Article NDIFF_2, MML version 4.99.1005

:: NDIFF_2:funcnot 1 => NDIFF_2:func 1
definition
  let a1, a2, a3 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a4 be Element of BoundedLinearOperators(a1,a2);
  let a5 be Element of BoundedLinearOperators(a2,a3);
  func A5 * A4 -> Element of BoundedLinearOperators(a1,a3) equals
    (modetrans(a5,a2,a3)) * modetrans(a4,a1,a2);
end;

:: NDIFF_2:def 1
theorem
for b1, b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Element of BoundedLinearOperators(b1,b2)
for b5 being Element of BoundedLinearOperators(b2,b3) holds
   b5 * b4 = (modetrans(b5,b2,b3)) * modetrans(b4,b1,b2);

:: NDIFF_2:funcnot 2 => NDIFF_2:func 2
definition
  let a1, a2, a3 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a4 be Element of the carrier of R_NormSpace_of_BoundedLinearOperators(a1,a2);
  let a5 be Element of the carrier of R_NormSpace_of_BoundedLinearOperators(a2,a3);
  func A5 * A4 -> Element of the carrier of R_NormSpace_of_BoundedLinearOperators(a1,a3) equals
    (modetrans(a5,a2,a3)) * modetrans(a4,a1,a2);
end;

:: NDIFF_2:def 2
theorem
for b1, b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Element of the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2)
for b5 being Element of the carrier of R_NormSpace_of_BoundedLinearOperators(b2,b3) holds
   b5 * b4 = (modetrans(b5,b2,b3)) * modetrans(b4,b1,b2);

:: NDIFF_2:th 1
theorem
for b1, b2 being non empty non trivial 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 the carrier of b2
      st b3 is_differentiable_in b4
   holds ex b5 being Neighbourhood of b4 st
      b5 c= dom b3 &
       (for b6 being Element of the carrier of b2
       for b7 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
       for b8 being Function-like constant quasi_total Relation of NAT,the carrier of b2
             st rng b8 = {b4} & rng ((b7 * b6) + b8) c= b5
          holds b7 " (#) ((b3 * ((b7 * b6) + b8)) - (b3 * b8)) is convergent(b1) &
           (diff(b3,b4)) . b6 = lim (b7 " (#) ((b3 * ((b7 * b6) + b8)) - (b3 * b8))));

:: NDIFF_2:th 2
theorem
for b1, b2 being non empty non trivial 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 the carrier of b2
   st b3 is_differentiable_in b4
for b5 being Element of the carrier of b2
for b6 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
for b7 being Function-like constant quasi_total Relation of NAT,the carrier of b2
      st rng b7 = {b4} & rng ((b6 * b5) + b7) c= dom b3
   holds b6 " (#) ((b3 * ((b6 * b5) + b7)) - (b3 * b7)) is convergent(b1) &
    (diff(b3,b4)) . b5 = lim (b6 " (#) ((b3 * ((b6 * b5) + b7)) - (b3 * b7)));

:: NDIFF_2:th 3
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1
for b5 being Neighbourhood of b4
   st b5 c= dom b3
for b6 being Element of the carrier of b1
for b7 being Element of the carrier of b2 holds
      for b8 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
      for b9 being Function-like constant quasi_total Relation of NAT,the carrier of b1
            st rng b9 = {b4} & rng ((b8 * b6) + b9) c= b5
         holds b8 " (#) ((b3 * ((b8 * b6) + b9)) - (b3 * b9)) is convergent(b2) &
          b7 = lim (b8 " (#) ((b3 * ((b8 * b6) + b9)) - (b3 * b9)))
   iff
      for b8 being Element of REAL
            st 0 < b8
         holds ex b9 being Element of REAL st
            0 < b9 &
             (for b10 being Element of REAL
                   st abs b10 < b9 & b10 <> 0 & (b10 * b6) + b4 in b5
                holds ||.(b10 " * ((b3 /. ((b10 * b6) + b4)) - (b3 /. b4))) - b7.|| < b8);

:: NDIFF_2:prednot 1 => NDIFF_2:pred 1
definition
  let a1, a2 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
  let a4, a5 be Element of the carrier of a1;
  pred A3 is_Gateaux_differentiable_in A4,A5 means
    ex b1 being Neighbourhood of a4 st
       b1 c= dom a3 &
        (ex b2 being Element of the carrier of a2 st
           for b3 being Element of REAL
                 st 0 < b3
              holds ex b4 being Element of REAL st
                 0 < b4 &
                  (for b5 being Element of REAL
                        st abs b5 < b4 & b5 <> 0 & (b5 * a5) + a4 in b1
                     holds ||.(b5 " * ((a3 /. ((b5 * a5) + a4)) - (a3 /. a4))) - b2.|| < b3));
end;

:: NDIFF_2:dfs 3
definiens
  let a1, a2 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
  let a4, a5 be Element of the carrier of a1;
To prove
     a3 is_Gateaux_differentiable_in a4,a5
it is sufficient to prove
  thus ex b1 being Neighbourhood of a4 st
       b1 c= dom a3 &
        (ex b2 being Element of the carrier of a2 st
           for b3 being Element of REAL
                 st 0 < b3
              holds ex b4 being Element of REAL st
                 0 < b4 &
                  (for b5 being Element of REAL
                        st abs b5 < b4 & b5 <> 0 & (b5 * a5) + a4 in b1
                     holds ||.(b5 " * ((a3 /. ((b5 * a5) + a4)) - (a3 /. a4))) - b2.|| < b3));

:: NDIFF_2:def 3
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4, b5 being Element of the carrier of b1 holds
   b3 is_Gateaux_differentiable_in b4,b5
iff
   ex b6 being Neighbourhood of b4 st
      b6 c= dom b3 &
       (ex b7 being Element of the carrier of b2 st
          for b8 being Element of REAL
                st 0 < b8
             holds ex b9 being Element of REAL st
                0 < b9 &
                 (for b10 being Element of REAL
                       st abs b10 < b9 & b10 <> 0 & (b10 * b5) + b4 in b6
                    holds ||.(b10 " * ((b3 /. ((b10 * b5) + b4)) - (b3 /. b4))) - b7.|| < b8));

:: NDIFF_2:th 4
theorem
(for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
 for b2, b3 being Element of the carrier of b1 holds
    0 < ||.b2 - b3.||
 iff
    b2 <> b3) &
 (for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
 for b2, b3 being Element of the carrier of b1 holds
 ||.b2 - b3.|| = ||.b3 - b2.||) &
 (for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
 for b2, b3 being Element of the carrier of b1 holds
    ||.b2 - b3.|| = 0
 iff
    b2 = b3) &
 (for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
 for b2, b3 being Element of the carrier of b1 holds
    ||.b2 - b3.|| <> 0
 iff
    b2 <> b3) &
 (for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
 for b2, b3, b4 being Element of the carrier of b1
 for b5 being Element of REAL
       st 0 < b5 &
          ||.b2 - b4.|| < b5 / 2 &
          ||.b4 - b3.|| < b5 / 2
    holds ||.b2 - b3.|| < b5) &
 (for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
 for b2, b3, b4 being Element of the carrier of b1
 for b5 being Element of REAL
       st 0 < b5 &
          ||.b2 - b4.|| < b5 / 2 &
          ||.b3 - b4.|| < b5 / 2
    holds ||.b2 - b3.|| < b5) &
 (for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
 for b2 being Element of the carrier of b1
       st for b3 being Element of REAL
               st 0 < b3
            holds ||.b2.|| < b3
    holds b2 = 0. b1) &
 (for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
 for b2, b3 being Element of the carrier of b1
       st for b4 being Element of REAL
               st 0 < b4
            holds ||.b2 - b3.|| < b4
    holds b2 = b3);

:: NDIFF_2:funcnot 3 => NDIFF_2:func 3
definition
  let a1, a2 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
  let a4, a5 be Element of the carrier of a1;
  assume a3 is_Gateaux_differentiable_in a4,a5;
  func Gateaux_diff(A3,A4,A5) -> Element of the carrier of a2 means
    ex b1 being Neighbourhood of a4 st
       b1 c= dom a3 &
        (for b2 being Element of REAL
              st 0 < b2
           holds ex b3 being Element of REAL st
              0 < b3 &
               (for b4 being Element of REAL
                     st abs b4 < b3 & b4 <> 0 & (b4 * a5) + a4 in b1
                  holds ||.(b4 " * ((a3 /. ((b4 * a5) + a4)) - (a3 /. a4))) - it.|| < b2));
end;

:: NDIFF_2:def 4
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4, b5 being Element of the carrier of b1
   st b3 is_Gateaux_differentiable_in b4,b5
for b6 being Element of the carrier of b2 holds
      b6 = Gateaux_diff(b3,b4,b5)
   iff
      ex b7 being Neighbourhood of b4 st
         b7 c= dom b3 &
          (for b8 being Element of REAL
                st 0 < b8
             holds ex b9 being Element of REAL st
                0 < b9 &
                 (for b10 being Element of REAL
                       st abs b10 < b9 & b10 <> 0 & (b10 * b5) + b4 in b7
                    holds ||.(b10 " * ((b3 /. ((b10 * b5) + b4)) - (b3 /. b4))) - b6.|| < b8));

:: NDIFF_2:th 5
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4, b5 being Element of the carrier of b1 holds
   b3 is_Gateaux_differentiable_in b4,b5
iff
   ex b6 being Neighbourhood of b4 st
      b6 c= dom b3 &
       (ex b7 being Element of the carrier of b2 st
          for b8 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
          for b9 being Function-like constant quasi_total Relation of NAT,the carrier of b1
                st rng b9 = {b4} & rng ((b8 * b5) + b9) c= b6
             holds b8 " (#) ((b3 * ((b8 * b5) + b9)) - (b3 * b9)) is convergent(b2) &
              b7 = lim (b8 " (#) ((b3 * ((b8 * b5) + b9)) - (b3 * b9))));

:: NDIFF_2:th 6
theorem
for b1, b2 being non empty non trivial 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 the carrier of b2
   st b3 is_differentiable_in b4
for b5 being Element of the carrier of b2 holds
   b3 is_Gateaux_differentiable_in b4,b5 &
    Gateaux_diff(b3,b4,b5) = (diff(b3,b4)) . b5 &
    (ex b6 being Neighbourhood of b4 st
       b6 c= dom b3 &
        (for b7 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
        for b8 being Function-like constant quasi_total Relation of NAT,the carrier of b2
              st rng b8 = {b4} & rng ((b7 * b5) + b8) c= b6
           holds b7 " (#) ((b3 * ((b7 * b5) + b8)) - (b3 * b8)) is convergent(b1) &
            Gateaux_diff(b3,b4,b5) = lim (b7 " (#) ((b3 * ((b7 * b5) + b8)) - (b3 * b8)))));

:: NDIFF_2:th 7
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like REST-like Relation of the carrier of b1,the carrier of b2
   st b3 /. 0. b1 = 0. b2
for b4 being Element of REAL
      st 0 < b4
   holds ex b5 being Element of REAL st
      0 < b5 &
       (for b6 being Element of the carrier of b1
             st ||.b6.|| < b5
          holds ||.b3 /. b6.|| <= b4 * ||.b6.||);

:: NDIFF_2:th 8
theorem
for b1, b2, b3 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like REST-like Relation of the carrier of b1,the carrier of b3
   st b4 /. 0. b1 = 0. b3
for b5 being Function-like quasi_total additive homogeneous bounded Relation of the carrier of b2,the carrier of b1 holds
   b4 * b5 is Function-like REST-like Relation of the carrier of b2,the carrier of b3;

:: NDIFF_2:th 9
theorem
for b1, b2, b3 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like REST-like Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total additive homogeneous bounded Relation of the carrier of b2,the carrier of b3 holds
   b5 * b4 is Function-like REST-like Relation of the carrier of b1,the carrier of b3;

:: NDIFF_2:th 10
theorem
for b1, b2, b3 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like REST-like Relation of the carrier of b1,the carrier of b2
   st b4 /. 0. b1 = 0. b2
for b5 being Function-like REST-like Relation of the carrier of b2,the carrier of b3
      st b5 /. 0. b2 = 0. b3
   holds b5 * b4 is Function-like REST-like Relation of the carrier of b1,the carrier of b3;

:: NDIFF_2:th 11
theorem
for b1, b2, b3 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like REST-like Relation of the carrier of b1,the carrier of b2
   st b4 /. 0. b1 = 0. b2
for b5 being Function-like REST-like Relation of the carrier of b2,the carrier of b3
   st b5 /. 0. b2 = 0. b3
for b6 being Function-like quasi_total additive homogeneous bounded Relation of the carrier of b1,the carrier of b2 holds
   b5 * (b6 + b4) is Function-like REST-like Relation of the carrier of b1,the carrier of b3;

:: NDIFF_2:th 12
theorem
for b1, b2, b3 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like REST-like Relation of the carrier of b1,the carrier of b2
   st b4 /. 0. b1 = 0. b2
for b5 being Function-like REST-like Relation of the carrier of b2,the carrier of b3
   st b5 /. 0. b2 = 0. b3
for b6 being Function-like quasi_total additive homogeneous bounded Relation of the carrier of b1,the carrier of b2
for b7 being Function-like quasi_total additive homogeneous bounded Relation of the carrier of b2,the carrier of b3 holds
   (b7 * b4) + (b5 * (b6 + b4)) is Function-like REST-like Relation of the carrier of b1,the carrier of b3;

:: NDIFF_2:th 13
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Element of the carrier of b1
for b4 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b5 being Function-like Relation of the carrier of b1,the carrier of b2
   st b5 is_differentiable_in b3
for b6 being Function-like Relation of the carrier of b2,the carrier of b4
      st b6 is_differentiable_in b5 /. b3
   holds b6 * b5 is_differentiable_in b3 &
    diff(b6 * b5,b3) = (diff(b6,b5 /. b3)) * diff(b5,b3);