:: SIN_COS2 semantic presentation

Lemma24: ( 0 < PI / 2 & PI / 2 < PI & PI < (3 / 2) * PI & (3 / 2) * PI < 2 * PI )
proof end;

theorem Th1: :: SIN_COS2:1
for p, r being real number st p >= 0 & r >= 0 holds
p + r >= 2 * (sqrt (p * r))
proof end;

theorem Th2: :: SIN_COS2:2
sin is_increasing_on ].0,(PI / 2).[
proof end;

Lemma30: for th being real number st th in ].0,(PI / 2).[ holds
0 < sin . th
proof end;

theorem Th3: :: SIN_COS2:3
sin is_decreasing_on ].(PI / 2),PI .[
proof end;

theorem Th4: :: SIN_COS2:4
cos is_decreasing_on ].0,(PI / 2).[
proof end;

theorem Th5: :: SIN_COS2:5
cos is_decreasing_on ].(PI / 2),PI .[
proof end;

theorem Th6: :: SIN_COS2:6
sin is_decreasing_on ].PI ,((3 / 2) * PI ).[
proof end;

theorem Th7: :: SIN_COS2:7
sin is_increasing_on ].((3 / 2) * PI ),(2 * PI ).[
proof end;

theorem Th8: :: SIN_COS2:8
cos is_increasing_on ].PI ,((3 / 2) * PI ).[
proof end;

theorem Th9: :: SIN_COS2:9
cos is_increasing_on ].((3 / 2) * PI ),(2 * PI ).[
proof end;

theorem Th10: :: SIN_COS2:10
for th being real number
for n being Nat holds sin . th = sin . (((2 * PI ) * n) + th)
proof end;

theorem Th11: :: SIN_COS2:11
for th being real number
for n being Nat holds cos . th = cos . (((2 * PI ) * n) + th)
proof end;

definition
func sinh -> PartFunc of REAL , REAL means :Def1: :: SIN_COS2:def 1
( dom it = REAL & ( for d being real number holds it . d = ((exp_R . d) - (exp_R . (- d))) / 2 ) );
existence
ex b1 being PartFunc of REAL , REAL st
( dom b1 = REAL & ( for d being real number holds b1 . d = ((exp_R . d) - (exp_R . (- d))) / 2 ) )
proof end;
uniqueness
for b1, b2 being PartFunc of REAL , REAL st dom b1 = REAL & ( for d being real number holds b1 . d = ((exp_R . d) - (exp_R . (- d))) / 2 ) & dom b2 = REAL & ( for d being real number holds b2 . d = ((exp_R . d) - (exp_R . (- d))) / 2 ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines sinh SIN_COS2:def 1 :
for b1 being PartFunc of REAL , REAL holds
( b1 = sinh iff ( dom b1 = REAL & ( for d being real number holds b1 . d = ((exp_R . d) - (exp_R . (- d))) / 2 ) ) );

definition
let d be number ;
func sinh c1 -> set equals :: SIN_COS2:def 2
sinh . d;
coherence
sinh . d is set
;
end;

:: deftheorem Def2 defines sinh SIN_COS2:def 2 :
for d being number holds sinh d = sinh . d;

registration
let d be number ;
cluster sinh a1 -> real ;
coherence
sinh d is real
;
end;

definition
let d be number ;
redefine func sinh as sinh c1 -> Real;
coherence
sinh d is Real
;
end;

definition
func cosh -> PartFunc of REAL , REAL means :Def3: :: SIN_COS2:def 3
( dom it = REAL & ( for d being real number holds it . d = ((exp_R . d) + (exp_R . (- d))) / 2 ) );
existence
ex b1 being PartFunc of REAL , REAL st
( dom b1 = REAL & ( for d being real number holds b1 . d = ((exp_R . d) + (exp_R . (- d))) / 2 ) )
proof end;
uniqueness
for b1, b2 being PartFunc of REAL , REAL st dom b1 = REAL & ( for d being real number holds b1 . d = ((exp_R . d) + (exp_R . (- d))) / 2 ) & dom b2 = REAL & ( for d being real number holds b2 . d = ((exp_R . d) + (exp_R . (- d))) / 2 ) holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines cosh SIN_COS2:def 3 :
for b1 being PartFunc of REAL , REAL holds
( b1 = cosh iff ( dom b1 = REAL & ( for d being real number holds b1 . d = ((exp_R . d) + (exp_R . (- d))) / 2 ) ) );

definition
let d be number ;
func cosh c1 -> set equals :: SIN_COS2:def 4
cosh . d;
coherence
cosh . d is set
;
end;

:: deftheorem Def4 defines cosh SIN_COS2:def 4 :
for d being number holds cosh d = cosh . d;

registration
let d be number ;
cluster cosh a1 -> real ;
coherence
cosh d is real
;
end;

definition
let d be number ;
redefine func cosh as cosh c1 -> Real;
coherence
cosh d is Real
;
end;

definition
func tanh -> PartFunc of REAL , REAL means :Def5: :: SIN_COS2:def 5
( dom it = REAL & ( for d being real number holds it . d = ((exp_R . d) - (exp_R . (- d))) / ((exp_R . d) + (exp_R . (- d))) ) );
existence
ex b1 being PartFunc of REAL , REAL st
( dom b1 = REAL & ( for d being real number holds b1 . d = ((exp_R . d) - (exp_R . (- d))) / ((exp_R . d) + (exp_R . (- d))) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of REAL , REAL st dom b1 = REAL & ( for d being real number holds b1 . d = ((exp_R . d) - (exp_R . (- d))) / ((exp_R . d) + (exp_R . (- d))) ) & dom b2 = REAL & ( for d being real number holds b2 . d = ((exp_R . d) - (exp_R . (- d))) / ((exp_R . d) + (exp_R . (- d))) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def5 defines tanh SIN_COS2:def 5 :
for b1 being PartFunc of REAL , REAL holds
( b1 = tanh iff ( dom b1 = REAL & ( for d being real number holds b1 . d = ((exp_R . d) - (exp_R . (- d))) / ((exp_R . d) + (exp_R . (- d))) ) ) );

definition
let d be number ;
func tanh c1 -> set equals :: SIN_COS2:def 6
tanh . d;
coherence
tanh . d is set
;
end;

:: deftheorem Def6 defines tanh SIN_COS2:def 6 :
for d being number holds tanh d = tanh . d;

registration
let d be number ;
cluster tanh a1 -> real ;
coherence
tanh d is real
;
end;

definition
let d be number ;
redefine func tanh as tanh c1 -> Real;
coherence
tanh d is Real
;
end;

theorem Th12: :: SIN_COS2:12
for p, q being real number holds exp_R . (p + q) = (exp_R . p) * (exp_R . q)
proof end;

theorem Th13: :: SIN_COS2:13
exp_R . 0 = 1 by SIN_COS:56, SIN_COS:def 27;

theorem Th14: :: SIN_COS2:14
for p being real number holds
( ((cosh . p) ^2 ) - ((sinh . p) ^2 ) = 1 & ((cosh . p) * (cosh . p)) - ((sinh . p) * (sinh . p)) = 1 )
proof end;

Lemma46: for p, r being real number holds cosh . (p + r) = ((cosh . p) * (cosh . r)) + ((sinh . p) * (sinh . r))
proof end;

Lemma47: for p, r being real number holds sinh . (p + r) = ((sinh . p) * (cosh . r)) + ((cosh . p) * (sinh . r))
proof end;

theorem Th15: :: SIN_COS2:15
for p being real number holds
( cosh . p <> 0 & cosh . p > 0 & cosh . 0 = 1 )
proof end;

theorem Th16: :: SIN_COS2:16
sinh . 0 = 0
proof end;

theorem Th17: :: SIN_COS2:17
for p being real number holds tanh . p = (sinh . p) / (cosh . p)
proof end;

Lemma51: for r, q, p, a1 being real number st r <> 0 & q <> 0 & (r * q) + (p * a1) <> 0 holds
((p * q) + (r * a1)) / ((r * q) + (p * a1)) = ((p / r) + (a1 / q)) / (1 + ((p / r) * (a1 / q)))
proof end;

Lemma53: for p, r being real number holds tanh . (p + r) = ((tanh . p) + (tanh . r)) / (1 + ((tanh . p) * (tanh . r)))
proof end;

theorem Th18: :: SIN_COS2:18
for p being real number holds
( (sinh . p) ^2 = (1 / 2) * ((cosh . (2 * p)) - 1) & (cosh . p) ^2 = (1 / 2) * ((cosh . (2 * p)) + 1) )
proof end;

Lemma55: for p being real number holds
( sinh . (2 * p) = (2 * (sinh . p)) * (cosh . p) & cosh . (2 * p) = (2 * ((cosh . p) ^2 )) - 1 )
proof end;

theorem Th19: :: SIN_COS2:19
for p being real number holds
( cosh . (- p) = cosh . p & sinh . (- p) = - (sinh . p) & tanh . (- p) = - (tanh . p) )
proof end;

Lemma57: for p, r being real number holds cosh . (p - r) = ((cosh . p) * (cosh . r)) - ((sinh . p) * (sinh . r))
proof end;

theorem Th20: :: SIN_COS2:20
for p, r being real number holds
( cosh . (p + r) = ((cosh . p) * (cosh . r)) + ((sinh . p) * (sinh . r)) & cosh . (p - r) = ((cosh . p) * (cosh . r)) - ((sinh . p) * (sinh . r)) ) by , ;

Lemma58: for p, r being real number holds sinh . (p - r) = ((sinh . p) * (cosh . r)) - ((cosh . p) * (sinh . r))
proof end;

theorem Th21: :: SIN_COS2:21
for p, r being real number holds
( sinh . (p + r) = ((sinh . p) * (cosh . r)) + ((cosh . p) * (sinh . r)) & sinh . (p - r) = ((sinh . p) * (cosh . r)) - ((cosh . p) * (sinh . r)) ) by , ;

Lemma59: for p, r being real number holds tanh . (p - r) = ((tanh . p) - (tanh . r)) / (1 - ((tanh . p) * (tanh . r)))
proof end;

theorem Th22: :: SIN_COS2:22
for p, r being real number holds
( tanh . (p + r) = ((tanh . p) + (tanh . r)) / (1 + ((tanh . p) * (tanh . r))) & tanh . (p - r) = ((tanh . p) - (tanh . r)) / (1 - ((tanh . p) * (tanh . r))) ) by , ;

theorem Th23: :: SIN_COS2:23
for p being real number holds
( sinh . (2 * p) = (2 * (sinh . p)) * (cosh . p) & cosh . (2 * p) = (2 * ((cosh . p) ^2 )) - 1 & tanh . (2 * p) = (2 * (tanh . p)) / (1 + ((tanh . p) ^2 )) )
proof end;

theorem Th24: :: SIN_COS2:24
for p, q being real number holds
( ((sinh . p) ^2 ) - ((sinh . q) ^2 ) = (sinh . (p + q)) * (sinh . (p - q)) & (sinh . (p + q)) * (sinh . (p - q)) = ((cosh . p) ^2 ) - ((cosh . q) ^2 ) & ((sinh . p) ^2 ) - ((sinh . q) ^2 ) = ((cosh . p) ^2 ) - ((cosh . q) ^2 ) )
proof end;

theorem Th25: :: SIN_COS2:25
for p, q being real number holds
( ((sinh . p) ^2 ) + ((cosh . q) ^2 ) = (cosh . (p + q)) * (cosh . (p - q)) & (cosh . (p + q)) * (cosh . (p - q)) = ((cosh . p) ^2 ) + ((sinh . q) ^2 ) & ((sinh . p) ^2 ) + ((cosh . q) ^2 ) = ((cosh . p) ^2 ) + ((sinh . q) ^2 ) )
proof end;

theorem Th26: :: SIN_COS2:26
for p, r being real number holds
( (sinh . p) + (sinh . r) = (2 * (sinh . ((p / 2) + (r / 2)))) * (cosh . ((p / 2) - (r / 2))) & (sinh . p) - (sinh . r) = (2 * (sinh . ((p / 2) - (r / 2)))) * (cosh . ((p / 2) + (r / 2))) )
proof end;

theorem Th27: :: SIN_COS2:27
for p, r being real number holds
( (cosh . p) + (cosh . r) = (2 * (cosh . ((p / 2) + (r / 2)))) * (cosh . ((p / 2) - (r / 2))) & (cosh . p) - (cosh . r) = (2 * (sinh . ((p / 2) - (r / 2)))) * (sinh . ((p / 2) + (r / 2))) )
proof end;

theorem Th28: :: SIN_COS2:28
for p, r being real number holds
( (tanh . p) + (tanh . r) = (sinh . (p + r)) / ((cosh . p) * (cosh . r)) & (tanh . p) - (tanh . r) = (sinh . (p - r)) / ((cosh . p) * (cosh . r)) )
proof end;

theorem Th29: :: SIN_COS2:29
for p being real number
for n being Element of NAT holds ((cosh . p) + (sinh . p)) |^ n = (cosh . (n * p)) + (sinh . (n * p))
proof end;

registration
cluster sinh -> total ;
coherence
sinh is total
proof end;
cluster cosh -> total ;
coherence
cosh is total
proof end;
cluster tanh -> total ;
coherence
tanh is total
proof end;
end;

theorem Th30: :: SIN_COS2:30
( dom sinh = REAL & dom cosh = REAL & dom tanh = REAL ) by , , ;

Lemma63: for d being real number holds compreal . d = (- 1) * d
proof end;

Lemma64: ( dom compreal = REAL & rng compreal c= REAL )
by FUNCT_2:def 1;

Lemma65: for f being PartFunc of REAL , REAL st f = compreal holds
for p being real number holds
( f is_differentiable_in p & diff f,p = - 1 )
proof end;

Lemma85: for p being real number
for f being PartFunc of REAL , REAL st f = compreal holds
( exp_R * f is_differentiable_in p & diff (exp_R * f),p = (- 1) * (exp_R . (f . p)) )
proof end;

Lemma86: for p being real number
for f being PartFunc of REAL , REAL st f = compreal holds
exp_R . ((- 1) * p) = (exp_R * f) . p
proof end;

Lemma87: for p being real number
for f being PartFunc of REAL , REAL st f = compreal holds
( exp_R - (exp_R * f) is_differentiable_in p & exp_R + (exp_R * f) is_differentiable_in p & diff (exp_R - (exp_R * f)),p = (exp_R . p) + (exp_R . (- p)) & diff (exp_R + (exp_R * f)),p = (exp_R . p) - (exp_R . (- p)) )
proof end;

Lemma88: for p being real number
for f being PartFunc of REAL , REAL st f = compreal holds
( (1 / 2) (#) (exp_R - (exp_R * f)) is_differentiable_in p & diff ((1 / 2) (#) (exp_R - (exp_R * f))),p = (1 / 2) * (diff (exp_R - (exp_R * f)),p) )
proof end;

Lemma89: for p being real number
for ff being PartFunc of REAL , REAL st ff = compreal holds
sinh . p = ((1 / 2) (#) (exp_R - (exp_R * ff))) . p
proof end;

Lemma91: for ff being PartFunc of REAL , REAL st ff = compreal holds
sinh = (1 / 2) (#) (exp_R - (exp_R * ff))
proof end;

theorem Th31: :: SIN_COS2:31
for p being real number holds
( sinh is_differentiable_in p & diff sinh ,p = cosh . p )
proof end;

Lemma93: for p being real number
for ff being PartFunc of REAL , REAL st ff = compreal holds
( (1 / 2) (#) (exp_R + (exp_R * ff)) is_differentiable_in p & diff ((1 / 2) (#) (exp_R + (exp_R * ff))),p = (1 / 2) * (diff (exp_R + (exp_R * ff)),p) )
proof end;

Lemma94: for p being real number
for ff being PartFunc of REAL , REAL st ff = compreal holds
cosh . p = ((1 / 2) (#) (exp_R + (exp_R * ff))) . p
proof end;

Lemma95: for ff being PartFunc of REAL , REAL st ff = compreal holds
cosh = (1 / 2) (#) (exp_R + (exp_R * ff))
proof end;

theorem Th32: :: SIN_COS2:32
for p being real number holds
( cosh is_differentiable_in p & diff cosh ,p = sinh . p )
proof end;

Lemma97: for p being real number holds
( sinh / cosh is_differentiable_in p & diff (sinh / cosh ),p = 1 / ((cosh . p) ^2 ) )
proof end;

Lemma98: tanh = sinh / cosh
proof end;

theorem Th33: :: SIN_COS2:33
for p being real number holds
( tanh is_differentiable_in p & diff tanh ,p = 1 / ((cosh . p) ^2 ) ) by , Def3;

theorem Th34: :: SIN_COS2:34
for p being real number holds
( sinh is_differentiable_on REAL & diff sinh ,p = cosh . p )
proof end;

theorem Th35: :: SIN_COS2:35
for p being real number holds
( cosh is_differentiable_on REAL & diff cosh ,p = sinh . p )
proof end;

theorem Th36: :: SIN_COS2:36
for p being real number holds
( tanh is_differentiable_on REAL & diff tanh ,p = 1 / ((cosh . p) ^2 ) )
proof end;

Lemma102: for p being real number holds (exp_R . p) + (exp_R . (- p)) >= 2
proof end;

theorem Th37: :: SIN_COS2:37
for p being real number holds cosh . p >= 1
proof end;

theorem Th38: :: SIN_COS2:38
for p being real number holds sinh is_continuous_in p
proof end;

theorem Th39: :: SIN_COS2:39
for p being real number holds cosh is_continuous_in p
proof end;

theorem Th40: :: SIN_COS2:40
for p being real number holds tanh is_continuous_in p
proof end;

theorem Th41: :: SIN_COS2:41
sinh is_continuous_on REAL by Def5, FDIFF_1:33;

theorem Th42: :: SIN_COS2:42
cosh is_continuous_on REAL by Th12, FDIFF_1:33;

theorem Th43: :: SIN_COS2:43
tanh is_continuous_on REAL by Th13, FDIFF_1:33;

theorem Th44: :: SIN_COS2:44
for p being real number holds
( tanh . p < 1 & tanh . p > - 1 )
proof end;