:: VECTSP_6 semantic presentation

definition

let GF be non empty ZeroStr ;
let V be non empty VectSpStr of GF;
canceled;
canceled;
canceled;
mode Linear_Combination of V -> Element of Funcs the carrier of V,the carrier of GF means :Def4: :: VECTSP_6:def 4
ex T being finite Subset of V st
for v being Element of V st not v in T holds
it . v = 0. GF;
existence

proof end;

end;

:: deftheorem VECTSP_6:def 1 :

:: deftheorem VECTSP_6:def 2 :

:: deftheorem VECTSP_6:def 3 :

:: deftheorem Def4 defines Linear_Combination VECTSP_6:def 4 :

definition

let GF be non empty ZeroStr ;
let V be non empty VectSpStr of GF;
let L be Linear_Combination of V;
func Carrier L -> finite Subset of V equals :: VECTSP_6:def 5
{ v where v is Element of V : L . v <> 0. GF } ;
coherence

proof end;

end;

:: deftheorem defines Carrier VECTSP_6:def 5 :

theorem :: VECTSP_6:1

canceled;

theorem :: VECTSP_6:2

canceled;

theorem :: VECTSP_6:3

canceled;

theorem :: VECTSP_6:4

canceled;

theorem :: VECTSP_6:5

canceled;

theorem :: VECTSP_6:6

canceled;

theorem :: VECTSP_6:7

canceled;

theorem :: VECTSP_6:8

canceled;

theorem :: VECTSP_6:9

canceled;

theorem :: VECTSP_6:10

canceled;

theorem :: VECTSP_6:11

canceled;

theorem :: VECTSP_6:12

canceled;

theorem :: VECTSP_6:13

canceled;

theorem :: VECTSP_6:14

canceled;

theorem :: VECTSP_6:15

canceled;

theorem :: VECTSP_6:16

canceled;

theorem :: VECTSP_6:17

canceled;

theorem :: VECTSP_6:18

canceled;

theorem :: VECTSP_6:19

for x being set
for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for L being Linear_Combination of V holds
( x in Carrier L iff ex v being Element of V st
( x = v & L . v <> 0. GF ) ) ;

theorem :: VECTSP_6:20

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for v being Element of V
for L being Linear_Combination of V holds
( L . v = 0. GF iff not v in Carrier L )

proof end;

definition

let GF be non empty ZeroStr ;
let V be non empty VectSpStr of GF;
func ZeroLC V -> Linear_Combination of V means :Def6: :: VECTSP_6:def 6
Carrier it = {} ;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def6 defines ZeroLC VECTSP_6:def 6 :

theorem :: VECTSP_6:21

canceled;

theorem Th22: :: VECTSP_6:22

proof end;

definition

let GF be non empty ZeroStr ;
let V be non empty VectSpStr of GF;
let A be Subset of V;
mode Linear_Combination of A -> Linear_Combination of V means :Def7: :: VECTSP_6:def 7
Carrier it c= A;
existence

proof end;

end;

:: deftheorem Def7 defines Linear_Combination VECTSP_6:def 7 :

theorem :: VECTSP_6:23

canceled;

theorem :: VECTSP_6:24

canceled;

theorem :: VECTSP_6:25

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
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 Th26: :: VECTSP_6:26

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for A being Subset of V holds ZeroLC V is Linear_Combination of A

proof end;

theorem Th27: :: VECTSP_6:27

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for l being Linear_Combination of {} the carrier of V holds l = ZeroLC V

proof end;

theorem :: VECTSP_6:28

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for L being Linear_Combination of V holds L is Linear_Combination of Carrier L by Def7;

definition

let GF be non empty LoopStr ;
let V be non empty VectSpStr of GF;
let F be FinSequence of the carrier of V;
let f be Function of the carrier of V,the carrier of GF;
func f (#) F -> FinSequence of the carrier of V means :Def8: :: VECTSP_6:def 8
( len it = len F & ( for i being Nat st i in dom it holds
it . i = (f . (F /. i)) * (F /. i) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def8 defines (#) VECTSP_6:def 8 :

theorem :: VECTSP_6:29

canceled;

theorem :: VECTSP_6:30

canceled;

theorem :: VECTSP_6:31

canceled;

theorem Th32: :: VECTSP_6:32

for i being Nat
for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for v being Element of V
for F being FinSequence of the carrier of V
for f being Function of the carrier of V,the carrier of GF st i in dom F & v = F . i holds
(f (#) F) . i = (f . v) * v

proof end;

theorem :: VECTSP_6:33

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for f being Function of the carrier of V,the carrier of GF holds f (#) (<*> the carrier of V) = <*> the carrier of V

proof end;

theorem Th34: :: VECTSP_6:34

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for v being Element of V
for f being Function of the carrier of V,the carrier of GF holds f (#) <*v*> = <*((f . v) * v)*>

proof end;

theorem Th35: :: VECTSP_6:35

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for v1, v2 being Element of V
for f being Function of the carrier of V,the carrier of GF holds f (#) <*v1,v2*> = <*((f . v1) * v1),((f . v2) * v2)*>

proof end;

theorem :: VECTSP_6:36

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for v1, v2, v3 being Element of V
for f being Function of the carrier of V,the carrier of GF holds f (#) <*v1,v2,v3*> = <*((f . v1) * v1),((f . v2) * v2),((f . v3) * v3)*>

proof end;

theorem Th37: :: VECTSP_6:37

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for F, G being FinSequence of the carrier of V
for f being Function of the carrier of V,the carrier of GF holds f (#) (F ^ G) = (f (#) F) ^ (f (#) G)

proof end;

definition

let GF be non empty LoopStr ;
let V be non empty VectSpStr of GF;
let L be Linear_Combination of V;
assume A1: ( V is Abelian & V is add-associative & V is right_zeroed & V is right_complementable ) ;
func Sum L -> Element of V means :Def9: :: VECTSP_6:def 9
ex F being FinSequence of the carrier of V st
( F is one-to-one & rng F = Carrier L & it = Sum (L (#) F) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def9 defines Sum VECTSP_6:def 9 :

Lm1: for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF holds Sum (ZeroLC V) = 0. V

proof end;

theorem :: VECTSP_6:38

canceled;

theorem :: VECTSP_6:39

canceled;

theorem :: VECTSP_6:40

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for A being Subset of V st 0. GF <> 1. GF holds
( ( A <> {} & A is lineary-closed ) iff for l being Linear_Combination of A holds Sum l in A )

proof end;

theorem :: VECTSP_6:41

theorem :: VECTSP_6:42

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for l being Linear_Combination of {} the carrier of V holds Sum l = 0. V

proof end;

theorem Th43: :: VECTSP_6:43

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for v being Element of V
for l being Linear_Combination of {v} holds Sum l = (l . v) * v

proof end;

theorem Th44: :: VECTSP_6:44

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for v1, v2 being Element 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 :: VECTSP_6:45

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for L being Linear_Combination of V st Carrier L = {} holds
Sum L = 0. V

proof end;

theorem :: VECTSP_6:46

proof end;

theorem :: VECTSP_6:47

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for v1, v2 being Element 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 GF be non empty ZeroStr ;
let V be non empty VectSpStr of GF;
let L1, L2 be Linear_Combination of V;
:: original: =
redefine pred L1 = L2 means :: VECTSP_6:def 10
for v being Element of V holds L1 . v = L2 . v;
compatibility by FUNCT_2:113;

end;

:: deftheorem defines = VECTSP_6:def 10 :

definition

let GF be non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let V be non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF;
let L1, L2 be Linear_Combination of V;
func L1 + L2 -> Linear_Combination of V means :Def11: :: VECTSP_6:def 11
for v being Element of V holds it . v = (L1 . v) + (L2 . v);
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def11 defines + VECTSP_6:def 11 :

theorem :: VECTSP_6:48

canceled;

theorem :: VECTSP_6:49

canceled;

theorem :: VECTSP_6:50

canceled;

theorem Th51: :: VECTSP_6:51

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for L1, L2 being Linear_Combination of V holds Carrier (L1 + L2) c= (Carrier L1) \/ (Carrier L2)

proof end;

theorem Th52: :: VECTSP_6:52

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
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 Th53: :: VECTSP_6:53

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for L1, L2 being Linear_Combination of V holds L1 + L2 = L2 + L1

proof end;

theorem :: VECTSP_6:54

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for L1, L2, L3 being Linear_Combination of V holds L1 + (L2 + L3) = (L1 + L2) + L3

proof end;

theorem :: VECTSP_6:55

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for L being Linear_Combination of V holds
( L + (ZeroLC V) = L & (ZeroLC V) + L = L )

proof end;

definition

let GF be non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let V be non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF;
let a be Element of GF;
let L be Linear_Combination of V;
func a * L -> Linear_Combination of V means :Def12: :: VECTSP_6:def 12
for v being Element of V holds it . v = a * (L . v);
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def12 defines * VECTSP_6:def 12 :

theorem :: VECTSP_6:56

canceled;

theorem :: VECTSP_6:57

canceled;

theorem Th58: :: VECTSP_6:58

for GF being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty Abelian add-associative right_zeroed right_complementable VectSp-like VectSpStr of GF
for a being Element of GF
for L being Linear_Combination of V holds Carrier (a * L) c= Carrier L

proof end;

theorem Th59: :: VECTSP_6:59

for GF being Field
for V being VectSp of GF
for a being Element of GF
for L being Linear_Combination of V st a <> 0. GF holds
Carrier (a * L) = Carrier L

proof end;

theorem Th60: :: VECTSP_6:60