Journal of Formalized Mathematics
Volume 2, 1990
University of Bialystok
Copyright (c) 1990 Association of Mizar Users

### Linear Independence in Left Module over Domain

by
Michal Muzalewski, and
Wojciech Skaba

MML identifier: LMOD_5
[ Mizar article, MML identifier index ]

```environ

vocabulary FUNCSDOM, VECTSP_2, VECTSP_1, RLVECT_3, RLVECT_2, RLVECT_1, BOOLE,
FUNCT_1, FUNCT_2, FINSET_1, RLSUB_1, FINSEQ_1, RELAT_1, SEQ_1, FINSEQ_4;
notation TARSKI, XBOOLE_0, NAT_1, SUBSET_1, FINSET_1, RELAT_1, FUNCT_1,
FUNCT_2, FRAENKEL, FINSEQ_1, FINSEQ_4, STRUCT_0, RLVECT_1, VECTSP_1,
FUNCSDOM, VECTSP_2, VECTSP_4, VECTSP_5, VECTSP_6;
constructors RLVECT_2, VECTSP_2, VECTSP_5, VECTSP_6, MEMBERED, XBOOLE_0,
FINSEQ_4, PARTFUN1;
clusters VECTSP_1, VECTSP_4, STRUCT_0, RLVECT_2, RELSET_1, SUBSET_1, MEMBERED,
ZFMISC_1, XBOOLE_0, FUNCT_1, ARYTM_3, FINSET_1;
requirements SUBSET, BOOLE;

begin

reserve x for set,
R for Ring,
V for LeftMod of R,
v,v1,v2 for Vector of V,
A,B for Subset of V;

definition let R be non empty doubleLoopStr;
let V be non empty VectSpStr over R;
let IT be Subset of V;
attr IT is linearly-independent means
:: LMOD_5:def 1
for l be Linear_Combination of IT
st Sum(l) = 0.V holds Carrier(l) = {};
antonym IT is linearly-dependent;
end;

canceled;

theorem :: LMOD_5:2
A c= B & B is linearly-independent implies A is linearly-independent;

theorem :: LMOD_5:3
0.R <> 1_ R & A is linearly-independent
implies not 0.V in A;

theorem :: LMOD_5:4
{}(the carrier of V) is linearly-independent;

theorem :: LMOD_5:5
0.R <> 1_ R & {v1,v2} is linearly-independent implies v1 <> 0.V & v2 <> 0.V;

theorem :: LMOD_5:6
0.R <> 1_ R implies
{v,0.V} is linearly-dependent & {0.V,v} is linearly-dependent;

theorem :: LMOD_5:7
for R being domRing,
V being LeftMod of R,
L being Linear_Combination of V,
a being Scalar of R holds
a <> 0.R implies Carrier(a * L) = Carrier(L);

theorem :: LMOD_5:8
for R being domRing,
V being LeftMod of R,
L being Linear_Combination of V,
a being Scalar of R holds
Sum(a * L) = a * Sum(L);

reserve R for domRing,
V for LeftMod of R,
A,B for Subset of V,
l for Linear_Combination of A,
f,g for Function of the carrier of V, the carrier of R;

definition let R; let V; let A;
func Lin(A) -> strict Subspace of V means
:: LMOD_5:def 2
the carrier of it = {Sum(l) : not contradiction};
end;

theorem :: LMOD_5:9
x in Lin(A) iff ex l st x = Sum(l);

theorem :: LMOD_5:10
x in A implies x in Lin(A);

theorem :: LMOD_5:11
Lin({}(the carrier of V)) = (0).V;

theorem :: LMOD_5:12
Lin(A) = (0).V implies A = {} or A = {0.V};

theorem :: LMOD_5:13
for W being strict Subspace of V st 0.R <> 1_ R &
A = the carrier of W holds Lin(A) = W;

theorem :: LMOD_5:14
for V being strict LeftMod of R, A being Subset of V st
0.R <> 1_ R &
A = the carrier of V holds Lin(A) = V;

theorem :: LMOD_5:15
A c= B implies Lin(A) is Subspace of Lin(B);

theorem :: LMOD_5:16
for V being strict LeftMod of R, A,B being Subset of V
st Lin(A) = V & A c= B holds Lin(B) = V;

theorem :: LMOD_5:17
Lin(A \/ B) = Lin(A) + Lin(B);

theorem :: LMOD_5:18
Lin(A /\ B) is Subspace of Lin(A) /\ Lin(B);
```