Article INTEGRA4, MML version 4.99.1005
:: INTEGRA4:th 1
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of divs b1
st vol b1 = 0
holds len b2 = 1;
:: INTEGRA4:th 2
theorem
for b1 being closed-interval Element of bool REAL holds
chi(b1,b1) is_integrable_on b1 & integral chi(b1,b1) = vol b1;
:: INTEGRA4:th 3
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of b1,REAL
for b3 being Element of REAL holds
b2 is total(b1, REAL) & rng b2 = {b3}
iff
b2 = b3 (#) chi(b1,b1);
:: INTEGRA4:th 4
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
for b3 being Element of REAL
st rng b2 = {b3}
holds b2 is_integrable_on b1 & integral b2 = b3 * vol b1;
:: INTEGRA4:th 5
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of REAL holds
ex b3 being Function-like quasi_total Relation of b1,REAL st
rng b3 = {b2} & b3 is_bounded_on b1;
:: INTEGRA4:th 6
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of b1,REAL
for b3 being Element of divs b1
st vol b1 = 0
holds b2 is_integrable_on b1 & integral b2 = 0;
:: INTEGRA4:th 7
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
holds ex b3 being Element of REAL st
inf rng b2 <= b3 & b3 <= sup rng b2 & integral b2 = b3 * vol b1;
:: INTEGRA4:th 8
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
for b3 being Function-like quasi_total Relation of NAT,divs b1
st b2 is_bounded_on b1 & delta b3 is convergent & lim delta b3 = 0
holds lower_sum(b2,b3) is convergent & lim lower_sum(b2,b3) = lower_integral b2;
:: INTEGRA4:th 9
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
for b3 being Function-like quasi_total Relation of NAT,divs b1
st b2 is_bounded_on b1 & delta b3 is convergent & lim delta b3 = 0
holds upper_sum(b2,b3) is convergent & lim upper_sum(b2,b3) = upper_integral b2;
:: INTEGRA4:th 10
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 b2 is_upper_integrable_on b1 & b2 is_lower_integrable_on b1;
:: INTEGRA4:prednot 1 => INTEGRA4:pred 1
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Element of divs a1;
let a3 be Element of NAT;
pred A2 divide_into_equal A3 means
len a2 = a3 &
(for b1 being Element of NAT
st b1 in dom a2
holds a2 . b1 = (inf a1) + (((vol a1) / len a2) * b1));
end;
:: INTEGRA4:dfs 1
definiens
let a1 be closed-interval Element of bool REAL;
let a2 be Element of divs a1;
let a3 be Element of NAT;
To prove
a2 divide_into_equal a3
it is sufficient to prove
thus len a2 = a3 &
(for b1 being Element of NAT
st b1 in dom a2
holds a2 . b1 = (inf a1) + (((vol a1) / len a2) * b1));
:: INTEGRA4:def 1
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of divs b1
for b3 being Element of NAT holds
b2 divide_into_equal b3
iff
len b2 = b3 &
(for b4 being Element of NAT
st b4 in dom b2
holds b2 . b4 = (inf b1) + (((vol b1) / len b2) * b4));
:: INTEGRA4:th 11
theorem
for b1 being closed-interval Element of bool REAL holds
ex b2 being Function-like quasi_total Relation of NAT,divs b1 st
delta b2 is convergent & lim delta b2 = 0;
:: INTEGRA4:th 12
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 b2 is_integrable_on b1
iff
for b3 being Function-like quasi_total Relation of NAT,divs b1
st delta b3 is convergent & lim delta b3 = 0
holds (lim upper_sum(b2,b3)) - lim lower_sum(b2,b3) = 0;
:: INTEGRA4:th 13
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of b1,REAL holds
max+ b2 is total(b1, REAL) & max- b2 is total(b1, REAL);
:: INTEGRA4:th 14
theorem
for b1 being non empty set
for b2 being set
for b3 being Function-like Relation of b1,REAL
st b3 is_bounded_above_on b2
holds max+ b3 is_bounded_above_on b2;
:: INTEGRA4:th 15
theorem
for b1 being non empty set
for b2 being set
for b3 being Function-like Relation of b1,REAL holds
max+ b3 is_bounded_below_on b2;
:: INTEGRA4:th 16
theorem
for b1 being non empty set
for b2 being set
for b3 being Function-like Relation of b1,REAL
st b3 is_bounded_below_on b2
holds max- b3 is_bounded_above_on b2;
:: INTEGRA4:th 17
theorem
for b1 being non empty set
for b2 being set
for b3 being Function-like Relation of b1,REAL holds
max- b3 is_bounded_below_on b2;
:: INTEGRA4:th 18
theorem
for b1 being closed-interval Element of bool REAL
for b2 being set
for b3 being Function-like Relation of b1,REAL
st b3 is_bounded_above_on b1
holds rng (b3 | b2) is bounded_above;
:: INTEGRA4:th 19
theorem
for b1 being closed-interval Element of bool REAL
for b2 being set
for b3 being Function-like Relation of b1,REAL
st b3 is_bounded_below_on b1
holds rng (b3 | b2) is bounded_below;
:: INTEGRA4:th 20
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
holds max+ b2 is_integrable_on b1;
:: INTEGRA4:th 21
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
max- b2 = max+ - b2;
:: INTEGRA4:th 22
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
holds max- b2 is_integrable_on b1;
:: INTEGRA4:th 23
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
holds abs b2 is_integrable_on b1 & abs integral b2 <= integral abs b2;
:: INTEGRA4: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
st for b4, b5 being Element of REAL
st b4 in b2 & b5 in b2
holds abs ((b3 . b4) - (b3 . b5)) <= b1
holds (sup rng b3) - inf rng b3 <= b1;
:: INTEGRA4:th 25
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3, b4 being Function-like quasi_total Relation of b2,REAL
st b3 is_bounded_on b2 &
0 <= b1 &
(for b5, b6 being Element of REAL
st b5 in b2 & b6 in b2
holds abs ((b4 . b5) - (b4 . b6)) <= b1 * abs ((b3 . b5) - (b3 . b6)))
holds (sup rng b4) - inf rng b4 <= b1 * ((sup rng b3) - inf rng b3);
:: INTEGRA4:th 26
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3, b4, b5 being Function-like quasi_total Relation of b2,REAL
st b3 is_bounded_on b2 &
b4 is_bounded_on b2 &
0 <= b1 &
(for b6, b7 being Element of REAL
st b6 in b2 & b7 in b2
holds abs ((b5 . b6) - (b5 . b7)) <= b1 * ((abs ((b3 . b6) - (b3 . b7))) + abs ((b4 . b6) - (b4 . b7))))
holds (sup rng b5) - inf rng b5 <= b1 * (((sup rng b3) - inf rng b3) + ((sup rng b4) - inf rng b4));
:: INTEGRA4:th 27
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3, b4 being Function-like quasi_total Relation of b2,REAL
st b3 is_bounded_on b2 &
b3 is_integrable_on b2 &
b4 is_bounded_on b2 &
0 < b1 &
(for b5, b6 being Element of REAL
st b5 in b2 & b6 in b2
holds abs ((b4 . b5) - (b4 . b6)) <= b1 * abs ((b3 . b5) - (b3 . b6)))
holds b4 is_integrable_on b2;
:: INTEGRA4:th 28
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3, b4, b5 being Function-like quasi_total Relation of b2,REAL
st b3 is_bounded_on b2 &
b3 is_integrable_on b2 &
b4 is_bounded_on b2 &
b4 is_integrable_on b2 &
b5 is_bounded_on b2 &
0 < b1 &
(for b6, b7 being Element of REAL
st b6 in b2 & b7 in b2
holds abs ((b5 . b6) - (b5 . b7)) <= b1 * ((abs ((b3 . b6) - (b3 . b7))) + abs ((b4 . b6) - (b4 . b7))))
holds b5 is_integrable_on b2;
:: INTEGRA4:th 29
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;
:: INTEGRA4:th 30
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 & not 0 in rng b2 & b2 ^ is_bounded_on b1
holds b2 ^ is_integrable_on b1;