Article SIN_COS7, MML version 4.99.1005
:: SIN_COS7:th 1
theorem
for b1 being real set
st 0 < b1
holds 1 / b1 = b1 to_power - 1;
:: SIN_COS7:th 2
theorem
for b1 being real set
st 1 < b1
holds ((sqrt (b1 ^2 - 1)) / b1) ^2 < 1;
:: SIN_COS7:th 3
theorem
for b1 being real set holds
(b1 / sqrt (b1 ^2 + 1)) ^2 < 1;
:: SIN_COS7:th 4
theorem
for b1 being real set holds
0 < sqrt (b1 ^2 + 1);
:: SIN_COS7:th 5
theorem
for b1 being real set holds
0 < (sqrt (b1 ^2 + 1)) + b1;
:: SIN_COS7:th 6
theorem
for b1, b2 being real set
st 0 <= b1 & 1 <= b2
holds 0 <= (b2 + 1) / b1;
:: SIN_COS7:th 7
theorem
for b1, b2 being real set
st 0 <= b1 & 1 <= b2
holds 0 <= (b2 - 1) / b1;
:: SIN_COS7:th 8
theorem
for b1 being real set
st 1 <= b1
holds 1 <= sqrt ((b1 + 1) / 2);
:: SIN_COS7:th 9
theorem
for b1, b2 being real set
st 0 <= b1 & 1 <= b2
holds 0 <= (b2 ^2 - 1) / b1;
:: SIN_COS7:th 10
theorem
for b1 being real set
st 1 <= b1
holds 0 < (sqrt ((b1 + 1) / 2)) + sqrt ((b1 - 1) / 2);
:: SIN_COS7:th 11
theorem
for b1 being real set
st b1 ^2 < 1
holds 0 < b1 + 1 & 0 < 1 - b1;
:: SIN_COS7:th 12
theorem
for b1 being real set
st b1 <> 1
holds 0 < (1 - b1) ^2;
:: SIN_COS7:th 13
theorem
for b1 being real set
st b1 ^2 < 1
holds 0 <= (b1 ^2 + 1) / (1 - (b1 ^2));
:: SIN_COS7:th 14
theorem
for b1 being real set
st b1 ^2 < 1
holds ((2 * b1) / (1 + (b1 ^2))) ^2 < 1;
:: SIN_COS7:th 15
theorem
for b1 being real set
st 0 < b1 & b1 < 1
holds 0 < (1 + b1) / (1 - b1);
:: SIN_COS7:th 16
theorem
for b1 being real set
st 0 < b1 & b1 < 1
holds b1 ^2 < 1;
:: SIN_COS7:th 17
theorem
for b1 being real set
st 0 < b1 & b1 < 1
holds 1 < 1 / sqrt (1 - (b1 ^2));
:: SIN_COS7:th 18
theorem
for b1 being real set
st 0 < b1 & b1 < 1
holds 0 < (2 * b1) / (1 - (b1 ^2));
:: SIN_COS7:th 19
theorem
for b1 being real set
st 0 < b1 & b1 < 1
holds 0 < (1 - (b1 ^2)) ^2;
:: SIN_COS7:th 20
theorem
for b1 being real set
st 0 < b1 & b1 < 1
holds 1 < (1 + (b1 ^2)) / (1 - (b1 ^2));
:: SIN_COS7:th 21
theorem
for b1 being real set
st 1 < b1 ^2
holds (1 / b1) ^2 < 1;
:: SIN_COS7:th 22
theorem
for b1 being real set
st 0 < b1 & b1 <= 1
holds 0 <= 1 - (b1 ^2);
:: SIN_COS7:th 23
theorem
for b1 being real set
st 1 <= b1
holds 0 < b1 + sqrt (b1 ^2 - 1);
:: SIN_COS7:th 24
theorem
for b1, b2 being real set
st 1 <= b1 & 1 <= b2
holds 0 <= (b1 * sqrt (b2 ^2 - 1)) + (b2 * sqrt (b1 ^2 - 1));
:: SIN_COS7:th 25
theorem
for b1, b2 being real set
st 1 <= b1 & 1 <= b2 & abs b2 <= abs b1
holds 0 < b2 - sqrt (b2 ^2 - 1);
:: SIN_COS7:th 26
theorem
for b1, b2 being real set
st 1 <= b1 & 1 <= b2 & abs b2 <= abs b1
holds 0 <= (b2 * sqrt (b1 ^2 - 1)) - (b1 * sqrt (b2 ^2 - 1));
:: SIN_COS7:th 27
theorem
for b1, b2 being real set
st b1 ^2 < 1 & b2 ^2 < 1
holds b1 * b2 <> - 1;
:: SIN_COS7:th 28
theorem
for b1, b2 being real set
st b1 ^2 < 1 & b2 ^2 < 1
holds b1 * b2 <> 1;
:: SIN_COS7:th 29
theorem
for b1 being real set
st b1 <> 0
holds exp_R b1 <> 1;
:: SIN_COS7:th 30
theorem
for b1 being real set
st 0 <> b1
holds (exp_R b1) ^2 - 1 <> 0;
:: SIN_COS7:th 31
theorem
for b1 being real set
st 0 < b1
holds (b1 ^2 - 1) / (b1 ^2 + 1) < 1;
:: SIN_COS7:th 32
theorem
for b1 being real set
st - 1 < b1 & b1 < 1
holds 0 < (b1 + 1) / (1 - b1);
:: SIN_COS7:funcnot 1 => SIN_COS7:func 1
definition
let a1 be real set;
func sinh" A1 -> real set equals
log(number_e,a1 + sqrt (a1 ^2 + 1));
end;
:: SIN_COS7:def 1
theorem
for b1 being real set holds
sinh" b1 = log(number_e,b1 + sqrt (b1 ^2 + 1));
:: SIN_COS7:funcnot 2 => SIN_COS7:func 2
definition
let a1 be real set;
func cosh1" A1 -> real set equals
log(number_e,a1 + sqrt (a1 ^2 - 1));
end;
:: SIN_COS7:def 2
theorem
for b1 being real set holds
cosh1" b1 = log(number_e,b1 + sqrt (b1 ^2 - 1));
:: SIN_COS7:funcnot 3 => SIN_COS7:func 3
definition
let a1 be real set;
func cosh2" A1 -> real set equals
- log(number_e,a1 + sqrt (a1 ^2 - 1));
end;
:: SIN_COS7:def 3
theorem
for b1 being real set holds
cosh2" b1 = - log(number_e,b1 + sqrt (b1 ^2 - 1));
:: SIN_COS7:funcnot 4 => SIN_COS7:func 4
definition
let a1 be real set;
func tanh" A1 -> real set equals
(1 / 2) * log(number_e,(1 + a1) / (1 - a1));
end;
:: SIN_COS7:def 4
theorem
for b1 being real set holds
tanh" b1 = (1 / 2) * log(number_e,(1 + b1) / (1 - b1));
:: SIN_COS7:funcnot 5 => SIN_COS7:func 5
definition
let a1 be real set;
func coth" A1 -> real set equals
(1 / 2) * log(number_e,(a1 + 1) / (a1 - 1));
end;
:: SIN_COS7:def 5
theorem
for b1 being real set holds
coth" b1 = (1 / 2) * log(number_e,(b1 + 1) / (b1 - 1));
:: SIN_COS7:funcnot 6 => SIN_COS7:func 6
definition
let a1 be real set;
func sech1" A1 -> real set equals
log(number_e,(1 + sqrt (1 - (a1 ^2))) / a1);
end;
:: SIN_COS7:def 6
theorem
for b1 being real set holds
sech1" b1 = log(number_e,(1 + sqrt (1 - (b1 ^2))) / b1);
:: SIN_COS7:funcnot 7 => SIN_COS7:func 7
definition
let a1 be real set;
func sech2" A1 -> real set equals
- log(number_e,(1 + sqrt (1 - (a1 ^2))) / a1);
end;
:: SIN_COS7:def 7
theorem
for b1 being real set holds
sech2" b1 = - log(number_e,(1 + sqrt (1 - (b1 ^2))) / b1);
:: SIN_COS7:funcnot 8 => SIN_COS7:func 8
definition
let a1 be real set;
func csch" A1 -> real set equals
log(number_e,(1 + sqrt (1 + (a1 ^2))) / a1)
if 0 < a1
otherwise case a1 < 0;
thus log(number_e,(1 - sqrt (1 + (a1 ^2))) / a1);
end;
;
end;
:: SIN_COS7:def 8
theorem
for b1 being real set holds
(b1 <= 0 or csch" b1 = log(number_e,(1 + sqrt (1 + (b1 ^2))) / b1)) &
(0 <= b1 or csch" b1 = log(number_e,(1 - sqrt (1 + (b1 ^2))) / b1));
:: SIN_COS7:th 33
theorem
for b1 being real set
st 0 <= b1
holds sinh" b1 = cosh1" sqrt (b1 ^2 + 1);
:: SIN_COS7:th 34
theorem
for b1 being real set
st b1 < 0
holds sinh" b1 = cosh2" sqrt (b1 ^2 + 1);
:: SIN_COS7:th 35
theorem
for b1 being real set holds
sinh" b1 = tanh" (b1 / sqrt (b1 ^2 + 1));
:: SIN_COS7:th 36
theorem
for b1 being real set
st 1 <= b1
holds cosh1" b1 = sinh" sqrt (b1 ^2 - 1);
:: SIN_COS7:th 37
theorem
for b1 being real set
st 1 < b1
holds cosh1" b1 = tanh" ((sqrt (b1 ^2 - 1)) / b1);
:: SIN_COS7:th 38
theorem
for b1 being real set
st 1 <= b1
holds cosh1" b1 = 2 * cosh1" sqrt ((b1 + 1) / 2);
:: SIN_COS7:th 39
theorem
for b1 being real set
st 1 <= b1
holds cosh2" b1 = 2 * cosh2" sqrt ((b1 + 1) / 2);
:: SIN_COS7:th 40
theorem
for b1 being real set
st 1 <= b1
holds cosh1" b1 = 2 * sinh" sqrt ((b1 - 1) / 2);
:: SIN_COS7:th 41
theorem
for b1 being real set
st b1 ^2 < 1
holds tanh" b1 = sinh" (b1 / sqrt (1 - (b1 ^2)));
:: SIN_COS7:th 42
theorem
for b1 being real set
st 0 < b1 & b1 < 1
holds tanh" b1 = cosh1" (1 / sqrt (1 - (b1 ^2)));
:: SIN_COS7:th 43
theorem
for b1 being real set
st b1 ^2 < 1
holds tanh" b1 = (1 / 2) * sinh" ((2 * b1) / (1 - (b1 ^2)));
:: SIN_COS7:th 44
theorem
for b1 being real set
st 0 < b1 & b1 < 1
holds tanh" b1 = (1 / 2) * cosh1" ((1 + (b1 ^2)) / (1 - (b1 ^2)));
:: SIN_COS7:th 45
theorem
for b1 being real set
st b1 ^2 < 1
holds tanh" b1 = (1 / 2) * tanh" ((2 * b1) / (1 + (b1 ^2)));
:: SIN_COS7:th 46
theorem
for b1 being real set
st 1 < b1 ^2
holds coth" b1 = tanh" (1 / b1);
:: SIN_COS7:th 47
theorem
for b1 being real set
st 0 < b1 & b1 <= 1
holds sech1" b1 = cosh1" (1 / b1);
:: SIN_COS7:th 48
theorem
for b1 being real set
st 0 < b1 & b1 <= 1
holds sech2" b1 = cosh2" (1 / b1);
:: SIN_COS7:th 49
theorem
for b1 being real set
st 0 < b1
holds csch" b1 = sinh" (1 / b1);
:: SIN_COS7:th 50
theorem
for b1, b2 being real set
st 0 <= (b1 * b2) + ((sqrt (b1 ^2 + 1)) * sqrt (b2 ^2 + 1))
holds (sinh" b1) + sinh" b2 = sinh" ((b1 * sqrt (1 + (b2 ^2))) + (b2 * sqrt (1 + (b1 ^2))));
:: SIN_COS7:th 51
theorem
for b1, b2 being real set holds
(sinh" b1) - sinh" b2 = sinh" ((b1 * sqrt (1 + (b2 ^2))) - (b2 * sqrt (1 + (b1 ^2))));
:: SIN_COS7:th 52
theorem
for b1, b2 being real set
st 1 <= b1 & 1 <= b2
holds (cosh1" b1) + cosh1" b2 = cosh1" ((b1 * b2) + sqrt ((b1 ^2 - 1) * (b2 ^2 - 1)));
:: SIN_COS7:th 53
theorem
for b1, b2 being real set
st 1 <= b1 & 1 <= b2
holds (cosh2" b1) + cosh2" b2 = cosh2" ((b1 * b2) + sqrt ((b1 ^2 - 1) * (b2 ^2 - 1)));
:: SIN_COS7:th 54
theorem
for b1, b2 being real set
st 1 <= b1 & 1 <= b2 & abs b2 <= abs b1
holds (cosh1" b1) - cosh1" b2 = cosh1" ((b1 * b2) - sqrt ((b1 ^2 - 1) * (b2 ^2 - 1)));
:: SIN_COS7:th 55
theorem
for b1, b2 being real set
st 1 <= b1 & 1 <= b2 & abs b2 <= abs b1
holds (cosh2" b1) - cosh2" b2 = cosh2" ((b1 * b2) - sqrt ((b1 ^2 - 1) * (b2 ^2 - 1)));
:: SIN_COS7:th 56
theorem
for b1, b2 being real set
st b1 ^2 < 1 & b2 ^2 < 1
holds (tanh" b1) + tanh" b2 = tanh" ((b1 + b2) / (1 + (b1 * b2)));
:: SIN_COS7:th 57
theorem
for b1, b2 being real set
st b1 ^2 < 1 & b2 ^2 < 1
holds (tanh" b1) - tanh" b2 = tanh" ((b1 - b2) / (1 - (b1 * b2)));
:: SIN_COS7:th 58
theorem
for b1 being real set
st 0 < b1 &
((b1 - 1) / (b1 + 1)) ^2 < 1
holds log(number_e,b1) = 2 * tanh" ((b1 - 1) / (b1 + 1));
:: SIN_COS7:th 59
theorem
for b1 being real set
st 0 < b1 &
((b1 ^2 - 1) / (b1 ^2 + 1)) ^2 < 1
holds log(number_e,b1) = tanh" ((b1 ^2 - 1) / (b1 ^2 + 1));
:: SIN_COS7:th 60
theorem
for b1 being real set
st 1 < b1 &
1 <= (b1 ^2 + 1) / (2 * b1)
holds log(number_e,b1) = cosh1" ((b1 ^2 + 1) / (2 * b1));
:: SIN_COS7:th 61
theorem
for b1 being real set
st 0 < b1 &
b1 < 1 &
1 <= (b1 ^2 + 1) / (2 * b1)
holds log(number_e,b1) = cosh2" ((b1 ^2 + 1) / (2 * b1));
:: SIN_COS7:th 62
theorem
for b1 being real set
st 0 < b1
holds log(number_e,b1) = sinh" ((b1 ^2 - 1) / (2 * b1));
:: SIN_COS7:th 63
theorem
for b1, b2 being real set
st b1 = (1 / 2) * ((exp_R b2) - exp_R - b2)
holds b2 = log(number_e,b1 + sqrt (b1 ^2 + 1));
:: SIN_COS7:th 64
theorem
for b1, b2 being real set
st b1 = (1 / 2) * ((exp_R b2) + exp_R - b2) &
1 <= b1 &
b2 <> log(number_e,b1 + sqrt (b1 ^2 - 1))
holds b2 = - log(number_e,b1 + sqrt (b1 ^2 - 1));
:: SIN_COS7:th 65
theorem
for b1, b2 being real set
st b1 = ((exp_R b2) - exp_R - b2) / ((exp_R b2) + exp_R - b2)
holds b2 = (1 / 2) * log(number_e,(1 + b1) / (1 - b1));
:: SIN_COS7:th 66
theorem
for b1, b2 being real set
st b1 = ((exp_R b2) + exp_R - b2) / ((exp_R b2) - exp_R - b2) &
b2 <> 0
holds b2 = (1 / 2) * log(number_e,(b1 + 1) / (b1 - 1));
:: SIN_COS7:th 67
theorem
for b1, b2 being real set
st b1 = 1 / (((exp_R b2) + exp_R - b2) / 2) &
b2 <> log(number_e,(1 + sqrt (1 - (b1 ^2))) / b1)
holds b2 = - log(number_e,(1 + sqrt (1 - (b1 ^2))) / b1);
:: SIN_COS7:th 68
theorem
for b1, b2 being real set
st b1 = 1 / (((exp_R b2) - exp_R - b2) / 2) &
b2 <> 0 &
b2 <> log(number_e,(1 + sqrt (1 + (b1 ^2))) / b1)
holds b2 = log(number_e,(1 - sqrt (1 + (b1 ^2))) / b1);
:: SIN_COS7:th 69
theorem
for b1 being real set holds
cosh . (2 * b1) = 1 + (2 * ((sinh . b1) ^2));
:: SIN_COS7:th 70
theorem
for b1 being real set holds
(cosh . b1) ^2 = 1 + ((sinh . b1) ^2);
:: SIN_COS7:th 71
theorem
for b1 being real set holds
(sinh . b1) ^2 = (cosh . b1) ^2 - 1;
:: SIN_COS7:th 72
theorem
for b1 being real set holds
sinh (5 * b1) = ((5 * sinh b1) + (20 * ((sinh b1) |^ 3))) + (16 * ((sinh b1) |^ 5));
:: SIN_COS7:th 73
theorem
for b1 being real set holds
cosh (5 * b1) = ((5 * cosh b1) - (20 * ((cosh b1) |^ 3))) + (16 * ((cosh b1) |^ 5));