:: Linear Combinations in Real Linear Space
:: by Wojciech A. Trybulec
::
:: Received April 8, 1990
:: Copyright (c) 1990 Association of Mizar Users

begin

definition

let S be 1-sorted ;
let x be set ;
assume A1: x in S ;
func vector (S,x) -> Element of S equals :Def1: :: RLVECT_2:def 1
x;
coherence
x is Element of S by A1, STRUCT_0:def 5;

end;

:: deftheorem Def1 defines vector RLVECT_2:def 1 :
for S being 1-sorted
for x being set st x in S holds
vector (S,x) = x;

theorem :: RLVECT_2:1

canceled;

theorem :: RLVECT_2:2

canceled;

theorem :: RLVECT_2:3

for S being non empty 1-sorted
for v being Element of S holds vector (S,v) = v by Def1, RLVECT_1:3;

theorem Th4: :: RLVECT_2:4

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for F, G, H being FinSequence of the carrier of V st len F = len G & len F = len H & ( for k being Element of NAT st k in dom F holds
H . k = (F /. k) + (G /. k) ) holds
Sum H = (Sum F) + (Sum G)

proof end;

theorem :: RLVECT_2:5

for V being RealLinearSpace
for a being Real
for F, G being FinSequence of V st len F = len G & ( for k being Element of NAT st k in dom F holds
G . k = a * (F /. k) ) holds
Sum G = a * (Sum F)

proof end;

theorem Th6: :: RLVECT_2:6

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for F, G being FinSequence of the carrier of V st len F = len G & ( for k being Element of NAT st k in dom F holds
G . k = - (F /. k) ) holds
Sum G = - (Sum F)

proof end;

theorem :: RLVECT_2:7

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for F, G, H being FinSequence of the carrier of V st len F = len G & len F = len H & ( for k being Element of NAT st k in dom F holds
H . k = (F /. k) - (G /. k) ) holds
Sum H = (Sum F) - (Sum G)

proof end;

theorem Th8: :: RLVECT_2:8

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for F, G being FinSequence of the carrier of V
for f being Permutation of (dom F) st len F = len G & ( for i being Element of NAT st i in dom G holds
G . i = F . (f . i) ) holds
Sum F = Sum G

proof end;

theorem :: RLVECT_2:9

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for F, G being FinSequence of the carrier of V
for f being Permutation of (dom F) st G = F * f holds
Sum F = Sum G

proof end;

definition

let V be non empty addLoopStr ;
let T be finite Subset of V;
assume A1: ( V is Abelian & V is add-associative & V is right_zeroed ) ;
canceled;
canceled;
func Sum T -> Element of V means :Def4: :: RLVECT_2:def 4
ex F being FinSequence of the carrier of V st
( rng F = T & F is one-to-one & it = Sum F );
existence
ex b₁ being Element of V ex F being FinSequence of the carrier of V st
( rng F = T & F is one-to-one & b₁ = Sum F )

proof end;

uniqueness
for b₁, b₂ being Element of V st ex F being FinSequence of the carrier of V st
( rng F = T & F is one-to-one & b₁ = Sum F ) & ex F being FinSequence of the carrier of V st
( rng F = T & F is one-to-one & b₂ = Sum F ) holds
b₁ = b₂ by A1, RLVECT_1:59;

end;

:: deftheorem RLVECT_2:def 2 :
canceled;

:: deftheorem RLVECT_2:def 3 :
canceled;

:: deftheorem Def4 defines Sum RLVECT_2:def 4 :
for V being non empty addLoopStr
for T being finite Subset of V st V is Abelian & V is add-associative & V is right_zeroed holds
for b₃ being Element of V holds
( b₃ = Sum T iff ex F being FinSequence of the carrier of V st
( rng F = T & F is one-to-one & b₃ = Sum F ) );

theorem :: RLVECT_2:10

canceled;

theorem :: RLVECT_2:11

canceled;

theorem :: RLVECT_2:12

canceled;

theorem :: RLVECT_2:13

canceled;

theorem Th14: :: RLVECT_2:14

for V being non empty Abelian add-associative right_zeroed addLoopStr holds Sum ({} V) = 0. V

proof end;

theorem :: RLVECT_2:15

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v being Element of V holds Sum {v} = v

proof end;

theorem :: RLVECT_2:16

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v1, v2 being Element of V st v1 <> v2 holds
Sum {v1,v2} = v1 + v2

proof end;

theorem :: RLVECT_2:17

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v1, v2, v3 being Element of V st v1 <> v2 & v2 <> v3 & v1 <> v3 holds
Sum {v1,v2,v3} = (v1 + v2) + v3

proof end;

theorem Th18: :: RLVECT_2:18

for V being non empty Abelian add-associative right_zeroed addLoopStr
for S, T being finite Subset of V st T misses S holds
Sum (T \/ S) = (Sum T) + (Sum S)

proof end;

theorem Th19: :: RLVECT_2:19

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for S, T being finite Subset of V holds Sum (T \/ S) = ((Sum T) + (Sum S)) - (Sum (T /\ S))

proof end;

theorem :: RLVECT_2:20

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for S, T being finite Subset of V holds Sum (T /\ S) = ((Sum T) + (Sum S)) - (Sum (T \/ S))

proof end;

theorem Th21: :: RLVECT_2:21

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for S, T being finite Subset of V holds Sum (T \ S) = (Sum (T \/ S)) - (Sum S)

proof end;

theorem Th22: :: RLVECT_2:22

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for S, T being finite Subset of V holds Sum (T \ S) = (Sum T) - (Sum (T /\ S))

proof end;

theorem :: RLVECT_2:23

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for S, T being finite Subset of V holds Sum (T \+\ S) = (Sum (T \/ S)) - (Sum (T /\ S))

proof end;

theorem :: RLVECT_2:24

for V being non empty Abelian add-associative right_zeroed addLoopStr
for S, T being finite Subset of V holds Sum (T \+\ S) = (Sum (T \ S)) + (Sum (S \ T)) by Th18, XBOOLE_1:82;

definition

let V be non empty ZeroStr ;
mode Linear_Combination of V -> Element of Funcs ( the carrier of V,REAL) means :Def5: :: RLVECT_2:def 5
ex T being finite Subset of V st
for v being Element of V st not v in T holds
it . v = 0 ;
existence
ex b₁ being Element of Funcs ( the carrier of V,REAL) ex T being finite Subset of V st
for v being Element of V st not v in T holds
b₁ . v = 0

proof end;

end;

:: deftheorem Def5 defines Linear_Combination RLVECT_2:def 5 :
for V being non empty ZeroStr
for b₂ being Element of Funcs ( the carrier of V,REAL) holds
( b₂ is Linear_Combination of V iff ex T being finite Subset of V st
for v being Element of V st not v in T holds
b₂ . v = 0 );

notation

let V be non empty addLoopStr ;
let L be Element of Funcs ( the carrier of V,REAL);
synonym Carrier L for support V;

end;

Lm1: now

let V be non empty addLoopStr ; :: thesis: for L being Element of Funcs ( the carrier of V,REAL) holds Carrier c= the carrier of V
let L be Element of Funcs ( the carrier of V,REAL); :: thesis: Carrier c= the carrier of V
A1: support L c= dom L by PRE_POLY:37;
thus Carrier c= the carrier of V :: thesis: verum

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in Carrier or x in the carrier of V )
assume x in support L ; :: thesis: x in the carrier of V
then x in dom L by A1;
hence x in the carrier of V ; :: thesis: verum

end;

end;

definition

let V be non empty addLoopStr ;
let L be Element of Funcs ( the carrier of V,REAL);
:: original: Carrier
redefine func Carrier L -> Subset of V equals :: RLVECT_2:def 6
{ v where v is Element of V : L . v <> 0 } ;
coherence
Carrier is Subset of V by Lm1;
compatibility
for b₁ being Subset of V holds
( b₁ = Carrier iff b₁ = { v where v is Element of V : L . v <> 0 } )

proof end;

end;

:: deftheorem defines Carrier RLVECT_2:def 6 :
for V being non empty addLoopStr
for L being Element of Funcs ( the carrier of V,REAL) holds Carrier L = { v where v is Element of V : L . v <> 0 } ;

registration

let V be non empty addLoopStr ;
let L be Linear_Combination of V;
cluster Carrier -> finite ;
coherence
Carrier L is finite

proof end;

end;

theorem :: RLVECT_2:25

canceled;

theorem :: RLVECT_2:26

canceled;

theorem :: RLVECT_2:27

canceled;

theorem :: RLVECT_2:28

for V being non empty addLoopStr
for L being Linear_Combination of V
for v being Element of V holds
( L . v = 0 iff not v in Carrier L )

proof end;

definition

let V be non empty addLoopStr ;
func ZeroLC V -> Linear_Combination of V means :Def7: :: RLVECT_2:def 7
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 Def7 defines ZeroLC RLVECT_2:def 7 :
for V being non empty addLoopStr
for b₂ being Linear_Combination of V holds
( b₂ = ZeroLC V iff Carrier b₂ = {} );

theorem :: RLVECT_2:29

canceled;

theorem Th30: :: RLVECT_2:30

for V being non empty addLoopStr
for v being Element of V holds (ZeroLC V) . v = 0

proof end;

definition

let V be non empty addLoopStr ;
let A be Subset of V;
mode Linear_Combination of A -> Linear_Combination of V means :Def8: :: RLVECT_2:def 8
Carrier it c= A;
existence
ex b₁ being Linear_Combination of V st Carrier b₁ c= A

proof end;

end;

:: deftheorem Def8 defines Linear_Combination RLVECT_2:def 8 :
for V being non empty addLoopStr
for A being Subset of V
for b₃ being Linear_Combination of V holds
( b₃ is Linear_Combination of A iff Carrier b₃ c= A );

theorem :: RLVECT_2:31

canceled;

theorem :: RLVECT_2:32

canceled;

theorem :: RLVECT_2:33

for V being RealLinearSpace
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 Th34: :: RLVECT_2:34

for V being RealLinearSpace
for A being Subset of V holds ZeroLC V is Linear_Combination of A

proof end;

theorem Th35: :: RLVECT_2:35

for V being RealLinearSpace
for l being Linear_Combination of {} the carrier of V holds l = ZeroLC V

proof end;

definition

let V be RealLinearSpace;
let F be FinSequence of V;
let f be Function of the carrier of V,REAL;
func f (#) F -> FinSequence of the carrier of V means :Def9: :: RLVECT_2:def 9
( len it = len F & ( for i being Element of NAT st i in dom it holds
it . i = (f . (F /. i)) * (F /. i) ) );
existence
ex b₁ being FinSequence of the carrier of V st
( len b₁ = len F & ( for i being Element of NAT st i in dom b₁ holds
b₁ . i = (f . (F /. i)) * (F /. i) ) )

proof end;

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

proof end;

end;

:: deftheorem Def9 defines (#) RLVECT_2:def 9 :
for V being RealLinearSpace
for F being FinSequence of V
for f being Function of the carrier of V,REAL
for b₄ being FinSequence of the carrier of V holds
( b₄ = f (#) F iff ( len b₄ = len F & ( for i being Element of NAT st i in dom b₄ holds
b₄ . i = (f . (F /. i)) * (F /. i) ) ) );

theorem :: RLVECT_2:36

canceled;

theorem :: RLVECT_2:37

canceled;

theorem :: RLVECT_2:38

canceled;

theorem :: RLVECT_2:39

canceled;

theorem Th40: :: RLVECT_2:40

for i being Element of NAT
for V being RealLinearSpace
for v being VECTOR of V
for F being FinSequence of V
for f being Function of the carrier of V,REAL st i in dom F & v = F . i holds
(f (#) F) . i = (f . v) * v

proof end;

theorem :: RLVECT_2:41

for V being RealLinearSpace
for f being Function of the carrier of V,REAL holds f (#) (<*> the carrier of V) = <*> the carrier of V

proof end;

theorem Th42: :: RLVECT_2:42

for V being RealLinearSpace
for v being VECTOR of V
for f being Function of the carrier of V,REAL holds f (#) <*v*> = <*((f . v) * v)*>

proof end;

theorem Th43: :: RLVECT_2:43

for V being RealLinearSpace
for v1, v2 being VECTOR of V
for f being Function of the carrier of V,REAL holds f (#) <*v1,v2*> = <*((f . v1) * v1),((f . v2) * v2)*>

proof end;

theorem :: RLVECT_2:44

for V being RealLinearSpace
for v1, v2, v3 being VECTOR of V
for f being Function of the carrier of V,REAL holds f (#) <*v1,v2,v3*> = <*((f . v1) * v1),((f . v2) * v2),((f . v3) * v3)*>

proof end;

definition

let V be RealLinearSpace;
let L be Linear_Combination of V;
func Sum L -> Element of V means :Def10: :: RLVECT_2:def 10
ex F being FinSequence of V st
( F is one-to-one & rng F = Carrier L & it = Sum (L (#) F) );
existence
ex b₁ being Element 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 Element 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 Def10 defines Sum RLVECT_2:def 10 :
for V being RealLinearSpace
for L being Linear_Combination of V
for b₃ being Element of V holds
( b₃ = Sum L iff ex F being FinSequence of V st
( F is one-to-one & rng F = Carrier L & b₃ = Sum (L (#) F) ) );

Lm2: for V being RealLinearSpace holds Sum (ZeroLC V) = 0. V

proof end;

theorem :: RLVECT_2:45

canceled;

theorem :: RLVECT_2:46

canceled;

theorem :: RLVECT_2:47

for V being RealLinearSpace
for A being Subset of V holds
( ( A <> {} & A is linearly-closed ) iff for l being Linear_Combination of A holds Sum l in A )

proof end;

theorem :: RLVECT_2:48

for V being RealLinearSpace holds Sum (ZeroLC V) = 0. V by Lm2;

theorem :: RLVECT_2:49

for V being RealLinearSpace
for l being Linear_Combination of {} the carrier of V holds Sum l = 0. V

proof end;

theorem Th50: :: RLVECT_2:50

for V being RealLinearSpace
for v being VECTOR of V
for l being Linear_Combination of {v} holds Sum l = (l . v) * v

proof end;

theorem Th51: :: RLVECT_2:51

for V being RealLinearSpace
for v1, v2 being VECTOR of V st v1 <> v2 holds
for l being Linear_Combination of {v1,v2} holds Sum l = ((l . v1) * v1) + ((l . v2) * v2)

proof end;

theorem :: RLVECT_2:52

for V being RealLinearSpace
for L being Linear_Combination of V st Carrier L = {} holds
Sum L = 0. V

proof end;

theorem :: RLVECT_2:53

for V being RealLinearSpace
for v being VECTOR of V
for L being Linear_Combination of V st Carrier L = {v} holds
Sum L = (L . v) * v

proof end;

theorem :: RLVECT_2:54

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

proof end;

definition

let V be non empty addLoopStr ;
let L1, L2 be Linear_Combination of V;
:: original: =
redefine pred L1 = L2 means :: RLVECT_2:def 11
for v being Element of V holds L1 . v = L2 . v;
compatibility
( L1 = L2 iff for v being Element of V holds L1 . v = L2 . v ) by FUNCT_2:113;

end;

:: deftheorem defines = RLVECT_2:def 11 :
for V being non empty addLoopStr
for L1, L2 being Linear_Combination of V holds
( L1 = L2 iff for v being Element of V holds L1 . v = L2 . v );

definition

let V be non empty addLoopStr ;
let L1, L2 be Linear_Combination of V;
:: original: +
redefine func L1 + L2 -> Linear_Combination of V means :Def12: :: RLVECT_2:def 12
for v being Element of V holds it . v = (L1 . v) + (L2 . v);
coherence
L1 + L2 is Linear_Combination of V

proof end;

compatibility
for b₁ being Linear_Combination of V holds
( b₁ = L1 + L2 iff for v being Element of V holds b₁ . v = (L1 . v) + (L2 . v) )

proof end;

end;

:: deftheorem Def12 defines + RLVECT_2:def 12 :
for V being non empty addLoopStr
for L1, L2, b₄ being Linear_Combination of V holds
( b₄ = L1 + L2 iff for v being Element of V holds b₄ . v = (L1 . v) + (L2 . v) );

theorem :: RLVECT_2:55

canceled;

theorem :: RLVECT_2:56

canceled;

theorem :: RLVECT_2:57

canceled;

theorem Th58: :: RLVECT_2:58

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

proof end;

theorem Th59: :: RLVECT_2:59

for V being RealLinearSpace
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 :: RLVECT_2:60

for V being non empty addLoopStr
for L1, L2 being Linear_Combination of V holds L1 + L2 = L2 + L1 ;

theorem Th61: :: RLVECT_2:61

for V being RealLinearSpace
for L1, L2, L3 being Linear_Combination of V holds L1 + (L2 + L3) = (L1 + L2) + L3

proof end;

theorem Th62: :: RLVECT_2:62

for V being RealLinearSpace
for L being Linear_Combination of V holds
( L + (ZeroLC V) = L & (ZeroLC V) + L = L )

proof end;

definition

let V be RealLinearSpace;
let a be Real;
let L be Linear_Combination of V;
func a * L -> Linear_Combination of V means :Def13: :: RLVECT_2:def 13
for v being VECTOR of V holds it . v = a * (L . v);
existence
ex b₁ being Linear_Combination of V st
for v being VECTOR of V holds b₁ . v = a * (L . v)

proof end;

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

proof end;

end;

:: deftheorem Def13 defines * RLVECT_2:def 13 :
for V being RealLinearSpace
for a being Real
for L, b₄ being Linear_Combination of V holds
( b₄ = a * L iff for v being VECTOR of V holds b₄ . v = a * (L . v) );

theorem :: RLVECT_2:63

canceled;

theorem :: RLVECT_2:64

canceled;

theorem Th65: :: RLVECT_2:65

for V being RealLinearSpace
for a being Real
for L being Linear_Combination of V st a <> 0 holds
Carrier (a * L) = Carrier L

proof end;

theorem Th66: :: RLVECT_2:66

for V being RealLinearSpace
for L being Linear_Combination of V holds 0 * L = ZeroLC V

proof end;

theorem Th67: :: RLVECT_2:67

for V being RealLinearSpace
for a being Real
for A being Subset of V
for L being Linear_Combination of V st L is Linear_Combination of A holds
a * L is Linear_Combination of A

proof end;

theorem Th68: :: RLVECT_2:68

for V being RealLinearSpace
for a, b being Real
for L being Linear_Combination of V holds (a + b) * L = (a * L) + (b * L)

proof end;

theorem Th69: :: RLVECT_2:69

for V being RealLinearSpace
for a being Real
for L1, L2 being Linear_Combination of V holds a * (L1 + L2) = (a * L1) + (a * L2)

proof end;

theorem Th70: :: RLVECT_2:70

for V being RealLinearSpace
for a, b being Real
for L being Linear_Combination of V holds a * (b * L) = (a * b) * L

proof end;

theorem Th71: :: RLVECT_2:71

for V being RealLinearSpace
for L being Linear_Combination of V holds 1 * L = L

proof end;

definition

let V be RealLinearSpace;
let L be Linear_Combination of V;
func - L -> Linear_Combination of V equals :: RLVECT_2:def 14
(- 1) * L;
correctness
coherence
(- 1) * L is Linear_Combination of V;
;

end;

:: deftheorem defines - RLVECT_2:def 14 :
for V being RealLinearSpace
for L being Linear_Combination of V holds - L = (- 1) * L;

theorem :: RLVECT_2:72

canceled;

theorem Th73: :: RLVECT_2:73

for V being RealLinearSpace
for v being VECTOR of V
for L being Linear_Combination of V holds (- L) . v = - (L . v)

proof end;

theorem :: RLVECT_2:74

for V being RealLinearSpace
for L1, L2 being Linear_Combination of V st L1 + L2 = ZeroLC V holds
L2 = - L1

proof end;

theorem :: RLVECT_2:75

for V being RealLinearSpace
for L being Linear_Combination of V holds Carrier (- L) = Carrier L by Th65;

theorem :: RLVECT_2:76

for V being RealLinearSpace
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 Th67;

theorem :: RLVECT_2:77

for V being RealLinearSpace
for L being Linear_Combination of V holds - (- L) = L

proof end;

definition

let V be RealLinearSpace;
let L1, L2 be Linear_Combination of V;
func L1 - L2 -> Linear_Combination of V equals :: RLVECT_2:def 15
L1 + (- L2);
correctness
coherence
L1 + (- L2) is Linear_Combination of V;
;

end;

:: deftheorem defines - RLVECT_2:def 15 :
for V being RealLinearSpace
for L1, L2 being Linear_Combination of V holds L1 - L2 = L1 + (- L2);

theorem :: RLVECT_2:78

canceled;

theorem Th79: :: RLVECT_2:79

for V being RealLinearSpace
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 :: RLVECT_2:80

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

proof end;

theorem :: RLVECT_2:81

for V being RealLinearSpace
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 Th82: :: RLVECT_2:82

for V being RealLinearSpace
for L being Linear_Combination of V holds L - L = ZeroLC V

proof end;

definition

let V be RealLinearSpace;
func LinComb V -> set means :Def16: :: RLVECT_2:def 16
for x being set holds
( x in it iff x is Linear_Combination of V );
existence
ex b₁ being set st
for x being set holds
( x in b₁ iff x is Linear_Combination of V )

proof end;

uniqueness
for b₁, b₂ being set st ( for x being set holds
( x in b₁ iff x is Linear_Combination of V ) ) & ( for x being set holds
( x in b₂ iff x is Linear_Combination of V ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def16 defines LinComb RLVECT_2:def 16 :
for V being RealLinearSpace
for b₂ being set holds
( b₂ = LinComb V iff for x being set holds
( x in b₂ iff x is Linear_Combination of V ) );

registration

let V be RealLinearSpace;
cluster LinComb V -> non empty ;
coherence
not LinComb V is empty

proof end;

end;

definition

let V be RealLinearSpace;
let e be Element of LinComb V;
func @ e -> Linear_Combination of V equals :: RLVECT_2:def 17
e;
coherence
e is Linear_Combination of V by Def16;

end;

:: deftheorem defines @ RLVECT_2:def 17 :
for V being RealLinearSpace
for e being Element of LinComb V holds @ e = e;

definition

let V be RealLinearSpace;
let L be Linear_Combination of V;
func @ L -> Element of LinComb V equals :: RLVECT_2:def 18
L;
coherence
L is Element of LinComb V by Def16;

end;

:: deftheorem defines @ RLVECT_2:def 18 :
for V being RealLinearSpace
for L being Linear_Combination of V holds @ L = L;

definition

let V be RealLinearSpace;
func LCAdd V -> BinOp of (LinComb V) means :Def19: :: RLVECT_2:def 19
for e1, e2 being Element of LinComb V holds it . (e1,e2) = (@ e1) + (@ e2);
existence
ex b₁ being BinOp of (LinComb V) st
for e1, e2 being Element of LinComb V holds b₁ . (e1,e2) = (@ e1) + (@ e2)

proof end;

uniqueness
for b₁, b₂ being BinOp of (LinComb V) st ( for e1, e2 being Element of LinComb V holds b₁ . (e1,e2) = (@ e1) + (@ e2) ) & ( for e1, e2 being Element of LinComb V holds b₂ . (e1,e2) = (@ e1) + (@ e2) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def19 defines LCAdd RLVECT_2:def 19 :
for V being RealLinearSpace
for b₂ being BinOp of (LinComb V) holds
( b₂ = LCAdd V iff for e1, e2 being Element of LinComb V holds b₂ . (e1,e2) = (@ e1) + (@ e2) );

definition

let V be RealLinearSpace;
func LCMult V -> Function of [:REAL,(LinComb V):],(LinComb V) means :Def20: :: RLVECT_2:def 20
for a being Real
for e being Element of LinComb V holds it . [a,e] = a * (@ e);
existence
ex b₁ being Function of [:REAL,(LinComb V):],(LinComb V) st
for a being Real
for e being Element of LinComb V holds b₁ . [a,e] = a * (@ e)

proof end;

uniqueness
for b₁, b₂ being Function of [:REAL,(LinComb V):],(LinComb V) st ( for a being Real
for e being Element of LinComb V holds b₁ . [a,e] = a * (@ e) ) & ( for a being Real
for e being Element of LinComb V holds b₂ . [a,e] = a * (@ e) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def20 defines LCMult RLVECT_2:def 20 :
for V being RealLinearSpace
for b₂ being Function of [:REAL,(LinComb V):],(LinComb V) holds
( b₂ = LCMult V iff for a being Real
for e being Element of LinComb V holds b₂ . [a,e] = a * (@ e) );

definition

let V be RealLinearSpace;
func LC_RLSpace V -> RLSStruct equals :: RLVECT_2:def 21
RLSStruct(# (LinComb V),(@ (ZeroLC V)),(LCAdd V),(LCMult V) #);
coherence
RLSStruct(# (LinComb V),(@ (ZeroLC V)),(LCAdd V),(LCMult V) #) is RLSStruct ;

end;

:: deftheorem defines LC_RLSpace RLVECT_2:def 21 :
for V being RealLinearSpace holds LC_RLSpace V = RLSStruct(# (LinComb V),(@ (ZeroLC V)),(LCAdd V),(LCMult V) #);

registration

let V be RealLinearSpace;
cluster LC_RLSpace V -> non empty strict ;
coherence
( LC_RLSpace V is strict & not LC_RLSpace V is empty ) ;

end;

registration

let V be RealLinearSpace;
cluster LC_RLSpace V -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;
coherence
( LC_RLSpace V is Abelian & LC_RLSpace V is add-associative & LC_RLSpace V is right_zeroed & LC_RLSpace V is right_complementable & LC_RLSpace V is vector-distributive & LC_RLSpace V is scalar-distributive & LC_RLSpace V is scalar-associative & LC_RLSpace V is scalar-unital )

proof end;

end;

theorem :: RLVECT_2:83

canceled;

theorem :: RLVECT_2:84

canceled;

theorem :: RLVECT_2:85

canceled;

theorem :: RLVECT_2:86

canceled;

theorem :: RLVECT_2:87

canceled;

theorem :: RLVECT_2:88

canceled;

theorem :: RLVECT_2:89

canceled;

theorem :: RLVECT_2:90

canceled;

theorem :: RLVECT_2:91

canceled;

theorem :: RLVECT_2:92

for V being RealLinearSpace holds the carrier of (LC_RLSpace V) = LinComb V ;

theorem :: RLVECT_2:93

for V being RealLinearSpace holds 0. (LC_RLSpace V) = ZeroLC V ;

theorem :: RLVECT_2:94

for V being RealLinearSpace holds the addF of (LC_RLSpace V) = LCAdd V ;

theorem :: RLVECT_2:95

for V being RealLinearSpace holds the Mult of (LC_RLSpace V) = LCMult V ;

theorem Th96: :: RLVECT_2:96

for V being RealLinearSpace
for L1, L2 being Linear_Combination of V holds (vector ((LC_RLSpace V),L1)) + (vector ((LC_RLSpace V),L2)) = L1 + L2

proof end;

theorem Th97: :: RLVECT_2:97

for V being RealLinearSpace
for a being Real
for L being Linear_Combination of V holds a * (vector ((LC_RLSpace V),L)) = a * L

proof end;

theorem Th98: :: RLVECT_2:98

for V being RealLinearSpace
for L being Linear_Combination of V holds - (vector ((LC_RLSpace V),L)) = - L

proof end;

theorem :: RLVECT_2:99

for V being RealLinearSpace
for L1, L2 being Linear_Combination of V holds (vector ((LC_RLSpace V),L1)) - (vector ((LC_RLSpace V),L2)) = L1 - L2

proof end;

definition

let V be RealLinearSpace;
let A be Subset of V;
func LC_RLSpace A -> strict Subspace of LC_RLSpace V means :: RLVECT_2:def 22
the carrier of it = { l where l is Linear_Combination of A : verum } ;
existence
ex b₁ being strict Subspace of LC_RLSpace V st the carrier of b₁ = { l where l is Linear_Combination of A : verum }

proof end;

uniqueness
for b₁, b₂ being strict Subspace of LC_RLSpace V st the carrier of b₁ = { l where l is Linear_Combination of A : verum } & the carrier of b₂ = { l where l is Linear_Combination of A : verum } holds
b₁ = b₂ by RLSUB_1:38;

end;

:: deftheorem defines LC_RLSpace RLVECT_2:def 22 :
for V being RealLinearSpace
for A being Subset of V
for b₃ being strict Subspace of LC_RLSpace V holds
( b₃ = LC_RLSpace A iff the carrier of b₃ = { l where l is Linear_Combination of A : verum } );

theorem Th100: :: RLVECT_2:100

for R being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for a being Element of R
for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr of R
for F, G being FinSequence of V st len F = len G & ( for k being Element of NAT
for v being Element of V st k in dom F & v = G . k holds
F . k = a * v ) holds
Sum F = a * (Sum G)

proof end;

theorem :: RLVECT_2:101

for R being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for a being Element of R
for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr of R
for F, G being FinSequence of V st len F = len G & ( for k being Element of NAT st k in dom F holds
G . k = a * (F /. k) ) holds
Sum G = a * (Sum F)

proof end;

theorem :: RLVECT_2:102

for R being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for V being non empty right_complementable Abelian add-associative right_zeroed VectSpStr of R
for F, G, H being FinSequence of V st len F = len G & len F = len H & ( for k being Element of NAT st k in dom F holds
H . k = (F /. k) - (G /. k) ) holds
Sum H = (Sum F) - (Sum G)

proof end;

theorem :: RLVECT_2:103

for R being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for a being Element of R
for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr of R holds a * (Sum (<*> the carrier of V)) = 0. V

proof end;

theorem :: RLVECT_2:104

for R being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for a being Element of R
for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr of R
for v, u being Element of V holds a * (Sum <*v,u*>) = (a * v) + (a * u)

proof end;

theorem :: RLVECT_2:105

for R being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for a being Element of R
for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr of R
for v, u, w being Element of V holds a * (Sum <*v,u,w*>) = ((a * v) + (a * u)) + (a * w)

proof end;