Article FDIFF_1, MML version 4.99.1005

:: FDIFF_1:th 1
theorem
for b1 being Element of bool REAL holds
      for b2 being Element of REAL holds
            b2 in b1
         iff
            b2 in REAL
   iff
      b1 = REAL;

:: FDIFF_1:attrnot 1 => FDIFF_1:attr 1
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  attr a1 is convergent_to_0 means
    a1 is non-empty & a1 is convergent & lim a1 = 0;
end;

:: FDIFF_1:dfs 1
definiens
  let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
     a1 is convergent_to_0
it is sufficient to prove
  thus a1 is non-empty & a1 is convergent & lim a1 = 0;

:: FDIFF_1:def 1
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is convergent_to_0
   iff
      b1 is non-empty & b1 is convergent & lim b1 = 0;

:: FDIFF_1:exreg 1
registration
  cluster Relation-like Function-like non empty total complex-valued ext-real-valued real-valued quasi_total convergent_to_0 Relation of NAT,REAL;
end;

:: FDIFF_1:exreg 2
registration
  cluster Relation-like Function-like constant non empty total complex-valued ext-real-valued real-valued quasi_total Relation of NAT,REAL;
end;

:: FDIFF_1:condreg 1
registration
  let a1 be Function-like constant quasi_total Relation of NAT,REAL;
  cluster -> constant (subsequence of a1);
end;

:: FDIFF_1:attrnot 2 => FDIFF_1:attr 2
definition
  let a1 be Function-like Relation of REAL,REAL;
  attr a1 is REST-like means
    a1 is total(REAL, REAL) &
     (for b1 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL holds
        b1 " (#) (a1 * b1) is convergent &
         lim (b1 " (#) (a1 * b1)) = 0);
end;

:: FDIFF_1:dfs 2
definiens
  let a1 be Function-like Relation of REAL,REAL;
To prove
     a1 is REST-like
it is sufficient to prove
  thus a1 is total(REAL, REAL) &
     (for b1 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL holds
        b1 " (#) (a1 * b1) is convergent &
         lim (b1 " (#) (a1 * b1)) = 0);

:: FDIFF_1:def 3
theorem
for b1 being Function-like Relation of REAL,REAL holds
      b1 is REST-like
   iff
      b1 is total(REAL, REAL) &
       (for b2 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL holds
          b2 " (#) (b1 * b2) is convergent &
           lim (b2 " (#) (b1 * b2)) = 0);

:: FDIFF_1:exreg 3
registration
  cluster Relation-like Function-like complex-valued ext-real-valued real-valued REST-like Relation of REAL,REAL;
end;

:: FDIFF_1:modenot 1
definition
  mode REST is Function-like REST-like Relation of REAL,REAL;
end;

:: FDIFF_1:attrnot 3 => FDIFF_1:attr 3
definition
  let a1 be Function-like Relation of REAL,REAL;
  attr a1 is linear means
    a1 is total(REAL, REAL) &
     (ex b1 being Element of REAL st
        for b2 being Element of REAL holds
           a1 . b2 = b1 * b2);
end;

:: FDIFF_1:dfs 3
definiens
  let a1 be Function-like Relation of REAL,REAL;
To prove
     a1 is linear
it is sufficient to prove
  thus a1 is total(REAL, REAL) &
     (ex b1 being Element of REAL st
        for b2 being Element of REAL holds
           a1 . b2 = b1 * b2);

:: FDIFF_1:def 4
theorem
for b1 being Function-like Relation of REAL,REAL holds
      b1 is linear
   iff
      b1 is total(REAL, REAL) &
       (ex b2 being Element of REAL st
          for b3 being Element of REAL holds
             b1 . b3 = b2 * b3);

:: FDIFF_1:exreg 4
registration
  cluster Relation-like Function-like complex-valued ext-real-valued real-valued linear Relation of REAL,REAL;
end;

:: FDIFF_1:modenot 2
definition
  mode LINEAR is Function-like linear Relation of REAL,REAL;
end;

:: FDIFF_1:th 6
theorem
for b1, b2 being Function-like linear Relation of REAL,REAL holds
b1 + b2 is Function-like linear Relation of REAL,REAL &
 b1 - b2 is Function-like linear Relation of REAL,REAL;

:: FDIFF_1:th 7
theorem
for b1 being Element of REAL
for b2 being Function-like linear Relation of REAL,REAL holds
   b1 (#) b2 is Function-like linear Relation of REAL,REAL;

:: FDIFF_1:th 8
theorem
for b1, b2 being Function-like REST-like Relation of REAL,REAL holds
b1 + b2 is Function-like REST-like Relation of REAL,REAL &
 b1 - b2 is Function-like REST-like Relation of REAL,REAL &
 b1 (#) b2 is Function-like REST-like Relation of REAL,REAL;

:: FDIFF_1:th 9
theorem
for b1 being Element of REAL
for b2 being Function-like REST-like Relation of REAL,REAL holds
   b1 (#) b2 is Function-like REST-like Relation of REAL,REAL;

:: FDIFF_1:th 10
theorem
for b1, b2 being Function-like linear Relation of REAL,REAL holds
b1 (#) b2 is REST-like;

:: FDIFF_1:th 11
theorem
for b1 being Function-like REST-like Relation of REAL,REAL
for b2 being Function-like linear Relation of REAL,REAL holds
   b1 (#) b2 is Function-like REST-like Relation of REAL,REAL &
    b2 (#) b1 is Function-like REST-like Relation of REAL,REAL;

:: FDIFF_1:prednot 1 => FDIFF_1:pred 1
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be real set;
  pred A1 is_differentiable_in A2 means
    ex b1 being Neighbourhood of a2 st
       b1 c= dom a1 &
        (ex b2 being Function-like linear Relation of REAL,REAL st
           ex b3 being Function-like REST-like Relation of REAL,REAL st
              for b4 being Element of REAL
                    st b4 in b1
                 holds (a1 . b4) - (a1 . a2) = (b2 . (b4 - a2)) + (b3 . (b4 - a2)));
end;

:: FDIFF_1:dfs 4
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be real set;
To prove
     a1 is_differentiable_in a2
it is sufficient to prove
  thus ex b1 being Neighbourhood of a2 st
       b1 c= dom a1 &
        (ex b2 being Function-like linear Relation of REAL,REAL st
           ex b3 being Function-like REST-like Relation of REAL,REAL st
              for b4 being Element of REAL
                    st b4 in b1
                 holds (a1 . b4) - (a1 . a2) = (b2 . (b4 - a2)) + (b3 . (b4 - a2)));

:: FDIFF_1:def 5
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being real set holds
      b1 is_differentiable_in b2
   iff
      ex b3 being Neighbourhood of b2 st
         b3 c= dom b1 &
          (ex b4 being Function-like linear Relation of REAL,REAL st
             ex b5 being Function-like REST-like Relation of REAL,REAL st
                for b6 being Element of REAL
                      st b6 in b3
                   holds (b1 . b6) - (b1 . b2) = (b4 . (b6 - b2)) + (b5 . (b6 - b2)));

:: FDIFF_1:funcnot 1 => FDIFF_1:func 1
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be real set;
  assume a1 is_differentiable_in a2;
  func diff(A1,A2) -> Element of REAL means
    ex b1 being Neighbourhood of a2 st
       b1 c= dom a1 &
        (ex b2 being Function-like linear Relation of REAL,REAL st
           ex b3 being Function-like REST-like Relation of REAL,REAL st
              it = b2 . 1 &
               (for b4 being Element of REAL
                     st b4 in b1
                  holds (a1 . b4) - (a1 . a2) = (b2 . (b4 - a2)) + (b3 . (b4 - a2))));
end;

:: FDIFF_1:def 6
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being real set
   st b1 is_differentiable_in b2
for b3 being Element of REAL holds
      b3 = diff(b1,b2)
   iff
      ex b4 being Neighbourhood of b2 st
         b4 c= dom b1 &
          (ex b5 being Function-like linear Relation of REAL,REAL st
             ex b6 being Function-like REST-like Relation of REAL,REAL st
                b3 = b5 . 1 &
                 (for b7 being Element of REAL
                       st b7 in b4
                    holds (b1 . b7) - (b1 . b2) = (b5 . (b7 - b2)) + (b6 . (b7 - b2))));

:: FDIFF_1:prednot 2 => FDIFF_1:pred 2
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be set;
  pred A1 is_differentiable_on A2 means
    a2 c= dom a1 &
     (for b1 being Element of REAL
           st b1 in a2
        holds a1 | a2 is_differentiable_in b1);
end;

:: FDIFF_1:dfs 6
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be set;
To prove
     a1 is_differentiable_on a2
it is sufficient to prove
  thus a2 c= dom a1 &
     (for b1 being Element of REAL
           st b1 in a2
        holds a1 | a2 is_differentiable_in b1);

:: FDIFF_1:def 7
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set holds
      b1 is_differentiable_on b2
   iff
      b2 c= dom b1 &
       (for b3 being Element of REAL
             st b3 in b2
          holds b1 | b2 is_differentiable_in b3);

:: FDIFF_1:th 15
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on b1
   holds b1 is Element of bool REAL;

:: FDIFF_1:th 16
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL holds
      b2 is_differentiable_on b1
   iff
      b1 c= dom b2 &
       (for b3 being Element of REAL
             st b3 in b1
          holds b2 is_differentiable_in b3);

:: FDIFF_1:th 17
theorem
for b1 being Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on b1
   holds b1 is open;

:: FDIFF_1:funcnot 2 => FDIFF_1:func 2
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be set;
  assume a1 is_differentiable_on a2;
  func A1 `| A2 -> Function-like Relation of REAL,REAL means
    dom it = a2 &
     (for b1 being Element of REAL
           st b1 in a2
        holds it . b1 = diff(a1,b1));
end;

:: FDIFF_1:def 8
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set
   st b1 is_differentiable_on b2
for b3 being Function-like Relation of REAL,REAL holds
      b3 = b1 `| b2
   iff
      dom b3 = b2 &
       (for b4 being Element of REAL
             st b4 in b2
          holds b3 . b4 = diff(b1,b4));

:: FDIFF_1:th 19
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= dom b2 &
         (ex b3 being Element of REAL st
            rng b2 = {b3})
   holds b2 is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds (b2 `| b1) . b3 = 0);

:: FDIFF_1:funcreg 1
registration
  let a1 be Function-like quasi_total convergent_to_0 Relation of NAT,REAL;
  let a2 be Element of NAT;
  cluster a1 ^\ a2 -> Function-like quasi_total convergent_to_0;
end;

:: FDIFF_1:funcreg 2
registration
  let a1 be Function-like constant quasi_total Relation of NAT,REAL;
  let a2 be Element of NAT;
  cluster a1 ^\ a2 -> Function-like constant quasi_total;
end;

:: FDIFF_1:th 20
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being real set
for b3 being Neighbourhood of b2
   st b1 is_differentiable_in b2 & b3 c= dom b1
for b4 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
for b5 being Function-like constant quasi_total Relation of NAT,REAL
      st rng b5 = {b2} & rng (b4 + b5) c= b3
   holds b4 " (#) ((b1 * (b4 + b5)) - (b1 * b5)) is convergent &
    diff(b1,b2) = lim (b4 " (#) ((b1 * (b4 + b5)) - (b1 * b5)));

:: FDIFF_1:th 21
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_in b1 & b3 is_differentiable_in b1
   holds b2 + b3 is_differentiable_in b1 &
    diff(b2 + b3,b1) = (diff(b2,b1)) + diff(b3,b1);

:: FDIFF_1:th 22
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_in b1 & b3 is_differentiable_in b1
   holds b2 - b3 is_differentiable_in b1 &
    diff(b2 - b3,b1) = (diff(b2,b1)) - diff(b3,b1);

:: FDIFF_1:th 23
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
      st b3 is_differentiable_in b1
   holds b2 (#) b3 is_differentiable_in b1 &
    diff(b2 (#) b3,b1) = b2 * diff(b3,b1);

:: FDIFF_1:th 24
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_in b1 & b3 is_differentiable_in b1
   holds b2 (#) b3 is_differentiable_in b1 &
    diff(b2 (#) b3,b1) = ((b3 . b1) * diff(b2,b1)) + ((b2 . b1) * diff(b3,b1));

:: FDIFF_1:th 25
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= dom b2 & b2 | b1 = id b1
   holds b2 is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds (b2 `| b1) . b3 = 1);

:: FDIFF_1:th 26
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b1 c= dom (b2 + b3) & b2 is_differentiable_on b1 & b3 is_differentiable_on b1
   holds b2 + b3 is_differentiable_on b1 &
    (for b4 being Element of REAL
          st b4 in b1
       holds ((b2 + b3) `| b1) . b4 = (diff(b2,b4)) + diff(b3,b4));

:: FDIFF_1:th 27
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b1 c= dom (b2 - b3) & b2 is_differentiable_on b1 & b3 is_differentiable_on b1
   holds b2 - b3 is_differentiable_on b1 &
    (for b4 being Element of REAL
          st b4 in b1
       holds ((b2 - b3) `| b1) . b4 = (diff(b2,b4)) - diff(b3,b4));

:: FDIFF_1:th 28
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3 being Function-like Relation of REAL,REAL
      st b2 c= dom (b1 (#) b3) & b3 is_differentiable_on b2
   holds b1 (#) b3 is_differentiable_on b2 &
    (for b4 being Element of REAL
          st b4 in b2
       holds ((b1 (#) b3) `| b2) . b4 = b1 * diff(b3,b4));

:: FDIFF_1:th 29
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b1 c= dom (b2 (#) b3) & b2 is_differentiable_on b1 & b3 is_differentiable_on b1
   holds b2 (#) b3 is_differentiable_on b1 &
    (for b4 being Element of REAL
          st b4 in b1
       holds ((b2 (#) b3) `| b1) . b4 = ((b3 . b4) * diff(b2,b4)) + ((b2 . b4) * diff(b3,b4)));

:: FDIFF_1:th 30
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= dom b2 & b2 is_constant_on b1
   holds b2 is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds (b2 `| b1) . b3 = 0);

:: FDIFF_1:th 31
theorem
for b1, b2 being Element of REAL
for b3 being open Element of bool REAL
for b4 being Function-like Relation of REAL,REAL
      st b3 c= dom b4 &
         (for b5 being Element of REAL
               st b5 in b3
            holds b4 . b5 = (b1 * b5) + b2)
   holds b4 is_differentiable_on b3 &
    (for b5 being Element of REAL
          st b5 in b3
       holds (b4 `| b3) . b5 = b1);

:: FDIFF_1:th 32
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being real set
      st b1 is_differentiable_in b2
   holds b1 is_continuous_in b2;

:: FDIFF_1:th 33
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on b1
   holds b2 is_continuous_on b1;

:: FDIFF_1:th 34
theorem
for b1 being set
for b2 being open Element of bool REAL
for b3 being Function-like Relation of REAL,REAL
      st b3 is_differentiable_on b1 & b2 c= b1
   holds b3 is_differentiable_on b2;

:: FDIFF_1:th 35
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_in b1
   holds ex b3 being Function-like REST-like Relation of REAL,REAL st
      b3 . 0 = 0 & b3 is_continuous_in 0;

:: FDIFF_1:attrnot 4 => FDIFF_1:attr 4
definition
  let a1 be Function-like Relation of REAL,REAL;
  attr a1 is differentiable means
    a1 is_differentiable_on dom a1;
end;

:: FDIFF_1:dfs 8
definiens
  let a1 be Function-like Relation of REAL,REAL;
To prove
     a1 is differentiable
it is sufficient to prove
  thus a1 is_differentiable_on dom a1;

:: FDIFF_1:def 9
theorem
for b1 being Function-like Relation of REAL,REAL holds
      b1 is differentiable
   iff
      b1 is_differentiable_on dom b1;

:: FDIFF_1:exreg 5
registration
  cluster Relation-like Function-like non empty total complex-valued ext-real-valued real-valued quasi_total differentiable Relation of REAL,REAL;
end;

:: FDIFF_1:th 36
theorem
for b1 being open Element of bool REAL
for b2 being Function-like differentiable Relation of REAL,REAL
      st b1 c= dom b2
   holds b2 is_differentiable_on b1;