Article FDIFF_2, MML version 4.99.1005

:: FDIFF_2:funcnot 1 => FDIFF_2:func 1
definition
  let a1 be Function-like quasi_total convergent_to_0 Relation of NAT,REAL;
  redefine func - a1 -> Function-like quasi_total convergent_to_0 Relation of NAT,REAL;
  involutiveness;
::  for a1 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL holds
::     - - a1 = a1;
end;

:: FDIFF_2:th 1
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL
      st b1 is convergent &
         b2 is convergent &
         lim b1 = lim b2 &
         (for b4 being Element of NAT holds
            b3 . (2 * b4) = b1 . b4 &
             b3 . ((2 * b4) + 1) = b2 . b4)
   holds b3 is convergent & lim b3 = lim b1;

:: FDIFF_2:th 2
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st for b2 being Element of NAT holds
           b1 . b2 = 2 * b2
   holds b1 is increasing & b1 is natural-valued;

:: FDIFF_2:th 3
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st for b2 being Element of NAT holds
           b1 . b2 = (2 * b2) + 1
   holds b1 is increasing & b1 is natural-valued;

:: FDIFF_2:th 4
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
for b3 being Function-like constant quasi_total Relation of NAT,REAL
      st rng b3 = {b1}
   holds b3 is convergent & lim b3 = b1 & b2 + b3 is convergent & lim (b2 + b3) = b1;

:: FDIFF_2:th 5
theorem
for b1 being Element of REAL
for b2, b3 being Function-like quasi_total Relation of NAT,REAL
      st rng b2 = {b1} & rng b3 = {b1}
   holds b2 = b3;

:: FDIFF_2:th 6
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
      st b1 is subsequence of b2
   holds b1 is Function-like quasi_total convergent_to_0 Relation of NAT,REAL;

:: FDIFF_2:th 7
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
   st for b3 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
     for b4 being Function-like constant quasi_total Relation of NAT,REAL
           st rng b4 = {b1} & rng (b3 + b4) c= dom b2 & {b1} c= dom b2
        holds b3 " (#) ((b2 * (b3 + b4)) - (b2 * b4)) is convergent
for b3, 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 = {b1} & rng (b3 + b5) c= dom b2 & rng (b4 + b5) c= dom b2 & {b1} c= dom b2
   holds lim (b3 " (#) ((b2 * (b3 + b5)) - (b2 * b5))) = lim (b4 " (#) ((b2 * (b4 + b5)) - (b2 * b5)));

:: FDIFF_2:th 8
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st ex b3 being Neighbourhood of b1 st
           b3 c= dom b2
   holds ex b3 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL st
      ex b4 being Function-like constant quasi_total Relation of NAT,REAL st
         rng b4 = {b1} & rng (b3 + b4) c= dom b2 & {b1} c= dom b2;

:: FDIFF_2:th 9
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st rng b1 c= dom (b2 * b3)
   holds rng b1 c= dom b3 & rng (b3 * b1) c= dom b2;

:: FDIFF_2:sch 1
scheme FDIFF_2:sch 1
{F1 -> Function-like quasi_total Relation of NAT,REAL}:
ex b1 being Function-like natural-valued quasi_total increasing Relation of NAT,REAL st
   (for b2 being Element of NAT holds
       P1[(F1() * b1) . b2]) &
    (for b2 being Element of NAT
          st for b3 being Element of REAL
                  st b3 = F1() . b2
               holds P1[b3]
       holds ex b3 being Element of NAT st
          b2 = b1 . b3)
provided
   for b1 being Element of NAT holds
      ex b2 being Element of NAT st
         b1 <= b2 & P1[F1() . b2];


:: FDIFF_2:th 10
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
      st b3 . b1 <> b2 & b3 is_differentiable_in b1
   holds ex b4 being Neighbourhood of b1 st
      b4 c= dom b3 &
       (for b5 being Element of REAL
             st b5 in b4
          holds b3 . b5 <> b2);

:: FDIFF_2:th 11
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL holds
      b2 is_differentiable_in b1
   iff
      (ex b3 being Neighbourhood of b1 st
          b3 c= dom b2) &
       (for b3 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
       for b4 being Function-like constant quasi_total Relation of NAT,REAL
             st rng b4 = {b1} & rng (b3 + b4) c= dom b2
          holds b3 " (#) ((b2 * (b3 + b4)) - (b2 * b4)) is convergent);

:: FDIFF_2:th 12
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL holds
      b3 is_differentiable_in b1 & diff(b3,b1) = b2
   iff
      (ex b4 being Neighbourhood of b1 st
          b4 c= dom b3) &
       (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 = {b1} & rng (b4 + b5) c= dom b3
          holds b4 " (#) ((b3 * (b4 + b5)) - (b3 * b5)) is convergent &
           lim (b4 " (#) ((b3 * (b4 + b5)) - (b3 * b5))) = b2);

:: FDIFF_2:th 13
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 b2 . b1
   holds b3 * b2 is_differentiable_in b1 &
    diff(b3 * b2,b1) = (diff(b3,b2 . b1)) * diff(b2,b1);

:: FDIFF_2:th 14
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 . b1 <> 0 & b3 is_differentiable_in b1 & b2 is_differentiable_in b1
   holds b3 / b2 is_differentiable_in b1 &
    diff(b3 / b2,b1) = (((diff(b3,b1)) * (b2 . b1)) - ((diff(b2,b1)) * (b3 . b1))) / ((b2 . b1) ^2);

:: FDIFF_2:th 15
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 . b1 <> 0 & b2 is_differentiable_in b1
   holds b2 ^ is_differentiable_in b1 &
    diff(b2 ^,b1) = - ((diff(b2,b1)) / ((b2 . b1) ^2));

:: FDIFF_2:th 16
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on b1
   holds b2 | b1 is_differentiable_on b1 & b2 `| b1 = (b2 | b1) `| b1;

:: FDIFF_2:th 17
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on b1 & b3 is_differentiable_on b1
   holds b2 + b3 is_differentiable_on b1 &
    (b2 + b3) `| b1 = (b2 `| b1) + (b3 `| b1);

:: FDIFF_2:th 18
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on b1 & b3 is_differentiable_on b1
   holds b2 - b3 is_differentiable_on b1 &
    (b2 - b3) `| b1 = (b2 `| b1) - (b3 `| b1);

:: FDIFF_2:th 19
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 b3 is_differentiable_on b2
   holds b1 (#) b3 is_differentiable_on b2 &
    (b1 (#) b3) `| b2 = b1 (#) (b3 `| b2);

:: FDIFF_2:th 20
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on b1 & b3 is_differentiable_on b1
   holds b2 (#) b3 is_differentiable_on b1 &
    (b2 (#) b3) `| b1 = ((b2 `| b1) (#) b3) + (b2 (#) (b3 `| b1));

:: FDIFF_2:th 21
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on b1 &
         b3 is_differentiable_on b1 &
         (for b4 being Element of REAL
               st b4 in b1
            holds b3 . b4 <> 0)
   holds b2 / b3 is_differentiable_on b1 &
    (b2 / b3) `| b1 = (((b2 `| b1) (#) b3) - ((b3 `| b1) (#) b2)) / (b3 (#) b3);

:: FDIFF_2:th 22
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on b1 &
         (for b3 being Element of REAL
               st b3 in b1
            holds b2 . b3 <> 0)
   holds b2 ^ is_differentiable_on b1 &
    b2 ^ `| b1 = - ((b2 `| b1) / (b2 (#) b2));

:: FDIFF_2:th 23
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on b1 & b2 .: b1 is open Element of bool REAL & b3 is_differentiable_on b2 .: b1
   holds b3 * b2 is_differentiable_on b1 &
    (b3 * b2) `| b1 = ((b3 `| (b2 .: b1)) * b2) (#) (b2 `| b1);

:: FDIFF_2:th 24
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
      st b1 c= dom b2 &
         (for b3, b4 being Element of REAL
               st b3 in b1 & b4 in b1
            holds abs ((b2 . b3) - (b2 . b4)) <= (b3 - b4) ^2)
   holds b2 is_differentiable_on b1 &
    (for b3 being Element of REAL
          st b3 in b1
       holds diff(b2,b3) = 0);

:: FDIFF_2:th 25
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
      st (for b4, b5 being Element of REAL
               st b4 in ].b1,b2.[ & b5 in ].b1,b2.[
            holds abs ((b3 . b4) - (b3 . b5)) <= (b4 - b5) ^2) &
         b1 < b2 &
         ].b1,b2.[ c= dom b3
   holds b3 is_differentiable_on ].b1,b2.[ & b3 is_constant_on ].b1,b2.[;

:: FDIFF_2:th 26
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st halfline b1 c= dom b2 &
         (for b3, b4 being Element of REAL
               st b3 in halfline b1 & b4 in halfline b1
            holds abs ((b2 . b3) - (b2 . b4)) <= (b3 - b4) ^2)
   holds b2 is_differentiable_on halfline b1 & b2 is_constant_on halfline b1;

:: FDIFF_2:th 27
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st right_open_halfline b1 c= dom b2 &
         (for b3, b4 being Element of REAL
               st b3 in right_open_halfline b1 & b4 in right_open_halfline b1
            holds abs ((b2 . b3) - (b2 . b4)) <= (b3 - b4) ^2)
   holds b2 is_differentiable_on right_open_halfline b1 & b2 is_constant_on right_open_halfline b1;

:: FDIFF_2:th 28
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is total(REAL, REAL) &
         (for b2, b3 being Element of REAL holds
         abs ((b1 . b2) - (b1 . b3)) <= (b2 - b3) ^2)
   holds b1 is_differentiable_on [#] REAL & b1 is_constant_on [#] REAL;

:: FDIFF_2:th 29
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on halfline b1 &
         (for b3 being Element of REAL
               st b3 in halfline b1
            holds 0 < diff(b2,b3))
   holds b2 is_increasing_on halfline b1 & b2 | halfline b1 is one-to-one;

:: FDIFF_2:th 30
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on halfline b1 &
         (for b3 being Element of REAL
               st b3 in halfline b1
            holds diff(b2,b3) < 0)
   holds b2 is_decreasing_on halfline b1 & b2 | halfline b1 is one-to-one;

:: FDIFF_2:th 31
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on halfline b1 &
         (for b3 being Element of REAL
               st b3 in halfline b1
            holds 0 <= diff(b2,b3))
   holds b2 is_non_decreasing_on halfline b1;

:: FDIFF_2:th 32
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on halfline b1 &
         (for b3 being Element of REAL
               st b3 in halfline b1
            holds diff(b2,b3) <= 0)
   holds b2 is_non_increasing_on halfline b1;

:: FDIFF_2:th 33
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on right_open_halfline b1 &
         (for b3 being Element of REAL
               st b3 in right_open_halfline b1
            holds 0 < diff(b2,b3))
   holds b2 is_increasing_on right_open_halfline b1 & b2 | right_open_halfline b1 is one-to-one;

:: FDIFF_2:th 34
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on right_open_halfline b1 &
         (for b3 being Element of REAL
               st b3 in right_open_halfline b1
            holds diff(b2,b3) < 0)
   holds b2 is_decreasing_on right_open_halfline b1 & b2 | right_open_halfline b1 is one-to-one;

:: FDIFF_2:th 35
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on right_open_halfline b1 &
         (for b3 being Element of REAL
               st b3 in right_open_halfline b1
            holds 0 <= diff(b2,b3))
   holds b2 is_non_decreasing_on right_open_halfline b1;

:: FDIFF_2:th 36
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on right_open_halfline b1 &
         (for b3 being Element of REAL
               st b3 in right_open_halfline b1
            holds diff(b2,b3) <= 0)
   holds b2 is_non_increasing_on right_open_halfline b1;

:: FDIFF_2:th 37
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is_differentiable_on [#] REAL &
         (for b2 being Element of REAL holds
            0 < diff(b1,b2))
   holds b1 is_increasing_on [#] REAL & b1 is one-to-one;

:: FDIFF_2:th 38
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is_differentiable_on [#] REAL &
         (for b2 being Element of REAL holds
            diff(b1,b2) < 0)
   holds b1 is_decreasing_on [#] REAL & b1 is one-to-one;

:: FDIFF_2:th 39
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is_differentiable_on [#] REAL &
         (for b2 being Element of REAL holds
            0 <= diff(b1,b2))
   holds b1 is_non_decreasing_on [#] REAL;

:: FDIFF_2:th 40
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is_differentiable_on [#] REAL &
         (for b2 being Element of REAL holds
            diff(b1,b2) <= 0)
   holds b1 is_non_increasing_on [#] REAL;

:: FDIFF_2:th 41
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
      st b3 is_differentiable_on ].b1,b2.[ &
         (for b4 being Element of REAL
               st b4 in ].b1,b2.[
            holds 0 < diff(b3,b4) or for b4 being Element of REAL
               st b4 in ].b1,b2.[
            holds diff(b3,b4) < 0)
   holds rng (b3 | ].b1,b2.[) is open;

:: FDIFF_2:th 42
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on halfline b1 &
         (for b3 being Element of REAL
               st b3 in halfline b1
            holds 0 < diff(b2,b3) or for b3 being Element of REAL
               st b3 in halfline b1
            holds diff(b2,b3) < 0)
   holds rng (b2 | halfline b1) is open;

:: FDIFF_2:th 43
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on right_open_halfline b1 &
         (for b3 being Element of REAL
               st b3 in right_open_halfline b1
            holds 0 < diff(b2,b3) or for b3 being Element of REAL
               st b3 in right_open_halfline b1
            holds diff(b2,b3) < 0)
   holds rng (b2 | right_open_halfline b1) is open;

:: FDIFF_2:th 44
theorem
for b1 being Function-like Relation of REAL,REAL
      st b1 is_differentiable_on [#] REAL &
         (for b2 being Element of REAL holds
            0 < diff(b1,b2) or for b2 being Element of REAL holds
            diff(b1,b2) < 0)
   holds rng b1 is open;

:: FDIFF_2:th 45
theorem
for b1 being Function-like one-to-one Relation of REAL,REAL
      st b1 is_differentiable_on [#] REAL &
         (for b2 being Element of REAL holds
            0 < diff(b1,b2) or for b2 being Element of REAL holds
            diff(b1,b2) < 0)
   holds b1 is one-to-one &
    b1 " is_differentiable_on dom (b1 ") &
    (for b2 being Element of REAL
          st b2 in dom (b1 ")
       holds diff(b1 ",b2) = 1 / diff(b1,b1 " . b2));

:: FDIFF_2:th 46
theorem
for b1 being Element of REAL
for b2 being Function-like one-to-one Relation of REAL,REAL
      st b2 is_differentiable_on halfline b1 &
         (for b3 being Element of REAL
               st b3 in halfline b1
            holds 0 < diff(b2,b3) or for b3 being Element of REAL
               st b3 in halfline b1
            holds diff(b2,b3) < 0)
   holds b2 | halfline b1 is one-to-one &
    (b2 | halfline b1) " is_differentiable_on dom ((b2 | halfline b1) ") &
    (for b3 being Element of REAL
          st b3 in dom ((b2 | halfline b1) ")
       holds diff((b2 | halfline b1) ",b3) = 1 / diff(b2,(b2 | halfline b1) " . b3));

:: FDIFF_2:th 47
theorem
for b1 being Element of REAL
for b2 being Function-like one-to-one Relation of REAL,REAL
      st b2 is_differentiable_on right_open_halfline b1 &
         (for b3 being Element of REAL
               st b3 in right_open_halfline b1
            holds 0 < diff(b2,b3) or for b3 being Element of REAL
               st b3 in right_open_halfline b1
            holds diff(b2,b3) < 0)
   holds b2 | right_open_halfline b1 is one-to-one &
    (b2 | right_open_halfline b1) " is_differentiable_on dom ((b2 | right_open_halfline b1) ") &
    (for b3 being Element of REAL
          st b3 in dom ((b2 | right_open_halfline b1) ")
       holds diff((b2 | right_open_halfline b1) ",b3) = 1 / diff(b2,(b2 | right_open_halfline b1) " . b3));

:: FDIFF_2:th 48
theorem
for b1, b2 being Element of REAL
for b3 being Function-like one-to-one Relation of REAL,REAL
      st b3 is_differentiable_on ].b1,b2.[ &
         (for b4 being Element of REAL
               st b4 in ].b1,b2.[
            holds 0 < diff(b3,b4) or for b4 being Element of REAL
               st b4 in ].b1,b2.[
            holds diff(b3,b4) < 0)
   holds b3 | ].b1,b2.[ is one-to-one &
    (b3 | ].b1,b2.[) " is_differentiable_on dom ((b3 | ].b1,b2.[) ") &
    (for b4 being Element of REAL
          st b4 in dom ((b3 | ].b1,b2.[) ")
       holds diff((b3 | ].b1,b2.[) ",b4) = 1 / diff(b3,(b3 | ].b1,b2.[) " . b4));

:: FDIFF_2:th 49
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
   st b2 is_differentiable_in b1
for b3 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
for b4 being Function-like constant quasi_total Relation of NAT,REAL
      st rng b4 = {b1} & rng (b3 + b4) c= dom b2 & rng ((- b3) + b4) c= dom b2
   holds (2 (#) b3) " (#) ((b2 * (b4 + b3)) - (b2 * (b4 - b3))) is convergent &
    lim ((2 (#) b3) " (#) ((b2 * (b4 + b3)) - (b2 * (b4 - b3)))) = diff(b2,b1);