:: VECTSP10 semantic presentation

definition

let K be doubleLoopStr ;
func StructVectSp K -> strict VectSpStr of K equals :: VECTSP10:def 1
VectSpStr(# the carrier of K,the add of K,the Zero of K,the mult of K #);
coherence ;

end;

:: deftheorem defines StructVectSp VECTSP10:def 1 :

registration

let K be non empty doubleLoopStr ;
cluster StructVectSp K -> non empty strict ;
coherence ;

end;

registration

let K be non empty Abelian doubleLoopStr ;
cluster StructVectSp K -> non empty Abelian strict ;
coherence

proof end;

end;

registration

let K be non empty add-associative doubleLoopStr ;
cluster StructVectSp K -> non empty add-associative strict ;
coherence

proof end;

end;

registration

let K be non empty right_zeroed doubleLoopStr ;
cluster StructVectSp K -> non empty right_zeroed strict ;
coherence

proof end;

end;

registration

let K be non empty right_complementable doubleLoopStr ;
cluster StructVectSp K -> non empty right_complementable strict ;
coherence

proof end;

end;

registration

let K be non empty associative distributive left_unital doubleLoopStr ;
cluster StructVectSp K -> non empty strict VectSp-like ;
coherence

proof end;

end;

registration

let K be non empty non degenerated doubleLoopStr ;
cluster StructVectSp K -> non empty strict non trivial ;
coherence

proof end;

end;

registration

let K be non empty non degenerated doubleLoopStr ;
cluster non empty non trivial VectSpStr of K;
existence

proof end;

end;

registration

let K be non empty add-associative right_zeroed right_complementable doubleLoopStr ;
cluster non empty add-associative right_zeroed right_complementable strict VectSpStr of K;
correctness

proof end;

end;

registration

let K be non empty add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
cluster non empty add-associative right_zeroed right_complementable strict VectSp-like VectSpStr of K;
correctness

proof end;

end;

registration

let K be non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital non degenerated doubleLoopStr ;
cluster non empty Abelian add-associative right_zeroed right_complementable strict VectSp-like non trivial VectSpStr of K;
existence

proof end;

end;

theorem :: VECTSP10:1

canceled;

theorem Th2: :: VECTSP10:2

for K being non empty add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for a being Element of K
for V being non empty add-associative right_zeroed right_complementable VectSp-like VectSpStr of K
for v being Vector of V holds
( (0. K) * v = 0. V & a * (0. V) = 0. V )

proof end;

theorem :: VECTSP10:3

for K being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being VectSp of K
for S, T being Subspace of V
for v being Vector of V st S /\ T = (0). V & v in S & v in T holds
v = 0. V

proof end;

theorem Th4: :: VECTSP10:4

for K being Field
for V being VectSp of K
for x being set
for v being Vector of V holds
( x in Lin {v} iff ex a being Element of K st x = a * v )

proof end;

theorem Th5: :: VECTSP10:5

for K being Field
for V being VectSp of K
for v being Vector of V
for a, b being Scalar of st v <> 0. V & a * v = b * v holds
a = b

proof end;

theorem Th6: :: VECTSP10:6

for K being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being VectSp of K
for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
for v, v1, v2 being Vector of V st v1 in W1 & v2 in W2 & v = v1 + v2 holds
v |-- W1,W2 = [v1,v2]

proof end;

theorem Th7: :: VECTSP10:7

proof end;

theorem Th8: :: VECTSP10:8

proof end;

theorem Th9: :: VECTSP10:9

proof end;

theorem Th10: :: VECTSP10:10

for K being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being VectSp of K
for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
for v being Vector of V st v in W1 holds
v |-- W1,W2 = [v,(0. V)]

proof end;

theorem Th11: :: VECTSP10:11

for K being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being VectSp of K
for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
for v being Vector of V st v in W2 holds
v |-- W1,W2 = [(0. V),v]

proof end;

theorem Th12: :: VECTSP10:12

for K being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being VectSp of K
for V1 being Subspace of V
for W1 being Subspace of V1
for v being Vector of V st v in W1 holds
v is Vector of V1

proof end;

theorem Th13: :: VECTSP10:13

for K being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being VectSp of K
for V1, V2, W being Subspace of V
for W1, W2 being Subspace of W st W1 = V1 & W2 = V2 holds
W1 + W2 = V1 + V2

proof end;

theorem Th14: :: VECTSP10:14

for K being Field
for V being VectSp of K
for W being Subspace of V
for v being Vector of V
for w being Vector of W st v = w holds
Lin {w} = Lin {v}

proof end;

theorem Th15: :: VECTSP10:15

for K being Field
for V being VectSp of K
for v being Vector of V
for X being Subspace of V st not v in X holds
for y being Vector of (X + (Lin {v}))
for W being Subspace of X + (Lin {v}) st v = y & W = X holds
X + (Lin {v}) is_the_direct_sum_of W, Lin {y}

proof end;

theorem Th16: :: VECTSP10:16

proof end;

theorem Th17: :: VECTSP10:17

proof end;

theorem Th18: :: VECTSP10:18

for K being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being VectSp of K
for v being Vector of V
for W1, W2 being Subspace of V ex v1, v2 being Vector of V st v |-- W1,W2 = [v1,v2]

proof end;

theorem Th19: :: VECTSP10:19

for K being Field
for V being VectSp of K
for v being Vector of V
for X being Subspace of V
for y being Vector of (X + (Lin {v}))
for W being Subspace of X + (Lin {v}) st v = y & X = W & not v in X holds
for w being Vector of (X + (Lin {v})) ex x being Vector of X ex r being Element of K st w |-- W,(Lin {y}) = [x,(r * v)]

proof end;

theorem Th20: :: VECTSP10:20

for K being Field
for V being VectSp of K
for v being Vector of V
for X being Subspace of V
for y being Vector of (X + (Lin {v}))
for W being Subspace of X + (Lin {v}) st v = y & X = W & not v in X holds
for w1, w2 being Vector of (X + (Lin {v}))
for x1, x2 being Vector of X
for r1, r2 being Element of K st w1 |-- W,(Lin {y}) = [x1,(r1 * v)] & w2 |-- W,(Lin {y}) = [x2,(r2 * v)] holds
(w1 + w2) |-- W,(Lin {y}) = [(x1 + x2),((r1 + r2) * v)]

proof end;

theorem Th21: :: VECTSP10:21

for K being Field
for V being VectSp of K
for v being Vector of V
for X being Subspace of V
for y being Vector of (X + (Lin {v}))
for W being Subspace of X + (Lin {v}) st v = y & X = W & not v in X holds
for w being Vector of (X + (Lin {v}))
for x being Vector of X
for t, r being Element of K st w |-- W,(Lin {y}) = [x,(r * v)] holds
(t * w) |-- W,(Lin {y}) = [(t * x),((t * r) * v)]

proof end;

definition

let K be non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let V be VectSp of K;
let W be Subspace of V;
func CosetSet V,W -> non empty Subset-Family of V equals :: VECTSP10:def 2
{ A where A is Coset of W : verum } ;
correctness

proof end;

end;

:: deftheorem defines CosetSet VECTSP10:def 2 :

definition

let K be non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let V be VectSp of K;
let W be Subspace of V;
func addCoset V,W -> BinOp of CosetSet V,W means :Def3: :: VECTSP10:def 3
for A, B being Element of CosetSet V,W
for a, b being Vector of V st A = a + W & B = b + W holds
it . A,B = (a + b) + W;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def3 defines addCoset VECTSP10:def 3 :

definition

proof end;

end;

:: deftheorem defines zeroCoset VECTSP10:def 4 :

definition

let K be non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let V be VectSp of K;
let W be Subspace of V;
func lmultCoset V,W -> Function of [:the carrier of K,(CosetSet V,W):], CosetSet V,W means :Def5: :: VECTSP10:def 5
for z being Element of K
for A being Element of CosetSet V,W
for a being Vector of V st A = a + W holds
it . z,A = (z * a) + W;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def5 defines lmultCoset VECTSP10:def 5 :

definition

let K be non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let V be VectSp of K;
let W be Subspace of V;
func VectQuot V,W -> non empty Abelian add-associative right_zeroed right_complementable strict VectSp-like VectSpStr of K means :Def6: :: VECTSP10:def 6
( the carrier of it = CosetSet V,W & the add of it = addCoset V,W & the Zero of it = zeroCoset V,W & the lmult of it = lmultCoset V,W );
existence

proof end;

uniqueness ;

end;

:: deftheorem Def6 defines VectQuot VECTSP10:def 6 :

theorem Th22: :: VECTSP10:22

for K being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being VectSp of K
for W being Subspace of V holds
( zeroCoset V,W = (0. V) + W & 0. (VectQuot V,W) = zeroCoset V,W )

proof end;

theorem Th23: :: VECTSP10:23

for K being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being VectSp of K
for W being Subspace of V
for w being Vector of (VectQuot V,W) holds
( w is Coset of W & ex v being Vector of V st w = v + W )

proof end;

theorem Th24: :: VECTSP10:24

for K being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being VectSp of K
for W being Subspace of V
for v being Vector of V holds
( v + W is Coset of W & v + W is Vector of (VectQuot V,W) )

proof end;

theorem :: VECTSP10:25

for K being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being VectSp of K
for W being Subspace of V
for A being Coset of W holds A is Vector of (VectQuot V,W)

proof end;

theorem :: VECTSP10:26

for K being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being VectSp of K
for W being Subspace of V
for A being Vector of (VectQuot V,W)
for v being Vector of V
for a being Scalar of st A = v + W holds
a * A = (a * v) + W

proof end;

theorem :: VECTSP10:27

for K being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being VectSp of K
for W being Subspace of V
for A1, A2 being Vector of (VectQuot V,W)
for v1, v2 being Vector of V st A1 = v1 + W & A2 = v2 + W holds
A1 + A2 = (v1 + v2) + W

proof end;

theorem Th28: :: VECTSP10:28

for K being Field
for V being VectSp of K
for X being Subspace of V
for fi being linear-Functional of X
for v being Vector of V
for y being Vector of (X + (Lin {v})) st v = y & not v in X holds
for r being Element of K ex psi being linear-Functional of (X + (Lin {v})) st
( psi | the carrier of X = fi & psi . y = r )

proof end;

registration

let K be non empty right_zeroed LoopStr ;
let V be non empty VectSpStr of K;
cluster additive 0-preserving Relation of the carrier of V,the carrier of K;
existence

proof end;

end;

registration

let K be non empty add-associative right_zeroed right_complementable doubleLoopStr ;
let V be non empty right_zeroed VectSpStr of K;
cluster additive -> 0-preserving Relation of the carrier of V,the carrier of K;
coherence

proof end;

end;

registration

let K be non empty add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let V be non empty add-associative right_zeroed right_complementable VectSp-like VectSpStr of K;
cluster homogeneous -> 0-preserving Relation of the carrier of V,the carrier of K;
coherence

proof end;

end;

registration

let K be non empty ZeroStr ;
let V be non empty VectSpStr of K;
cluster 0Functional V -> constant ;
coherence

proof end;

end;

registration

let K be non empty ZeroStr ;
let V be non empty VectSpStr of K;
cluster constant Relation of the carrier of V,the carrier of K;
existence

proof end;

end;

definition

let K be non empty add-associative right_zeroed right_complementable doubleLoopStr ;
let V be non empty right_zeroed VectSpStr of K;
let f be 0-preserving Functional of V;
:: original: constant
redefine attr f is constant means :Def7: :: VECTSP10:def 7
f = 0Functional V;
compatibility

proof end;

end;

:: deftheorem Def7 defines constant VECTSP10:def 7 :

registration

let K be non empty add-associative right_zeroed right_complementable doubleLoopStr ;
let V be non empty right_zeroed VectSpStr of K;
cluster constant additive 0-preserving Relation of the carrier of V,the carrier of K;
existence

proof end;

end;

registration

let K be non empty 1-sorted ;
let V be non empty VectSpStr of K;
cluster non constant -> non trivial Relation of the carrier of V,the carrier of K;
coherence

proof end;

end;

registration

let K be Field;
let V be non trivial VectSp of K;
cluster non constant additive homogeneous 0-preserving non trivial Relation of the carrier of V,the carrier of K;
existence

proof end;

end;

registration

let K be Field;
let V be non trivial VectSp of K;
cluster trivial -> constant Relation of the carrier of V,the carrier of K;
coherence

proof end;

end;

definition

let K be Field;
let V be non trivial VectSp of K;
let v be Vector of V;
let W be Linear_Compl of Lin {v};
assume A1: v <> 0. V ;
func coeffFunctional v,W -> V71 non trivial linear-Functional of V means :Def8: :: VECTSP10:def 8
( it . v = 1. K & it | the carrier of W = 0Functional W );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def8 defines coeffFunctional VECTSP10:def 8 :

theorem Th29: :: VECTSP10:29

for K being Field
for V being non trivial VectSp of K
for f being non constant 0-preserving Functional of V ex v being Vector of V st
( v <> 0. V & f . v <> 0. K )

proof end;

theorem Th30: :: VECTSP10:30

for K being Field
for V being non trivial VectSp of K
for v being Vector of V
for a being Scalar of
for W being Linear_Compl of Lin {v} st v <> 0. V holds
(coeffFunctional v,W) . (a * v) = a

proof end;

theorem Th31: :: VECTSP10:31

for K being Field
for V being non trivial VectSp of K
for v, w being Vector of V
for W being Linear_Compl of Lin {v} st v <> 0. V & w in W holds
(coeffFunctional v,W) . w = 0. K

proof end;

theorem :: VECTSP10:32

for K being Field
for V being non trivial VectSp of K
for v, w being Vector of V
for a being Scalar of
for W being Linear_Compl of Lin {v} st v <> 0. V & w in W holds
(coeffFunctional v,W) . ((a * v) + w) = a

proof end;

theorem :: VECTSP10:33

for K being non empty LoopStr
for V being non empty VectSpStr of K
for f, g being Functional of V
for v being Vector of V holds (f - g) . v = (f . v) - (g . v)

proof end;

registration

let K be Field;
let V be non trivial VectSp of K;
cluster V *' -> non trivial ;
coherence

proof end;

end;

definition

let K be non empty ZeroStr ;
let V be non empty VectSpStr of K;
let f be Functional of V;
func ker f -> Subset of V equals :: VECTSP10:def 9
{ v where v is Vector of V : f . v = 0. K } ;
coherence

proof end;

end;

:: deftheorem defines ker VECTSP10:def 9 :

registration

let K be non empty right_zeroed LoopStr ;
let V be non empty VectSpStr of K;
let f be 0-preserving Functional of V;
cluster ker f -> non empty ;
coherence

proof end;

end;

theorem Th34: :: VECTSP10:34

for K being non empty add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being non empty add-associative right_zeroed right_complementable VectSp-like VectSpStr of K
for f being linear-Functional of V holds ker f is lineary-closed

proof end;

definition

let K be non empty ZeroStr ;
let V be non empty VectSpStr of K;
let f be Functional of V;
attr f is degenerated means :Def10: :: VECTSP10:def 10
ker f <> {(0. V)};

end;

:: deftheorem Def10 defines degenerated VECTSP10:def 10 :

registration

let K be non empty non degenerated doubleLoopStr ;
let V be non empty non trivial VectSpStr of K;
cluster 0-preserving non degenerated -> non constant non trivial Relation of the carrier of V,the carrier of K;
coherence

proof end;

end;

definition

let K be non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let V be VectSp of K;
let f be linear-Functional of V;
func Ker f -> non empty strict Subspace of V means :Def11: :: VECTSP10:def 11
the carrier of it = ker f;
existence

proof end;

uniqueness by VECTSP_4:37;

end;

:: deftheorem Def11 defines Ker VECTSP10:def 11 :

definition

let K be non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let V be VectSp of K;
let W be Subspace of V;
let f be additive Functional of V;
assume A1: the carrier of W c= ker f ;
func QFunctional f,W -> additive Functional of (VectQuot V,W) means :Def12: :: VECTSP10:def 12
for A being Vector of (VectQuot V,W)
for a being Vector of V st A = a + W holds
it . A = f . a;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def12 defines QFunctional VECTSP10:def 12 :

theorem Th35: :: VECTSP10:35

for K being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being VectSp of K
for W being Subspace of V
for f being linear-Functional of V st the carrier of W c= ker f holds
QFunctional f,W is homogeneous

proof end;

definition

let K be non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let V be VectSp of K;
let f be linear-Functional of V;
func CQFunctional f -> linear-Functional of (VectQuot V,(Ker f)) equals :: VECTSP10:def 13
QFunctional f,(Ker f);
correctness

proof end;

end;

:: deftheorem defines CQFunctional VECTSP10:def 13 :

theorem Th36: :: VECTSP10:36

for K being non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr
for V being VectSp of K
for f being linear-Functional of V
for A being Vector of (VectQuot V,(Ker f))
for v being Vector of V st A = v + (Ker f) holds
(CQFunctional f) . A = f . v

proof end;

registration

let K be Field;
let V be non trivial VectSp of K;
let f be V71 linear-Functional of V;
cluster CQFunctional f -> V71 0-preserving non trivial ;
coherence

proof end;

end;

registration

let K be non empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let V be VectSp of K;
let f be linear-Functional of V;
cluster CQFunctional f -> 0-preserving non degenerated ;
coherence

proof end;

end;