let X be non empty set ; for Y being RealLinearSpace holds RLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) is RealLinearSpace
let Y be RealLinearSpace; RLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) is RealLinearSpace
A1:
for a, b being real number
for v being VECTOR of RLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) holds (a * b) * v = a * (b * v)
proof
let a,
b be
real number ;
for v being VECTOR of RLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) holds (a * b) * v = a * (b * v)
reconsider a =
a,
b =
b as
Real by XREAL_0:def 1;
for
v being
VECTOR of
RLSStruct(#
(Funcs X,the carrier of Y),
(FuncZero X,Y),
(FuncAdd X,Y),
(FuncExtMult X,Y) #) holds
(a * b) * v = a * (b * v)
by Th11;
hence
for
v being
VECTOR of
RLSStruct(#
(Funcs X,the carrier of Y),
(FuncZero X,Y),
(FuncAdd X,Y),
(FuncExtMult X,Y) #) holds
(a * b) * v = a * (b * v)
;
verum
end;
A2:
for a, b being real number
for v being VECTOR of RLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) holds (a + b) * v = (a * v) + (b * v)
proof
let a,
b be
real number ;
for v being VECTOR of RLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) holds (a + b) * v = (a * v) + (b * v)
reconsider a =
a,
b =
b as
Real by XREAL_0:def 1;
for
v being
VECTOR of
RLSStruct(#
(Funcs X,the carrier of Y),
(FuncZero X,Y),
(FuncAdd X,Y),
(FuncExtMult X,Y) #) holds
(a + b) * v = (a * v) + (b * v)
by Th12;
hence
for
v being
VECTOR of
RLSStruct(#
(Funcs X,the carrier of Y),
(FuncZero X,Y),
(FuncAdd X,Y),
(FuncExtMult X,Y) #) holds
(a + b) * v = (a * v) + (b * v)
;
verum
end;
set IT = RLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #);
A3:
for u, v, w being VECTOR of RLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) holds (u + v) + w = u + (v + w)
by Th7;
A4:
RLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) is right_complementable
proof
let u be
VECTOR of
RLSStruct(#
(Funcs X,the carrier of Y),
(FuncZero X,Y),
(FuncAdd X,Y),
(FuncExtMult X,Y) #);
ALGSTR_0:def 16 u is right_complementable
reconsider u9 =
u as
Element of
Funcs X,the
carrier of
Y ;
reconsider w =
(FuncExtMult X,Y) . [(- 1),u9] as
VECTOR of
RLSStruct(#
(Funcs X,the carrier of Y),
(FuncZero X,Y),
(FuncAdd X,Y),
(FuncExtMult X,Y) #) ;
take
w
;
ALGSTR_0:def 11 u + w = 0. RLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #)
thus
u + w = 0. RLSStruct(#
(Funcs X,the carrier of Y),
(FuncZero X,Y),
(FuncAdd X,Y),
(FuncExtMult X,Y) #)
by Th9;
verum
end;
A5:
for a being real number
for u, v being VECTOR of RLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) holds a * (u + v) = (a * u) + (a * v)
proof
let a be
real number ;
for u, v being VECTOR of RLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) holds a * (u + v) = (a * u) + (a * v)
reconsider a =
a as
Real by XREAL_0:def 1;
for
u,
v being
VECTOR of
RLSStruct(#
(Funcs X,the carrier of Y),
(FuncZero X,Y),
(FuncAdd X,Y),
(FuncExtMult X,Y) #) holds
a * (u + v) = (a * u) + (a * v)
by Lm2;
hence
for
u,
v being
VECTOR of
RLSStruct(#
(Funcs X,the carrier of Y),
(FuncZero X,Y),
(FuncAdd X,Y),
(FuncExtMult X,Y) #) holds
a * (u + v) = (a * u) + (a * v)
;
verum
end;
A6:
for v being VECTOR of RLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) holds 1 * v = v
by Th10;
A7:
for u being VECTOR of RLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) holds u + (0. RLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #)) = u
proof
let u be
VECTOR of
RLSStruct(#
(Funcs X,the carrier of Y),
(FuncZero X,Y),
(FuncAdd X,Y),
(FuncExtMult X,Y) #);
u + (0. RLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #)) = u
reconsider u9 =
u as
Element of
Funcs X,the
carrier of
Y ;
thus u + (0. RLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #)) =
(FuncAdd X,Y) . (FuncZero X,Y),
u9
by Th6
.=
u
by Th8
;
verum
end;
for u, v being VECTOR of RLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) holds u + v = v + u
by Th6;
hence
RLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) is RealLinearSpace
by A3, A7, A4, A5, A2, A1, A6, RLVECT_1:def 5, RLVECT_1:def 6, RLVECT_1:def 7, RLVECT_1:def 9; verum