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

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

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 a be Element of GF; :: thesis: for v being Element of V st v in V1 holds
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;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

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