Article MOD_3, MML version 4.99.1005

:: MOD_3:th 2
theorem
for b1 being non empty non degenerated right_complementable add-associative right_zeroed doubleLoopStr holds
   0. b1 <> - 1. b1;

:: MOD_3:th 6
theorem
for b1 being non empty right_complementable associative 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 Linear_Combination of b2
for b4 being finite Element of bool the carrier of b2
      st Carrier b3 c= b4
   holds ex b5 being FinSequence of the carrier of b2 st
      b5 is one-to-one & proj2 b5 = b4 & Sum b3 = Sum (b3 (#) b5);

:: MOD_3:th 7
theorem
for b1 being non empty right_complementable associative 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 Linear_Combination of b2
for b4 being Element of the carrier of b1 holds
   Sum (b4 * b3) = b4 * Sum b3;

:: MOD_3:funcnot 1 => MOD_3:func 1
definition
  let a1 be non empty right_complementable associative 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;

:: MOD_3:def 1
theorem
for b1 being non empty right_complementable associative 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};

:: MOD_3:th 11
theorem
for b1 being set
for b2 being non empty right_complementable associative 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;

:: MOD_3:th 12
theorem
for b1 being set
for b2 being non empty right_complementable associative 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;

:: MOD_3:th 13
theorem
for b1 being non empty right_complementable associative 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;

:: MOD_3:th 14
theorem
for b1 being non empty right_complementable associative 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};

:: MOD_3:th 15
theorem
for b1 being non empty right_complementable associative 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 0. b1 <> 1. b1 & b3 = the carrier of b4
   holds Lin b3 = b4;

:: MOD_3:th 16
theorem
for b1 being non empty right_complementable associative 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 0. b1 <> 1. b1 & b3 = the carrier of b2
   holds Lin b3 = b2;

:: MOD_3:th 17
theorem
for b1 being non empty right_complementable associative 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;

:: MOD_3:th 18
theorem
for b1 being non empty right_complementable associative 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 Lin b3 = b2 & b3 c= b4
   holds Lin b4 = b2;

:: MOD_3:th 19
theorem
for b1 being non empty right_complementable associative 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;

:: MOD_3:th 20
theorem
for b1 being non empty right_complementable associative 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;

:: MOD_3:attrnot 1 => MOD_3:attr 1
definition
  let a1 be non empty right_complementable associative 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 base means
    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#);
end;

:: MOD_3:dfs 2
definiens
  let a1 be non empty right_complementable associative 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 base
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#);

:: MOD_3:def 2
theorem
for b1 being non empty right_complementable associative 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 base(b1, 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#);

:: MOD_3:attrnot 2 => MOD_3:attr 2
definition
  let a1 be non empty right_complementable associative 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;
  attr a2 is free means
    ex b1 being Element of bool the carrier of a2 st
       b1 is base(a1, a2);
end;

:: MOD_3:dfs 3
definiens
  let a1 be non empty right_complementable associative 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;
To prove
     a2 is free
it is sufficient to prove
  thus ex b1 being Element of bool the carrier of a2 st
       b1 is base(a1, a2);

:: MOD_3:def 3
theorem
for b1 being non empty right_complementable associative 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
      b2 is free(b1)
   iff
      ex b3 being Element of bool the carrier of b2 st
         b3 is base(b1, b2);

:: MOD_3:th 21
theorem
for b1 being non empty right_complementable associative 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
   (0). b2 is free(b1);

:: MOD_3:exreg 1
registration
  let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  cluster non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed free VectSpStr over a1;
end;

:: MOD_3:th 23
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative 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;

:: MOD_3:th 24
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative 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);

:: MOD_3:th 25
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative 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;

:: MOD_3:th 26
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative 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 base(b1, b2);

:: MOD_3:th 27
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative 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 base(b1, b2);

:: MOD_3:th 28
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative 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
   b2 is free(b1);

:: MOD_3:modenot 1 => MOD_3:mode 1
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative 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 base(a1, a2);
end;

:: MOD_3:dfs 4
definiens
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative 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 base(a1, a2);

:: MOD_3:def 5
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative 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 base(b1, b2);

:: MOD_3:th 29
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative 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;

:: MOD_3:th 30
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative 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;