Article COMPLFLD, MML version 4.99.1005

:: COMPLFLD:funcnot 1 => COMPLFLD:func 1
definition
  func F_Complex -> strict doubleLoopStr means
    the carrier of it = COMPLEX & the addF of it = addcomplex & the multF of it = multcomplex & 1. it = 1r & 0. it = 0c;
end;

:: COMPLFLD:def 1
theorem
for b1 being strict doubleLoopStr holds
      b1 = F_Complex
   iff
      the carrier of b1 = COMPLEX & the addF of b1 = addcomplex & the multF of b1 = multcomplex & 1. b1 = 1r & 0. b1 = 0c;

:: COMPLFLD:funcreg 1
registration
  cluster F_Complex -> non empty strict;
end;

:: COMPLFLD:condreg 1
registration
  cluster -> complex (Element of the carrier of F_Complex);
end;

:: COMPLFLD:funcreg 2
registration
  cluster F_Complex -> strict well-unital;
end;

:: COMPLFLD:funcreg 3
registration
  cluster F_Complex -> non degenerated right_complementable almost_left_invertible strict associative commutative right_unital distributive left_unital Abelian add-associative right_zeroed;
end;

:: COMPLFLD:th 3
theorem
for b1, b2 being Element of the carrier of F_Complex
for b3, b4 being complex set
      st b1 = b3 & b2 = b4
   holds b1 + b2 = b3 + b4;

:: COMPLFLD:th 4
theorem
for b1 being Element of the carrier of F_Complex
for b2 being complex set
      st b1 = b2
   holds - b1 = - b2;

:: COMPLFLD:th 5
theorem
for b1, b2 being Element of the carrier of F_Complex
for b3, b4 being complex set
      st b1 = b3 & b2 = b4
   holds b1 - b2 = b3 - b4;

:: COMPLFLD:th 6
theorem
for b1, b2 being Element of the carrier of F_Complex
for b3, b4 being complex set
      st b1 = b3 & b2 = b4
   holds b1 * b2 = b3 * b4;

:: COMPLFLD:th 7
theorem
for b1 being Element of the carrier of F_Complex
for b2 being complex set
      st b1 = b2 & b1 <> 0. F_Complex
   holds b1 " = b2 ";

:: COMPLFLD:th 8
theorem
for b1, b2 being Element of the carrier of F_Complex
for b3, b4 being complex set
      st b1 = b3 & b2 = b4 & b2 <> 0. F_Complex
   holds b1 / b2 = b3 / b4;

:: COMPLFLD:th 9
theorem
0. F_Complex = 0c;

:: COMPLFLD:th 10
theorem
1_ F_Complex = 1r;

:: COMPLFLD:th 11
theorem
(1_ F_Complex) + 1_ F_Complex <> 0. F_Complex;

:: COMPLFLD:funcnot 2 => COMPLFLD:func 2
definition
  let a1 be Element of the carrier of F_Complex;
  redefine func a1 *' -> Element of the carrier of F_Complex;
  involutiveness;
::  for a1 being Element of the carrier of F_Complex holds
::     a1 *' *' = a1;
end;

:: COMPLFLD:funcnot 3 => COMPLFLD:func 3
definition
  let a1 be Element of the carrier of F_Complex;
  redefine func |.a1.| -> Element of REAL;
  projectivity;
::  for a1 being Element of the carrier of F_Complex holds
::     |.|.a1.|.| = |.a1.|;
end;

:: COMPLFLD:th 29
theorem
for b1 being Element of the carrier of F_Complex holds
   - b1 = (- 1_ F_Complex) * b1;

:: COMPLFLD:th 35
theorem
for b1, b2 being Element of the carrier of F_Complex holds
b1 - - b2 = b1 + b2;

:: COMPLFLD:th 41
theorem
for b1, b2 being Element of the carrier of F_Complex holds
b1 = (b1 + b2) - b2;

:: COMPLFLD:th 42
theorem
for b1, b2 being Element of the carrier of F_Complex holds
b1 = (b1 - b2) + b2;

:: COMPLFLD:th 47
theorem
for b1, b2 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex & b2 <> 0. F_Complex & b1 " = b2 "
   holds b1 = b2;

:: COMPLFLD:th 48
theorem
for b1, b2 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex &
         (b2 * b1 = 1. F_Complex or b1 * b2 = 1. F_Complex)
   holds b2 = b1 ";

:: COMPLFLD:th 49
theorem
for b1, b2, b3 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex & (b2 * b1 = b3 or b1 * b2 = b3)
   holds b2 = b3 * (b1 ") & b2 = b1 " * b3;

:: COMPLFLD:th 50
theorem
(1. F_Complex) " = 1. F_Complex;

:: COMPLFLD:th 51
theorem
for b1, b2 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex & b2 <> 0. F_Complex
   holds (b1 * b2) " = b1 " * (b2 ");

:: COMPLFLD:th 53
theorem
for b1 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex
   holds (- b1) " = - (b1 ");

:: COMPLFLD:th 55
theorem
for b1, b2 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex & b2 <> 0. F_Complex
   holds b1 " + (b2 ") = (b1 + b2) * ((b1 * b2) ");

:: COMPLFLD:th 56
theorem
for b1, b2 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex & b2 <> 0. F_Complex
   holds b1 " - (b2 ") = (b2 - b1) * ((b1 * b2) ");

:: COMPLFLD:th 58
theorem
for b1 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex
   holds b1 " = (1. F_Complex) / b1;

:: COMPLFLD:th 59
theorem
for b1 being Element of the carrier of F_Complex holds
   b1 / 1. F_Complex = b1;

:: COMPLFLD:th 60
theorem
for b1 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex
   holds b1 / b1 = 1. F_Complex;

:: COMPLFLD:th 61
theorem
for b1 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex
   holds (0. F_Complex) / b1 = 0. F_Complex;

:: COMPLFLD:th 62
theorem
for b1, b2 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex & b2 / b1 = 0. F_Complex
   holds b2 = 0. F_Complex;

:: COMPLFLD:th 63
theorem
for b1, b2, b3, b4 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex & b2 <> 0. F_Complex
   holds (b3 / b1) * (b4 / b2) = (b3 * b4) / (b1 * b2);

:: COMPLFLD:th 64
theorem
for b1, b2, b3 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex
   holds b2 * (b3 / b1) = (b2 * b3) / b1;

:: COMPLFLD:th 65
theorem
for b1, b2 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex & b2 / b1 = 1. F_Complex
   holds b2 = b1;

:: COMPLFLD:th 66
theorem
for b1, b2 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex
   holds b2 = (b2 * b1) / b1;

:: COMPLFLD:th 67
theorem
for b1, b2 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex & b2 <> 0. F_Complex
   holds (b1 / b2) " = b2 / b1;

:: COMPLFLD:th 68
theorem
for b1, b2 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex & b2 <> 0. F_Complex
   holds b1 " / (b2 ") = b2 / b1;

:: COMPLFLD:th 69
theorem
for b1, b2 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex
   holds b2 / (b1 ") = b2 * b1;

:: COMPLFLD:th 70
theorem
for b1, b2 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex & b2 <> 0. F_Complex
   holds b1 " / b2 = (b1 * b2) ";

:: COMPLFLD:th 71
theorem
for b1, b2, b3 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex & b2 <> 0. F_Complex
   holds b1 " * (b3 / b2) = b3 / (b1 * b2);

:: COMPLFLD:th 72
theorem
for b1, b2, b3 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex & b2 <> 0. F_Complex
   holds b3 / b2 = (b3 * b1) / (b2 * b1) &
    b3 / b2 = (b1 * b3) / (b1 * b2);

:: COMPLFLD:th 73
theorem
for b1, b2, b3 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex & b2 <> 0. F_Complex
   holds b3 / (b1 * b2) = (b3 / b1) / b2;

:: COMPLFLD:th 74
theorem
for b1, b2, b3 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex & b2 <> 0. F_Complex
   holds (b3 * b2) / b1 = b3 / (b1 / b2);

:: COMPLFLD:th 75
theorem
for b1, b2, b3, b4 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex & b2 <> 0. F_Complex & b3 <> 0. F_Complex
   holds (b4 / b1) / (b2 / b3) = (b4 * b3) / (b1 * b2);

:: COMPLFLD:th 76
theorem
for b1, b2, b3, b4 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex & b2 <> 0. F_Complex
   holds (b3 / b1) + (b4 / b2) = ((b3 * b2) + (b4 * b1)) / (b1 * b2);

:: COMPLFLD:th 77
theorem
for b1, b2, b3 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex
   holds (b2 / b1) + (b3 / b1) = (b2 + b3) / b1;

:: COMPLFLD:th 78
theorem
for b1, b2 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex
   holds - (b2 / b1) = (- b2) / b1 & - (b2 / b1) = b2 / - b1;

:: COMPLFLD:th 79
theorem
for b1, b2 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex
   holds b2 / b1 = (- b2) / - b1;

:: COMPLFLD:th 80
theorem
for b1, b2, b3, b4 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex & b2 <> 0. F_Complex
   holds (b3 / b1) - (b4 / b2) = ((b3 * b2) - (b4 * b1)) / (b1 * b2);

:: COMPLFLD:th 81
theorem
for b1, b2, b3 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex
   holds (b2 / b1) - (b3 / b1) = (b2 - b3) / b1;

:: COMPLFLD:th 82
theorem
for b1, b2, b3 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex & (b2 * b1 = b3 or b1 * b2 = b3)
   holds b2 = b3 / b1;

:: COMPLFLD:th 83
theorem
(0. F_Complex) *' = 0. F_Complex;

:: COMPLFLD:th 84
theorem
for b1 being Element of the carrier of F_Complex
      st b1 *' = 0. F_Complex
   holds b1 = 0. F_Complex;

:: COMPLFLD:th 85
theorem
(1. F_Complex) *' = 1. F_Complex;

:: COMPLFLD:th 86
theorem
for b1 being Element of the carrier of F_Complex holds
   b1 *' *' = b1;

:: COMPLFLD:th 87
theorem
for b1, b2 being Element of the carrier of F_Complex holds
(b1 + b2) *' = b1 *' + (b2 *');

:: COMPLFLD:th 88
theorem
for b1 being Element of the carrier of F_Complex holds
   (- b1) *' = - (b1 *');

:: COMPLFLD:th 89
theorem
for b1, b2 being Element of the carrier of F_Complex holds
(b1 - b2) *' = b1 *' - (b2 *');

:: COMPLFLD:th 90
theorem
for b1, b2 being Element of the carrier of F_Complex holds
(b1 * b2) *' = b1 *' * (b2 *');

:: COMPLFLD:th 91
theorem
for b1 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex
   holds b1 " *' = b1 *' ";

:: COMPLFLD:th 92
theorem
for b1, b2 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex
   holds (b2 / b1) *' = b2 *' / (b1 *');

:: COMPLFLD:th 93
theorem
|.0. F_Complex.| = 0;

:: COMPLFLD:th 94
theorem
for b1 being Element of the carrier of F_Complex
      st |.b1.| = 0
   holds b1 = 0. F_Complex;

:: COMPLFLD:th 96
theorem
for b1 being Element of the carrier of F_Complex holds
      b1 <> 0. F_Complex
   iff
      0 < |.b1.|;

:: COMPLFLD:th 97
theorem
|.1. F_Complex.| = 1;

:: COMPLFLD:th 98
theorem
for b1 being Element of the carrier of F_Complex holds
   |.- b1.| = |.b1.|;

:: COMPLFLD:th 100
theorem
for b1, b2 being Element of the carrier of F_Complex holds
|.b1 + b2.| <= |.b1.| + |.b2.|;

:: COMPLFLD:th 101
theorem
for b1, b2 being Element of the carrier of F_Complex holds
|.b1 - b2.| <= |.b1.| + |.b2.|;

:: COMPLFLD:th 102
theorem
for b1, b2 being Element of the carrier of F_Complex holds
|.b1.| - |.b2.| <= |.b1 + b2.|;

:: COMPLFLD:th 103
theorem
for b1, b2 being Element of the carrier of F_Complex holds
|.b1.| - |.b2.| <= |.b1 - b2.|;

:: COMPLFLD:th 104
theorem
for b1, b2 being Element of the carrier of F_Complex holds
|.b1 - b2.| = |.b2 - b1.|;

:: COMPLFLD:th 105
theorem
for b1, b2 being Element of the carrier of F_Complex holds
   |.b1 - b2.| = 0
iff
   b1 = b2;

:: COMPLFLD:th 106
theorem
for b1, b2 being Element of the carrier of F_Complex holds
   b1 <> b2
iff
   0 < |.b1 - b2.|;

:: COMPLFLD:th 107
theorem
for b1, b2, b3 being Element of the carrier of F_Complex holds
|.b1 - b2.| <= |.b1 - b3.| + |.b3 - b2.|;

:: COMPLFLD:th 108
theorem
for b1, b2 being Element of the carrier of F_Complex holds
abs (|.b1.| - |.b2.|) <= |.b1 - b2.|;

:: COMPLFLD:th 109
theorem
for b1, b2 being Element of the carrier of F_Complex holds
|.b1 * b2.| = |.b1.| * |.b2.|;

:: COMPLFLD:th 110
theorem
for b1 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex
   holds |.b1 ".| = |.b1.| ";

:: COMPLFLD:th 111
theorem
for b1, b2 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex
   holds |.b2.| / |.b1.| = |.b2 / b1.|;

:: COMPLFLD:sch 1
scheme COMPLFLD:sch 1
P1[1]
provided
   ex b1 being non empty Element of NAT st
      P1[b1]
and
   for b1 being non empty Element of NAT
         st b1 <> 1 & P1[b1]
      holds ex b2 being non empty Element of NAT st
         b2 < b1 & P1[b2];


:: COMPLFLD:th 112
theorem
for b1 being Element of the carrier of F_Complex
for b2 being non empty natural set holds
   (power F_Complex) .(b1,b2) = b1 |^ b2;

:: COMPLFLD:modenot 1 => COMPLFLD:mode 1
definition
  let a1 be Element of the carrier of F_Complex;
  let a2 be non empty Element of NAT;
  redefine mode CRoot of A2,A1 -> Element of the carrier of F_Complex means
    (power F_Complex) .(it,a2) = a1;
end;

:: COMPLFLD:dfs 2
definiens
  let a1 be Element of the carrier of F_Complex;
  let a2 be non empty Element of NAT;
  let a3 be Element of the carrier of F_Complex;
To prove
     a3 is CRoot of a2,a1
it is sufficient to prove
  thus (power F_Complex) .(a3,a2) = a1;

:: COMPLFLD:def 2
theorem
for b1 being Element of the carrier of F_Complex
for b2 being non empty Element of NAT
for b3 being Element of the carrier of F_Complex holds
      b3 is CRoot of b2,b1
   iff
      (power F_Complex) .(b3,b2) = b1;

:: COMPLFLD:th 113
theorem
for b1 being Element of the carrier of F_Complex
for b2 being CRoot of 1,b1 holds
   b2 = b1;

:: COMPLFLD:th 114
theorem
for b1 being non empty Element of NAT
for b2 being CRoot of b1,0. F_Complex holds
   b2 = 0. F_Complex;

:: COMPLFLD:th 115
theorem
for b1 being non empty Element of NAT
for b2 being Element of the carrier of F_Complex
for b3 being CRoot of b1,b2
      st b3 = 0. F_Complex
   holds b2 = 0. F_Complex;