Article SIN_COS3, MML version 4.99.1005

:: SIN_COS3:funcnot 1 => SIN_COS3:func 1
definition
  func sin_C -> Function-like quasi_total Relation of COMPLEX,COMPLEX means
    for b1 being Element of COMPLEX holds
       it . b1 = ((exp (<i> * b1)) - exp - (<i> * b1)) / (2 * <i>);
end;

:: SIN_COS3:def 1
theorem
for b1 being Function-like quasi_total Relation of COMPLEX,COMPLEX holds
      b1 = sin_C
   iff
      for b2 being Element of COMPLEX holds
         b1 . b2 = ((exp (<i> * b2)) - exp - (<i> * b2)) / (2 * <i>);

:: SIN_COS3:funcnot 2 => SIN_COS3:func 2
definition
  func cos_C -> Function-like quasi_total Relation of COMPLEX,COMPLEX means
    for b1 being Element of COMPLEX holds
       it . b1 = ((exp (<i> * b1)) + exp - (<i> * b1)) / 2;
end;

:: SIN_COS3:def 2
theorem
for b1 being Function-like quasi_total Relation of COMPLEX,COMPLEX holds
      b1 = cos_C
   iff
      for b2 being Element of COMPLEX holds
         b1 . b2 = ((exp (<i> * b2)) + exp - (<i> * b2)) / 2;

:: SIN_COS3:funcnot 3 => SIN_COS3:func 3
definition
  func sinh_C -> Function-like quasi_total Relation of COMPLEX,COMPLEX means
    for b1 being Element of COMPLEX holds
       it . b1 = ((exp b1) - exp - b1) / 2;
end;

:: SIN_COS3:def 3
theorem
for b1 being Function-like quasi_total Relation of COMPLEX,COMPLEX holds
      b1 = sinh_C
   iff
      for b2 being Element of COMPLEX holds
         b1 . b2 = ((exp b2) - exp - b2) / 2;

:: SIN_COS3:funcnot 4 => SIN_COS3:func 4
definition
  func cosh_C -> Function-like quasi_total Relation of COMPLEX,COMPLEX means
    for b1 being Element of COMPLEX holds
       it . b1 = ((exp b1) + exp - b1) / 2;
end;

:: SIN_COS3:def 4
theorem
for b1 being Function-like quasi_total Relation of COMPLEX,COMPLEX holds
      b1 = cosh_C
   iff
      for b2 being Element of COMPLEX holds
         b1 . b2 = ((exp b2) + exp - b2) / 2;

:: SIN_COS3:th 1
theorem
for b1 being Element of COMPLEX holds
   ((sin_C /. b1) * (sin_C /. b1)) + ((cos_C /. b1) * (cos_C /. b1)) = 1;

:: SIN_COS3:th 2
theorem
for b1 being Element of COMPLEX holds
   - (sin_C /. b1) = sin_C /. - b1;

:: SIN_COS3:th 3
theorem
for b1 being Element of COMPLEX holds
   cos_C /. b1 = cos_C /. - b1;

:: SIN_COS3:th 4
theorem
for b1, b2 being Element of COMPLEX holds
sin_C /. (b1 + b2) = ((sin_C /. b1) * (cos_C /. b2)) + ((cos_C /. b1) * (sin_C /. b2));

:: SIN_COS3:th 5
theorem
for b1, b2 being Element of COMPLEX holds
sin_C /. (b1 - b2) = ((sin_C /. b1) * (cos_C /. b2)) - ((cos_C /. b1) * (sin_C /. b2));

:: SIN_COS3:th 6
theorem
for b1, b2 being Element of COMPLEX holds
cos_C /. (b1 + b2) = ((cos_C /. b1) * (cos_C /. b2)) - ((sin_C /. b1) * (sin_C /. b2));

:: SIN_COS3:th 7
theorem
for b1, b2 being Element of COMPLEX holds
cos_C /. (b1 - b2) = ((cos_C /. b1) * (cos_C /. b2)) + ((sin_C /. b1) * (sin_C /. b2));

:: SIN_COS3:th 8
theorem
for b1 being Element of COMPLEX holds
   ((cosh_C /. b1) * (cosh_C /. b1)) - ((sinh_C /. b1) * (sinh_C /. b1)) = 1;

:: SIN_COS3:th 9
theorem
for b1 being Element of COMPLEX holds
   - (sinh_C /. b1) = sinh_C /. - b1;

:: SIN_COS3:th 10
theorem
for b1 being Element of COMPLEX holds
   cosh_C /. b1 = cosh_C /. - b1;

:: SIN_COS3:th 11
theorem
for b1, b2 being Element of COMPLEX holds
sinh_C /. (b1 + b2) = ((sinh_C /. b1) * (cosh_C /. b2)) + ((cosh_C /. b1) * (sinh_C /. b2));

:: SIN_COS3:th 12
theorem
for b1, b2 being Element of COMPLEX holds
sinh_C /. (b1 - b2) = ((sinh_C /. b1) * (cosh_C /. b2)) - ((cosh_C /. b1) * (sinh_C /. b2));

:: SIN_COS3:th 13
theorem
for b1, b2 being Element of COMPLEX holds
cosh_C /. (b1 - b2) = ((cosh_C /. b1) * (cosh_C /. b2)) - ((sinh_C /. b1) * (sinh_C /. b2));

:: SIN_COS3:th 14
theorem
for b1, b2 being Element of COMPLEX holds
cosh_C /. (b1 + b2) = ((cosh_C /. b1) * (cosh_C /. b2)) + ((sinh_C /. b1) * (sinh_C /. b2));

:: SIN_COS3:th 15
theorem
for b1 being Element of COMPLEX holds
   sin_C /. (<i> * b1) = <i> * (sinh_C /. b1);

:: SIN_COS3:th 16
theorem
for b1 being Element of COMPLEX holds
   cos_C /. (<i> * b1) = cosh_C /. b1;

:: SIN_COS3:th 17
theorem
for b1 being Element of COMPLEX holds
   sinh_C /. (<i> * b1) = <i> * (sin_C /. b1);

:: SIN_COS3:th 18
theorem
for b1 being Element of COMPLEX holds
   cosh_C /. (<i> * b1) = cos_C /. b1;

:: SIN_COS3:th 19
theorem
for b1, b2 being Element of REAL holds
exp [*b1,b2*] = [*(exp_R . b1) * (cos . b2),(exp_R . b1) * (sin . b2)*];

:: SIN_COS3:th 20
theorem
exp 0c = 1;

:: SIN_COS3:th 21
theorem
sin_C /. 0c = 0;

:: SIN_COS3:th 22
theorem
sinh_C /. 0c = 0;

:: SIN_COS3:th 23
theorem
cos_C /. 0c = 1;

:: SIN_COS3:th 24
theorem
cosh_C /. 0c = 1;

:: SIN_COS3:th 25
theorem
for b1 being Element of COMPLEX holds
   exp b1 = (cosh_C /. b1) + (sinh_C /. b1);

:: SIN_COS3:th 26
theorem
for b1 being Element of COMPLEX holds
   exp - b1 = (cosh_C /. b1) - (sinh_C /. b1);

:: SIN_COS3:th 27
theorem
for b1 being Element of COMPLEX holds
   exp (b1 + ([*2 * PI,0*] * <i>)) = exp b1 &
    exp (b1 + [*0,2 * PI*]) = exp b1;

:: SIN_COS3:th 28
theorem
for b1 being Element of NAT holds
   exp [*0,(2 * PI) * b1*] = 1 &
    exp ([*(2 * PI) * b1,0*] * <i>) = 1;

:: SIN_COS3:th 29
theorem
for b1 being Element of NAT holds
   exp [*0,- ((2 * PI) * b1)*] = 1 &
    exp ([*- ((2 * PI) * b1),0*] * <i>) = 1;

:: SIN_COS3:th 30
theorem
for b1 being Element of NAT holds
   exp [*0,((2 * b1) + 1) * PI*] = [*- 1,0*] &
    exp ([*((2 * b1) + 1) * PI,0*] * <i>) = [*- 1,0*];

:: SIN_COS3:th 31
theorem
for b1 being Element of NAT holds
   exp [*0,- (((2 * b1) + 1) * PI)*] = [*- 1,0*] &
    exp ([*- (((2 * b1) + 1) * PI),0*] * <i>) = [*- 1,0*];

:: SIN_COS3:th 32
theorem
for b1 being Element of NAT holds
   exp [*0,((2 * b1) + (1 / 2)) * PI*] = [*0,1*] &
    exp ([*((2 * b1) + (1 / 2)) * PI,0*] * <i>) = [*0,1*];

:: SIN_COS3:th 33
theorem
for b1 being Element of NAT holds
   exp [*0,- (((2 * b1) + (1 / 2)) * PI)*] = [*0,- 1*] &
    exp ([*- (((2 * b1) + (1 / 2)) * PI),0*] * <i>) = [*0,- 1*];

:: SIN_COS3:th 34
theorem
for b1 being Element of COMPLEX
for b2 being Element of NAT holds
   sin_C /. (b1 + ((2 * b2) * PI)) = sin_C /. b1;

:: SIN_COS3:th 35
theorem
for b1 being Element of COMPLEX
for b2 being Element of NAT holds
   cos_C /. (b1 + ((2 * b2) * PI)) = cos_C /. b1;

:: SIN_COS3:th 36
theorem
for b1 being Element of COMPLEX holds
   exp (<i> * b1) = (cos_C /. b1) + (<i> * (sin_C /. b1));

:: SIN_COS3:th 37
theorem
for b1 being Element of COMPLEX holds
   exp - (<i> * b1) = (cos_C /. b1) - (<i> * (sin_C /. b1));

:: SIN_COS3:th 38
theorem
for b1 being Element of REAL holds
   sin_C /. [*b1,0*] = sin . b1;

:: SIN_COS3:th 39
theorem
for b1 being Element of REAL holds
   cos_C /. [*b1,0*] = cos . b1;

:: SIN_COS3:th 40
theorem
for b1 being Element of REAL holds
   sinh_C /. [*b1,0*] = sinh . b1;

:: SIN_COS3:th 41
theorem
for b1 being Element of REAL holds
   cosh_C /. [*b1,0*] = cosh . b1;

:: SIN_COS3:th 42
theorem
for b1, b2 being Element of REAL holds
[*b1,0*] + (<i> * [*b2,0*]) = [*b1,b2*];

:: SIN_COS3:th 43
theorem
for b1, b2 being Element of REAL holds
sin_C /. [*b1,b2*] = [*(sin . b1) * (cosh . b2),(cos . b1) * (sinh . b2)*];

:: SIN_COS3:th 44
theorem
for b1, b2 being Element of REAL holds
sin_C /. [*b1,- b2*] = [*(sin . b1) * (cosh . b2),- ((cos . b1) * (sinh . b2))*];

:: SIN_COS3:th 45
theorem
for b1, b2 being Element of REAL holds
cos_C /. [*b1,b2*] = [*(cos . b1) * (cosh . b2),- ((sin . b1) * (sinh . b2))*];

:: SIN_COS3:th 46
theorem
for b1, b2 being Element of REAL holds
cos_C /. [*b1,- b2*] = [*(cos . b1) * (cosh . b2),(sin . b1) * (sinh . b2)*];

:: SIN_COS3:th 47
theorem
for b1, b2 being Element of REAL holds
sinh_C /. [*b1,b2*] = [*(sinh . b1) * (cos . b2),(cosh . b1) * (sin . b2)*];

:: SIN_COS3:th 48
theorem
for b1, b2 being Element of REAL holds
sinh_C /. [*b1,- b2*] = [*(sinh . b1) * (cos . b2),- ((cosh . b1) * (sin . b2))*];

:: SIN_COS3:th 49
theorem
for b1, b2 being Element of REAL holds
cosh_C /. [*b1,b2*] = [*(cosh . b1) * (cos . b2),(sinh . b1) * (sin . b2)*];

:: SIN_COS3:th 50
theorem
for b1, b2 being Element of REAL holds
cosh_C /. [*b1,- b2*] = [*(cosh . b1) * (cos . b2),- ((sinh . b1) * (sin . b2))*];

:: SIN_COS3:th 51
theorem
for b1 being Element of NAT
for b2 being Element of COMPLEX holds
   ((cos_C /. b2) + (<i> * (sin_C /. b2))) #N b1 = (cos_C /. ([*b1,0*] * b2)) + (<i> * (sin_C /. ([*b1,0*] * b2)));

:: SIN_COS3:th 52
theorem
for b1 being Element of NAT
for b2 being Element of COMPLEX holds
   ((cos_C /. b2) - (<i> * (sin_C /. b2))) #N b1 = (cos_C /. ([*b1,0*] * b2)) - (<i> * (sin_C /. ([*b1,0*] * b2)));

:: SIN_COS3:th 53
theorem
for b1 being Element of NAT
for b2 being Element of COMPLEX holds
   exp ((<i> * [*b1,0*]) * b2) = ((cos_C /. b2) + (<i> * (sin_C /. b2))) #N b1;

:: SIN_COS3:th 54
theorem
for b1 being Element of NAT
for b2 being Element of COMPLEX holds
   exp - ((<i> * [*b1,0*]) * b2) = ((cos_C /. b2) - (<i> * (sin_C /. b2))) #N b1;

:: SIN_COS3:th 55
theorem
for b1, b2 being Element of REAL holds
(([*1,- 1*] / 2) * (sinh_C /. [*b1,b2*])) + (([*1,1*] / 2) * (sinh_C /. [*b1,- b2*])) = [*((sinh . b1) * (cos . b2)) + ((cosh . b1) * (sin . b2)),0*];

:: SIN_COS3:th 56
theorem
for b1, b2 being Element of REAL holds
(([*1,- 1*] / 2) * (cosh_C /. [*b1,b2*])) + (([*1,1*] / 2) * (cosh_C /. [*b1,- b2*])) = ((sinh . b1) * (sin . b2)) + ((cosh . b1) * (cos . b2));

:: SIN_COS3:th 57
theorem
for b1 being Element of COMPLEX holds
   (sinh_C /. b1) * (sinh_C /. b1) = ((cosh_C /. (2 * b1)) - 1) / 2;

:: SIN_COS3:th 58
theorem
for b1 being Element of COMPLEX holds
   (cosh_C /. b1) * (cosh_C /. b1) = ((cosh_C /. (2 * b1)) + 1) / 2;

:: SIN_COS3:th 59
theorem
for b1 being Element of COMPLEX holds
   sinh_C /. (2 * b1) = (2 * (sinh_C /. b1)) * (cosh_C /. b1) &
    cosh_C /. (2 * b1) = ((2 * (cosh_C /. b1)) * (cosh_C /. b1)) - 1;

:: SIN_COS3:th 60
theorem
for b1, b2 being Element of COMPLEX holds
((sinh_C /. b1) * (sinh_C /. b1)) - ((sinh_C /. b2) * (sinh_C /. b2)) = (sinh_C /. (b1 + b2)) * (sinh_C /. (b1 - b2)) &
 ((cosh_C /. b1) * (cosh_C /. b1)) - ((cosh_C /. b2) * (cosh_C /. b2)) = (sinh_C /. (b1 + b2)) * (sinh_C /. (b1 - b2)) &
 ((sinh_C /. b1) * (sinh_C /. b1)) - ((sinh_C /. b2) * (sinh_C /. b2)) = ((cosh_C /. b1) * (cosh_C /. b1)) - ((cosh_C /. b2) * (cosh_C /. b2));

:: SIN_COS3:th 61
theorem
for b1, b2 being Element of COMPLEX holds
(cosh_C /. (b1 + b2)) * (cosh_C /. (b1 - b2)) = ((sinh_C /. b1) * (sinh_C /. b1)) + ((cosh_C /. b2) * (cosh_C /. b2)) &
 (cosh_C /. (b1 + b2)) * (cosh_C /. (b1 - b2)) = ((cosh_C /. b1) * (cosh_C /. b1)) + ((sinh_C /. b2) * (sinh_C /. b2)) &
 ((sinh_C /. b1) * (sinh_C /. b1)) + ((cosh_C /. b2) * (cosh_C /. b2)) = ((cosh_C /. b1) * (cosh_C /. b1)) + ((sinh_C /. b2) * (sinh_C /. b2));

:: SIN_COS3:th 62
theorem
for b1, b2 being Element of COMPLEX holds
(sinh_C /. (2 * b1)) + (sinh_C /. (2 * b2)) = (2 * (sinh_C /. (b1 + b2))) * (cosh_C /. (b1 - b2)) &
 (sinh_C /. (2 * b1)) - (sinh_C /. (2 * b2)) = (2 * (sinh_C /. (b1 - b2))) * (cosh_C /. (b1 + b2));

:: SIN_COS3:th 63
theorem
for b1, b2 being Element of COMPLEX holds
(cosh_C /. (2 * b1)) + (cosh_C /. (2 * b2)) = (2 * (cosh_C /. (b1 + b2))) * (cosh_C /. (b1 - b2)) &
 (cosh_C /. (2 * b1)) - (cosh_C /. (2 * b2)) = (2 * (sinh_C /. (b1 + b2))) * (sinh_C /. (b1 - b2));