:: RMOD_3 semantic presentation

definition

let R be Ring;
let V be RightMod of R;
let W1, W2 be Submodule of V;
func W1 + W2 -> strict Submodule of V means :Def1: :: RMOD_3:def 1
the carrier of it = { (v + u) where v, u is Vector of V : ( v in W1 & u in W2 ) } ;
existence

proof end;

uniqueness by RMOD_2:37;

end;

:: deftheorem Def1 defines + RMOD_3:def 1 :

definition

let R be Ring;
let V be RightMod of R;
let W1, W2 be Submodule of V;
func W1 /\ W2 -> strict Submodule of V means :Def2: :: RMOD_3:def 2
the carrier of it = the carrier of W1 /\ the carrier of W2;
existence

proof end;

uniqueness by RMOD_2:37;

end;

:: deftheorem Def2 defines /\ RMOD_3:def 2 :

theorem :: RMOD_3:1

canceled;

theorem :: RMOD_3:2

canceled;

theorem :: RMOD_3:3

canceled;

theorem :: RMOD_3:4

canceled;

theorem Th5: :: RMOD_3:5

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V
for x being set holds
( x in W1 + W2 iff ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) )

proof end;

theorem :: RMOD_3:6

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

proof end;

theorem Th7: :: RMOD_3:7

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V
for x being set holds
( x in W1 /\ W2 iff ( x in W1 & x in W2 ) )

proof end;

Lm1: for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds W1 + W2 = W2 + W1

proof end;

Lm2: for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds the carrier of W1 c= the carrier of (W1 + W2)

proof end;

Lm3: for R being Ring
for V being RightMod of R
for W1 being Submodule of V
for W2 being strict Submodule of V st the carrier of W1 c= the carrier of W2 holds
W1 + W2 = W2

proof end;

theorem :: RMOD_3:8

for R being Ring
for V being RightMod of R
for W being strict Submodule of V holds W + W = W by Lm3;

theorem :: RMOD_3:9

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds W1 + W2 = W2 + W1 by Lm1;

theorem Th10: :: RMOD_3:10

for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V holds W1 + (W2 + W3) = (W1 + W2) + W3

proof end;

theorem Th11: :: RMOD_3:11

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds
( W1 is Submodule of W1 + W2 & W2 is Submodule of W1 + W2 )

proof end;

theorem Th12: :: RMOD_3:12

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

proof end;

theorem Th13: :: RMOD_3:13

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

proof end;

Lm4: for R being Ring
for V being RightMod of R
for W, W', W1 being Submodule of V st the carrier of W = the carrier of W' holds
( W1 + W = W1 + W' & W + W1 = W' + W1 )

proof end;

Lm5: for R being Ring
for V being RightMod of R
for W being Submodule of V holds W is Submodule of (Omega). V

proof end;

theorem Th14: :: RMOD_3:14

for R being Ring
for V being strict RightMod of R holds
( ((0). V) + ((Omega). V) = V & ((Omega). V) + ((0). V) = V ) by Th13;

theorem Th15: :: RMOD_3:15

for R being Ring
for V being RightMod of R
for W being Submodule of V holds
( ((Omega). V) + W = RightModStr(# the carrier of V,the add of V,the Zero of V,the rmult of V #) & W + ((Omega). V) = RightModStr(# the carrier of V,the add of V,the Zero of V,the rmult of V #) )

proof end;

theorem :: RMOD_3:16

for R being Ring
for V being strict RightMod of R holds ((Omega). V) + ((Omega). V) = V by Th15;

theorem :: RMOD_3:17

for R being Ring
for V being RightMod of R
for W being strict Submodule of V holds W /\ W = W

proof end;

theorem Th18: :: RMOD_3:18

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds W1 /\ W2 = W2 /\ W1

proof end;

theorem Th19: :: RMOD_3:19

for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V holds W1 /\ (W2 /\ W3) = (W1 /\ W2) /\ W3

proof end;

Lm6: for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds the carrier of (W1 /\ W2) c= the carrier of W1

proof end;

theorem Th20: :: RMOD_3:20

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds
( W1 /\ W2 is Submodule of W1 & W1 /\ W2 is Submodule of W2 )

proof end;

theorem Th21: :: RMOD_3:21

for R being Ring
for V being RightMod of R
for W2 being Submodule of V holds
( ( for W1 being strict Submodule of V st W1 is Submodule of W2 holds
W1 /\ W2 = W1 ) & ( for W1 being Submodule of V st W1 /\ W2 = W1 holds
W1 is Submodule of W2 ) )

proof end;

theorem :: RMOD_3:22

for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 holds
W1 /\ W3 is Submodule of W2 /\ W3

proof end;

theorem :: RMOD_3:23

for R being Ring
for V being RightMod of R
for W1, W3, W2 being Submodule of V st W1 is Submodule of W3 holds
W1 /\ W2 is Submodule of W3

proof end;

theorem :: RMOD_3:24

for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 & W1 is Submodule of W3 holds
W1 is Submodule of W2 /\ W3

proof end;

theorem Th25: :: RMOD_3:25

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

proof end;

theorem :: RMOD_3:26

canceled;

theorem Th27: :: RMOD_3:27

for R being Ring
for V being RightMod of R
for W being strict Submodule of V holds
( ((Omega). V) /\ W = W & W /\ ((Omega). V) = W )

proof end;

theorem :: RMOD_3:28

for R being Ring
for V being strict RightMod of R holds ((Omega). V) /\ ((Omega). V) = V by Th27;

Lm7: for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds the carrier of (W1 /\ W2) c= the carrier of (W1 + W2)

proof end;

theorem :: RMOD_3:29

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds W1 /\ W2 is Submodule of W1 + W2

proof end;

Lm8: for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds the carrier of ((W1 /\ W2) + W2) = the carrier of W2

proof end;

theorem Th30: :: RMOD_3:30

for R being Ring
for V being RightMod of R
for W1 being Submodule of V
for W2 being strict Submodule of V holds (W1 /\ W2) + W2 = W2

proof end;

Lm9: for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds the carrier of (W1 /\ (W1 + W2)) = the carrier of W1

proof end;

theorem Th31: :: RMOD_3:31

for R being Ring
for V being RightMod of R
for W2 being Submodule of V
for W1 being strict Submodule of V holds W1 /\ (W1 + W2) = W1

proof end;

Lm10: for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V holds the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + W3))

proof end;

theorem :: RMOD_3:32

for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V holds (W1 /\ W2) + (W2 /\ W3) is Submodule of W2 /\ (W1 + W3)

proof end;

Lm11: for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 holds
the carrier of (W2 /\ (W1 + W3)) = the carrier of ((W1 /\ W2) + (W2 /\ W3))

proof end;

theorem Th33: :: RMOD_3:33

for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 holds
W2 /\ (W1 + W3) = (W1 /\ W2) + (W2 /\ W3)

proof end;

Lm12: for R being Ring
for V being RightMod of R
for W2, W1, W3 being Submodule of V holds the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3))

proof end;

theorem :: RMOD_3:34

for R being Ring
for V being RightMod of R
for W2, W1, W3 being Submodule of V holds W2 + (W1 /\ W3) is Submodule of (W1 + W2) /\ (W2 + W3)

proof end;

Lm13: for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 holds
the carrier of (W2 + (W1 /\ W3)) = the carrier of ((W1 + W2) /\ (W2 + W3))

proof end;

theorem :: RMOD_3:35

for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 holds
W2 + (W1 /\ W3) = (W1 + W2) /\ (W2 + W3)

proof end;

theorem Th36: :: RMOD_3:36

for R being Ring
for V being RightMod of R
for W3, W2 being Submodule of V
for W1 being strict Submodule of V st W1 is Submodule of W3 holds
W1 + (W2 /\ W3) = (W1 + W2) /\ W3

proof end;

theorem :: RMOD_3:37

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

proof end;

theorem :: RMOD_3:38

for R being Ring
for V being RightMod of R
for W1 being Submodule of V
for W2, W3 being strict Submodule of V st W1 is Submodule of W2 holds
W1 + W3 is Submodule of W2 + W3

proof end;

theorem :: RMOD_3:39

for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 holds
W1 is Submodule of W2 + W3

proof end;

theorem :: RMOD_3:40

for R being Ring
for V being RightMod of R
for W1, W3, W2 being Submodule of V st W1 is Submodule of W3 & W2 is Submodule of W3 holds
W1 + W2 is Submodule of W3

proof end;

theorem :: RMOD_3:41

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds
( ( W1 is Submodule of W2 or W2 is Submodule of W1 ) iff ex W being Submodule of V st the carrier of W = the carrier of W1 \/ the carrier of W2 )

proof end;

definition

let R be Ring;
let V be RightMod of R;
func Submodules V -> set means :Def3: :: RMOD_3:def 3
for x being set holds
( x in it iff ex W being strict Submodule of V st W = x );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def3 defines Submodules RMOD_3:def 3 :

registration

let R be Ring;
let V be RightMod of R;
cluster Submodules V -> non empty ;
coherence

proof end;

end;

theorem :: RMOD_3:42

canceled;

theorem :: RMOD_3:43

canceled;

theorem :: RMOD_3:44

for R being Ring
for V being strict RightMod of R holds V in Submodules V

proof end;

definition

let R be Ring;
let V be RightMod of R;
let W1, W2 be Submodule of V;
pred V is_the_direct_sum_of W1,W2 means :Def4: :: RMOD_3:def 4
( RightModStr(# the carrier of V,the add of V,the Zero of V,the rmult of V #) = W1 + W2 & W1 /\ W2 = (0). V );

end;

:: deftheorem Def4 defines is_the_direct_sum_of RMOD_3:def 4 :

Lm14: for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds
( W1 + W2 = RightModStr(# the carrier of V,the add of V,the Zero of V,the rmult of V #) iff for v being Vector of V ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) )

proof end;

Lm15: for R being Ring
for V being RightMod of R
for v, v1, v2 being Vector of V holds
( v = v1 + v2 iff v1 = v - v2 )

proof end;

theorem :: RMOD_3:45

canceled;

theorem Th46: :: RMOD_3:46

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V st V is_the_direct_sum_of W1,W2 holds
V is_the_direct_sum_of W2,W1

proof end;

theorem :: RMOD_3:47

for R being Ring
for V being strict RightMod of R holds
( V is_the_direct_sum_of (0). V, (Omega). V & V is_the_direct_sum_of (Omega). V, (0). V )

proof end;

theorem Th48: :: RMOD_3:48

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V
for C1 being Coset of W1
for C2 being Coset of W2 st C1 meets C2 holds
C1 /\ C2 is Coset of W1 /\ W2

proof end;

theorem Th49: :: RMOD_3:49

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds
( V is_the_direct_sum_of W1,W2 iff for C1 being Coset of W1
for C2 being Coset of W2 ex v being Vector of V st C1 /\ C2 = {v} )

proof end;

theorem :: RMOD_3:50

for R being Ring
for V being strict RightMod of R
for W1, W2 being Submodule of V holds
( W1 + W2 = V iff for v being Vector of V ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) ) by Lm14;

theorem Th51: :: RMOD_3:51

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V
for v, v1, v2, u1, u2 being Vector of V st V is_the_direct_sum_of W1,W2 & v = v1 + v2 & v = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds
( v1 = u1 & v2 = u2 )

proof end;

theorem :: RMOD_3:52

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V st V = W1 + W2 & ex v being Vector of V st
for v1, v2, u1, u2 being Vector of V st v = v1 + v2 & v = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds
( v1 = u1 & v2 = u2 ) holds
V is_the_direct_sum_of W1,W2

proof end;

definition

let R be Ring;
let V be RightMod of R;
let v be Vector of V;
let W1, W2 be Submodule of V;
assume A1: V is_the_direct_sum_of W1,W2 ;
func v |-- W1,W2 -> Element of [:the carrier of V,the carrier of V:] means :Def5: :: RMOD_3:def 5
( v = (it `1 ) + (it `2 ) & it `1 in W1 & it `2 in W2 );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def5 defines |-- RMOD_3:def 5 :

theorem :: RMOD_3:53

canceled;

theorem :: RMOD_3:54

canceled;

theorem :: RMOD_3:55

canceled;

theorem :: RMOD_3:56

canceled;

theorem :: RMOD_3:57

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V
for v being Vector of V st V is_the_direct_sum_of W1,W2 holds
(v |-- W1,W2) `1 = (v |-- W2,W1) `2

proof end;

theorem :: RMOD_3:58

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V
for v being Vector of V st V is_the_direct_sum_of W1,W2 holds
(v |-- W1,W2) `2 = (v |-- W2,W1) `1

proof end;

definition

let R be Ring;
let V be RightMod of R;
func SubJoin V -> BinOp of Submodules V means :Def6: :: RMOD_3:def 6
for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
it . A1,A2 = W1 + W2;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def6 defines SubJoin RMOD_3:def 6 :

definition

let R be Ring;
let V be RightMod of R;
func SubMeet V -> BinOp of Submodules V means :Def7: :: RMOD_3:def 7
for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
it . A1,A2 = W1 /\ W2;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def7 defines SubMeet RMOD_3:def 7 :

theorem :: RMOD_3:59

canceled;

theorem :: RMOD_3:60

canceled;

theorem :: RMOD_3:61

canceled;

theorem :: RMOD_3:62

canceled;

theorem Th63: :: RMOD_3:63

for R being Ring
for V being RightMod of R holds LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is Lattice

proof end;

theorem Th64: :: RMOD_3:64

for R being Ring
for V being RightMod of R holds LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is 0_Lattice

proof end;

theorem Th65: :: RMOD_3:65

for R being Ring
for V being RightMod of R holds LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is 1_Lattice

proof end;

theorem :: RMOD_3:66

for R being Ring
for V being RightMod of R holds LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is 01_Lattice

proof end;

theorem :: RMOD_3:67

for R being Ring
for V being RightMod of R holds LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is M_Lattice

proof end;