Article FCONT_2, MML version 4.99.1005

:: FCONT_2:prednot 1 => FCONT_2:pred 1
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be set;
  pred A1 is_uniformly_continuous_on A2 means
    a2 c= dom a1 &
     (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 REAL
                  st b3 in a2 & b4 in a2 & abs (b3 - b4) < b2
               holds abs ((a1 . b3) - (a1 . b4)) < b1));
end;

:: FCONT_2:dfs 1
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be set;
To prove
     a1 is_uniformly_continuous_on a2
it is sufficient to prove
  thus a2 c= dom a1 &
     (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 REAL
                  st b3 in a2 & b4 in a2 & abs (b3 - b4) < b2
               holds abs ((a1 . b3) - (a1 . b4)) < b1));

:: FCONT_2:def 1
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set holds
      b1 is_uniformly_continuous_on b2
   iff
      b2 c= dom b1 &
       (for b3 being Element of REAL
             st 0 < b3
          holds ex b4 being Element of REAL st
             0 < b4 &
              (for b5, b6 being Element of REAL
                    st b5 in b2 & b6 in b2 & abs (b5 - b6) < b4
                 holds abs ((b1 . b5) - (b1 . b6)) < b3));

:: FCONT_2:th 2
theorem
for b1, b2 being set
for b3 being Function-like Relation of REAL,REAL
      st b3 is_uniformly_continuous_on b1 & b2 c= b1
   holds b3 is_uniformly_continuous_on b2;

:: FCONT_2:th 3
theorem
for b1, b2 being set
for b3, b4 being Function-like Relation of REAL,REAL
      st b3 is_uniformly_continuous_on b1 & b4 is_uniformly_continuous_on b2
   holds b3 + b4 is_uniformly_continuous_on b1 /\ b2;

:: FCONT_2:th 4
theorem
for b1, b2 being set
for b3, b4 being Function-like Relation of REAL,REAL
      st b3 is_uniformly_continuous_on b1 & b4 is_uniformly_continuous_on b2
   holds b3 - b4 is_uniformly_continuous_on b1 /\ b2;

:: FCONT_2:th 5
theorem
for b1 being set
for b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
      st b3 is_uniformly_continuous_on b1
   holds b2 (#) b3 is_uniformly_continuous_on b1;

:: FCONT_2:th 6
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b2 is_uniformly_continuous_on b1
   holds - b2 is_uniformly_continuous_on b1;

:: FCONT_2:th 7
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b2 is_uniformly_continuous_on b1
   holds abs b2 is_uniformly_continuous_on b1;

:: FCONT_2:th 8
theorem
for b1, b2, b3, b4 being set
for b5, b6 being Function-like Relation of REAL,REAL
      st b5 is_uniformly_continuous_on b1 & b6 is_uniformly_continuous_on b2 & b5 is_bounded_on b3 & b6 is_bounded_on b4
   holds b5 (#) b6 is_uniformly_continuous_on ((b1 /\ b3) /\ b2) /\ b4;

:: FCONT_2:th 9
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b2 is_uniformly_continuous_on b1
   holds b2 is_continuous_on b1;

:: FCONT_2:th 10
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b2 is_Lipschitzian_on b1
   holds b2 is_uniformly_continuous_on b1;

:: FCONT_2:th 11
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of bool REAL
      st b2 is compact & b1 is_continuous_on b2
   holds b1 is_uniformly_continuous_on b2;

:: FCONT_2:th 13
theorem
for b1 being Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= dom b2 & b1 is compact & b2 is_uniformly_continuous_on b1
   holds b2 .: b1 is compact;

:: FCONT_2:th 14
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of bool REAL
      st b2 <> {} & b2 c= dom b1 & b2 is compact & b1 is_uniformly_continuous_on b2
   holds ex b3, b4 being Element of REAL st
      b3 in b2 & b4 in b2 & b1 . b3 = upper_bound (b1 .: b2) & b1 . b4 = lower_bound (b1 .: b2);

:: FCONT_2:th 15
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b1 c= dom b2 & b2 is_constant_on b1
   holds b2 is_uniformly_continuous_on b1;

:: FCONT_2:th 16
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
   st b1 <= b2 & b3 is_continuous_on [.b1,b2.]
for b4 being Element of REAL
      st b4 in [.b3 . b1,b3 . b2.] \/ [.b3 . b2,b3 . b1.]
   holds ex b5 being Element of REAL st
      b5 in [.b1,b2.] & b4 = b3 . b5;

:: FCONT_2:th 17
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
   st b1 <= b2 & b3 is_continuous_on [.b1,b2.]
for b4 being Element of REAL
      st b4 in [.lower_bound (b3 .: [.b1,b2.]),upper_bound (b3 .: [.b1,b2.]).]
   holds ex b5 being Element of REAL st
      b5 in [.b1,b2.] & b4 = b3 . b5;

:: FCONT_2:th 18
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
      st b3 is one-to-one & b1 <= b2 & b3 is_continuous_on [.b1,b2.] & not b3 is_increasing_on [.b1,b2.]
   holds b3 is_decreasing_on [.b1,b2.];

:: FCONT_2:th 19
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
      st b3 is one-to-one &
         b1 <= b2 &
         b3 is_continuous_on [.b1,b2.] &
         (lower_bound (b3 .: [.b1,b2.]) = b3 . b1 implies upper_bound (b3 .: [.b1,b2.]) <> b3 . b2)
   holds lower_bound (b3 .: [.b1,b2.]) = b3 . b2 &
    upper_bound (b3 .: [.b1,b2.]) = b3 . b1;

:: FCONT_2:th 20
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
      st b1 <= b2 & b3 is_continuous_on [.b1,b2.]
   holds b3 .: [.b1,b2.] = [.lower_bound (b3 .: [.b1,b2.]),upper_bound (b3 .: [.b1,b2.]).];

:: FCONT_2:th 21
theorem
for b1, b2 being Element of REAL
for b3 being Function-like one-to-one Relation of REAL,REAL
      st b1 <= b2 & b3 is_continuous_on [.b1,b2.]
   holds b3 " is_continuous_on [.lower_bound (b3 .: [.b1,b2.]),upper_bound (b3 .: [.b1,b2.]).];