Article SUPINF_1, MML version 4.99.1005
:: SUPINF_1:th 1
theorem
not +infty in REAL;
:: SUPINF_1:attrnot 1 => SUPINF_1:attr 1
definition
let a1 be set;
attr a1 is +Inf-like means
a1 = +infty;
end;
:: SUPINF_1:dfs 1
definiens
let a1 be set;
To prove
a1 is +Inf-like
it is sufficient to prove
thus a1 = +infty;
:: SUPINF_1:def 2
theorem
for b1 being set holds
b1 is +Inf-like
iff
b1 = +infty;
:: SUPINF_1:exreg 1
registration
cluster +Inf-like set;
end;
:: SUPINF_1:modenot 1
definition
mode +Inf is +Inf-like set;
end;
:: SUPINF_1:th 4
theorem
+infty is +Inf-like set;
:: SUPINF_1:th 6
theorem
not -infty in REAL;
:: SUPINF_1:attrnot 2 => SUPINF_1:attr 2
definition
let a1 be set;
attr a1 is -Inf-like means
a1 = -infty;
end;
:: SUPINF_1:dfs 2
definiens
let a1 be set;
To prove
a1 is -Inf-like
it is sufficient to prove
thus a1 = -infty;
:: SUPINF_1:def 4
theorem
for b1 being set holds
b1 is -Inf-like
iff
b1 = -infty;
:: SUPINF_1:exreg 2
registration
cluster -Inf-like set;
end;
:: SUPINF_1:modenot 2
definition
mode -Inf is -Inf-like set;
end;
:: SUPINF_1:th 8
theorem
-infty is -Inf-like set;
:: SUPINF_1:modenot 3
definition
mode R_eal is Element of ExtREAL;
end;
:: SUPINF_1:th 11
theorem
-infty is ext-real set & +infty is ext-real set;
:: SUPINF_1:funcnot 1 => SUPINF_1:func 1
definition
redefine func +infty -> Element of ExtREAL;
end;
:: SUPINF_1:funcnot 2 => SUPINF_1:func 2
definition
redefine func -infty -> Element of ExtREAL;
end;
:: SUPINF_1: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
b1 = a1 & b2 = a2 & b1 <= b2
if a1 in REAL & a2 in REAL
otherwise (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;
:: SUPINF_1:dfs 3
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
b1 = a1 & b2 = a2 & b1 <= b2;
end;
case (a1 in REAL implies not a2 in REAL);
thus (a1 <> -infty) implies a2 = +infty;
end;
:: SUPINF_1:def 7
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
b3 = b1 & b4 = b2 & b3 <= b4)) &
(b1 in REAL & b2 in REAL or (b1 <= b2
iff
(b1 = -infty or b2 = +infty)));
:: SUPINF_1:sch 1
scheme SUPINF_1:sch 1
ex b1 being Element of bool ExtREAL st
for b2 being Element of ExtREAL holds
b2 in b1
iff
P1[b2]
:: SUPINF_1:modenot 4 => SUPINF_1:mode 1
definition
let a1 be ext-real-membered set;
mode majorant of A1 -> ext-real set means
for b1 being ext-real set
st b1 in a1
holds b1 <= it;
end;
:: SUPINF_1:dfs 4
definiens
let a1 be ext-real-membered set;
let a2 be ext-real set;
To prove
a2 is majorant of a1
it is sufficient to prove
thus for b1 being ext-real set
st b1 in a1
holds b1 <= a2;
:: SUPINF_1:def 9
theorem
for b1 being ext-real-membered set
for b2 being ext-real set holds
b2 is majorant of b1
iff
for b3 being ext-real set
st b3 in b1
holds b3 <= b2;
:: SUPINF_1:th 33
theorem
for b1 being ext-real-membered set holds
+infty is majorant of b1;
:: SUPINF_1:th 34
theorem
for b1, b2 being ext-real-membered set
st b1 c= b2
for b3 being ext-real set
st b3 is majorant of b2
holds b3 is majorant of b1;
:: SUPINF_1:modenot 5 => SUPINF_1:mode 2
definition
let a1 be ext-real-membered set;
mode minorant of A1 -> ext-real set means
for b1 being ext-real set
st b1 in a1
holds it <= b1;
end;
:: SUPINF_1:dfs 5
definiens
let a1 be ext-real-membered set;
let a2 be ext-real set;
To prove
a2 is minorant of a1
it is sufficient to prove
thus for b1 being ext-real set
st b1 in a1
holds a2 <= b1;
:: SUPINF_1:def 10
theorem
for b1 being ext-real-membered set
for b2 being ext-real set holds
b2 is minorant of b1
iff
for b3 being ext-real set
st b3 in b1
holds b2 <= b3;
:: SUPINF_1:th 36
theorem
for b1 being ext-real-membered set holds
-infty is minorant of b1;
:: SUPINF_1:th 39
theorem
for b1, b2 being ext-real-membered set
st b1 c= b2
for b3 being ext-real set
st b3 is minorant of b2
holds b3 is minorant of b1;
:: SUPINF_1:funcnot 3 => SUPINF_1:func 3
definition
redefine func REAL -> non empty ext-real-membered set;
end;
:: SUPINF_1:th 41
theorem
+infty is majorant of REAL;
:: SUPINF_1:th 42
theorem
-infty is minorant of REAL;
:: SUPINF_1:attrnot 3 => SUPINF_1:attr 3
definition
let a1 be ext-real-membered set;
attr a1 is bounded_above means
ex b1 being majorant of a1 st
b1 in REAL;
end;
:: SUPINF_1:dfs 6
definiens
let a1 be ext-real-membered set;
To prove
a1 is bounded_above
it is sufficient to prove
thus ex b1 being majorant of a1 st
b1 in REAL;
:: SUPINF_1:def 11
theorem
for b1 being ext-real-membered set holds
b1 is bounded_above
iff
ex b2 being majorant of b1 st
b2 in REAL;
:: SUPINF_1:th 44
theorem
for b1, b2 being ext-real-membered set
st b1 c= b2 & b2 is bounded_above
holds b1 is bounded_above;
:: SUPINF_1:th 45
theorem
REAL is not bounded_above;
:: SUPINF_1:attrnot 4 => SUPINF_1:attr 4
definition
let a1 be ext-real-membered set;
attr a1 is bounded_below means
ex b1 being minorant of a1 st
b1 in REAL;
end;
:: SUPINF_1:dfs 7
definiens
let a1 be ext-real-membered set;
To prove
a1 is bounded_below
it is sufficient to prove
thus ex b1 being minorant of a1 st
b1 in REAL;
:: SUPINF_1:def 12
theorem
for b1 being ext-real-membered set holds
b1 is bounded_below
iff
ex b2 being minorant of b1 st
b2 in REAL;
:: SUPINF_1:th 47
theorem
for b1, b2 being ext-real-membered set
st b1 c= b2 & b2 is bounded_below
holds b1 is bounded_below;
:: SUPINF_1:th 48
theorem
REAL is not bounded_below;
:: SUPINF_1:attrnot 5 => SUPINF_1:attr 5
definition
let a1 be ext-real-membered set;
attr a1 is bounded means
a1 is bounded_above & a1 is bounded_below;
end;
:: SUPINF_1:dfs 8
definiens
let a1 be ext-real-membered set;
To prove
a1 is bounded
it is sufficient to prove
thus a1 is bounded_above & a1 is bounded_below;
:: SUPINF_1:def 13
theorem
for b1 being ext-real-membered set holds
b1 is bounded
iff
b1 is bounded_above & b1 is bounded_below;
:: SUPINF_1:th 50
theorem
for b1, b2 being ext-real-membered set
st b1 c= b2 & b2 is bounded
holds b1 is bounded;
:: SUPINF_1:th 51
theorem
for b1 being ext-real-membered set holds
ex b2 being non empty ext-real-membered set st
for b3 being ext-real set holds
b3 in b2
iff
b3 is majorant of b1;
:: SUPINF_1:funcnot 4 => SUPINF_1:func 4
definition
let a1 be ext-real-membered set;
func SetMajorant A1 -> ext-real-membered set means
for b1 being ext-real set holds
b1 in it
iff
b1 is majorant of a1;
end;
:: SUPINF_1:def 14
theorem
for b1, b2 being ext-real-membered set holds
b2 = SetMajorant b1
iff
for b3 being ext-real set holds
b3 in b2
iff
b3 is majorant of b1;
:: SUPINF_1:funcreg 1
registration
let a1 be ext-real-membered set;
cluster SetMajorant a1 -> non empty ext-real-membered;
end;
:: SUPINF_1:th 54
theorem
for b1, b2 being ext-real-membered set
st b1 c= b2
for b3 being ext-real set
st b3 in SetMajorant b2
holds b3 in SetMajorant b1;
:: SUPINF_1:th 55
theorem
for b1 being ext-real-membered set holds
ex b2 being non empty ext-real-membered set st
for b3 being ext-real set holds
b3 in b2
iff
b3 is minorant of b1;
:: SUPINF_1:funcnot 5 => SUPINF_1:func 5
definition
let a1 be ext-real-membered set;
func SetMinorant A1 -> ext-real-membered set means
for b1 being ext-real set holds
b1 in it
iff
b1 is minorant of a1;
end;
:: SUPINF_1:def 15
theorem
for b1, b2 being ext-real-membered set holds
b2 = SetMinorant b1
iff
for b3 being ext-real set holds
b3 in b2
iff
b3 is minorant of b1;
:: SUPINF_1:funcreg 2
registration
let a1 be ext-real-membered set;
cluster SetMinorant a1 -> non empty ext-real-membered;
end;
:: SUPINF_1:th 58
theorem
for b1, b2 being ext-real-membered set
st b1 c= b2
for b3 being ext-real set
st b3 in SetMinorant b2
holds b3 in SetMinorant b1;
:: SUPINF_1:th 59
theorem
for b1 being non empty ext-real-membered set
st b1 is bounded_above & b1 <> {-infty}
holds ex b2 being Element of REAL st
b2 in b1 & b2 <> -infty;
:: SUPINF_1:th 60
theorem
for b1 being non empty ext-real-membered set
st b1 is bounded_below & b1 <> {+infty}
holds ex b2 being Element of REAL st
b2 in b1 & b2 <> +infty;
:: SUPINF_1:th 62
theorem
for b1 being non empty ext-real-membered set
st b1 is bounded_above & b1 <> {-infty}
holds ex b2 being ext-real set st
b2 is majorant of b1 &
(for b3 being ext-real set
st b3 is majorant of b1
holds b2 <= b3);
:: SUPINF_1:th 63
theorem
for b1 being non empty ext-real-membered set
st b1 is bounded_below & b1 <> {+infty}
holds ex b2 being ext-real set st
b2 is minorant of b1 &
(for b3 being ext-real set
st b3 is minorant of b1
holds b3 <= b2);
:: SUPINF_1:th 64
theorem
for b1 being ext-real-membered set
st b1 = {-infty}
holds b1 is bounded_above;
:: SUPINF_1:th 65
theorem
for b1 being ext-real-membered set
st b1 = {+infty}
holds b1 is bounded_below;
:: SUPINF_1:th 66
theorem
for b1 being ext-real-membered set
st b1 = {-infty}
holds ex b2 being ext-real set st
b2 is majorant of b1 &
(for b3 being ext-real set
st b3 is majorant of b1
holds b2 <= b3);
:: SUPINF_1:th 67
theorem
for b1 being ext-real-membered set
st b1 = {+infty}
holds ex b2 being ext-real set st
b2 is minorant of b1 &
(for b3 being ext-real set
st b3 is minorant of b1
holds b3 <= b2);
:: SUPINF_1:th 68
theorem
for b1 being non empty ext-real-membered set
st b1 is bounded_above
holds ex b2 being ext-real set st
b2 is majorant of b1 &
(for b3 being ext-real set
st b3 is majorant of b1
holds b2 <= b3);
:: SUPINF_1:th 69
theorem
for b1 being non empty ext-real-membered set
st b1 is bounded_below
holds ex b2 being ext-real set st
b2 is minorant of b1 &
(for b3 being ext-real set
st b3 is minorant of b1
holds b3 <= b2);
:: SUPINF_1:th 70
theorem
for b1 being non empty ext-real-membered set
st b1 is not bounded_above
for b2 being ext-real set
st b2 is majorant of b1
holds b2 = +infty;
:: SUPINF_1:th 71
theorem
for b1 being non empty ext-real-membered set
st b1 is not bounded_below
for b2 being ext-real set
st b2 is minorant of b1
holds b2 = -infty;
:: SUPINF_1:th 72
theorem
for b1 being non empty ext-real-membered set holds
ex b2 being ext-real set st
b2 is majorant of b1 &
(for b3 being ext-real set
st b3 is majorant of b1
holds b2 <= b3);
:: SUPINF_1:th 73
theorem
for b1 being non empty ext-real-membered set holds
ex b2 being ext-real set st
b2 is minorant of b1 &
(for b3 being ext-real set
st b3 is minorant of b1
holds b3 <= b2);
:: SUPINF_1:funcnot 6 => SUPINF_1:func 6
definition
let a1 be non empty ext-real-membered set;
func sup A1 -> ext-real set means
it is majorant of a1 &
(for b1 being ext-real set
st b1 is majorant of a1
holds it <= b1);
end;
:: SUPINF_1:def 16
theorem
for b1 being non empty ext-real-membered set
for b2 being ext-real set holds
b2 = sup b1
iff
b2 is majorant of b1 &
(for b3 being ext-real set
st b3 is majorant of b1
holds b2 <= b3);
:: SUPINF_1:th 76
theorem
for b1 being non empty ext-real-membered set
for b2 being ext-real set
st b2 in b1
holds b2 <= sup b1;
:: SUPINF_1:funcnot 7 => SUPINF_1:func 7
definition
let a1 be non empty ext-real-membered set;
func inf A1 -> ext-real set means
it is minorant of a1 &
(for b1 being ext-real set
st b1 is minorant of a1
holds b1 <= it);
end;
:: SUPINF_1:def 17
theorem
for b1 being non empty ext-real-membered set
for b2 being ext-real set holds
b2 = inf b1
iff
b2 is minorant of b1 &
(for b3 being ext-real set
st b3 is minorant of b1
holds b3 <= b2);
:: SUPINF_1:th 79
theorem
for b1 being non empty ext-real-membered set
for b2 being ext-real set
st b2 in b1
holds inf b1 <= b2;
:: SUPINF_1:th 80
theorem
for b1 being non empty ext-real-membered set
for b2 being majorant of b1
st b2 in b1
holds b2 = sup b1;
:: SUPINF_1:th 81
theorem
for b1 being non empty ext-real-membered set
for b2 being minorant of b1
st b2 in b1
holds b2 = inf b1;
:: SUPINF_1:th 82
theorem
for b1 being non empty ext-real-membered set holds
sup b1 = inf SetMajorant b1 & inf b1 = sup SetMinorant b1;
:: SUPINF_1:th 83
theorem
for b1 being non empty ext-real-membered set
st b1 is bounded_above & b1 <> {-infty}
holds sup b1 in REAL;
:: SUPINF_1:th 84
theorem
for b1 being non empty ext-real-membered set
st b1 is bounded_below & b1 <> {+infty}
holds inf b1 in REAL;
:: SUPINF_1:th 85
theorem
for b1 being ext-real set holds
sup {b1} = b1;
:: SUPINF_1:th 86
theorem
for b1 being ext-real set holds
inf {b1} = b1;
:: SUPINF_1:th 91
theorem
for b1, b2 being non empty ext-real-membered set
st b1 c= b2
holds sup b1 <= sup b2;
:: SUPINF_1:th 92
theorem
for b1, b2, b3 being ext-real set
st b1 <= b3 & b2 <= b3
holds sup {b1,b2} <= b3;
:: SUPINF_1:th 93
theorem
for b1, b2 being ext-real set holds
(b1 <= b2 implies sup {b1,b2} = b2) &
(b2 <= b1 implies sup {b1,b2} = b1);
:: SUPINF_1:th 94
theorem
for b1, b2 being non empty ext-real-membered set
st b1 c= b2
holds inf b2 <= inf b1;
:: SUPINF_1:th 95
theorem
for b1, b2, b3 being ext-real set
st b3 <= b1 & b3 <= b2
holds b3 <= inf {b1,b2};
:: SUPINF_1:th 96
theorem
for b1, b2 being ext-real set holds
(b1 <= b2 implies inf {b1,b2} = b1) &
(b2 <= b1 implies inf {b1,b2} = b2);
:: SUPINF_1:th 97
theorem
for b1 being non empty ext-real-membered set
for b2 being ext-real set
st ex b3 being ext-real set st
b3 in b1 & b2 <= b3
holds b2 <= sup b1;
:: SUPINF_1:th 98
theorem
for b1 being non empty ext-real-membered set
for b2 being ext-real set
st ex b3 being ext-real set st
b3 in b1 & b3 <= b2
holds inf b1 <= b2;
:: SUPINF_1:th 99
theorem
for b1, b2 being non empty ext-real-membered set
st for b3 being ext-real set
st b3 in b1
holds ex b4 being ext-real set st
b4 in b2 & b3 <= b4
holds sup b1 <= sup b2;
:: SUPINF_1:th 100
theorem
for b1, b2 being non empty ext-real-membered set
st for b3 being ext-real set
st b3 in b2
holds ex b4 being ext-real set st
b4 in b1 & b4 <= b3
holds inf b1 <= inf b2;
:: SUPINF_1:th 101
theorem
for b1, b2 being ext-real-membered set
for b3 being majorant of b1
for b4 being majorant of b2 holds
sup {b3,b4} is majorant of b1 \/ b2;
:: SUPINF_1:th 102
theorem
for b1, b2 being ext-real-membered set
for b3 being minorant of b1
for b4 being minorant of b2 holds
inf {b3,b4} is minorant of b1 \/ b2;
:: SUPINF_1:th 103
theorem
for b1, b2, b3 being ext-real-membered set
for b4 being majorant of b1
for b5 being majorant of b2
st b3 = b1 /\ b2
holds inf {b4,b5} is majorant of b3;
:: SUPINF_1:th 104
theorem
for b1, b2, b3 being ext-real-membered set
for b4 being minorant of b1
for b5 being minorant of b2
st b3 = b1 /\ b2
holds sup {b4,b5} is minorant of b3;
:: SUPINF_1:th 105
theorem
for b1, b2 being non empty ext-real-membered set holds
sup (b1 \/ b2) = sup {sup b1,sup b2};
:: SUPINF_1:th 106
theorem
for b1, b2 being non empty ext-real-membered set holds
inf (b1 \/ b2) = inf {inf b1,inf b2};
:: SUPINF_1:th 107
theorem
for b1, b2, b3 being non empty ext-real-membered set
st b3 = b1 /\ b2
holds sup b3 <= inf {sup b1,sup b2};
:: SUPINF_1:th 108
theorem
for b1, b2, b3 being non empty ext-real-membered set
st b3 = b1 /\ b2
holds sup {inf b1,inf b2} <= inf b3;
:: SUPINF_1:exreg 3
registration
let a1 be non empty set;
cluster non empty with_non-empty_elements Element of bool bool a1;
end;
:: SUPINF_1:modenot 6
definition
let a1 be non empty set;
mode bool_DOMAIN of a1 is non empty with_non-empty_elements Element of bool bool a1;
end;
:: SUPINF_1:funcnot 8 => SUPINF_1:func 8
definition
let a1 be non empty with_non-empty_elements Element of bool bool ExtREAL;
func SUP A1 -> ext-real-membered set means
for b1 being ext-real set holds
b1 in it
iff
ex b2 being non empty ext-real-membered set st
b2 in a1 & b1 = sup b2;
end;
:: SUPINF_1:def 19
theorem
for b1 being non empty with_non-empty_elements Element of bool bool ExtREAL
for b2 being ext-real-membered set holds
b2 = SUP b1
iff
for b3 being ext-real set holds
b3 in b2
iff
ex b4 being non empty ext-real-membered set st
b4 in b1 & b3 = sup b4;
:: SUPINF_1:funcreg 3
registration
let a1 be non empty with_non-empty_elements Element of bool bool ExtREAL;
cluster SUP a1 -> non empty ext-real-membered;
end;
:: SUPINF_1:th 112
theorem
for b1 being non empty with_non-empty_elements Element of bool bool ExtREAL
for b2 being non empty ext-real-membered set
st b2 = union b1
holds sup b2 is majorant of SUP b1;
:: SUPINF_1:th 113
theorem
for b1 being non empty with_non-empty_elements Element of bool bool ExtREAL
for b2 being ext-real-membered set
st b2 = union b1
holds sup SUP b1 is majorant of b2;
:: SUPINF_1:th 114
theorem
for b1 being non empty with_non-empty_elements Element of bool bool ExtREAL
for b2 being non empty ext-real-membered set
st b2 = union b1
holds sup b2 = sup SUP b1;
:: SUPINF_1:funcnot 9 => SUPINF_1:func 9
definition
let a1 be non empty with_non-empty_elements Element of bool bool ExtREAL;
func INF A1 -> ext-real-membered set means
for b1 being ext-real set holds
b1 in it
iff
ex b2 being non empty ext-real-membered set st
b2 in a1 & b1 = inf b2;
end;
:: SUPINF_1:def 20
theorem
for b1 being non empty with_non-empty_elements Element of bool bool ExtREAL
for b2 being ext-real-membered set holds
b2 = INF b1
iff
for b3 being ext-real set holds
b3 in b2
iff
ex b4 being non empty ext-real-membered set st
b4 in b1 & b3 = inf b4;
:: SUPINF_1:funcreg 4
registration
let a1 be non empty with_non-empty_elements Element of bool bool ExtREAL;
cluster INF a1 -> non empty ext-real-membered;
end;
:: SUPINF_1:th 117
theorem
for b1 being non empty with_non-empty_elements Element of bool bool ExtREAL
for b2 being non empty ext-real-membered set
st b2 = union b1
holds inf b2 is minorant of INF b1;
:: SUPINF_1:th 118
theorem
for b1 being non empty with_non-empty_elements Element of bool bool ExtREAL
for b2 being ext-real-membered set
st b2 = union b1
holds inf INF b1 is minorant of b2;
:: SUPINF_1:th 119
theorem
for b1 being non empty with_non-empty_elements Element of bool bool ExtREAL
for b2 being non empty ext-real-membered set
st b2 = union b1
holds inf b2 = inf INF b1;