Article XXREAL_0, MML version 4.99.1005
:: XXREAL_0:attrnot 1 => XXREAL_0:attr 1
definition
let a1 be set;
attr a1 is ext-real means
a1 in ExtREAL;
end;
:: XXREAL_0:dfs 1
definiens
let a1 be set;
To prove
a1 is ext-real
it is sufficient to prove
thus a1 in ExtREAL;
:: XXREAL_0:def 1
theorem
for b1 being set holds
b1 is ext-real
iff
b1 in ExtREAL;
:: XXREAL_0:exreg 1
registration
cluster ext-real set;
end;
:: XXREAL_0:condreg 1
registration
cluster -> ext-real (Element of ExtREAL);
end;
:: XXREAL_0:funcnot 1 => XXREAL_0:func 1
definition
func +infty -> set equals
REAL;
end;
:: XXREAL_0:def 2
theorem
+infty = REAL;
:: XXREAL_0:funcnot 2 => XXREAL_0:func 2
definition
func -infty -> set equals
[0,REAL];
end;
:: XXREAL_0:def 3
theorem
-infty = [0,REAL];
:: XXREAL_0:funcnot 3 => NUMBERS:func 7
definition
func ExtREAL -> set equals
REAL \/ {+infty,-infty};
end;
:: XXREAL_0:def 4
theorem
ExtREAL = REAL \/ {+infty,-infty};
:: XXREAL_0:funcreg 1
registration
cluster +infty -> ext-real;
end;
:: XXREAL_0:funcreg 2
registration
cluster -infty -> ext-real;
end;
:: XXREAL_0:prednot 1 => XXREAL_0:pred 1
definition
let a1, a2 be ext-real set;
pred A1 <= A2 means
ex b1, b2 being Element of REAL+ st
a1 = b1 & a2 = b2 & b1 <=' b2
if a1 in REAL+ & a2 in REAL+,
ex b1, b2 being Element of REAL+ st
a1 = [0,b1] & a2 = [0,b2] & b2 <=' b1
if a1 in [:{0},REAL+:] &
a2 in [:{0},REAL+:]
otherwise ((a2 in REAL+ implies not a1 in [:{0},REAL+:]) &
a1 <> -infty) implies a2 = +infty;
reflexivity;
:: for a1 being ext-real set holds
:: a1 <= a1;
connectedness;
:: for a1, a2 being ext-real set
:: st a2 < a1
:: holds a2 <= a1;
end;
:: XXREAL_0:dfs 5
definiens
let a1, a2 be ext-real set;
To prove
a1 <= a2
it is sufficient to prove
per cases;
case a1 in REAL+ & a2 in REAL+;
thus ex b1, b2 being Element of REAL+ st
a1 = b1 & a2 = b2 & b1 <=' b2;
end;
case a1 in [:{0},REAL+:] &
a2 in [:{0},REAL+:];
thus ex b1, b2 being Element of REAL+ st
a1 = [0,b1] & a2 = [0,b2] & b2 <=' b1;
end;
case (a1 in REAL+ implies not a2 in REAL+) &
(a1 in [:{0},REAL+:] implies not a2 in [:{0},REAL+:]);
thus ((a2 in REAL+ implies not a1 in [:{0},REAL+:]) &
a1 <> -infty) implies a2 = +infty;
end;
:: XXREAL_0:def 5
theorem
for b1, b2 being ext-real set holds
(b1 in REAL+ & b2 in REAL+ implies (b1 <= b2
iff
ex b3, b4 being Element of REAL+ st
b1 = b3 & b2 = b4 & b3 <=' b4)) &
(b1 in [:{0},REAL+:] &
b2 in [:{0},REAL+:] implies (b1 <= b2
iff
ex b3, b4 being Element of REAL+ st
b1 = [0,b3] & b2 = [0,b4] & b4 <=' b3)) &
((b1 in REAL+ implies not b2 in REAL+) &
(b1 in [:{0},REAL+:] implies not b2 in [:{0},REAL+:]) implies (b1 <= b2
iff
((b2 in REAL+ implies not b1 in [:{0},REAL+:]) &
b1 <> -infty implies b2 = +infty)));
:: XXREAL_0:prednot 2 => XXREAL_0:pred 1
notation
let a1, a2 be ext-real set;
synonym a2 >= a1 for a1 <= a2;
end;
:: XXREAL_0:prednot 3 => not XXREAL_0:pred 1
notation
let a1, a2 be ext-real set;
antonym a2 < a1 for a1 <= a2;
end;
:: XXREAL_0:prednot 4 => not XXREAL_0:pred 1
notation
let a1, a2 be ext-real set;
antonym a1 > a2 for a1 <= a2;
end;
:: XXREAL_0:th 1
theorem
for b1, b2 being ext-real set
st b1 <= b2 & b2 <= b1
holds b1 = b2;
:: XXREAL_0:th 2
theorem
for b1, b2, b3 being ext-real set
st b1 <= b2 & b2 <= b3
holds b1 <= b3;
:: XXREAL_0:th 3
theorem
for b1 being ext-real set holds
b1 <= +infty;
:: XXREAL_0:th 4
theorem
for b1 being ext-real set
st +infty <= b1
holds b1 = +infty;
:: XXREAL_0:th 5
theorem
for b1 being ext-real set holds
-infty <= b1;
:: XXREAL_0:th 6
theorem
for b1 being ext-real set
st b1 <= -infty
holds b1 = -infty;
:: XXREAL_0:th 7
theorem
-infty < +infty;
:: XXREAL_0:th 8
theorem
not +infty in REAL;
:: XXREAL_0:th 9
theorem
for b1 being ext-real set
st b1 in REAL
holds b1 < +infty;
:: XXREAL_0:th 10
theorem
for b1, b2 being ext-real set
st b1 in REAL & b1 <= b2 & not b2 in REAL
holds b2 = +infty;
:: XXREAL_0:th 11
theorem
not -infty in REAL;
:: XXREAL_0:th 12
theorem
for b1 being ext-real set
st b1 in REAL
holds -infty < b1;
:: XXREAL_0:th 13
theorem
for b1, b2 being ext-real set
st b1 in REAL & b2 <= b1 & not b2 in REAL
holds b2 = -infty;
:: XXREAL_0:th 14
theorem
for b1 being ext-real set
st not b1 in REAL & b1 <> +infty
holds b1 = -infty;
:: XXREAL_0:condreg 2
registration
cluster natural -> ext-real (set);
end;
:: XXREAL_0:attrnot 2 => XXREAL_0:attr 2
definition
let a1 be ext-real set;
attr a1 is positive means
0 < a1;
end;
:: XXREAL_0:dfs 6
definiens
let a1 be ext-real set;
To prove
a1 is positive
it is sufficient to prove
thus 0 < a1;
:: XXREAL_0:def 6
theorem
for b1 being ext-real set holds
b1 is positive
iff
0 < b1;
:: XXREAL_0:attrnot 3 => XXREAL_0:attr 3
definition
let a1 be ext-real set;
attr a1 is negative means
a1 < 0;
end;
:: XXREAL_0:dfs 7
definiens
let a1 be ext-real set;
To prove
a1 is negative
it is sufficient to prove
thus a1 < 0;
:: XXREAL_0:def 7
theorem
for b1 being ext-real set holds
b1 is negative
iff
b1 < 0;
:: XXREAL_0:attrnot 4 => XBOOLE_0:attr 1
notation
let a1 be set;
synonym zero for empty;
end;
:: XXREAL_0:condreg 3
registration
cluster ext-real positive -> non empty non negative (set);
end;
:: XXREAL_0:condreg 4
registration
cluster non empty ext-real non negative -> positive (set);
end;
:: XXREAL_0:condreg 5
registration
cluster ext-real negative -> non empty non positive (set);
end;
:: XXREAL_0:condreg 6
registration
cluster non empty ext-real non positive -> negative (set);
end;
:: XXREAL_0:condreg 7
registration
cluster empty ext-real -> non positive non negative (set);
end;
:: XXREAL_0:condreg 8
registration
cluster ext-real non positive non negative -> empty (set);
end;
:: XXREAL_0:funcreg 3
registration
cluster +infty -> positive;
end;
:: XXREAL_0:funcreg 4
registration
cluster -infty -> negative;
end;
:: XXREAL_0:exreg 2
registration
cluster ext-real positive set;
end;
:: XXREAL_0:exreg 3
registration
cluster ext-real negative set;
end;
:: XXREAL_0:exreg 4
registration
cluster empty ext-real set;
end;
:: XXREAL_0:funcnot 4 => XXREAL_0:func 3
definition
let a1, a2 be ext-real set;
func min(A1,A2) -> set equals
a1
if a1 <= a2
otherwise a2;
commutativity;
:: for a1, a2 being ext-real set holds
:: min(a1,a2) = min(a2,a1);
idempotence;
:: for a1 being ext-real set holds
:: min(a1,a1) = a1;
end;
:: XXREAL_0:def 8
theorem
for b1, b2 being ext-real set holds
(b1 <= b2 implies min(b1,b2) = b1) & (b1 <= b2 or min(b1,b2) = b2);
:: XXREAL_0:funcnot 5 => XXREAL_0:func 4
definition
let a1, a2 be ext-real set;
func max(A1,A2) -> set equals
a1
if a2 <= a1
otherwise a2;
commutativity;
:: for a1, a2 being ext-real set holds
:: max(a1,a2) = max(a2,a1);
idempotence;
:: for a1 being ext-real set holds
:: max(a1,a1) = a1;
end;
:: XXREAL_0:def 9
theorem
for b1, b2 being ext-real set holds
(b2 <= b1 implies max(b1,b2) = b1) & (b2 <= b1 or max(b1,b2) = b2);
:: XXREAL_0:th 15
theorem
for b1, b2 being ext-real set
st min(b1,b2) <> b1
holds min(b1,b2) = b2;
:: XXREAL_0:th 16
theorem
for b1, b2 being ext-real set
st max(b1,b2) <> b1
holds max(b1,b2) = b2;
:: XXREAL_0:funcreg 5
registration
let a1, a2 be ext-real set;
cluster min(a1,a2) -> ext-real;
end;
:: XXREAL_0:funcreg 6
registration
let a1, a2 be ext-real set;
cluster max(a1,a2) -> ext-real;
end;
:: XXREAL_0:th 17
theorem
for b1, b2 being ext-real set holds
min(b1,b2) <= b1;
:: XXREAL_0:th 18
theorem
for b1, b2, b3, b4 being ext-real set
st b1 <= b2 & b3 <= b4
holds min(b1,b3) <= min(b2,b4);
:: XXREAL_0:th 19
theorem
for b1, b2, b3, b4 being ext-real set
st b1 < b2 & b3 < b4
holds min(b1,b3) < min(b2,b4);
:: XXREAL_0:th 20
theorem
for b1, b2, b3 being ext-real set
st b1 <= b2 & b1 <= b3
holds b1 <= min(b2,b3);
:: XXREAL_0:th 21
theorem
for b1, b2, b3 being ext-real set
st b1 < b2 & b1 < b3
holds b1 < min(b2,b3);
:: XXREAL_0:th 22
theorem
for b1, b2, b3 being ext-real set
st b1 <= min(b2,b3)
holds b1 <= b2;
:: XXREAL_0:th 23
theorem
for b1, b2, b3 being ext-real set
st b1 < min(b2,b3)
holds b1 < b2;
:: XXREAL_0:th 24
theorem
for b1, b2, b3 being ext-real set
st b1 <= b2 &
b1 <= b3 &
(for b4 being ext-real set
st b4 <= b2 & b4 <= b3
holds b4 <= b1)
holds b1 = min(b2,b3);
:: XXREAL_0:th 25
theorem
for b1, b2 being ext-real set holds
b1 <= max(b1,b2);
:: XXREAL_0:th 26
theorem
for b1, b2, b3, b4 being ext-real set
st b1 <= b2 & b3 <= b4
holds max(b1,b3) <= max(b2,b4);
:: XXREAL_0:th 27
theorem
for b1, b2, b3, b4 being ext-real set
st b1 < b2 & b3 < b4
holds max(b1,b3) < max(b2,b4);
:: XXREAL_0:th 28
theorem
for b1, b2, b3 being ext-real set
st b1 <= b2 & b3 <= b2
holds max(b1,b3) <= b2;
:: XXREAL_0:th 29
theorem
for b1, b2, b3 being ext-real set
st b1 < b2 & b3 < b2
holds max(b1,b3) < b2;
:: XXREAL_0:th 30
theorem
for b1, b2, b3 being ext-real set
st max(b1,b2) <= b3
holds b1 <= b3;
:: XXREAL_0:th 31
theorem
for b1, b2, b3 being ext-real set
st max(b1,b2) < b3
holds b1 < b3;
:: XXREAL_0:th 32
theorem
for b1, b2, b3 being ext-real set
st b1 <= b2 &
b3 <= b2 &
(for b4 being ext-real set
st b1 <= b4 & b3 <= b4
holds b2 <= b4)
holds b2 = max(b1,b3);
:: XXREAL_0:th 33
theorem
for b1, b2, b3 being ext-real set holds
min(min(b1,b2),b3) = min(b1,min(b2,b3));
:: XXREAL_0:th 34
theorem
for b1, b2, b3 being ext-real set holds
max(max(b1,b2),b3) = max(b1,max(b2,b3));
:: XXREAL_0:th 35
theorem
for b1, b2 being ext-real set holds
min(max(b1,b2),b2) = b2;
:: XXREAL_0:th 36
theorem
for b1, b2 being ext-real set holds
max(min(b1,b2),b2) = b2;
:: XXREAL_0:th 37
theorem
for b1, b2, b3 being ext-real set
st b1 <= b2
holds max(b1,min(b3,b2)) = min(max(b1,b3),b2);
:: XXREAL_0:th 38
theorem
for b1, b2, b3 being ext-real set holds
min(b1,max(b2,b3)) = max(min(b1,b2),min(b1,b3));
:: XXREAL_0:th 39
theorem
for b1, b2, b3 being ext-real set holds
max(b1,min(b2,b3)) = min(max(b1,b2),max(b1,b3));
:: XXREAL_0:th 40
theorem
for b1, b2, b3 being ext-real set holds
max(max(min(b1,b2),min(b2,b3)),min(b3,b1)) = min(min(max(b1,b2),max(b2,b3)),max(b3,b1));
:: XXREAL_0:th 41
theorem
for b1 being ext-real set holds
max(b1,+infty) = +infty;
:: XXREAL_0:th 42
theorem
for b1 being ext-real set holds
min(b1,+infty) = b1;
:: XXREAL_0:th 43
theorem
for b1 being ext-real set holds
max(b1,-infty) = b1;
:: XXREAL_0:th 44
theorem
for b1 being ext-real set holds
min(b1,-infty) = -infty;
:: XXREAL_0:th 45
theorem
for b1, b2, b3 being ext-real set
st b1 in REAL & b2 in REAL & b1 <= b3 & b3 <= b2
holds b3 in REAL;
:: XXREAL_0:th 46
theorem
for b1, b2, b3 being ext-real set
st b1 in REAL & b1 <= b2 & b2 < b3
holds b2 in REAL;
:: XXREAL_0:th 47
theorem
for b1, b2, b3 being ext-real set
st b1 in REAL & b2 < b3 & b3 <= b1
holds b3 in REAL;
:: XXREAL_0:th 48
theorem
for b1, b2, b3 being ext-real set
st b1 < b2 & b2 < b3
holds b2 in REAL;
:: XXREAL_0:funcnot 6 => XXREAL_0:func 5
definition
let a1, a2 be ext-real set;
let a3, a4 be set;
func IFGT(A1,A2,A3,A4) -> set equals
a3
if a2 < a1
otherwise a4;
end;
:: XXREAL_0:def 10
theorem
for b1, b2 being ext-real set
for b3, b4 being set holds
(b1 <= b2 or IFGT(b1,b2,b3,b4) = b3) & (b1 <= b2 implies IFGT(b1,b2,b3,b4) = b4);
:: XXREAL_0:funcreg 7
registration
let a1, a2 be ext-real set;
let a3, a4 be natural set;
cluster IFGT(a1,a2,a3,a4) -> natural;
end;