let l be RLSStruct ; :: thesis: ( RLSStruct(# the carrier of l, the ZeroF of l, the addF of l, the Mult of l #) is RealLinearSpace implies l is RealLinearSpace )
assume A1: RLSStruct(# the carrier of l, the ZeroF of l, the addF of l, the Mult of l #) is RealLinearSpace ; :: thesis:
reconsider l = l as non empty RLSStruct by A1;
reconsider l0 = RLSStruct(# the carrier of l,(0. l), the addF of l, the Mult of l #) as RealLinearSpace by A1;
A2: l is Abelian
proof
let v, w be VECTOR of l; :: according to RLVECT_1:def 2 :: thesis: v + w = w + v
reconsider v1 = v as VECTOR of l0 ;
reconsider w1 = w as VECTOR of l0 ;
thus v + w = v1 + w1
.= w1 + v1
.= w + v ; :: thesis: verum
end;
A3: l is right_zeroed
proof
let v be VECTOR of l; :: according to RLVECT_1:def 4 :: thesis: v + (0. l) = v
reconsider v1 = v as VECTOR of l0 ;
thus v + (0. l) = v1 + (0. l0)
.= v ; :: thesis: verum
end;
A4: l is right_complementable
proof
let v be VECTOR of l; :: according to ALGSTR_0:def 16 :: thesis:
reconsider v1 = v as VECTOR of l0 ;
consider w1 being VECTOR of l0 such that
A5: v1 + w1 = 0. l0 by ALGSTR_0:def 11;
reconsider w = w1 as VECTOR of l ;
take w ; :: according to ALGSTR_0:def 11 :: thesis: v + w = 0. l
thus v + w = 0. l by A5; :: thesis: verum
end;
A6: for v being VECTOR of l holds 1 * v = v
proof
let v be VECTOR of l; :: thesis: 1 * v = v
reconsider v1 = v as VECTOR of l0 ;
thus 1 * v = 1 * v1
.= v by RLVECT_1:def 8 ; :: thesis: verum
end;
A7: for a, b being Real
for v being VECTOR of l holds (a * b) * v = a * (b * v)
proof
let a, b be Real; :: thesis: for v being VECTOR of l holds (a * b) * v = a * (b * v)
let v be VECTOR of l; :: thesis: (a * b) * v = a * (b * v)
reconsider v1 = v as VECTOR of l0 ;
thus (a * b) * v = (a * b) * v1
.= a * (b * v1) by RLVECT_1:def 7
.= a * (b * v) ; :: thesis: verum
end;
A8: for a, b being Real
for v being VECTOR of l holds (a + b) * v = (a * v) + (b * v)
proof
let a, b be Real; :: thesis: for v being VECTOR of l holds (a + b) * v = (a * v) + (b * v)
let v be VECTOR of l; :: thesis: (a + b) * v = (a * v) + (b * v)
reconsider v1 = v as VECTOR of l0 ;
thus (a + b) * v = (a + b) * v1
.= (a * v1) + (b * v1) by RLVECT_1:def 6
.= (a * v) + (b * v) ; :: thesis: verum
end;
A9: for a being Real
for v, w being VECTOR of l holds a * (v + w) = (a * v) + (a * w)
proof
let a be Real; :: thesis: for v, w being VECTOR of l holds a * (v + w) = (a * v) + (a * w)
let v, w be VECTOR of l; :: thesis: a * (v + w) = (a * v) + (a * w)
reconsider v1 = v, w1 = w as VECTOR of l0 ;
thus a * (v + w) = a * (v1 + w1)
.= (a * v1) + (a * w1) by RLVECT_1:def 5
.= (a * v) + (a * w) ; :: thesis: verum
end;