:: Operations on 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 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 u, v is Vector of V : ( v in W1 & u in W2 ) } ;

existence
ex b₁ being strict Submodule of V st the carrier of b₁ = { (v + u) where u, v is Vector of V : ( v in W1 & u in W2 ) }

proof end;

uniqueness
for b₁, b₂ being strict Submodule of V st the carrier of b₁ = { (v + u) where u, v is Vector of V : ( v in W1 & u in W2 ) } & the carrier of b₂ = { (v + u) where u, v is Vector of V : ( v in W1 & u in W2 ) } holds
b₁ = b₂ by RMOD_2:29;

end;

:: deftheorem Def1 defines + RMOD_3:def 1 :
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V
for b₅ being strict Submodule of V holds
( b₅ = W1 + W2 iff the carrier of b₅ = { (v + u) where u, v is Vector of V : ( v in W1 & u in W2 ) } );

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
ex b₁ being strict Submodule of V st the carrier of b₁ = the carrier of W1 /\ the carrier of W2

proof end;

uniqueness
for b₁, b₂ being strict Submodule of V st the carrier of b₁ = the carrier of W1 /\ the carrier of W2 & the carrier of b₂ = the carrier of W1 /\ the carrier of W2 holds
b₁ = b₂ by RMOD_2:29;

end;

:: deftheorem Def2 defines /\ RMOD_3:def 2 :
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V
for b₅ being strict Submodule of V holds
( b₅ = W1 /\ W2 iff the carrier of b₅ = the carrier of W1 /\ the carrier of W2 );

theorem Th1: :: RMOD_3:1

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V
for x being object 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_3:2

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 Th3: :: RMOD_3:3

for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V
for x being object 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_3:4

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_3:5

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 Th6: :: RMOD_3:6

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 Th7: :: RMOD_3:7

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 Th8: :: RMOD_3:8

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 Th9: :: RMOD_3:9

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, W9, W1 being Submodule of V st the carrier of W = the carrier of W9 holds
( W1 + W = W1 + W9 & W + W1 = W9 + 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 :: RMOD_3:10

for R being Ring
for V being strict RightMod of R holds
( ((0). V) + ((Omega). V) = V & ((Omega). V) + ((0). V) = V ) by Th9;

theorem Th11: :: RMOD_3:11

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 U7 of V, the ZeroF of V, the rmult of V #) & W + ((Omega). V) = RightModStr(# the carrier of V, the U7 of V, the ZeroF of V, the rmult of V #) )

proof end;

theorem :: RMOD_3:12

for R being Ring
for V being strict RightMod of R holds ((Omega). V) + ((Omega). V) = V by Th11;

theorem :: RMOD_3:13

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

proof end;

theorem Th14: :: RMOD_3:14

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 Th15: :: RMOD_3:15

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 Th16: :: RMOD_3:16

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 Th17: :: RMOD_3:17

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_3:18

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_3:19

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

proof end;

theorem :: RMOD_3:20

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 Th21: :: RMOD_3:21

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 Th22: :: RMOD_3:22

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_3:23

for R being Ring
for V being strict RightMod of R holds ((Omega). V) /\ ((Omega). V) = V by Th22;

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_3:24

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 by Lm7, RMOD_2:27;

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 :: RMOD_3:25

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 by Lm8, RMOD_2:29;

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 :: RMOD_3:26

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 by Lm9, RMOD_2:29;

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_3:27

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) by Lm10, RMOD_2:27;

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 :: RMOD_3:28

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) by Lm11, RMOD_2:29;

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

proof end;

theorem :: RMOD_3:29

for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V holds W2 + (W1 /\ W3) is Submodule of (W1 + W2) /\ (W2 + W3) by Lm12, RMOD_2:27;

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_3:30

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) by Lm13, RMOD_2:29;

theorem Th31: :: RMOD_3:31

for R being Ring
for V being RightMod of R
for W2, W3 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_3:32

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_3:33

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_3:34

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_3:35

for R being Ring
for V being RightMod of R
for W1, W2, W3 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_3:36

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 object holds
( x in it iff ex W being strict Submodule of V st W = x );

existence
ex b₁ being set st
for x being object holds
( x in b₁ iff ex W being strict Submodule of V st W = x )

proof end;

uniqueness
for b₁, b₂ being set st ( for x being object holds
( x in b₁ iff ex W being strict Submodule of V st W = x ) ) & ( for x being object holds
( x in b₂ iff ex W being strict Submodule of V st W = x ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def3 defines Submodules RMOD_3:def 3 :
for R being Ring
for V being RightMod of R
for b₃ being set holds
( b₃ = Submodules V iff for x being object holds
( x in b₃ iff ex W being strict Submodule of V st W = x ) );

registration

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

cluster Submodules V -> non empty ;

coherence
not Submodules V is empty

proof end;

end;

theorem :: RMOD_3:37

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 :: RMOD_3:def 4
( RightModStr(# the carrier of V, the U7 of V, the ZeroF of V, the rmult of V #) = W1 + W2 & W1 /\ W2 = (0). V );

end;

:: deftheorem defines is_the_direct_sum_of RMOD_3:def 4 :
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 ( RightModStr(# the carrier of V, the U7 of V, the ZeroF of V, the rmult of V #) = W1 + W2 & W1 /\ W2 = (0). V ) );

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 U7 of V, the ZeroF 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 Th38: :: RMOD_3:38

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 by Th14, Lm1;

theorem :: RMOD_3:39

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 ) by Th9, Th21;

theorem Th40: :: RMOD_3:40

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 Th41: :: RMOD_3:41

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_3:42

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 Th43: :: RMOD_3:43

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_3:44

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
ex b₁ being Element of [: the carrier of V, the carrier of V:] st
( v = (b₁ `1) + (b<sub>1</sub> `2) & b₁ `1 in W1 & b<sub>1</sub> `2 in W2 )

proof end;

uniqueness
for b₁, b₂ being Element of [: the carrier of V, the carrier of V:] st v = (b₁ `1) + (b<sub>1</sub> `2) & b₁ `1 in W1 & b<sub>1</sub> `2 in W2 & v = (b₂ `1) + (b<sub>2</sub> `2) & b₂ `1 in W1 & b<sub>2</sub> `2 in W2 holds
b₁ = b₂

proof end;

end;

:: deftheorem Def5 defines |-- RMOD_3:def 5 :
for R being Ring
for V being RightMod of R
for v being Vector of V
for W1, W2 being Submodule of V st V is_the_direct_sum_of W1,W2 holds
for b₆ being Element of [: the carrier of V, the carrier of V:] holds
( b₆ = v |-- (W1,W2) iff ( v = (b₆ `1) + (b<sub>6</sub> `2) & b₆ `1 in W1 & b<sub>6</sub> `2 in W2 ) );

theorem :: RMOD_3:45

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_3:46

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
ex b₁ being BinOp of (Submodules V) st
for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b₁ . (A1,A2) = W1 + W2

proof end;

uniqueness
for b₁, b₂ being BinOp of (Submodules V) st ( for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b₁ . (A1,A2) = W1 + W2 ) & ( for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b₂ . (A1,A2) = W1 + W2 ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines SubJoin RMOD_3:def 6 :
for R being Ring
for V being RightMod of R
for b₃ being BinOp of (Submodules V) holds
( b₃ = SubJoin V iff for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b₃ . (A1,A2) = W1 + W2 );

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
ex b₁ being BinOp of (Submodules V) st
for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b₁ . (A1,A2) = W1 /\ W2

proof end;

uniqueness
for b₁, b₂ being BinOp of (Submodules V) st ( for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b₁ . (A1,A2) = W1 /\ W2 ) & ( for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b₂ . (A1,A2) = W1 /\ W2 ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def7 defines SubMeet RMOD_3:def 7 :
for R being Ring
for V being RightMod of R
for b₃ being BinOp of (Submodules V) holds
( b₃ = SubMeet V iff for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b₃ . (A1,A2) = W1 /\ W2 );

theorem Th47: :: RMOD_3:47

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

proof end;

theorem Th48: :: RMOD_3:48

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

proof end;

theorem Th49: :: RMOD_3:49

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_3:50

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_3:51

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

proof end;