Article EXTREAL1, MML version 4.99.1005
:: EXTREAL1:th 1
theorem
for b1 being Element of ExtREAL
st b1 <> +infty & b1 <> -infty
holds b1 is Element of REAL;
:: EXTREAL1:th 4
theorem
for b1 being Element of ExtREAL holds
(b1 = +infty implies - b1 = -infty) & (- b1 = -infty implies b1 = +infty) & (b1 = -infty implies - b1 = +infty) & (- b1 = +infty implies b1 = -infty);
:: EXTREAL1:th 5
theorem
for b1, b2 being Element of ExtREAL holds
b1 - - b2 = b1 + b2;
:: EXTREAL1:th 7
theorem
for b1, b2 being Element of ExtREAL
st b1 <> -infty & b2 <> +infty & b1 <= b2
holds b1 <> +infty & b2 <> -infty;
:: EXTREAL1:th 8
theorem
for b1, b2, b3 being Element of ExtREAL
st (b1 = +infty implies b2 <> -infty) & (b1 = -infty implies b2 <> +infty) & (b2 = +infty implies b3 <> -infty) & (b2 = -infty implies b3 <> +infty) & (b1 = +infty implies b3 <> -infty) & (b1 = -infty implies b3 <> +infty)
holds (b1 + b2) + b3 = b1 + (b2 + b3);
:: EXTREAL1:th 9
theorem
for b1 being Element of ExtREAL holds
b1 + - b1 = 0.;
:: EXTREAL1:th 11
theorem
for b1, b2, b3 being Element of ExtREAL
st (b1 = +infty implies b2 <> -infty) & (b1 = -infty implies b2 <> +infty) & (b2 = +infty implies b3 <> +infty) & (b2 = -infty implies b3 <> -infty) & (b1 = +infty implies b3 <> +infty) & (b1 = -infty implies b3 <> -infty)
holds (b1 + b2) - b3 = b1 + (b2 - b3);
:: EXTREAL1:funcnot 1 => EXTREAL1:func 1
definition
let a1, a2 be Element of ExtREAL;
func A1 * A2 -> Element of ExtREAL means
((for b1, b2 being Element of REAL
st a1 = b1 & a2 = b2
holds it <> b1 * b2) &
(((0. < a1 implies a2 <> +infty) & (0. < a2 implies a1 <> +infty) & (a1 < 0. implies a2 <> -infty) implies a2 < 0. & a1 = -infty) implies it <> +infty) &
(((a1 < 0. implies a2 <> +infty) & (a2 < 0. implies a1 <> +infty) & (0. < a1 implies a2 <> -infty) implies 0. < a2 & a1 = -infty) implies it <> -infty)) implies (a1 = 0. or a2 = 0.) & it = 0.;
end;
:: EXTREAL1:def 1
theorem
for b1, b2, b3 being Element of ExtREAL holds
b3 = b1 * b2
iff
((for b4, b5 being Element of REAL
st b1 = b4 & b2 = b5
holds b3 <> b4 * b5) &
(((0. < b1 implies b2 <> +infty) & (0. < b2 implies b1 <> +infty) & (b1 < 0. implies b2 <> -infty) implies b2 < 0. & b1 = -infty) implies b3 <> +infty) &
(((b1 < 0. implies b2 <> +infty) & (b2 < 0. implies b1 <> +infty) & (0. < b1 implies b2 <> -infty) implies 0. < b2 & b1 = -infty) implies b3 <> -infty) implies (b1 = 0. or b2 = 0.) & b3 = 0.);
:: EXTREAL1:th 13
theorem
for b1, b2 being Element of ExtREAL
for b3, b4 being Element of REAL
st b1 = b3 & b2 = b4
holds b1 * b2 = b3 * b4;
:: EXTREAL1:th 14
theorem
for b1, b2 being Element of ExtREAL
st (0. <= b1 & 0. < b2 or 0. < b1 & 0. <= b2)
holds 0. < b1 + b2;
:: EXTREAL1:th 15
theorem
for b1, b2 being Element of ExtREAL
st (b1 <= 0. & b2 < 0. or b1 < 0. & b2 <= 0.)
holds b1 + b2 < 0.;
:: EXTREAL1:th 16
theorem
for b1 being Element of ExtREAL
st b1 in REAL & (b1 <= -infty or 0. <= b1) & b1 <> 0.
holds 0. < b1 & b1 < +infty;
:: EXTREAL1:th 17
theorem
for b1, b2 being Element of ExtREAL holds
b1 * b2 = b2 * b1;
:: EXTREAL1:funcnot 2 => EXTREAL1:func 2
definition
let a1, a2 be Element of ExtREAL;
redefine func a1 * a2 -> Element of ExtREAL;
commutativity;
:: for a1, a2 being Element of ExtREAL holds
:: a1 * a2 = a2 * a1;
end;
:: EXTREAL1:th 20
theorem
for b1, b2 being Element of ExtREAL
st (0. < b1 & 0. < b2 or b1 < 0. & b2 < 0.)
holds 0. < b1 * b2;
:: EXTREAL1:th 21
theorem
for b1, b2 being Element of ExtREAL
st (0. < b1 & b2 < 0. or b1 < 0. & 0. < b2)
holds b1 * b2 < 0.;
:: EXTREAL1:th 22
theorem
for b1, b2 being Element of ExtREAL holds
b1 * b2 = 0.
iff
(b1 = 0. or b2 = 0.);
:: EXTREAL1:th 23
theorem
for b1, b2, b3 being Element of ExtREAL holds
(b1 * b2) * b3 = b1 * (b2 * b3);
:: EXTREAL1:th 24
theorem
- 0. = 0.;
:: EXTREAL1:th 25
theorem
for b1 being Element of ExtREAL holds
(0. < b1 implies - b1 < 0.) &
(- b1 < 0. implies 0. < b1) &
(b1 < 0. implies 0. < - b1) &
(0. < - b1 implies b1 < 0.);
:: EXTREAL1:th 26
theorem
for b1, b2 being Element of ExtREAL holds
- (b1 * b2) = b1 * - b2 & - (b1 * b2) = (- b1) * b2;
:: EXTREAL1:th 27
theorem
for b1, b2 being Element of ExtREAL
st b1 <> +infty & b1 <> -infty & b1 * b2 = +infty & b2 <> +infty
holds b2 = -infty;
:: EXTREAL1:th 28
theorem
for b1, b2 being Element of ExtREAL
st b1 <> +infty & b1 <> -infty & b1 * b2 = -infty & b2 <> +infty
holds b2 = -infty;
:: EXTREAL1:th 29
theorem
for b1, b2, b3 being Element of ExtREAL
st ((b1 = +infty or b1 = -infty) & (b2 = -infty implies b3 <> +infty) & (0. <= b2 or 0. <= b3) & b2 <> 0. & b3 <> 0. & (b2 <= 0. or b3 <= 0.) implies b2 = +infty & b3 = -infty)
holds b1 * (b2 + b3) = (b1 * b2) + (b1 * b3);
:: EXTREAL1:th 30
theorem
for b1, b2, b3 being Element of ExtREAL
st (b1 = +infty implies b2 <> +infty) & (b1 = -infty implies b2 <> -infty) & b3 <> +infty & b3 <> -infty
holds b3 * (b1 - b2) = (b3 * b1) - (b3 * b2);
:: EXTREAL1:funcnot 3 => EXTREAL1:func 3
definition
let a1, a2 be Element of ExtREAL;
assume (a1 <> -infty & a1 <> +infty or a2 <> -infty & a2 <> +infty) &
a2 <> 0.;
func A1 / A2 -> Element of ExtREAL means
((for b1, b2 being Element of REAL
st a1 = b1 & a2 = b2
holds it <> b1 / b2) &
((a1 = +infty & 0. < a2 or a1 = -infty & a2 < 0.) implies it <> +infty) &
((a1 = -infty & 0. < a2 or a1 = +infty & a2 < 0.) implies it <> -infty)) implies (a2 = -infty or a2 = +infty) & it = 0.;
end;
:: EXTREAL1:def 2
theorem
for b1, b2 being Element of ExtREAL
st (b1 <> -infty & b1 <> +infty or b2 <> -infty & b2 <> +infty) &
b2 <> 0.
for b3 being Element of ExtREAL holds
b3 = b1 / b2
iff
((for b4, b5 being Element of REAL
st b1 = b4 & b2 = b5
holds b3 <> b4 / b5) &
((b1 = +infty & 0. < b2 or b1 = -infty & b2 < 0.) implies b3 <> +infty) &
((b1 = -infty & 0. < b2 or b1 = +infty & b2 < 0.) implies b3 <> -infty) implies (b2 = -infty or b2 = +infty) & b3 = 0.);
:: EXTREAL1:th 32
theorem
for b1, b2 being Element of ExtREAL
st b2 <> 0.
for b3, b4 being Element of REAL
st b1 = b3 & b2 = b4
holds b1 / b2 = b3 / b4;
:: EXTREAL1:th 33
theorem
for b1, b2 being Element of ExtREAL
st b1 <> -infty & b1 <> +infty & (b2 = -infty or b2 = +infty)
holds b1 / b2 = 0.;
:: EXTREAL1:th 34
theorem
for b1 being Element of ExtREAL
st b1 <> -infty & b1 <> +infty & b1 <> 0.
holds b1 / b1 = 1;
:: EXTREAL1:funcnot 4 => EXTREAL1:func 4
definition
let a1 be Element of ExtREAL;
func |.A1.| -> Element of ExtREAL equals
a1
if 0. <= a1
otherwise - a1;
end;
:: EXTREAL1:def 3
theorem
for b1 being Element of ExtREAL holds
(0. <= b1 implies |.b1.| = b1) &
(0. <= b1 or |.b1.| = - b1);
:: EXTREAL1:th 36
theorem
for b1 being Element of ExtREAL
st 0. < b1
holds |.b1.| = b1;
:: EXTREAL1:th 37
theorem
for b1 being Element of ExtREAL
st b1 < 0.
holds |.b1.| = - b1;
:: EXTREAL1:th 38
theorem
for b1, b2 being Element of REAL holds
b1 * b2 = (R_EAL b1) * R_EAL b2;
:: EXTREAL1:th 39
theorem
for b1, b2 being Element of REAL
st b2 <> 0
holds b1 / b2 = (R_EAL b1) / R_EAL b2;
:: EXTREAL1:th 40
theorem
for b1, b2 being Element of ExtREAL
st b1 <= b2 & b1 < +infty & -infty < b2
holds 0. <= b2 - b1;
:: EXTREAL1:th 41
theorem
for b1, b2 being Element of ExtREAL
st b1 < b2 & b1 < +infty & -infty < b2
holds 0. < b2 - b1;
:: EXTREAL1:th 42
theorem
for b1, b2, b3 being Element of ExtREAL
st b1 <= b2 & 0. <= b3
holds b1 * b3 <= b2 * b3;
:: EXTREAL1:th 43
theorem
for b1, b2, b3 being Element of ExtREAL
st b1 <= b2 & b3 <= 0.
holds b2 * b3 <= b1 * b3;
:: EXTREAL1:th 44
theorem
for b1, b2, b3 being Element of ExtREAL
st b1 < b2 & 0. < b3 & b3 <> +infty
holds b1 * b3 < b2 * b3;
:: EXTREAL1:th 45
theorem
for b1, b2, b3 being Element of ExtREAL
st b1 < b2 & b3 < 0. & b3 <> -infty
holds b2 * b3 < b1 * b3;
:: EXTREAL1:th 46
theorem
for b1, b2 being Element of ExtREAL
st b1 is Element of REAL & b2 is Element of REAL
holds b1 < b2
iff
ex b3, b4 being Element of REAL st
b3 = b1 & b4 = b2 & b3 < b4;
:: EXTREAL1:th 47
theorem
for b1, b2, b3 being Element of ExtREAL
st b1 <> -infty & b2 <> +infty & b1 <= b2 & 0. < b3
holds b1 / b3 <= b2 / b3;
:: EXTREAL1:th 48
theorem
for b1, b2, b3 being Element of ExtREAL
st b1 <= b2 & 0. < b3 & b3 <> +infty
holds b1 / b3 <= b2 / b3;
:: EXTREAL1:th 49
theorem
for b1, b2, b3 being Element of ExtREAL
st b1 <> -infty & b2 <> +infty & b1 <= b2 & b3 < 0.
holds b2 / b3 <= b1 / b3;
:: EXTREAL1:th 50
theorem
for b1, b2, b3 being Element of ExtREAL
st b1 <= b2 & b3 < 0. & b3 <> -infty
holds b2 / b3 <= b1 / b3;
:: EXTREAL1:th 51
theorem
for b1, b2, b3 being Element of ExtREAL
st b1 < b2 & 0. < b3 & b3 <> +infty
holds b1 / b3 < b2 / b3;
:: EXTREAL1:th 52
theorem
for b1, b2, b3 being Element of ExtREAL
st b1 < b2 & b3 < 0. & b3 <> -infty
holds b2 / b3 < b1 / b3;