let R be Ring; :: thesis: for V being RightMod of R
for V1 being Subset of V
for W being Submodule of V st the carrier of W = V1 holds
V1 is linearly-closed
let V be RightMod of R; :: thesis: for V1 being Subset of V
for W being Submodule of V st the carrier of W = V1 holds
V1 is linearly-closed
let V1 be Subset of V; :: thesis: for W being Submodule of V st the carrier of W = V1 holds
V1 is linearly-closed
let W be Submodule of V; :: thesis: ( the carrier of W = V1 implies V1 is linearly-closed )
set VW = the carrier of W;
reconsider WW = W as RightMod of R ;
assume A1: the carrier of W = V1 ; :: thesis: V1 is linearly-closed
thus for v, u being Vector of V st v in V1 & u in V1 holds
v + u in V1 :: according to RMOD_2:def 1 :: thesis: for a being Scalar of R
for v being Vector of V st v in V1 holds
v * a in V1
proof
let v, u be Vector 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 Vector 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 Scalar of R; :: thesis: for v being Vector of V st v in V1 holds
v * a in V1
let v be Vector of V; :: thesis: ( v in V1 implies v * a in V1 )
assume v in V1 ; :: thesis: v * a in V1
then reconsider vv = v as Vector of WW by A1;
reconsider vw = vv * a as Element of the carrier of W ;
vw in V1 by A1;
hence v * a in V1 by Th14; :: thesis: verum