let X be non empty set ; for Y being ComplexLinearSpace holds CLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) is ComplexLinearSpace
let Y be ComplexLinearSpace; CLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) is ComplexLinearSpace
set IT = CLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #);
A1:
for u, v, w being VECTOR of CLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) holds (u + v) + w = u + (v + w)
by Th5;
A2:
for u being VECTOR of CLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) holds u is right_complementable
proof
let u be
VECTOR of
CLSStruct(#
(Funcs X,the carrier of Y),
(FuncZero X,Y),
(FuncAdd X,Y),
(FuncExtMult X,Y) #);
u is right_complementable
reconsider u9 =
u as
Element of
Funcs X,the
carrier of
Y ;
reconsider w =
(FuncExtMult X,Y) . [(- 1r ),u9] as
VECTOR of
CLSStruct(#
(Funcs X,the carrier of Y),
(FuncZero X,Y),
(FuncAdd X,Y),
(FuncExtMult X,Y) #) ;
take
w
;
ALGSTR_0:def 11 u + w = 0. CLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #)
thus
u + w = 0. CLSStruct(#
(Funcs X,the carrier of Y),
(FuncZero X,Y),
(FuncAdd X,Y),
(FuncExtMult X,Y) #)
by Th7;
verum
end;
A3:
for a being Complex
for u, v being VECTOR of CLSStruct(# (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
Complex;
for u, v being VECTOR of CLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) holds a * (u + v) = (a * u) + (a * v)let u,
v be
VECTOR of
CLSStruct(#
(Funcs X,the carrier of Y),
(FuncZero X,Y),
(FuncAdd X,Y),
(FuncExtMult X,Y) #);
a * (u + v) = (a * u) + (a * v)
a * (u + v) = (a * u) + (a * v)
proof
reconsider v9 =
v,
u9 =
u as
Element of
Funcs X,the
carrier of
Y ;
reconsider w =
(FuncExtMult X,Y) . [a,u9],
w9 =
(FuncExtMult X,Y) . [a,v9] as
VECTOR of
CLSStruct(#
(Funcs X,the carrier of Y),
(FuncZero X,Y),
(FuncAdd X,Y),
(FuncExtMult X,Y) #) ;
a * (u + v) =
(FuncExtMult X,Y) . [a,((FuncAdd X,Y) . u9,v9)]
by CLVECT_1:def 1
.=
(FuncAdd X,Y) . w,
w9
by Lm1
.=
w + (a * v)
by CLVECT_1:def 1
.=
(a * u) + (a * v)
by CLVECT_1:def 1
;
hence
a * (u + v) = (a * u) + (a * v)
;
verum
end;
hence
a * (u + v) = (a * u) + (a * v)
;
verum
end;
A4:
for a, b being Complex
for v being VECTOR of CLSStruct(# (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
Complex;
for v being VECTOR of CLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) holds (a * b) * v = a * (b * v)let v be
VECTOR of
CLSStruct(#
(Funcs X,the carrier of Y),
(FuncZero X,Y),
(FuncAdd X,Y),
(FuncExtMult X,Y) #);
(a * b) * v = a * (b * v)
reconsider v9 =
v as
Element of
Funcs X,the
carrier of
Y ;
thus (a * b) * v =
(FuncExtMult X,Y) . [(a * b),v9]
by CLVECT_1:def 1
.=
(FuncExtMult X,Y) . [a,((FuncExtMult X,Y) . [b,v9])]
by Th9
.=
(FuncExtMult X,Y) . [a,(b * v)]
by CLVECT_1:def 1
.=
a * (b * v)
by CLVECT_1:def 1
;
verum
end;
A5:
for a, b being Complex
for v being VECTOR of CLSStruct(# (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
Complex;
for v being VECTOR of CLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) holds (a + b) * v = (a * v) + (b * v)let v be
VECTOR of
CLSStruct(#
(Funcs X,the carrier of Y),
(FuncZero X,Y),
(FuncAdd X,Y),
(FuncExtMult X,Y) #);
(a + b) * v = (a * v) + (b * v)
reconsider v9 =
v as
Element of
Funcs X,the
carrier of
Y ;
reconsider w =
(FuncExtMult X,Y) . [a,v9],
w9 =
(FuncExtMult X,Y) . [b,v9] as
VECTOR of
CLSStruct(#
(Funcs X,the carrier of Y),
(FuncZero X,Y),
(FuncAdd X,Y),
(FuncExtMult X,Y) #) ;
thus (a + b) * v =
(FuncExtMult X,Y) . [(a + b),v9]
by CLVECT_1:def 1
.=
(FuncAdd X,Y) . w,
w9
by Th10
.=
w + (b * v)
by CLVECT_1:def 1
.=
(a * v) + (b * v)
by CLVECT_1:def 1
;
verum
end;
A6:
for u being VECTOR of CLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) holds u + (0. CLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #)) = u
proof
let u be
VECTOR of
CLSStruct(#
(Funcs X,the carrier of Y),
(FuncZero X,Y),
(FuncAdd X,Y),
(FuncExtMult X,Y) #);
u + (0. CLSStruct(# (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. CLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #)) =
(FuncAdd X,Y) . (FuncZero X,Y),
u9
by Th4
.=
u
by Th6
;
verum
end;
A7:
for v being VECTOR of CLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) holds 1r * v = v
proof
let v be
VECTOR of
CLSStruct(#
(Funcs X,the carrier of Y),
(FuncZero X,Y),
(FuncAdd X,Y),
(FuncExtMult X,Y) #);
1r * v = v
reconsider v9 =
v as
Element of
Funcs X,the
carrier of
Y ;
thus 1r * v =
(FuncExtMult X,Y) . [1r ,v9]
by CLVECT_1:def 1
.=
v
by Th8
;
verum
end;
for u, v being VECTOR of CLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) holds u + v = v + u
by Th4;
hence
CLSStruct(# (Funcs X,the carrier of Y),(FuncZero X,Y),(FuncAdd X,Y),(FuncExtMult X,Y) #) is ComplexLinearSpace
by A1, A6, A2, A3, A5, A4, A7, ALGSTR_0:def 16, CLVECT_1:def 2, RLVECT_1:def 5, RLVECT_1:def 6, RLVECT_1:def 7; verum