let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W being Subspace of V
for V1 being Subset of V st the carrier of W = V1 holds
V1 is linearly-closed
let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for W being Subspace of V
for V1 being Subset of V st the carrier of W = V1 holds
V1 is linearly-closed
let W be Subspace of V; :: thesis: for V1 being Subset of V st the carrier of W = V1 holds
V1 is linearly-closed
let V1 be Subset of V; :: thesis: ( the carrier of W = V1 implies V1 is linearly-closed )
set VW = the carrier of W;
reconsider WW = W as non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF ;
assume A1: the carrier of W = V1 ; :: thesis: V1 is linearly-closed
thus for v, u being Element of V st v in V1 & u in V1 holds
v + u in V1 :: according to VECTSP_4:def 1 :: thesis: for a being Element of GF
for v being Element of V st v in V1 holds
a * v in V1
proof
let v, u be Element of V; :: thesis: ( v in V1 & u in V1 implies v + u in V1 )
assume ( v in V1 & u in V1 ) ; :: thesis: v + u in V1
then reconsider vv = v, uu = u as Element of WW by A1;
reconsider vw = vv + uu as Element of the carrier of W ;
vw in V1 by A1;
hence v + u in V1 by Th13; :: thesis: verum
end;
let a be Element of GF; :: thesis: for v being Element of V st v in V1 holds
a * v in V1
let v be Element of V; :: thesis: ( v in V1 implies a * v in V1 )
assume v in V1 ; :: thesis: a * v in V1
then reconsider vv = v as Element of WW by A1;
reconsider vw = a * vv as Element of the carrier of W ;
vw in V1 by A1;
hence a * v in V1 by Th14; :: thesis: verum