:: RMOD_2 semantic presentation

definition

let R be Ring;
let V be RightMod of R;
let V1 be Subset of V;
attr V1 is lineary-closed means :Def1: :: 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 Def1 defines lineary-closed RMOD_2:def 1 :

theorem :: RMOD_2:1

canceled;

theorem :: RMOD_2:2

canceled;

theorem :: RMOD_2:3

canceled;

theorem Th4: :: RMOD_2:4

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

proof end;

theorem Th5: :: RMOD_2:5

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

proof end;

theorem :: RMOD_2:6

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

proof end;

theorem Th7: :: RMOD_2:7

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

proof end;

theorem :: RMOD_2:8

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 lineary-closed

proof end;

theorem :: RMOD_2:9

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

proof end;

theorem :: RMOD_2:10

for R being Ring
for V being RightMod of R
for V1, V2 being Subset of V st V1 is lineary-closed & V2 is lineary-closed holds
V1 /\ V2 is lineary-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 & the Zero of it = the Zero of V & the add of it = the add 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

proof end;

end;

:: deftheorem Def2 defines Submodule RMOD_2:def 2 :

theorem :: RMOD_2:11

canceled;

theorem :: RMOD_2:12

canceled;

theorem :: RMOD_2:13

canceled;

theorem :: RMOD_2:14

canceled;

theorem :: RMOD_2:15

canceled;

theorem :: RMOD_2:16

for x being set
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 Th17: :: RMOD_2:17

for x being set
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 Th18: :: RMOD_2:18

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 Th19: :: RMOD_2:19

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_2:20

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 Th21: :: RMOD_2:21

for R being Ring
for V being RightMod of R
for v, u 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 Th22: :: RMOD_2:22

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 Th23: :: RMOD_2:23

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 Th24: :: RMOD_2:24

for R being Ring
for V being RightMod of R
for v, u 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 lineary-closed

proof end;

theorem Th25: :: RMOD_2:25

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_2:26

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_2:27

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

proof end;

theorem Th28: :: RMOD_2:28

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 Th29: :: RMOD_2:29

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 Th30: :: RMOD_2:30

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 Th31: :: RMOD_2:31

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 Th32: :: RMOD_2:32

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

proof end;

theorem Th33: :: RMOD_2:33

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 strict Submodule of V;
existence

proof end;

end;

theorem Th34: :: RMOD_2:34

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 Th35: :: RMOD_2:35

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_2:36

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 Th37: :: RMOD_2:37

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 Th38: :: RMOD_2:38

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_2:39

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_2:40

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_2:41

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 lineary-closed by Lm1;

theorem Th42: :: RMOD_2:42

for R being Ring
for V being RightMod of R
for V1 being Subset of V st V1 <> {} & V1 is lineary-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

proof end;

end;

:: deftheorem Def3 defines (0). RMOD_2:def 3 :

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 add of V,the Zero of V,the rmult of V #);
coherence

proof end;

end;

:: deftheorem defines (Omega). RMOD_2:def 4 :

theorem :: RMOD_2:43

canceled;

theorem :: RMOD_2:44

canceled;

theorem :: RMOD_2:45

canceled;

theorem :: RMOD_2:46

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

proof end;

theorem Th47: :: RMOD_2:47

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

proof end;

theorem Th48: :: RMOD_2:48

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_2:49

for R being Ring
for V being RightMod of R
for W being Submodule of V holds (0). W is Submodule of V by Th34;

theorem :: RMOD_2:50

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_2:51

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_2:52

canceled;

theorem :: RMOD_2:53

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

proof end;

end;

:: deftheorem defines + RMOD_2:def 5 :

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

proof end;

end;

:: deftheorem Def6 defines Coset RMOD_2:def 6 :

theorem :: RMOD_2:54

canceled;

theorem :: RMOD_2:55

canceled;

theorem :: RMOD_2:56

canceled;

theorem Th57: :: RMOD_2:57

for x being set
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 Th58: :: RMOD_2:58

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 Th59: :: RMOD_2:59

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_2:60

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 Th61: :: RMOD_2:61

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 Th62: :: RMOD_2:62

for R being Ring
for V being RightMod of R
for v being Vector of V holds v + ((Omega). V) = the carrier of V

proof end;

theorem Th63: :: RMOD_2:63

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 )

proof end;

theorem :: RMOD_2:64

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_2:65

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

proof end;

theorem Th66: :: RMOD_2:66

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_2:67

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 Th68: :: RMOD_2:68

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

proof end;

theorem Th69: :: RMOD_2:69

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 Th70: :: RMOD_2:70

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_2:71

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 Th72: :: RMOD_2:72

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_2:73

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

proof end;

theorem :: RMOD_2:74

canceled;

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 holds
( u in v + W iff ex v1 being Vector of V st
( v1 in W & u = v - v1 ) )

proof end;

theorem Th76: :: RMOD_2:76

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 Th77: :: RMOD_2:77

for R being Ring
for V being RightMod of R
for v, u 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 Th78: :: RMOD_2:78

for R being Ring
for V being RightMod of R
for v, u 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 Th79: :: RMOD_2:79

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 Th80: :: RMOD_2:80

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

proof end;

theorem :: RMOD_2:81

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_2:82

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 lineary-closed iff C = the carrier of W )

proof end;

theorem :: RMOD_2:83

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_2:84

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_2:85

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_2:86

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_2:87

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

proof end;

theorem :: RMOD_2:88

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_2:89

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 Th90: :: RMOD_2:90

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_2:91

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_2:92

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_2:93

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_2:94

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;