Article METRIC_6, MML version 4.99.1005
:: METRIC_6:th 1
theorem
for b1 being Reflexive discerning symmetric triangle MetrStruct
for b2, b3, b4 being Element of the carrier of b1 holds
abs ((dist(b2,b3)) - dist(b4,b3)) <= dist(b2,b4);
:: METRIC_6:th 2
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL
st b2 is_metric_of b1
for b3, b4 being Element of b1 holds
0 <= b2 .(b3,b4);
:: METRIC_6:th 3
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL holds
b2 is_metric_of b1
iff
b2 is Reflexive(b1) & b2 is discerning(b1) & b2 is symmetric(b1) & b2 is triangle(b1);
:: METRIC_6:th 4
theorem
for b1 being non empty strict Reflexive discerning symmetric triangle MetrStruct holds
the distance of b1 is Reflexive(the carrier of b1) & the distance of b1 is discerning(the carrier of b1) & the distance of b1 is symmetric(the carrier of b1) & the distance of b1 is triangle(the carrier of b1);
:: METRIC_6:th 5
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL holds
b2 is_metric_of b1
iff
b2 is Reflexive(b1) &
b2 is discerning(b1) &
(for b3, b4, b5 being Element of b1 holds
b2 .(b4,b5) <= (b2 .(b3,b4)) + (b2 .(b3,b5)));
:: METRIC_6:funcnot 1 => METRIC_6:func 1
definition
let a1 be non empty set;
let a2 be Function-like quasi_total Relation of [:a1,a1:],REAL;
func bounded_metric(A1,A2) -> Function-like quasi_total Relation of [:a1,a1:],REAL means
for b1, b2 being Element of a1 holds
it .(b1,b2) = (a2 .(b1,b2)) / (1 + (a2 .(b1,b2)));
end;
:: METRIC_6:def 4
theorem
for b1 being non empty set
for b2, b3 being Function-like quasi_total Relation of [:b1,b1:],REAL holds
b3 = bounded_metric(b1,b2)
iff
for b4, b5 being Element of b1 holds
b3 .(b4,b5) = (b2 .(b4,b5)) / (1 + (b2 .(b4,b5)));
:: METRIC_6:th 6
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL
st b2 is_metric_of b1
holds bounded_metric(b1,b2) is_metric_of b1;
:: METRIC_6:th 10
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1 holds
ex b3 being Function-like quasi_total Relation of NAT,the carrier of b1 st
proj2 b3 = {b2};
:: METRIC_6:prednot 1 => METRIC_6:pred 1
definition
let a1 be non empty MetrStruct;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
let a3 be Element of the carrier of a1;
pred A2 is_convergent_in_metrspace_to A3 means
for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of NAT st
for b3 being Element of NAT
st b2 <= b3
holds dist(a2 . b3,a3) < b1;
end;
:: METRIC_6:dfs 2
definiens
let a1 be non empty MetrStruct;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
let a3 be Element of the carrier of a1;
To prove
a2 is_convergent_in_metrspace_to a3
it is sufficient to prove
thus for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of NAT st
for b3 being Element of NAT
st b2 <= b3
holds dist(a2 . b3,a3) < b1;
:: METRIC_6:def 8
theorem
for b1 being non empty MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1 holds
b2 is_convergent_in_metrspace_to b3
iff
for b4 being Element of REAL
st 0 < b4
holds ex b5 being Element of NAT st
for b6 being Element of NAT
st b5 <= b6
holds dist(b2 . b6,b3) < b4;
:: METRIC_6:attrnot 1 => TBSP_1:attr 6
definition
let a1 be non empty MetrStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is bounded means
ex b1 being Element of REAL st
ex b2 being Element of the carrier of a1 st
0 < b1 & a2 c= Ball(b2,b1);
end;
:: METRIC_6:dfs 3
definiens
let a1 be non empty symmetric triangle MetrStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is bounded
it is sufficient to prove
thus ex b1 being Element of REAL st
ex b2 being Element of the carrier of a1 st
0 < b1 & a2 c= Ball(b2,b1);
:: METRIC_6:def 10
theorem
for b1 being non empty symmetric triangle MetrStruct
for b2 being Element of bool the carrier of b1 holds
b2 is bounded(b1)
iff
ex b3 being Element of REAL st
ex b4 being Element of the carrier of b1 st
0 < b3 & b2 c= Ball(b4,b3);
:: METRIC_6:attrnot 2 => METRIC_6:attr 1
definition
let a1 be non empty MetrStruct;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
attr a2 is bounded means
ex b1 being Element of REAL st
ex b2 being Element of the carrier of a1 st
0 < b1 & proj2 a2 c= Ball(b2,b1);
end;
:: METRIC_6:dfs 4
definiens
let a1 be non empty MetrStruct;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
a2 is bounded
it is sufficient to prove
thus ex b1 being Element of REAL st
ex b2 being Element of the carrier of a1 st
0 < b1 & proj2 a2 c= Ball(b2,b1);
:: METRIC_6:def 11
theorem
for b1 being non empty MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is bounded(b1)
iff
ex b3 being Element of REAL st
ex b4 being Element of the carrier of b1 st
0 < b3 & proj2 b2 c= Ball(b4,b3);
:: METRIC_6:prednot 2 => METRIC_6:pred 2
definition
let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
let a2 be Element of bool the carrier of a1;
let a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
pred A2 contains_almost_all_sequence A3 means
ex b1 being Element of NAT st
for b2 being Element of NAT
st b1 <= b2
holds a3 . b2 in a2;
end;
:: METRIC_6:dfs 5
definiens
let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
let a2 be Element of bool the carrier of a1;
let a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
a2 contains_almost_all_sequence a3
it is sufficient to prove
thus ex b1 being Element of NAT st
for b2 being Element of NAT
st b1 <= b2
holds a3 . b2 in a2;
:: METRIC_6:def 12
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of bool the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 contains_almost_all_sequence b3
iff
ex b4 being Element of NAT st
for b5 being Element of NAT
st b4 <= b5
holds b3 . b5 in b2;
:: METRIC_6:th 20
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is bounded(b1)
iff
ex b3 being Element of REAL st
ex b4 being Element of the carrier of b1 st
0 < b3 &
(for b5 being Element of NAT holds
b2 . b5 in Ball(b4,b3));
:: METRIC_6:th 21
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is_convergent_in_metrspace_to b2
holds b3 is convergent(b1);
:: METRIC_6:th 22
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1)
holds ex b3 being Element of the carrier of b1 st
b2 is_convergent_in_metrspace_to b3;
:: METRIC_6:funcnot 2 => METRIC_6:func 2
definition
let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
let a3 be Element of the carrier of a1;
func dist_to_point(A2,A3) -> Function-like quasi_total Relation of NAT,REAL means
for b1 being Element of NAT holds
it . b1 = dist(a2 . b1,a3);
end;
:: METRIC_6:def 14
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,REAL holds
b4 = dist_to_point(b2,b3)
iff
for b5 being Element of NAT holds
b4 . b5 = dist(b2 . b5,b3);
:: METRIC_6:funcnot 3 => METRIC_6:func 3
definition
let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
let a2, a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
func sequence_of_dist(A2,A3) -> Function-like quasi_total Relation of NAT,REAL means
for b1 being Element of NAT holds
it . b1 = dist(a2 . b1,a3 . b1);
end;
:: METRIC_6:def 15
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,REAL holds
b4 = sequence_of_dist(b2,b3)
iff
for b5 being Element of NAT holds
b4 . b5 = dist(b2 . b5,b3 . b5);
:: METRIC_6:th 26
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is_convergent_in_metrspace_to b2
holds lim b3 = b2;
:: METRIC_6:th 27
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b3 is_convergent_in_metrspace_to b2
iff
b3 is convergent(b1) & lim b3 = b2;
:: METRIC_6:th 28
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1)
holds ex b3 being Element of the carrier of b1 st
b2 is_convergent_in_metrspace_to b3 & lim b2 = b3;
:: METRIC_6:th 29
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b3 is_convergent_in_metrspace_to b2
iff
dist_to_point(b3,b2) is convergent & lim dist_to_point(b3,b2) = 0;
:: METRIC_6:th 30
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is_convergent_in_metrspace_to b2
for b4 being Element of REAL
st 0 < b4
holds Ball(b2,b4) contains_almost_all_sequence b3;
:: METRIC_6:th 31
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st for b4 being Element of REAL
st 0 < b4
holds Ball(b2,b4) contains_almost_all_sequence b3
for b4 being Element of bool the carrier of b1
st b2 in b4 & b4 in Family_open_set b1
holds b4 contains_almost_all_sequence b3;
:: METRIC_6:th 32
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st for b4 being Element of bool the carrier of b1
st b2 in b4 & b4 in Family_open_set b1
holds b4 contains_almost_all_sequence b3
holds b3 is_convergent_in_metrspace_to b2;
:: METRIC_6:th 33
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b3 is_convergent_in_metrspace_to b2
iff
for b4 being Element of REAL
st 0 < b4
holds Ball(b2,b4) contains_almost_all_sequence b3;
:: METRIC_6:th 34
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b3 is_convergent_in_metrspace_to b2
iff
for b4 being Element of bool the carrier of b1
st b2 in b4 & b4 in Family_open_set b1
holds b4 contains_almost_all_sequence b3;
:: METRIC_6:th 35
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
for b4 being Element of REAL
st 0 < b4
holds Ball(b2,b4) contains_almost_all_sequence b3
iff
for b4 being Element of bool the carrier of b1
st b2 in b4 & b4 in Family_open_set b1
holds b4 contains_almost_all_sequence b3;
:: METRIC_6:th 36
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1) & b3 is convergent(b1)
holds dist(lim b2,lim b3) = lim sequence_of_dist(b2,b3);
:: METRIC_6:th 37
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st b4 is_convergent_in_metrspace_to b2 & b4 is_convergent_in_metrspace_to b3
holds b2 = b3;
:: METRIC_6:th 38
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is constant
holds b2 is convergent(b1);
:: METRIC_6:th 39
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is_convergent_in_metrspace_to b2 & b4 is subsequence of b3
holds b4 is_convergent_in_metrspace_to b2;
:: METRIC_6:th 40
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is Cauchy(b1) & b3 is subsequence of b2
holds b3 is Cauchy(b1);
:: METRIC_6:th 42
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is constant
holds b2 is Cauchy(b1);
:: METRIC_6:th 43
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1)
holds b2 is bounded(b1);
:: METRIC_6:th 44
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is Cauchy(b1)
holds b2 is bounded(b1);
:: METRIC_6:condreg 1
registration
let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
cluster Function-like constant quasi_total -> convergent (Relation of NAT,the carrier of a1);
end;
:: METRIC_6:condreg 2
registration
let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
cluster Function-like quasi_total Cauchy -> bounded (Relation of NAT,the carrier of a1);
end;
:: METRIC_6:exreg 1
registration
let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
cluster Relation-like Function-like constant quasi_total convergent Cauchy bounded Relation of NAT,the carrier of a1;
end;