Article ANPROJ_1, MML version 4.99.1005

:: ANPROJ_1:prednot 1 => RLVECT_1:attr 6
notation
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Element of the carrier of a1;
  synonym a2 is_Prop_Vect for non-zero;
end;

:: ANPROJ_1:prednot 2 => ANPROJ_1:pred 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3 be Element of the carrier of a1;
  pred are_Prop A2,A3 means
    ex b1, b2 being Element of REAL st
       b1 * a2 = b2 * a3 & b1 <> 0 & b2 <> 0;
  symmetry;
::  for a1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
::  for a2, a3 being Element of the carrier of a1
::        st are_Prop a2,a3
::     holds are_Prop a3,a2;
  reflexivity;
::  for a1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
::  for a2 being Element of the carrier of a1 holds
::     are_Prop a2,a2;
end;

:: ANPROJ_1:dfs 1
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3 be Element of the carrier of a1;
To prove
     are_Prop a2,a3
it is sufficient to prove
  thus ex b1, b2 being Element of REAL st
       b1 * a2 = b2 * a3 & b1 <> 0 & b2 <> 0;

:: ANPROJ_1:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1 holds
   are_Prop b2,b3
iff
   ex b4, b5 being Element of REAL st
      b4 * b2 = b5 * b3 & b4 <> 0 & b5 <> 0;

:: ANPROJ_1:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1 holds
   are_Prop b2,b3
iff
   ex b4 being Element of REAL st
      b4 <> 0 & b2 = b4 * b3;

:: ANPROJ_1:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
      st are_Prop b2,b3 & are_Prop b3,b4
   holds are_Prop b2,b4;

:: ANPROJ_1:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1 holds
      are_Prop b2,0. b1
   iff
      b2 = 0. b1;

:: ANPROJ_1:prednot 3 => ANPROJ_1:pred 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3, a4 be Element of the carrier of a1;
  pred A2,A3,A4 are_LinDep means
    ex b1, b2, b3 being Element of REAL st
       ((b1 * a2) + (b2 * a3)) + (b3 * a4) = 0. a1 &
        (b1 = 0 & b2 = 0 implies b3 <> 0);
end;

:: ANPROJ_1:dfs 2
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3, a4 be Element of the carrier of a1;
To prove
     a2,a3,a4 are_LinDep
it is sufficient to prove
  thus ex b1, b2, b3 being Element of REAL st
       ((b1 * a2) + (b2 * a3)) + (b3 * a4) = 0. a1 &
        (b1 = 0 & b2 = 0 implies b3 <> 0);

:: ANPROJ_1:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1 holds
   b2,b3,b4 are_LinDep
iff
   ex b5, b6, b7 being Element of REAL st
      ((b5 * b2) + (b6 * b3)) + (b7 * b4) = 0. b1 &
       (b5 = 0 & b6 = 0 implies b7 <> 0);

:: ANPROJ_1:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st are_Prop b2,b3 & are_Prop b4,b5 & are_Prop b6,b7 & b2,b4,b6 are_LinDep
   holds b3,b5,b7 are_LinDep;

:: ANPROJ_1:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
      st b2,b3,b4 are_LinDep
   holds b2,b4,b3 are_LinDep & b3,b2,b4 are_LinDep & b4,b3,b2 are_LinDep & b4,b2,b3 are_LinDep & b3,b4,b2 are_LinDep;

:: ANPROJ_1:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
      st b2 is non-zero(b1) & b3 is non-zero(b1) & not are_Prop b2,b3
   holds    b2,b3,b4 are_LinDep
   iff
      ex b5, b6 being Element of REAL st
         b4 = (b5 * b2) + (b6 * b3);

:: ANPROJ_1:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
for b4, b5, b6, b7 being Element of REAL
      st not are_Prop b2,b3 &
         (b4 * b2) + (b5 * b3) = (b6 * b2) + (b7 * b3) &
         b2 is non-zero(b1) &
         b3 is non-zero(b1)
   holds b4 = b6 & b5 = b7;

:: ANPROJ_1:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
for b5, b6, b7, b8, b9, b10 being Element of REAL
      st not b2,b3,b4 are_LinDep &
         ((b5 * b2) + (b6 * b3)) + (b7 * b4) = ((b8 * b2) + (b9 * b3)) + (b10 * b4)
   holds b5 = b8 & b6 = b9 & b7 = b10;

:: ANPROJ_1:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6, b7, b8, b9 being Element of REAL
      st not are_Prop b2,b3 & b4 = (b6 * b2) + (b7 * b3) & b5 = (b8 * b2) + (b9 * b3) & (b6 * b9) - (b8 * b7) = 0 & b2 is non-zero(b1) & b3 is non-zero(b1) & not are_Prop b4,b5 & b4 <> 0. b1
   holds b5 = 0. b1;

:: ANPROJ_1:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
      st (b2 <> 0. b1 & b3 <> 0. b1 implies b4 = 0. b1)
   holds b2,b3,b4 are_LinDep;

:: ANPROJ_1:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
      st (not are_Prop b2,b3 & not are_Prop b4,b2 implies are_Prop b3,b4)
   holds b4,b2,b3 are_LinDep;

:: ANPROJ_1:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
      st not b2,b3,b4 are_LinDep
   holds b2 is non-zero(b1) & b3 is non-zero(b1) & b4 is non-zero(b1) & not are_Prop b2,b3 & not are_Prop b3,b4 & not are_Prop b4,b2;

:: ANPROJ_1:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
      st b2 + b3 = 0. b1
   holds are_Prop b2,b3;

:: ANPROJ_1:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st not are_Prop b2,b3 & b2,b3,b4 are_LinDep & b2,b3,b5 are_LinDep & b2,b3,b6 are_LinDep & b2 is non-zero(b1) & b3 is non-zero(b1)
   holds b4,b5,b6 are_LinDep;

:: ANPROJ_1:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st not b2,b3,b4 are_LinDep & b2,b3,b5 are_LinDep & b3,b4,b6 are_LinDep
   holds ex b7 being Element of the carrier of b1 st
      b2,b4,b7 are_LinDep & b5,b6,b7 are_LinDep & b7 is non-zero(b1);

:: ANPROJ_1:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
   st not are_Prop b2,b3 & b2 is non-zero(b1) & b3 is non-zero(b1)
for b4, b5 being Element of the carrier of b1 holds
ex b6 being Element of the carrier of b1 st
   b6 is non-zero(b1) & b4,b5,b6 are_LinDep & not are_Prop b4,b6 & not are_Prop b5,b6;

:: ANPROJ_1:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
   st not b2,b3,b4 are_LinDep
for b5, b6 being Element of the carrier of b1
      st b5 is non-zero(b1) & b6 is non-zero(b1) & not are_Prop b5,b6
   holds ex b7 being Element of the carrier of b1 st
      b7 is non-zero(b1) & not b5,b6,b7 are_LinDep;

:: ANPROJ_1:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
      st b2,b3,b4 are_LinDep & b5,b6,b4 are_LinDep & b2,b5,b7 are_LinDep & b3,b6,b7 are_LinDep & b2,b6,b8 are_LinDep & b3,b5,b8 are_LinDep & b7,b4,b8 are_LinDep & b7 is non-zero(b1) & b4 is non-zero(b1) & b8 is non-zero(b1) & not b2,b3,b6 are_LinDep & not b2,b3,b5 are_LinDep & not b2,b5,b6 are_LinDep
   holds b3,b5,b6 are_LinDep;

:: ANPROJ_1:funcnot 1 => ANPROJ_1:func 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  func Proper_Vectors_of A1 -> set means
    for b1 being set holds
          b1 in it
       iff
          b1 <> 0. a1 & b1 is Element of the carrier of a1;
end;

:: ANPROJ_1:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being set holds
      b2 = Proper_Vectors_of b1
   iff
      for b3 being set holds
            b3 in b2
         iff
            b3 <> 0. b1 & b3 is Element of the carrier of b1;

:: ANPROJ_1:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1 holds
      b2 in Proper_Vectors_of b1
   iff
      b2 is non-zero(b1);

:: ANPROJ_1:funcnot 2 => ANPROJ_1:func 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  func Proportionality_as_EqRel_of A1 -> symmetric transitive total Relation of Proper_Vectors_of a1,Proper_Vectors_of a1 means
    for b1, b2 being set holds
       [b1,b2] in it
    iff
       b1 in Proper_Vectors_of a1 &
        b2 in Proper_Vectors_of a1 &
        (ex b3, b4 being Element of the carrier of a1 st
           b1 = b3 & b2 = b4 & are_Prop b3,b4);
end;

:: ANPROJ_1:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being symmetric transitive total Relation of Proper_Vectors_of b1,Proper_Vectors_of b1 holds
      b2 = Proportionality_as_EqRel_of b1
   iff
      for b3, b4 being set holds
         [b3,b4] in b2
      iff
         b3 in Proper_Vectors_of b1 &
          b4 in Proper_Vectors_of b1 &
          (ex b5, b6 being Element of the carrier of b1 st
             b3 = b5 & b4 = b6 & are_Prop b5,b6);

:: ANPROJ_1:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being set
      st [b2,b3] in Proportionality_as_EqRel_of b1
   holds b2 is Element of the carrier of b1 & b3 is Element of the carrier of b1;

:: ANPROJ_1:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1 holds
   [b2,b3] in Proportionality_as_EqRel_of b1
iff
   b2 is non-zero(b1) & b3 is non-zero(b1) & are_Prop b2,b3;

:: ANPROJ_1:funcnot 3 => ANPROJ_1:func 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Element of the carrier of a1;
  func Dir A2 -> Element of bool Proper_Vectors_of a1 equals
    Class(Proportionality_as_EqRel_of a1,a2);
end;

:: ANPROJ_1:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1 holds
   Dir b2 = Class(Proportionality_as_EqRel_of b1,b2);

:: ANPROJ_1:funcnot 4 => ANPROJ_1:func 4
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  func ProjectivePoints A1 -> set means
    ex b1 being Element of bool bool Proper_Vectors_of a1 st
       b1 = Class Proportionality_as_EqRel_of a1 & it = b1;
end;

:: ANPROJ_1:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being set holds
      b2 = ProjectivePoints b1
   iff
      ex b3 being Element of bool bool Proper_Vectors_of b1 st
         b3 = Class Proportionality_as_EqRel_of b1 & b2 = b3;

:: ANPROJ_1:attrnot 1 => STRUCT_0:attr 7
definition
  let a1 be 1-sorted;
  attr a1 is trivial means
    for b1 being Element of the carrier of a1 holds
       b1 = 0. a1;
end;

:: ANPROJ_1:dfs 7
definiens
  let a1 be non empty ZeroStr;
To prove
     a1 is trivial
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       b1 = 0. a1;

:: ANPROJ_1:def 8
theorem
for b1 being non empty ZeroStr holds
      b1 is trivial
   iff
      for b2 being Element of the carrier of b1 holds
         b2 = 0. b1;

:: ANPROJ_1:exreg 1
registration
  cluster non empty non trivial left_complementable right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like zeroed RLSStruct;
end;

:: ANPROJ_1:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
      b1 is non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
   iff
      ex b2 being Element of the carrier of b1 st
         b2 in Proper_Vectors_of b1;

:: ANPROJ_1:funcreg 1
registration
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster Proper_Vectors_of a1 -> non empty;
end;

:: ANPROJ_1:funcreg 2
registration
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster ProjectivePoints a1 -> non empty;
end;

:: ANPROJ_1:th 34
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1
      st b2 is non-zero(b1)
   holds Dir b2 is Element of ProjectivePoints b1;

:: ANPROJ_1:th 35
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
      st b2 is non-zero(b1) & b3 is non-zero(b1)
   holds    Dir b2 = Dir b3
   iff
      are_Prop b2,b3;

:: ANPROJ_1:funcnot 5 => ANPROJ_1:func 5
definition
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  func ProjectiveCollinearity A1 -> Relation3 of ProjectivePoints a1 means
    for b1, b2, b3 being set holds
       [b1,b2,b3] in it
    iff
       ex b4, b5, b6 being Element of the carrier of a1 st
          b1 = Dir b4 & b2 = Dir b5 & b3 = Dir b6 & b4 is non-zero(a1) & b5 is non-zero(a1) & b6 is non-zero(a1) & b4,b5,b6 are_LinDep;
end;

:: ANPROJ_1:def 9
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Relation3 of ProjectivePoints b1 holds
      b2 = ProjectiveCollinearity b1
   iff
      for b3, b4, b5 being set holds
         [b3,b4,b5] in b2
      iff
         ex b6, b7, b8 being Element of the carrier of b1 st
            b3 = Dir b6 & b4 = Dir b7 & b5 = Dir b8 & b6 is non-zero(b1) & b7 is non-zero(b1) & b8 is non-zero(b1) & b6,b7,b8 are_LinDep;

:: ANPROJ_1:funcnot 6 => ANPROJ_1:func 6
definition
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  func ProjectiveSpace A1 -> strict CollStr equals
    CollStr(#ProjectivePoints a1,ProjectiveCollinearity a1#);
end;

:: ANPROJ_1:def 10
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   ProjectiveSpace b1 = CollStr(#ProjectivePoints b1,ProjectiveCollinearity b1#);

:: ANPROJ_1:funcreg 3
registration
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster ProjectiveSpace a1 -> non empty strict;
end;

:: ANPROJ_1:th 39
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   the carrier of ProjectiveSpace b1 = ProjectivePoints b1 & the Collinearity of ProjectiveSpace b1 = ProjectiveCollinearity b1;

:: ANPROJ_1:th 40
theorem
for b1, b2, b3 being set
for b4 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
      st [b1,b2,b3] in the Collinearity of ProjectiveSpace b4
   holds ex b5, b6, b7 being Element of the carrier of b4 st
      b1 = Dir b5 & b2 = Dir b6 & b3 = Dir b7 & b5 is non-zero(b4) & b6 is non-zero(b4) & b7 is non-zero(b4) & b5,b6,b7 are_LinDep;

:: ANPROJ_1:th 41
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
      st b2 is non-zero(b1) & b3 is non-zero(b1) & b4 is non-zero(b1)
   holds    [Dir b2,Dir b3,Dir b4] in the Collinearity of ProjectiveSpace b1
   iff
      b2,b3,b4 are_LinDep;

:: ANPROJ_1:th 42
theorem
for b1 being set
for b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
      b1 is Element of the carrier of ProjectiveSpace b2
   iff
      ex b3 being Element of the carrier of b2 st
         b3 is non-zero(b2) & b1 = Dir b3;