Article TBSP_1, MML version 4.99.1005
:: TBSP_1:th 1
theorem
for b1 being Element of REAL
st 0 < b1 & b1 < 1
for b2, b3 being Element of NAT
st b2 <= b3
holds b1 to_power b3 <= b1 to_power b2;
:: TBSP_1:th 2
theorem
for b1 being Element of REAL
st 0 < b1 & b1 < 1
for b2 being Element of NAT holds
b1 to_power b2 <= 1 & 0 < b1 to_power b2;
:: TBSP_1:th 3
theorem
for b1 being Element of REAL
st 0 < b1 & b1 < 1
for b2 being Element of REAL
st 0 < b2
holds ex b3 being Element of NAT st
b1 to_power b3 < b2;
:: TBSP_1:attrnot 1 => TBSP_1:attr 1
definition
let a1 be non empty MetrStruct;
attr a1 is totally_bounded means
for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of bool bool the carrier of a1 st
b2 is finite &
the carrier of a1 = union b2 &
(for b3 being Element of bool the carrier of a1
st b3 in b2
holds ex b4 being Element of the carrier of a1 st
b3 = Ball(b4,b1));
end;
:: TBSP_1:dfs 1
definiens
let a1 be non empty MetrStruct;
To prove
a1 is totally_bounded
it is sufficient to prove
thus for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of bool bool the carrier of a1 st
b2 is finite &
the carrier of a1 = union b2 &
(for b3 being Element of bool the carrier of a1
st b3 in b2
holds ex b4 being Element of the carrier of a1 st
b3 = Ball(b4,b1));
:: TBSP_1:def 1
theorem
for b1 being non empty MetrStruct holds
b1 is totally_bounded
iff
for b2 being Element of REAL
st 0 < b2
holds ex b3 being Element of bool bool the carrier of b1 st
b3 is finite &
the carrier of b1 = union b3 &
(for b4 being Element of bool the carrier of b1
st b4 in b3
holds ex b5 being Element of the carrier of b1 st
b4 = Ball(b5,b2));
:: TBSP_1:th 5
theorem
for b1 being non empty MetrStruct
for b2 being Relation-like Function-like set holds
b2 is Function-like quasi_total Relation of NAT,the carrier of b1
iff
proj1 b2 = NAT &
(for b3 being Element of NAT holds
b2 . b3 is Element of the carrier of b1);
:: TBSP_1:attrnot 2 => TBSP_1:attr 2
definition
let a1 be non empty MetrStruct;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
attr a2 is convergent means
ex b1 being Element of the carrier of a1 st
for b2 being Element of REAL
st 0 < b2
holds ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds dist(a2 . b4,b1) < b2;
end;
:: TBSP_1:dfs 2
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 convergent
it is sufficient to prove
thus ex b1 being Element of the carrier of a1 st
for b2 being Element of REAL
st 0 < b2
holds ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds dist(a2 . b4,b1) < b2;
:: TBSP_1:def 3
theorem
for b1 being non empty MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is convergent(b1)
iff
ex b3 being Element of the carrier of b1 st
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;
:: TBSP_1:funcnot 1 => TBSP_1:func 1
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;
assume a2 is convergent(a1);
func lim A2 -> Element of the carrier of a1 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,it) < b1;
end;
:: TBSP_1:def 4
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)
for b3 being Element of the carrier of b1 holds
b3 = lim b2
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;
:: TBSP_1:attrnot 3 => TBSP_1:attr 3
definition
let a1 be non empty MetrStruct;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
attr a2 is Cauchy means
for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of NAT st
for b3, b4 being Element of NAT
st b2 <= b3 & b2 <= b4
holds dist(a2 . b3,a2 . b4) < b1;
end;
:: TBSP_1: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 Cauchy
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, b4 being Element of NAT
st b2 <= b3 & b2 <= b4
holds dist(a2 . b3,a2 . b4) < b1;
:: TBSP_1:def 5
theorem
for b1 being non empty MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is Cauchy(b1)
iff
for b3 being Element of REAL
st 0 < b3
holds ex b4 being Element of NAT st
for b5, b6 being Element of NAT
st b4 <= b5 & b4 <= b6
holds dist(b2 . b5,b2 . b6) < b3;
:: TBSP_1:attrnot 4 => TBSP_1:attr 4
definition
let a1 be non empty MetrStruct;
attr a1 is complete means
for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st b1 is Cauchy(a1)
holds b1 is convergent(a1);
end;
:: TBSP_1:dfs 5
definiens
let a1 be non empty MetrStruct;
To prove
a1 is complete
it is sufficient to prove
thus for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st b1 is Cauchy(a1)
holds b1 is convergent(a1);
:: TBSP_1:def 6
theorem
for b1 being non empty MetrStruct holds
b1 is complete
iff
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is Cauchy(b1)
holds b2 is convergent(b1);
:: TBSP_1:th 7
theorem
for b1 being non empty MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b1 is triangle & b1 is symmetric & b2 is convergent(b1)
holds b2 is Cauchy(b1);
:: TBSP_1:condreg 1
registration
let a1 be non empty symmetric triangle MetrStruct;
cluster Function-like quasi_total convergent -> Cauchy (Relation of NAT,the carrier of a1);
end;
:: TBSP_1:th 8
theorem
for b1 being non empty MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b1 is symmetric
holds b2 is Cauchy(b1)
iff
for b3 being Element of REAL
st 0 < b3
holds ex b4 being Element of NAT st
for b5, b6 being Element of NAT
st b4 <= b5
holds dist(b2 . (b5 + b6),b2 . b5) < b3;
:: TBSP_1:th 9
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being contraction of b1
st b1 is complete
holds ex b3 being Element of the carrier of b1 st
b2 . b3 = b3 &
(for b4 being Element of the carrier of b1
st b2 . b4 = b4
holds b4 = b3);
:: TBSP_1:th 10
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
st TopSpaceMetr b1 is compact
holds b1 is complete;
:: TBSP_1:th 12
theorem
for b1 being non empty MetrStruct
st b1 is Reflexive & b1 is triangle & TopSpaceMetr b1 is compact
holds b1 is totally_bounded;
:: TBSP_1:attrnot 5 => TBSP_1:attr 5
definition
let a1 be non empty MetrStruct;
attr a1 is bounded means
ex b1 being Element of REAL st
0 < b1 &
(for b2, b3 being Element of the carrier of a1 holds
dist(b2,b3) <= b1);
end;
:: TBSP_1:dfs 6
definiens
let a1 be non empty MetrStruct;
To prove
a1 is bounded
it is sufficient to prove
thus ex b1 being Element of REAL st
0 < b1 &
(for b2, b3 being Element of the carrier of a1 holds
dist(b2,b3) <= b1);
:: TBSP_1:def 8
theorem
for b1 being non empty MetrStruct holds
b1 is bounded
iff
ex b2 being Element of REAL st
0 < b2 &
(for b3, b4 being Element of the carrier of b1 holds
dist(b3,b4) <= b2);
:: TBSP_1:attrnot 6 => 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
0 < b1 &
(for b2, b3 being Element of the carrier of a1
st b2 in a2 & b3 in a2
holds dist(b2,b3) <= b1);
end;
:: TBSP_1:dfs 7
definiens
let a1 be non empty 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
0 < b1 &
(for b2, b3 being Element of the carrier of a1
st b2 in a2 & b3 in a2
holds dist(b2,b3) <= b1);
:: TBSP_1:def 9
theorem
for b1 being non empty MetrStruct
for b2 being Element of bool the carrier of b1 holds
b2 is bounded(b1)
iff
ex b3 being Element of REAL st
0 < b3 &
(for b4, b5 being Element of the carrier of b1
st b4 in b2 & b5 in b2
holds dist(b4,b5) <= b3);
:: TBSP_1:funcreg 1
registration
let a1 be non empty set;
cluster DiscreteSpace a1 -> strict bounded;
end;
:: TBSP_1:exreg 1
registration
cluster non empty Reflexive discerning symmetric triangle bounded MetrStruct;
end;
:: TBSP_1:th 14
theorem
for b1 being non empty MetrStruct holds
{} b1 is bounded(b1);
:: TBSP_1:funcreg 2
registration
let a1 be non empty MetrStruct;
cluster {} a1 -> bounded;
end;
:: TBSP_1:exreg 2
registration
let a1 be non empty MetrStruct;
cluster bounded Element of bool the carrier of a1;
end;
:: TBSP_1:th 15
theorem
for b1 being non empty MetrStruct
for b2 being Element of bool the carrier of b1 holds
(b2 <> {} & b2 is bounded(b1) implies ex b3 being Element of REAL st
ex b4 being Element of the carrier of b1 st
0 < b3 &
b4 in b2 &
(for b5 being Element of the carrier of b1
st b5 in b2
holds dist(b4,b5) <= b3)) &
(b1 is symmetric &
b1 is triangle &
(ex b3 being Element of REAL st
ex b4 being Element of the carrier of b1 st
0 < b3 &
b4 in b2 &
(for b5 being Element of the carrier of b1
st b5 in b2
holds dist(b4,b5) <= b3)) implies b2 is bounded(b1));
:: TBSP_1:th 16
theorem
for b1 being non empty MetrStruct
for b2 being Element of the carrier of b1
for b3 being real set
st b1 is Reflexive & 0 < b3
holds b2 in Ball(b2,b3);
:: TBSP_1:th 17
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being real set
st b3 <= 0
holds Ball(b2,b3) = {};
:: TBSP_1:th 19
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being real set holds
Ball(b2,b3) is bounded(b1);
:: TBSP_1:th 20
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is bounded(b1) & b3 is bounded(b1)
holds b2 \/ b3 is bounded(b1);
:: TBSP_1:th 21
theorem
for b1 being non empty MetrStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is bounded(b1) & b3 c= b2
holds b3 is bounded(b1);
:: TBSP_1:th 22
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
st b3 = {b2}
holds b3 is bounded(b1);
:: TBSP_1:th 23
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2 being Element of bool the carrier of b1
st b2 is finite
holds b2 is bounded(b1);
:: TBSP_1:condreg 2
registration
let a1 be non empty Reflexive symmetric triangle MetrStruct;
cluster finite -> bounded (Element of bool the carrier of a1);
end;
:: TBSP_1:exreg 3
registration
let a1 be non empty Reflexive symmetric triangle MetrStruct;
cluster non empty finite Element of bool the carrier of a1;
end;
:: TBSP_1:th 24
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2 being Element of bool bool the carrier of b1
st b2 is finite &
(for b3 being Element of bool the carrier of b1
st b3 in b2
holds b3 is bounded(b1))
holds union b2 is bounded(b1);
:: TBSP_1:th 25
theorem
for b1 being non empty MetrStruct holds
b1 is bounded
iff
[#] b1 is bounded(b1);
:: TBSP_1:funcreg 3
registration
let a1 be non empty bounded MetrStruct;
cluster [#] a1 -> bounded;
end;
:: TBSP_1:th 26
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
st b1 is totally_bounded
holds b1 is bounded;
:: TBSP_1:funcnot 2 => TBSP_1:func 2
definition
let a1 be non empty Reflexive MetrStruct;
let a2 be Element of bool the carrier of a1;
assume a2 is bounded(a1);
func diameter A2 -> Element of REAL means
(for b1, b2 being Element of the carrier of a1
st b1 in a2 & b2 in a2
holds dist(b1,b2) <= it) &
(for b1 being Element of REAL
st for b2, b3 being Element of the carrier of a1
st b2 in a2 & b3 in a2
holds dist(b2,b3) <= b1
holds it <= b1)
if a2 <> {}
otherwise it = 0;
end;
:: TBSP_1:def 10
theorem
for b1 being non empty Reflexive MetrStruct
for b2 being Element of bool the carrier of b1
st b2 is bounded(b1)
for b3 being Element of REAL holds
(b2 = {} or (b3 = diameter b2
iff
(for b4, b5 being Element of the carrier of b1
st b4 in b2 & b5 in b2
holds dist(b4,b5) <= b3) &
(for b4 being Element of REAL
st for b5, b6 being Element of the carrier of b1
st b5 in b2 & b6 in b2
holds dist(b5,b6) <= b4
holds b3 <= b4))) &
(b2 = {} implies (b3 = diameter b2
iff
b3 = 0));
:: TBSP_1:th 28
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2 being set
for b3 being Element of bool the carrier of b1
st b3 = {b2}
holds diameter b3 = 0;
:: TBSP_1:th 29
theorem
for b1 being non empty Reflexive MetrStruct
for b2 being Element of bool the carrier of b1
st b2 is bounded(b1)
holds 0 <= diameter b2;
:: TBSP_1:th 30
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of bool the carrier of b1
st b2 <> {} & b2 is bounded(b1) & diameter b2 = 0
holds ex b3 being Element of the carrier of b1 st
b2 = {b3};
:: TBSP_1:th 31
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Element of REAL
st 0 < b3
holds diameter Ball(b2,b3) <= 2 * b3;
:: TBSP_1:th 32
theorem
for b1 being non empty Reflexive MetrStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is bounded(b1) & b3 c= b2
holds diameter b3 <= diameter b2;
:: TBSP_1:th 33
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is bounded(b1) & b3 is bounded(b1) & b2 meets b3
holds diameter (b2 \/ b3) <= (diameter b2) + diameter b3;
:: TBSP_1:th 34
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is Cauchy(b1)
holds rng b2 is bounded(b1);