:: Vectors in Real Linear Space
:: by Wojciech A. Trybulec
::
:: Received July 24, 1989
:: Copyright (c) 1990 Association of Mizar Users

begin

definition

attr c₁ is strict ;
struct RLSStruct -> addLoopStr ;
aggr RLSStruct(# carrier, ZeroF, addF, Mult #) -> RLSStruct ;
sel Mult c₁ -> Function of [:REAL, the carrier of c₁:], the carrier of c₁;

end;

registration

cluster non empty RLSStruct ;
existence
not for b₁ being RLSStruct holds b₁ is empty

proof end;

end;

definition

let V be RLSStruct ;
mode VECTOR of V is Element of V;

end;

theorem :: RLVECT_1:1

canceled;

theorem :: RLVECT_1:2

canceled;

theorem :: RLVECT_1:3

for V being non empty 1-sorted
for v being Element of V holds v in V by STRUCT_0:def 5;

definition

let V be non empty RLSStruct ;
let v be VECTOR of V;
let a be real number ;
canceled;
canceled;
canceled;
func a * v -> Element of V equals :: RLVECT_1:def 4
the Mult of V . (a,v);
coherence
the Mult of V . (a,v) is Element of V

proof end;

end;

:: deftheorem RLVECT_1:def 1 :
canceled;

:: deftheorem RLVECT_1:def 2 :
canceled;

:: deftheorem RLVECT_1:def 3 :
canceled;

:: deftheorem defines * RLVECT_1:def 4 :
for V being non empty RLSStruct
for v being VECTOR of V
for a being real number holds a * v = the Mult of V . (a,v);

theorem :: RLVECT_1:4

canceled;

theorem :: RLVECT_1:5

for V being non empty addLoopStr
for v, w being Element of V holds v + w = the addF of V . (v,w) ;

registration

let ZS be non empty set ;
let O be Element of ZS;
let F be BinOp of ZS;
let G be Function of [:REAL,ZS:],ZS;
cluster RLSStruct(# ZS,O,F,G #) -> non empty ;
coherence
not RLSStruct(# ZS,O,F,G #) is empty ;

end;

definition

let IT be addLoopStr ;
attr IT is Abelian means :Def5: :: RLVECT_1:def 5
for v, w being Element of IT holds v + w = w + v;
attr IT is add-associative means :Def6: :: RLVECT_1:def 6
for u, v, w being Element of IT holds (u + v) + w = u + (v + w);
attr IT is right_zeroed means :Def7: :: RLVECT_1:def 7
for v being Element of IT holds v + (0. IT) = v;

end;

:: deftheorem Def5 defines Abelian RLVECT_1:def 5 :
for IT being addLoopStr holds
( IT is Abelian iff for v, w being Element of IT holds v + w = w + v );

:: deftheorem Def6 defines add-associative RLVECT_1:def 6 :
for IT being addLoopStr holds
( IT is add-associative iff for u, v, w being Element of IT holds (u + v) + w = u + (v + w) );

:: deftheorem Def7 defines right_zeroed RLVECT_1:def 7 :
for IT being addLoopStr holds
( IT is right_zeroed iff for v being Element of IT holds v + (0. IT) = v );

definition

let IT be non empty RLSStruct ;
attr IT is vector-distributive means :Def8: :: RLVECT_1:def 8
for a being real number
for v, w being VECTOR of IT holds a * (v + w) = (a * v) + (a * w);
attr IT is scalar-distributive means :Def9: :: RLVECT_1:def 9
for a, b being real number
for v being VECTOR of IT holds (a + b) * v = (a * v) + (b * v);
attr IT is scalar-associative means :Def10: :: RLVECT_1:def 10
for a, b being real number
for v being VECTOR of IT holds (a * b) * v = a * (b * v);
attr IT is scalar-unital means :Def11: :: RLVECT_1:def 11
for v being VECTOR of IT holds 1 * v = v;

end;

:: deftheorem Def8 defines vector-distributive RLVECT_1:def 8 :
for IT being non empty RLSStruct holds
( IT is vector-distributive iff for a being real number
for v, w being VECTOR of IT holds a * (v + w) = (a * v) + (a * w) );

:: deftheorem Def9 defines scalar-distributive RLVECT_1:def 9 :
for IT being non empty RLSStruct holds
( IT is scalar-distributive iff for a, b being real number
for v being VECTOR of IT holds (a + b) * v = (a * v) + (b * v) );

:: deftheorem Def10 defines scalar-associative RLVECT_1:def 10 :
for IT being non empty RLSStruct holds
( IT is scalar-associative iff for a, b being real number
for v being VECTOR of IT holds (a * b) * v = a * (b * v) );

:: deftheorem Def11 defines scalar-unital RLVECT_1:def 11 :
for IT being non empty RLSStruct holds
( IT is scalar-unital iff for v being VECTOR of IT holds 1 * v = v );

definition

func Trivial-RLSStruct -> strict RLSStruct equals :: RLVECT_1:def 12
RLSStruct(# 1,op0,op2,(pr2 (REAL,1)) #);
coherence
RLSStruct(# 1,op0,op2,(pr2 (REAL,1)) #) is strict RLSStruct ;

end;

:: deftheorem defines Trivial-RLSStruct RLVECT_1:def 12 :
Trivial-RLSStruct = RLSStruct(# 1,op0,op2,(pr2 (REAL,1)) #);

registration

cluster Trivial-RLSStruct -> non empty trivial strict ;
coherence
( Trivial-RLSStruct is trivial & not Trivial-RLSStruct is empty ) by CARD_1:87;

end;

registration

cluster non empty strict right_complementable Abelian add-associative right_zeroed addLoopStr ;
existence
ex b₁ being non empty addLoopStr st
( b₁ is strict & b₁ is Abelian & b₁ is add-associative & b₁ is right_zeroed & b₁ is right_complementable )

proof end;

end;

registration

cluster non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct ;
existence
ex b₁ being non empty RLSStruct st
( b₁ is strict & b₁ is Abelian & b₁ is add-associative & b₁ is right_zeroed & b₁ is right_complementable & b₁ is scalar-distributive & b₁ is vector-distributive & b₁ is scalar-associative & b₁ is scalar-unital )

proof end;

end;

definition

mode RealLinearSpace is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct ;

end;

definition

let V be non empty Abelian addLoopStr ;
let v, w be Element of V;
:: original: +
redefine func v + w -> Element of the carrier of V;
commutativity
for v, w being Element of V holds v + w = w + v by Def5;

end;

theorem :: RLVECT_1:6

canceled;

theorem :: RLVECT_1:7

canceled;

Lm1: for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v, w being Element of V st v + w = 0. V holds
w + v = 0. V

proof end;

theorem :: RLVECT_1:8

canceled;

theorem Th9: :: RLVECT_1:9

for V being right_complementable add-associative right_zeroed addLoopStr holds V is right_add-cancelable

proof end;

theorem Th10: :: RLVECT_1:10

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v being Element of V holds
( v + (0. V) = v & (0. V) + v = v )

proof end;

definition

let V be non empty addLoopStr ;
let v be Element of V;
assume A1: ( V is add-associative & V is right_zeroed & V is right_complementable ) ;
redefine func - v means :Def13: :: RLVECT_1:def 13
v + it = 0. V;
compatibility
for b₁ being Element of the carrier of V holds
( b₁ = - v iff v + b₁ = 0. V )

proof end;

end;

:: deftheorem Def13 defines - RLVECT_1:def 13 :
for V being non empty addLoopStr
for v being Element of V st V is add-associative & V is right_zeroed & V is right_complementable holds
for b₃ being Element of the carrier of V holds
( b₃ = - v iff v + b₃ = 0. V );

Lm2: for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v, u being Element of V ex w being Element of V st v + w = u

proof end;

definition

let V be addLoopStr ;
let v, w be Element of V;
redefine func v - w equals :Def14: :: RLVECT_1:def 14
v + (- w);
correctness
compatibility
for b₁ being Element of the carrier of V holds
( b₁ = v - w iff b₁ = v + (- w) );
;

end;

:: deftheorem Def14 defines - RLVECT_1:def 14 :
for V being addLoopStr
for v, w being Element of V holds v - w = v + (- w);

theorem :: RLVECT_1:11

canceled;

theorem :: RLVECT_1:12

canceled;

theorem :: RLVECT_1:13

canceled;

theorem :: RLVECT_1:14

canceled;

theorem :: RLVECT_1:15

canceled;

theorem Th16: :: RLVECT_1:16

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v being Element of V holds
( v + (- v) = 0. V & (- v) + v = 0. V )

proof end;

theorem :: RLVECT_1:17

canceled;

theorem :: RLVECT_1:18

canceled;

theorem Th19: :: RLVECT_1:19

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v, w being Element of V st v + w = 0. V holds
v = - w

proof end;

theorem :: RLVECT_1:20

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v, u being Element of V ex w being Element of V st v + w = u by Lm2;

theorem Th21: :: RLVECT_1:21

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for w, u, v1, v2 being Element of V st ( w + v1 = w + v2 or v1 + w = v2 + w ) holds
v1 = v2

proof end;

theorem :: RLVECT_1:22

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v, w being Element of V st ( v + w = v or w + v = v ) holds
w = 0. V

proof end;

theorem Th23: :: RLVECT_1:23

for a being real number
for V being RealLinearSpace
for v being VECTOR of V st ( a = 0 or v = 0. V ) holds
a * v = 0. V

proof end;

theorem Th24: :: RLVECT_1:24

for a being real number
for V being RealLinearSpace
for v being VECTOR of V holds
( not a * v = 0. V or a = 0 or v = 0. V )

proof end;

theorem Th25: :: RLVECT_1:25

for V being non empty right_complementable add-associative right_zeroed addLoopStr holds - (0. V) = 0. V

proof end;

theorem :: RLVECT_1:26

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v being Element of V holds v - (0. V) = v

proof end;

theorem :: RLVECT_1:27

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v being Element of V holds (0. V) - v = - v by Th10;

theorem :: RLVECT_1:28

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v being Element of V holds v - v = 0. V by Def13;

theorem Th29: :: RLVECT_1:29

for V being RealLinearSpace
for v being VECTOR of V holds - v = (- 1) * v

proof end;

theorem Th30: :: RLVECT_1:30

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v being Element of V holds - (- v) = v

proof end;

theorem Th31: :: RLVECT_1:31

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v, w being Element of V st - v = - w holds
v = w

proof end;

theorem :: RLVECT_1:32

canceled;

theorem Th33: :: RLVECT_1:33

for V being RealLinearSpace
for v being VECTOR of V st v = - v holds
v = 0. V

proof end;

theorem :: RLVECT_1:34

for V being RealLinearSpace
for v being VECTOR of V st v + v = 0. V holds
v = 0. V

proof end;

theorem Th35: :: RLVECT_1:35

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v, w being Element of V st v - w = 0. V holds
v = w

proof end;

theorem :: RLVECT_1:36

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for u, v being Element of V ex w being Element of V st v - w = u

proof end;

theorem :: RLVECT_1:37

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for w, v1, v2 being Element of V st w - v1 = w - v2 holds
v1 = v2

proof end;

theorem Th38: :: RLVECT_1:38

for a being real number
for V being RealLinearSpace
for v being VECTOR of V holds a * (- v) = (- a) * v

proof end;

theorem Th39: :: RLVECT_1:39

for a being real number
for V being RealLinearSpace
for v being VECTOR of V holds a * (- v) = - (a * v)

proof end;

theorem :: RLVECT_1:40

for a being real number
for V being RealLinearSpace
for v being VECTOR of V holds (- a) * (- v) = a * v

proof end;

Lm3: for V being non empty right_complementable add-associative right_zeroed addLoopStr
for u, w being Element of V holds - (u + w) = (- w) + (- u)

proof end;

theorem Th41: :: RLVECT_1:41

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v, u, w being Element of V holds v - (u + w) = (v - w) - u

proof end;

theorem :: RLVECT_1:42

for V being non empty add-associative addLoopStr
for v, u, w being Element of V holds (v + u) - w = v + (u - w) by Def6;

theorem :: RLVECT_1:43

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v, u, w being Element of V holds v - (u - w) = (v - u) + w

proof end;

theorem Th44: :: RLVECT_1:44

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v, w being Element of V holds - (v + w) = (- w) - v

proof end;

theorem Th45: :: RLVECT_1:45

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v, w being Element of V holds - (v + w) = (- w) + (- v) by Lm3;

theorem :: RLVECT_1:46

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v, w being Element of V holds (- v) - w = (- w) - v

proof end;

theorem Th47: :: RLVECT_1:47

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v, w being Element of V holds - (v - w) = w + (- v)

proof end;

theorem Th48: :: RLVECT_1:48

for a being real number
for V being RealLinearSpace
for v, w being VECTOR of V holds a * (v - w) = (a * v) - (a * w)

proof end;

theorem Th49: :: RLVECT_1:49

for a, b being real number
for V being RealLinearSpace
for v being VECTOR of V holds (a - b) * v = (a * v) - (b * v)

proof end;

theorem :: RLVECT_1:50

for a being real number
for V being RealLinearSpace
for v, w being VECTOR of V st a <> 0 & a * v = a * w holds
v = w

proof end;

theorem :: RLVECT_1:51

for a, b being real number
for V being RealLinearSpace
for v being VECTOR of V st v <> 0. V & a * v = b * v holds
a = b

proof end;

definition

let V be non empty 1-sorted ;
let v, u be Element of V;
:: original: <*
redefine func <*v,u*> -> FinSequence of the carrier of V;
coherence
<*v,u*> is FinSequence of the carrier of V

proof end;

end;

definition

let V be non empty 1-sorted ;
let v, u, w be Element of V;
:: original: <*
redefine func <*v,u,w*> -> FinSequence of the carrier of V;
coherence
<*v,u,w*> is FinSequence of the carrier of V

proof end;

end;

definition

let V be non empty addLoopStr ;
let F be FinSequence of the carrier of V;
func Sum F -> Element of V means :Def15: :: RLVECT_1:def 15
ex f being Function of NAT, the carrier of V st
( it = f . (len F) & f . 0 = 0. V & ( for j being Element of NAT
for v being Element of V st j < len F & v = F . (j + 1) holds
f . (j + 1) = (f . j) + v ) );
existence
ex b₁ being Element of V ex f being Function of NAT, the carrier of V st
( b₁ = f . (len F) & f . 0 = 0. V & ( for j being Element of NAT
for v being Element of V st j < len F & v = F . (j + 1) holds
f . (j + 1) = (f . j) + v ) )

proof end;

uniqueness
for b₁, b₂ being Element of V st ex f being Function of NAT, the carrier of V st
( b₁ = f . (len F) & f . 0 = 0. V & ( for j being Element of NAT
for v being Element of V st j < len F & v = F . (j + 1) holds
f . (j + 1) = (f . j) + v ) ) & ex f being Function of NAT, the carrier of V st
( b₂ = f . (len F) & f . 0 = 0. V & ( for j being Element of NAT
for v being Element of V st j < len F & v = F . (j + 1) holds
f . (j + 1) = (f . j) + v ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def15 defines Sum RLVECT_1:def 15 :
for V being non empty addLoopStr
for F being FinSequence of the carrier of V
for b₃ being Element of V holds
( b₃ = Sum F iff ex f being Function of NAT, the carrier of V st
( b₃ = f . (len F) & f . 0 = 0. V & ( for j being Element of NAT
for v being Element of V st j < len F & v = F . (j + 1) holds
f . (j + 1) = (f . j) + v ) ) );

Lm4: for V being non empty addLoopStr holds Sum (<*> the carrier of V) = 0. V

proof end;

Lm5: for V being non empty addLoopStr
for F being FinSequence of the carrier of V st len F = 0 holds
Sum F = 0. V

proof end;

theorem :: RLVECT_1:52

canceled;

theorem :: RLVECT_1:53

canceled;

theorem :: RLVECT_1:54

canceled;

theorem Th55: :: RLVECT_1:55

for V being non empty addLoopStr
for F, G being FinSequence of the carrier of V
for v being Element of V st len F = (len G) + 1 & G = F | (dom G) & v = F . (len F) holds
Sum F = (Sum G) + v

proof end;

theorem :: RLVECT_1:56

for a being real number
for V being RealLinearSpace
for F, G being FinSequence of the carrier of V st len F = len G & ( for k being Element of NAT
for v being VECTOR 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_1:57

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
for v being Element of V st k in dom F & v = G . k holds
F . k = - v ) holds
Sum F = - (Sum G)

proof end;

theorem Th58: :: RLVECT_1:58

for V being non empty add-associative right_zeroed addLoopStr
for F, G being FinSequence of the carrier of V holds Sum (F ^ G) = (Sum F) + (Sum G)

proof end;

Lm6: for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v being Element of V holds Sum <*v*> = v

proof end;

theorem :: RLVECT_1:59

for V being non empty Abelian add-associative right_zeroed addLoopStr
for F, G being FinSequence of the carrier of V st rng F = rng G & F is one-to-one & G is one-to-one holds
Sum F = Sum G

proof end;

theorem Th60: :: RLVECT_1:60

for V being non empty addLoopStr holds Sum (<*> the carrier of V) = 0. V by Lm4;

theorem :: RLVECT_1:61

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v being Element of V holds Sum <*v*> = v by Lm6;

theorem Th62: :: RLVECT_1:62

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v, u being Element of V holds Sum <*v,u*> = v + u

proof end;

theorem Th63: :: RLVECT_1:63

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v, u, w being Element of V holds Sum <*v,u,w*> = (v + u) + w

proof end;

theorem :: RLVECT_1:64

for V being RealLinearSpace
for a being Real holds a * (Sum (<*> the carrier of V)) = 0. V by Lm4, Th23;

theorem :: RLVECT_1:65

canceled;

theorem :: RLVECT_1:66

for V being RealLinearSpace
for a being Real
for v, u being VECTOR of V holds a * (Sum <*v,u*>) = (a * v) + (a * u)

proof end;

theorem :: RLVECT_1:67

for V being RealLinearSpace
for a being Real
for v, u, w being VECTOR of V holds a * (Sum <*v,u,w*>) = ((a * v) + (a * u)) + (a * w)

proof end;

theorem :: RLVECT_1:68

for V being non empty right_complementable add-associative right_zeroed addLoopStr holds - (Sum (<*> the carrier of V)) = 0. V

proof end;

theorem :: RLVECT_1:69

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v being Element of V holds - (Sum <*v*>) = - v by Lm6;

theorem :: RLVECT_1:70

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v, u being Element of V holds - (Sum <*v,u*>) = (- v) - u

proof end;

theorem :: RLVECT_1:71

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v, u, w being Element of V holds - (Sum <*v,u,w*>) = ((- v) - u) - w

proof end;

theorem Th72: :: RLVECT_1:72

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v, w being Element of V holds Sum <*v,w*> = Sum <*w,v*>

proof end;

theorem :: RLVECT_1:73

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v, w being Element of V holds Sum <*v,w*> = (Sum <*v*>) + (Sum <*w*>)

proof end;

theorem :: RLVECT_1:74

for V being non empty right_complementable add-associative right_zeroed addLoopStr holds Sum <*(0. V),(0. V)*> = 0. V

proof end;

theorem :: RLVECT_1:75

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v being Element of V holds
( Sum <*(0. V),v*> = v & Sum <*v,(0. V)*> = v )

proof end;

theorem :: RLVECT_1:76

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v being Element of V holds
( Sum <*v,(- v)*> = 0. V & Sum <*(- v),v*> = 0. V )

proof end;

theorem :: RLVECT_1:77

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v, w being Element of V holds Sum <*v,(- w)*> = v - w by Th62;

theorem Th78: :: RLVECT_1:78

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v, w being Element of V holds Sum <*(- v),(- w)*> = - (w + v)

proof end;

theorem Th79: :: RLVECT_1:79

for V being RealLinearSpace
for v being VECTOR of V holds Sum <*v,v*> = 2 * v

proof end;

theorem :: RLVECT_1:80

for V being RealLinearSpace
for v being VECTOR of V holds Sum <*(- v),(- v)*> = (- 2) * v

proof end;

theorem :: RLVECT_1:81

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for u, v, w being Element of V holds Sum <*u,v,w*> = ((Sum <*u*>) + (Sum <*v*>)) + (Sum <*w*>)

proof end;

theorem :: RLVECT_1:82

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for u, v, w being Element of V holds Sum <*u,v,w*> = (Sum <*u,v*>) + w

proof end;

theorem :: RLVECT_1:83

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v, u, w being Element of V holds Sum <*u,v,w*> = (Sum <*v,w*>) + u

proof end;

theorem Th84: :: RLVECT_1:84

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v, u, w being Element of V holds Sum <*u,v,w*> = (Sum <*u,w*>) + v

proof end;

theorem Th85: :: RLVECT_1:85

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v, u, w being Element of V holds Sum <*u,v,w*> = Sum <*u,w,v*>

proof end;

theorem Th86: :: RLVECT_1:86

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v, u, w being Element of V holds Sum <*u,v,w*> = Sum <*v,u,w*>

proof end;

theorem Th87: :: RLVECT_1:87

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v, u, w being Element of V holds Sum <*u,v,w*> = Sum <*v,w,u*>

proof end;

theorem :: RLVECT_1:88

canceled;

theorem :: RLVECT_1:89

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v, u, w being Element of V holds Sum <*u,v,w*> = Sum <*w,v,u*>

proof end;

theorem :: RLVECT_1:90

for V being non empty right_complementable add-associative right_zeroed addLoopStr holds Sum <*(0. V),(0. V),(0. V)*> = 0. V

proof end;

theorem :: RLVECT_1:91

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v being Element of V holds
( Sum <*(0. V),(0. V),v*> = v & Sum <*(0. V),v,(0. V)*> = v & Sum <*v,(0. V),(0. V)*> = v )

proof end;

theorem :: RLVECT_1:92

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for u, v being Element of V holds
( Sum <*(0. V),u,v*> = u + v & Sum <*u,v,(0. V)*> = u + v & Sum <*u,(0. V),v*> = u + v )

proof end;

theorem :: RLVECT_1:93

for V being RealLinearSpace
for v being VECTOR of V holds Sum <*v,v,v*> = 3 * v

proof end;

theorem :: RLVECT_1:94

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for F being FinSequence of the carrier of V st len F = 0 holds
Sum F = 0. V by Lm5;

theorem :: RLVECT_1:95

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for F being FinSequence of the carrier of V st len F = 1 holds
Sum F = F . 1

proof end;

theorem :: RLVECT_1:96

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for F being FinSequence of the carrier of V
for v1, v2 being Element of V st len F = 2 & v1 = F . 1 & v2 = F . 2 holds
Sum F = v1 + v2

proof end;

theorem :: RLVECT_1:97

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for F being FinSequence of the carrier of V
for v1, v2, v being Element of V st len F = 3 & v1 = F . 1 & v2 = F . 2 & v = F . 3 holds
Sum F = (v1 + v2) + v

proof end;

begin

definition

let L be non empty addLoopStr ;
attr L is zeroed means :Def16: :: RLVECT_1:def 16
for a being Element of L holds
( a + (0. L) = a & (0. L) + a = a );

end;

:: deftheorem Def16 defines zeroed RLVECT_1:def 16 :
for L being non empty addLoopStr holds
( L is zeroed iff for a being Element of L holds
( a + (0. L) = a & (0. L) + a = a ) );

registration

cluster non empty zeroed -> non empty right_zeroed addLoopStr ;
coherence
for b₁ being non empty addLoopStr st b₁ is zeroed holds
b₁ is right_zeroed

proof end;

end;

registration

cluster non empty Abelian right_zeroed -> non empty zeroed addLoopStr ;
coherence
for b₁ being non empty addLoopStr st b₁ is Abelian & b₁ is right_zeroed holds
b₁ is zeroed

proof end;

cluster non empty right_complementable Abelian -> non empty left_complementable addLoopStr ;
coherence
for b₁ being non empty addLoopStr st b₁ is Abelian & b₁ is right_complementable holds
b₁ is left_complementable

proof end;

end;

theorem :: RLVECT_1:98

for a being real number
for V being RealLinearSpace
for v being VECTOR of V holds (- a) * v = - (a * v)

proof end;

begin

theorem :: RLVECT_1:99

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr holds - (Sum (<*> the carrier of V)) = 0. V

proof end;

theorem :: RLVECT_1:100

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v, u being Element of V holds - (Sum <*v,u*>) = (- v) - u

proof end;

theorem :: RLVECT_1:101

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v, u, w being Element of V holds - (Sum <*v,u,w*>) = ((- v) - u) - w

proof end;

theorem :: RLVECT_1:102

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v being Element of V holds
( Sum <*v,(- v)*> = 0. V & Sum <*(- v),v*> = 0. V )

proof end;

theorem :: RLVECT_1:103

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v, w being Element of V holds
( Sum <*v,(- w)*> = v - w & Sum <*(- w),v*> = v - w )

proof end;

theorem :: RLVECT_1:104

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v, w being Element of V holds
( Sum <*(- v),(- w)*> = - (v + w) & Sum <*(- w),(- v)*> = - (v + w) )

proof end;