Article COMPLSP2, MML version 4.99.1005

:: COMPLSP2:funcnot 1 => COMPLSP2:func 1
definition
  let a1 be FinSequence of COMPLEX;
  func A1 *' -> FinSequence of COMPLEX means
    len it = len a1 &
     (for b1 being natural set
           st 1 <= b1 & b1 <= len a1
        holds it . b1 = (a1 . b1) *');
end;

:: COMPLSP2:def 1
theorem
for b1, b2 being FinSequence of COMPLEX holds
   b2 = b1 *'
iff
   len b2 = len b1 &
    (for b3 being natural set
          st 1 <= b3 & b3 <= len b1
       holds b2 . b3 = (b1 . b3) *');

:: COMPLSP2:th 1
theorem
for b1 being Element of NAT
for b2, b3 being FinSequence of COMPLEX
      st b1 in dom (b2 + b3)
   holds (b2 + b3) . b1 = (b2 . b1) + (b3 . b1);

:: COMPLSP2:th 2
theorem
for b1 being Element of NAT
for b2, b3 being FinSequence of COMPLEX
      st b1 in dom (b2 - b3)
   holds (b2 - b3) . b1 = (b2 . b1) - (b3 . b1);

:: COMPLSP2:funcnot 2 => COMPLSP2:func 2
definition
  let a1 be Element of NAT;
  let a2, a3 be Element of a1 -tuples_on COMPLEX;
  redefine func a2 - a3 -> Element of a1 -tuples_on COMPLEX;
end;

:: COMPLSP2:funcnot 3 => COMPLSP2:func 3
definition
  let a1 be Element of NAT;
  let a2, a3 be Element of a1 -tuples_on COMPLEX;
  redefine func a2 + a3 -> Element of a1 -tuples_on COMPLEX;
  commutativity;
::  for a1 being Element of NAT
::  for a2, a3 being Element of a1 -tuples_on COMPLEX holds
::  a2 + a3 = a3 + a2;
end;

:: COMPLSP2:funcnot 4 => COMPLSP2:func 4
definition
  let a1 be Element of NAT;
  let a2 be Element of a1 -tuples_on COMPLEX;
  let a3 be complex set;
  redefine func a3 * a2 -> Element of a1 -tuples_on COMPLEX;
end;

:: COMPLSP2:th 3
theorem
for b1 being complex set
for b2 being FinSequence of COMPLEX holds
   len (b1 * b2) = len b2;

:: COMPLSP2:th 4
theorem
for b1 being FinSequence of COMPLEX holds
   dom b1 = dom - b1;

:: COMPLSP2:th 5
theorem
for b1 being FinSequence of COMPLEX holds
   len - b1 = len b1;

:: COMPLSP2:th 6
theorem
for b1, b2 being FinSequence of COMPLEX
      st len b1 = len b2
   holds len (b1 + b2) = len b1;

:: COMPLSP2:th 7
theorem
for b1, b2 being FinSequence of COMPLEX
      st len b1 = len b2
   holds len (b1 - b2) = len b1;

:: COMPLSP2:th 8
theorem
for b1 being FinSequence of COMPLEX holds
   b1 is Element of COMPLEX len b1;

:: COMPLSP2:th 9
theorem
for b1 being Element of NAT
for b2, b3 being Element of b1 -tuples_on COMPLEX holds
b2 - b3 = b2 + - b3;

:: COMPLSP2:th 10
theorem
for b1, b2 being FinSequence of COMPLEX
      st len b1 = len b2
   holds b1 - b2 = b1 + - b2;

:: COMPLSP2:th 11
theorem
for b1 being Element of NAT
for b2 being Element of b1 -tuples_on COMPLEX holds
   (- 1) * b2 = - b2;

:: COMPLSP2:th 12
theorem
for b1 being FinSequence of COMPLEX holds
   (- 1) * b1 = - b1;

:: COMPLSP2:th 13
theorem
for b1 being Element of NAT
for b2 being FinSequence of COMPLEX holds
   (- b2) . b1 = - (b2 . b1);

:: COMPLSP2:funcnot 5 => COMPLSP2:func 5
definition
  let a1 be Element of NAT;
  let a2 be Element of a1 -tuples_on COMPLEX;
  redefine func - a2 -> Element of a1 -tuples_on COMPLEX;
  involutiveness;
::  for a1 being Element of NAT
::  for a2 being Element of a1 -tuples_on COMPLEX holds
::     - - a2 = a2;
end;

:: COMPLSP2:th 14
theorem
for b1, b2 being Element of NAT
for b3 being Element of COMPLEX
for b4 being Element of b1 -tuples_on COMPLEX
      st b3 = b4 . b2
   holds (- b4) . b2 = - b3;

:: COMPLSP2:th 15
theorem
for b1 being FinSequence of COMPLEX
for b2 being complex set holds
   dom (b2 * b1) = dom b1;

:: COMPLSP2:th 16
theorem
for b1 being FinSequence of COMPLEX
for b2 being natural set
for b3 being complex set holds
   (b3 * b1) . b2 = b3 * (b1 . b2);

:: COMPLSP2:th 17
theorem
for b1 being FinSequence of COMPLEX
for b2 being complex set holds
   (b2 * b1) *' = b2 *' * (b1 *');

:: COMPLSP2:th 18
theorem
for b1, b2 being Element of NAT
for b3, b4 being Element of b1 -tuples_on COMPLEX holds
(b3 + b4) . b2 = (b3 . b2) + (b4 . b2);

:: COMPLSP2:th 19
theorem
for b1, b2 being FinSequence of COMPLEX
      st len b1 = len b2
   holds (b1 + b2) *' = b1 *' + (b2 *');

:: COMPLSP2:th 20
theorem
for b1, b2 being Element of NAT
for b3, b4 being Element of b1 -tuples_on COMPLEX holds
(b3 - b4) . b2 = (b3 . b2) - (b4 . b2);

:: COMPLSP2:th 21
theorem
for b1, b2 being FinSequence of COMPLEX
      st len b1 = len b2
   holds (b1 - b2) *' = b1 *' - (b2 *');

:: COMPLSP2:th 22
theorem
for b1 being FinSequence of COMPLEX holds
   b1 *' *' = b1;

:: COMPLSP2:th 23
theorem
for b1 being FinSequence of COMPLEX holds
   (- b1) *' = - (b1 *');

:: COMPLSP2:th 24
theorem
for b1 being complex set holds
   b1 + (b1 *') = 2 * Re b1;

:: COMPLSP2:th 25
theorem
for b1 being Element of NAT
for b2, b3 being FinSequence of COMPLEX
      st len b2 = len b3
   holds (b2 - b3) . b1 = (b2 . b1) - (b3 . b1);

:: COMPLSP2:th 26
theorem
for b1 being Element of NAT
for b2, b3 being FinSequence of COMPLEX
      st len b2 = len b3
   holds (b2 + b3) . b1 = (b2 . b1) + (b3 . b1);

:: COMPLSP2:funcnot 6 => COMPLSP2:func 6
definition
  let a1 be FinSequence of COMPLEX;
  func Re A1 -> FinSequence of REAL equals
    (1 / 2) * (a1 + (a1 *'));
end;

:: COMPLSP2:def 2
theorem
for b1 being FinSequence of COMPLEX holds
   Re b1 = (1 / 2) * (b1 + (b1 *'));

:: COMPLSP2:th 27
theorem
for b1 being complex set holds
   b1 - (b1 *') = (2 * Im b1) * <i>;

:: COMPLSP2:funcnot 7 => COMPLSP2:func 7
definition
  let a1 be FinSequence of COMPLEX;
  func Im A1 -> FinSequence of REAL equals
    (- ((1 / 2) * <i>)) * (a1 - (a1 *'));
end;

:: COMPLSP2:def 3
theorem
for b1 being FinSequence of COMPLEX holds
   Im b1 = (- ((1 / 2) * <i>)) * (b1 - (b1 *'));

:: COMPLSP2:funcnot 8 => COMPLSP2:func 8
definition
  let a1, a2 be FinSequence of COMPLEX;
  func |(A1,A2)| -> Element of COMPLEX equals
    ((|(Re a1,Re a2)| - (<i> * |(Re a1,Im a2)|)) + (<i> * |(Im a1,Re a2)|)) + |(Im a1,Im a2)|;
end;

:: COMPLSP2:def 4
theorem
for b1, b2 being FinSequence of COMPLEX holds
|(b1,b2)| = ((|(Re b1,Re b2)| - (<i> * |(Re b1,Im b2)|)) + (<i> * |(Im b1,Re b2)|)) + |(Im b1,Im b2)|;

:: COMPLSP2:th 28
theorem
for b1, b2, b3 being FinSequence of COMPLEX
      st len b1 = len b2 & len b2 = len b3
   holds b1 + (b2 + b3) = (b1 + b2) + b3;

:: COMPLSP2:th 29
theorem
for b1, b2 being FinSequence of COMPLEX
      st len b1 = len b2
   holds b1 + b2 = b2 + b1;

:: COMPLSP2:th 30
theorem
for b1 being complex set
for b2, b3 being FinSequence of COMPLEX
      st len b2 = len b3
   holds b1 * (b2 + b3) = (b1 * b2) + (b1 * b3);

:: COMPLSP2:th 31
theorem
for b1, b2 being FinSequence of COMPLEX
      st len b1 = len b2
   holds b1 - b2 = b1 + - b2;

:: COMPLSP2:funcnot 9 => COMPLSP2:func 9
definition
  let a1 be Element of NAT;
  let a2 be Element of COMPLEX;
  redefine func a1 |-> a2 -> Element of a1 -tuples_on COMPLEX;
end;

:: COMPLSP2:th 32
theorem
for b1, b2 being FinSequence of COMPLEX
      st len b1 = len b2 & b1 + b2 = 0c len b1
   holds b1 = - b2 & b2 = - b1;

:: COMPLSP2:th 33
theorem
for b1 being FinSequence of COMPLEX holds
   b1 + 0c len b1 = b1;

:: COMPLSP2:th 34
theorem
for b1 being FinSequence of COMPLEX holds
   b1 + - b1 = 0c len b1;

:: COMPLSP2:th 35
theorem
for b1, b2 being FinSequence of COMPLEX
      st len b1 = len b2
   holds - (b1 + b2) = (- b1) + - b2;

:: COMPLSP2:th 36
theorem
for b1, b2, b3 being FinSequence of COMPLEX
      st len b1 = len b2 & len b2 = len b3
   holds (b1 - b2) - b3 = b1 - (b2 + b3);

:: COMPLSP2:th 37
theorem
for b1, b2, b3 being FinSequence of COMPLEX
      st len b1 = len b2 & len b2 = len b3
   holds b1 + (b2 - b3) = (b1 + b2) - b3;

:: COMPLSP2:th 38
theorem
for b1 being FinSequence of COMPLEX holds
   - - b1 = b1;

:: COMPLSP2:th 39
theorem
for b1, b2 being FinSequence of COMPLEX
      st len b1 = len b2
   holds - (b1 - b2) = (- b1) + b2;

:: COMPLSP2:th 40
theorem
for b1, b2, b3 being FinSequence of COMPLEX
      st len b1 = len b2 & len b2 = len b3
   holds b1 - (b2 - b3) = (b1 - b2) + b3;

:: COMPLSP2:th 41
theorem
for b1 being FinSequence of COMPLEX
for b2 being complex set holds
   b2 * 0c len b1 = 0c len b1;

:: COMPLSP2:th 42
theorem
for b1 being FinSequence of COMPLEX
for b2 being complex set holds
   - (b2 * b1) = b2 * - b1;

:: COMPLSP2:th 43
theorem
for b1 being complex set
for b2, b3 being FinSequence of COMPLEX
      st len b2 = len b3
   holds b1 * (b2 - b3) = (b1 * b2) - (b1 * b3);

:: COMPLSP2:th 44
theorem
for b1, b2 being Element of COMPLEX
for b3, b4 being Element of REAL
      st b1 = b3 & b2 = b4
   holds addcomplex .(b1,b2) = addreal .(b3,b4);

:: COMPLSP2:th 45
theorem
for b1 being Function-like quasi_total Relation of [:COMPLEX,COMPLEX:],COMPLEX
for b2 being Function-like quasi_total Relation of [:REAL,REAL:],REAL
for b3, b4 being FinSequence of COMPLEX
for b5, b6 being FinSequence of REAL
      st b3 = b5 &
         b4 = b6 &
         len b3 = len b6 &
         (for b7 being Element of NAT
               st b7 in dom b3
            holds b1 .(b3 . b7,b4 . b7) = b2 .(b5 . b7,b6 . b7))
   holds b1 .:(b3,b4) = b2 .:(b5,b6);

:: COMPLSP2:th 46
theorem
for b1, b2 being FinSequence of COMPLEX
for b3, b4 being FinSequence of REAL
      st b1 = b3 & b2 = b4 & len b1 = len b4
   holds addcomplex .:(b1,b2) = addreal .:(b3,b4);

:: COMPLSP2:th 47
theorem
for b1, b2 being FinSequence of COMPLEX
for b3, b4 being FinSequence of REAL
      st b1 = b3 & b2 = b4 & len b1 = len b4
   holds b1 + b2 = b3 + b4;

:: COMPLSP2:th 48
theorem
for b1 being FinSequence of COMPLEX holds
   len Re b1 = len b1 & len Im b1 = len b1;

:: COMPLSP2:th 49
theorem
for b1, b2 being FinSequence of COMPLEX
      st len b1 = len b2
   holds Re (b1 + b2) = (Re b1) + Re b2 &
    Im (b1 + b2) = (Im b1) + Im b2;

:: COMPLSP2:th 50
theorem
for b1, b2 being FinSequence of COMPLEX
for b3, b4 being FinSequence of REAL
      st b1 = b3 & b2 = b4 & len b1 = len b4
   holds diffcomplex .:(b1,b2) = diffreal .:(b3,b4);

:: COMPLSP2:th 51
theorem
for b1, b2 being FinSequence of COMPLEX
for b3, b4 being FinSequence of REAL
      st b1 = b3 & b2 = b4 & len b1 = len b4
   holds b1 - b2 = b3 - b4;

:: COMPLSP2:th 52
theorem
for b1, b2 being FinSequence of COMPLEX
      st len b1 = len b2
   holds Re (b1 - b2) = (Re b1) - Re b2 &
    Im (b1 - b2) = (Im b1) - Im b2;

:: COMPLSP2:th 53
theorem
for b1 being FinSequence of COMPLEX
for b2, b3 being complex set holds
b2 * (b3 * b1) = (b2 * b3) * b1;

:: COMPLSP2:th 54
theorem
for b1 being FinSequence of COMPLEX
for b2 being complex set holds
   (- b2) * b1 = - (b2 * b1);

:: COMPLSP2:th 55
theorem
for b1 being Function-like quasi_total Relation of COMPLEX,COMPLEX
for b2 being Function-like quasi_total Relation of REAL,REAL
for b3 being FinSequence of COMPLEX
for b4 being FinSequence of REAL
      st len b3 = len b4 &
         (for b5 being Element of NAT
               st b5 in dom b3
            holds b1 . (b3 . b5) = b2 . (b4 . b5))
   holds b1 * b3 = b2 * b4;

:: COMPLSP2:th 56
theorem
for b1 being FinSequence of COMPLEX
for b2 being FinSequence of REAL
      st b1 = b2 & len b1 = len b2
   holds compcomplex * b1 = compreal * b2;

:: COMPLSP2:th 57
theorem
for b1 being FinSequence of COMPLEX
for b2 being FinSequence of REAL
      st b1 = b2 & len b1 = len b2
   holds - b1 = - b2;

:: COMPLSP2:th 58
theorem
for b1 being FinSequence of COMPLEX holds
   Re (<i> * b1) = - Im b1 &
    Im (<i> * b1) = Re b1;

:: COMPLSP2:th 59
theorem
for b1, b2 being FinSequence of COMPLEX
      st len b1 = len b2
   holds |(<i> * b1,b2)| = <i> * |(b1,b2)|;

:: COMPLSP2:th 60
theorem
for b1, b2 being FinSequence of COMPLEX
      st len b1 = len b2
   holds |(b1,<i> * b2)| = - (<i> * |(b1,b2)|);

:: COMPLSP2:th 61
theorem
for b1 being Element of COMPLEX
for b2 being FinSequence of COMPLEX
for b3 being Element of REAL
for b4 being FinSequence of REAL
      st b1 = b3 & b2 = b4 & len b2 = len b4
   holds b1 multcomplex * b2 = b3 multreal * b4;

:: COMPLSP2:th 62
theorem
for b1 being complex set
for b2 being FinSequence of COMPLEX
for b3 being Element of REAL
for b4 being FinSequence of REAL
      st b1 = b3 & b2 = b4 & len b2 = len b4
   holds b1 * b2 = b3 * b4;

:: COMPLSP2:th 63
theorem
for b1 being FinSequence of COMPLEX
for b2, b3 being complex set holds
(b2 + b3) * b1 = (b2 * b1) + (b3 * b1);

:: COMPLSP2:th 64
theorem
for b1 being FinSequence of COMPLEX
for b2, b3 being complex set holds
(b2 - b3) * b1 = (b2 * b1) - (b3 * b1);

:: COMPLSP2:th 65
theorem
for b1 being Element of COMPLEX
for b2 being FinSequence of COMPLEX holds
   Re (b1 * b2) = ((Re b1) * Re b2) - ((Im b1) * Im b2) &
    Im (b1 * b2) = ((Im b1) * Re b2) + ((Re b1) * Im b2);

:: COMPLSP2:th 66
theorem
for b1, b2, b3 being FinSequence of COMPLEX
      st len b1 = len b2 & len b2 = len b3
   holds |(b1 + b2,b3)| = |(b1,b3)| + |(b2,b3)|;

:: COMPLSP2:th 67
theorem
for b1, b2 being FinSequence of COMPLEX
      st len b1 = len b2
   holds |(- b1,b2)| = - |(b1,b2)|;

:: COMPLSP2:th 68
theorem
for b1, b2 being FinSequence of COMPLEX
      st len b1 = len b2
   holds |(b1,- b2)| = - |(b1,b2)|;

:: COMPLSP2:th 69
theorem
for b1, b2 being FinSequence of COMPLEX
      st len b1 = len b2
   holds |(- b1,- b2)| = |(b1,b2)|;

:: COMPLSP2:th 70
theorem
for b1, b2, b3 being FinSequence of COMPLEX
      st len b1 = len b2 & len b2 = len b3
   holds |(b1 - b2,b3)| = |(b1,b3)| - |(b2,b3)|;

:: COMPLSP2:th 71
theorem
for b1, b2, b3 being FinSequence of COMPLEX
      st len b1 = len b2 & len b2 = len b3
   holds |(b1,b2 + b3)| = |(b1,b2)| + |(b1,b3)|;

:: COMPLSP2:th 72
theorem
for b1, b2, b3 being FinSequence of COMPLEX
      st len b1 = len b2 & len b2 = len b3
   holds |(b1,b2 - b3)| = |(b1,b2)| - |(b1,b3)|;

:: COMPLSP2:th 73
theorem
for b1, b2, b3, b4 being FinSequence of COMPLEX
      st len b1 = len b2 & len b2 = len b3 & len b3 = len b4
   holds |(b1 + b2,b3 + b4)| = ((|(b1,b3)| + |(b1,b4)|) + |(b2,b3)|) + |(b2,b4)|;

:: COMPLSP2:th 74
theorem
for b1, b2, b3, b4 being FinSequence of COMPLEX
      st len b1 = len b2 & len b2 = len b3 & len b3 = len b4
   holds |(b1 - b2,b3 - b4)| = ((|(b1,b3)| - |(b1,b4)|) - |(b2,b3)|) + |(b2,b4)|;

:: COMPLSP2:th 75
theorem
for b1, b2 being FinSequence of COMPLEX
      st len b1 = len b2
   holds |(b1,b2)| = |(b2,b1)| *';

:: COMPLSP2:th 76
theorem
for b1, b2 being FinSequence of COMPLEX
      st len b1 = len b2
   holds |(b1 + b2,b1 + b2)| = (|(b1,b1)| + (2 * Re |(b1,b2)|)) + |(b2,b2)|;

:: COMPLSP2:th 77
theorem
for b1, b2 being FinSequence of COMPLEX
      st len b1 = len b2
   holds |(b1 - b2,b1 - b2)| = (|(b1,b1)| - (2 * Re |(b1,b2)|)) + |(b2,b2)|;

:: COMPLSP2:th 78
theorem
for b1 being Element of COMPLEX
for b2, b3 being FinSequence of COMPLEX
      st len b2 = len b3
   holds |(b1 * b2,b3)| = b1 * |(b2,b3)|;

:: COMPLSP2:th 79
theorem
for b1 being Element of COMPLEX
for b2, b3 being FinSequence of COMPLEX
      st len b2 = len b3
   holds |(b2,b1 * b3)| = b1 *' * |(b2,b3)|;

:: COMPLSP2:th 80
theorem
for b1, b2 being Element of COMPLEX
for b3, b4, b5 being FinSequence of COMPLEX
      st len b3 = len b4 & len b4 = len b5
   holds |((b1 * b3) + (b2 * b4),b5)| = (b1 * |(b3,b5)|) + (b2 * |(b4,b5)|);

:: COMPLSP2:th 81
theorem
for b1, b2 being Element of COMPLEX
for b3, b4, b5 being FinSequence of COMPLEX
      st len b3 = len b4 & len b4 = len b5
   holds |(b3,(b1 * b4) + (b2 * b5))| = (b1 *' * |(b3,b4)|) + (b2 *' * |(b3,b5)|);