Article MOD_1, MML version 4.99.1005

:: MOD_1:th 13
theorem
for b1 being non empty right_complementable right-distributive right_unital add-associative right_zeroed doubleLoopStr
for b2 being Element of the carrier of b1 holds
   b2 * - 1. b1 = - b2;

:: MOD_1:th 14
theorem
for b1 being non empty right_complementable left-distributive left_unital add-associative right_zeroed doubleLoopStr
for b2 being Element of the carrier of b1 holds
   (- 1. b1) * b2 = - b2;

:: MOD_1: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 Element of the carrier of b1
for b3 being non empty right_complementable VectSp-like add-associative right_zeroed VectSpStr over b1
for b4 being Element of the carrier of b3 holds
      b2 * b4 = 0. b3
   iff
      (b2 = 0. b1 or b4 = 0. b3);

:: MOD_1: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 Element of the carrier of b1
for b3 being non empty right_complementable VectSp-like add-associative right_zeroed VectSpStr over b1
for b4 being Element of the carrier of b3
      st b2 <> 0. b1
   holds b2 " * (b2 * b4) = b4;

:: MOD_1:th 37
theorem
for b1 being non empty right_complementable associative right_unital well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2 holds
   b4 * 0. b1 = 0. b2 & b4 * - 1_ b1 = - b4 & (0. b2) * b3 = 0. b2;

:: MOD_1:th 38
theorem
for b1 being non empty right_complementable associative right_unital well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b1
for b4, b5 being Element of the carrier of b2 holds
- (b4 * b3) = b4 * - b3 &
 b5 - (b4 * b3) = b5 + (b4 * - b3);

:: MOD_1:th 39
theorem
for b1 being non empty right_complementable associative right_unital well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2 holds
   (- b4) * b3 = - (b4 * b3);

:: MOD_1:th 40
theorem
for b1 being non empty right_complementable associative right_unital well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b1
for b4, b5 being Element of the carrier of b2 holds
(b4 - b5) * b3 = (b4 * b3) - (b5 * b3);

:: MOD_1:th 42
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 Element of the carrier of b1
for b3 being non empty right_complementable add-associative right_zeroed RightMod-like RightModStr over b1
for b4 being Element of the carrier of b3 holds
      b4 * b2 = 0. b3
   iff
      (b2 = 0. b1 or b4 = 0. b3);

:: MOD_1:th 43
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 Element of the carrier of b1
for b3 being non empty right_complementable add-associative right_zeroed RightMod-like RightModStr over b1
for b4 being Element of the carrier of b3
      st b2 <> 0. b1
   holds (b4 * b2) * (b2 ") = b4;