:: Construction of Rings and Left-, Right-, and Bi-Modules over a Ring
:: by Micha{\l} Muzalewski
::
:: Received June 20, 1990
:: Copyright (c) 1990-2011 Association of Mizar Users

begin

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

proof end;

registration

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

proof end;

end;

definition

let IT be non empty multLoopStr_0 ;
canceled;
canceled;
canceled;
canceled;
attr IT is domRing-like means :Def5: :: VECTSP_2:def 5
for x, y being Element of IT holds
( not x * y = 0. IT or x = 0. IT or y = 0. IT );

end;

:: deftheorem VECTSP_2:def 1 :
canceled;

:: deftheorem VECTSP_2:def 2 :
canceled;

:: deftheorem VECTSP_2:def 3 :
canceled;

:: deftheorem VECTSP_2:def 4 :
canceled;

:: deftheorem Def5 defines domRing-like VECTSP_2:def 5 :
for IT being non empty multLoopStr_0 holds
( IT is domRing-like iff for x, y being Element of IT holds
( not x * y = 0. IT or x = 0. IT or y = 0. IT ) );

registration

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

proof end;

end;

definition

mode comRing is commutative Ring;

end;

definition

mode domRing is non degenerated domRing-like comRing;

end;

theorem :: VECTSP_2:1

canceled;

theorem :: VECTSP_2:2

canceled;

theorem :: VECTSP_2:3

canceled;

theorem :: VECTSP_2:4

canceled;

theorem :: VECTSP_2:5

canceled;

theorem :: VECTSP_2:6

canceled;

theorem :: VECTSP_2:7

canceled;

theorem :: VECTSP_2:8

canceled;

theorem :: VECTSP_2:9

canceled;

theorem :: VECTSP_2:10

canceled;

theorem :: VECTSP_2:11

canceled;

theorem :: VECTSP_2:12

canceled;

theorem :: VECTSP_2:13

for F being Field holds F is domRing

proof end;

definition

mode Skew-Field is non degenerated almost_left_invertible Ring;

end;

theorem :: VECTSP_2:14

canceled;

theorem :: VECTSP_2:15

canceled;

theorem :: VECTSP_2:16

canceled;

registration

cluster non empty commutative left_unital -> non empty well-unital multLoopStr ;
coherence
for b₁ being non empty multLoopStr st b₁ is commutative & b₁ is left_unital holds
b₁ is well-unital

proof end;

cluster non empty commutative right_unital -> non empty well-unital multLoopStr ;
coherence
for b₁ being non empty multLoopStr st b₁ is commutative & b₁ is right_unital holds
b₁ is well-unital

proof end;

end;

Lm2: for R being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for x, y, z being Scalar of R st x + y = z holds
x = z - y

proof end;

Lm3: for R being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for x, z, y being Scalar of R st x = z - y holds
x + y = z

proof end;

theorem :: VECTSP_2:17

canceled;

theorem :: VECTSP_2:18

canceled;

theorem :: VECTSP_2:19

canceled;

theorem :: VECTSP_2:20

canceled;

theorem :: VECTSP_2:21

canceled;

theorem :: VECTSP_2:22

for R being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for x, y, z being Scalar of R holds
( ( x + y = z implies x = z - y ) & ( x = z - y implies x + y = z ) & ( x + y = z implies y = z - x ) & ( y = z - x implies x + y = z ) ) by Lm2, Lm3;

theorem :: VECTSP_2:23

canceled;

theorem :: VECTSP_2:24

canceled;

theorem :: VECTSP_2:25

canceled;

theorem :: VECTSP_2:26

canceled;

theorem :: VECTSP_2:27

canceled;

theorem :: VECTSP_2:28

canceled;

theorem :: VECTSP_2:29

canceled;

theorem :: VECTSP_2:30

canceled;

theorem :: VECTSP_2:31

canceled;

theorem :: VECTSP_2:32

canceled;

theorem :: VECTSP_2:33

canceled;

theorem Th34: :: VECTSP_2:34

for R being non empty right_complementable add-associative right_zeroed addLoopStr
for x being Element of R holds
( x = 0. R iff - x = 0. R )

proof end;

theorem :: VECTSP_2:35

canceled;

theorem :: VECTSP_2:36

canceled;

theorem :: VECTSP_2:37

canceled;

theorem :: VECTSP_2:38

for R being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for x, y being Element of R ex z being Element of R st
( x = y + z & x = z + y )

proof end;

theorem :: VECTSP_2:39

for F being non empty non degenerated right_complementable distributive add-associative right_zeroed doubleLoopStr
for x, y being Element of F st x * y = 1. F holds
( x <> 0. F & y <> 0. F ) by VECTSP_1:36, VECTSP_1:39;

theorem Th40: :: VECTSP_2:40

for SF being non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive add-associative right_zeroed doubleLoopStr
for x being Element of SF st x <> 0. SF holds
ex y being Element of SF st x * y = 1. SF

proof end;

theorem Th41: :: VECTSP_2:41

for SF being non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive add-associative right_zeroed doubleLoopStr
for x, y being Element of SF st y * x = 1. SF holds
x * y = 1. SF

proof end;

theorem Th42: :: VECTSP_2:42

for SF being non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for x, y, z being Element of SF st x * y = x * z & x <> 0. SF holds
y = z

proof end;

definition

let SF be non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive add-associative right_zeroed doubleLoopStr ;
let x be Element of SF;
assume A1: x <> 0. SF ;
canceled;
redefine func x " means :Def7: :: VECTSP_2:def 7
it * x = 1. SF;
compatibility
for b₁ being Element of the carrier of SF holds
( b₁ = x " iff b₁ * x = 1. SF )

proof end;

end;

:: deftheorem VECTSP_2:def 6 :
canceled;

:: deftheorem Def7 defines " VECTSP_2:def 7 :
for SF being non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive add-associative right_zeroed doubleLoopStr
for x being Element of SF st x <> 0. SF holds
for b₃ being Element of the carrier of SF holds
( b₃ = x " iff b₃ * x = 1. SF );

definition

let SF be Skew-Field;
let x, y be Scalar of SF;
func x / y -> Scalar of SF equals :: VECTSP_2:def 8
x * (y ");
correctness
coherence
x * (y ") is Scalar of SF;
;

end;

:: deftheorem defines / VECTSP_2:def 8 :
for SF being Skew-Field
for x, y being Scalar of SF holds x / y = x * (y ");

theorem Th43: :: VECTSP_2:43

for SF being Skew-Field
for x being Scalar of SF st x <> 0. SF holds
( x * (x ") = 1. SF & (x ") * x = 1. SF )

proof end;

theorem :: VECTSP_2:44

canceled;

theorem Th45: :: VECTSP_2:45

for SF being Skew-Field
for y, x being Scalar of SF st y * x = 1_ SF holds
( x = y " & y = x " )

proof end;

theorem Th46: :: VECTSP_2:46

for SF being Skew-Field
for x, y being Scalar of SF st x <> 0. SF & y <> 0. SF holds
(x ") * (y ") = (y * x) "

proof end;

theorem :: VECTSP_2:47

for SF being Skew-Field
for x, y being Scalar of SF holds
( not x * y = 0. SF or x = 0. SF or y = 0. SF )

proof end;

theorem Th48: :: VECTSP_2:48

for SF being Skew-Field
for x being Scalar of SF st x <> 0. SF holds
x " <> 0. SF

proof end;

theorem Th49: :: VECTSP_2:49

for SF being Skew-Field
for x being Scalar of SF st x <> 0. SF holds
(x ") " = x

proof end;

theorem Th50: :: VECTSP_2:50

for SF being Skew-Field
for x being Scalar of SF st x <> 0. SF holds
( (1_ SF) / x = x " & (1_ SF) / (x ") = x )

proof end;

theorem :: VECTSP_2:51

for SF being Skew-Field
for x being Scalar of SF st x <> 0. SF holds
( x * ((1_ SF) / x) = 1_ SF & ((1_ SF) / x) * x = 1_ SF )

proof end;

theorem :: VECTSP_2:52

for SF being Skew-Field
for x being Scalar of SF st x <> 0. SF holds
x / x = 1_ SF by Th43;

theorem Th53: :: VECTSP_2:53

for SF being Skew-Field
for y, z, x being Scalar of SF st y <> 0. SF & z <> 0. SF holds
x / y = (x * z) / (y * z)

proof end;

theorem Th54: :: VECTSP_2:54

for SF being Skew-Field
for y, x being Scalar of SF st y <> 0. SF holds
( - (x / y) = (- x) / y & x / (- y) = - (x / y) )

proof end;

theorem :: VECTSP_2:55

for SF being Skew-Field
for z, x, y being Scalar of SF st z <> 0. SF holds
( (x / z) + (y / z) = (x + y) / z & (x / z) - (y / z) = (x - y) / z )

proof end;

theorem :: VECTSP_2:56

for SF being Skew-Field
for y, z, x being Scalar of SF st y <> 0. SF & z <> 0. SF holds
x / (y / z) = (x * z) / y

proof end;

theorem :: VECTSP_2:57

for SF being Skew-Field
for y, x being Scalar of SF st y <> 0. SF holds
(x / y) * y = x

proof end;

definition

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

end;

registration

let FS be 1-sorted ;
cluster non empty RightModStr of FS;
existence
not for b₁ being RightModStr of FS holds b₁ is empty

proof end;

end;

registration

let FS be 1-sorted ;
let A be non empty set ;
let a be BinOp of A;
let Z be Element of A;
let r be Function of [:A, the carrier of FS:],A;
cluster RightModStr(# A,a,Z,r #) -> non empty ;
coherence
not RightModStr(# A,a,Z,r #) is empty ;

end;

definition

let FS be non empty doubleLoopStr ;
let RMS be non empty RightModStr of FS;
mode Scalar of RMS is Element of FS;
mode Vector of RMS is Element of RMS;

end;

definition

let FS1, FS2 be 1-sorted ;
attr c₃ is strict ;
struct BiModStr of FS1,FS2 -> VectSpStr of FS1, RightModStr of FS2;
aggr BiModStr(# carrier, addF, ZeroF, lmult, rmult #) -> BiModStr of FS1,FS2;

end;

registration

let FS1, FS2 be 1-sorted ;
cluster non empty BiModStr of FS1,FS2;
existence
not for b₁ being BiModStr of FS1,FS2 holds b₁ is empty

proof end;

end;

registration

let FS1, FS2 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 FS1,A:],A;
let r be Function of [:A, the carrier of FS2:],A;
cluster BiModStr(# A,a,Z,l,r #) -> non empty ;
coherence
not BiModStr(# A,a,Z,l,r #) is empty ;

end;

definition

let R be non empty right_complementable Abelian add-associative right_zeroed addLoopStr ;
func AbGr R -> strict AbGroup equals :: VECTSP_2:def 9
addLoopStr(# the carrier of R, the addF of R,(0. R) #);
coherence
addLoopStr(# the carrier of R, the addF of R,(0. R) #) is strict AbGroup

proof end;

end;

:: deftheorem defines AbGr VECTSP_2:def 9 :
for R being non empty right_complementable Abelian add-associative right_zeroed addLoopStr holds AbGr R = addLoopStr(# the carrier of R, the addF of R,(0. R) #);

deffunc H₁( Ring) -> VectSpStr of $1 = VectSpStr(# the carrier of $1, the addF of $1,(0. $1), the multF of $1 #);

Lm4: for R being Ring holds
( H₁(R) is Abelian & H₁(R) is add-associative & H₁(R) is right_zeroed & H₁(R) is right_complementable )

proof end;

registration

let R be Ring;
cluster non empty right_complementable strict Abelian add-associative right_zeroed VectSpStr of R;
existence
ex b₁ being non empty VectSpStr of R st
( b₁ is Abelian & b₁ is add-associative & b₁ is right_zeroed & b₁ is right_complementable & b₁ is strict )

proof end;

end;

definition

let R be Ring;
canceled;
func LeftModule R -> non empty right_complementable strict Abelian add-associative right_zeroed VectSpStr of R equals :: VECTSP_2:def 11
VectSpStr(# the carrier of R, the addF of R,(0. R), the multF of R #);
coherence
VectSpStr(# the carrier of R, the addF of R,(0. R), the multF of R #) is non empty right_complementable strict Abelian add-associative right_zeroed VectSpStr of R by Lm4;

end;

:: deftheorem VECTSP_2:def 10 :
canceled;

:: deftheorem defines LeftModule VECTSP_2:def 11 :
for R being Ring holds LeftModule R = VectSpStr(# the carrier of R, the addF of R,(0. R), the multF of R #);

deffunc H₂( Ring) -> RightModStr of $1 = RightModStr(# the carrier of $1, the addF of $1,(0. $1), the multF of $1 #);

Lm5: for R being Ring holds
( H₂(R) is Abelian & H₂(R) is add-associative & H₂(R) is right_zeroed & H₂(R) is right_complementable )

proof end;

registration

let R be Ring;
cluster non empty right_complementable Abelian add-associative right_zeroed strict RightModStr of R;
existence
ex b₁ being non empty RightModStr of R st
( b₁ is Abelian & b₁ is add-associative & b₁ is right_zeroed & b₁ is right_complementable & b₁ is strict )

proof end;

end;

definition

let R be Ring;
canceled;
canceled;
func RightModule R -> non empty right_complementable Abelian add-associative right_zeroed strict RightModStr of R equals :: VECTSP_2:def 14
RightModStr(# the carrier of R, the addF of R,(0. R), the multF of R #);
coherence
RightModStr(# the carrier of R, the addF of R,(0. R), the multF of R #) is non empty right_complementable Abelian add-associative right_zeroed strict RightModStr of R by Lm5;

end;

:: deftheorem VECTSP_2:def 12 :
canceled;

:: deftheorem VECTSP_2:def 13 :
canceled;

:: deftheorem defines RightModule VECTSP_2:def 14 :
for R being Ring holds RightModule R = RightModStr(# the carrier of R, the addF of R,(0. R), the multF of R #);

definition

let R be non empty 1-sorted ;
let V be non empty RightModStr of R;
let x be Element of R;
let v be Element of V;
func v * x -> Element of V equals :: VECTSP_2:def 15
the rmult of V . (v,x);
coherence
the rmult of V . (v,x) is Element of V ;

end;

:: deftheorem defines * VECTSP_2:def 15 :
for R being non empty 1-sorted
for V being non empty RightModStr of R
for x being Element of R
for v being Element of V holds v * x = the rmult of V . (v,x);

deffunc H₃( Ring, Ring) -> BiModStr of $1,$2 = BiModStr(# 1,op2,op0,(pr2 ( the carrier of $1,1)),(pr1 (1, the carrier of $2)) #);

Lm6: for R1, R2 being Ring holds
( H₃(R1,R2) is Abelian & H₃(R1,R2) is add-associative & H₃(R1,R2) is right_zeroed & H₃(R1,R2) is right_complementable )

proof end;

registration

let R1, R2 be Ring;
cluster non empty right_complementable Abelian add-associative right_zeroed strict BiModStr of R1,R2;
existence
ex b₁ being non empty BiModStr of R1,R2 st
( b₁ is Abelian & b₁ is add-associative & b₁ is right_zeroed & b₁ is right_complementable & b₁ is strict )

proof end;

end;

definition

let R1, R2 be Ring;
canceled;
canceled;
canceled;
canceled;
canceled;
func BiModule (R1,R2) -> non empty right_complementable Abelian add-associative right_zeroed strict BiModStr of R1,R2 equals :: VECTSP_2:def 21
BiModStr(# 1,op2,op0,(pr2 ( the carrier of R1,1)),(pr1 (1, the carrier of R2)) #);
coherence
BiModStr(# 1,op2,op0,(pr2 ( the carrier of R1,1)),(pr1 (1, the carrier of R2)) #) is non empty right_complementable Abelian add-associative right_zeroed strict BiModStr of R1,R2 by Lm6;

end;

:: deftheorem VECTSP_2:def 16 :
canceled;

:: deftheorem VECTSP_2:def 17 :
canceled;

:: deftheorem VECTSP_2:def 18 :
canceled;

:: deftheorem VECTSP_2:def 19 :
canceled;

:: deftheorem VECTSP_2:def 20 :
canceled;

:: deftheorem defines BiModule VECTSP_2:def 21 :
for R1, R2 being Ring holds BiModule (R1,R2) = BiModStr(# 1,op2,op0,(pr2 ( the carrier of R1,1)),(pr1 (1, the carrier of R2)) #);

theorem :: VECTSP_2:58

canceled;

theorem :: VECTSP_2:59

canceled;

theorem :: VECTSP_2:60

canceled;

theorem :: VECTSP_2:61

canceled;

theorem :: VECTSP_2:62

canceled;

theorem :: VECTSP_2:63

canceled;

theorem :: VECTSP_2:64

canceled;

theorem :: VECTSP_2:65

canceled;

theorem :: VECTSP_2:66

canceled;

theorem :: VECTSP_2:67

canceled;

theorem :: VECTSP_2:68

canceled;

theorem :: VECTSP_2:69

canceled;

theorem :: VECTSP_2:70

canceled;

theorem Th71: :: VECTSP_2:71

for R being Ring
for x, y being Scalar of R
for v, w being Vector of (LeftModule R) holds
( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1. R) * v = v )

proof end;

registration

let R be Ring;
cluster non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of R;
existence
ex b₁ being non empty VectSpStr of R st
( b₁ is vector-distributive & b₁ is scalar-distributive & b₁ is scalar-associative & b₁ is scalar-unital & b₁ is Abelian & b₁ is add-associative & b₁ is right_zeroed & b₁ is right_complementable & b₁ is strict )

proof end;

end;

definition

let R be Ring;
mode LeftMod of R is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of R;

end;

Lm7: for R being Ring holds
( LeftModule R is vector-distributive & LeftModule R is scalar-distributive & LeftModule R is scalar-associative & LeftModule R is scalar-unital )

proof end;

registration

let R be Ring;
cluster LeftModule R -> non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ;
coherence
( LeftModule R is Abelian & LeftModule R is add-associative & LeftModule R is right_zeroed & LeftModule R is right_complementable & LeftModule R is strict & LeftModule R is vector-distributive & LeftModule R is scalar-distributive & LeftModule R is scalar-associative & LeftModule R is scalar-unital ) by Lm7;

end;

theorem :: VECTSP_2:72

canceled;

theorem :: VECTSP_2:73

canceled;

theorem :: VECTSP_2:74

canceled;

theorem :: VECTSP_2:75

canceled;

theorem :: VECTSP_2:76

canceled;

theorem Th77: :: VECTSP_2:77

for R being Ring
for x, y being Scalar of R
for v, w being Vector of (RightModule R) holds
( (v + w) * x = (v * x) + (w * x) & v * (x + y) = (v * x) + (v * y) & v * (y * x) = (v * y) * x & v * (1_ R) = v )

proof end;

definition

let R be non empty doubleLoopStr ;
let IT be non empty RightModStr of R;
canceled;
attr IT is RightMod-like means :Def23: :: VECTSP_2:def 23
for x, y being Scalar of R
for v, w being Vector of IT holds
( (v + w) * x = (v * x) + (w * x) & v * (x + y) = (v * x) + (v * y) & v * (y * x) = (v * y) * x & v * (1_ R) = v );

end;

:: deftheorem VECTSP_2:def 22 :
canceled;

:: deftheorem Def23 defines RightMod-like VECTSP_2:def 23 :
for R being non empty doubleLoopStr
for IT being non empty RightModStr of R holds
( IT is RightMod-like iff for x, y being Scalar of R
for v, w being Vector of IT holds
( (v + w) * x = (v * x) + (w * x) & v * (x + y) = (v * x) + (v * y) & v * (y * x) = (v * y) * x & v * (1_ R) = v ) );

registration

let R be Ring;
cluster non empty right_complementable Abelian add-associative right_zeroed strict RightMod-like RightModStr of R;
existence
ex b₁ being non empty RightModStr of R st
( b₁ is Abelian & b₁ is add-associative & b₁ is right_zeroed & b₁ is right_complementable & b₁ is RightMod-like & b₁ is strict )

proof end;

end;

definition

let R be Ring;
mode RightMod of R is non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr of R;

end;

Lm8: for R being Ring holds RightModule R is RightMod-like

proof end;

registration

let R be Ring;
cluster RightModule R -> non empty right_complementable Abelian add-associative right_zeroed strict RightMod-like ;
coherence
( RightModule R is Abelian & RightModule R is add-associative & RightModule R is right_zeroed & RightModule R is right_complementable & RightModule R is RightMod-like ) by Lm8;

end;

Lm9: for R1, R2 being Ring
for x, y being Scalar of R1
for p, q being Scalar of R2
for v, w being Vector of (BiModule (R1,R2)) holds
( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1_ R1) * v = v & (v + w) * p = (v * p) + (w * p) & v * (p + q) = (v * p) + (v * q) & v * (q * p) = (v * q) * p & v * (1_ R2) = v & x * (v * p) = (x * v) * p )

proof end;

definition

let R1, R2 be Ring;
let IT be non empty BiModStr of R1,R2;
attr IT is BiMod-like means :Def24: :: VECTSP_2:def 24
for x being Scalar of R1
for p being Scalar of R2
for v being Vector of IT holds x * (v * p) = (x * v) * p;

end;

:: deftheorem Def24 defines BiMod-like VECTSP_2:def 24 :
for R1, R2 being Ring
for IT being non empty BiModStr of R1,R2 holds
( IT is BiMod-like iff for x being Scalar of R1
for p being Scalar of R2
for v being Vector of IT holds x * (v * p) = (x * v) * p );

registration

let R1, R2 be Ring;
cluster non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed strict RightMod-like BiMod-like BiModStr of R1,R2;
existence
ex b₁ being non empty BiModStr of R1,R2 st
( b₁ is Abelian & b₁ is add-associative & b₁ is right_zeroed & b₁ is right_complementable & b₁ is RightMod-like & b₁ is vector-distributive & b₁ is scalar-distributive & b₁ is scalar-associative & b₁ is scalar-unital & b₁ is BiMod-like & b₁ is strict )

proof end;

end;

definition

let R1, R2 be Ring;
mode BiMod of R1,R2 is non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed RightMod-like BiMod-like BiModStr of R1,R2;

end;

theorem :: VECTSP_2:78

canceled;

theorem :: VECTSP_2:79

canceled;

theorem :: VECTSP_2:80

canceled;

theorem :: VECTSP_2:81

canceled;

theorem :: VECTSP_2:82

canceled;

theorem :: VECTSP_2:83

for R1, R2 being Ring
for V being non empty BiModStr of R1,R2 holds
( ( for x, y being Scalar of R1
for p, q being Scalar of R2
for v, w being Vector of V holds
( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1_ R1) * v = v & (v + w) * p = (v * p) + (w * p) & v * (p + q) = (v * p) + (v * q) & v * (q * p) = (v * q) * p & v * (1_ R2) = v & x * (v * p) = (x * v) * p ) ) iff ( V is RightMod-like & V is vector-distributive & V is scalar-distributive & V is scalar-associative & V is scalar-unital & V is BiMod-like ) ) by Def23, Def24, VECTSP_1:def 26, VECTSP_1:def 27, VECTSP_1:def 28, VECTSP_1:def 29;

theorem Th84: :: VECTSP_2:84

for R1, R2 being Ring holds BiModule (R1,R2) is BiMod of R1,R2

proof end;

registration

let R1, R2 be Ring;
cluster BiModule (R1,R2) -> non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed strict RightMod-like BiMod-like ;
coherence
( BiModule (R1,R2) is Abelian & BiModule (R1,R2) is add-associative & BiModule (R1,R2) is right_zeroed & BiModule (R1,R2) is right_complementable & BiModule (R1,R2) is RightMod-like & BiModule (R1,R2) is vector-distributive & BiModule (R1,R2) is scalar-distributive & BiModule (R1,R2) is scalar-associative & BiModule (R1,R2) is scalar-unital & BiModule (R1,R2) is BiMod-like ) by Th84;

end;

theorem :: VECTSP_2:85

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

begin

theorem :: VECTSP_2:86

for K being non empty right_complementable right-distributive right_unital add-associative right_zeroed doubleLoopStr
for a being Element of K holds a * (- (1. K)) = - a

proof end;

theorem :: VECTSP_2:87

for K being non empty right_complementable left-distributive left_unital add-associative right_zeroed doubleLoopStr
for a being Element of K holds (- (1. K)) * a = - a

proof end;

theorem :: VECTSP_2:88

for F being non degenerated almost_left_invertible Ring
for x being Scalar of F
for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital add-associative right_zeroed VectSpStr of F
for v being Vector of V holds
( x * v = 0. V iff ( x = 0. F or v = 0. V ) )

proof end;

theorem :: VECTSP_2:89

for F being non degenerated almost_left_invertible Ring
for x being Scalar of F
for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital add-associative right_zeroed VectSpStr of F
for v being Vector of V st x <> 0. F holds
(x ") * (x * v) = v

proof end;

theorem Th90: :: VECTSP_2:90

for R being non empty right_complementable associative right_unital well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for V being non empty right_complementable add-associative right_zeroed RightMod-like RightModStr of R
for x being Scalar of R
for v being Vector of V holds
( v * (0. R) = 0. V & v * (- (1_ R)) = - v & (0. V) * x = 0. V )

proof end;

theorem Th91: :: VECTSP_2:91

for R being non empty right_complementable associative right_unital well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for V being non empty right_complementable add-associative right_zeroed RightMod-like RightModStr of R
for x being Scalar of R
for v, w being Vector of V holds
( - (v * x) = v * (- x) & w - (v * x) = w + (v * (- x)) )

proof end;

theorem Th92: :: VECTSP_2:92

for R being non empty right_complementable associative right_unital well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for V being non empty right_complementable add-associative right_zeroed RightMod-like RightModStr of R
for x being Scalar of R
for v being Vector of V holds (- v) * x = - (v * x)

proof end;

theorem :: VECTSP_2:93

for R being non empty right_complementable associative right_unital well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for V being non empty right_complementable add-associative right_zeroed RightMod-like RightModStr of R
for x being Scalar of R
for v, w being Vector of V holds (v - w) * x = (v * x) - (w * x)

proof end;

theorem :: VECTSP_2:94

for F being non degenerated almost_left_invertible Ring
for x being Scalar of F
for V being non empty right_complementable add-associative right_zeroed RightMod-like RightModStr of F
for v being Vector of V holds
( v * x = 0. V iff ( x = 0. F or v = 0. V ) )

proof end;

theorem :: VECTSP_2:95

for F being non degenerated almost_left_invertible Ring
for x being Scalar of F
for V being non empty right_complementable add-associative right_zeroed RightMod-like RightModStr of F
for v being Vector of V st x <> 0. F holds
(v * x) * (x ") = v

proof end;