Article VECTSP_7, MML version 4.99.1005

:: VECTSP_7:attrnot 1 => VECTSP_7:attr 1
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
  let a3 be Element of bool the carrier of a2;
  attr a3 is linearly-independent means
    for b1 being Linear_Combination of a3
          st Sum b1 = 0. a2
       holds Carrier b1 = {};
end;

:: VECTSP_7:dfs 1
definiens
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
  let a3 be Element of bool the carrier of a2;
To prove
     a3 is linearly-independent
it is sufficient to prove
  thus for b1 being Linear_Combination of a3
          st Sum b1 = 0. a2
       holds Carrier b1 = {};

:: VECTSP_7:def 1
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2 holds
      b3 is linearly-independent(b1, b2)
   iff
      for b4 being Linear_Combination of b3
            st Sum b4 = 0. b2
         holds Carrier b4 = {};

:: VECTSP_7:attrnot 2 => VECTSP_7:attr 1
notation
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
  let a3 be Element of bool the carrier of a2;
  antonym linearly-dependent for linearly-independent;
end;

:: VECTSP_7:th 2
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of bool the carrier of b2
      st b3 c= b4 & b4 is linearly-independent(b1, b2)
   holds b3 is linearly-independent(b1, b2);

:: VECTSP_7:th 3
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
      st b3 is linearly-independent(b1, b2)
   holds not 0. b2 in b3;

:: VECTSP_7:th 4
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
   {} the carrier of b2 is linearly-independent(b1, b2);

:: VECTSP_7:exreg 1
registration
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
  cluster linearly-independent Element of bool the carrier of a2;
end;

:: VECTSP_7:th 5
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2 holds
      {b3} is linearly-independent(b1, b2)
   iff
      b3 <> 0. b2;

:: VECTSP_7:th 6
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
      st {b3,b4} is linearly-independent(b1, b2)
   holds b3 <> 0. b2 & b4 <> 0. b2;

:: VECTSP_7:th 7
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2 holds
   {b3,0. b2} is linearly-dependent(b1, b2) & {0. b2,b3} is linearly-dependent(b1, b2);

:: VECTSP_7:th 8
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2 holds
   b3 <> b4 & {b3,b4} is linearly-independent(b1, b2)
iff
   b4 <> 0. b2 &
    (for b5 being Element of the carrier of b1 holds
       b3 <> b5 * b4);

:: VECTSP_7:th 9
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2 holds
   b3 <> b4 & {b3,b4} is linearly-independent(b1, b2)
iff
   for b5, b6 being Element of the carrier of b1
         st (b5 * b3) + (b6 * b4) = 0. b2
      holds b5 = 0. b1 & b6 = 0. b1;

:: VECTSP_7:funcnot 1 => VECTSP_7:func 1
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
  let a3 be Element of bool the carrier of a2;
  func Lin A3 -> strict Subspace of a2 means
    the carrier of it = {Sum b1 where b1 is Linear_Combination of a3: TRUE};
end;

:: VECTSP_7:def 2
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
for b4 being strict Subspace of b2 holds
      b4 = Lin b3
   iff
      the carrier of b4 = {Sum b5 where b5 is Linear_Combination of b3: TRUE};

:: VECTSP_7:th 12
theorem
for b1 being set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b2
for b4 being Element of bool the carrier of b3 holds
      b1 in Lin b4
   iff
      ex b5 being Linear_Combination of b4 st
         b1 = Sum b5;

:: VECTSP_7:th 13
theorem
for b1 being set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b2
for b4 being Element of bool the carrier of b3
      st b1 in b4
   holds b1 in Lin b4;

:: VECTSP_7:th 14
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
   Lin {} the carrier of b2 = (0). b2;

:: VECTSP_7:th 15
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
      st Lin b3 = (0). b2 & b3 <> {}
   holds b3 = {0. b2};

:: VECTSP_7:th 16
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
for b4 being strict Subspace of b2
      st b3 = the carrier of b4
   holds Lin b3 = b4;

:: VECTSP_7:th 17
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
      st b3 = the carrier of b2
   holds Lin b3 = b2;

:: VECTSP_7:th 18
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of bool the carrier of b2
      st b3 c= b4
   holds Lin b3 is Subspace of Lin b4;

:: VECTSP_7:th 19
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of bool the carrier of b2
      st Lin b3 = b2 & b3 c= b4
   holds Lin b4 = b2;

:: VECTSP_7:th 20
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of bool the carrier of b2 holds
Lin (b3 \/ b4) = (Lin b3) + Lin b4;

:: VECTSP_7:th 21
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of bool the carrier of b2 holds
Lin (b3 /\ b4) is Subspace of (Lin b3) /\ Lin b4;

:: VECTSP_7:th 22
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
      st b3 is linearly-independent(b1, b2)
   holds ex b4 being Element of bool the carrier of b2 st
      b3 c= b4 &
       b4 is linearly-independent(b1, b2) &
       Lin b4 = VectSpStr(#the carrier of b2,the addF of b2,the ZeroF of b2,the lmult of b2#);

:: VECTSP_7:th 23
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
      st Lin b3 = b2
   holds ex b4 being Element of bool the carrier of b2 st
      b4 c= b3 & b4 is linearly-independent(b1, b2) & Lin b4 = b2;

:: VECTSP_7:modenot 1 => VECTSP_7:mode 1
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
  mode Basis of A2 -> Element of bool the carrier of a2 means
    it is linearly-independent(a1, a2) &
     Lin it = VectSpStr(#the carrier of a2,the addF of a2,the ZeroF of a2,the lmult of a2#);
end;

:: VECTSP_7:dfs 3
definiens
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
  let a3 be Element of bool the carrier of a2;
To prove
     a3 is Basis of a2
it is sufficient to prove
  thus a3 is linearly-independent(a1, a2) &
     Lin a3 = VectSpStr(#the carrier of a2,the addF of a2,the ZeroF of a2,the lmult of a2#);

:: VECTSP_7:def 3
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2 holds
      b3 is Basis of b2
   iff
      b3 is linearly-independent(b1, b2) &
       Lin b3 = VectSpStr(#the carrier of b2,the addF of b2,the ZeroF of b2,the lmult of b2#);

:: VECTSP_7:th 27
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
      st b3 is linearly-independent(b1, b2)
   holds ex b4 being Basis of b2 st
      b3 c= b4;

:: VECTSP_7:th 28
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
      st Lin b3 = b2
   holds ex b4 being Basis of b2 st
      b4 c= b3;