Article NFCONT_2, MML version 4.99.1005
:: NFCONT_2:prednot 1 => NFCONT_2:pred 1
definition
let a1 be set;
let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a4 be Function-like Relation of the carrier of a2,the carrier of a3;
pred A4 is_uniformly_continuous_on A1 means
a1 c= dom a4 &
(for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of REAL st
0 < b2 &
(for b3, b4 being Element of the carrier of a2
st b3 in a1 & b4 in a1 & ||.b3 - b4.|| < b2
holds ||.(a4 /. b3) - (a4 /. b4).|| < b1));
end;
:: NFCONT_2:dfs 1
definiens
let a1 be set;
let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a4 be Function-like Relation of the carrier of a2,the carrier of a3;
To prove
a4 is_uniformly_continuous_on a1
it is sufficient to prove
thus a1 c= dom a4 &
(for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of REAL st
0 < b2 &
(for b3, b4 being Element of the carrier of a2
st b3 in a1 & b4 in a1 & ||.b3 - b4.|| < b2
holds ||.(a4 /. b3) - (a4 /. b4).|| < b1));
:: NFCONT_2:def 1
theorem
for b1 being set
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3 holds
b4 is_uniformly_continuous_on b1
iff
b1 c= dom b4 &
(for b5 being Element of REAL
st 0 < b5
holds ex b6 being Element of REAL st
0 < b6 &
(for b7, b8 being Element of the carrier of b2
st b7 in b1 & b8 in b1 & ||.b7 - b8.|| < b6
holds ||.(b4 /. b7) - (b4 /. b8).|| < b5));
:: NFCONT_2:prednot 2 => NFCONT_2:pred 2
definition
let a1 be set;
let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a3 be Function-like Relation of the carrier of a2,REAL;
pred A3 is_uniformly_continuous_on A1 means
a1 c= dom a3 &
(for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of REAL st
0 < b2 &
(for b3, b4 being Element of the carrier of a2
st b3 in a1 & b4 in a1 & ||.b3 - b4.|| < b2
holds abs ((a3 /. b3) - (a3 /. b4)) < b1));
end;
:: NFCONT_2:dfs 2
definiens
let a1 be set;
let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a3 be Function-like Relation of the carrier of a2,REAL;
To prove
a3 is_uniformly_continuous_on a1
it is sufficient to prove
thus a1 c= dom a3 &
(for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of REAL st
0 < b2 &
(for b3, b4 being Element of the carrier of a2
st b3 in a1 & b4 in a1 & ||.b3 - b4.|| < b2
holds abs ((a3 /. b3) - (a3 /. b4)) < b1));
:: NFCONT_2:def 2
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,REAL holds
b3 is_uniformly_continuous_on b1
iff
b1 c= dom b3 &
(for b4 being Element of REAL
st 0 < b4
holds ex b5 being Element of REAL st
0 < b5 &
(for b6, b7 being Element of the carrier of b2
st b6 in b1 & b7 in b1 & ||.b6 - b7.|| < b5
holds abs ((b3 /. b6) - (b3 /. b7)) < b4));
:: NFCONT_2:th 1
theorem
for b1, b2 being set
for b3, b4 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b5 being Function-like Relation of the carrier of b3,the carrier of b4
st b5 is_uniformly_continuous_on b1 & b2 c= b1
holds b5 is_uniformly_continuous_on b2;
:: NFCONT_2:th 2
theorem
for b1, b2 being set
for b3, b4 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b5, b6 being Function-like Relation of the carrier of b3,the carrier of b4
st b5 is_uniformly_continuous_on b1 & b6 is_uniformly_continuous_on b2
holds b5 + b6 is_uniformly_continuous_on b1 /\ b2;
:: NFCONT_2:th 3
theorem
for b1, b2 being set
for b3, b4 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b5, b6 being Function-like Relation of the carrier of b3,the carrier of b4
st b5 is_uniformly_continuous_on b1 & b6 is_uniformly_continuous_on b2
holds b5 - b6 is_uniformly_continuous_on b1 /\ b2;
:: NFCONT_2:th 4
theorem
for b1 being set
for b2 being Element of REAL
for b3, b4 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b5 being Function-like Relation of the carrier of b3,the carrier of b4
st b5 is_uniformly_continuous_on b1
holds b2 (#) b5 is_uniformly_continuous_on b1;
:: NFCONT_2:th 5
theorem
for b1 being set
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
st b4 is_uniformly_continuous_on b1
holds - b4 is_uniformly_continuous_on b1;
:: NFCONT_2:th 6
theorem
for b1 being set
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
st b4 is_uniformly_continuous_on b1
holds ||.b4.|| is_uniformly_continuous_on b1;
:: NFCONT_2:th 7
theorem
for b1 being set
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
st b4 is_uniformly_continuous_on b1
holds b4 is_continuous_on b1;
:: NFCONT_2:th 8
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,REAL
st b3 is_uniformly_continuous_on b1
holds b3 is_continuous_on b1;
:: NFCONT_2:th 9
theorem
for b1 being set
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
st b4 is_Lipschitzian_on b1
holds b4 is_uniformly_continuous_on b1;
:: NFCONT_2:th 10
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4 being Element of bool the carrier of b2
st b4 is compact(b2) & b3 is_continuous_on b4
holds b3 is_uniformly_continuous_on b4;
:: NFCONT_2:th 11
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4 being Element of bool the carrier of b2
st b4 c= dom b3 & b4 is compact(b2) & b3 is_uniformly_continuous_on b4
holds b3 .: b4 is compact(b1);
:: NFCONT_2:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
for b3 being Element of bool the carrier of b1
st b3 <> {} & b3 c= dom b2 & b3 is compact(b1) & b2 is_uniformly_continuous_on b3
holds ex b4, b5 being Element of the carrier of b1 st
b4 in b3 & b5 in b3 & b2 /. b4 = upper_bound (b2 .: b3) & b2 /. b5 = lower_bound (b2 .: b3);
:: NFCONT_2:th 13
theorem
for b1 being set
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
st b1 c= dom b4 & b4 is_constant_on b1
holds b4 is_uniformly_continuous_on b1;
:: NFCONT_2:modenot 1 => NFCONT_2:mode 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR;
mode contraction of A1 -> Function-like quasi_total Relation of the carrier of a1,the carrier of a1 means
ex b1 being Element of REAL st
0 < b1 &
b1 < 1 &
(for b2, b3 being Element of the carrier of a1 holds
||.(it . b2) - (it . b3).|| <= b1 * ||.b2 - b3.||);
end;
:: NFCONT_2:dfs 3
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR;
let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
To prove
a2 is contraction of a1
it is sufficient to prove
thus ex b1 being Element of REAL st
0 < b1 &
b1 < 1 &
(for b2, b3 being Element of the carrier of a1 holds
||.(a2 . b2) - (a2 . b3).|| <= b1 * ||.b2 - b3.||);
:: NFCONT_2:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
b2 is contraction of b1
iff
ex b3 being Element of REAL st
0 < b3 &
b3 < 1 &
(for b4, b5 being Element of the carrier of b1 holds
||.(b2 . b4) - (b2 . b5).|| <= b3 * ||.b4 - b5.||);
:: NFCONT_2:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
st b2 is contraction of b1
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 b3 = b4);
:: NFCONT_2:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
st ex b3 being Element of NAT st
iter(b2,b3) is contraction of b1
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 b3 = b4);
:: NFCONT_2:th 16
theorem
for b1, b2, b3 being real set
st 0 < b1 & b1 < 1 & 0 < b3
holds ex b4 being Element of NAT st
abs (b2 * (b1 to_power b4)) < b3;