:: Abelian Groups, Fields and Vector Spaces
:: by Eugeniusz Kusak, Wojciech Leo\'nczuk and Micha{\l} Muzalewski
::
:: Received November 23, 1989
:: Copyright (c) 1990-2021 Association of Mizar Users

definition

func G_Real -> strict addLoopStr equals :: VECTSP_1:def 1
addLoopStr(# REAL,addreal,(In (0,REAL)) #);

coherence
addLoopStr(# REAL,addreal,(In (0,REAL)) #) is strict addLoopStr ;

end;

:: deftheorem defines G_Real VECTSP_1:def 1 :
G_Real = addLoopStr(# REAL,addreal,(In (0,REAL)) #);

registration

cluster G_Real -> non empty strict ;

coherence
not G_Real is empty ;

end;

registration

cluster the carrier of G_Real -> real-membered ;

coherence
the carrier of G_Real is real-membered ;

end;

registration

let a, b be Element of G_Real;
let x, y be Real;

identify a + b with x + y when a = x, b = y;

compatibility
( a = x & b = y implies a + b = x + y ) by BINOP_2:def 9;

end;

registration

cluster G_Real -> strict right_complementable Abelian add-associative right_zeroed ;

coherence
( G_Real is Abelian & G_Real is add-associative & G_Real is right_zeroed & G_Real is right_complementable )

proof end;

end;

registration

let a be Element of G_Real;
let x be Real;

identify - a with - x when a = x;

compatibility
( a = x implies - a = - x )

proof end;

end;

registration

let a, b be Element of G_Real;
let x, y be Real;

identify a - b with x - y when a = x, b = y;

compatibility
( a = x & b = y implies a - b = x - y ) ;

end;

registration

cluster non empty strict right_complementable Abelian add-associative right_zeroed for addLoopStr ;

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

proof end;

end;

definition

mode AddGroup is non empty right_complementable add-associative right_zeroed addLoopStr ;

end;

definition

mode AbGroup is Abelian AddGroup;

end;

:: 4. FIELD STRUCTURE

definition

let IT be non empty doubleLoopStr ;

attr IT is right-distributive means :Def2: :: VECTSP_1:def 2
for a, b, c being Element of IT holds a * (b + c) = (a * b) + (a * c);

attr IT is left-distributive means :Def3: :: VECTSP_1:def 3
for a, b, c being Element of IT holds (b + c) * a = (b * a) + (c * a);

end;

:: deftheorem Def2 defines right-distributive VECTSP_1:def 2 :
for IT being non empty doubleLoopStr holds
( IT is right-distributive iff for a, b, c being Element of IT holds a * (b + c) = (a * b) + (a * c) );

:: deftheorem Def3 defines left-distributive VECTSP_1:def 3 :
for IT being non empty doubleLoopStr holds
( IT is left-distributive iff for a, b, c being Element of IT holds (b + c) * a = (b * a) + (c * a) );

definition

let IT be non empty multLoopStr ;

attr IT is right_unital means :Def4: :: VECTSP_1:def 4
for x being Element of IT holds x * (1. IT) = x;

end;

:: deftheorem Def4 defines right_unital VECTSP_1:def 4 :
for IT being non empty multLoopStr holds
( IT is right_unital iff for x being Element of IT holds x * (1. IT) = x );

definition

func F_Real -> strict doubleLoopStr equals :: VECTSP_1:def 5
doubleLoopStr(# REAL,addreal,multreal,(In (1,REAL)),(In (0,REAL)) #);

correctness
coherence
doubleLoopStr(# REAL,addreal,multreal,(In (1,REAL)),(In (0,REAL)) #) is strict doubleLoopStr ;
;

end;

:: deftheorem defines F_Real VECTSP_1:def 5 :
F_Real = doubleLoopStr(# REAL,addreal,multreal,(In (1,REAL)),(In (0,REAL)) #);

registration

cluster F_Real -> non empty strict ;

coherence
not F_Real is empty ;

end;

registration

cluster the carrier of F_Real -> real-membered ;

coherence
the carrier of F_Real is real-membered ;

end;

registration

let a, b be Element of F_Real;
let x, y be Real;

identify a + b with x + y when a = x, b = y;

compatibility
( a = x & b = y implies a + b = x + y ) by BINOP_2:def 9;

end;

registration

let a, b be Element of F_Real;
let x, y be Real;

identify a * b with x * y when a = x, b = y;

compatibility
( a = x & b = y implies a * b = x * y ) by BINOP_2:def 11;

end;

definition

let IT be non empty multLoopStr ;

attr IT is well-unital means :Def6: :: VECTSP_1:def 6
for x being Element of IT holds
( x * (1. IT) = x & (1. IT) * x = x );

end;

:: deftheorem Def6 defines well-unital VECTSP_1:def 6 :
for IT being non empty multLoopStr holds
( IT is well-unital iff for x being Element of IT holds
( x * (1. IT) = x & (1. IT) * x = x ) );

Lm1: for L being non empty multLoopStr st L is well-unital holds
1. L = 1_ L

proof end;

registration

cluster F_Real -> strict well-unital ;

coherence
F_Real is well-unital ;

end;

registration

cluster non empty well-unital for multLoopStr_0 ;

existence
ex b₁ being non empty multLoopStr_0 st b₁ is well-unital

proof end;

end;

registration

let L be non empty well-unital multLoopStr_0 ;
let x be Element of L;

reduce x * (1. L) to x;

reducibility
x * (1. L) = x by Def6;

reduce (1. L) * x to x;

reducibility
(1. L) * x = x by Def6;

end;

definition

let IT be non empty doubleLoopStr ;

attr IT is distributive means :Def7: :: VECTSP_1:def 7
for x, y, z being Element of IT holds
( x * (y + z) = (x * y) + (x * z) & (y + z) * x = (y * x) + (z * x) );

end;

:: deftheorem Def7 defines distributive VECTSP_1:def 7 :
for IT being non empty doubleLoopStr holds
( IT is distributive iff for x, y, z being Element of IT holds
( x * (y + z) = (x * y) + (x * z) & (y + z) * x = (y * x) + (z * x) ) );

definition

let IT be non empty multLoopStr ;

attr IT is left_unital means :Def8: :: VECTSP_1:def 8
for x being Element of IT holds (1. IT) * x = x;

end;

:: deftheorem Def8 defines left_unital VECTSP_1:def 8 :
for IT being non empty multLoopStr holds
( IT is left_unital iff for x being Element of IT holds (1. IT) * x = x );

definition

let IT be non empty multLoopStr_0 ;

redefine attr IT is almost_left_invertible means :Def9: :: VECTSP_1:def 9
for x being Element of IT st x <> 0. IT holds
ex y being Element of IT st y * x = 1. IT;

compatibility
( IT is almost_left_invertible iff for x being Element of IT st x <> 0. IT holds
ex y being Element of IT st y * x = 1. IT )

proof end;

end;

:: deftheorem Def9 defines almost_left_invertible VECTSP_1:def 9 :
for IT being non empty multLoopStr_0 holds
( IT is almost_left_invertible iff for x being Element of IT st x <> 0. IT holds
ex y being Element of IT st y * x = 1. IT );

set FR = F_Real ;

reconsider jj = 1 as Real ;

registration

cluster F_Real -> strict unital ;

coherence
F_Real is unital

proof end;

end;

registration

cluster F_Real -> non degenerated right_complementable almost_left_invertible strict Abelian add-associative right_zeroed associative commutative right_unital distributive left_unital ;

coherence
( F_Real is add-associative & F_Real is right_zeroed & F_Real is right_complementable & F_Real is Abelian & F_Real is commutative & F_Real is associative & F_Real is left_unital & F_Real is right_unital & F_Real is distributive & F_Real is almost_left_invertible & not F_Real is degenerated )

proof end;

end;

registration

let a be Element of F_Real;
let x be Real;

identify - a with - x when a = x;

compatibility
( a = x implies - a = - x )

proof end;

end;

registration

let a, b be Element of F_Real;
let x, y be Real;

identify a - b with x - y when a = x, b = y;

compatibility
( a = x & b = y implies a - b = x - y ) ;

end;

registration

cluster non empty distributive -> non empty right-distributive left-distributive for doubleLoopStr ;

coherence
for b₁ being non empty doubleLoopStr st b₁ is distributive holds
( b₁ is left-distributive & b₁ is right-distributive ) ;

cluster non empty right-distributive left-distributive -> non empty distributive for doubleLoopStr ;

coherence
for b₁ being non empty doubleLoopStr st b₁ is left-distributive & b₁ is right-distributive holds
b₁ is distributive ;

end;

registration

cluster non empty well-unital -> non empty right_unital left_unital for multLoopStr ;

coherence
for b₁ being non empty multLoopStr st b₁ is well-unital holds
( b₁ is left_unital & b₁ is right_unital ) ;

cluster non empty right_unital left_unital -> non empty unital for multLoopStr ;

coherence
for b₁ being non empty multLoopStr st b₁ is left_unital & b₁ is right_unital holds
b₁ is unital ;

end;

registration

cluster non empty associative commutative for multMagma ;

existence
ex b₁ being non empty multMagma st
( b₁ is commutative & b₁ is associative )

proof end;

end;

registration

cluster non empty unital associative commutative for multLoopStr ;

existence
ex b₁ being non empty multLoopStr st
( b₁ is commutative & b₁ is associative & b₁ is unital )

proof end;

end;

registration

cluster non empty non degenerated right_complementable almost_left_invertible strict Abelian add-associative right_zeroed associative commutative right_unital well-unital distributive left_unital for doubleLoopStr ;

existence
ex b₁ being non empty doubleLoopStr st
( b₁ is add-associative & b₁ is right_zeroed & b₁ is right_complementable & b₁ is Abelian & b₁ is commutative & b₁ is associative & b₁ is left_unital & b₁ is right_unital & b₁ is distributive & b₁ is almost_left_invertible & not b₁ is degenerated & b₁ is well-unital & b₁ is strict )

proof end;

end;

definition

mode Ring is non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr ;

end;

definition

mode Skew-Field is non degenerated almost_left_invertible Ring;

end;

definition

mode Field is commutative Skew-Field;

end;

:: 6. AXIOMS OF FIELD

registration

cluster non empty well-unital -> non empty unital for multLoopStr ;

coherence
for b₁ being non empty multLoopStr st b₁ is well-unital holds
b₁ is unital ;

end;

theorem :: VECTSP_1:1

canceled;

theorem :: VECTSP_1:2

canceled;

theorem :: VECTSP_1:3

canceled;

theorem :: VECTSP_1:4

canceled;

::$CT 4

theorem :: VECTSP_1:5

for F being non empty almost_left_invertible associative commutative well-unital distributive doubleLoopStr
for x, y, z being Element of F st x <> 0. F & x * y = x * z holds
y = z

proof end;

notation

let F be non empty almost_left_invertible associative commutative well-unital doubleLoopStr ;
let x be Element of F;
synonym x " for / x;

end;

definition

let F be non empty almost_left_invertible associative commutative well-unital doubleLoopStr ;
let x be Element of F;
assume A1: x <> 0. F ;

redefine func / x means :Def10: :: VECTSP_1:def 10
it * x = 1. F;

compatibility
for b₁ being Element of the carrier of F holds
( b₁ = x " iff b₁ * x = 1. F )

proof end;

end;

:: deftheorem Def10 defines " VECTSP_1:def 10 :
for F being non empty almost_left_invertible associative commutative well-unital doubleLoopStr
for x being Element of F st x <> 0. F holds
for b₃ being Element of the carrier of F holds
( b₃ = x " iff b₃ * x = 1. F );

registration

let a be non zero Element of F_Real;
let x be Real;

identify a " with x " when a = x;

compatibility
( a = x implies a " = x " )

proof end;

end;

registration

let a, b be non zero Element of F_Real;
let x, y be Real;

identify a / b with x / y when a = x, b = y;

compatibility
( a = x & b = y implies a / b = x / y ) ;

end;

canceled;

:: definition
:: let F be associative commutative well-unital distributive
:: almost_left_invertible non empty doubleLoopStr, x,y be Element of F;
:: func x/y ->Element of F equals
:: x*y";
:: coherence;
:: end;
::$CD

theorem Th2: :: VECTSP_1:6

for F being non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr
for x being Element of F holds x * (0. F) = 0. F

proof end;

registration

let F be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr ;
let x be Element of F;
let y be zero Element of F;

reduce x * y to y;

reducibility
x * y = y

proof end;

end;

theorem Th3: :: VECTSP_1:7

for F being non empty right_complementable add-associative right_zeroed left-distributive doubleLoopStr
for x being Element of F holds (0. F) * x = 0. F

proof end;

registration

let F be non empty right_complementable add-associative right_zeroed left-distributive doubleLoopStr ;
let x be zero Element of F;
let y be Element of F;

reduce x * y to x;

reducibility
x * y = x

proof end;

end;

theorem Th4: :: VECTSP_1:8

for F being non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr
for x, y being Element of F holds x * (- y) = - (x * y)

proof end;

theorem Th5: :: VECTSP_1:9

for F being non empty right_complementable add-associative right_zeroed left-distributive doubleLoopStr
for x, y being Element of F holds (- x) * y = - (x * y)

proof end;

theorem Th6: :: VECTSP_1:10

for F being non empty right_complementable add-associative right_zeroed distributive doubleLoopStr
for x, y being Element of F holds (- x) * (- y) = x * y

proof end;

theorem :: VECTSP_1:11

for F being non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr
for x, y, z being Element of F holds x * (y - z) = (x * y) - (x * z)

proof end;

theorem Th8: :: VECTSP_1:12

for F being non empty right_complementable almost_left_invertible add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for x, y being Element of F holds
( x * y = 0. F iff ( x = 0. F or y = 0. F ) )

proof end;

theorem :: VECTSP_1:13

for K being non empty right_complementable add-associative right_zeroed left-distributive doubleLoopStr
for a, b, c being Element of K holds (a - b) * c = (a * c) - (b * c)

proof end;

:: 8. VECTOR SPACE STRUCTURE

definition

let F be 1-sorted ;
attr c₂ is strict ;
struct ModuleStr over F -> addLoopStr ;
aggr ModuleStr(# carrier, addF, ZeroF, lmult #) -> ModuleStr over F;
sel lmult c₂ -> Function of [: the carrier of F, the carrier of c₂:], the carrier of c₂;

end;

registration

let F be 1-sorted ;

cluster non empty strict for ModuleStr over F;

existence
ex b₁ being ModuleStr over F st
( not b₁ is empty & b₁ is strict )

proof end;

end;

registration

let F be 1-sorted ;
let A be non empty set ;
let a be BinOp of A;
let Z be Element of A;
let l be Function of [: the carrier of F,A:],A;

cluster ModuleStr(# A,a,Z,l #) -> non empty ;

coherence
not ModuleStr(# A,a,Z,l #) is empty ;

end;

definition

let F be 1-sorted ;
mode Scalar of F is Element of F;

end;

definition

let F be 1-sorted ;
let VS be ModuleStr over F;
mode Scalar of VS is Scalar of F;
mode Vector of VS is Element of VS;

end;

definition

let F be non empty 1-sorted ;
let V be non empty ModuleStr over F;
let x be Element of F;
let v be Element of V;

func x * v -> Element of V equals :: VECTSP_1:def 11
the lmult of V . (x,v);

coherence
the lmult of V . (x,v) is Element of V ;

end;

:: deftheorem defines * VECTSP_1:def 11 :
for F being non empty 1-sorted
for V being non empty ModuleStr over F
for x being Element of F
for v being Element of V holds x * v = the lmult of V . (x,v);

definition

let F be non empty addLoopStr ;

func comp F -> UnOp of the carrier of F means :: VECTSP_1:def 12
for x being Element of F holds it . x = - x;

existence
ex b₁ being UnOp of the carrier of F st
for x being Element of F holds b₁ . x = - x

proof end;

uniqueness
for b₁, b₂ being UnOp of the carrier of F st ( for x being Element of F holds b₁ . x = - x ) & ( for x being Element of F holds b₂ . x = - x ) holds
b₁ = b₂

proof end;

end;

:: deftheorem defines comp VECTSP_1:def 12 :
for F being non empty addLoopStr
for b₂ being UnOp of the carrier of F holds
( b₂ = comp F iff for x being Element of F holds b₂ . x = - x );

Lm2: now :: thesis: for F being non empty right_complementable Abelian add-associative right_zeroed associative distributive left_unital doubleLoopStr
for MLT being Function of [: the carrier of F, the carrier of F:], the carrier of F holds
( ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is Abelian & ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is add-associative & ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is right_zeroed & ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is right_complementable )

let F be non empty right_complementable Abelian add-associative right_zeroed associative distributive left_unital doubleLoopStr ; :: thesis: for MLT being Function of [: the carrier of F, the carrier of F:], the carrier of F holds
( ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is Abelian & ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is add-associative & ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is right_zeroed & ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is right_complementable )
let MLT be Function of [: the carrier of F, the carrier of F:], the carrier of F; :: thesis: ( ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is Abelian & ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is add-associative & ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is right_zeroed & ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is right_complementable )
set GF = ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #);
thus ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is Abelian :: thesis: ( ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is add-associative & ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is right_zeroed & ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is right_complementable )

proof

let x, y be Element of ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #); :: according to RLVECT_1:def 2 :: thesis: x + y = y + x
reconsider x9 = x, y9 = y as Element of F ;
thus x + y = y9 + x9 by RLVECT_1:2
.= y + x ; :: thesis: verum

end;

thus ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is add-associative :: thesis: ( ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is right_zeroed & ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is right_complementable )

proof

let x, y, z be Element of ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #); :: according to RLVECT_1:def 3 :: thesis: (x + y) + z = x + (y + z)
reconsider x9 = x, y9 = y, z9 = z as Element of F ;
thus (x + y) + z = (x9 + y9) + z9
.= x9 + (y9 + z9) by RLVECT_1:def 3
.= x + (y + z) ; :: thesis: verum

end;

thus ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is right_zeroed :: thesis: ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is right_complementable

proof

let x be Element of ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #); :: according to RLVECT_1:def 4 :: thesis: x + (0. ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #)) = x
reconsider x9 = x as Element of F ;
thus x + (0. ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #)) = x9 + (0. F)
.= x by RLVECT_1:4 ; :: thesis: verum

end;

thus ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) is right_complementable :: thesis: verum

proof

let x be Element of ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #); :: according to ALGSTR_0:def 16 :: thesis: x is right_complementable
reconsider x9 = x as Element of F ;
consider t being Element of F such that
A1: x9 + t = 0. F by ALGSTR_0:def 11;
reconsider t9 = t as Element of ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) ;
take t9 ; :: according to ALGSTR_0:def 11 :: thesis: x + t9 = 0. ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #)
thus x + t9 = 0. ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) by A1; :: thesis: verum

end;

end;

Lm3: now :: thesis: for F being non empty right_complementable add-associative right_zeroed associative well-unital distributive doubleLoopStr
for MLT being Function of [: the carrier of F, the carrier of F:], the carrier of F st MLT = the multF of F holds
for x, y being Element of F
for v, w being Element of ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) holds
( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1. F) * v = v )

let F be non empty right_complementable add-associative right_zeroed associative well-unital distributive doubleLoopStr ; :: thesis: for MLT being Function of [: the carrier of F, the carrier of F:], the carrier of F st MLT = the multF of F holds
for x, y being Element of F
for v, w being Element of ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) holds
( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1. F) * v = v )
let MLT be Function of [: the carrier of F, the carrier of F:], the carrier of F; :: thesis: ( MLT = the multF of F implies for x, y being Element of F
for v, w being Element of ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) holds
( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1. F) * v = v ) )
assume A1: MLT = the multF of F ; :: thesis: for x, y being Element of F
for v, w being Element of ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) holds
( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1. F) * v = v )
let x, y be Element of F; :: thesis: for v, w being Element of ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #) holds
( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1. F) * v = v )
set LS = ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #);
let v, w be Element of ModuleStr(# the carrier of F, the addF of F,(0. F),MLT #); :: thesis: ( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1. F) * v = v )
reconsider v9 = v, w9 = w as Element of F ;
thus x * (v + w) = x * (v9 + w9) by A1
.= (x * v9) + (x * w9) by Def7
.= (x * v) + (x * w) by A1 ; :: thesis: ( (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1. F) * v = v )
thus (x + y) * v = (x + y) * v9 by A1
.= (x * v9) + (y * v9) by Def7
.= (x * v) + (y * v) by A1 ; :: thesis: ( (x * y) * v = x * (y * v) & (1. F) * v = v )
thus (x * y) * v = (x * y) * v9 by A1
.= x * (y * v9) by GROUP_1:def 3
.= x * (y * v) by A1 ; :: thesis: (1. F) * v = v
thus (1. F) * v = (1. F) * v9 by A1
.= v ; :: thesis: verum

end;

definition

let F be non empty doubleLoopStr ;
let IT be non empty ModuleStr over F;

attr IT is vector-distributive means :Def13: :: VECTSP_1:def 13
for x being Element of F
for v, w being Element of IT holds x * (v + w) = (x * v) + (x * w);

attr IT is scalar-distributive means :Def14: :: VECTSP_1:def 14
for x, y being Element of F
for v being Element of IT holds (x + y) * v = (x * v) + (y * v);

attr IT is scalar-associative means :Def15: :: VECTSP_1:def 15
for x, y being Element of F
for v being Element of IT holds (x * y) * v = x * (y * v);

attr IT is scalar-unital means :Def16: :: VECTSP_1:def 16
for v being Element of IT holds (1. F) * v = v;

end;

:: deftheorem Def13 defines vector-distributive VECTSP_1:def 13 :
for F being non empty doubleLoopStr
for IT being non empty ModuleStr over F holds
( IT is vector-distributive iff for x being Element of F
for v, w being Element of IT holds x * (v + w) = (x * v) + (x * w) );

:: deftheorem Def14 defines scalar-distributive VECTSP_1:def 14 :
for F being non empty doubleLoopStr
for IT being non empty ModuleStr over F holds
( IT is scalar-distributive iff for x, y being Element of F
for v being Element of IT holds (x + y) * v = (x * v) + (y * v) );

:: deftheorem Def15 defines scalar-associative VECTSP_1:def 15 :
for F being non empty doubleLoopStr
for IT being non empty ModuleStr over F holds
( IT is scalar-associative iff for x, y being Element of F
for v being Element of IT holds (x * y) * v = x * (y * v) );

:: deftheorem Def16 defines scalar-unital VECTSP_1:def 16 :
for F being non empty doubleLoopStr
for IT being non empty ModuleStr over F holds
( IT is scalar-unital iff for v being Element of IT holds (1. F) * v = v );

registration

let F be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr ;

cluster non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital for ModuleStr over F;

existence
ex b₁ being non empty ModuleStr over F st
( b₁ is scalar-distributive & b₁ is vector-distributive & b₁ is scalar-associative & b₁ is scalar-unital & b₁ is add-associative & b₁ is right_zeroed & b₁ is right_complementable & b₁ is Abelian & b₁ is strict )

proof end;

end;

definition

let F be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr ;
mode VectSp of F is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over F;

end;

theorem Th10: :: VECTSP_1:14

for F being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for x being Element of F
for V being non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over F
for v being Element of V holds
( (0. F) * v = 0. V & (- (1. F)) * v = - v & x * (0. V) = 0. V )

proof end;

theorem :: VECTSP_1:15

for F being Field
for x being Element of F
for V being VectSp of F
for v being Element of V holds
( x * v = 0. V iff ( x = 0. F or v = 0. V ) )

proof end;

:: 13. APPENDIX

theorem :: VECTSP_1:16

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

proof end;

Lm4: for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v, w being Element of V holds - (w + (- v)) = v - w

proof end;

Lm5: 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 :: VECTSP_1:17

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for u, v, w being Element of V holds
( - (v + w) = (- w) - v & - (w + (- v)) = v - w & - (v - w) = w + (- v) & - ((- v) - w) = w + v & u - (w + v) = (u - v) - w ) by Lm4, Lm5, RLVECT_1:27, RLVECT_1:30, RLVECT_1:33;

theorem :: VECTSP_1:18

for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v being Element of V holds
( (0. V) - v = - v & v - (0. V) = v ) by RLVECT_1:13, RLVECT_1:14;

theorem Th15: :: VECTSP_1:19

for F being non empty right_complementable add-associative right_zeroed addLoopStr
for x, y being Element of F holds
( ( x + (- y) = 0. F implies x = y ) & ( x = y implies x + (- y) = 0. F ) & ( x - y = 0. F implies x = y ) & ( x = y implies x - y = 0. F ) )

proof end;

theorem :: VECTSP_1:20

for F being Field
for x being Element of F
for V being VectSp of F
for v being Element of V st x <> 0. F holds
(x ") * (x * v) = v

proof end;

theorem Th17: :: VECTSP_1:21

for F being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for V being non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over F
for x being Element of F
for v, w being Element of V holds
( - (x * v) = (- x) * v & w - (x * v) = w + ((- x) * v) )

proof end;

registration

cluster non empty commutative left_unital -> non empty right_unital for multLoopStr ;

coherence
for b₁ being non empty multLoopStr st b₁ is commutative & b₁ is left_unital holds
b₁ is right_unital

proof end;

end;

theorem Th18: :: VECTSP_1:22

for F being non empty right_complementable Abelian add-associative right_zeroed associative right_unital well-unital distributive doubleLoopStr
for V being non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over F
for x being Element of F
for v being Element of V holds x * (- v) = - (x * v)

proof end;

theorem :: VECTSP_1:23

for F being non empty right_complementable Abelian add-associative right_zeroed associative right_unital well-unital distributive doubleLoopStr
for V being non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over F
for x being Element of F
for v, w being Element of V holds x * (v - w) = (x * v) - (x * w)

proof end;

theorem Th20: :: VECTSP_1:24

for F being non empty non degenerated right_complementable almost_left_invertible add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for x being Element of F st x <> 0. F holds
(x ") " = x

proof end;

registration

let F be non degenerated right_complementable almost_left_invertible add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr ;
let x be non zero Element of F;

reduce (x ") " to x;

reducibility
(x ") " = x

proof end;

end;

theorem :: VECTSP_1:25

for F being Field
for x being Element of F st x <> 0. F holds
( x " <> 0. F & - (x ") <> 0. F )

proof end;

theorem Th22: :: VECTSP_1:26

(1. F_Real) + (1. F_Real) <> 0. F_Real ;

definition

let IT be non empty addLoopStr ;

attr IT is Fanoian means :: VECTSP_1:def 17
for a being Element of IT st a + a = 0. IT holds
a = 0. IT;

end;

:: deftheorem defines Fanoian VECTSP_1:def 17 :
for IT being non empty addLoopStr holds
( IT is Fanoian iff for a being Element of IT st a + a = 0. IT holds
a = 0. IT );

registration

cluster non empty Fanoian for addLoopStr ;

existence
ex b₁ being non empty addLoopStr st b₁ is Fanoian

proof end;

end;

definition

let F be non empty non degenerated right_complementable almost_left_invertible add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr ;

redefine attr F is Fanoian means :: VECTSP_1:def 18
(1. F) + (1. F) <> 0. F;

compatibility
( F is Fanoian iff (1. F) + (1. F) <> 0. F )

proof end;

end;

:: deftheorem defines Fanoian VECTSP_1:def 18 :
for F being non empty non degenerated right_complementable almost_left_invertible add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr holds
( F is Fanoian iff (1. F) + (1. F) <> 0. F );

registration

cluster non empty non degenerated non trivial right_complementable almost_left_invertible strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Fanoian for doubleLoopStr ;

existence
ex b₁ being Field st
( b₁ is strict & b₁ is Fanoian )

proof end;

end;

theorem :: VECTSP_1:27

for F being non empty right_complementable add-associative right_zeroed addLoopStr
for a, b being Element of F st a - b = 0. F holds
a = b by Th15;

theorem Th24: :: VECTSP_1:28

for F being non empty right_complementable add-associative right_zeroed addLoopStr
for a being Element of F st - a = 0. F holds
a = 0. F

proof end;

theorem :: VECTSP_1:29

for F being non empty right_complementable add-associative right_zeroed addLoopStr
for a, b being Element of F st a - b = 0. F holds
b - a = 0. F

proof end;

theorem :: VECTSP_1:30

for F being Field
for a, b, c being Element of F holds
( ( a <> 0. F & (a * c) - b = 0. F implies c = b * (a ") ) & ( a <> 0. F & b - (c * a) = 0. F implies c = b * (a ") ) )

proof end;

theorem :: VECTSP_1:31

for F being non empty right_complementable add-associative right_zeroed addLoopStr
for a, b being Element of F holds a + b = - ((- b) + (- a))

proof end;

theorem :: VECTSP_1:32

for F being non empty right_complementable add-associative right_zeroed addLoopStr
for a, b, c being Element of F holds (b + a) - (c + a) = b - c

proof end;

theorem :: VECTSP_1:33

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

proof end;

theorem :: VECTSP_1:34

for G being non empty Abelian add-associative addLoopStr
for u, v, w being Element of G holds (u - v) - w = (u - w) - v

proof end;

theorem :: VECTSP_1:35

for B being AbGroup holds multMagma(# the carrier of B, the addF of B #) is commutative Group

proof end;

:: from COMPTRIG, 2006.08.12, A.T.

theorem :: VECTSP_1:36

for L being non empty right_complementable add-associative right_zeroed unital right-distributive doubleLoopStr
for n being Element of NAT st n > 0 holds
(power L) . ((0. L),n) = 0. L

proof end;

:: 2007.02.14, A.T.

registration

cluster non empty well-unital for multLoopStr ;

existence
ex b₁ being non empty multLoopStr st b₁ is well-unital

proof end;

end;

registration

let S be non empty well-unital multLoopStr ;

identify 1_ S with 1. S;

compatibility
1_ S = 1. S by Lm1;

end;

theorem :: VECTSP_1:37

for L being non empty multLoopStr st L is well-unital holds
1. L = 1_ L by Lm1;

definition

let G, H be non empty addMagma ;
let f be Function of G,H;

attr f is additive means :: VECTSP_1:def 19
for x, y being Element of G holds f . (x + y) = (f . x) + (f . y);

end;

:: deftheorem defines additive VECTSP_1:def 19 :
for G, H being non empty addMagma
for f being Function of G,H holds
( f is additive iff for x, y being Element of G holds f . (x + y) = (f . x) + (f . y) );

registration

let L be non empty right_unital multLoopStr ;
let x be Element of L;

reduce x * (1. L) to x;

reducibility
x * (1. L) = x by Def4;

end;

registration

let L be non empty left_unital multLoopStr ;
let x be Element of L;

reduce (1. L) * x to x;

reducibility
(1. L) * x = x by Def8;

end;