Article INTEGRA2, MML version 4.99.1005
:: INTEGRA2:th 1
theorem
for b1 being closed-interval Element of bool REAL
for b2 being real set holds
b2 in b1
iff
inf b1 <= b2 & b2 <= sup b1;
:: INTEGRA2:attrnot 1 => INTEGRA2:attr 1
definition
let a1 be FinSequence of REAL;
attr a1 is non-decreasing means
for b1 being Element of NAT
st b1 in dom a1 & b1 + 1 in dom a1
holds a1 . b1 <= a1 . (b1 + 1);
end;
:: INTEGRA2:dfs 1
definiens
let a1 be FinSequence of REAL;
To prove
a1 is non-decreasing
it is sufficient to prove
thus for b1 being Element of NAT
st b1 in dom a1 & b1 + 1 in dom a1
holds a1 . b1 <= a1 . (b1 + 1);
:: INTEGRA2:def 1
theorem
for b1 being FinSequence of REAL holds
b1 is non-decreasing
iff
for b2 being Element of NAT
st b2 in dom b1 & b2 + 1 in dom b1
holds b1 . b2 <= b1 . (b2 + 1);
:: INTEGRA2:exreg 1
registration
cluster Relation-like Function-like complex-valued ext-real-valued real-valued finite FinSequence-like non-decreasing FinSequence of REAL;
end;
:: INTEGRA2:th 2
theorem
for b1 being non-decreasing FinSequence of REAL
for b2, b3 being Element of NAT
st b2 in dom b1 & b3 in dom b1 & b2 <= b3
holds b1 . b2 <= b1 . b3;
:: INTEGRA2:th 3
theorem
for b1 being FinSequence of REAL holds
ex b2 being non-decreasing FinSequence of REAL st
b1,b2 are_fiberwise_equipotent;
:: INTEGRA2:th 4
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4, b5 being Element of NAT
st 1 <= b3 & b5 <= len b2 & b3 <= b4 & b4 < b5
holds (mid(b2,b3,b4)) ^ mid(b2,b4 + 1,b5) = mid(b2,b3,b5);
:: INTEGRA2:funcnot 1 => INTEGRA2:func 1
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;
:: INTEGRA2:def 2
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;
:: INTEGRA2:th 5
theorem
for b1, b2 being non empty set
for b3 being Function-like Relation of b1,REAL
st b3 is_bounded_above_on b1 & b2 c= b1
holds b3 | b2 is_bounded_above_on b2;
:: INTEGRA2:th 6
theorem
for b1, b2 being non empty set
for b3 being Function-like Relation of b1,REAL
st b3 is_bounded_below_on b1 & b2 c= b1
holds b3 | b2 is_bounded_below_on b2;
:: INTEGRA2:th 7
theorem
for b1 being real-membered set
for b2 being real set holds
b1 is empty
iff
b2 ** b1 is empty;
:: INTEGRA2:th 8
theorem
for b1 being Element of REAL
for b2 being Element of bool REAL holds
b1 ** b2 = {b1 * b3 where b3 is Element of REAL: b3 in b2};
:: INTEGRA2:th 9
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool REAL
st b2 is bounded_above & 0 <= b1
holds b1 ** b2 is bounded_above;
:: INTEGRA2:th 10
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool REAL
st b2 is bounded_above & b1 <= 0
holds b1 ** b2 is bounded_below;
:: INTEGRA2:th 11
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool REAL
st b2 is bounded_below & 0 <= b1
holds b1 ** b2 is bounded_below;
:: INTEGRA2:th 12
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool REAL
st b2 is bounded_below & b1 <= 0
holds b1 ** b2 is bounded_above;
:: INTEGRA2:th 13
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool REAL
st b2 is bounded_above & 0 <= b1
holds sup (b1 ** b2) = b1 * sup b2;
:: INTEGRA2:th 14
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool REAL
st b2 is bounded_above & b1 <= 0
holds inf (b1 ** b2) = b1 * sup b2;
:: INTEGRA2:th 15
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool REAL
st b2 is bounded_below & 0 <= b1
holds inf (b1 ** b2) = b1 * inf b2;
:: INTEGRA2:th 16
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool REAL
st b2 is bounded_below & b1 <= 0
holds sup (b1 ** b2) = b1 * inf b2;
:: INTEGRA2:th 17
theorem
for b1 being Element of REAL
for b2 being non empty set
for b3 being Function-like quasi_total Relation of b2,REAL holds
rng (b1 (#) b3) = b1 ** rng b3;
:: INTEGRA2:th 18
theorem
for b1 being Element of REAL
for b2, b3 being non empty set
for b4 being Function-like Relation of b2,REAL holds
rng (b1 (#) (b4 | b3)) = b1 ** rng (b4 | b3);
:: INTEGRA2:th 19
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3 being Function-like quasi_total Relation of b2,REAL
for b4 being Element of divs b2
st b3 is_bounded_on b2 & 0 <= b1
holds (b1 * inf rng b3) * vol b2 <= (upper_sum_set (b1 (#) b3)) . b4;
:: INTEGRA2:th 20
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3 being Function-like quasi_total Relation of b2,REAL
for b4 being Element of divs b2
st b3 is_bounded_on b2 & b1 <= 0
holds (b1 * sup rng b3) * vol b2 <= (upper_sum_set (b1 (#) b3)) . b4;
:: INTEGRA2:th 21
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3 being Function-like quasi_total Relation of b2,REAL
for b4 being Element of divs b2
st b3 is_bounded_on b2 & 0 <= b1
holds (lower_sum_set (b1 (#) b3)) . b4 <= (b1 * sup rng b3) * vol b2;
:: INTEGRA2:th 22
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3 being Function-like quasi_total Relation of b2,REAL
for b4 being Element of divs b2
st b3 is_bounded_on b2 & b1 <= 0
holds (lower_sum_set (b1 (#) b3)) . b4 <= (b1 * inf rng b3) * vol b2;
:: INTEGRA2:th 23
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3 being Function-like quasi_total Relation of b2,REAL
for b4 being non empty Division of b2
for b5 being Element of b4
for b6 being Element of NAT
st b6 in dom b5 & b3 is_bounded_above_on b2 & 0 <= b1
holds (upper_volume(b1 (#) b3,b5)) . b6 = b1 * ((upper_volume(b3,b5)) . b6);
:: INTEGRA2:th 24
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3 being Function-like quasi_total Relation of b2,REAL
for b4 being non empty Division of b2
for b5 being Element of b4
for b6 being Element of NAT
st b6 in dom b5 & b3 is_bounded_above_on b2 & b1 <= 0
holds (lower_volume(b1 (#) b3,b5)) . b6 = b1 * ((upper_volume(b3,b5)) . b6);
:: INTEGRA2:th 25
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3 being Function-like quasi_total Relation of b2,REAL
for b4 being non empty Division of b2
for b5 being Element of b4
for b6 being Element of NAT
st b6 in dom b5 & b3 is_bounded_below_on b2 & 0 <= b1
holds (lower_volume(b1 (#) b3,b5)) . b6 = b1 * ((lower_volume(b3,b5)) . b6);
:: INTEGRA2:th 26
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3 being Function-like quasi_total Relation of b2,REAL
for b4 being non empty Division of b2
for b5 being Element of b4
for b6 being Element of NAT
st b6 in dom b5 & b3 is_bounded_below_on b2 & b1 <= 0
holds (upper_volume(b1 (#) b3,b5)) . b6 = b1 * ((lower_volume(b3,b5)) . b6);
:: INTEGRA2:th 27
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3 being Function-like quasi_total Relation of b2,REAL
for b4 being non empty Division of b2
for b5 being Element of b4
st b3 is_bounded_above_on b2 & 0 <= b1
holds upper_sum(b1 (#) b3,b5) = b1 * upper_sum(b3,b5);
:: INTEGRA2:th 28
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3 being Function-like quasi_total Relation of b2,REAL
for b4 being non empty Division of b2
for b5 being Element of b4
st b3 is_bounded_above_on b2 & b1 <= 0
holds lower_sum(b1 (#) b3,b5) = b1 * upper_sum(b3,b5);
:: INTEGRA2:th 29
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3 being Function-like quasi_total Relation of b2,REAL
for b4 being non empty Division of b2
for b5 being Element of b4
st b3 is_bounded_below_on b2 & 0 <= b1
holds lower_sum(b1 (#) b3,b5) = b1 * lower_sum(b3,b5);
:: INTEGRA2:th 30
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3 being Function-like quasi_total Relation of b2,REAL
for b4 being non empty Division of b2
for b5 being Element of b4
st b3 is_bounded_below_on b2 & b1 <= 0
holds upper_sum(b1 (#) b3,b5) = b1 * lower_sum(b3,b5);
:: INTEGRA2:th 31
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3 being Function-like quasi_total Relation of b2,REAL
st b3 is_bounded_on b2 & b3 is_integrable_on b2
holds b1 (#) b3 is_integrable_on b2 & integral (b1 (#) b3) = b1 * integral b3;
:: INTEGRA2:th 32
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
st b2 is_bounded_on b1 &
b2 is_integrable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds 0 <= b2 . b3)
holds 0 <= integral b2;
:: INTEGRA2:th 33
theorem
for b1 being closed-interval Element of bool REAL
for b2, b3 being Function-like quasi_total Relation of b1,REAL
st b2 is_bounded_on b1 & b2 is_integrable_on b1 & b3 is_bounded_on b1 & b3 is_integrable_on b1
holds b2 - b3 is_integrable_on b1 &
integral (b2 - b3) = (integral b2) - integral b3;
:: INTEGRA2:th 34
theorem
for b1 being closed-interval Element of bool REAL
for b2, b3 being Function-like quasi_total Relation of b1,REAL
st b2 is_bounded_on b1 &
b2 is_integrable_on b1 &
b3 is_bounded_on b1 &
b3 is_integrable_on b1 &
(for b4 being Element of REAL
st b4 in b1
holds b3 . b4 <= b2 . b4)
holds integral b3 <= integral b2;
:: INTEGRA2:th 35
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
st b2 is_bounded_on b1
holds rng upper_sum_set b2 is bounded_below;
:: INTEGRA2:th 36
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
st b2 is_bounded_on b1
holds rng lower_sum_set b2 is bounded_above;
:: INTEGRA2:modenot 1
definition
let a1 be closed-interval Element of bool REAL;
mode DivSequence of a1 is Function-like quasi_total Relation of NAT,divs a1;
end;
:: INTEGRA2:funcnot 2 => INTEGRA2:func 2
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like quasi_total Relation of NAT,divs a1;
func delta A2 -> Function-like quasi_total Relation of NAT,REAL means
for b1 being Element of NAT holds
it . b1 = delta (a2 . b1);
end;
:: INTEGRA2:def 3
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of NAT,divs b1
for b3 being Function-like quasi_total Relation of NAT,REAL holds
b3 = delta b2
iff
for b4 being Element of NAT holds
b3 . b4 = delta (b2 . b4);
:: INTEGRA2:funcnot 3 => INTEGRA2:func 3
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of a1,REAL;
let a3 be Function-like quasi_total Relation of NAT,divs a1;
func upper_sum(A2,A3) -> Function-like quasi_total Relation of NAT,REAL means
for b1 being Element of NAT holds
it . b1 = upper_sum(a2,a3 . b1);
end;
:: INTEGRA2:def 4
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of b1,REAL
for b3 being Function-like quasi_total Relation of NAT,divs b1
for b4 being Function-like quasi_total Relation of NAT,REAL holds
b4 = upper_sum(b2,b3)
iff
for b5 being Element of NAT holds
b4 . b5 = upper_sum(b2,b3 . b5);
:: INTEGRA2:funcnot 4 => INTEGRA2:func 4
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of a1,REAL;
let a3 be Function-like quasi_total Relation of NAT,divs a1;
func lower_sum(A2,A3) -> Function-like quasi_total Relation of NAT,REAL means
for b1 being Element of NAT holds
it . b1 = lower_sum(a2,a3 . b1);
end;
:: INTEGRA2:def 5
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of b1,REAL
for b3 being Function-like quasi_total Relation of NAT,divs b1
for b4 being Function-like quasi_total Relation of NAT,REAL holds
b4 = lower_sum(b2,b3)
iff
for b5 being Element of NAT holds
b4 . b5 = lower_sum(b2,b3 . b5);
:: INTEGRA2:th 37
theorem
for b1 being closed-interval Element of bool REAL
for b2, b3 being Element of divs b1
st b2 <= b3
for b4 being Element of NAT
st b4 in dom b3
holds ex b5 being Element of NAT st
b5 in dom b2 & divset(b3,b4) c= divset(b2,b5);
:: INTEGRA2:th 38
theorem
for b1, b2 being non empty finite Element of bool REAL
st b1 c= b2
holds max b1 <= max b2;
:: INTEGRA2:th 39
theorem
for b1, b2 being non empty finite Element of bool REAL
st ex b3 being Element of REAL st
b3 in b2 & max b1 <= b3
holds max b1 <= max b2;
:: INTEGRA2:th 40
theorem
for b1, b2 being closed-interval Element of bool REAL
st b1 c= b2
holds vol b1 <= vol b2;
:: INTEGRA2:th 41
theorem
for b1 being closed-interval Element of bool REAL
for b2, b3 being Element of divs b1
st b2 <= b3
holds delta b3 <= delta b2;