:: Submodules and Cosets of Submodules in Right Module over Associative Ring
:: by Michal Muzalewski and Wojciech Skaba
::
:: Received October 22, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users

definition

let R be Ring;
let V be RightMod of R;
let V1 be Subset of V;

attr V1 is linearly-closed means :: RMOD_2:def 1
( ( for v, u being Vector of V st v in V1 & u in V1 holds
v + u in V1 ) & ( for a being Scalar of R
for v being Vector of V st v in V1 holds
v * a in V1 ) );

end;

:: deftheorem defines linearly-closed RMOD_2:def 1 :
for R being Ring
for V being RightMod of R
for V1 being Subset of V holds
( V1 is linearly-closed iff ( ( for v, u being Vector of V st v in V1 & u in V1 holds
v + u in V1 ) & ( for a being Scalar of R
for v being Vector of V st v in V1 holds
v * a in V1 ) ) );

theorem Th1: :: RMOD_2:1

for R being Ring
for V being RightMod of R
for V1 being Subset of V st V1 <> {} & V1 is linearly-closed holds
0. V in V1

proof end;

theorem Th2: :: RMOD_2:2

for R being Ring
for V being RightMod of R
for V1 being Subset of V st V1 is linearly-closed holds
for v being Vector of V st v in V1 holds
- v in V1

proof end;

theorem :: RMOD_2:3

for R being Ring
for V being RightMod of R
for V1 being Subset of V st V1 is linearly-closed holds
for v, u being Vector of V st v in V1 & u in V1 holds
v - u in V1

proof end;

theorem Th4: :: RMOD_2:4

for R being Ring
for V being RightMod of R holds {(0. V)} is linearly-closed

proof end;

theorem :: RMOD_2:5

for R being Ring
for V being RightMod of R
for V1 being Subset of V st the carrier of V = V1 holds
V1 is linearly-closed ;

theorem :: RMOD_2:6

for R being Ring
for V being RightMod of R
for V1, V2, V3 being Subset of V st V1 is linearly-closed & V2 is linearly-closed & V3 = { (v + u) where u, v is Vector of V : ( v in V1 & u in V2 ) } holds
V3 is linearly-closed

proof end;

theorem :: RMOD_2:7

for R being Ring
for V being RightMod of R
for V1, V2 being Subset of V st V1 is linearly-closed & V2 is linearly-closed holds
V1 /\ V2 is linearly-closed

proof end;

definition

let R be Ring;
let V be RightMod of R;

mode Submodule of V -> RightMod of R means :Def2: :: RMOD_2:def 2
( the carrier of it c= the carrier of V & 0. it = 0. V & the addF of it = the addF 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:] );

existence
ex b₁ being RightMod of R st
( the carrier of b₁ c= the carrier of V & 0. b₁ = 0. V & the addF of b₁ = the addF of V | [: the carrier of b₁, the carrier of b₁:] & the rmult of b₁ = the rmult of V | [: the carrier of b₁, the carrier of R:] )

proof end;

end;

:: deftheorem Def2 defines Submodule RMOD_2:def 2 :
for R being Ring
for V, b₃ being RightMod of R holds
( b₃ is Submodule of V iff ( the carrier of b₃ c= the carrier of V & 0. b₃ = 0. V & the addF of b₃ = the addF of V | [: the carrier of b₃, the carrier of b₃:] & the rmult of b₃ = the rmult of V | [: the carrier of b₃, the carrier of R:] ) );

theorem :: RMOD_2:8

for x being object
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V st x in W1 & W1 is Submodule of W2 holds
x in W2

proof end;

theorem Th9: :: RMOD_2:9

for x being object
for R being Ring
for V being RightMod of R
for W being Submodule of V st x in W holds
x in V

proof end;

theorem Th10: :: RMOD_2:10

for R being Ring
for V being RightMod of R
for W being Submodule of V
for w being Vector of W holds w is Vector of V

proof end;

theorem :: RMOD_2:11

for R being Ring
for V being RightMod of R
for W being Submodule of V holds 0. W = 0. V by Def2;

theorem :: RMOD_2:12

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds 0. W1 = 0. W2

proof end;

theorem Th13: :: RMOD_2:13

for R being Ring
for V being RightMod of R
for u, v being Vector of V
for W being Submodule of V
for w1, w2 being Vector of W st w1 = v & w2 = u holds
w1 + w2 = v + u

proof end;

theorem Th14: :: RMOD_2:14

for R being Ring
for a being Scalar of R
for V being RightMod of R
for v being Vector of V
for W being Submodule of V
for w being Vector of W st w = v holds
w * a = v * a

proof end;

theorem Th15: :: RMOD_2:15

for R being Ring
for V being RightMod of R
for v being Vector of V
for W being Submodule of V
for w being Vector of W st w = v holds
- v = - w

proof end;

theorem Th16: :: RMOD_2:16

for R being Ring
for V being RightMod of R
for u, v being Vector of V
for W being Submodule of V
for w1, w2 being Vector of W st w1 = v & w2 = u holds
w1 - w2 = v - u

proof end;

Lm1: for R being Ring
for V being RightMod of R
for V1 being Subset of V
for W being Submodule of V st the carrier of W = V1 holds
V1 is linearly-closed

proof end;

theorem Th17: :: RMOD_2:17

for R being Ring
for V being RightMod of R
for W being Submodule of V holds 0. V in W

proof end;

theorem :: RMOD_2:18

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds 0. W1 in W2

proof end;

theorem :: RMOD_2:19

for R being Ring
for V being RightMod of R
for W being Submodule of V holds 0. W in V by Th9, RLVECT_1:1;

theorem Th20: :: RMOD_2:20

for R being Ring
for V being RightMod of R
for u, v being Vector of V
for W being Submodule of V st u in W & v in W holds
u + v in W

proof end;

theorem Th21: :: RMOD_2:21

for R being Ring
for a being Scalar of R
for V being RightMod of R
for v being Vector of V
for W being Submodule of V st v in W holds
v * a in W

proof end;

theorem Th22: :: RMOD_2:22

for R being Ring
for V being RightMod of R
for v being Vector of V
for W being Submodule of V st v in W holds
- v in W

proof end;

theorem Th23: :: RMOD_2:23

for R being Ring
for V being RightMod of R
for u, v being Vector of V
for W being Submodule of V st u in W & v in W holds
u - v in W

proof end;

theorem Th24: :: RMOD_2:24

for R being Ring
for V being RightMod of R holds V is Submodule of V

proof end;

theorem Th25: :: RMOD_2:25

for R being Ring
for X, V being strict RightMod of R st V is Submodule of X & X is Submodule of V holds
V = X

proof end;

registration

let R be Ring;
let V be RightMod of R;

cluster non empty right_complementable Abelian add-associative right_zeroed strict RightMod-like for Submodule of V;

existence
ex b₁ being Submodule of V st b₁ is strict

proof end;

end;

theorem Th26: :: RMOD_2:26

for R being Ring
for V, X, Y being RightMod of R st V is Submodule of X & X is Submodule of Y holds
V is Submodule of Y

proof end;

theorem Th27: :: RMOD_2:27

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V st the carrier of W1 c= the carrier of W2 holds
W1 is Submodule of W2

proof end;

theorem :: RMOD_2:28

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V st ( for v being Vector of V st v in W1 holds
v in W2 ) holds
W1 is Submodule of W2

proof end;

theorem Th29: :: RMOD_2:29

for R being Ring
for V being RightMod of R
for W1, W2 being strict Submodule of V st the carrier of W1 = the carrier of W2 holds
W1 = W2

proof end;

theorem Th30: :: RMOD_2:30

for R being Ring
for V being RightMod of R
for W1, W2 being strict Submodule of V st ( for v being Vector of V holds
( v in W1 iff v in W2 ) ) holds
W1 = W2

proof end;

theorem :: RMOD_2:31

for R being Ring
for V being strict RightMod of R
for W being strict Submodule of V st the carrier of W = the carrier of V holds
W = V

proof end;

theorem :: RMOD_2:32

for R being Ring
for V being strict RightMod of R
for W being strict Submodule of V st ( for v being Vector of V holds v in W ) holds
W = V

proof end;

theorem :: RMOD_2:33

for R being Ring
for V being RightMod of R
for V1 being Subset of V
for W being Submodule of V st the carrier of W = V1 holds
V1 is linearly-closed by Lm1;

theorem Th34: :: RMOD_2:34

for R being Ring
for V being RightMod of R
for V1 being Subset of V st V1 <> {} & V1 is linearly-closed holds
ex W being strict Submodule of V st V1 = the carrier of W

proof end;

definition

let R be Ring;
let V be RightMod of R;

func (0). V -> strict Submodule of V means :Def3: :: RMOD_2:def 3
the carrier of it = {(0. V)};

correctness
existence
ex b₁ being strict Submodule of V st the carrier of b₁ = {(0. V)};
uniqueness
for b₁, b₂ being strict Submodule of V st the carrier of b₁ = {(0. V)} & the carrier of b₂ = {(0. V)} holds
b₁ = b₂;
by Th4, Th29, Th34;

end;

:: deftheorem Def3 defines (0). RMOD_2:def 3 :
for R being Ring
for V being RightMod of R
for b₃ being strict Submodule of V holds
( b₃ = (0). V iff the carrier of b₃ = {(0. V)} );

definition

let R be Ring;
let V be RightMod of R;

func (Omega). V -> strict Submodule of V equals :: RMOD_2:def 4
RightModStr(# the carrier of V, the addF of V, the ZeroF of V, the rmult of V #);

coherence
RightModStr(# the carrier of V, the addF of V, the ZeroF of V, the rmult of V #) is strict Submodule of V

proof end;

end;

:: deftheorem defines (Omega). RMOD_2:def 4 :
for R being Ring
for V being RightMod of R holds (Omega). V = RightModStr(# the carrier of V, the addF of V, the ZeroF of V, the rmult of V #);

theorem :: RMOD_2:35

for x being object
for R being Ring
for V being RightMod of R holds
( x in (0). V iff x = 0. V )

proof end;

theorem Th36: :: RMOD_2:36

for R being Ring
for V being RightMod of R
for W being Submodule of V holds (0). W = (0). V

proof end;

theorem Th37: :: RMOD_2:37

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds (0). W1 = (0). W2

proof end;

theorem :: RMOD_2:38

for R being Ring
for V being RightMod of R
for W being Submodule of V holds (0). W is Submodule of V by Th26;

theorem :: RMOD_2:39

for R being Ring
for V being RightMod of R
for W being Submodule of V holds (0). V is Submodule of W

proof end;

theorem :: RMOD_2:40

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds (0). W1 is Submodule of W2

proof end;

theorem :: RMOD_2:41

for R being Ring
for V being strict RightMod of R holds V is Submodule of (Omega). V ;

definition

let R be Ring;
let V be RightMod of R;
let v be Vector of V;
let W be Submodule of V;

func v + W -> Subset of V equals :: RMOD_2:def 5
{ (v + u) where u is Vector of V : u in W } ;

coherence
{ (v + u) where u is Vector of V : u in W } is Subset of V

proof end;

end;

:: deftheorem defines + RMOD_2:def 5 :
for R being Ring
for V being RightMod of R
for v being Vector of V
for W being Submodule of V holds v + W = { (v + u) where u is Vector of V : u in W } ;

Lm2: for R being Ring
for V being RightMod of R
for W being Submodule of V holds (0. V) + W = the carrier of W

proof end;

definition

let R be Ring;
let V be RightMod of R;
let W be Submodule of V;

mode Coset of W -> Subset of V means :Def6: :: RMOD_2:def 6
ex v being Vector of V st it = v + W;

existence
ex b₁ being Subset of V ex v being Vector of V st b₁ = v + W

proof end;

end;

:: deftheorem Def6 defines Coset RMOD_2:def 6 :
for R being Ring
for V being RightMod of R
for W being Submodule of V
for b₄ being Subset of V holds
( b₄ is Coset of W iff ex v being Vector of V st b₄ = v + W );

theorem Th42: :: RMOD_2:42

for x being object
for R being Ring
for V being RightMod of R
for v being Vector of V
for W being Submodule of V holds
( x in v + W iff ex u being Vector of V st
( u in W & x = v + u ) )

proof end;

theorem Th43: :: RMOD_2:43

for R being Ring
for V being RightMod of R
for v being Vector of V
for W being Submodule of V holds
( 0. V in v + W iff v in W )

proof end;

theorem Th44: :: RMOD_2:44

for R being Ring
for V being RightMod of R
for v being Vector of V
for W being Submodule of V holds v in v + W

proof end;

theorem :: RMOD_2:45

for R being Ring
for V being RightMod of R
for W being Submodule of V holds (0. V) + W = the carrier of W by Lm2;

theorem Th46: :: RMOD_2:46

for R being Ring
for V being RightMod of R
for v being Vector of V holds v + ((0). V) = {v}

proof end;

Lm3: for R being Ring
for V being RightMod of R
for v being Vector of V
for W being Submodule of V holds
( v in W iff v + W = the carrier of W )

proof end;

theorem Th47: :: RMOD_2:47

for R being Ring
for V being RightMod of R
for v being Vector of V holds v + ((Omega). V) = the carrier of V by RLVECT_1:1, Lm3;

theorem Th48: :: RMOD_2:48

for R being Ring
for V being RightMod of R
for v being Vector of V
for W being Submodule of V holds
( 0. V in v + W iff v + W = the carrier of W ) by Th43, Lm3;

theorem :: RMOD_2:49

for R being Ring
for V being RightMod of R
for v being Vector of V
for W being Submodule of V holds
( v in W iff v + W = the carrier of W ) by Lm3;

theorem :: RMOD_2:50

for R being Ring
for a being Scalar of R
for V being RightMod of R
for v being Vector of V
for W being Submodule of V st v in W holds
(v * a) + W = the carrier of W by Th21, Lm3;

theorem Th51: :: RMOD_2:51

for R being Ring
for V being RightMod of R
for u, v being Vector of V
for W being Submodule of V holds
( u in W iff v + W = (v + u) + W )

proof end;

theorem :: RMOD_2:52

for R being Ring
for V being RightMod of R
for u, v being Vector of V
for W being Submodule of V holds
( u in W iff v + W = (v - u) + W )

proof end;

theorem Th53: :: RMOD_2:53

for R being Ring
for V being RightMod of R
for u, v being Vector of V
for W being Submodule of V holds
( v in u + W iff u + W = v + W )

proof end;

theorem Th54: :: RMOD_2:54

for R being Ring
for V being RightMod of R
for u, v1, v2 being Vector of V
for W being Submodule of V st u in v1 + W & u in v2 + W holds
v1 + W = v2 + W

proof end;

theorem Th55: :: RMOD_2:55

for R being Ring
for a being Scalar of R
for V being RightMod of R
for v being Vector of V
for W being Submodule of V st v in W holds
v * a in v + W

proof end;

theorem :: RMOD_2:56

for R being Ring
for V being RightMod of R
for v being Vector of V
for W being Submodule of V st v in W holds
- v in v + W

proof end;

theorem Th57: :: RMOD_2:57

for R being Ring
for V being RightMod of R
for u, v being Vector of V
for W being Submodule of V holds
( u + v in v + W iff u in W )

proof end;

theorem :: RMOD_2:58

for R being Ring
for V being RightMod of R
for u, v being Vector of V
for W being Submodule of V holds
( v - u in v + W iff u in W )

proof end;

theorem :: RMOD_2:59

for R being Ring
for V being RightMod of R
for u, v being Vector of V
for W being Submodule of V holds
( u in v + W iff ex v1 being Vector of V st
( v1 in W & u = v - v1 ) )

proof end;

theorem Th60: :: RMOD_2:60

for R being Ring
for V being RightMod of R
for v1, v2 being Vector of V
for W being Submodule of V holds
( ex v being Vector of V st
( v1 in v + W & v2 in v + W ) iff v1 - v2 in W )

proof end;

theorem Th61: :: RMOD_2:61

for R being Ring
for V being RightMod of R
for u, v being Vector of V
for W being Submodule of V st v + W = u + W holds
ex v1 being Vector of V st
( v1 in W & v + v1 = u )

proof end;

theorem Th62: :: RMOD_2:62

for R being Ring
for V being RightMod of R
for u, v being Vector of V
for W being Submodule of V st v + W = u + W holds
ex v1 being Vector of V st
( v1 in W & v - v1 = u )

proof end;

theorem Th63: :: RMOD_2:63

for R being Ring
for V being RightMod of R
for v being Vector of V
for W1, W2 being strict Submodule of V holds
( v + W1 = v + W2 iff W1 = W2 )

proof end;

theorem Th64: :: RMOD_2:64

for R being Ring
for V being RightMod of R
for u, v being Vector of V
for W1, W2 being strict Submodule of V st v + W1 = u + W2 holds
W1 = W2

proof end;

theorem :: RMOD_2:65

for R being Ring
for V being RightMod of R
for v being Vector of V
for W being Submodule of V ex C being Coset of W st v in C

proof end;

theorem :: RMOD_2:66

for R being Ring
for V being RightMod of R
for W being Submodule of V
for C being Coset of W holds
( C is linearly-closed iff C = the carrier of W )

proof end;

theorem :: RMOD_2:67

for R being Ring
for V being RightMod of R
for W1, W2 being strict Submodule of V
for C1 being Coset of W1
for C2 being Coset of W2 st C1 = C2 holds
W1 = W2

proof end;

theorem :: RMOD_2:68

for R being Ring
for V being RightMod of R
for v being Vector of V holds {v} is Coset of (0). V

proof end;

theorem :: RMOD_2:69

for R being Ring
for V being RightMod of R
for V1 being Subset of V st V1 is Coset of (0). V holds
ex v being Vector of V st V1 = {v}

proof end;

theorem :: RMOD_2:70

for R being Ring
for V being RightMod of R
for W being Submodule of V holds the carrier of W is Coset of W

proof end;

theorem :: RMOD_2:71

for R being Ring
for V being RightMod of R holds the carrier of V is Coset of (Omega). V

proof end;

theorem :: RMOD_2:72

for R being Ring
for V being RightMod of R
for V1 being Subset of V st V1 is Coset of (Omega). V holds
V1 = the carrier of V

proof end;

theorem :: RMOD_2:73

for R being Ring
for V being RightMod of R
for W being Submodule of V
for C being Coset of W holds
( 0. V in C iff C = the carrier of W )

proof end;

theorem Th74: :: RMOD_2:74

for R being Ring
for V being RightMod of R
for u being Vector of V
for W being Submodule of V
for C being Coset of W holds
( u in C iff C = u + W )

proof end;

theorem :: RMOD_2:75

for R being Ring
for V being RightMod of R
for u, v being Vector of V
for W being Submodule of V
for C being Coset of W st u in C & v in C holds
ex v1 being Vector of V st
( v1 in W & u + v1 = v )

proof end;

theorem :: RMOD_2:76

for R being Ring
for V being RightMod of R
for u, v being Vector of V
for W being Submodule of V
for C being Coset of W st u in C & v in C holds
ex v1 being Vector of V st
( v1 in W & u - v1 = v )

proof end;

theorem :: RMOD_2:77

for R being Ring
for V being RightMod of R
for v1, v2 being Vector of V
for W being Submodule of V holds
( ex C being Coset of W st
( v1 in C & v2 in C ) iff v1 - v2 in W )

proof end;

theorem :: RMOD_2:78

for R being Ring
for V being RightMod of R
for u being Vector of V
for W being Submodule of V
for B, C being Coset of W st u in B & u in C holds
B = C

proof end;