Article MATRIX_9, MML version 4.99.1005

:: MATRIX_9:th 1
theorem
for b1, b2 being set
      st b1 in b2
   holds {b1} in Fin b2;

:: MATRIX_9:exreg 1
registration
  let a1 be natural set;
  cluster non empty finite Element of Fin Permutations a1;
end;

:: MATRIX_9:sch 1
scheme MATRIX_9:sch 1
{F1 -> Element of NAT,
  F2 -> non empty Element of Fin Permutations F1()}:
P1[F2()]
provided
   for b1 being Element of Permutations F1()
         st b1 in F2()
      holds P1[{b1}]
and
   for b1 being Element of Permutations F1()
   for b2 being non empty Element of Fin Permutations F1()
         st b1 in F2() & b2 c= F2() & not b1 in b2 & P1[b2]
      holds P1[b2 \/ {b1}];


:: MATRIX_9:exreg 2
registration
  let a1 be natural set;
  cluster Relation-like Function-like one-to-one quasi_total finite FinSequence-like Relation of Seg a1,Seg a1;
end;

:: MATRIX_9:funcreg 1
registration
  let a1 be natural set;
  cluster id Seg a1 -> Relation-like FinSequence-like;
end;

:: MATRIX_9:th 2
theorem
(Rev idseq 2) . 1 = 2 & (Rev idseq 2) . 2 = 1;

:: MATRIX_9:th 3
theorem
for b1 being Relation-like Function-like one-to-one set
      st proj1 b1 = Seg 2 & proj2 b1 = Seg 2 & b1 <> id Seg 2
   holds b1 = Rev id Seg 2;

:: MATRIX_9:th 4
theorem
for b1 being natural set holds
   Rev idseq b1 in Permutations b1;

:: MATRIX_9:th 5
theorem
for b1 being natural set
for b2 being Relation-like Function-like FinSequence-like set
      st b1 <> 0 & b2 in Permutations b1
   holds Rev b2 in Permutations b1;

:: MATRIX_9:th 6
theorem
Permutations 2 = {idseq 2,Rev idseq 2};

:: MATRIX_9:funcnot 1 => MATRIX_9: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 Matrix of a1,a1,the carrier of a2;
  func PPath_product A3 -> Function-like quasi_total Relation of Permutations a1,the carrier of a2 means
    for b1 being Element of Permutations a1 holds
       it . b1 = (the multF of a2) "**" Path_matrix(b1,a3);
end;

:: MATRIX_9: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 Matrix of b1,b1,the carrier of b2
for b4 being Function-like quasi_total Relation of Permutations b1,the carrier of b2 holds
      b4 = PPath_product b3
   iff
      for b5 being Element of Permutations b1 holds
         b4 . b5 = (the multF of b2) "**" Path_matrix(b5,b3);

:: MATRIX_9:funcnot 2 => MATRIX_9: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 Matrix of a1,a1,the carrier of a2;
  func Per A3 -> Element of the carrier of a2 equals
    (the addF of a2) $$(FinOmega Permutations a1,PPath_product a3);
end;

:: MATRIX_9: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 Matrix of b1,b1,the carrier of b2 holds
   Per b3 = (the addF of b2) $$(FinOmega Permutations b1,PPath_product b3);

:: MATRIX_9:th 7
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 Element of the carrier of b1 holds
   Per <*<*b2*>*> = b2;

:: MATRIX_9:th 8
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 Element of NAT
      st 1 <= b2
   holds Per 0.(b1,b2,b2) = 0. b1;

:: MATRIX_9:th 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, b3, b4, b5 being Element of the carrier of b1
for b6 being Element of Permutations 2
      st b6 = idseq 2
   holds Path_matrix(b6,(b2,b3)][(b4,b5)) = <*b2,b5*>;

:: MATRIX_9:th 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, b3, b4, b5 being Element of the carrier of b1
for b6 being Element of Permutations 2
      st b6 = Rev idseq 2
   holds Path_matrix(b6,(b2,b3)][(b4,b5)) = <*b3,b4*>;

:: MATRIX_9:th 11
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 Element of the carrier of b1 holds
(the multF of b1) "**" <*b2,b3*> = b2 * b3;

:: MATRIX_9:exreg 3
registration
  cluster Relation-like Function-like one-to-one non empty total quasi_total onto bijective finite odd Relation of Seg 2,Seg 2;
end;

:: MATRIX_9:exreg 4
registration
  let a1 be natural set;
  cluster Relation-like Function-like one-to-one quasi_total onto bijective finite even Relation of Seg a1,Seg a1;
end;

:: MATRIX_9:th 12
theorem
<*2,1*> is Function-like quasi_total bijective odd Relation of Seg 2,Seg 2;

:: MATRIX_9:th 13
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, b5 being Element of the carrier of b1 holds
Det ((b2,b3)][(b4,b5)) = (b2 * b5) - (b3 * b4);

:: MATRIX_9:th 14
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, b5 being Element of the carrier of b1 holds
Per ((b2,b3)][(b4,b5)) = (b2 * b5) + (b3 * b4);

:: MATRIX_9:th 15
theorem
Rev idseq 3 = <*3,2,1*>;

:: MATRIX_9:th 16
theorem
for b1 being non empty set
for b2, b3, b4 being Element of b1
for b5 being FinSequence of b1
      st b5 = <*b2,b3,b4*>
   holds Rev b5 = <*b4,b3,b2*>;

:: MATRIX_9:th 17
theorem
for b1 being natural set
for b2, b3 being Relation-like Function-like FinSequence-like set
      st b2 ^ b3 in Permutations b1
   holds b2 ^ Rev b3 in Permutations b1;

:: MATRIX_9:th 18
theorem
for b1 being natural set
for b2, b3 being Relation-like Function-like FinSequence-like set
      st b2 ^ b3 in Permutations b1
   holds b3 ^ b2 in Permutations b1;

:: MATRIX_9:th 19
theorem
Permutations 3 = {<*1,2,3*>,<*3,2,1*>,<*1,3,2*>,<*2,3,1*>,<*2,1,3*>,<*3,1,2*>};

:: MATRIX_9:th 20
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, b5, b6, b7, b8, b9, b10 being Element of the carrier of b1
for b11 being Matrix of 3,3,the carrier of b1
   st b11 = <*<*b2,b3,b4*>,<*b5,b6,b7*>,<*b8,b9,b10*>*>
for b12 being Element of Permutations 3
      st b12 = <*1,2,3*>
   holds Path_matrix(b12,b11) = <*b2,b6,b10*>;

:: MATRIX_9:th 21
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, b5, b6, b7, b8, b9, b10 being Element of the carrier of b1
for b11 being Matrix of 3,3,the carrier of b1
   st b11 = <*<*b2,b3,b4*>,<*b5,b6,b7*>,<*b8,b9,b10*>*>
for b12 being Element of Permutations 3
      st b12 = <*3,2,1*>
   holds Path_matrix(b12,b11) = <*b4,b6,b8*>;

:: MATRIX_9: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
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of b1
for b11 being Matrix of 3,3,the carrier of b1
   st b11 = <*<*b2,b3,b4*>,<*b5,b6,b7*>,<*b8,b9,b10*>*>
for b12 being Element of Permutations 3
      st b12 = <*1,3,2*>
   holds Path_matrix(b12,b11) = <*b2,b7,b9*>;

:: MATRIX_9:th 23
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, b5, b6, b7, b8, b9, b10 being Element of the carrier of b1
for b11 being Matrix of 3,3,the carrier of b1
   st b11 = <*<*b2,b3,b4*>,<*b5,b6,b7*>,<*b8,b9,b10*>*>
for b12 being Element of Permutations 3
      st b12 = <*2,3,1*>
   holds Path_matrix(b12,b11) = <*b3,b7,b8*>;

:: MATRIX_9:th 24
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, b5, b6, b7, b8, b9, b10 being Element of the carrier of b1
for b11 being Matrix of 3,3,the carrier of b1
   st b11 = <*<*b2,b3,b4*>,<*b5,b6,b7*>,<*b8,b9,b10*>*>
for b12 being Element of Permutations 3
      st b12 = <*2,1,3*>
   holds Path_matrix(b12,b11) = <*b3,b5,b10*>;

:: MATRIX_9:th 25
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, b5, b6, b7, b8, b9, b10 being Element of the carrier of b1
for b11 being Matrix of 3,3,the carrier of b1
   st b11 = <*<*b2,b3,b4*>,<*b5,b6,b7*>,<*b8,b9,b10*>*>
for b12 being Element of Permutations 3
      st b12 = <*3,1,2*>
   holds Path_matrix(b12,b11) = <*b4,b5,b9*>;

:: MATRIX_9:th 26
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 Element of the carrier of b1 holds
(the multF of b1) "**" <*b2,b3,b4*> = (b2 * b3) * b4;

:: MATRIX_9:th 27
theorem
<*1,3,2*> in Permutations 3 &
 <*2,3,1*> in Permutations 3 &
 <*2,1,3*> in Permutations 3 &
 <*3,1,2*> in Permutations 3 &
 <*1,2,3*> in Permutations 3 &
 <*3,2,1*> in Permutations 3;

:: MATRIX_9:th 28
theorem
<*2,3,1*> " = <*3,1,2*>;

:: MATRIX_9:th 29
theorem
for b1 being Element of the carrier of Group_of_Perm 3
      st b1 = <*2,3,1*>
   holds b1 " = <*3,1,2*>;

:: MATRIX_9:th 30
theorem
for b1 being Function-like quasi_total bijective Relation of Seg 3,Seg 3
      st b1 = <*1,3,2*>
   holds b1 is being_transposition(3);

:: MATRIX_9:th 31
theorem
for b1 being Function-like quasi_total bijective Relation of Seg 3,Seg 3
      st b1 = <*2,1,3*>
   holds b1 is being_transposition(3);

:: MATRIX_9:th 32
theorem
for b1 being Function-like quasi_total bijective Relation of Seg 3,Seg 3
      st b1 = <*3,2,1*>
   holds b1 is being_transposition(3);

:: MATRIX_9:th 33
theorem
for b1 being natural set
for b2 being Function-like quasi_total bijective Relation of Seg b1,Seg b1
      st b2 = id Seg b1
   holds b2 is not being_transposition(b1);

:: MATRIX_9:th 34
theorem
for b1 being Function-like quasi_total bijective Relation of Seg 3,Seg 3
      st b1 = <*3,1,2*>
   holds b1 is not being_transposition(3);

:: MATRIX_9:th 35
theorem
for b1 being Function-like quasi_total bijective Relation of Seg 3,Seg 3
      st b1 = <*2,3,1*>
   holds b1 is not being_transposition(3);

:: MATRIX_9:th 36
theorem
for b1 being natural set
for b2 being Function-like quasi_total bijective Relation of Seg b1,Seg b1 holds
   b2 is FinSequence of Seg b1;

:: MATRIX_9:th 37
theorem
<*2,1,3*> * <*1,3,2*> = <*2,3,1*> &
 <*1,3,2*> * <*2,1,3*> = <*3,1,2*> &
 <*2,1,3*> * <*3,2,1*> = <*3,1,2*> &
 <*3,2,1*> * <*2,1,3*> = <*2,3,1*> &
 <*3,2,1*> * <*3,2,1*> = <*1,2,3*> &
 <*2,1,3*> * <*2,1,3*> = <*1,2,3*> &
 <*1,3,2*> * <*1,3,2*> = <*1,2,3*> &
 <*1,3,2*> * <*2,3,1*> = <*3,2,1*> &
 <*2,3,1*> * <*2,3,1*> = <*3,1,2*> &
 <*2,3,1*> * <*3,1,2*> = <*1,2,3*> &
 <*3,1,2*> * <*2,3,1*> = <*1,2,3*> &
 <*3,1,2*> * <*3,1,2*> = <*2,3,1*> &
 <*1,3,2*> * <*3,2,1*> = <*2,3,1*> &
 <*3,2,1*> * <*1,3,2*> = <*3,1,2*>;

:: MATRIX_9:th 38
theorem
for b1 being Function-like quasi_total bijective Relation of Seg 3,Seg 3
      st b1 is being_transposition(3) & b1 <> <*2,1,3*> & b1 <> <*1,3,2*>
   holds b1 = <*3,2,1*>;

:: MATRIX_9:th 39
theorem
for b1 being natural set
for b2, b3 being Element of Permutations b1 holds
b2 * b3 in Permutations b1;

:: MATRIX_9:th 40
theorem
for b1 being natural set
for b2 being FinSequence of the carrier of Group_of_Perm b1
      st (len b2) mod 2 = 0 &
         (for b3 being Element of NAT
               st b3 in dom b2
            holds ex b4 being Element of Permutations b1 st
               b2 . b3 = b4 & b4 is being_transposition(len Permutations b1))
   holds Product b2 is Function-like quasi_total bijective even Relation of Seg b1,Seg b1;

:: MATRIX_9:th 41
theorem
for b1 being FinSequence of the carrier of Group_of_Perm 3
      st (len b1) mod 2 = 0 &
         (for b2 being Element of NAT
               st b2 in dom b1
            holds ex b3 being Element of Permutations 3 st
               b1 . b2 = b3 & b3 is being_transposition(len Permutations 3)) &
         Product b1 <> <*1,2,3*> &
         Product b1 <> <*2,3,1*>
   holds Product b1 = <*3,1,2*>;

:: MATRIX_9:exreg 5
registration
  cluster Relation-like Function-like one-to-one non empty total quasi_total onto bijective finite odd Relation of Seg 3,Seg 3;
end;

:: MATRIX_9:th 42
theorem
<*3,2,1*> is Function-like quasi_total bijective odd Relation of Seg 3,Seg 3;

:: MATRIX_9:th 43
theorem
<*2,1,3*> is Function-like quasi_total bijective odd Relation of Seg 3,Seg 3;

:: MATRIX_9:th 44
theorem
<*1,3,2*> is Function-like quasi_total bijective odd Relation of Seg 3,Seg 3;

:: MATRIX_9:th 45
theorem
for b1 being Function-like quasi_total bijective odd Relation of Seg 3,Seg 3
      st b1 <> <*3,2,1*> & b1 <> <*1,3,2*>
   holds b1 = <*2,1,3*>;

:: MATRIX_9:th 46
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, b5, b6, b7, b8, b9, b10 being Element of the carrier of b1
for b11 being Matrix of 3,3,the carrier of b1
      st b11 = <*<*b2,b3,b4*>,<*b5,b6,b7*>,<*b8,b9,b10*>*>
   holds Det b11 = ((((((b2 * b6) * b10) - ((b4 * b6) * b8)) - ((b2 * b7) * b9)) + ((b3 * b7) * b8)) - ((b3 * b5) * b10)) + ((b4 * b5) * b9);

:: MATRIX_9:th 47
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, b5, b6, b7, b8, b9, b10 being Element of the carrier of b1
for b11 being Matrix of 3,3,the carrier of b1
      st b11 = <*<*b2,b3,b4*>,<*b5,b6,b7*>,<*b8,b9,b10*>*>
   holds Per b11 = ((((((b2 * b6) * b10) + ((b4 * b6) * b8)) + ((b2 * b7) * b9)) + ((b3 * b7) * b8)) + ((b3 * b5) * b10)) + ((b4 * b5) * b9);

:: MATRIX_9:th 48
theorem
for b1, b2 being Element of NAT
for b3 being Element of Permutations b2
      st b1 in Seg b2
   holds ex b4 being Element of NAT st
      b4 in Seg b2 & b1 = b3 . b4;

:: MATRIX_9: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 being Matrix of b1,b1,the carrier of b2
   st ex b4 being Element of NAT st
        b4 in Seg b1 &
         (for b5 being Element of NAT
               st b5 in Seg b1
            holds (Col(b3,b4)) . b5 = 0. b2)
for b4 being Element of Permutations b1 holds
   ex b5 being Element of NAT st
      b5 in Seg b1 & (Path_matrix(b4,b3)) . b5 = 0. b2;

:: MATRIX_9: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 being Element of Permutations b1
for b4 being Matrix of b1,b1,the carrier of b2
      st ex b5 being Element of NAT st
           b5 in Seg b1 &
            (for b6 being Element of NAT
                  st b6 in Seg b1
               holds (Col(b4,b5)) . b6 = 0. b2)
   holds (Path_product b4) . b3 = 0. b2;

:: MATRIX_9: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
      st ex b4 being Element of NAT st
           b4 in Seg b1 &
            (for b5 being Element of NAT
                  st b5 in Seg b1
               holds (Col(b3,b4)) . b5 = 0. b2)
   holds (the addF of b2) $$(FinOmega Permutations b1,Path_product b3) = 0. b2;

:: MATRIX_9: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 Permutations b1
for b4 being Matrix of b1,b1,the carrier of b2
      st ex b5 being Element of NAT st
           b5 in Seg b1 &
            (for b6 being Element of NAT
                  st b6 in Seg b1
               holds (Col(b4,b5)) . b6 = 0. b2)
   holds (PPath_product b4) . b3 = 0. b2;

:: MATRIX_9: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 Matrix of b1,b1,the carrier of b2
      st ex b4 being Element of NAT st
           b4 in Seg b1 &
            (for b5 being Element of NAT
                  st b5 in Seg b1
               holds (Col(b3,b4)) . b5 = 0. b2)
   holds Det b3 = 0. b2;

:: MATRIX_9: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 Matrix of b1,b1,the carrier of b2
      st ex b4 being Element of NAT st
           b4 in Seg b1 &
            (for b5 being Element of NAT
                  st b5 in Seg b1
               holds (Col(b3,b4)) . b5 = 0. b2)
   holds Per b3 = 0. b2;

:: MATRIX_9:th 55
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 Matrix of b1,b1,the carrier of b3
   st b2 in Seg b1
for b6 being Element of Permutations b1 holds
   ex b7 being Element of NAT st
      b7 in Seg b1 &
       b2 = b6 . b7 &
       (Col(b5,b2)) /. b7 = (Path_matrix(b6,b5)) /. b7;

:: MATRIX_9: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 being Element of the carrier of b2
for b4, b5 being Matrix of b1,b1,the carrier of b2
   st ex b6 being Element of NAT st
        b6 in Seg b1 &
         (for b7 being Element of NAT
               st b7 in Seg b1
            holds (Col(b4,b6)) . b7 = b3 * ((Col(b5,b6)) /. b7)) &
         (for b7 being Element of NAT
               st b7 <> b6 & b7 in Seg b1
            holds Col(b4,b7) = Col(b5,b7))
for b6 being Element of Permutations b1 holds
   ex b7 being Element of NAT st
      b7 in Seg b1 &
       (Path_matrix(b6,b4)) /. b7 = b3 * ((Path_matrix(b6,b5)) /. b7);