Article NORMSP_1, MML version 4.99.1005

:: NORMSP_1:structnot 1 => NORMSP_1:struct 1
definition
  struct(RLSStruct) NORMSTR(#
    carrier -> set,
    ZeroF -> Element of the carrier of it,
    addF -> Function-like quasi_total Relation of [:the carrier of it,the carrier of it:],the carrier of it,
    Mult -> Function-like quasi_total Relation of [:REAL,the carrier of it:],the carrier of it,
    norm -> Function-like quasi_total Relation of the carrier of it,REAL
  #);
end;

:: NORMSP_1:attrnot 1 => NORMSP_1:attr 1
definition
  let a1 be NORMSTR;
  attr a1 is strict;
end;

:: NORMSP_1:exreg 1
registration
  cluster strict NORMSTR;
end;

:: NORMSP_1:aggrnot 1 => NORMSP_1:aggr 1
definition
  let a1 be set;
  let a2 be Element of a1;
  let a3 be Function-like quasi_total Relation of [:a1,a1:],a1;
  let a4 be Function-like quasi_total Relation of [:REAL,a1:],a1;
  let a5 be Function-like quasi_total Relation of a1,REAL;
  aggr NORMSTR(#a1,a2,a3,a4,a5#) -> strict NORMSTR;
end;

:: NORMSP_1:selnot 1 => NORMSP_1:sel 1
definition
  let a1 be NORMSTR;
  sel the norm of a1 -> Function-like quasi_total Relation of the carrier of a1,REAL;
end;

:: NORMSP_1:exreg 2
registration
  cluster non empty strict NORMSTR;
end;

:: NORMSP_1:funcnot 1 => NORMSP_1:func 1
definition
  let a1 be non empty NORMSTR;
  let a2 be Element of the carrier of a1;
  func ||.A2.|| -> Element of REAL equals
    (the norm of a1) . a2;
end;

:: NORMSP_1:def 1
theorem
for b1 being non empty NORMSTR
for b2 being Element of the carrier of b1 holds
   ||.b2.|| = (the norm of b1) . b2;

:: NORMSP_1:attrnot 2 => NORMSP_1:attr 2
definition
  let a1 be non empty NORMSTR;
  attr a1 is RealNormSpace-like means
    for b1, b2 being Element of the carrier of a1
    for b3 being Element of REAL holds
       (||.b1.|| = 0 implies b1 = 0. a1) &
        (b1 = 0. a1 implies ||.b1.|| = 0) &
        ||.b3 * b1.|| = (abs b3) * ||.b1.|| &
        ||.b1 + b2.|| <= ||.b1.|| + ||.b2.||;
end;

:: NORMSP_1:dfs 2
definiens
  let a1 be non empty NORMSTR;
To prove
     a1 is RealNormSpace-like
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1
    for b3 being Element of REAL holds
       (||.b1.|| = 0 implies b1 = 0. a1) &
        (b1 = 0. a1 implies ||.b1.|| = 0) &
        ||.b3 * b1.|| = (abs b3) * ||.b1.|| &
        ||.b1 + b2.|| <= ||.b1.|| + ||.b2.||;

:: NORMSP_1:def 2
theorem
for b1 being non empty NORMSTR holds
      b1 is RealNormSpace-like
   iff
      for b2, b3 being Element of the carrier of b1
      for b4 being Element of REAL holds
         (||.b2.|| = 0 implies b2 = 0. b1) &
          (b2 = 0. b1 implies ||.b2.|| = 0) &
          ||.b4 * b2.|| = (abs b4) * ||.b2.|| &
          ||.b2 + b3.|| <= ||.b2.|| + ||.b3.||;

:: NORMSP_1:exreg 3
registration
  cluster non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like strict RealNormSpace-like NORMSTR;
end;

:: NORMSP_1:modenot 1
definition
  mode RealNormSpace is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
end;

:: NORMSP_1:funcreg 1
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster ||.0. a1.|| -> empty;
end;

:: NORMSP_1:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
   ||.0. b1.|| = 0;

:: NORMSP_1:th 6
theorem
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 holds
   ||.- b2.|| = ||.b2.||;

:: NORMSP_1:th 7
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
||.b2 - b3.|| <= ||.b2.|| + ||.b3.||;

:: NORMSP_1:th 8
theorem
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 holds
   0 <= ||.b2.||;

:: NORMSP_1:th 9
theorem
for b1, b2 being Element of REAL
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4, b5 being Element of the carrier of b3 holds
||.(b1 * b4) + (b2 * b5).|| <= ((abs b1) * ||.b4.||) + ((abs b2) * ||.b5.||);

:: NORMSP_1:th 10
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
   ||.b2 - b3.|| = 0
iff
   b2 = b3;

:: NORMSP_1:th 11
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
||.b2 - b3.|| = ||.b3 - b2.||;

:: NORMSP_1:th 12
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
||.b2.|| - ||.b3.|| <= ||.b2 - b3.||;

:: NORMSP_1:th 13
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
abs (||.b2.|| - ||.b3.||) <= ||.b2 - b3.||;

:: NORMSP_1:th 14
theorem
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 holds
||.b2 - b3.|| <= ||.b2 - b4.|| + ||.b4 - b3.||;

:: NORMSP_1:th 15
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
      st b2 <> b3
   holds ||.b2 - b3.|| <> 0;

:: NORMSP_1:th 17
theorem
for b1 being Relation-like Function-like set
for b2 being non empty 1-sorted
for b3 being Element of the carrier of b2 holds
      b1 is Function-like quasi_total Relation of NAT,the carrier of b2
   iff
      proj1 b1 = NAT &
       (for b4 being set
             st b4 in NAT
          holds b1 . b4 is Element of the carrier of b2);

:: NORMSP_1:th 19
theorem
for b1 being non empty 1-sorted
for b2 being Element of the carrier of b1 holds
   ex b3 being Function-like quasi_total Relation of NAT,the carrier of b1 st
      proj2 b3 = {b2};

:: NORMSP_1:th 20
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st ex b3 being Element of the carrier of b1 st
           for b4 being Element of NAT holds
              b2 . b4 = b3
   holds ex b3 being Element of the carrier of b1 st
      proj2 b2 = {b3};

:: NORMSP_1:th 21
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
   st ex b3 being Element of the carrier of b1 st
        proj2 b2 = {b3}
for b3 being Element of NAT holds
   b2 . b3 = b2 . (b3 + 1);

:: NORMSP_1:th 22
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
   st for b3 being Element of NAT holds
        b2 . b3 = b2 . (b3 + 1)
for b3, b4 being Element of NAT holds
b2 . b3 = b2 . (b3 + b4);

:: NORMSP_1:th 23
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
   st for b3, b4 being Element of NAT holds
     b2 . b3 = b2 . (b3 + b4)
for b3, b4 being Element of NAT holds
b2 . b3 = b2 . b4;

:: NORMSP_1:th 24
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st for b3, b4 being Element of NAT holds
        b2 . b3 = b2 . b4
   holds ex b3 being Element of the carrier of b1 st
      for b4 being Element of NAT holds
         b2 . b4 = b3;

:: NORMSP_1:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
   ex b2 being Function-like quasi_total Relation of NAT,the carrier of b1 st
      proj2 b2 = {0. b1};

:: NORMSP_1:attrnot 3 => NORMSP_1:attr 3
definition
  let a1 be non empty 1-sorted;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  redefine attr a2 is constant means
    ex b1 being Element of the carrier of a1 st
       for b2 being Element of NAT holds
          a2 . b2 = b1;
end;

:: NORMSP_1:dfs 3
definiens
  let a1 be non empty 1-sorted;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
     a1 is constant
it is sufficient to prove
  thus ex b1 being Element of the carrier of a1 st
       for b2 being Element of NAT holds
          a2 . b2 = b1;

:: NORMSP_1:def 4
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is constant
   iff
      ex b3 being Element of the carrier of b1 st
         for b4 being Element of NAT holds
            b2 . b4 = b3;

:: NORMSP_1:th 27
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is constant
   iff
      ex b3 being Element of the carrier of b1 st
         proj2 b2 = {b3};

:: NORMSP_1:funcnot 2 => NORMSP_1:func 2
definition
  let a1 be non empty 1-sorted;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  let a3 be Element of NAT;
  redefine func a2 . a3 -> Element of the carrier of a1;
end;

:: NORMSP_1:funcnot 3 => NORMSP_1:func 3
definition
  let a1 be non empty addLoopStr;
  let a2, a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
  func A2 + A3 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
    for b1 being Element of NAT holds
       it . b1 = (a2 . b1) + (a3 . b1);
end;

:: NORMSP_1:def 5
theorem
for b1 being non empty addLoopStr
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   b4 = b2 + b3
iff
   for b5 being Element of NAT holds
      b4 . b5 = (b2 . b5) + (b3 . b5);

:: NORMSP_1:funcnot 4 => NORMSP_1:func 4
definition
  let a1 be non empty addLoopStr;
  let a2, a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
  func A2 - A3 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
    for b1 being Element of NAT holds
       it . b1 = (a2 . b1) - (a3 . b1);
end;

:: NORMSP_1:def 6
theorem
for b1 being non empty addLoopStr
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   b4 = b2 - b3
iff
   for b5 being Element of NAT holds
      b4 . b5 = (b2 . b5) - (b3 . b5);

:: NORMSP_1:funcnot 5 => NORMSP_1:func 5
definition
  let a1 be non empty addLoopStr;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  let a3 be Element of the carrier of a1;
  func A2 - A3 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
    for b1 being Element of NAT holds
       it . b1 = (a2 . b1) - a3;
end;

:: NORMSP_1:def 7
theorem
for b1 being non empty addLoopStr
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b4 = b2 - b3
   iff
      for b5 being Element of NAT holds
         b4 . b5 = (b2 . b5) - b3;

:: NORMSP_1:funcnot 6 => NORMSP_1:func 6
definition
  let a1 be non empty RLSStruct;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  let a3 be Element of REAL;
  func A3 * A2 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
    for b1 being Element of NAT holds
       it . b1 = a3 * (a2 . b1);
end;

:: NORMSP_1:def 8
theorem
for b1 being non empty RLSStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of REAL
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b4 = b3 * b2
   iff
      for b5 being Element of NAT holds
         b4 . b5 = b3 * (b2 . b5);

:: NORMSP_1:attrnot 4 => NORMSP_1:attr 4
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  attr a2 is convergent means
    ex b1 being Element of the carrier of a1 st
       for b2 being Element of REAL
             st 0 < b2
          holds ex b3 being Element of NAT st
             for b4 being Element of NAT
                   st b3 <= b4
                holds ||.(a2 . b4) - b1.|| < b2;
end;

:: NORMSP_1:dfs 8
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
     a2 is convergent
it is sufficient to prove
  thus ex b1 being Element of the carrier of a1 st
       for b2 being Element of REAL
             st 0 < b2
          holds ex b3 being Element of NAT st
             for b4 being Element of NAT
                   st b3 <= b4
                holds ||.(a2 . b4) - b1.|| < b2;

:: NORMSP_1:def 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is convergent(b1)
   iff
      ex b3 being Element of the carrier of b1 st
         for b4 being Element of REAL
               st 0 < b4
            holds ex b5 being Element of NAT st
               for b6 being Element of NAT
                     st b5 <= b6
                  holds ||.(b2 . b6) - b3.|| < b4;

:: NORMSP_1:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is convergent(b1) & b3 is convergent(b1)
   holds b2 + b3 is convergent(b1);

:: NORMSP_1:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is convergent(b1) & b3 is convergent(b1)
   holds b2 - b3 is convergent(b1);

:: NORMSP_1:th 36
theorem
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
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b3 is convergent(b1)
   holds b3 - b2 is convergent(b1);

:: NORMSP_1:th 37
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
      st b3 is convergent(b2)
   holds b1 * b3 is convergent(b2);

:: NORMSP_1:funcnot 7 => NORMSP_1:func 7
definition
  let a1 be non empty NORMSTR;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  func ||.A2.|| -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = ||.a2 . b1.||;
end;

:: NORMSP_1:def 10
theorem
for b1 being non empty NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,REAL holds
      b3 = ||.b2.||
   iff
      for b4 being Element of NAT holds
         b3 . b4 = ||.b2 . b4.||;

:: NORMSP_1:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is convergent(b1)
   holds ||.b2.|| is convergent;

:: NORMSP_1:funcnot 8 => NORMSP_1:func 8
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  assume a2 is convergent(a1);
  func lim A2 -> Element of the carrier of a1 means
    for b1 being Element of REAL
          st 0 < b1
       holds ex b2 being Element of NAT st
          for b3 being Element of NAT
                st b2 <= b3
             holds ||.(a2 . b3) - it.|| < b1;
end;

:: NORMSP_1:def 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
   st b2 is convergent(b1)
for b3 being Element of the carrier of b1 holds
      b3 = lim b2
   iff
      for b4 being Element of REAL
            st 0 < b4
         holds ex b5 being Element of NAT st
            for b6 being Element of NAT
                  st b5 <= b6
               holds ||.(b2 . b6) - b3.|| < b4;

:: NORMSP_1:th 41
theorem
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
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b3 is convergent(b1) & lim b3 = b2
   holds ||.b3 - b2.|| is convergent & lim ||.b3 - b2.|| = 0;

:: NORMSP_1:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is convergent(b1) & b3 is convergent(b1)
   holds lim (b2 + b3) = (lim b2) + lim b3;

:: NORMSP_1:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is convergent(b1) & b3 is convergent(b1)
   holds lim (b2 - b3) = (lim b2) - lim b3;

:: NORMSP_1:th 44
theorem
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
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b3 is convergent(b1)
   holds lim (b3 - b2) = (lim b3) - b2;

:: NORMSP_1:th 45
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
      st b3 is convergent(b2)
   holds lim (b1 * b3) = b1 * lim b3;