = Line(b5,b4)
for b6, b7, b8 being Element of Permutations b1
st b7 = b6 * b8 & b8 is being_transposition(len Permutations b1) & b8 . b3 = b4
holds (Path_product b5) . b7 = - ((Path_product b5) . b6);
:: MATRIX11:th 50
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 & b3 < b4
for b5 being Matrix of b1,b1,the carrier of b2
st Line(b5,b3) = Line(b5,b4)
holds Det b5 = 0. b2;
:: MATRIX11:th 51
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 & b4 <> b5
holds Det ReplaceLine(b3,b4,Line(b3,b5)) = 0. b2;
:: MATRIX11:th 52
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 Element of the carrier of b2
for b4 being Matrix of b1,b1,the carrier of b2
for b5, b6 being natural set
st b5 in Seg b1 & b6 in Seg b1 & b5 <> b6
holds Det ReplaceLine(b4,b5,b3 * Line(b4,b6)) = 0. b2;
:: MATRIX11:th 53
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 Element of the carrier of b2
for b4 being Matrix of b1,b1,the carrier of b2
for b5, b6 being natural set
st b5 in Seg b1 & b6 in Seg b1 & b5 <> b6
holds Det b4 = Det ReplaceLine(b4,b5,(Line(b4,b5)) + (b3 * Line(b4,b6)));
:: MATRIX11:th 54
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 Function-like quasi_total Relation of Seg b1,Seg b1
for b4 being Matrix of b1,b1,the carrier of b2
st not b3 in Permutations b1
holds Det (b4 * b3) = 0. b2;
:: MATRIX11:funcnot 7 => MATRIX11:func 5
definition
let a1 be non empty addLoopStr;
func addFinS A1 -> Function-like quasi_total Relation of [:(the carrier of a1) *,(the carrier of a1) *:],(the carrier of a1) * means
for b1, b2 being Element of (the carrier of a1) * holds
it .(b1,b2) = b1 + b2;
end;
:: MATRIX11:def 5
theorem
for b1 being non empty addLoopStr
for b2 being Function-like quasi_total Relation of [:(the carrier of b1) *,(the carrier of b1) *:],(the carrier of b1) * holds
b2 = addFinS b1
iff
for b3, b4 being Element of (the carrier of b1) * holds
b2 .(b3,b4) = b3 + b4;
:: MATRIX11:funcreg 5
registration
let a1 be non empty Abelian addLoopStr;
cluster addFinS a1 -> Function-like quasi_total commutative;
end;
:: MATRIX11:funcreg 6
registration
let a1 be non empty add-associative addLoopStr;
cluster addFinS a1 -> Function-like quasi_total associative;
end;
:: MATRIX11:th 55
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 being tabular FinSequence of (the carrier of b1) *
st width b2 = len b3 & {} < len b3
for b4 being natural set
st b4 in Seg len b2
holds ex b5 being FinSequence of (the carrier of b1) * st
len b5 = len b3 &
Line(b2 * b3,b4) = (addFinS b1) "**" b5 &
(for b6 being natural set
st b6 in Seg len b3
holds b5 . b6 = (b2 *(b4,b6)) * Line(b3,b6));
:: MATRIX11:th 56
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, b5 being Matrix of b1,b1,the carrier of b2
for b6 being natural set
st b6 in Seg b1
holds ex b7 being FinSequence of the carrier of b2 st
len b7 = b1 &
Det ReplaceLine(b5,b6,Line(b3 * b4,b6)) = (the addF of b2) "**" b7 &
(for b8 being natural set
st b8 in Seg b1
holds b7 . b8 = (b3 *(b6,b8)) * Det ReplaceLine(b5,b6,Line(b4,b8)));
:: MATRIX11:th 57
theorem
for b1 being set
for b2 being non empty set
for b3 being set
st not b3 in b1
holds ex b4 being Function-like quasi_total Relation of [:Funcs(b1,b2),b2:],Funcs(b1 \/ {b3},b2) st
b4 is bijective([:Funcs(b1,b2),b2:], Funcs(b1 \/ {b3},b2)) &
(for b5 being Function-like quasi_total Relation of b1,b2
for b6 being Function-like quasi_total Relation of b1 \/ {b3},b2
st b6 | b1 = b5
holds b4 .(b5,b6 . b3) = b6);
:: MATRIX11:th 58
theorem
for b1 being non empty set
for b2 being finite set
for b3 being non empty finite set
for b4 being set
st not b4 in b2
for b5 being Function-like quasi_total Relation of [:b1,b1:],b1
st b5 is having_a_unity(b1) & b5 is commutative(b1) & b5 is associative(b1) & b5 is having_an_inverseOp(b1)
for b6 being Function-like quasi_total Relation of Funcs(b2,b3),b1
for b7 being Function-like quasi_total Relation of Funcs(b2 \/ {b4},b3),b1
st for b8 being Function-like quasi_total Relation of b2,b3
for b9 being Element of Fin Funcs(b2 \/ {b4},b3)
st b9 = {b10 where b10 is Function-like quasi_total Relation of b2 \/ {b4},b3: b10 | b2 = b8}
holds b5 $$(b9,b7) = b6 . b8
holds b5 $$(FinOmega Funcs(b2,b3),b6) = b5 $$(FinOmega Funcs(b2 \/ {b4},b3),b7);
:: MATRIX11:th 59
theorem
for b1, b2 being natural set
for b3 being non empty set
for b4, b5 being Matrix of b1,b2,b3
for b6 being natural set
st b6 <= b1 & {} < b1
for b7 being Function-like quasi_total Relation of Seg b6,Seg b1 holds
ex b8 being Matrix of b1,b2,b3 st
b8 = b4 +* ((((idseq b1) +* b7) * b5) | Seg b6) &
(for b9 being natural set holds
(b9 in Seg b6 implies b8 . b9 = b5 . (b7 . b9)) &
(b9 in Seg b6 or b8 . b9 = b4 . b9));
:: MATRIX11:th 60
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
st {} < b1
holds ex b5 being Function-like quasi_total Relation of Funcs(Seg b1,Seg b1),the carrier of b2 st
(for b6 being Function-like quasi_total Relation of Seg b1,Seg b1 holds
ex b7 being FinSequence of the carrier of b2 st
len b7 = b1 &
(for b8, b9 being natural set
st b9 in Seg b1 & b8 = b6 . b9
holds b7 . b9 = b3 *(b9,b8)) &
b5 . b6 = ((the multF of b2) "**" b7) * Det (b4 * b6)) &
Det (b3 * b4) = (the addF of b2) $$(FinOmega Funcs(Seg b1,Seg b1),b5);
:: MATRIX11:th 61
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
st {} < b1
holds ex b5 being Function-like quasi_total Relation of Permutations b1,the carrier of b2 st
Det (b3 * b4) = (the addF of b2) $$(FinOmega Permutations b1,b5) &
(for b6 being Element of Permutations b1 holds
b5 . b6 = ((the multF of b2) "**" Path_matrix(b6,b3)) * -(Det b4,b6));
:: MATRIX11:th 62
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
st {} < b1
holds Det (b3 * b4) = (Det b3) * Det b4;