Article LMOD_6, MML version 4.99.1005

:: LMOD_6:funcnot 1 => VECTSP_5:func 3
notation
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
  synonym Submodules a2 for Subspaces a2;
end;

:: LMOD_6:th 1
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b3, b4 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b2
      st b3 = VectSpStr(#the carrier of b4,the addF of b4,the ZeroF of b4,the lmult of b4#)
   holds    b1 in b3
   iff
      b1 in b4;

:: LMOD_6:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being Element of the carrier of b1
for b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Element of the carrier of b3
for b5 being Element of the carrier of VectSpStr(#the carrier of b3,the addF of b3,the ZeroF of b3,the lmult of b3#)
      st b4 = b5
   holds b2 * b4 = b2 * b5;

:: LMOD_6:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1 holds
   VectSpStr(#the carrier of b2,the addF of b2,the ZeroF of b2,the lmult of b2#) is strict Subspace of b2;

:: LMOD_6:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1 holds
   b2 is Subspace of (Omega). b2;

:: LMOD_6:attrnot 1 => STRUCT_0:attr 7
definition
  let a1 be 1-sorted;
  attr a1 is trivial means
    0. a1 = 1_ a1;
end;

:: LMOD_6:dfs 1
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
To prove
     a1 is trivial
it is sufficient to prove
  thus 0. a1 = 1_ a1;

:: LMOD_6:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr holds
      b1 is trivial
   iff
      0. b1 = 1_ b1;

:: LMOD_6:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
      st b1 is trivial
   holds (for b3 being Element of the carrier of b1 holds
       b3 = 0. b1) &
    (for b3 being Element of the carrier of b2 holds
       b3 = 0. b2);

:: LMOD_6:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
      st b1 is trivial
   holds b2 is trivial;

:: LMOD_6:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1 holds
      b2 is trivial
   iff
      VectSpStr(#the carrier of b2,the addF of b2,the ZeroF of b2,the lmult of b2#) = (0). b2;

:: LMOD_6:funcnot 2 => LMOD_6:func 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
  let a3 be strict Subspace of a2;
  func @ A3 -> Element of Subspaces a2 equals
    a3;
end;

:: LMOD_6:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being strict Subspace of b2 holds
   @ b3 = b3;

:: LMOD_6:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Subspace of b2
for b4 being Element of bool the carrier of b3 holds
   b4 is Element of bool the carrier of b2;

:: LMOD_6:funcnot 3 => LMOD_6:func 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
  let a3 be Subspace of a2;
  let a4 be Element of bool the carrier of a3;
  func @ A4 -> Element of bool the carrier of a2 equals
    a4;
end;

:: LMOD_6:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Subspace of b2
for b4 being Element of bool the carrier of b3 holds
   @ b4 = b4;

:: LMOD_6:funcreg 1
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
  let a3 be Subspace of a2;
  let a4 be non empty Element of bool the carrier of a3;
  cluster @ a4 -> non empty;
end;

:: LMOD_6:th 10
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b2 holds
      b1 in [#] b3
   iff
      b1 in b3;

:: LMOD_6:th 11
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b2
for b4 being Subspace of b3 holds
      b1 in @ [#] b4
   iff
      b1 in b4;

:: LMOD_6:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Element of bool the carrier of b2 holds
   b3 c= [#] Lin b3;

:: LMOD_6:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Element of bool the carrier of b2
for b4 being Linear_Combination of b3
      st b3 <> {} & b3 is linearly-closed(b1, b2)
   holds Sum b4 in b3;

:: LMOD_6:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Element of bool the carrier of b2
      st 0. b2 in b3 & b3 is linearly-closed(b1, b2)
   holds b3 = [#] Lin b3;

:: LMOD_6:funcnot 4 => LMOD_6:func 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
  let a3 be Element of the carrier of a2;
  func <:A3:> -> strict Subspace of a2 equals
    Lin {a3};
end;

:: LMOD_6:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Element of the carrier of b2 holds
   <:b3:> = Lin {b3};

:: LMOD_6:prednot 1 => LMOD_6:pred 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
  pred A2 c= A3 means
    a2 is Subspace of a3;
  reflexivity;
::  for a1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
::  for a2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1 holds
::     a2 c= a2;
end;

:: LMOD_6:dfs 5
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
To prove
     a2 c= a3
it is sufficient to prove
  thus a2 is Subspace of a3;

:: LMOD_6:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1 holds
   b2 c= b3
iff
   b2 is Subspace of b3;

:: LMOD_6:th 16
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b3, b4 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b2
      st b3 c= b4
   holds (b1 in b3 implies b1 in b4) &
    (b1 is Element of the carrier of b3 implies b1 is Element of the carrier of b4);

:: LMOD_6:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being Element of the carrier of b1
for b3, b4 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b5, b6, b7 being Element of the carrier of b3
for b8, b9, b10 being Element of the carrier of b4
      st b3 c= b4
   holds 0. b3 = 0. b4 &
    (b5 = b8 & b6 = b9 implies b5 + b6 = b8 + b9) &
    (b7 = b10 implies b2 * b7 = b2 * b10) &
    (b7 = b10 implies - b10 = - b7) &
    (b5 = b8 & b6 = b9 implies b5 - b6 = b8 - b9) &
    0. b4 in b3 &
    0. b3 in b4 &
    (b8 in b3 & b9 in b3 implies b8 + b9 in b3) &
    (b10 in b3 implies b2 * b10 in b3) &
    (b10 in b3 implies - b10 in b3) &
    (b8 in b3 & b9 in b3 implies b8 - b9 in b3);

:: LMOD_6:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3, b4 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
      st b2 c= b3 & b4 c= b3
   holds 0. b2 = 0. b4 &
    0. b2 in b4 &
    (the carrier of b2 c= the carrier of b4 implies b2 c= b4) &
    (for b5 being Element of the carrier of b3
          st b5 in b2
       holds b5 in b4 implies b2 c= b4) &
    (the carrier of b2 = the carrier of b4 & b2 is strict(b1) & b4 is strict(b1) implies b2 = b4) &
    (0). b2 c= b4;

:: LMOD_6:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed strict VectSp-like VectSpStr over b1
      st b2 c= b3 & b3 c= b2
   holds b2 = b3;

:: LMOD_6:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3, b4 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
      st b2 c= b3 & b3 c= b4
   holds b2 c= b4;

:: LMOD_6:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
      st b2 c= b3
   holds (0). b2 c= b3;

:: LMOD_6:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
      st b2 c= b3
   holds (0). b3 c= b2;

:: LMOD_6:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
      st b2 c= b3
   holds b2 c= (Omega). b3;

:: LMOD_6:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4 being Subspace of b2 holds
b3 c= b3 + b4 & b4 c= b3 + b4;

:: LMOD_6:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4 being Subspace of b2 holds
b3 /\ b4 c= b3 & b3 /\ b4 c= b4;

:: LMOD_6:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4, b5 being Subspace of b2
      st b3 c= b4
   holds b3 /\ b5 c= b4 /\ b5;

:: LMOD_6:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4, b5 being Subspace of b2
      st b3 c= b4
   holds b3 /\ b5 c= b4;

:: LMOD_6:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4, b5 being Subspace of b2
      st b3 c= b4 & b3 c= b5
   holds b3 c= b4 /\ b5;

:: LMOD_6:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4 being Subspace of b2 holds
b3 /\ b4 c= b3 + b4;

:: LMOD_6:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4, b5 being Subspace of b2 holds
(b3 /\ b4) + (b4 /\ b5) c= b4 /\ (b3 + b5);

:: LMOD_6:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4, b5 being Subspace of b2
      st b3 c= b4
   holds b4 /\ (b3 + b5) = (b3 /\ b4) + (b4 /\ b5);

:: LMOD_6:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4, b5 being Subspace of b2 holds
b3 + (b4 /\ b5) c= (b4 + b3) /\ (b3 + b5);

:: LMOD_6:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4, b5 being Subspace of b2
      st b3 c= b4
   holds b4 + (b3 /\ b5) = (b3 + b4) /\ (b4 + b5);

:: LMOD_6:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4, b5 being Subspace of b2
      st b3 c= b4
   holds b3 c= b4 + b5;

:: LMOD_6:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4, b5 being Subspace of b2
      st b3 c= b4 & b5 c= b4
   holds b3 + b5 c= b4;

:: LMOD_6:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4 being Element of bool the carrier of b2
      st b3 c= b4
   holds Lin b3 c= Lin b4;

:: LMOD_6:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4 being Element of bool the carrier of b2 holds
Lin (b3 /\ b4) c= (Lin b3) /\ Lin b4;

:: LMOD_6:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
      st b2 c= b3
   holds [#] b2 c= [#] b3;

:: LMOD_6:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4 being Subspace of b2 holds
   b3 c= b4
iff
   for b5 being Element of the carrier of b2
         st b5 in b3
      holds b5 in b4;

:: LMOD_6:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4 being Subspace of b2 holds
   b3 c= b4
iff
   [#] b3 c= [#] b4;

:: LMOD_6:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4 being Subspace of b2 holds
   b3 c= b4
iff
   @ [#] b3 c= @ [#] b4;

:: LMOD_6:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4, b5 being Subspace of b2 holds
(0). b3 c= b2 & (0). b2 c= b3 & (0). b4 c= b5;