Article SIN_COS2, MML version 4.99.1005

:: SIN_COS2:th 1
theorem
for b1, b2 being real set
      st 0 <= b1 & 0 <= b2
   holds 2 * sqrt (b1 * b2) <= b1 + b2;

:: SIN_COS2:th 2
theorem
sin is_increasing_on ].0,PI / 2.[;

:: SIN_COS2:th 3
theorem
sin is_decreasing_on ].PI / 2,PI.[;

:: SIN_COS2:th 4
theorem
cos is_decreasing_on ].0,PI / 2.[;

:: SIN_COS2:th 5
theorem
cos is_decreasing_on ].PI / 2,PI.[;

:: SIN_COS2:th 6
theorem
sin is_decreasing_on ].PI,(3 / 2) * PI.[;

:: SIN_COS2:th 7
theorem
sin is_increasing_on ].(3 / 2) * PI,2 * PI.[;

:: SIN_COS2:th 8
theorem
cos is_increasing_on ].PI,(3 / 2) * PI.[;

:: SIN_COS2:th 9
theorem
cos is_increasing_on ].(3 / 2) * PI,2 * PI.[;

:: SIN_COS2:th 10
theorem
for b1 being real set
for b2 being natural set holds
   sin . b1 = sin . (((2 * PI) * b2) + b1);

:: SIN_COS2:th 11
theorem
for b1 being real set
for b2 being natural set holds
   cos . b1 = cos . (((2 * PI) * b2) + b1);

:: SIN_COS2:funcnot 1 => SIN_COS2:func 1
definition
  func sinh -> Function-like quasi_total Relation of REAL,REAL means
    for b1 being real set holds
       it . b1 = ((exp_R . b1) - (exp_R . - b1)) / 2;
end;

:: SIN_COS2:def 1
theorem
for b1 being Function-like quasi_total Relation of REAL,REAL holds
      b1 = sinh
   iff
      for b2 being real set holds
         b1 . b2 = ((exp_R . b2) - (exp_R . - b2)) / 2;

:: SIN_COS2:funcnot 2 => SIN_COS2:func 2
definition
  let a1 be set;
  func sinh A1 -> set equals
    sinh . a1;
end;

:: SIN_COS2:def 2
theorem
for b1 being set holds
   sinh b1 = sinh . b1;

:: SIN_COS2:funcreg 1
registration
  let a1 be set;
  cluster sinh a1 -> real;
end;

:: SIN_COS2:funcnot 3 => SIN_COS2:func 3
definition
  let a1 be set;
  redefine func sinh a1 -> Element of REAL;
end;

:: SIN_COS2:funcnot 4 => SIN_COS2:func 4
definition
  func cosh -> Function-like quasi_total Relation of REAL,REAL means
    for b1 being real set holds
       it . b1 = ((exp_R . b1) + (exp_R . - b1)) / 2;
end;

:: SIN_COS2:def 3
theorem
for b1 being Function-like quasi_total Relation of REAL,REAL holds
      b1 = cosh
   iff
      for b2 being real set holds
         b1 . b2 = ((exp_R . b2) + (exp_R . - b2)) / 2;

:: SIN_COS2:funcnot 5 => SIN_COS2:func 5
definition
  let a1 be set;
  func cosh A1 -> set equals
    cosh . a1;
end;

:: SIN_COS2:def 4
theorem
for b1 being set holds
   cosh b1 = cosh . b1;

:: SIN_COS2:funcreg 2
registration
  let a1 be set;
  cluster cosh a1 -> real;
end;

:: SIN_COS2:funcnot 6 => SIN_COS2:func 6
definition
  let a1 be set;
  redefine func cosh a1 -> Element of REAL;
end;

:: SIN_COS2:funcnot 7 => SIN_COS2:func 7
definition
  func tanh -> Function-like quasi_total Relation of REAL,REAL means
    for b1 being real set holds
       it . b1 = ((exp_R . b1) - (exp_R . - b1)) / ((exp_R . b1) + (exp_R . - b1));
end;

:: SIN_COS2:def 5
theorem
for b1 being Function-like quasi_total Relation of REAL,REAL holds
      b1 = tanh
   iff
      for b2 being real set holds
         b1 . b2 = ((exp_R . b2) - (exp_R . - b2)) / ((exp_R . b2) + (exp_R . - b2));

:: SIN_COS2:funcnot 8 => SIN_COS2:func 8
definition
  let a1 be set;
  func tanh A1 -> set equals
    tanh . a1;
end;

:: SIN_COS2:def 6
theorem
for b1 being set holds
   tanh b1 = tanh . b1;

:: SIN_COS2:funcreg 3
registration
  let a1 be set;
  cluster tanh a1 -> real;
end;

:: SIN_COS2:funcnot 9 => SIN_COS2:func 9
definition
  let a1 be set;
  redefine func tanh a1 -> Element of REAL;
end;

:: SIN_COS2:th 12
theorem
for b1, b2 being real set holds
exp_R . (b1 + b2) = (exp_R . b1) * (exp_R . b2);

:: SIN_COS2:th 13
theorem
exp_R . 0 = 1;

:: SIN_COS2:th 14
theorem
for b1 being real set holds
   (cosh . b1) ^2 - ((sinh . b1) ^2) = 1 &
    ((cosh . b1) * (cosh . b1)) - ((sinh . b1) * (sinh . b1)) = 1;

:: SIN_COS2:th 15
theorem
for b1 being real set holds
   cosh . b1 <> 0 & 0 < cosh . b1 & cosh . 0 = 1;

:: SIN_COS2:th 16
theorem
sinh . 0 = 0;

:: SIN_COS2:th 17
theorem
for b1 being real set holds
   tanh . b1 = (sinh . b1) / (cosh . b1);

:: SIN_COS2:th 18
theorem
for b1 being real set holds
   (sinh . b1) ^2 = (1 / 2) * ((cosh . (2 * b1)) - 1) &
    (cosh . b1) ^2 = (1 / 2) * ((cosh . (2 * b1)) + 1);

:: SIN_COS2:th 19
theorem
for b1 being real set holds
   cosh . - b1 = cosh . b1 &
    sinh . - b1 = - (sinh . b1) &
    tanh . - b1 = - (tanh . b1);

:: SIN_COS2:th 20
theorem
for b1, b2 being real set holds
cosh . (b1 + b2) = ((cosh . b1) * (cosh . b2)) + ((sinh . b1) * (sinh . b2)) &
 cosh . (b1 - b2) = ((cosh . b1) * (cosh . b2)) - ((sinh . b1) * (sinh . b2));

:: SIN_COS2:th 21
theorem
for b1, b2 being real set holds
sinh . (b1 + b2) = ((sinh . b1) * (cosh . b2)) + ((cosh . b1) * (sinh . b2)) &
 sinh . (b1 - b2) = ((sinh . b1) * (cosh . b2)) - ((cosh . b1) * (sinh . b2));

:: SIN_COS2:th 22
theorem
for b1, b2 being real set holds
tanh . (b1 + b2) = ((tanh . b1) + (tanh . b2)) / (1 + ((tanh . b1) * (tanh . b2))) &
 tanh . (b1 - b2) = ((tanh . b1) - (tanh . b2)) / (1 - ((tanh . b1) * (tanh . b2)));

:: SIN_COS2:th 23
theorem
for b1 being real set holds
   sinh . (2 * b1) = (2 * (sinh . b1)) * (cosh . b1) &
    cosh . (2 * b1) = (2 * ((cosh . b1) ^2)) - 1 &
    tanh . (2 * b1) = (2 * (tanh . b1)) / (1 + ((tanh . b1) ^2));

:: SIN_COS2:th 24
theorem
for b1, b2 being real set holds
(sinh . b1) ^2 - ((sinh . b2) ^2) = (sinh . (b1 + b2)) * (sinh . (b1 - b2)) &
 (sinh . (b1 + b2)) * (sinh . (b1 - b2)) = (cosh . b1) ^2 - ((cosh . b2) ^2) &
 (sinh . b1) ^2 - ((sinh . b2) ^2) = (cosh . b1) ^2 - ((cosh . b2) ^2);

:: SIN_COS2:th 25
theorem
for b1, b2 being real set holds
(sinh . b1) ^2 + ((cosh . b2) ^2) = (cosh . (b1 + b2)) * (cosh . (b1 - b2)) &
 (cosh . (b1 + b2)) * (cosh . (b1 - b2)) = (cosh . b1) ^2 + ((sinh . b2) ^2) &
 (sinh . b1) ^2 + ((cosh . b2) ^2) = (cosh . b1) ^2 + ((sinh . b2) ^2);

:: SIN_COS2:th 26
theorem
for b1, b2 being real set holds
(sinh . b1) + (sinh . b2) = (2 * (sinh . ((b1 / 2) + (b2 / 2)))) * (cosh . ((b1 / 2) - (b2 / 2))) &
 (sinh . b1) - (sinh . b2) = (2 * (sinh . ((b1 / 2) - (b2 / 2)))) * (cosh . ((b1 / 2) + (b2 / 2)));

:: SIN_COS2:th 27
theorem
for b1, b2 being real set holds
(cosh . b1) + (cosh . b2) = (2 * (cosh . ((b1 / 2) + (b2 / 2)))) * (cosh . ((b1 / 2) - (b2 / 2))) &
 (cosh . b1) - (cosh . b2) = (2 * (sinh . ((b1 / 2) - (b2 / 2)))) * (sinh . ((b1 / 2) + (b2 / 2)));

:: SIN_COS2:th 28
theorem
for b1, b2 being real set holds
(tanh . b1) + (tanh . b2) = (sinh . (b1 + b2)) / ((cosh . b1) * (cosh . b2)) &
 (tanh . b1) - (tanh . b2) = (sinh . (b1 - b2)) / ((cosh . b1) * (cosh . b2));

:: SIN_COS2:th 29
theorem
for b1 being real set
for b2 being Element of NAT holds
   ((cosh . b1) + (sinh . b1)) |^ b2 = (cosh . (b2 * b1)) + (sinh . (b2 * b1));

:: SIN_COS2:th 30
theorem
dom sinh = REAL & dom cosh = REAL & dom tanh = REAL;

:: SIN_COS2:th 31
theorem
for b1 being real set holds
   sinh is_differentiable_in b1 & diff(sinh,b1) = cosh . b1;

:: SIN_COS2:th 32
theorem
for b1 being real set holds
   cosh is_differentiable_in b1 & diff(cosh,b1) = sinh . b1;

:: SIN_COS2:th 33
theorem
for b1 being real set holds
   tanh is_differentiable_in b1 &
    diff(tanh,b1) = 1 / ((cosh . b1) ^2);

:: SIN_COS2:th 34
theorem
for b1 being real set holds
   sinh is_differentiable_on REAL & diff(sinh,b1) = cosh . b1;

:: SIN_COS2:th 35
theorem
for b1 being real set holds
   cosh is_differentiable_on REAL & diff(cosh,b1) = sinh . b1;

:: SIN_COS2:th 36
theorem
for b1 being real set holds
   tanh is_differentiable_on REAL &
    diff(tanh,b1) = 1 / ((cosh . b1) ^2);

:: SIN_COS2:th 37
theorem
for b1 being real set holds
   1 <= cosh . b1;

:: SIN_COS2:th 38
theorem
for b1 being real set holds
   sinh is_continuous_in b1;

:: SIN_COS2:th 39
theorem
for b1 being real set holds
   cosh is_continuous_in b1;

:: SIN_COS2:th 40
theorem
for b1 being real set holds
   tanh is_continuous_in b1;

:: SIN_COS2:th 41
theorem
sinh is_continuous_on REAL;

:: SIN_COS2:th 42
theorem
cosh is_continuous_on REAL;

:: SIN_COS2:th 43
theorem
tanh is_continuous_on REAL;

:: SIN_COS2:th 44
theorem
for b1 being real set holds
   tanh . b1 < 1 & - 1 < tanh . b1;