Article VECTSP_3, MML version 4.99.1005

:: VECTSP_3:th 9
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, b5 being FinSequence of the carrier of b3
      st len b4 = len b5 &
         (for b6 being Element of NAT
         for b7 being Element of the carrier of b3
               st b6 in dom b4 & b7 = b5 . b6
            holds b4 . b6 = b2 * b7)
   holds Sum b4 = b2 * Sum b5;

:: VECTSP_3:th 10
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, b5 being FinSequence of the carrier of b3
      st len b4 = len b5 &
         (for b6 being Element of NAT
               st b6 in dom b4
            holds b5 . b6 = b2 * (b4 /. b6))
   holds Sum b5 = b2 * Sum b4;

:: VECTSP_3: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 VectSpStr over b1
for b3, b4, b5 being FinSequence of the carrier of b2
      st len b3 = len b4 &
         len b3 = len b5 &
         (for b6 being Element of NAT
               st b6 in dom b3
            holds b5 . b6 = (b3 /. b6) - (b4 /. b6))
   holds Sum b5 = (Sum b3) - Sum b4;

:: VECTSP_3:th 21
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 holds
   b2 * Sum <*> the carrier of b3 = 0. b3;

:: VECTSP_3:th 23
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, b5 being Element of the carrier of b3 holds
b2 * Sum <*b4,b5*> = (b2 * b4) + (b2 * b5);

:: VECTSP_3:th 24
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, b5, b6 being Element of the carrier of b3 holds
b2 * Sum <*b4,b5,b6*> = ((b2 * b4) + (b2 * b5)) + (b2 * b6);

:: VECTSP_3:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr holds
   - Sum <*> the carrier of b1 = 0. b1;

:: VECTSP_3:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1 holds
- Sum <*b2,b3*> = (- b2) - b3;

:: VECTSP_3:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3, b4 being Element of the carrier of b1 holds
- Sum <*b2,b3,b4*> = ((- b2) - b3) - b4;

:: VECTSP_3:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2 being Element of the carrier of b1 holds
   Sum <*b2,- b2*> = 0. b1 &
    Sum <*- b2,b2*> = 0. b1;

:: VECTSP_3:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1 holds
Sum <*b2,- b3*> = b2 - b3 &
 Sum <*- b3,b2*> = b2 - b3;

:: VECTSP_3:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1 holds
Sum <*- b2,- b3*> = - (b2 + b3) &
 Sum <*- b3,- b2*> = - (b2 + b3);