let GF be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF
for A being Subset of V st 0. GF <> 1. GF holds
( ( A <> {} & A is linearly-closed ) iff for l being Linear_Combination of A holds Sum l in A )
let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF; for A being Subset of V st 0. GF <> 1. GF 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; ( 0. GF <> 1. GF implies ( ( A <> {} & A is linearly-closed ) iff for l being Linear_Combination of A holds Sum l in A ) )
assume A1:
0. GF <> 1. GF
; ( ( 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[
Element of
NAT ]
means for
l being
Linear_Combination of
A st
card (Carrier l) = $1 holds
Sum l in A;
assume that A2:
A <> {}
and A3:
A is
linearly-closed
;
for l being Linear_Combination of A holds Sum l in A
A4:
S1[
0 ]
A5:
for
k being
Element of
NAT st
S1[
k] holds
S1[
k + 1]
proof
let k be
Element of
NAT ;
( S1[k] implies S1[k + 1] )
assume A6:
S1[
k]
;
S1[k + 1]
let l be
Linear_Combination of
A;
( card (Carrier l) = k + 1 implies Sum l in A )
deffunc H1(
Element of
V)
-> Element of the
carrier of
GF =
l . $1;
consider F being
FinSequence of
V such that A7:
F is
one-to-one
and A8:
rng F = Carrier l
and A9:
Sum l = Sum (l (#) F)
by Def9;
reconsider G =
F | (Seg k) as
FinSequence of the
carrier of
V by FINSEQ_1:18;
assume A10:
card (Carrier l) = k + 1
;
Sum l in A
then A11:
len F = k + 1
by A7, A8, FINSEQ_4:62;
then A12:
len (l (#) F) = k + 1
by Def8;
A13:
k + 1
in Seg (k + 1)
by FINSEQ_1:4;
then A14:
k + 1
in dom F
by A11, FINSEQ_1:def 3;
k + 1
in dom F
by A11, A13, FINSEQ_1:def 3;
then reconsider v =
F . (k + 1) as
Element of
V by FINSEQ_2:11;
consider f being
Function of the
carrier of
V, the
carrier of
GF such that A15:
f . v = 0. GF
and A16:
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, the
carrier of
GF)
by FUNCT_2:8;
A17:
v in Carrier l
by A8, A14, FUNCT_1:def 3;
then reconsider f =
f as
Linear_Combination of
V by Def4;
A19:
A \ {v} c= A
by XBOOLE_1:36;
A20:
Carrier l c= A
by Def7;
then A21:
(l . v) * v in A
by A3, A17, VECTSP_4:def 1;
A22:
Carrier f = (Carrier l) \ {v}
then
Carrier f c= A \ {v}
by A20, XBOOLE_1:33;
then
Carrier f c= A
by A19, XBOOLE_1:1;
then reconsider f =
f as
Linear_Combination of
A by Def7;
A29:
len G = k
by A11, FINSEQ_3:53;
then A30:
len (f (#) G) = k
by Def8;
A31:
rng G = Carrier f
(Seg (k + 1)) /\ (Seg k) =
Seg k
by FINSEQ_1:7, NAT_1:12
.=
dom (f (#) G)
by A30, FINSEQ_1:def 3
;
then A41:
dom (f (#) G) = (dom (l (#) F)) /\ (Seg k)
by A12, FINSEQ_1:def 3;
now let x be
set ;
( x in dom (f (#) G) implies (f (#) G) . x = (l (#) F) . x )A42:
rng F c= the
carrier of
V
by FINSEQ_1:def 4;
assume A43:
x in dom (f (#) G)
;
(f (#) G) . x = (l (#) F) . xthen reconsider n =
x as
Element of
NAT by FINSEQ_3:23;
n in dom (l (#) F)
by A41, A43, XBOOLE_0:def 4;
then A44:
n in dom F
by A11, A12, FINSEQ_3:29;
then
F . n in rng F
by FUNCT_1:def 3;
then reconsider w =
F . n as
Element of
V by A42;
A45:
n in dom G
by A29, A30, A43, FINSEQ_3:29;
then A46:
G . n in rng G
by FUNCT_1:def 3;
rng G c= the
carrier of
V
by FINSEQ_1:def 4;
then reconsider u =
G . n as
Element of
V by A46;
not
u in {v}
by A22, A31, A46, XBOOLE_0:def 5;
then A47:
u <> v
by TARSKI:def 1;
A48:
(f (#) G) . n =
(f . u) * u
by A45, Th32
.=
(l . u) * u
by A16, A47
;
w = u
by A45, FUNCT_1:47;
hence
(f (#) G) . x = (l (#) F) . x
by A48, A44, Th32;
verum end;
then
f (#) G = (l (#) F) | (Seg k)
by A41, FUNCT_1:46;
then A49:
f (#) G = (l (#) F) | (dom (f (#) G))
by A30, FINSEQ_1:def 3;
v in rng F
by A14, FUNCT_1:def 3;
then
{v} c= Carrier l
by A8, ZFMISC_1:31;
then card (Carrier f) =
(k + 1) - (card {v})
by A10, A22, CARD_2:44
.=
(k + 1) - 1
by CARD_1:30
.=
k
by XCMPLX_1:26
;
then A50:
Sum f in A
by A6;
G is
one-to-one
by A7, FUNCT_1:52;
then A51:
Sum (f (#) G) = Sum f
by A31, Def9;
(l (#) F) . (len F) = (l . v) * v
by A11, A14, Th32;
then
Sum (l (#) F) = (Sum (f (#) G)) + ((l . v) * v)
by A11, A12, A30, A49, RLVECT_1:38;
hence
Sum l in A
by A3, A9, A21, A51, A50, VECTSP_4:def 1;
verum
end;
let l be
Linear_Combination of
A;
Sum l in A
A52:
card (Carrier l) = card (Carrier l)
;
for
k being
Element of
NAT holds
S1[
k]
from NAT_1:sch 1(A4, A5);
hence
Sum l in A
by A52;
verum
end;
assume A53:
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 Lm1, Th26;
then A54:
0. V in A
by A53;
A55:
for a being Element of GF
for v being Element of V st v in A holds
a * v in A
thus
for v, u being Element of V st v in A & u in A holds
v + u in A
VECTSP_4:def 1 for b1 being Element of the carrier of GF
for b2 being Element of the carrier of V holds
( not b2 in A or b1 * b2 in A )proof
let v,
u be
Element of
V;
( v in A & u in A implies v + u in A )
assume that A65:
v in A
and A66:
u in A
;
v + u in A
now per cases
( u = v or v <> u )
;
suppose A67:
v <> u
;
v + u in Adeffunc H1(
set )
-> Element of the
carrier of
GF =
0. GF;
consider f being
Function of
V,
GF such that A68:
(
f . v = 1. GF &
f . u = 1. GF )
and A69:
for
w being
Element of
V st
w <> v &
w <> u holds
f . w = H1(
w)
from FUNCT_2:sch 7(A67);
reconsider f =
f as
Element of
Funcs ( the
carrier of
V, the
carrier of
GF)
by FUNCT_2:8;
then reconsider f =
f as
Linear_Combination of
V by Def4;
A70:
Carrier f = {v,u}
then A71:
Carrier f c= A
by A65, A66, ZFMISC_1:32;
A72:
(
(1. GF) * u = u &
(1. GF) * v = v )
by VECTSP_1:def 17;
reconsider f =
f as
Linear_Combination of
A by A71, Def7;
consider F being
FinSequence of
V such that A73:
(
F is
one-to-one &
rng F = Carrier f )
and A74:
Sum (f (#) F) = Sum f
by Def9;
(
F = <*v,u*> or
F = <*u,v*> )
by A67, A70, A73, FINSEQ_3:99;
then
(
f (#) F = <*((1. GF) * v),((1. GF) * u)*> or
f (#) F = <*((1. GF) * u),((1. GF) * v)*> )
by A68, Th35;
then
Sum f = v + u
by A74, A72, RLVECT_1:45;
hence
v + u in A
by A53;
verum end;
hence
v + u in A
;
verum
end;
thus
for b1 being Element of the carrier of GF
for b2 being Element of the carrier of V holds
( not b2 in A or b1 * b2 in A )
by A55; verum