:: Rings and Modules - Part II
:: by Michal Muzalewski
::
:: Received October 18, 1991
:: Copyright (c) 1991-2016 Association of Mizar Users

Lm1: for R being Ring holds ModuleStr(# {0},op2,op0,(pr2 ( the carrier of R,{0})) #) is strict LeftMod of R

proof end;

definition

let R be Ring;

func TrivialLMod R -> strict LeftMod of R equals :: MOD_2:def 1
ModuleStr(# {0},op2,op0,(pr2 ( the carrier of R,{0})) #);

coherence
ModuleStr(# {0},op2,op0,(pr2 ( the carrier of R,{0})) #) is strict LeftMod of R by Lm1;

end;

:: deftheorem defines TrivialLMod MOD_2:def 1 :
for R being Ring holds TrivialLMod R = ModuleStr(# {0},op2,op0,(pr2 ( the carrier of R,{0})) #);

theorem :: MOD_2:1

for R being Ring
for x being Vector of (TrivialLMod R) holds x = 0. (TrivialLMod R) by GRCAT_1:4;

definition

let R be non empty multMagma ;
let G, H be non empty ModuleStr over R;
let f be Function of G,H;

attr f is homogeneous means :: MOD_2:def 2
for a being Scalar of R
for x being Vector of G holds f . (a * x) = a * (f . x);

end;

:: deftheorem defines homogeneous MOD_2:def 2 :
for R being non empty multMagma
for G, H being non empty ModuleStr over R
for f being Function of G,H holds
( f is homogeneous iff for a being Scalar of R
for x being Vector of G holds f . (a * x) = a * (f . x) );

theorem Th2: :: MOD_2:2

for R being Ring
for G, H, S being non empty ModuleStr over R
for f being Function of G,H
for g being Function of H,S st f is homogeneous & g is homogeneous holds
g * f is homogeneous

proof end;

registration

let R be Ring;
let G, H be LeftMod of R;

cluster ZeroMap (G,H) -> homogeneous ;

coherence
ZeroMap (G,H) is homogeneous

proof end;

end;

definition

let R be Ring;
attr c₂ is strict ;
struct LModMorphismStr over R -> ;
aggr LModMorphismStr(# Dom, Cod, Fun #) -> LModMorphismStr over R;
sel Dom c₂ -> LeftMod of R;
sel Cod c₂ -> LeftMod of R;
sel Fun c₂ -> Function of the Dom of c₂, the Cod of c₂;

end;

definition

let R be Ring;
let f be LModMorphismStr over R;

func dom f -> LeftMod of R equals :: MOD_2:def 3
the Dom of f;

correctness
coherence
the Dom of f is LeftMod of R;
;

func cod f -> LeftMod of R equals :: MOD_2:def 4
the Cod of f;

correctness
coherence
the Cod of f is LeftMod of R;
;

end;

:: deftheorem defines dom MOD_2:def 3 :
for R being Ring
for f being LModMorphismStr over R holds dom f = the Dom of f;

:: deftheorem defines cod MOD_2:def 4 :
for R being Ring
for f being LModMorphismStr over R holds cod f = the Cod of f;

definition

let R be Ring;
let f be LModMorphismStr over R;

func fun f -> Function of (dom f),(cod f) equals :: MOD_2:def 5
the Fun of f;

coherence
the Fun of f is Function of (dom f),(cod f) ;

end;

:: deftheorem defines fun MOD_2:def 5 :
for R being Ring
for f being LModMorphismStr over R holds fun f = the Fun of f;

theorem :: MOD_2:3

for R being Ring
for G1, G2 being LeftMod of R
for f being LModMorphismStr over R
for f0 being Function of G1,G2 st f = LModMorphismStr(# G1,G2,f0 #) holds
( dom f = G1 & cod f = G2 & fun f = f0 ) ;

definition

let R be Ring;
let G, H be LeftMod of R;

func ZERO (G,H) -> strict LModMorphismStr over R equals :: MOD_2:def 6
LModMorphismStr(# G,H,(ZeroMap (G,H)) #);

correctness
coherence
LModMorphismStr(# G,H,(ZeroMap (G,H)) #) is strict LModMorphismStr over R;
;

end;

:: deftheorem defines ZERO MOD_2:def 6 :
for R being Ring
for G, H being LeftMod of R holds ZERO (G,H) = LModMorphismStr(# G,H,(ZeroMap (G,H)) #);

definition

let R be Ring;
let IT be LModMorphismStr over R;

attr IT is LModMorphism-like means :Def7: :: MOD_2:def 7
( fun IT is additive & fun IT is homogeneous );

end;

:: deftheorem Def7 defines LModMorphism-like MOD_2:def 7 :
for R being Ring
for IT being LModMorphismStr over R holds
( IT is LModMorphism-like iff ( fun IT is additive & fun IT is homogeneous ) );

registration

let R be Ring;

cluster strict LModMorphism-like for LModMorphismStr over R;

existence
ex b₁ being LModMorphismStr over R st
( b₁ is strict & b₁ is LModMorphism-like )

proof end;

end;

definition

let R be Ring;
mode LModMorphism of R is LModMorphism-like LModMorphismStr over R;

end;

theorem Th4: :: MOD_2:4

for R being Ring
for F being LModMorphism of R holds
( the Fun of F is additive & the Fun of F is homogeneous )

proof end;

registration

let R be Ring;
let G, H be LeftMod of R;

cluster ZERO (G,H) -> strict LModMorphism-like ;

coherence
ZERO (G,H) is LModMorphism-like ;

end;

definition

let R be Ring;
let G, H be LeftMod of R;

mode Morphism of G,H -> LModMorphism of R means :Def8: :: MOD_2:def 8
( dom it = G & cod it = H );

existence
ex b₁ being LModMorphism of R st
( dom b₁ = G & cod b₁ = H )

proof end;

end;

:: deftheorem Def8 defines Morphism MOD_2:def 8 :
for R being Ring
for G, H being LeftMod of R
for b₄ being LModMorphism of R holds
( b₄ is Morphism of G,H iff ( dom b₄ = G & cod b₄ = H ) );

registration

let R be Ring;
let G, H be LeftMod of R;

cluster strict LModMorphism-like for Morphism of G,H;

existence
ex b₁ being Morphism of G,H st b₁ is strict

proof end;

end;

theorem Th5: :: MOD_2:5

for R being Ring
for G, H being LeftMod of R
for f being LModMorphismStr over R st dom f = G & cod f = H & fun f is additive & fun f is homogeneous holds
f is Morphism of G,H

proof end;

theorem Th6: :: MOD_2:6

for R being Ring
for G, H being LeftMod of R
for f being Function of G,H st f is additive & f is homogeneous holds
LModMorphismStr(# G,H,f #) is strict Morphism of G,H

proof end;

registration

let R be Ring;
let G be LeftMod of R;

cluster id G -> homogeneous ;

coherence
id G is homogeneous

proof end;

end;

definition

let R be Ring;
let G be LeftMod of R;

func ID G -> strict Morphism of G,G equals :: MOD_2:def 9
LModMorphismStr(# G,G,(id G) #);

coherence
LModMorphismStr(# G,G,(id G) #) is strict Morphism of G,G

proof end;

end;

:: deftheorem defines ID MOD_2:def 9 :
for R being Ring
for G being LeftMod of R holds ID G = LModMorphismStr(# G,G,(id G) #);

definition

let R be Ring;
let G, H be LeftMod of R;
:: original: ZERO
redefine func ZERO (G,H) -> strict Morphism of G,H;
coherence
ZERO (G,H) is strict Morphism of G,H

proof end;

end;

theorem Th7: :: MOD_2:7

for R being Ring
for G, H being LeftMod of R
for F being Morphism of G,H ex f being Function of G,H st
( LModMorphismStr(# the Dom of F, the Cod of F, the Fun of F #) = LModMorphismStr(# G,H,f #) & f is additive & f is homogeneous )

proof end;

theorem Th8: :: MOD_2:8

for R being Ring
for G, H being LeftMod of R
for F being strict Morphism of G,H ex f being Function of G,H st F = LModMorphismStr(# G,H,f #)

proof end;

theorem Th9: :: MOD_2:9

for R being Ring
for F being LModMorphism of R ex G, H being LeftMod of R st F is Morphism of G,H

proof end;

theorem :: MOD_2:10

for R being Ring
for F being strict LModMorphism of R ex G, H being LeftMod of R ex f being Function of G,H st
( F is strict Morphism of G,H & F = LModMorphismStr(# G,H,f #) & f is additive & f is homogeneous )

proof end;

theorem Th11: :: MOD_2:11

for R being Ring
for g, f being LModMorphism of R st dom g = cod f holds
ex G1, G2, G3 being LeftMod of R st
( g is Morphism of G2,G3 & f is Morphism of G1,G2 )

proof end;

definition

let R be Ring;
let G, F be LModMorphism of R;
assume A1: dom G = cod F ;

func G * F -> strict LModMorphism of R means :Def10: :: MOD_2:def 10
for G1, G2, G3 being LeftMod of R
for g being Function of G2,G3
for f being Function of G1,G2 st LModMorphismStr(# the Dom of G, the Cod of G, the Fun of G #) = LModMorphismStr(# G2,G3,g #) & LModMorphismStr(# the Dom of F, the Cod of F, the Fun of F #) = LModMorphismStr(# G1,G2,f #) holds
it = LModMorphismStr(# G1,G3,(g * f) #);

existence
ex b₁ being strict LModMorphism of R st
for G1, G2, G3 being LeftMod of R
for g being Function of G2,G3
for f being Function of G1,G2 st LModMorphismStr(# the Dom of G, the Cod of G, the Fun of G #) = LModMorphismStr(# G2,G3,g #) & LModMorphismStr(# the Dom of F, the Cod of F, the Fun of F #) = LModMorphismStr(# G1,G2,f #) holds
b₁ = LModMorphismStr(# G1,G3,(g * f) #)

proof end;

uniqueness
for b₁, b₂ being strict LModMorphism of R st ( for G1, G2, G3 being LeftMod of R
for g being Function of G2,G3
for f being Function of G1,G2 st LModMorphismStr(# the Dom of G, the Cod of G, the Fun of G #) = LModMorphismStr(# G2,G3,g #) & LModMorphismStr(# the Dom of F, the Cod of F, the Fun of F #) = LModMorphismStr(# G1,G2,f #) holds
b₁ = LModMorphismStr(# G1,G3,(g * f) #) ) & ( for G1, G2, G3 being LeftMod of R
for g being Function of G2,G3
for f being Function of G1,G2 st LModMorphismStr(# the Dom of G, the Cod of G, the Fun of G #) = LModMorphismStr(# G2,G3,g #) & LModMorphismStr(# the Dom of F, the Cod of F, the Fun of F #) = LModMorphismStr(# G1,G2,f #) holds
b₂ = LModMorphismStr(# G1,G3,(g * f) #) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def10 defines * MOD_2:def 10 :
for R being Ring
for G, F being LModMorphism of R st dom G = cod F holds
for b₄ being strict LModMorphism of R holds
( b₄ = G * F iff for G1, G2, G3 being LeftMod of R
for g being Function of G2,G3
for f being Function of G1,G2 st LModMorphismStr(# the Dom of G, the Cod of G, the Fun of G #) = LModMorphismStr(# G2,G3,g #) & LModMorphismStr(# the Dom of F, the Cod of F, the Fun of F #) = LModMorphismStr(# G1,G2,f #) holds
b₄ = LModMorphismStr(# G1,G3,(g * f) #) );

theorem Th12: :: MOD_2:12

for R being Ring
for G1, G2, G3 being LeftMod of R
for G being Morphism of G2,G3
for F being Morphism of G1,G2 holds G * F is strict Morphism of G1,G3

proof end;

definition

let R be Ring;
let G1, G2, G3 be LeftMod of R;
let G be Morphism of G2,G3;
let F be Morphism of G1,G2;

func G *' F -> strict Morphism of G1,G3 equals :: MOD_2:def 11
G * F;

coherence
G * F is strict Morphism of G1,G3 by Th12;

end;

:: deftheorem defines *' MOD_2:def 11 :
for R being Ring
for G1, G2, G3 being LeftMod of R
for G being Morphism of G2,G3
for F being Morphism of G1,G2 holds G *' F = G * F;

theorem Th13: :: MOD_2:13

for R being Ring
for G1, G2, G3 being LeftMod of R
for G being Morphism of G2,G3
for F being Morphism of G1,G2
for g being Function of G2,G3
for f being Function of G1,G2 st G = LModMorphismStr(# G2,G3,g #) & F = LModMorphismStr(# G1,G2,f #) holds
( G *' F = LModMorphismStr(# G1,G3,(g * f) #) & G * F = LModMorphismStr(# G1,G3,(g * f) #) )

proof end;

theorem Th14: :: MOD_2:14

for R being Ring
for f, g being strict LModMorphism of R st dom g = cod f holds
ex G1, G2, G3 being LeftMod of R ex f0 being Function of G1,G2 ex g0 being Function of G2,G3 st
( f = LModMorphismStr(# G1,G2,f0 #) & g = LModMorphismStr(# G2,G3,g0 #) & g * f = LModMorphismStr(# G1,G3,(g0 * f0) #) )

proof end;

theorem :: MOD_2:15

for R being Ring
for f, g being strict LModMorphism of R st dom g = cod f holds
( dom (g * f) = dom f & cod (g * f) = cod g )

proof end;

theorem Th16: :: MOD_2:16

for R being Ring
for G1, G2, G3, G4 being LeftMod of R
for f being strict Morphism of G1,G2
for g being strict Morphism of G2,G3
for h being strict Morphism of G3,G4 holds h * (g * f) = (h * g) * f

proof end;

theorem :: MOD_2:17

for R being Ring
for f, g, h being strict LModMorphism of R st dom h = cod g & dom g = cod f holds
h * (g * f) = (h * g) * f

proof end;

theorem :: MOD_2:18

for R being Ring
for G being LeftMod of R holds
( dom (ID G) = G & cod (ID G) = G & ( for f being strict LModMorphism of R st cod f = G holds
(ID G) * f = f ) & ( for g being strict LModMorphism of R st dom g = G holds
g * (ID G) = g ) )

proof end;

theorem Th19: :: MOD_2:19

for UN being Universe
for u, v, w being Element of UN holds {u,v,w} is Element of UN

proof end;

theorem Th20: :: MOD_2:20

for UN being Universe
for u being Element of UN holds succ u is Element of UN

proof end;

theorem Th21: :: MOD_2:21

for UN being Universe holds
( 0 is Element of UN & 1 is Element of UN & 2 is Element of UN )

proof end;

Lm2: ex c being Element of {0,1,2} st c = 0

proof end;

Lm3: ex c being Element of {0,1,2} st c = 1

proof end;

Lm4: ex c being Element of {0,1,2} st c = 2

proof end;

definition

let a be Element of {0,1,2};

func - a -> Element of {0,1,2} equals :Def12: :: MOD_2:def 12
0 if a = 0
2 if a = 1
1 if a = 2
;

coherence
( ( a = 0 implies 0 is Element of {0,1,2} ) & ( a = 1 implies 2 is Element of {0,1,2} ) & ( a = 2 implies 1 is Element of {0,1,2} ) ) by Lm3, Lm4;
consistency
for b₁ being Element of {0,1,2} holds
( ( a = 0 & a = 1 implies ( b₁ = 0 iff b₁ = 2 ) ) & ( a = 0 & a = 2 implies ( b₁ = 0 iff b₁ = 1 ) ) & ( a = 1 & a = 2 implies ( b₁ = 2 iff b₁ = 1 ) ) ) ;
let b be Element of {0,1,2};

func a + b -> Element of {0,1,2} equals :Def13: :: MOD_2:def 13
b if a = 0
a if b = 0
2 if ( a = 1 & b = 1 )
0 if ( a = 1 & b = 2 )
0 if ( a = 2 & b = 1 )
1 if ( a = 2 & b = 2 )
;

coherence
( ( a = 0 implies b is Element of {0,1,2} ) & ( b = 0 implies a is Element of {0,1,2} ) & ( a = 1 & b = 1 implies 2 is Element of {0,1,2} ) & ( a = 1 & b = 2 implies 0 is Element of {0,1,2} ) & ( a = 2 & b = 1 implies 0 is Element of {0,1,2} ) & ( a = 2 & b = 2 implies 1 is Element of {0,1,2} ) ) by Lm2, Lm3, Lm4;
consistency
for b₁ being Element of {0,1,2} holds
( ( a = 0 & b = 0 implies ( b₁ = b iff b₁ = a ) ) & ( a = 0 & a = 1 & b = 1 implies ( b₁ = b iff b₁ = 2 ) ) & ( a = 0 & a = 1 & b = 2 implies ( b₁ = b iff b₁ = 0 ) ) & ( a = 0 & a = 2 & b = 1 implies ( b₁ = b iff b₁ = 0 ) ) & ( a = 0 & a = 2 & b = 2 implies ( b₁ = b iff b₁ = 1 ) ) & ( b = 0 & a = 1 & b = 1 implies ( b₁ = a iff b₁ = 2 ) ) & ( b = 0 & a = 1 & b = 2 implies ( b₁ = a iff b₁ = 0 ) ) & ( b = 0 & a = 2 & b = 1 implies ( b₁ = a iff b₁ = 0 ) ) & ( b = 0 & a = 2 & b = 2 implies ( b₁ = a iff b₁ = 1 ) ) & ( a = 1 & b = 1 & a = 1 & b = 2 implies ( b₁ = 2 iff b₁ = 0 ) ) & ( a = 1 & b = 1 & a = 2 & b = 1 implies ( b₁ = 2 iff b₁ = 0 ) ) & ( a = 1 & b = 1 & a = 2 & b = 2 implies ( b₁ = 2 iff b₁ = 1 ) ) & ( a = 1 & b = 2 & a = 2 & b = 1 implies ( b₁ = 0 iff b₁ = 0 ) ) & ( a = 1 & b = 2 & a = 2 & b = 2 implies ( b₁ = 0 iff b₁ = 1 ) ) & ( a = 2 & b = 1 & a = 2 & b = 2 implies ( b₁ = 0 iff b₁ = 1 ) ) ) ;

func a * b -> Element of {0,1,2} equals :Def14: :: MOD_2:def 14
0 if b = 0
0 if a = 0
a if b = 1
b if a = 1
1 if ( a = 2 & b = 2 )
;

coherence
( ( b = 0 implies 0 is Element of {0,1,2} ) & ( a = 0 implies 0 is Element of {0,1,2} ) & ( b = 1 implies a is Element of {0,1,2} ) & ( a = 1 implies b is Element of {0,1,2} ) & ( a = 2 & b = 2 implies 1 is Element of {0,1,2} ) ) by Lm3;
consistency
for b₁ being Element of {0,1,2} holds
( ( b = 0 & a = 0 implies ( b₁ = 0 iff b₁ = 0 ) ) & ( b = 0 & b = 1 implies ( b₁ = 0 iff b₁ = a ) ) & ( b = 0 & a = 1 implies ( b₁ = 0 iff b₁ = b ) ) & ( b = 0 & a = 2 & b = 2 implies ( b₁ = 0 iff b₁ = 1 ) ) & ( a = 0 & b = 1 implies ( b₁ = 0 iff b₁ = a ) ) & ( a = 0 & a = 1 implies ( b₁ = 0 iff b₁ = b ) ) & ( a = 0 & a = 2 & b = 2 implies ( b₁ = 0 iff b₁ = 1 ) ) & ( b = 1 & a = 1 implies ( b₁ = a iff b₁ = b ) ) & ( b = 1 & a = 2 & b = 2 implies ( b₁ = a iff b₁ = 1 ) ) & ( a = 1 & a = 2 & b = 2 implies ( b₁ = b iff b₁ = 1 ) ) ) ;

end;

:: deftheorem Def12 defines - MOD_2:def 12 :
for a being Element of {0,1,2} holds
( ( a = 0 implies - a = 0 ) & ( a = 1 implies - a = 2 ) & ( a = 2 implies - a = 1 ) );

:: deftheorem Def13 defines + MOD_2:def 13 :
for a, b being Element of {0,1,2} holds
( ( a = 0 implies a + b = b ) & ( b = 0 implies a + b = a ) & ( a = 1 & b = 1 implies a + b = 2 ) & ( a = 1 & b = 2 implies a + b = 0 ) & ( a = 2 & b = 1 implies a + b = 0 ) & ( a = 2 & b = 2 implies a + b = 1 ) );

:: deftheorem Def14 defines * MOD_2:def 14 :
for a, b being Element of {0,1,2} holds
( ( b = 0 implies a * b = 0 ) & ( a = 0 implies a * b = 0 ) & ( b = 1 implies a * b = a ) & ( a = 1 implies a * b = b ) & ( a = 2 & b = 2 implies a * b = 1 ) );

definition

func add3 -> BinOp of {0,1,2} means :Def15: :: MOD_2:def 15
for a, b being Element of {0,1,2} holds it . (a,b) = a + b;

existence
ex b₁ being BinOp of {0,1,2} st
for a, b being Element of {0,1,2} holds b₁ . (a,b) = a + b

proof end;

uniqueness
for b₁, b₂ being BinOp of {0,1,2} st ( for a, b being Element of {0,1,2} holds b₁ . (a,b) = a + b ) & ( for a, b being Element of {0,1,2} holds b₂ . (a,b) = a + b ) holds
b₁ = b₂

proof end;

func mult3 -> BinOp of {0,1,2} means :Def16: :: MOD_2:def 16
for a, b being Element of {0,1,2} holds it . (a,b) = a * b;

existence
ex b₁ being BinOp of {0,1,2} st
for a, b being Element of {0,1,2} holds b₁ . (a,b) = a * b

proof end;

uniqueness
for b₁, b₂ being BinOp of {0,1,2} st ( for a, b being Element of {0,1,2} holds b₁ . (a,b) = a * b ) & ( for a, b being Element of {0,1,2} holds b₂ . (a,b) = a * b ) holds
b₁ = b₂

proof end;

func compl3 -> UnOp of {0,1,2} means :Def17: :: MOD_2:def 17
for a being Element of {0,1,2} holds it . a = - a;

existence
ex b₁ being UnOp of {0,1,2} st
for a being Element of {0,1,2} holds b₁ . a = - a

proof end;

uniqueness
for b₁, b₂ being UnOp of {0,1,2} st ( for a being Element of {0,1,2} holds b₁ . a = - a ) & ( for a being Element of {0,1,2} holds b₂ . a = - a ) holds
b₁ = b₂

proof end;

func unit3 -> Element of {0,1,2} equals :: MOD_2:def 18
1;

coherence
1 is Element of {0,1,2} by ENUMSET1:def 1;

func zero3 -> Element of {0,1,2} equals :: MOD_2:def 19
0 ;

coherence
0 is Element of {0,1,2} by ENUMSET1:def 1;

end;

:: deftheorem Def15 defines add3 MOD_2:def 15 :
for b₁ being BinOp of {0,1,2} holds
( b₁ = add3 iff for a, b being Element of {0,1,2} holds b₁ . (a,b) = a + b );

:: deftheorem Def16 defines mult3 MOD_2:def 16 :
for b₁ being BinOp of {0,1,2} holds
( b₁ = mult3 iff for a, b being Element of {0,1,2} holds b₁ . (a,b) = a * b );

:: deftheorem Def17 defines compl3 MOD_2:def 17 :
for b₁ being UnOp of {0,1,2} holds
( b₁ = compl3 iff for a being Element of {0,1,2} holds b₁ . a = - a );

:: deftheorem defines unit3 MOD_2:def 18 :
unit3 = 1;

:: deftheorem defines zero3 MOD_2:def 19 :
zero3 = 0 ;

definition

func Z_3 -> strict doubleLoopStr equals :Def20: :: MOD_2:def 20
doubleLoopStr(# {0,1,2},add3,mult3,unit3,zero3 #);

coherence
doubleLoopStr(# {0,1,2},add3,mult3,unit3,zero3 #) is strict doubleLoopStr ;

end;

:: deftheorem Def20 defines Z_3 MOD_2:def 20 :
Z_3 = doubleLoopStr(# {0,1,2},add3,mult3,unit3,zero3 #);

registration

cluster Z_3 -> non empty strict ;

coherence
not Z_3 is empty ;

end;

Lm5: now :: thesis: for h, e being Element of Z_3 st e = 1 holds
( h * e = h & e * h = h )

let h, e be Element of Z_3; :: thesis: ( e = 1 implies ( h * e = h & e * h = h ) )
assume A1: e = 1 ; :: thesis: ( h * e = h & e * h = h )
reconsider a = e, b = h as Element of {0,1,2} ;
thus h * e = b * a by Def16
.= h by A1, Def14 ; :: thesis: e * h = h
thus e * h = a * b by Def16
.= h by A1, Def14 ; :: thesis: verum

end;

registration

cluster Z_3 -> strict well-unital ;

coherence
Z_3 is well-unital by Lm5;

end;

theorem Th22: :: MOD_2:22

( 0. Z_3 = 0 & 1. Z_3 = 1 & 0. Z_3 is Element of {0,1,2} & 1. Z_3 is Element of {0,1,2} & the addF of Z_3 = add3 & the multF of Z_3 = mult3 ) ;

Lm6: for x, y, z being Scalar of Z_3
for X, Y, Z being Element of {0,1,2} st X = x & Y = y & Z = z holds
( (x + y) + z = (X + Y) + Z & x + (y + z) = X + (Y + Z) & (x * y) * z = (X * Y) * Z & x * (y * z) = X * (Y * Z) )

proof end;

Lm7: for x, y, z, a being Element of {0,1,2} st a = 0 holds
( x + (- x) = a & x + a = x & (x + y) + z = x + (y + z) )

proof end;

registration

cluster Z_3 -> right_complementable strict add-associative right_zeroed ;

coherence
( Z_3 is add-associative & Z_3 is right_zeroed & Z_3 is right_complementable )

proof end;

end;

theorem Th23: :: MOD_2:23

for x, y being Scalar of Z_3
for X, Y being Element of {0,1,2} st X = x & Y = y holds
( x + y = X + Y & x * y = X * Y & - x = - X )

proof end;

theorem Th24: :: MOD_2:24

for x, y, z being Scalar of Z_3
for X, Y, Z being Element of {0,1,2} st X = x & Y = y & Z = z holds
( (x + y) + z = (X + Y) + Z & x + (y + z) = X + (Y + Z) & (x * y) * z = (X * Y) * Z & x * (y * z) = X * (Y * Z) )

proof end;

theorem Th25: :: MOD_2:25

for x, y, z, a, b being Element of {0,1,2} st a = 0 & b = 1 holds
( x + y = y + x & (x + y) + z = x + (y + z) & x + a = x & x + (- x) = a & x * y = y * x & (x * y) * z = x * (y * z) & b * x = x & ( x <> a implies ex y being Element of {0,1,2} st x * y = b ) & a <> b & x * (y + z) = (x * y) + (x * z) )

proof end;

theorem Th26: :: MOD_2:26

for F being non empty doubleLoopStr st ( for x, y, z being Scalar of F holds
( x + y = y + x & (x + y) + z = x + (y + z) & x + (0. F) = x & x + (- x) = 0. F & x * y = y * x & (x * y) * z = x * (y * z) & (1. F) * x = x & x * (1. F) = x & ( x <> 0. F implies ex y being Scalar of F st x * y = 1. F ) & 0. F <> 1. F & x * (y + z) = (x * y) + (x * z) ) ) holds
F is Field

proof end;

theorem Th27: :: MOD_2:27

Z_3 is Fanoian Field

proof end;

registration

cluster Z_3 -> right_complementable almost_left_invertible strict well-unital distributive Fanoian Abelian add-associative right_zeroed associative commutative ;

coherence
( Z_3 is Fanoian & Z_3 is add-associative & Z_3 is right_zeroed & Z_3 is right_complementable & Z_3 is Abelian & Z_3 is commutative & Z_3 is associative & Z_3 is well-unital & Z_3 is distributive & Z_3 is almost_left_invertible ) by Th27;

end;

Lm8: for UN being Universe holds the carrier of Z_3 in UN

proof end;

theorem Th28: :: MOD_2:28

for D, D9 being non empty set
for UN being Universe
for f being Function of D,D9 st D in UN & D9 in UN holds
f in UN

proof end;

Lm9: for D being non empty set
for UN being Universe holds
( ( for f being BinOp of D st D in UN holds
f in UN ) & ( for f being UnOp of D st D in UN holds
f in UN ) )

proof end;

theorem :: MOD_2:29

for UN being Universe holds
( the carrier of Z_3 in UN & the addF of Z_3 in UN & comp Z_3 in UN & 0. Z_3 in UN & the multF of Z_3 in UN & 1. Z_3 in UN )

proof end;