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

### Submodules and Cosets of Submodules in Right Module over Associative Ring

by
Michal Muzalewski, and
Wojciech Skaba

MML identifier: RMOD_2
[ Mizar article, MML identifier index ]

```environ

vocabulary FUNCSDOM, VECTSP_1, VECTSP_2, RLSUB_1, BOOLE, RLVECT_1, ARYTM_1,
LMOD_4, RELAT_1, FUNCT_1, BINOP_1;
notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, FUNCT_2,
STRUCT_0, DOMAIN_1, RLVECT_1, BINOP_1, VECTSP_1, FUNCSDOM, VECTSP_2;
constructors DOMAIN_1, BINOP_1, VECTSP_2, PARTFUN1, MEMBERED, XBOOLE_0;
clusters FUNCT_1, VECTSP_2, STRUCT_0, RELSET_1, SUBSET_1, MEMBERED, ZFMISC_1,
XBOOLE_0;
requirements SUBSET, BOOLE;

begin

reserve x,y,y1,y2 for set;
reserve R for Ring;
reserve a for Scalar of R;
reserve V,X,Y for RightMod of R;
reserve u,u1,u2,v,v1,v2 for Vector of V;
reserve V1,V2,V3 for Subset of V;

definition let R, V, V1;
attr V1 is lineary-closed means
:: RMOD_2:def 1
(for v,u st v in V1 & u in V1 holds v + u in V1) &
(for a,v st v in V1 holds v * a in V1);
end;

canceled 3;

theorem :: RMOD_2:4
V1 <> {} & V1 is lineary-closed implies 0.V in V1;

theorem :: RMOD_2:5
V1 is lineary-closed implies (for v st v in V1 holds - v in V1);

theorem :: RMOD_2:6
V1 is lineary-closed implies
(for v,u st v in V1 & u in V1 holds v - u in V1);

theorem :: RMOD_2:7
{0.V} is lineary-closed;

theorem :: RMOD_2:8
the carrier of V = V1 implies V1 is lineary-closed;

theorem :: RMOD_2:9
V1 is lineary-closed & V2 is lineary-closed &
V3 = {v + u : v in V1 & u in V2} implies V3 is lineary-closed;

theorem :: RMOD_2:10
V1 is lineary-closed & V2 is lineary-closed implies
V1 /\ V2 is lineary-closed;

definition let R; let V;
mode Submodule of V -> RightMod of R means
:: RMOD_2:def 2
the carrier of it c= the carrier of V &
the Zero of it = the Zero of V &
[:the carrier of it,the carrier of it:] &
the rmult of it =
(the rmult of V) | [:the carrier of it, the carrier of R:];
end;

reserve W,W1,W2 for Submodule of V;
reserve w,w1,w2 for Vector of W;

canceled 5;

theorem :: RMOD_2:16
x in W1 & W1 is Submodule of W2 implies x in W2;

theorem :: RMOD_2:17
x in W implies x in V;

theorem :: RMOD_2:18
w is Vector of V;

theorem :: RMOD_2:19
0.W = 0.V;

theorem :: RMOD_2:20
0.W1 = 0.W2;

theorem :: RMOD_2:21
w1 = v & w2 = u implies w1 + w2 = v + u;

theorem :: RMOD_2:22
w = v implies w * a = v * a;

theorem :: RMOD_2:23
w = v implies - v = - w;

theorem :: RMOD_2:24
w1 = v & w2 = u implies w1 - w2 = v - u;

theorem :: RMOD_2:25
0.V in W;

theorem :: RMOD_2:26
0.W1 in W2;

theorem :: RMOD_2:27
0.W in V;

theorem :: RMOD_2:28
u in W & v in W implies u + v in W;

theorem :: RMOD_2:29
v in W implies v * a in W;

theorem :: RMOD_2:30
v in W implies - v in W;

theorem :: RMOD_2:31
u in W & v in W implies u - v in W;

theorem :: RMOD_2:32
V is Submodule of V;

theorem :: RMOD_2:33
for X,V being strict RightMod of R
holds V is Submodule of X & X is Submodule of V implies V = X;

definition let R,V;
cluster strict Submodule of V;
end;

theorem :: RMOD_2:34
V is Submodule of X & X is Submodule of Y implies V is Submodule of Y;

theorem :: RMOD_2:35
the carrier of W1 c= the carrier of W2 implies
W1 is Submodule of W2;

theorem :: RMOD_2:36
(for v st v in W1 holds v in W2) implies W1 is Submodule of W2;

theorem :: RMOD_2:37
for W1,W2 being strict Submodule of V
holds the carrier of W1 = the carrier of W2 implies
W1 = W2;

theorem :: RMOD_2:38
for W1,W2 being strict Submodule of V
holds (for v being Vector of V holds v in W1 iff v in W2) implies W1 = W2;

theorem :: RMOD_2:39
for V being strict RightMod of R, W being strict Submodule of V
holds the carrier of W = the carrier of V
implies W = V;

theorem :: RMOD_2:40
for V being strict RightMod of R, W being strict Submodule of V
holds (for v being Vector of V holds v in W) implies W = V;

theorem :: RMOD_2:41
the carrier of W = V1 implies V1 is lineary-closed;

theorem :: RMOD_2:42
V1 <> {} & V1 is lineary-closed implies
(ex W being strict Submodule of V st V1 = the carrier of W);

definition let R; let V;
func (0).V -> strict Submodule of V means
:: RMOD_2:def 3
the carrier of it = {0.V};
end;

definition let R; let V;
func (Omega).V -> strict Submodule of V equals
:: RMOD_2:def 4
the RightModStr of V;
end;

canceled 3;

theorem :: RMOD_2:46
x in (0).V iff x = 0.V;

theorem :: RMOD_2:47
(0).W = (0).V;

theorem :: RMOD_2:48
(0).W1 = (0).W2;

theorem :: RMOD_2:49
(0).W is Submodule of V;

theorem :: RMOD_2:50
(0).V is Submodule of W;

theorem :: RMOD_2:51
(0).W1 is Submodule of W2;

canceled;

theorem :: RMOD_2:53
for V being strict RightMod of R
holds V is Submodule of (Omega).V;

definition let R; let V; let v,W;
func v + W -> Subset of V equals
:: RMOD_2:def 5
{v + u : u in W};
end;

definition let R; let V; let W;
mode Coset of W -> Subset of V means
:: RMOD_2:def 6
ex v st it = v + W;
end;

reserve B,C for Coset of W;

canceled 3;

theorem :: RMOD_2:57
x in v + W iff ex u st u in W & x = v + u;

theorem :: RMOD_2:58
0.V in v + W iff v in W;

theorem :: RMOD_2:59
v in v + W;

theorem :: RMOD_2:60
0.V + W = the carrier of W;

theorem :: RMOD_2:61
v + (0).V = {v};

theorem :: RMOD_2:62
v + (Omega).V = the carrier of V;

theorem :: RMOD_2:63
0.V in v + W iff v + W = the carrier of W;

theorem :: RMOD_2:64
v in W iff v + W = the carrier of W;

theorem :: RMOD_2:65
v in W implies (v * a) + W = the carrier of W;

theorem :: RMOD_2:66
u in W iff v + W = (v + u) + W;

theorem :: RMOD_2:67
u in W iff v + W = (v - u) + W;

theorem :: RMOD_2:68
v in u + W iff u + W = v + W;

theorem :: RMOD_2:69
u in v1 + W & u in v2 + W implies v1 + W = v2 + W;

theorem :: RMOD_2:70
v in W implies v * a in v + W;

theorem :: RMOD_2:71
v in W implies - v in v + W;

theorem :: RMOD_2:72
u + v in v + W iff u in W;

theorem :: RMOD_2:73
v - u in v + W iff u in W;

canceled;

theorem :: RMOD_2:75
u in v + W iff
(ex v1 st v1 in W & u = v - v1);

theorem :: RMOD_2:76
(ex v st v1 in v + W & v2 in v + W) iff v1 - v2 in W;

theorem :: RMOD_2:77
v + W = u + W implies
(ex v1 st v1 in W & v + v1 = u);

theorem :: RMOD_2:78
v + W = u + W implies
(ex v1 st v1 in W & v - v1 = u);

theorem :: RMOD_2:79
for W1,W2 being strict Submodule of V
holds v + W1 = v + W2 iff W1 = W2;

theorem :: RMOD_2:80
for W1,W2 being strict Submodule of V
holds v + W1 = u + W2 implies W1 = W2;

theorem :: RMOD_2:81
ex C st v in C;

theorem :: RMOD_2:82
C is lineary-closed iff C = the carrier of W;

theorem :: RMOD_2:83
for W1,W2 being strict Submodule of V
for C1 being Coset of W1, C2 being Coset of W2
holds C1 = C2 implies W1 = W2;

theorem :: RMOD_2:84
{v} is Coset of (0).V;

theorem :: RMOD_2:85
V1 is Coset of (0).V implies (ex v st V1 = {v});

theorem :: RMOD_2:86
the carrier of W is Coset of W;

theorem :: RMOD_2:87
the carrier of V is Coset of (Omega).V;

theorem :: RMOD_2:88
V1 is Coset of (Omega).V implies V1 = the carrier of V;

theorem :: RMOD_2:89
0.V in C iff C = the carrier of W;

theorem :: RMOD_2:90
u in C iff C = u + W;

theorem :: RMOD_2:91
u in C & v in C implies (ex v1 st v1 in W & u + v1 = v);

theorem :: RMOD_2:92
u in C & v in C implies (ex v1 st v1 in W & u - v1 = v);

theorem :: RMOD_2:93
(ex C st v1 in C & v2 in C) iff v1 - v2 in W;

theorem :: RMOD_2:94
u in B & u in C implies B = C;
```