Article MEASURE6, MML version 4.99.1005
:: MEASURE6:th 1
theorem
ex b1 being Function-like quasi_total Relation of NAT,[:NAT,NAT:] st
b1 is one-to-one & dom b1 = NAT & rng b1 = [:NAT,NAT:];
:: MEASURE6:th 2
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 is nonnegative
holds 0. <= SUM b1;
:: MEASURE6:th 3
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
for b2 being Element of ExtREAL
st (ex b3 being Element of NAT st
b2 <= b1 . b3) &
b1 is nonnegative
holds b2 <= SUM b1;
:: MEASURE6:th 8
theorem
for b1, b2 being Element of ExtREAL
st b1 is Element of REAL
holds (b2 - b1) + b1 = b2 & (b2 + b1) - b1 = b2;
:: MEASURE6:th 10
theorem
for b1, b2, b3 being Element of ExtREAL
st b3 in REAL & b2 < b1
holds (b3 + b1) - (b3 + b2) = b1 - b2;
:: MEASURE6:th 11
theorem
for b1, b2, b3 being Element of ExtREAL
st b3 in REAL & b1 <= b2
holds b3 + b1 <= b3 + b2 & b1 + b3 <= b2 + b3 & b1 - b3 <= b2 - b3;
:: MEASURE6:th 12
theorem
for b1, b2, b3 being Element of ExtREAL
st b3 in REAL & b1 < b2
holds b3 + b1 < b3 + b2 & b1 + b3 < b2 + b3 & b1 - b3 < b2 - b3;
:: MEASURE6:funcnot 1 => MEASURE6:func 1
definition
let a1 be ext-real set;
func R_EAL A1 -> Element of ExtREAL equals
a1;
end;
:: MEASURE6:def 1
theorem
for b1 being ext-real set holds
R_EAL b1 = b1;
:: MEASURE6:th 13
theorem
for b1, b2 being ext-real set holds
b1 <= b2
iff
R_EAL b1 <= R_EAL b2;
:: MEASURE6:th 14
theorem
for b1, b2 being ext-real set holds
b1 < b2
iff
R_EAL b1 < R_EAL b2;
:: MEASURE6:th 15
theorem
for b1, b2, b3 being ext-real set
st b1 < b2 & b2 < b3
holds b2 is Element of REAL;
:: MEASURE6:th 16
theorem
for b1, b2, b3 being ext-real set
st b1 is Element of REAL & b3 is Element of REAL & b1 <= b2 & b2 <= b3
holds b2 is Element of REAL;
:: MEASURE6:th 17
theorem
for b1, b2, b3 being ext-real set
st b1 is Element of REAL & b1 <= b2 & b2 < b3
holds b2 is Element of REAL;
:: MEASURE6:th 18
theorem
for b1, b2, b3 being ext-real set
st b1 < b2 & b2 <= b3 & b3 is Element of REAL
holds b2 is Element of REAL;
:: MEASURE6:th 19
theorem
for b1, b2 being Element of ExtREAL
st 0. < b1 & b1 < b2
holds 0. < b2 - b1;
:: MEASURE6:th 20
theorem
for b1, b2, b3 being Element of ExtREAL
st 0. <= b1 & 0. <= b3 & b3 + b1 < b2
holds b3 < b2 - b1;
:: MEASURE6:th 21
theorem
for b1 being Element of ExtREAL holds
b1 - 0. = b1;
:: MEASURE6:th 22
theorem
for b1, b2, b3 being Element of ExtREAL
st 0. <= b1 & 0. <= b3 & b3 + b1 < b2
holds b3 <= b2;
:: MEASURE6:th 23
theorem
for b1 being Element of ExtREAL
st 0. < b1
holds ex b2 being Element of ExtREAL st
0. < b2 & b2 < b1;
:: MEASURE6:th 24
theorem
for b1, b2 being Element of ExtREAL
st 0. < b1 & b1 < b2
holds ex b3 being Element of ExtREAL st
0. < b3 & b1 + b3 < b2 & b3 in REAL;
:: MEASURE6:th 25
theorem
for b1, b2 being Element of ExtREAL
st 0. <= b1 & b1 < b2
holds ex b3 being Element of ExtREAL st
0. < b3 & b1 + b3 < b2 & b3 in REAL;
:: MEASURE6:th 26
theorem
for b1 being Element of ExtREAL
st 0. < b1
holds ex b2 being Element of ExtREAL st
0. < b2 & b2 + b2 < b1;
:: MEASURE6:funcnot 2 => MEASURE6:func 2
definition
let a1 be Element of ExtREAL;
assume 0. < a1;
func Seg A1 -> non empty Element of bool ExtREAL means
for b1 being Element of ExtREAL holds
b1 in it
iff
0. < b1 & b1 + b1 < a1;
end;
:: MEASURE6:def 2
theorem
for b1 being Element of ExtREAL
st 0. < b1
for b2 being non empty Element of bool ExtREAL holds
b2 = Seg b1
iff
for b3 being Element of ExtREAL holds
b3 in b2
iff
0. < b3 & b3 + b3 < b1;
:: MEASURE6:funcnot 3 => MEASURE6:func 3
definition
let a1 be Element of ExtREAL;
func len A1 -> Element of ExtREAL equals
sup Seg a1;
end;
:: MEASURE6:def 3
theorem
for b1 being Element of ExtREAL holds
len b1 = sup Seg b1;
:: MEASURE6:th 27
theorem
for b1 being Element of ExtREAL
st 0. < b1
holds 0. < len b1;
:: MEASURE6:th 28
theorem
for b1 being Element of ExtREAL
st 0. < b1
holds len b1 <= b1;
:: MEASURE6:th 29
theorem
for b1 being Element of ExtREAL
st 0. < b1 & b1 < +infty
holds len b1 is Element of REAL;
:: MEASURE6:th 30
theorem
for b1 being Element of ExtREAL
st 0. < b1
holds (len b1) + len b1 <= b1;
:: MEASURE6:th 31
theorem
for b1 being Element of ExtREAL
st 0. < b1
holds ex b2 being Function-like quasi_total Relation of NAT,ExtREAL st
(for b3 being Element of NAT holds
0. < b2 . b3) &
SUM b2 < b1;
:: MEASURE6:th 32
theorem
for b1 being Element of ExtREAL
for b2 being non empty Element of bool ExtREAL
st 0. < b1 & inf b2 is Element of REAL
holds ex b3 being ext-real set st
b3 in b2 & b3 < (inf b2) + b1;
:: MEASURE6:th 33
theorem
for b1 being Element of ExtREAL
for b2 being non empty Element of bool ExtREAL
st 0. < b1 & sup b2 is Element of REAL
holds ex b3 being ext-real set st
b3 in b2 & (sup b2) - b1 < b3;
:: MEASURE6:th 34
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 is nonnegative & SUM b1 < +infty
for b2 being Element of NAT holds
b1 . b2 in REAL;
:: MEASURE6:th 35
theorem
REAL is interval Element of bool REAL & REAL = ].-infty,+infty.[ & REAL = [.-infty,+infty.] & REAL = [.-infty,+infty.[ & REAL = ].-infty,+infty.];
:: MEASURE6:th 36
theorem
for b1, b2 being Element of ExtREAL
st b2 = -infty
holds ].b1,b2.[ = {} & [.b1,b2.] = {} & [.b1,b2.[ = {} & ].b1,b2.] = {};
:: MEASURE6:th 37
theorem
for b1, b2 being Element of ExtREAL
st b1 = +infty
holds ].b1,b2.[ = {} & [.b1,b2.] = {} & [.b1,b2.[ = {} & ].b1,b2.] = {};
:: MEASURE6:th 38
theorem
for b1 being interval Element of bool REAL
for b2, b3 being Element of ExtREAL
for b4, b5, b6 being Element of REAL
st b1 = ].b2,b3.[ & b4 in b1 & b5 in b1 & b4 <= b6 & b6 <= b5
holds b6 in b1;
:: MEASURE6:th 39
theorem
for b1 being interval Element of bool REAL
for b2, b3 being Element of ExtREAL
for b4, b5, b6 being Element of REAL
st b1 = [.b2,b3.] & b4 in b1 & b5 in b1 & b4 <= b6 & b6 <= b5
holds b6 in b1;
:: MEASURE6:th 40
theorem
for b1 being interval Element of bool REAL
for b2, b3 being Element of ExtREAL
for b4, b5, b6 being Element of REAL
st b1 = ].b2,b3.] & b4 in b1 & b5 in b1 & b4 <= b6 & b6 <= b5
holds b6 in b1;
:: MEASURE6:th 41
theorem
for b1 being interval Element of bool REAL
for b2, b3 being Element of ExtREAL
for b4, b5, b6 being Element of REAL
st b1 = [.b2,b3.[ & b4 in b1 & b5 in b1 & b4 <= b6 & b6 <= b5
holds b6 in b1;
:: MEASURE6:th 42
theorem
for b1 being non empty Element of bool ExtREAL
for b2, b3 being Element of ExtREAL
st b2 = inf b1 &
b3 = sup b1 &
(for b4, b5 being Element of REAL
st b4 in b1 & b5 in b1
for b6 being Element of REAL
st b4 <= b6 & b6 <= b5
holds b6 in b1) &
not b2 in b1 &
not b3 in b1
holds b1 = ].b2,b3.[;
:: MEASURE6:th 43
theorem
for b1 being non empty Element of bool ExtREAL
for b2, b3 being Element of ExtREAL
st b2 = inf b1 &
b3 = sup b1 &
(for b4, b5 being Element of REAL
st b4 in b1 & b5 in b1
for b6 being Element of REAL
st b4 <= b6 & b6 <= b5
holds b6 in b1) &
b2 in b1 &
b3 in b1 &
b1 c= REAL
holds b1 = [.b2,b3.];
:: MEASURE6:th 44
theorem
for b1 being non empty Element of bool ExtREAL
for b2, b3 being Element of ExtREAL
st b2 = inf b1 &
b3 = sup b1 &
(for b4, b5 being Element of REAL
st b4 in b1 & b5 in b1
for b6 being Element of REAL
st b4 <= b6 & b6 <= b5
holds b6 in b1) &
b2 in b1 &
not b3 in b1 &
b1 c= REAL
holds b1 = [.b2,b3.[;
:: MEASURE6:th 45
theorem
for b1 being non empty Element of bool ExtREAL
for b2, b3 being Element of ExtREAL
st b2 = inf b1 &
b3 = sup b1 &
(for b4, b5 being Element of REAL
st b4 in b1 & b5 in b1
for b6 being Element of REAL
st b4 <= b6 & b6 <= b5
holds b6 in b1) &
not b2 in b1 &
b3 in b1 &
b1 c= REAL
holds b1 = ].b2,b3.];
:: MEASURE6:th 46
theorem
for b1 being Element of bool REAL holds
b1 is interval Element of bool REAL
iff
for b2, b3 being Element of REAL
st b2 in b1 & b3 in b1
for b4 being Element of REAL
st b2 <= b4 & b4 <= b3
holds b4 in b1;
:: MEASURE6:th 47
theorem
for b1, b2 being interval Element of bool REAL
st b1 meets b2
holds b1 \/ b2 is interval Element of bool REAL;
:: MEASURE6:funcnot 4 => MEASURE6:func 4
definition
let a1 be interval Element of bool REAL;
assume a1 <> {};
func ^^ A1 -> Element of ExtREAL means
ex b1 being Element of ExtREAL st
it <= b1 &
(a1 <> ].it,b1.[ & a1 <> ].it,b1.] & a1 <> [.it,b1.] implies a1 = [.it,b1.[);
end;
:: MEASURE6:def 4
theorem
for b1 being interval Element of bool REAL
st b1 <> {}
for b2 being Element of ExtREAL holds
b2 = ^^ b1
iff
ex b3 being Element of ExtREAL st
b2 <= b3 &
(b1 <> ].b2,b3.[ & b1 <> ].b2,b3.] & b1 <> [.b2,b3.] implies b1 = [.b2,b3.[);
:: MEASURE6:funcnot 5 => MEASURE6:func 5
definition
let a1 be interval Element of bool REAL;
assume a1 <> {};
func A1 ^^ -> Element of ExtREAL means
ex b1 being Element of ExtREAL st
b1 <= it &
(a1 <> ].b1,it.[ & a1 <> ].b1,it.] & a1 <> [.b1,it.] implies a1 = [.b1,it.[);
end;
:: MEASURE6:def 5
theorem
for b1 being interval Element of bool REAL
st b1 <> {}
for b2 being Element of ExtREAL holds
b2 = b1 ^^
iff
ex b3 being Element of ExtREAL st
b3 <= b2 &
(b1 <> ].b3,b2.[ & b1 <> ].b3,b2.] & b1 <> [.b3,b2.] implies b1 = [.b3,b2.[);
:: MEASURE6:th 48
theorem
for b1 being interval Element of bool REAL
st b1 is open_interval & b1 <> {}
holds ^^ b1 <= b1 ^^ & b1 = ].^^ b1,b1 ^^.[;
:: MEASURE6:th 49
theorem
for b1 being interval Element of bool REAL
st b1 is closed_interval & b1 <> {}
holds ^^ b1 <= b1 ^^ & b1 = [.^^ b1,b1 ^^.];
:: MEASURE6:th 50
theorem
for b1 being interval Element of bool REAL
st b1 is right_open_interval & b1 <> {}
holds ^^ b1 <= b1 ^^ & b1 = [.^^ b1,b1 ^^.[;
:: MEASURE6:th 51
theorem
for b1 being interval Element of bool REAL
st b1 is left_open_interval & b1 <> {}
holds ^^ b1 <= b1 ^^ & b1 = ].^^ b1,b1 ^^.];
:: MEASURE6:th 52
theorem
for b1 being interval Element of bool REAL
st b1 <> {}
holds ^^ b1 <= b1 ^^ &
(b1 <> ].^^ b1,b1 ^^.[ & b1 <> ].^^ b1,b1 ^^.] & b1 <> [.^^ b1,b1 ^^.] implies b1 = [.^^ b1,b1 ^^.[);
:: MEASURE6:th 54
theorem
for b1 being interval Element of bool REAL
for b2 being real set
st b2 in b1
holds ^^ b1 <= R_EAL b2 & R_EAL b2 <= b1 ^^;
:: MEASURE6:th 55
theorem
for b1, b2 being interval Element of bool REAL
for b3, b4 being real set
st b3 in b1 & b4 in b2 & b1 ^^ <= ^^ b2
holds b3 <= b4;
:: MEASURE6:th 56
theorem
for b1 being interval Element of bool REAL
for b2 being Element of ExtREAL
st b2 in b1
holds ^^ b1 <= b2 & b2 <= b1 ^^;
:: MEASURE6:th 57
theorem
for b1 being interval Element of bool REAL
st b1 <> {}
for b2 being Element of ExtREAL
st ^^ b1 < b2 & b2 < b1 ^^
holds b2 in b1;
:: MEASURE6:th 58
theorem
for b1, b2 being interval Element of bool REAL
st b1 ^^ = ^^ b2 & (b1 ^^ in b1 or ^^ b2 in b2)
holds b1 \/ b2 is interval Element of bool REAL;
:: MEASURE6:funcnot 6 => MEASURE6:func 6
definition
let a1 be real-membered set;
let a2 be real set;
func A2 ++ A1 -> Element of bool REAL means
for b1 being Element of REAL holds
b1 in it
iff
ex b2 being Element of REAL st
b2 in a1 & b1 = a2 + b2;
end;
:: MEASURE6:def 6
theorem
for b1 being real-membered set
for b2 being real set
for b3 being Element of bool REAL holds
b3 = b2 ++ b1
iff
for b4 being Element of REAL holds
b4 in b3
iff
ex b5 being Element of REAL st
b5 in b1 & b4 = b2 + b5;
:: MEASURE6:th 59
theorem
for b1 being Element of bool REAL
for b2 being real set holds
(- b2) ++ (b2 ++ b1) = b1;
:: MEASURE6:th 60
theorem
for b1 being real set
for b2 being Element of bool REAL
st b2 = REAL
holds b1 ++ b2 = b2;
:: MEASURE6:th 61
theorem
for b1 being real set holds
b1 ++ {} = {};
:: MEASURE6:th 62
theorem
for b1 being interval Element of bool REAL
for b2 being real set holds
b1 is open_interval
iff
b2 ++ b1 is open_interval;
:: MEASURE6:th 63
theorem
for b1 being interval Element of bool REAL
for b2 being real set holds
b1 is closed_interval
iff
b2 ++ b1 is closed_interval;
:: MEASURE6:th 64
theorem
for b1 being interval Element of bool REAL
for b2 being real set holds
b1 is right_open_interval
iff
b2 ++ b1 is right_open_interval;
:: MEASURE6:th 65
theorem
for b1 being interval Element of bool REAL
for b2 being real set holds
b1 is left_open_interval
iff
b2 ++ b1 is left_open_interval;
:: MEASURE6:th 66
theorem
for b1 being interval Element of bool REAL
for b2 being real set holds
b2 ++ b1 is interval Element of bool REAL;
:: MEASURE6:funcreg 1
registration
let a1 be interval Element of bool REAL;
let a2 be real set;
cluster a2 ++ a1 -> interval;
end;
:: MEASURE6:th 67
theorem
for b1 being interval Element of bool REAL
for b2 being real set holds
vol b1 = vol (b2 ++ b1);