theorem :: VECTSP_2:25

for R1, R2 being Ring
for V being non empty BiModStr over R1,R2 holds
( ( for x, y being Scalar of R1
for p, q being Scalar of R2
for v, w being Vector of V holds
( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1_ R1) * v = v & (v + w) * p = (v * p) + (w * p) & v * (p + q) = (v * p) + (v * q) & v * (q * p) = (v * q) * p & v * (1_ R2) = v & x * (v * p) = (x * v) * p ) ) iff ( V is RightMod-like & V is vector-distributive & V is scalar-distributive & V is scalar-associative & V is scalar-unital & V is BiMod-like ) ) ;