Article COMPTRIG, MML version 4.99.1005

:: COMPTRIG:sch 1
scheme COMPTRIG:sch 1
P1[1]
provided
   ex b1 being non empty natural set st
      P1[b1]
and
   for b1 being non empty natural set
         st b1 <> 1 & P1[b1]
      holds ex b2 being non empty natural set st
         b2 < b1 & P1[b2];


:: COMPTRIG:th 3
theorem
for b1 being complex set holds
   - |.b1.| <= Re b1;

:: COMPTRIG:th 4
theorem
for b1 being complex set holds
   - |.b1.| <= Im b1;

:: COMPTRIG:th 7
theorem
for b1 being complex set holds
   |.b1.| ^2 = (Re b1) ^2 + ((Im b1) ^2);

:: COMPTRIG:th 17
theorem
for b1 being real set
for b2 being natural set
      st 0 <= b1 & b2 <> 0
   holds (b2 -root b1) |^ b2 = b1;

:: COMPTRIG:funcreg 1
registration
  cluster PI -> real non negative;
end;

:: COMPTRIG:th 20
theorem
PI + (PI / 2) = (3 / 2) * PI &
 ((3 / 2) * PI) + (PI / 2) = 2 * PI &
 ((3 / 2) * PI) - PI = PI / 2;

:: COMPTRIG:th 21
theorem
0 < PI / 2 &
 PI / 2 < PI &
 0 < PI &
 - (PI / 2) < PI / 2 &
 PI < 2 * PI &
 PI / 2 < (3 / 2) * PI &
 - (PI / 2) < 0 &
 0 < 2 * PI &
 PI < (3 / 2) * PI &
 (3 / 2) * PI < 2 * PI &
 0 < (3 / 2) * PI;

:: COMPTRIG:th 22
theorem
for b1, b2, b3, b4 being real set
      st b4 in ].b1,b3.[ & not b4 in ].b1,b2.[ & b4 <> b2
   holds b4 in ].b2,b3.[;

:: COMPTRIG:th 23
theorem
for b1 being real set
      st b1 in ].0,PI.[
   holds 0 < sin . b1;

:: COMPTRIG:th 24
theorem
for b1 being real set
      st b1 in [.0,PI.]
   holds 0 <= sin . b1;

:: COMPTRIG:th 25
theorem
for b1 being real set
      st b1 in ].PI,2 * PI.[
   holds sin . b1 < 0;

:: COMPTRIG:th 26
theorem
for b1 being real set
      st b1 in [.PI,2 * PI.]
   holds sin . b1 <= 0;

:: COMPTRIG:th 27
theorem
for b1 being real set
      st b1 in ].- (PI / 2),PI / 2.[
   holds 0 < cos . b1;

:: COMPTRIG:th 28
theorem
for b1 being real set
      st b1 in [.- (PI / 2),PI / 2.]
   holds 0 <= cos . b1;

:: COMPTRIG:th 29
theorem
for b1 being real set
      st b1 in ].PI / 2,(3 / 2) * PI.[
   holds cos . b1 < 0;

:: COMPTRIG:th 30
theorem
for b1 being real set
      st b1 in [.PI / 2,(3 / 2) * PI.]
   holds cos . b1 <= 0;

:: COMPTRIG:th 31
theorem
for b1 being real set
      st b1 in ].(3 / 2) * PI,2 * PI.[
   holds 0 < cos . b1;

:: COMPTRIG:th 32
theorem
for b1 being real set
      st b1 in [.(3 / 2) * PI,2 * PI.]
   holds 0 <= cos . b1;

:: COMPTRIG:th 33
theorem
for b1 being real set
      st 0 <= b1 & b1 < 2 * PI & sin b1 = 0 & b1 <> 0
   holds b1 = PI;

:: COMPTRIG:th 34
theorem
for b1 being real set
      st 0 <= b1 & b1 < 2 * PI & cos b1 = 0 & b1 <> PI / 2
   holds b1 = (3 / 2) * PI;

:: COMPTRIG:th 35
theorem
sin is_increasing_on ].- (PI / 2),PI / 2.[;

:: COMPTRIG:th 36
theorem
sin is_decreasing_on ].PI / 2,(3 / 2) * PI.[;

:: COMPTRIG:th 37
theorem
cos is_decreasing_on ].0,PI.[;

:: COMPTRIG:th 38
theorem
cos is_increasing_on ].PI,2 * PI.[;

:: COMPTRIG:th 39
theorem
sin is_increasing_on [.- (PI / 2),PI / 2.];

:: COMPTRIG:th 40
theorem
sin is_decreasing_on [.PI / 2,(3 / 2) * PI.];

:: COMPTRIG:th 41
theorem
cos is_decreasing_on [.0,PI.];

:: COMPTRIG:th 42
theorem
cos is_increasing_on [.PI,2 * PI.];

:: COMPTRIG:th 43
theorem
sin is_continuous_on REAL &
 (for b1, b2 being real set holds
 sin is_continuous_on [.b1,b2.] & sin is_continuous_on ].b1,b2.[);

:: COMPTRIG:th 44
theorem
cos is_continuous_on REAL &
 (for b1, b2 being real set holds
 cos is_continuous_on [.b1,b2.] & cos is_continuous_on ].b1,b2.[);

:: COMPTRIG:th 45
theorem
for b1 being real set holds
   sin . b1 in [.- 1,1.] &
    cos . b1 in [.- 1,1.];

:: COMPTRIG:th 46
theorem
proj2 sin = [.- 1,1.];

:: COMPTRIG:th 47
theorem
proj2 cos = [.- 1,1.];

:: COMPTRIG:th 48
theorem
proj2 (sin | [.- (PI / 2),PI / 2.]) = [.- 1,1.];

:: COMPTRIG:th 49
theorem
proj2 (sin | [.PI / 2,(3 / 2) * PI.]) = [.- 1,1.];

:: COMPTRIG:th 50
theorem
proj2 (cos | [.0,PI.]) = [.- 1,1.];

:: COMPTRIG:th 51
theorem
proj2 (cos | [.PI,2 * PI.]) = [.- 1,1.];

:: COMPTRIG:funcnot 1 => COMPTRIG:func 1
definition
  let a1 be complex set;
  func Arg A1 -> Element of REAL means
    a1 = (|.a1.| * cos it) + ((|.a1.| * sin it) * <i>) &
     0 <= it &
     it < 2 * PI
    if a1 <> 0
    otherwise it = 0;
end;

:: COMPTRIG:def 1
theorem
for b1 being complex set
for b2 being Element of REAL holds
   (b1 = 0 or    (b2 = Arg b1
    iff
       b1 = (|.b1.| * cos b2) + ((|.b1.| * sin b2) * <i>) &
        0 <= b2 &
        b2 < 2 * PI)) &
    (b1 = 0 implies    (b2 = Arg b1
    iff
       b2 = 0));

:: COMPTRIG:th 52
theorem
for b1 being complex set holds
   0 <= Arg b1 & Arg b1 < 2 * PI;

:: COMPTRIG:th 53
theorem
for b1 being Element of REAL
      st 0 <= b1
   holds Arg b1 = 0;

:: COMPTRIG:th 54
theorem
for b1 being Element of REAL
      st b1 < 0
   holds Arg b1 = PI;

:: COMPTRIG:th 55
theorem
for b1 being Element of REAL
      st 0 < b1
   holds Arg (b1 * <i>) = PI / 2;

:: COMPTRIG:th 56
theorem
for b1 being Element of REAL
      st b1 < 0
   holds Arg (b1 * <i>) = (3 / 2) * PI;

:: COMPTRIG:th 57
theorem
Arg 1 = 0;

:: COMPTRIG:th 58
theorem
Arg <i> = PI / 2;

:: COMPTRIG:th 59
theorem
for b1 being complex set holds
      Arg b1 in ].0,PI / 2.[
   iff
      0 < Re b1 & 0 < Im b1;

:: COMPTRIG:th 60
theorem
for b1 being complex set holds
      Arg b1 in ].PI / 2,PI.[
   iff
      Re b1 < 0 & 0 < Im b1;

:: COMPTRIG:th 61
theorem
for b1 being complex set holds
      Arg b1 in ].PI,(3 / 2) * PI.[
   iff
      Re b1 < 0 & Im b1 < 0;

:: COMPTRIG:th 62
theorem
for b1 being complex set holds
      Arg b1 in ].(3 / 2) * PI,2 * PI.[
   iff
      0 < Re b1 & Im b1 < 0;

:: COMPTRIG:th 63
theorem
for b1 being complex set
      st 0 < Im b1
   holds 0 < sin Arg b1;

:: COMPTRIG:th 64
theorem
for b1 being complex set
      st Im b1 < 0
   holds sin Arg b1 < 0;

:: COMPTRIG:th 65
theorem
for b1 being complex set
      st 0 <= Im b1
   holds 0 <= sin Arg b1;

:: COMPTRIG:th 66
theorem
for b1 being complex set
      st Im b1 <= 0
   holds sin Arg b1 <= 0;

:: COMPTRIG:th 67
theorem
for b1 being complex set
      st 0 < Re b1
   holds 0 < cos Arg b1;

:: COMPTRIG:th 68
theorem
for b1 being complex set
      st Re b1 < 0
   holds cos Arg b1 < 0;

:: COMPTRIG:th 69
theorem
for b1 being complex set
      st 0 <= Re b1
   holds 0 <= cos Arg b1;

:: COMPTRIG:th 70
theorem
for b1 being complex set
      st Re b1 <= 0 & b1 <> 0
   holds cos Arg b1 <= 0;

:: COMPTRIG:th 71
theorem
for b1 being real set
for b2 being natural set holds
   ((cos b1) + ((sin b1) * <i>)) |^ b2 = (cos (b2 * b1)) + ((sin (b2 * b1)) * <i>);

:: COMPTRIG:th 72
theorem
for b1 being Element of COMPLEX
for b2 being natural set
      st (b1 = 0 implies b2 <> 0)
   holds b1 |^ b2 = ((|.b1.| |^ b2) * cos (b2 * Arg b1)) + (((|.b1.| |^ b2) * sin (b2 * Arg b1)) * <i>);

:: COMPTRIG:th 73
theorem
for b1 being Element of REAL
for b2, b3 being natural set
      st b2 <> 0
   holds ((cos ((b1 + ((2 * PI) * b3)) / b2)) + ((sin ((b1 + ((2 * PI) * b3)) / b2)) * <i>)) |^ b2 = (cos b1) + ((sin b1) * <i>);

:: COMPTRIG:th 74
theorem
for b1 being complex set
for b2, b3 being natural set
      st b2 <> 0
   holds b1 = (((b2 -root |.b1.|) * cos (((Arg b1) + ((2 * PI) * b3)) / b2)) + (((b2 -root |.b1.|) * sin (((Arg b1) + ((2 * PI) * b3)) / b2)) * <i>)) |^ b2;

:: COMPTRIG:modenot 1 => COMPTRIG:mode 1
definition
  let a1 be complex set;
  let a2 be non empty natural set;
  mode CRoot of A2,A1 -> complex set means
    it |^ a2 = a1;
end;

:: COMPTRIG:dfs 2
definiens
  let a1 be complex set;
  let a2 be non empty natural set;
  let a3 be complex set;
To prove
     a3 is CRoot of a2,a1
it is sufficient to prove
  thus a3 |^ a2 = a1;

:: COMPTRIG:def 2
theorem
for b1 being complex set
for b2 being non empty natural set
for b3 being complex set holds
      b3 is CRoot of b2,b1
   iff
      b3 |^ b2 = b1;

:: COMPTRIG:th 75
theorem
for b1 being Element of COMPLEX
for b2 being non empty natural set
for b3 being natural set holds
   ((b2 -root |.b1.|) * cos (((Arg b1) + ((2 * PI) * b3)) / b2)) + (((b2 -root |.b1.|) * sin (((Arg b1) + ((2 * PI) * b3)) / b2)) * <i>) is CRoot of b2,b1;

:: COMPTRIG:th 76
theorem
for b1 being Element of COMPLEX
for b2 being CRoot of 1,b1 holds
   b2 = b1;

:: COMPTRIG:th 77
theorem
for b1 being non empty natural set
for b2 being CRoot of b1,0 holds
   b2 = 0;

:: COMPTRIG:th 78
theorem
for b1 being non empty natural set
for b2 being Element of COMPLEX
for b3 being CRoot of b1,b2
      st b3 = 0
   holds b2 = 0;

:: COMPTRIG:th 79
theorem
for b1 being real set
      st 0 <= b1 & b1 < 2 * PI & cos b1 = 1
   holds b1 = 0;

:: COMPTRIG:th 80
theorem
for b1 being complex set holds
   b1 = (|.b1.| * cos Arg b1) + ((|.b1.| * sin Arg b1) * <i>);