Article RUSUB_4, MML version 4.99.1005

:: RUSUB_4:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being finite Element of bool the carrier of b1
for b4 being Element of the carrier of b1
      st b4 in Lin (b2 \/ b3) & not b4 in Lin b3
   holds ex b5 being Element of the carrier of b1 st
      b5 in b2 &
       b5 in Lin (((b2 \/ b3) \ {b5}) \/ {b4});

:: RUSUB_4:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being finite Element of bool the carrier of b1
      st UNITSTR(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1,the scalar of b1#) = Lin b2 &
         b3 is linearly-independent(b1)
   holds card b3 <= card b2 &
    (ex b4 being finite Element of bool the carrier of b1 st
       b4 c= b2 &
        card b4 = (card b2) - card b3 &
        UNITSTR(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1,the scalar of b1#) = Lin (b3 \/ b4));

:: RUSUB_4:attrnot 1 => RUSUB_4:attr 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  attr a1 is finite-dimensional means
    ex b1 being finite Element of bool the carrier of a1 st
       b1 is Basis of a1;
end;

:: RUSUB_4:dfs 1
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
To prove
     a1 is finite-dimensional
it is sufficient to prove
  thus ex b1 being finite Element of bool the carrier of a1 st
       b1 is Basis of a1;

:: RUSUB_4:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
      b1 is finite-dimensional
   iff
      ex b2 being finite Element of bool the carrier of b1 st
         b2 is Basis of b1;

:: RUSUB_4:exreg 1
registration
  cluster non empty left_complementable right_complementable Abelian add-associative right_zeroed RealLinearSpace-like zeroed strict RealUnitarySpace-like finite-dimensional UNITSTR;
end;

:: RUSUB_4:attrnot 2 => RUSUB_4:attr 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  attr a1 is finite-dimensional means
    ex b1 being finite Element of bool the carrier of a1 st
       b1 is Basis of a1;
end;

:: RUSUB_4:dfs 2
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
To prove
     a1 is finite-dimensional
it is sufficient to prove
  thus ex b1 being finite Element of bool the carrier of a1 st
       b1 is Basis of a1;

:: RUSUB_4:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
      b1 is finite-dimensional
   iff
      ex b2 being finite Element of bool the carrier of b1 st
         b2 is Basis of b1;

:: RUSUB_4:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
   st b1 is finite-dimensional
for b2 being Basis of b1 holds
   b2 is finite;

:: RUSUB_4:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
      st b1 is finite-dimensional & b2 is linearly-independent(b1)
   holds b2 is finite;

:: RUSUB_4:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Basis of b1
      st b1 is finite-dimensional
   holds Card b2 = Card b3;

:: RUSUB_4:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
   (0). b1 is finite-dimensional;

:: RUSUB_4:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
      st b1 is finite-dimensional
   holds b2 is finite-dimensional;

:: RUSUB_4:exreg 2
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  cluster non empty left_complementable right_complementable Abelian add-associative right_zeroed RealLinearSpace-like zeroed strict RealUnitarySpace-like finite-dimensional Subspace of a1;
end;

:: RUSUB_4:condreg 1
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR;
  cluster -> finite-dimensional (Subspace of a1);
end;

:: RUSUB_4:exreg 3
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR;
  cluster non empty left_complementable right_complementable Abelian add-associative right_zeroed RealLinearSpace-like zeroed strict RealUnitarySpace-like finite-dimensional Subspace of a1;
end;

:: RUSUB_4:funcnot 1 => RUSUB_4:func 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  assume a1 is finite-dimensional;
  func dim A1 -> Element of NAT means
    for b1 being Basis of a1 holds
       it = Card b1;
end;

:: RUSUB_4:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
   st b1 is finite-dimensional
for b2 being Element of NAT holds
      b2 = dim b1
   iff
      for b3 being Basis of b1 holds
         b2 = Card b3;

:: RUSUB_4:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2 being Subspace of b1 holds
   dim b2 <= dim b1;

:: RUSUB_4:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2 being Element of bool the carrier of b1
      st b2 is linearly-independent(b1)
   holds Card b2 = dim Lin b2;

:: RUSUB_4:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR holds
   dim b1 = dim (Omega). b1;

:: RUSUB_4:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2 being Subspace of b1 holds
      dim b1 = dim b2
   iff
      (Omega). b1 = (Omega). b2;

:: RUSUB_4:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR holds
      dim b1 = 0
   iff
      (Omega). b1 = (0). b1;

:: RUSUB_4:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR holds
      dim b1 = 1
   iff
      ex b2 being Element of the carrier of b1 st
         b2 <> 0. b1 & (Omega). b1 = Lin {b2};

:: RUSUB_4:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR holds
      dim b1 = 2
   iff
      ex b2, b3 being Element of the carrier of b1 st
         b2 <> b3 & {b2,b3} is linearly-independent(b1) & (Omega). b1 = Lin {b2,b3};

:: RUSUB_4:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2, b3 being Subspace of b1 holds
(dim (b2 + b3)) + dim (b2 /\ b3) = (dim b2) + dim b3;

:: RUSUB_4:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2, b3 being Subspace of b1 holds
((dim b2) + dim b3) - dim b1 <= dim (b2 /\ b3);

:: RUSUB_4:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2, b3 being Subspace of b1
      st b1 is_the_direct_sum_of b2,b3
   holds dim b1 = (dim b2) + dim b3;

:: RUSUB_4:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2 being Subspace of b1
for b3 being Element of NAT holds
      b3 <= dim b1
   iff
      ex b4 being strict Subspace of b1 st
         dim b4 = b3;

:: RUSUB_4:funcnot 2 => RUSUB_4:func 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR;
  let a2 be Element of NAT;
  func A2 Subspaces_of A1 -> set means
    for b1 being set holds
          b1 in it
       iff
          ex b2 being strict Subspace of a1 st
             b2 = b1 & dim b2 = a2;
end;

:: RUSUB_4:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2 being Element of NAT
for b3 being set holds
      b3 = b2 Subspaces_of b1
   iff
      for b4 being set holds
            b4 in b3
         iff
            ex b5 being strict Subspace of b1 st
               b5 = b4 & dim b5 = b2;

:: RUSUB_4:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2 being Element of NAT
      st b2 <= dim b1
   holds b2 Subspaces_of b1 is not empty;

:: RUSUB_4:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2 being Element of NAT
      st dim b1 < b2
   holds b2 Subspaces_of b1 = {};

:: RUSUB_4:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2 being Subspace of b1
for b3 being Element of NAT holds
   b3 Subspaces_of b2 c= b3 Subspaces_of b1;

:: RUSUB_4:attrnot 3 => RUSUB_4:attr 2
definition
  let a1 be non empty RLSStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is Affine means
    for b1, b2 being Element of the carrier of a1
    for b3 being Element of REAL
          st b1 in a2 & b2 in a2
       holds ((1 - b3) * b1) + (b3 * b2) in a2;
end;

:: RUSUB_4:dfs 5
definiens
  let a1 be non empty RLSStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is Affine
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1
    for b3 being Element of REAL
          st b1 in a2 & b2 in a2
       holds ((1 - b3) * b1) + (b3 * b2) in a2;

:: RUSUB_4:def 5
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is Affine(b1)
   iff
      for b3, b4 being Element of the carrier of b1
      for b5 being Element of REAL
            st b3 in b2 & b4 in b2
         holds ((1 - b5) * b3) + (b5 * b4) in b2;

:: RUSUB_4:th 22
theorem
for b1 being non empty RLSStruct holds
   [#] b1 is Affine(b1) & {} b1 is Affine(b1);

:: RUSUB_4:th 23
theorem
for b1 being non empty RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1 holds
   {b2} is Affine(b1);

:: RUSUB_4:exreg 4
registration
  let a1 be non empty RLSStruct;
  cluster non empty Affine Element of bool the carrier of a1;
end;

:: RUSUB_4:exreg 5
registration
  let a1 be non empty RLSStruct;
  cluster empty Affine Element of bool the carrier of a1;
end;

:: RUSUB_4:funcnot 3 => RUSUB_4:func 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Subspace of a1;
  func Up A2 -> non empty Element of bool the carrier of a1 equals
    the carrier of a2;
end;

:: RUSUB_4:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1 holds
   Up b2 = the carrier of b2;

:: RUSUB_4:funcnot 4 => RUSUB_4:func 4
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2 be Subspace of a1;
  func Up A2 -> non empty Element of bool the carrier of a1 equals
    the carrier of a2;
end;

:: RUSUB_4:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1 holds
   Up b2 = the carrier of b2;

:: RUSUB_4:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1 holds
   Up b2 is Affine(b1) & 0. b1 in the carrier of b2;

:: RUSUB_4:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Affine Element of bool the carrier of b1
   st 0. b1 in b2
for b3 being Element of the carrier of b1
for b4 being Element of REAL
      st b3 in b2
   holds b4 * b3 in b2;

:: RUSUB_4:attrnot 4 => RUSUB_4:attr 3
definition
  let a1 be non empty RLSStruct;
  let a2 be non empty Element of bool the carrier of a1;
  attr a2 is Subspace-like means
    0. a1 in a2 &
     (for b1, b2 being Element of the carrier of a1
     for b3 being Element of REAL
           st b1 in a2 & b2 in a2
        holds b1 + b2 in a2 & b3 * b1 in a2);
end;

:: RUSUB_4:dfs 8
definiens
  let a1 be non empty RLSStruct;
  let a2 be non empty Element of bool the carrier of a1;
To prove
     a2 is Subspace-like
it is sufficient to prove
  thus 0. a1 in a2 &
     (for b1, b2 being Element of the carrier of a1
     for b3 being Element of REAL
           st b1 in a2 & b2 in a2
        holds b1 + b2 in a2 & b3 * b1 in a2);

:: RUSUB_4:def 8
theorem
for b1 being non empty RLSStruct
for b2 being non empty Element of bool the carrier of b1 holds
      b2 is Subspace-like(b1)
   iff
      0. b1 in b2 &
       (for b3, b4 being Element of the carrier of b1
       for b5 being Element of REAL
             st b3 in b2 & b4 in b2
          holds b3 + b4 in b2 & b5 * b3 in b2);

:: RUSUB_4:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty Affine Element of bool the carrier of b1
      st 0. b1 in b2
   holds b2 is Subspace-like(b1) & b2 = the carrier of Lin b2;

:: RUSUB_4:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1 holds
   Up b2 is Subspace-like(b1);

:: RUSUB_4:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being non empty Affine Element of bool the carrier of b1
      st 0. b1 in b2
   holds b2 = the carrier of Lin b2;

:: RUSUB_4:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1 holds
   Up b2 is Subspace-like(b1);

:: RUSUB_4:funcnot 5 => RUSUB_4:func 5
definition
  let a1 be non empty addLoopStr;
  let a2 be Element of bool the carrier of a1;
  let a3 be Element of the carrier of a1;
  func A3 + A2 -> Element of bool the carrier of a1 equals
    {a3 + b1 where b1 is Element of the carrier of a1: b1 in a2};
end;

:: RUSUB_4:def 9
theorem
for b1 being non empty addLoopStr
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
   b3 + b2 = {b3 + b4 where b4 is Element of the carrier of b1: b4 in b2};

:: RUSUB_4:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being strict Subspace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of the carrier of b1
      st Up b2 = b3
   holds b4 + b2 = b4 + b3;

:: RUSUB_4:th 33
theorem
for b1 being non empty Abelian add-associative RealLinearSpace-like RLSStruct
for b2 being Affine Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
   b3 + b2 is Affine(b1);

:: RUSUB_4:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being strict Subspace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of the carrier of b1
      st Up b2 = b3
   holds b4 + b2 = b4 + b3;

:: RUSUB_4:funcnot 6 => RUSUB_4:func 6
definition
  let a1 be non empty addLoopStr;
  let a2, a3 be Element of bool the carrier of a1;
  func A2 + A3 -> Element of bool the carrier of a1 equals
    {b1 + b2 where b1 is Element of the carrier of a1, b2 is Element of the carrier of a1: b1 in a2 & b2 in a3};
end;

:: RUSUB_4:def 10
theorem
for b1 being non empty addLoopStr
for b2, b3 being Element of bool the carrier of b1 holds
b2 + b3 = {b4 + b5 where b4 is Element of the carrier of b1, b5 is Element of the carrier of b1: b4 in b2 & b5 in b3};

:: RUSUB_4:th 35
theorem
for b1 being non empty Abelian addLoopStr
for b2, b3 being Element of bool the carrier of b1 holds
b3 + b2 = b2 + b3;

:: RUSUB_4:funcnot 7 => RUSUB_4:func 7
definition
  let a1 be non empty Abelian addLoopStr;
  let a2, a3 be Element of bool the carrier of a1;
  redefine func a2 + a3 -> Element of bool the carrier of a1;
  commutativity;
::  for a1 being non empty Abelian addLoopStr
::  for a2, a3 being Element of bool the carrier of a1 holds
::  a2 + a3 = a3 + a2;
end;

:: RUSUB_4:th 36
theorem
for b1 being non empty addLoopStr
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
   {b3} + b2 = b3 + b2;

:: RUSUB_4:th 37
theorem
for b1 being non empty Abelian add-associative RealLinearSpace-like RLSStruct
for b2 being Affine Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
   {b3} + b2 is Affine(b1);

:: RUSUB_4:th 38
theorem
for b1 being non empty RLSStruct
for b2, b3 being Affine Element of bool the carrier of b1 holds
b2 /\ b3 is Affine(b1);

:: RUSUB_4:th 39
theorem
for b1 being non empty Abelian add-associative RealLinearSpace-like RLSStruct
for b2, b3 being Affine Element of bool the carrier of b1 holds
b2 + b3 is Affine(b1);