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

Lm1: for R being Ring
for V being RightMod of R
for v being Vector of V
for F, G being FinSequence of V st len F = (len G) + 1 & G = F | (Seg (len G)) & v = F . (len F) holds
Sum F = (Sum G) + v

proof end;

theorem Th1: :: RMOD_4:1

for R being Ring
for V being RightMod of R
for a being Scalar of R
for F, G being FinSequence of V st len F = len G & ( for k being Nat
for v being Vector of V st k in dom F & v = G . k holds
F . k = v * a ) holds
Sum F = (Sum G) * a

proof end;

Lm2: for R being Ring
for V being RightMod of R
for a being Scalar of R
for F, G being FinSequence of V st len F = len G & ( for k being Nat st k in dom F holds
G . k = (F /. k) * a ) holds
Sum G = (Sum F) * a

proof end;

theorem :: RMOD_4:2

for R being Ring
for V being RightMod of R
for a being Scalar of R holds (Sum (<*> the carrier of V)) * a = 0. V

proof end;

theorem :: RMOD_4:3

for R being Ring
for V being RightMod of R
for a being Scalar of R
for u, v being Vector of V holds (Sum <*v,u*>) * a = (v * a) + (u * a)

proof end;

theorem :: RMOD_4:4

for R being Ring
for V being RightMod of R
for a being Scalar of R
for u, v, w being Vector of V holds (Sum <*v,u,w*>) * a = ((v * a) + (u * a)) + (w * a)

proof end;

definition

let R be Ring;
let V be RightMod of R;
let T be finite Subset of V;

func Sum T -> Vector of V means :Def1: :: RMOD_4:def 1
ex F being FinSequence of V st
( rng F = T & F is one-to-one & it = Sum F );

existence
ex b₁ being Vector of V ex F being FinSequence of V st
( rng F = T & F is one-to-one & b₁ = Sum F )

proof end;

uniqueness
for b₁, b₂ being Vector of V st ex F being FinSequence of V st
( rng F = T & F is one-to-one & b₁ = Sum F ) & ex F being FinSequence of V st
( rng F = T & F is one-to-one & b₂ = Sum F ) holds
b₁ = b₂ by RLVECT_1:42;

end;

:: deftheorem Def1 defines Sum RMOD_4:def 1 :
for R being Ring
for V being RightMod of R
for T being finite Subset of V
for b₄ being Vector of V holds
( b₄ = Sum T iff ex F being FinSequence of V st
( rng F = T & F is one-to-one & b₄ = Sum F ) );

theorem Th5: :: RMOD_4:5

for R being Ring
for V being RightMod of R holds Sum ({} V) = 0. V

proof end;

theorem :: RMOD_4:6

for R being Ring
for V being RightMod of R
for v being Vector of V holds Sum {v} = v

proof end;

theorem :: RMOD_4:7

for R being Ring
for V being RightMod of R
for v1, v2 being Vector of V st v1 <> v2 holds
Sum {v1,v2} = v1 + v2

proof end;

theorem :: RMOD_4:8

for R being Ring
for V being RightMod of R
for v1, v2, v3 being Vector of V st v1 <> v2 & v2 <> v3 & v1 <> v3 holds
Sum {v1,v2,v3} = (v1 + v2) + v3

proof end;

theorem Th9: :: RMOD_4:9

for R being Ring
for V being RightMod of R
for S, T being finite Subset of V st T misses S holds
Sum (T \/ S) = (Sum T) + (Sum S)

proof end;

theorem Th10: :: RMOD_4:10

for R being Ring
for V being RightMod of R
for S, T being finite Subset of V holds Sum (T \/ S) = ((Sum T) + (Sum S)) - (Sum (T /\ S))

proof end;

theorem :: RMOD_4:11

for R being Ring
for V being RightMod of R
for S, T being finite Subset of V holds Sum (T /\ S) = ((Sum T) + (Sum S)) - (Sum (T \/ S))

proof end;

theorem Th12: :: RMOD_4:12

for R being Ring
for V being RightMod of R
for S, T being finite Subset of V holds Sum (T \ S) = (Sum (T \/ S)) - (Sum S)

proof end;

theorem Th13: :: RMOD_4:13

for R being Ring
for V being RightMod of R
for S, T being finite Subset of V holds Sum (T \ S) = (Sum T) - (Sum (T /\ S))

proof end;

theorem :: RMOD_4:14

for R being Ring
for V being RightMod of R
for S, T being finite Subset of V holds Sum (T \+\ S) = (Sum (T \/ S)) - (Sum (T /\ S))

proof end;

theorem :: RMOD_4:15

for R being Ring
for V being RightMod of R
for S, T being finite Subset of V holds Sum (T \+\ S) = (Sum (T \ S)) + (Sum (S \ T)) by Th9, XBOOLE_1:82;

definition

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

mode Linear_Combination of V -> Element of Funcs ( the carrier of V, the carrier of R) means :Def2: :: RMOD_4:def 2
ex T being finite Subset of V st
for v being Vector of V st not v in T holds
it . v = 0. R;

existence
ex b₁ being Element of Funcs ( the carrier of V, the carrier of R) ex T being finite Subset of V st
for v being Vector of V st not v in T holds
b₁ . v = 0. R

proof end;

end;

:: deftheorem Def2 defines Linear_Combination RMOD_4:def 2 :
for R being Ring
for V being RightMod of R
for b₃ being Element of Funcs ( the carrier of V, the carrier of R) holds
( b₃ is Linear_Combination of V iff ex T being finite Subset of V st
for v being Vector of V st not v in T holds
b₃ . v = 0. R );

definition

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

func Carrier L -> finite Subset of V equals :: RMOD_4:def 3
{ v where v is Vector of V : L . v <> 0. R } ;

coherence
{ v where v is Vector of V : L . v <> 0. R } is finite Subset of V

proof end;

end;

:: deftheorem defines Carrier RMOD_4:def 3 :
for R being Ring
for V being RightMod of R
for L being Linear_Combination of V holds Carrier L = { v where v is Vector of V : L . v <> 0. R } ;

theorem :: RMOD_4:16

for R being Ring
for V being RightMod of R
for x being set
for L being Linear_Combination of V holds
( x in Carrier L iff ex v being Vector of V st
( x = v & L . v <> 0. R ) ) ;

theorem :: RMOD_4:17

for R being Ring
for V being RightMod of R
for v being Vector of V
for L being Linear_Combination of V holds
( L . v = 0. R iff not v in Carrier L )

proof end;

definition

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

func ZeroLC V -> Linear_Combination of V means :Def4: :: RMOD_4:def 4
Carrier it = {} ;

existence
ex b₁ being Linear_Combination of V st Carrier b₁ = {}

proof end;

uniqueness
for b₁, b₂ being Linear_Combination of V st Carrier b₁ = {} & Carrier b₂ = {} holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines ZeroLC RMOD_4:def 4 :
for R being Ring
for V being RightMod of R
for b₃ being Linear_Combination of V holds
( b₃ = ZeroLC V iff Carrier b₃ = {} );

theorem Th18: :: RMOD_4:18

for R being Ring
for V being RightMod of R
for v being Vector of V holds (ZeroLC V) . v = 0. R

proof end;

definition

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

mode Linear_Combination of A -> Linear_Combination of V means :Def5: :: RMOD_4:def 5
Carrier it c= A;

existence
ex b₁ being Linear_Combination of V st Carrier b₁ c= A

proof end;

end;

:: deftheorem Def5 defines Linear_Combination RMOD_4:def 5 :
for R being Ring
for V being RightMod of R
for A being Subset of V
for b₄ being Linear_Combination of V holds
( b₄ is Linear_Combination of A iff Carrier b₄ c= A );

theorem Th19: :: RMOD_4:19

for R being Ring
for V being RightMod of R
for A, B being Subset of V
for l being Linear_Combination of A st A c= B holds
l is Linear_Combination of B

proof end;

theorem Th20: :: RMOD_4:20

for R being Ring
for V being RightMod of R
for A being Subset of V holds ZeroLC V is Linear_Combination of A

proof end;

theorem Th21: :: RMOD_4:21

for R being Ring
for V being RightMod of R
for l being Linear_Combination of {} the carrier of V holds l = ZeroLC V

proof end;

theorem :: RMOD_4:22

for R being Ring
for V being RightMod of R
for L being Linear_Combination of V holds L is Linear_Combination of Carrier L by Def5;

definition

let R be Ring;
let V be RightMod of R;
let F be FinSequence of V;
let f be Function of V,R;

func f (#) F -> FinSequence of V means :Def6: :: RMOD_4:def 6
( len it = len F & ( for i being Nat st i in dom it holds
it . i = (F /. i) * (f . (F /. i)) ) );

existence
ex b₁ being FinSequence of V st
( len b₁ = len F & ( for i being Nat st i in dom b₁ holds
b₁ . i = (F /. i) * (f . (F /. i)) ) )

proof end;

uniqueness
for b₁, b₂ being FinSequence of V st len b₁ = len F & ( for i being Nat st i in dom b₁ holds
b₁ . i = (F /. i) * (f . (F /. i)) ) & len b₂ = len F & ( for i being Nat st i in dom b₂ holds
b₂ . i = (F /. i) * (f . (F /. i)) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines (#) RMOD_4:def 6 :
for R being Ring
for V being RightMod of R
for F being FinSequence of V
for f being Function of V,R
for b₅ being FinSequence of V holds
( b₅ = f (#) F iff ( len b₅ = len F & ( for i being Nat st i in dom b₅ holds
b₅ . i = (F /. i) * (f . (F /. i)) ) ) );

theorem Th23: :: RMOD_4:23

for R being Ring
for V being RightMod of R
for i being Nat
for v being Vector of V
for F being FinSequence of V
for f being Function of V,R st i in dom F & v = F . i holds
(f (#) F) . i = v * (f . v)

proof end;

theorem :: RMOD_4:24

for R being Ring
for V being RightMod of R
for f being Function of V,R holds f (#) (<*> the carrier of V) = <*> the carrier of V

proof end;

theorem Th25: :: RMOD_4:25

for R being Ring
for V being RightMod of R
for v being Vector of V
for f being Function of V,R holds f (#) <*v*> = <*(v * (f . v))*>

proof end;

theorem Th26: :: RMOD_4:26

for R being Ring
for V being RightMod of R
for v1, v2 being Vector of V
for f being Function of V,R holds f (#) <*v1,v2*> = <*(v1 * (f . v1)),(v2 * (f . v2))*>

proof end;

theorem :: RMOD_4:27

for R being Ring
for V being RightMod of R
for v1, v2, v3 being Vector of V
for f being Function of V,R holds f (#) <*v1,v2,v3*> = <*(v1 * (f . v1)),(v2 * (f . v2)),(v3 * (f . v3))*>

proof end;

theorem Th28: :: RMOD_4:28

for R being Ring
for V being RightMod of R
for F, G being FinSequence of V
for f being Function of V,R holds f (#) (F ^ G) = (f (#) F) ^ (f (#) G)

proof end;

definition

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

func Sum L -> Vector of V means :Def7: :: RMOD_4:def 7
ex F being FinSequence of V st
( F is one-to-one & rng F = Carrier L & it = Sum (L (#) F) );

existence
ex b₁ being Vector of V ex F being FinSequence of V st
( F is one-to-one & rng F = Carrier L & b₁ = Sum (L (#) F) )

proof end;

uniqueness
for b₁, b₂ being Vector of V st ex F being FinSequence of V st
( F is one-to-one & rng F = Carrier L & b₁ = Sum (L (#) F) ) & ex F being FinSequence of V st
( F is one-to-one & rng F = Carrier L & b₂ = Sum (L (#) F) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def7 defines Sum RMOD_4:def 7 :
for R being Ring
for V being RightMod of R
for L being Linear_Combination of V
for b₄ being Vector of V holds
( b₄ = Sum L iff ex F being FinSequence of V st
( F is one-to-one & rng F = Carrier L & b₄ = Sum (L (#) F) ) );

Lm3: for R being Ring
for V being RightMod of R holds Sum (ZeroLC V) = 0. V

proof end;

theorem Th29: :: RMOD_4:29

for R being Ring
for V being RightMod of R
for A being Subset of V st 0. R <> 1_ R holds
( ( A <> {} & A is linearly-closed ) iff for l being Linear_Combination of A holds Sum l in A )

proof end;

theorem :: RMOD_4:30

for R being Ring
for V being RightMod of R holds Sum (ZeroLC V) = 0. V by Lm3;

theorem Th31: :: RMOD_4:31

for R being Ring
for V being RightMod of R
for l being Linear_Combination of {} the carrier of V holds Sum l = 0. V

proof end;

theorem Th32: :: RMOD_4:32

for R being Ring
for V being RightMod of R
for v being Vector of V
for l being Linear_Combination of {v} holds Sum l = v * (l . v)

proof end;

theorem Th33: :: RMOD_4:33

for R being Ring
for V being RightMod of R
for v1, v2 being Vector of V st v1 <> v2 holds
for l being Linear_Combination of {v1,v2} holds Sum l = (v1 * (l . v1)) + (v2 * (l . v2))

proof end;

theorem :: RMOD_4:34

for R being Ring
for V being RightMod of R
for L being Linear_Combination of V st Carrier L = {} holds
Sum L = 0. V

proof end;

theorem Th35: :: RMOD_4:35

for R being Ring
for V being RightMod of R
for v being Vector of V
for L being Linear_Combination of V st Carrier L = {v} holds
Sum L = v * (L . v)

proof end;

theorem :: RMOD_4:36

for R being Ring
for V being RightMod of R
for v1, v2 being Vector of V
for L being Linear_Combination of V st Carrier L = {v1,v2} & v1 <> v2 holds
Sum L = (v1 * (L . v1)) + (v2 * (L . v2))

proof end;

definition

let R be Ring;
let V be RightMod of R;
let L1, L2 be Linear_Combination of V;

:: original: =
redefine pred L1 = L2 means :: RMOD_4:def 8
for v being Vector of V holds L1 . v = L2 . v;

compatibility
( L1 = L2 iff for v being Vector of V holds L1 . v = L2 . v ) by FUNCT_2:63;

end;

:: deftheorem defines = RMOD_4:def 8 :
for R being Ring
for V being RightMod of R
for L1, L2 being Linear_Combination of V holds
( L1 = L2 iff for v being Vector of V holds L1 . v = L2 . v );

definition

let R be Ring;
let V be RightMod of R;
let L1, L2 be Linear_Combination of V;

func L1 + L2 -> Linear_Combination of V means :Def9: :: RMOD_4:def 9
for v being Vector of V holds it . v = (L1 . v) + (L2 . v);

existence
ex b₁ being Linear_Combination of V st
for v being Vector of V holds b₁ . v = (L1 . v) + (L2 . v)

proof end;

uniqueness
for b₁, b₂ being Linear_Combination of V st ( for v being Vector of V holds b₁ . v = (L1 . v) + (L2 . v) ) & ( for v being Vector of V holds b₂ . v = (L1 . v) + (L2 . v) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def9 defines + RMOD_4:def 9 :
for R being Ring
for V being RightMod of R
for L1, L2, b₅ being Linear_Combination of V holds
( b₅ = L1 + L2 iff for v being Vector of V holds b₅ . v = (L1 . v) + (L2 . v) );

theorem Th37: :: RMOD_4:37

for R being Ring
for V being RightMod of R
for L1, L2 being Linear_Combination of V holds Carrier (L1 + L2) c= (Carrier L1) \/ (Carrier L2)

proof end;

theorem Th38: :: RMOD_4:38

for R being Ring
for V being RightMod of R
for A being Subset of V
for L1, L2 being Linear_Combination of V st L1 is Linear_Combination of A & L2 is Linear_Combination of A holds
L1 + L2 is Linear_Combination of A

proof end;

theorem Th39: :: RMOD_4:39

for R being comRing
for V being RightMod of R
for L1, L2 being Linear_Combination of V holds L1 + L2 = L2 + L1

proof end;

theorem :: RMOD_4:40

for R being Ring
for V being RightMod of R
for L1, L2, L3 being Linear_Combination of V holds L1 + (L2 + L3) = (L1 + L2) + L3

proof end;

theorem :: RMOD_4:41

for R being comRing
for V being RightMod of R
for L being Linear_Combination of V holds
( L + (ZeroLC V) = L & (ZeroLC V) + L = L )

proof end;

definition

let R be Ring;
let V be RightMod of R;
let a be Scalar of R;
let L be Linear_Combination of V;

func L * a -> Linear_Combination of V means :Def10: :: RMOD_4:def 10
for v being Vector of V holds it . v = (L . v) * a;

existence
ex b₁ being Linear_Combination of V st
for v being Vector of V holds b₁ . v = (L . v) * a

proof end;

uniqueness
for b₁, b₂ being Linear_Combination of V st ( for v being Vector of V holds b₁ . v = (L . v) * a ) & ( for v being Vector of V holds b₂ . v = (L . v) * a ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def10 defines * RMOD_4:def 10 :
for R being Ring
for V being RightMod of R
for a being Scalar of R
for L, b₅ being Linear_Combination of V holds
( b₅ = L * a iff for v being Vector of V holds b₅ . v = (L . v) * a );

theorem Th42: :: RMOD_4:42

for R being Ring
for V being RightMod of R
for a being Scalar of R
for L being Linear_Combination of V holds Carrier (L * a) c= Carrier L

proof end;

theorem Th43: :: RMOD_4:43

for RR being domRing
for VV being RightMod of RR
for LL being Linear_Combination of VV
for aa being Scalar of RR st aa <> 0. RR holds
Carrier (LL * aa) = Carrier LL

proof end;

theorem Th44: :: RMOD_4:44

for R being Ring
for V being RightMod of R
for L being Linear_Combination of V holds L * (0. R) = ZeroLC V

proof end;

theorem Th45: :: RMOD_4:45

for R being Ring
for V being RightMod of R
for a being Scalar of R
for A being Subset of V
for L being Linear_Combination of V st L is Linear_Combination of A holds
L * a is Linear_Combination of A

proof end;

theorem :: RMOD_4:46

for R being Ring
for V being RightMod of R
for a, b being Scalar of R
for L being Linear_Combination of V holds L * (a + b) = (L * a) + (L * b)

proof end;

theorem :: RMOD_4:47

for R being Ring
for V being RightMod of R
for a being Scalar of R
for L1, L2 being Linear_Combination of V holds (L1 + L2) * a = (L1 * a) + (L2 * a)

proof end;

theorem Th48: :: RMOD_4:48

for R being Ring
for V being RightMod of R
for a, b being Scalar of R
for L being Linear_Combination of V holds (L * b) * a = L * (b * a)

proof end;

theorem :: RMOD_4:49

for R being Ring
for V being RightMod of R
for L being Linear_Combination of V holds L * (1_ R) = L

proof end;

definition

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

func - L -> Linear_Combination of V equals :: RMOD_4:def 11
L * (- (1_ R));

correctness
coherence
L * (- (1_ R)) is Linear_Combination of V;
;
involutiveness
for b₁, b₂ being Linear_Combination of V st b₁ = b₂ * (- (1_ R)) holds
b₂ = b₁ * (- (1_ R))

proof end;

end;

:: deftheorem defines - RMOD_4:def 11 :
for R being Ring
for V being RightMod of R
for L being Linear_Combination of V holds - L = L * (- (1_ R));

theorem Th50: :: RMOD_4:50

for R being Ring
for V being RightMod of R
for v being Vector of V
for L being Linear_Combination of V holds (- L) . v = - (L . v)

proof end;

theorem :: RMOD_4:51

for R being Ring
for V being RightMod of R
for L1, L2 being Linear_Combination of V st L1 + L2 = ZeroLC V holds
L2 = - L1

proof end;

theorem Th52: :: RMOD_4:52

for R being Ring
for V being RightMod of R
for L being Linear_Combination of V holds Carrier (- L) = Carrier L

proof end;

theorem :: RMOD_4:53

for R being Ring
for V being RightMod of R
for A being Subset of V
for L being Linear_Combination of V st L is Linear_Combination of A holds
- L is Linear_Combination of A by Th45;

definition

let R be Ring;
let V be RightMod of R;
let L1, L2 be Linear_Combination of V;

func L1 - L2 -> Linear_Combination of V equals :: RMOD_4:def 12
L1 + (- L2);

correctness
coherence
L1 + (- L2) is Linear_Combination of V;
;

end;

:: deftheorem defines - RMOD_4:def 12 :
for R being Ring
for V being RightMod of R
for L1, L2 being Linear_Combination of V holds L1 - L2 = L1 + (- L2);

theorem Th54: :: RMOD_4:54

for R being Ring
for V being RightMod of R
for v being Vector of V
for L1, L2 being Linear_Combination of V holds (L1 - L2) . v = (L1 . v) - (L2 . v)

proof end;

theorem :: RMOD_4:55

for R being Ring
for V being RightMod of R
for L1, L2 being Linear_Combination of V holds Carrier (L1 - L2) c= (Carrier L1) \/ (Carrier L2)

proof end;

theorem :: RMOD_4:56

for R being Ring
for V being RightMod of R
for A being Subset of V
for L1, L2 being Linear_Combination of V st L1 is Linear_Combination of A & L2 is Linear_Combination of A holds
L1 - L2 is Linear_Combination of A

proof end;

theorem :: RMOD_4:57

for R being Ring
for V being RightMod of R
for L being Linear_Combination of V holds L - L = ZeroLC V

proof end;

theorem Th58: :: RMOD_4:58

for R being Ring
for V being RightMod of R
for L1, L2 being Linear_Combination of V holds Sum (L1 + L2) = (Sum L1) + (Sum L2)

proof end;

theorem Th59: :: RMOD_4:59

for R being domRing
for V being RightMod of R
for L being Linear_Combination of V
for a being Scalar of R holds Sum (L * a) = (Sum L) * a

proof end;

theorem Th60: :: RMOD_4:60

for R being domRing
for V being RightMod of R
for L being Linear_Combination of V holds Sum (- L) = - (Sum L)

proof end;

theorem :: RMOD_4:61

for R being domRing
for V being RightMod of R
for L1, L2 being Linear_Combination of V holds Sum (L1 - L2) = (Sum L1) - (Sum L2)

proof end;

definition

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

attr IT is linearly-independent means :: RMOD_4:def 13
for l being Linear_Combination of IT st Sum l = 0. V holds
Carrier l = {} ;

end;

:: deftheorem defines linearly-independent RMOD_4:def 13 :
for R being Ring
for V being RightMod of R
for IT being Subset of V holds
( IT is linearly-independent iff for l being Linear_Combination of IT st Sum l = 0. V holds
Carrier l = {} );

notation

let R be Ring;
let V be RightMod of R;
let IT be Subset of V;
antonym linearly-dependent IT for linearly-independent ;

end;

theorem :: RMOD_4:62

for R being Ring
for V being RightMod of R
for A, B being Subset of V st A c= B & B is linearly-independent holds
A is linearly-independent

proof end;

theorem Th63: :: RMOD_4:63

for R being Ring
for V being RightMod of R
for A being Subset of V st 0. R <> 1_ R & A is linearly-independent holds
not 0. V in A

proof end;

theorem :: RMOD_4:64

for R being Ring
for V being RightMod of R holds {} the carrier of V is linearly-independent

proof end;

theorem Th65: :: RMOD_4:65

for R being Ring
for V being RightMod of R
for v1, v2 being Vector of V st 0. R <> 1_ R & {v1,v2} is linearly-independent holds
( v1 <> 0. V & v2 <> 0. V )

proof end;

theorem :: RMOD_4:66

for R being Ring
for V being RightMod of R
for v being Vector of V st 0. R <> 1_ R holds
( {v,(0. V)} is linearly-dependent & {(0. V),v} is linearly-dependent ) by Th65;

definition

let R be domRing;
let V be RightMod of R;
let A be Subset of V;

func Lin A -> strict Submodule of V means :Def14: :: RMOD_4:def 14
the carrier of it = { (Sum l) where l is Linear_Combination of A : verum } ;

existence
ex b₁ being strict Submodule of V st the carrier of b₁ = { (Sum l) where l is Linear_Combination of A : verum }

proof end;

uniqueness
for b₁, b₂ being strict Submodule of V st the carrier of b₁ = { (Sum l) where l is Linear_Combination of A : verum } & the carrier of b₂ = { (Sum l) where l is Linear_Combination of A : verum } holds
b₁ = b₂ by RMOD_2:29;

end;

:: deftheorem Def14 defines Lin RMOD_4:def 14 :
for R being domRing
for V being RightMod of R
for A being Subset of V
for b₄ being strict Submodule of V holds
( b₄ = Lin A iff the carrier of b₄ = { (Sum l) where l is Linear_Combination of A : verum } );

theorem Th67: :: RMOD_4:67

for x being set
for R being domRing
for V being RightMod of R
for A being Subset of V holds
( x in Lin A iff ex l being Linear_Combination of A st x = Sum l )

proof end;

theorem Th68: :: RMOD_4:68

for x being set
for R being domRing
for V being RightMod of R
for A being Subset of V st x in A holds
x in Lin A

proof end;

theorem :: RMOD_4:69

for R being domRing
for V being RightMod of R holds Lin ({} the carrier of V) = (0). V

proof end;

theorem :: RMOD_4:70

for R being domRing
for V being RightMod of R
for A being Subset of V holds
( not Lin A = (0). V or A = {} or A = {(0. V)} )

proof end;

theorem Th71: :: RMOD_4:71

for R being domRing
for V being RightMod of R
for A being Subset of V
for W being strict Submodule of V st 0. R <> 1_ R & A = the carrier of W holds
Lin A = W

proof end;

theorem :: RMOD_4:72

for R being domRing
for V being strict RightMod of R
for A being Subset of V st 0. R <> 1_ R & A = the carrier of V holds
Lin A = V

proof end;

theorem Th73: :: RMOD_4:73

for R being domRing
for V being RightMod of R
for A, B being Subset of V st A c= B holds
Lin A is Submodule of Lin B

proof end;

theorem :: RMOD_4:74

for R being domRing
for V being strict RightMod of R
for A, B being Subset of V st Lin A = V & A c= B holds
Lin B = V

proof end;

theorem :: RMOD_4:75

for R being domRing
for V being RightMod of R
for A, B being Subset of V holds Lin (A \/ B) = (Lin A) + (Lin B)

proof end;

theorem :: RMOD_4:76

for R being domRing
for V being RightMod of R
for A, B being Subset of V holds Lin (A /\ B) is Submodule of (Lin A) /\ (Lin B)

proof end;