Article MATRIX15, MML version 4.99.1005
:: MATRIX15:th 1
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 tabular FinSequence of (the carrier of b1) *
st width b3 = len b4
holds (b2 * b3) * b4 = b2 * (b3 * b4);
:: MATRIX15:th 2
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 Element of the carrier of b1
for b4 being tabular FinSequence of (the carrier of b1) * holds
(1_ b1) * b4 = b4 &
b2 * (b3 * b4) = (b2 * b3) * b4;
:: MATRIX15:th 3
theorem
for b1 being non empty addLoopStr
for b2, b3, b4, b5 being FinSequence of the carrier of b1
st len b2 = len b3 & len b4 = len b5
holds (b2 ^ b4) + (b3 ^ b5) = (b2 + b3) ^ (b4 + b5);
:: MATRIX15:th 4
theorem
for b1 being non empty multMagma
for b2, b3 being FinSequence of the carrier of b1
for b4 being Element of the carrier of b1 holds
b4 * (b2 ^ b3) = (b4 * b2) ^ (b4 * b3);
:: MATRIX15:th 5
theorem
for b1 being Relation-like Function-like set
for b2, b3, b4, b5 being Relation-like Function-like FinSequence-like set
st proj2 b2 c= proj1 b1 & proj2 b3 c= proj1 b1 & b4 = b2 * b1 & b5 = b3 * b1
holds (b2 ^ b3) * b1 = b4 ^ b5;
:: MATRIX15:th 6
theorem
for b1 being FinSequence of NAT
for b2 being natural set
st b1 is one-to-one &
proj2 b1 c= Seg b2 &
(for b3, b4 being natural set
st b3 in dom b1 & b4 in dom b1 & b3 < b4
holds b1 . b3 < b1 . b4)
holds Sgm proj2 b1 = b1;
:: MATRIX15:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2 being FinSequence of the carrier of b1
for b3, b4 being natural set
st b3 in dom b2 &
b4 in dom b2 &
b3 <> b4 &
(for b5 being natural set
st b5 in dom b2 & b5 <> b3 & b5 <> b4
holds b2 . b5 = 0. b1)
holds Sum b2 = (b2 /. b3) + (b2 /. b4);
:: MATRIX15:th 8
theorem
for b1, b2, b3 being natural set
st b1 in Seg b2
holds (Sgm ((Seg (b3 + b2)) \ Seg b3)) . b1 = b3 + b1;
:: MATRIX15:th 9
theorem
for b1 being non empty set
for b2 being tabular FinSequence of b1 *
for b3, b4, b5, b6 being finite with_non-empty_elements Element of bool NAT
st [:b3,b4:] c= Indices b2 & [:b5,b6:] c= Indices b2
for b7 being Matrix of card b3,card b4,b1
for b8 being Matrix of card b5,card b6,b1
st for b9, b10, b11, b12, b13, b14 being natural set
st [b9,b10] in [:b3,b4:] /\ [:b5,b6:] &
b11 = (Sgm b3) " . b9 &
b12 = (Sgm b4) " . b10 &
b13 = (Sgm b5) " . b9 &
b14 = (Sgm b6) " . b10
holds b7 *(b11,b12) = b8 *(b13,b14)
holds ex b9 being Matrix of len b2,width b2,b1 st
Segm(b9,b3,b4) = b7 &
Segm(b9,b5,b6) = b8 &
(for b10, b11 being natural set
st [b10,b11] in (Indices b9) \ ([:b3,b4:] \/ [:b5,b6:])
holds b9 *(b10,b11) = b2 *(b10,b11));
:: MATRIX15: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 tabular FinSequence of (the carrier of b1) *
for b3, b4, b5 being finite with_non-empty_elements Element of bool NAT
st [:b3,b5:] c= Indices b2
for b6, b7 being natural set
st b6 in (dom b2) \ b3 &
b7 in (Seg width b2) \ b4 &
b2 *(b6,b7) <> 0. b1 &
b4 c= b5 &
(Line(b2,b6)) * Sgm b5 = (card b5) |-> 0. b1
holds the_rank_of Segm(b2,b3,b4) < the_rank_of b2;
:: MATRIX15: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 tabular FinSequence of (the carrier of b1) *
for b3 being finite with_non-empty_elements Element of bool NAT
st b3 c= dom b2 &
(for b4 being natural set
st b4 in (dom b2) \ b3
holds Line(b2,b4) = (width b2) |-> 0. b1)
holds the_rank_of b2 = the_rank_of Segm(b2,b3,Seg width b2);
:: MATRIX15: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 tabular FinSequence of (the carrier of b1) *
for b3 being finite with_non-empty_elements Element of bool NAT
st b3 c= Seg width b2 &
(for b4 being natural set
st b4 in (Seg width b2) \ b3
holds Col(b2,b4) = (len b2) |-> 0. b1)
holds the_rank_of b2 = the_rank_of Segm(b2,Seg len b2,b3);
:: MATRIX15: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 VectSpStr over b1
for b3 being finite Element of bool the carrier of b2
for b4, b5 being Element of the carrier of b2
for b6 being Element of the carrier of b1
st b4 in b3 & b5 in b3
holds Lin ((b3 \ {b4}) \/ {b4 + (b6 * b5)}) is Subspace of Lin b3;
:: MATRIX15: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 VectSpStr over b1
for b3 being finite Element of bool the carrier of b2
for b4, b5 being Element of the carrier of b2
for b6 being Element of the carrier of b1
st b4 in b3 &
b5 in b3 &
(b4 = b5 & b6 = - 1_ b1 implies b4 = 0. b2)
holds Lin ((b3 \ {b4}) \/ {b4 + (b6 * b5)}) = Lin b3;
:: MATRIX15:funcnot 1 => MATRIX15:func 1
definition
let a1 be non empty set;
let a2, a3, a4 be natural set;
let a5 be Matrix of a2,a3,a1;
let a6 be Matrix of a2,a4,a1;
redefine func a5 ^^ a6 -> Matrix of a2,(width a5) + width a6,a1;
end;
:: MATRIX15:th 15
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 b1,b3,b4
for b7 being natural set
st b7 in Seg b1
holds Line(b5 ^^ b6,b7) = (Line(b5,b7)) ^ Line(b6,b7);
:: MATRIX15: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 Matrix of b1,b3,b4
for b7 being natural set
st b7 in Seg width b5
holds Col(b5 ^^ b6,b7) = Col(b5,b7);
:: MATRIX15: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 Matrix of b1,b3,b4
for b7 being natural set
st b7 in Seg width b6
holds Col(b5 ^^ b6,(width b5) + b7) = Col(b6,b7);
:: MATRIX15:th 18
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 b1,b3,b5
for b8, b9 being FinSequence of b5
st len b8 = width b6 & len b9 = width b7
holds ReplaceLine(b6 ^^ b7,b4,b8 ^ b9) = (ReplaceLine(b6,b4,b8)) ^^ ReplaceLine(b7,b4,b9);
:: MATRIX15:th 19
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 b1,b3,b4 holds
Segm(b5 ^^ b6,Seg b1,Seg width b5) = b5 &
Segm(b5 ^^ b6,Seg b1,(Seg ((width b5) + width b6)) \ Seg width b5) = b6;
:: MATRIX15: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 tabular FinSequence of (the carrier of b1) *
st len b2 = len b3
holds the_rank_of b2 <= the_rank_of (b2 ^^ b3) & the_rank_of b3 <= the_rank_of (b2 ^^ b3);
:: MATRIX15: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 tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b2 = the_rank_of b2
holds the_rank_of b2 = the_rank_of (b2 ^^ b3);
:: MATRIX15: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 tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & width b2 = {}
holds b2 ^^ b3 = b3 & b3 ^^ b2 = b3;
:: MATRIX15:th 23
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, b4 being tabular FinSequence of (the carrier of b2) *
st b4 = 0.(b2,len b3,b1)
holds the_rank_of b3 = the_rank_of (b3 ^^ b4);
:: MATRIX15: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, b3 being tabular FinSequence of (the carrier of b1) *
st the_rank_of b2 = the_rank_of (b2 ^^ b3) & len b2 = len b3
for b4 being finite with_non-empty_elements Element of bool NAT
st b4 c= dom b2 &
(for b5 being natural set
st b5 in b4
holds Line(b2,b5) = (width b2) |-> 0. b1)
for b5 being natural set
st b5 in b4
holds Line(b3,b5) = (width b3) |-> 0. b1;
:: MATRIX15:funcnot 2 => MATRIX15:func 2
definition
let a1 be non empty set;
let a2 be FinSequence of a1;
func LineVec2Mx A2 -> Matrix of 1,len a2,a1 equals
<*a2*>;
end;
:: MATRIX15:def 1
theorem
for b1 being non empty set
for b2 being FinSequence of b1 holds
LineVec2Mx b2 = <*b2*>;
:: MATRIX15:funcnot 3 => MATRIX15:func 3
definition
let a1 be non empty set;
let a2 be FinSequence of a1;
func ColVec2Mx A2 -> Matrix of len a2,1,a1 equals
<*a2*> @;
end;
:: MATRIX15:def 2
theorem
for b1 being non empty set
for b2 being FinSequence of b1 holds
ColVec2Mx b2 = <*b2*> @;
:: MATRIX15:th 25
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being tabular FinSequence of b1 * holds
b3 = LineVec2Mx b2
iff
Line(b3,1) = b2 & len b3 = 1;
:: MATRIX15:th 26
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being tabular FinSequence of b1 *
st (len b3 = {} implies len b2 <> {})
holds b3 = ColVec2Mx b2
iff
Col(b3,1) = b2 & width b3 = 1;
:: MATRIX15:th 27
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 FinSequence of the carrier of b1
st len b2 = len b3
holds (LineVec2Mx b2) + LineVec2Mx b3 = LineVec2Mx (b2 + b3);
:: MATRIX15:th 28
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 FinSequence of the carrier of b1
st len b2 = len b3
holds (ColVec2Mx b2) + ColVec2Mx b3 = ColVec2Mx (b2 + b3);
:: MATRIX15:th 29
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 being FinSequence of the carrier of b1 holds
b2 * LineVec2Mx b3 = LineVec2Mx (b2 * b3);
:: MATRIX15:th 30
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 being FinSequence of the carrier of b1 holds
b2 * ColVec2Mx b3 = ColVec2Mx (b2 * b3);
:: MATRIX15:th 31
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 holds
LineVec2Mx (b1 |-> 0. b2) = 0.(b2,1,b1);
:: MATRIX15:th 32
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 holds
ColVec2Mx (b1 |-> 0. b2) = 0.(b2,b1,1);
:: MATRIX15:funcnot 4 => MATRIX15:func 4
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 tabular FinSequence of (the carrier of a1) *;
func Solutions_of(A2,A3) -> set equals
{b1 where b1 is tabular FinSequence of (the carrier of a1) *: len b1 = width a2 & width b1 = width a3 & a2 * b1 = a3};
end;
:: MATRIX15: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, b3 being tabular FinSequence of (the carrier of b1) * holds
Solutions_of(b2,b3) = {b4 where b4 is tabular FinSequence of (the carrier of b1) *: len b4 = width b2 & width b4 = width b3 & b2 * b4 = b3};
:: MATRIX15: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, b3 being tabular FinSequence of (the carrier of b1) *
st Solutions_of(b2,b3) is not empty
holds len b2 = len b3;
:: MATRIX15:th 34
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, b4, b5 being tabular FinSequence of (the carrier of b2) *
st b3 in Solutions_of(b4,b5) & b1 in Seg width b3 & Col(b3,b1) = (len b3) |-> 0. b2
holds Col(b5,b1) = (len b5) |-> 0. b2;
:: MATRIX15: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 Element of the carrier of b1
for b3, b4, b5 being tabular FinSequence of (the carrier of b1) *
st b3 in Solutions_of(b4,b5)
holds b2 * b3 in Solutions_of(b4,b2 * b5) & b3 in Solutions_of(b2 * b4,b2 * b5);
:: MATRIX15:th 36
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 tabular FinSequence of (the carrier of b1) *
st b2 <> 0. b1
holds Solutions_of(b3,b4) = Solutions_of(b2 * b3,b2 * b4);
:: MATRIX15: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, b3, b4, b5, b6 being tabular FinSequence of (the carrier of b1) *
st b2 in Solutions_of(b3,b4) & b5 in Solutions_of(b3,b6) & width b4 = width b6
holds b2 + b5 in Solutions_of(b3,b4 + b6);
:: MATRIX15:th 38
theorem
for b1, b2, b3, b4 being natural set
for b5 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b6 being Element of the carrier of b5
for b7 being tabular FinSequence of (the carrier of b5) *
for b8 being Matrix of b2,b1,the carrier of b5
for b9 being Matrix of b2,b3,the carrier of b5
st b7 in Solutions_of(b8,b9)
holds b7 in Solutions_of(ReplaceLine(b8,b4,b6 * Line(b8,b4)),ReplaceLine(b9,b4,b6 * Line(b9,b4)));
:: MATRIX15:th 39
theorem
for b1, b2, b3, b4, b5 being natural set
for b6 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b7 being Element of the carrier of b6
for b8 being tabular FinSequence of (the carrier of b6) *
for b9 being Matrix of b4,b1,the carrier of b6
for b10 being Matrix of b4,b2,the carrier of b6
st b8 in Solutions_of(b9,b10) & b3 in Seg b4 & b5 <> b3
holds b8 in Solutions_of(ReplaceLine(b9,b5,(Line(b9,b5)) + (b7 * Line(b9,b3))),ReplaceLine(b10,b5,(Line(b10,b5)) + (b7 * Line(b10,b3))));
:: MATRIX15:th 40
theorem
for b1, b2, b3, b4, b5 being natural set
for b6 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b7 being Element of the carrier of b6
for b8 being Matrix of b4,b1,the carrier of b6
for b9 being Matrix of b4,b2,the carrier of b6
st b3 in Seg b4 & (b5 = b3 implies b7 <> - 1_ b6)
holds Solutions_of(b8,b9) = Solutions_of(ReplaceLine(b8,b5,(Line(b8,b5)) + (b7 * Line(b8,b3))),ReplaceLine(b9,b5,(Line(b9,b5)) + (b7 * Line(b9,b3))));
:: MATRIX15:th 41
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, b4, b5 being tabular FinSequence of (the carrier of b2) *
st b3 in Solutions_of(b4,b5) & b1 in dom b4 & Line(b4,b1) = (width b4) |-> 0. b2
holds Line(b5,b1) = (width b5) |-> 0. b2;
:: MATRIX15:th 42
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, b4 being tabular FinSequence of (the carrier of b2) *
for b5 being Element of b1 -tuples_on NAT
st proj2 b5 c= dom b3 & {} < b1
holds Solutions_of(b3,b4) c= Solutions_of(Segm(b3,b5,Sgm Seg width b3),Segm(b4,b5,Sgm Seg width b4));
:: MATRIX15:th 43
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, b4 being tabular FinSequence of (the carrier of b2) *
for b5 being Element of b1 -tuples_on NAT
st proj2 b5 c= dom b3 &
dom b3 = dom b4 &
{} < b1 &
(for b6 being natural set
st b6 in (dom b3) \ proj2 b5
holds Line(b3,b6) = (width b3) |-> 0. b2 & Line(b4,b6) = (width b4) |-> 0. b2)
holds Solutions_of(b3,b4) = Solutions_of(Segm(b3,b5,Sgm Seg width b3),Segm(b4,b5,Sgm Seg width b4));
:: MATRIX15: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 tabular FinSequence of (the carrier of b1) *
for b4 being finite with_non-empty_elements Element of bool NAT
st b4 c= dom b2 & b4 is not empty
holds Solutions_of(b2,b3) c= Solutions_of(Segm(b2,b4,Seg width b2),Segm(b3,b4,Seg width b3));
:: MATRIX15: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 tabular FinSequence of (the carrier of b1) *
for b4 being finite with_non-empty_elements Element of bool NAT
st b4 c= dom b2 &
b4 is not empty &
dom b2 = dom b3 &
(for b5 being natural set
st b5 in (dom b2) \ b4
holds Line(b2,b5) = (width b2) |-> 0. b1 & Line(b3,b5) = (width b3) |-> 0. b1)
holds Solutions_of(b2,b3) = Solutions_of(Segm(b2,b4,Seg width b2),Segm(b3,b4,Seg width b3));
:: MATRIX15:th 46
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, b4 being tabular FinSequence of (the carrier of b2) *
st b1 in dom b3 & 1 < len b3
holds Solutions_of(b3,b4) c= Solutions_of(DelLine(b3,b1),DelLine(b4,b1));
:: MATRIX15: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 tabular FinSequence of (the carrier of b1) *
for b4 being natural set
st b4 in dom b2 & 1 < len b2 & Line(b2,b4) = (width b2) |-> 0. b1 & b4 in dom b3 & Line(b3,b4) = (width b3) |-> 0. b1
holds Solutions_of(b2,b3) = Solutions_of(DelLine(b2,b4),DelLine(b3,b4));
:: MATRIX15:th 48
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 Matrix of b1,b2,the carrier of b4
for b6 being Matrix of b1,b3,the carrier of b4
for b7 being Function-like quasi_total Relation of Seg b1,Seg b1 holds
Solutions_of(b5,b6) c= Solutions_of(b5 * b7,b6 * b7) &
(b7 is one-to-one implies Solutions_of(b5,b6) = Solutions_of(b5 * b7,b6 * b7));
:: MATRIX15:th 49
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 Matrix of b1,b2,the carrier of b3
for b5 being finite with_non-empty_elements Element of bool NAT
st card b5 = b1 & b5 c= Seg b2 & Segm(b4,Seg b1,b5) = 1.(b3,b1) & {} < b1
holds ex b6 being Matrix of b2 -' b1,b2,the carrier of b3 st
Segm(b6,Seg (b2 -' b1),(Seg b2) \ b5) = 1.(b3,b2 -' b1) &
Segm(b6,Seg (b2 -' b1),b5) = - ((Segm(b4,Seg b1,(Seg b2) \ b5)) @) &
(for b7 being natural set
for b8 being Matrix of b2,b7,the carrier of b3
st for b9 being natural set
st b9 in Seg b7 &
(for b10 being natural set
st b10 in Seg (b2 -' b1)
holds Col(b8,b9) <> Line(b6,b10))
holds Col(b8,b9) = b2 |-> 0. b3
holds b8 in Solutions_of(b4,0.(b3,b1,b7)));
:: MATRIX15:th 50
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 Matrix of b1,b2,the carrier of b4
for b6 being Matrix of b1,b3,the carrier of b4
for b7 being finite with_non-empty_elements Element of bool NAT
st card b7 = b1 & b7 c= Seg b2 & {} < b1 & Segm(b5,Seg b1,b7) = 1.(b4,b1)
holds ex b8 being Matrix of b2,b3,the carrier of b4 st
Segm(b8,(Seg b2) \ b7,Seg b3) = 0.(b4,b2 -' b1,b3) &
Segm(b8,b7,Seg b3) = b6 &
b8 in Solutions_of(b5,b6);
:: MATRIX15:th 51
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 Matrix of {},b1,the carrier of b3
for b5 being Matrix of {},b2,the carrier of b3 holds
Solutions_of(b4,b5) = {{}};
:: MATRIX15:th 52
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 tabular FinSequence of (the carrier of b3) *
st Solutions_of(0.(b3,b1,b2),b4) is not empty
holds b4 = 0.(b3,b1,width b4);
:: MATRIX15:th 53
theorem
for b1 being set
for b2, b3, b4 being natural set
for b5 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b6 being Matrix of b2,b3,the carrier of b5
for b7 being Matrix of b2,b4,the carrier of b5
st {} < b2 & b1 in Solutions_of(b6,b7)
holds b1 is Matrix of b3,b4,the carrier of b5;
:: MATRIX15:th 54
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
st {} < b1 & {} < b2
holds Solutions_of(0.(b4,b1,b2),0.(b4,b1,b3)) = {b5 where b5 is Matrix of b2,b3,the carrier of b4: TRUE};
:: MATRIX15:th 55
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
st {} < b1 &
Solutions_of(0.(b3,b1,{}),0.(b3,b1,b2)) is not empty
holds b2 = {};
:: MATRIX15:th 56
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 holds
Solutions_of(0.(b2,b1,{}),0.(b2,b1,{})) = {{}};
:: MATRIX15:sch 1
scheme MATRIX15:sch 1
{F1 -> non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr,
F2 -> natural set,
F3 -> natural set,
F4 -> natural set,
F5 -> Matrix of F2(),F3(),the carrier of F1(),
F6 -> Matrix of F2(),F4(),the carrier of F1(),
F7 -> Matrix of F2(),F4(),the carrier of F1()}:
ex b1 being Matrix of F2(),F3(),the carrier of F1() st
ex b2 being Matrix of F2(),F4(),the carrier of F1() st
ex b3 being finite with_non-empty_elements Element of bool NAT st
b3 c= Seg F3() &
the_rank_of F5() = the_rank_of b1 &
the_rank_of F5() = card b3 &
P1[b1, b2] &
Segm(b1,Seg card b3,b3) is diagonal(F1()) &
(for b4 being natural set
st b4 in Seg card b3
holds b1 *(b4,(Sgm b3) /. b4) <> 0. F1()) &
(for b4 being natural set
st b4 in dom b1 & card b3 < b4
holds Line(b1,b4) = F3() |-> 0. F1()) &
(for b4, b5 being natural set
st b4 in Seg card b3 & b5 in Seg width b1 & b5 < (Sgm b3) . b4
holds b1 *(b4,b5) = 0. F1())
provided
P1[F5(), F6()]
and
for b1 being Matrix of F2(),F3(),the carrier of F1()
for b2 being Matrix of F2(),F4(),the carrier of F1()
st P1[b1, b2]
for b3, b4 being natural set
st b3 <> b4 & b4 in dom b1
for b5 being Element of the carrier of F1() holds
P1[ReplaceLine(b1,b3,(Line(b1,b3)) + (b5 * Line(b1,b4))), F7(b2, b3, b4, b5)];
:: MATRIX15:sch 2
scheme MATRIX15:sch 2
{F1 -> non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr,
F2 -> natural set,
F3 -> natural set,
F4 -> natural set,
F5 -> Matrix of F2(),F3(),the carrier of F1(),
F6 -> Matrix of F2(),F4(),the carrier of F1(),
F7 -> Matrix of F2(),F4(),the carrier of F1()}:
ex b1 being Matrix of F2(),F3(),the carrier of F1() st
ex b2 being Matrix of F2(),F4(),the carrier of F1() st
ex b3 being finite with_non-empty_elements Element of bool NAT st
b3 c= Seg F3() &
the_rank_of F5() = the_rank_of b1 &
the_rank_of F5() = card b3 &
P1[b1, b2] &
Segm(b1,Seg card b3,b3) = 1.(F1(),card b3) &
(for b4 being natural set
st b4 in dom b1 & card b3 < b4
holds Line(b1,b4) = F3() |-> 0. F1()) &
(for b4, b5 being natural set
st b4 in Seg card b3 & b5 in Seg width b1 & b5 < (Sgm b3) . b4
holds b1 *(b4,b5) = 0. F1())
provided
P1[F5(), F6()]
and
for b1 being Matrix of F2(),F3(),the carrier of F1()
for b2 being Matrix of F2(),F4(),the carrier of F1()
st P1[b1, b2]
for b3 being Element of the carrier of F1()
for b4, b5 being natural set
st b5 in dom b1 & (b4 = b5 implies b3 <> - 1_ F1())
holds P1[ReplaceLine(b1,b4,(Line(b1,b4)) + (b3 * Line(b1,b5))), F7(b2, b4, b5, b3)];
:: MATRIX15:th 57
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 tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 &
(width b2 = {} implies width b3 = {})
holds the_rank_of b2 = the_rank_of (b2 ^^ b3)
iff
Solutions_of(b2,b3) is not empty;
:: MATRIX15:funcnot 5 => MATRIX15:func 5
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 tabular FinSequence of (the carrier of a1) *;
let a3 be FinSequence of the carrier of a1;
func Solutions_of(A2,A3) -> set equals
{b1 where b1 is FinSequence of the carrier of a1: ColVec2Mx b1 in Solutions_of(a2,ColVec2Mx a3)};
end;
:: MATRIX15: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 tabular FinSequence of (the carrier of b1) *
for b3 being FinSequence of the carrier of b1 holds
Solutions_of(b2,b3) = {b4 where b4 is FinSequence of the carrier of b1: ColVec2Mx b4 in Solutions_of(b2,ColVec2Mx b3)};
:: MATRIX15:th 58
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 tabular FinSequence of (the carrier of b1) *
for b3 being FinSequence of the carrier of b1
for b4 being set
st b4 in Solutions_of(b2,ColVec2Mx b3)
holds ex b5 being FinSequence of the carrier of b1 st
b4 = ColVec2Mx b5 & len b5 = width b2;
:: MATRIX15:th 59
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 tabular FinSequence of (the carrier of b1) *
for b3, b4 being FinSequence of the carrier of b1
st ColVec2Mx b4 in Solutions_of(b2,ColVec2Mx b3)
holds len b4 = width b2;
:: MATRIX15:funcnot 6 => MATRIX15:func 6
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 tabular FinSequence of (the carrier of a1) *;
let a3 be FinSequence of the carrier of a1;
redefine func Solutions_of(a2,a3) -> Element of bool the carrier of (width a2) -VectSp_over a1;
end;
:: MATRIX15:funcreg 1
registration
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 tabular FinSequence of (the carrier of a1) *;
let a3 be Element of NAT;
cluster Solutions_of(a2,a3 |-> 0. a1) -> linearly-closed;
end;
:: MATRIX15:th 60
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 tabular FinSequence of (the carrier of b1) *
for b3 being FinSequence of the carrier of b1
st Solutions_of(b2,b3) is not empty & width b2 = {}
holds len b2 = {};
:: MATRIX15:th 61
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 tabular FinSequence of (the carrier of b1) *
st (width b2 = {} implies len b2 = {})
holds Solutions_of(b2,(len b2) |-> 0. b1) is not empty;
:: MATRIX15:funcnot 7 => MATRIX15: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 tabular FinSequence of (the carrier of a1) *;
assume (width a2 = {} implies len a2 = {});
func Space_of_Solutions_of A2 -> strict Subspace of (width a2) -VectSp_over a1 means
the carrier of it = Solutions_of(a2,(len a2) |-> 0. a1);
end;
:: MATRIX15:def 5
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 tabular FinSequence of (the carrier of b1) *
st (width b2 = {} implies len b2 = {})
for b3 being strict Subspace of (width b2) -VectSp_over b1 holds
b3 = Space_of_Solutions_of b2
iff
the carrier of b3 = Solutions_of(b2,(len b2) |-> 0. b1);
:: MATRIX15:th 62
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 tabular FinSequence of (the carrier of b1) *
for b3 being FinSequence of the carrier of b1
st Solutions_of(b2,b3) is not empty
holds Solutions_of(b2,b3) is Coset of Space_of_Solutions_of b2;
:: MATRIX15:th 63
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 tabular FinSequence of (the carrier of b1) *
st (width b2 = {} implies len b2 = {}) &
the_rank_of b2 = {}
holds Space_of_Solutions_of b2 = (width b2) -VectSp_over b1;
:: MATRIX15:th 64
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 tabular FinSequence of (the carrier of b1) *
st Space_of_Solutions_of b2 = (width b2) -VectSp_over b1
holds the_rank_of b2 = {};
:: MATRIX15:th 65
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 Element of the carrier of b3
for b5 being Matrix of b1,b2,the carrier of b3
for b6, b7 being natural set
st b7 in Seg b1 & {} < b2 & (b6 = b7 implies b4 <> - 1_ b3)
holds Space_of_Solutions_of b5 = Space_of_Solutions_of ReplaceLine(b5,b6,(Line(b5,b6)) + (b4 * Line(b5,b7)));
:: MATRIX15:th 66
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 tabular FinSequence of (the carrier of b1) *
for b3 being finite with_non-empty_elements Element of bool NAT
st b3 c= dom b2 &
b3 is not empty &
{} < width b2 &
(for b4 being natural set
st b4 in (dom b2) \ b3
holds Line(b2,b4) = (width b2) |-> 0. b1)
holds Space_of_Solutions_of b2 = Space_of_Solutions_of Segm(b2,b3,Seg width b2);
:: MATRIX15:th 67
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 Matrix of b1,b2,the carrier of b3
for b5 being finite with_non-empty_elements Element of bool NAT
st card b5 = b1 & b5 c= Seg b2 & Segm(b4,Seg b1,b5) = 1.(b3,b1) & {} < b1 & {} < b2 -' b1
holds ex b6 being Matrix of b2 -' b1,b2,the carrier of b3 st
Segm(b6,Seg (b2 -' b1),(Seg b2) \ b5) = 1.(b3,b2 -' b1) &
Segm(b6,Seg (b2 -' b1),b5) = - ((Segm(b4,Seg b1,(Seg b2) \ b5)) @) &
Lin lines b6 = Space_of_Solutions_of b4;
:: MATRIX15:th 68
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 tabular FinSequence of (the carrier of b1) *
st (width b2 = {} implies len b2 = {})
holds dim Space_of_Solutions_of b2 = (width b2) - the_rank_of b2;
:: MATRIX15:th 69
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 Matrix of b1,b2,the carrier of b3
for b5, b6 being natural set
for b7 being Element of the carrier of b3
st b4 is one-to-one & b6 in dom b4 & (b5 = b6 implies b7 <> - 1_ b3)
holds Lin lines b4 = Lin lines ReplaceLine(b4,b5,(Line(b4,b5)) + (b7 * Line(b4,b6)));
:: MATRIX15:th 70
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 Subspace of b1 -VectSp_over b2 holds
ex b4 being Matrix of dim b3,b1,the carrier of b2 st
ex b5 being finite with_non-empty_elements Element of bool NAT st
b5 c= Seg b1 &
dim b3 = card b5 &
Segm(b4,Seg dim b3,b5) = 1.(b2,dim b3) &
the_rank_of b4 = dim b3 &
lines b4 is Basis of b3;
:: MATRIX15:th 71
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 strict Subspace of b1 -VectSp_over b2
st dim b3 < b1
holds ex b4 being Matrix of b1 -' dim b3,b1,the carrier of b2 st
ex b5 being finite with_non-empty_elements Element of bool NAT st
card b5 = b1 -' dim b3 &
b5 c= Seg b1 &
Segm(b4,Seg (b1 -' dim b3),b5) = 1.(b2,b1 -' dim b3) &
b3 = Space_of_Solutions_of b4;
:: MATRIX15:th 72
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 tabular FinSequence of (the carrier of b1) *
st width b2 = len b3 &
(width b2 = {} implies len b2 = {}) &
(width b3 = {} implies len b3 = {})
holds Space_of_Solutions_of b3 is Subspace of Space_of_Solutions_of (b2 * b3);
:: MATRIX15:th 73
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 tabular FinSequence of (the carrier of b1) *
st width b2 = len b3
holds the_rank_of (b2 * b3) <= the_rank_of b2 & the_rank_of (b2 * b3) <= the_rank_of b3;
:: MATRIX15:th 74
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 Matrix of b1,b1,the carrier of b2
for b4 being tabular FinSequence of (the carrier of b2) *
st Det b3 <> 0. b2 &
width b3 = len b4 &
(width b4 = {} implies len b4 = {})
holds Space_of_Solutions_of b4 = Space_of_Solutions_of (b3 * b4);
:: MATRIX15:th 75
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 Matrix of b1,b1,the carrier of b2
for b4 being tabular FinSequence of (the carrier of b2) *
st width b3 = len b4 & Det b3 <> 0. b2
holds the_rank_of (b3 * b4) = the_rank_of b4;
:: MATRIX15:th 76
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 Matrix of b1,b1,the carrier of b2
for b4 being tabular FinSequence of (the carrier of b2) *
st len b3 = width b4 & Det b3 <> 0. b2
holds the_rank_of (b4 * b3) = the_rank_of b4;