Article ABSVALUE, MML version 4.99.1005
:: ABSVALUE:funcnot 1 => COMPLEX1:func 16
definition
let a1 be complex set;
func |.A1.| -> complex set equals
a1
if 0 <= a1
otherwise - a1;
projectivity;
:: for a1 being complex set holds
:: |.|.a1.|.| = |.a1.|;
end;
:: ABSVALUE:def 1
theorem
for b1 being real set holds
(0 <= b1 implies |.b1.| = b1) &
(0 <= b1 or |.b1.| = - b1);
:: ABSVALUE:th 1
theorem
for b1 being real set
st abs b1 <> b1
holds abs b1 = - b1;
:: ABSVALUE:th 7
theorem
for b1 being real set holds
b1 = 0
iff
abs b1 = 0;
:: ABSVALUE:th 9
theorem
for b1 being real set
st abs b1 = - b1 & b1 <> 0
holds b1 < 0;
:: ABSVALUE:th 11
theorem
for b1 being real set holds
- abs b1 <= b1 & b1 <= abs b1;
:: ABSVALUE:th 12
theorem
for b1, b2 being real set holds
- b1 <= b2 & b2 <= b1
iff
abs b2 <= b1;
:: ABSVALUE:th 14
theorem
for b1 being real set
st b1 <> 0
holds (abs b1) * abs (1 / b1) = 1;
:: ABSVALUE:th 15
theorem
for b1 being real set holds
abs (1 / b1) = 1 / abs b1;
:: ABSVALUE:th 20
theorem
for b1, b2 being real set
st 0 <= b1 * b2
holds sqrt (b1 * b2) = (sqrt abs b1) * sqrt abs b2;
:: ABSVALUE:th 21
theorem
for b1, b2, b3, b4 being real set
st abs b1 <= b2 & abs b3 <= b4
holds abs (b1 + b3) <= b2 + b4;
:: ABSVALUE:th 23
theorem
for b1, b2 being real set
st 0 < b1 / b2
holds sqrt (b1 / b2) = (sqrt abs b1) / sqrt abs b2;
:: ABSVALUE:th 24
theorem
for b1, b2 being real set
st 0 <= b1 * b2
holds abs (b1 + b2) = (abs b1) + abs b2;
:: ABSVALUE:th 25
theorem
for b1, b2 being real set
st abs (b1 + b2) = (abs b1) + abs b2
holds 0 <= b1 * b2;
:: ABSVALUE:th 26
theorem
for b1, b2 being real set holds
(abs (b1 + b2)) / (1 + abs (b1 + b2)) <= ((abs b1) / (1 + abs b1)) + ((abs b2) / (1 + abs b2));
:: ABSVALUE:funcnot 2 => ABSVALUE:func 1
definition
let a1 be real set;
func sgn A1 -> set equals
1
if 0 < a1,
- 1
if a1 < 0
otherwise 0;
end;
:: ABSVALUE:def 2
theorem
for b1 being real set holds
(b1 <= 0 or sgn b1 = 1) &
(0 <= b1 or sgn b1 = - 1) &
(b1 <= 0 & 0 <= b1 implies sgn b1 = 0);
:: ABSVALUE:funcreg 1
registration
let a1 be real set;
cluster sgn a1 -> real;
end;
:: ABSVALUE:funcnot 3 => ABSVALUE:func 2
definition
let a1 be Element of REAL;
redefine func sgn a1 -> Element of REAL;
end;
:: ABSVALUE:th 31
theorem
for b1 being real set
st sgn b1 = 1
holds 0 < b1;
:: ABSVALUE:th 32
theorem
for b1 being real set
st sgn b1 = - 1
holds b1 < 0;
:: ABSVALUE:th 33
theorem
for b1 being real set
st sgn b1 = 0
holds b1 = 0;
:: ABSVALUE:th 34
theorem
for b1 being real set holds
b1 = (abs b1) * sgn b1;
:: ABSVALUE:th 35
theorem
for b1, b2 being real set holds
sgn (b1 * b2) = (sgn b1) * sgn b2;
:: ABSVALUE:th 36
theorem
for b1 being real set holds
sgn sgn b1 = sgn b1;
:: ABSVALUE:th 37
theorem
for b1, b2 being real set holds
sgn (b1 + b2) <= ((sgn b1) + sgn b2) + 1;
:: ABSVALUE:th 38
theorem
for b1 being real set
st b1 <> 0
holds (sgn b1) * sgn (1 / b1) = 1;
:: ABSVALUE:th 39
theorem
for b1 being real set holds
1 / sgn b1 = sgn (1 / b1);
:: ABSVALUE:th 40
theorem
for b1, b2 being real set holds
((sgn b1) + sgn b2) - 1 <= sgn (b1 + b2);
:: ABSVALUE:th 41
theorem
for b1 being real set holds
sgn b1 = sgn (1 / b1);
:: ABSVALUE:th 42
theorem
for b1, b2 being real set holds
sgn (b1 / b2) = (sgn b1) / sgn b2;