let V be RealLinearSpace; for A being Subset of V holds
( ( A <> {} & A is linearly-closed ) iff for l being Linear_Combination of A holds Sum l in A )
let A be Subset of V; ( ( A <> {} & A is linearly-closed ) iff for l being Linear_Combination of A holds Sum l in A )
thus
( A <> {} & A is linearly-closed implies for l being Linear_Combination of A holds Sum l in A )
( ( for l being Linear_Combination of A holds Sum l in A ) implies ( A <> {} & A is linearly-closed ) )proof
defpred S1[
Nat]
means for
l being
Linear_Combination of
A st
card (Carrier l) = $1 holds
Sum l in A;
assume that A1:
A <> {}
and A2:
A is
linearly-closed
;
for l being Linear_Combination of A holds Sum l in A
then A3:
S1[
0 ]
;
now for k being Nat st ( for l being Linear_Combination of A st card (Carrier l) = k holds
Sum l in A ) holds
for l being Linear_Combination of A st card (Carrier l) = k + 1 holds
Sum l in Alet k be
Nat;
( ( for l being Linear_Combination of A st card (Carrier l) = k holds
Sum l in A ) implies for l being Linear_Combination of A st card (Carrier l) = k + 1 holds
Sum l in A )assume A4:
for
l being
Linear_Combination of
A st
card (Carrier l) = k holds
Sum l in A
;
for l being Linear_Combination of A st card (Carrier l) = k + 1 holds
Sum l in Alet l be
Linear_Combination of
A;
( card (Carrier l) = k + 1 implies Sum l in A )deffunc H1(
Element of
V)
-> Element of
REAL =
l . $1;
consider F being
FinSequence of
V such that A5:
F is
one-to-one
and A6:
rng F = Carrier l
and A7:
Sum l = Sum (l (#) F)
by Def8;
reconsider G =
F | (Seg k) as
FinSequence of the
carrier of
V by FINSEQ_1:18;
assume A8:
card (Carrier l) = k + 1
;
Sum l in Athen A9:
len F = k + 1
by A5, A6, FINSEQ_4:62;
then A10:
len (l (#) F) = k + 1
by Def7;
A11:
k + 1
in Seg (k + 1)
by FINSEQ_1:4;
then A12:
k + 1
in dom F
by A9, FINSEQ_1:def 3;
k + 1
in dom F
by A9, A11, FINSEQ_1:def 3;
then reconsider v =
F . (k + 1) as
VECTOR of
V by FUNCT_1:102;
consider f being
Function of the
carrier of
V,
REAL such that A13:
f . v = In (
0,
REAL)
and A14:
for
u being
Element of
V st
u <> v holds
f . u = H1(
u)
from FUNCT_2:sch 6();
reconsider f =
f as
Element of
Funcs ( the
carrier of
V,
REAL)
by FUNCT_2:8;
A15:
v in Carrier l
by A6, A12, FUNCT_1:def 3;
then reconsider f =
f as
Linear_Combination of
V by Def3;
A17:
A \ {v} c= A
by XBOOLE_1:36;
A18:
Carrier l c= A
by Def6;
then A19:
(l . v) * v in A
by A2, A15;
A20:
Carrier f = (Carrier l) \ {v}
then
Carrier f c= A \ {v}
by A18, XBOOLE_1:33;
then
Carrier f c= A
by A17;
then reconsider f =
f as
Linear_Combination of
A by Def6;
A27:
len G = k
by A9, FINSEQ_3:53;
then A28:
len (f (#) G) = k
by Def7;
A29:
rng G = Carrier f
(Seg (k + 1)) /\ (Seg k) =
Seg k
by FINSEQ_1:7, NAT_1:12
.=
dom (f (#) G)
by A28, FINSEQ_1:def 3
;
then A39:
dom (f (#) G) = (dom (l (#) F)) /\ (Seg k)
by A10, FINSEQ_1:def 3;
now for x being object st x in dom (f (#) G) holds
(f (#) G) . x = (l (#) F) . xlet x be
object ;
( x in dom (f (#) G) implies (f (#) G) . x = (l (#) F) . x )assume A40:
x in dom (f (#) G)
;
(f (#) G) . x = (l (#) F) . xthen reconsider n =
x as
Element of
NAT ;
n in dom (l (#) F)
by A39, A40, XBOOLE_0:def 4;
then A41:
n in dom F
by A9, A10, FINSEQ_3:29;
then
F . n in rng F
by FUNCT_1:def 3;
then reconsider w =
F . n as
VECTOR of
V ;
A42:
n in dom G
by A27, A28, A40, FINSEQ_3:29;
then A43:
G . n in rng G
by FUNCT_1:def 3;
then reconsider u =
G . n as
VECTOR of
V ;
not
u in {v}
by A20, A29, A43, XBOOLE_0:def 5;
then A44:
u <> v
by TARSKI:def 1;
A45:
(f (#) G) . n =
(f . u) * u
by A42, Th24
.=
(l . u) * u
by A14, A44
;
w = u
by A42, FUNCT_1:47;
hence
(f (#) G) . x = (l (#) F) . x
by A45, A41, Th24;
verum end; then A46:
f (#) G = (l (#) F) | (Seg k)
by A39, FUNCT_1:46;
v in rng F
by A12, FUNCT_1:def 3;
then
{v} c= Carrier l
by A6, ZFMISC_1:31;
then card (Carrier f) =
(k + 1) - (card {v})
by A8, A20, CARD_2:44
.=
(k + 1) - 1
by CARD_1:30
.=
k
;
then A47:
Sum f in A
by A4;
G is
one-to-one
by A5, FUNCT_1:52;
then A48:
Sum (f (#) G) = Sum f
by A29, Def8;
(
dom (f (#) G) = Seg (len (f (#) G)) &
(l (#) F) . (len F) = (l . v) * v )
by A9, A12, Th24, FINSEQ_1:def 3;
then
Sum (l (#) F) = (Sum (f (#) G)) + ((l . v) * v)
by A9, A10, A28, A46, RLVECT_1:38;
hence
Sum l in A
by A2, A7, A19, A48, A47;
verum end;
then A49:
for
k being
Nat st
S1[
k] holds
S1[
k + 1]
;
let l be
Linear_Combination of
A;
Sum l in A
A50:
card (Carrier l) = card (Carrier l)
;
for
k being
Nat holds
S1[
k]
from NAT_1:sch 2(A3, A49);
hence
Sum l in A
by A50;
verum
end;
assume A51:
for l being Linear_Combination of A holds Sum l in A
; ( A <> {} & A is linearly-closed )
hence
A <> {}
; A is linearly-closed
( ZeroLC V is Linear_Combination of A & Sum (ZeroLC V) = 0. V )
by Lm2, Th22;
then A52:
0. V in A
by A51;
A53:
for a being Real
for v being VECTOR of V st v in A holds
a * v in A
thus
for v, u being VECTOR of V st v in A & u in A holds
v + u in A
RLSUB_1:def 1 for b1 being object
for b2 being Element of the carrier of V holds
( not b2 in A or b1 * b2 in A )proof
let v,
u be
VECTOR of
V;
( v in A & u in A implies v + u in A )
assume that A63:
v in A
and A64:
u in A
;
v + u in A
now v + u in Aper cases
( u = v or v <> u )
;
suppose A65:
v <> u
;
v + u in Adeffunc H1(
Element of
V)
-> Element of
REAL =
zz;
reconsider jj = 1 as
Element of
REAL by XREAL_0:def 1;
consider f being
Function of the
carrier of
V,
REAL such that A66:
(
f . v = jj &
f . u = jj )
and A67:
for
w being
Element of
V st
w <> v &
w <> u holds
f . w = H1(
w)
from FUNCT_2:sch 7(A65);
reconsider f =
f as
Element of
Funcs ( the
carrier of
V,
REAL)
by FUNCT_2:8;
then reconsider f =
f as
Linear_Combination of
V by Def3;
A68:
Carrier f = {v,u}
then A69:
Carrier f c= A
by A63, A64, ZFMISC_1:32;
A70:
( 1
* u = u & 1
* v = v )
by RLVECT_1:def 8;
reconsider f =
f as
Linear_Combination of
A by A69, Def6;
consider F being
FinSequence of
V such that A71:
(
F is
one-to-one &
rng F = Carrier f )
and A72:
Sum (f (#) F) = Sum f
by Def8;
(
F = <*v,u*> or
F = <*u,v*> )
by A65, A68, A71, FINSEQ_3:99;
then
(
f (#) F = <*(1 * v),(1 * u)*> or
f (#) F = <*(1 * u),(1 * v)*> )
by A66, Th27;
then
Sum f = v + u
by A72, A70, RLVECT_1:45;
hence
v + u in A
by A51;
verum end;
hence
v + u in A
;
verum
end;
thus
for b1 being object
for b2 being Element of the carrier of V holds
( not b2 in A or b1 * b2 in A )
by A53; verum