:: MOD_2 semantic presentation

Lm1: op0 = {} by TARSKI:def 1
.= Extract {} by ALGSTR_1:def 3 ;

Lm2: for R being Ring holds VectSpStr(# {{} },op2 ,op0 ,(pr2 the carrier of R,{{} }) #) is strict LeftMod of R

proof end;

definition

let R be Ring;
canceled;
func TrivialLMod R -> strict LeftMod of R equals :: MOD_2:def 2
VectSpStr(# {{} },op2 ,op0 ,(pr2 the carrier of R,{{} }) #);
coherence by Lm2;

end;

:: deftheorem MOD_2:def 1 :

:: deftheorem defines TrivialLMod MOD_2:def 2 :

theorem :: MOD_2:1

for R being Ring
for x being Vector of (TrivialLMod R) holds x = 0. (TrivialLMod R) by Lm1, ALGSTR_1:def 4, GRCAT_1:8;

definition

let R be non empty doubleLoopStr ;
let G, H be non empty VectSpStr of R;
let f be Function of G,H;
canceled;
canceled;
attr f is linear means :Def5: :: MOD_2:def 5
( ( for x, y being Vector of G holds f . (x + y) = (f . x) + (f . y) ) & ( for a being Scalar of R
for x being Vector of G holds f . (a * x) = a * (f . x) ) );

end;

:: deftheorem MOD_2:def 3 :

:: deftheorem MOD_2:def 4 :

:: deftheorem Def5 defines linear MOD_2:def 5 :

theorem :: MOD_2:2

canceled;

theorem :: MOD_2:3

canceled;

theorem Th4: :: MOD_2:4

for R being Ring
for G, H being non empty VectSpStr of R
for f being Function of G,H st f is linear holds
f is additive

proof end;

theorem :: MOD_2:5

canceled;

theorem Th6: :: MOD_2:6

for R being Ring
for G, H, S being non empty VectSpStr of R
for f being Function of G,H
for g being Function of H,S st f is linear & g is linear holds
g * f is linear

proof end;

theorem :: MOD_2:7

canceled;

theorem Th8: :: MOD_2:8

for R being Ring
for G, H being LeftMod of R holds ZeroMap G,H is linear

proof end;

definition

let R be Ring;
attr c₂ is strict;
struct LModMorphismStr of R -> ;
aggr LModMorphismStr(# Dom, Cod, Fun #) -> LModMorphismStr of 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 of R;
func dom f -> LeftMod of R equals :: MOD_2:def 6
the Dom of f;
correctness ;
func cod f -> LeftMod of R equals :: MOD_2:def 7
the Cod of f;
correctness ;

end;

:: deftheorem defines dom MOD_2:def 6 :

:: deftheorem defines cod MOD_2:def 7 :

definition

let R be Ring;
let f be LModMorphismStr of R;
func fun f -> Function of (dom f),(cod f) equals :: MOD_2:def 8
the Fun of f;
coherence ;

end;

:: deftheorem defines fun MOD_2:def 8 :

theorem :: MOD_2:9

for R being Ring
for G1, G2 being LeftMod of R
for f being LModMorphismStr of 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 of R equals :: MOD_2:def 9
LModMorphismStr(# G,H,(ZeroMap G,H) #);
correctness ;

end;

:: deftheorem defines ZERO MOD_2:def 9 :

definition

let R be Ring;
let IT be LModMorphismStr of R;
attr IT is LModMorphism-like means :Def10: :: MOD_2:def 10
fun IT is linear;

end;

:: deftheorem Def10 defines LModMorphism-like MOD_2:def 10 :

registration

let R be Ring;
cluster strict LModMorphism-like LModMorphismStr of R;
existence

proof end;

end;

definition

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

end;

theorem Th10: :: MOD_2:10

for R being Ring
for F being LModMorphism of R holds the Fun of F is linear

proof end;

registration

let R be Ring;
let G, H be LeftMod of R;
cluster ZERO G,H -> strict LModMorphism-like ;
coherence

proof end;

end;

definition

let R be Ring;
let G, H be LeftMod of R;
mode Morphism of G,H -> LModMorphism of R means :Def11: :: MOD_2:def 11
( dom it = G & cod it = H );
existence

proof end;

end;

:: deftheorem Def11 defines Morphism MOD_2:def 11 :

registration

let R be Ring;
let G, H be LeftMod of R;
cluster strict Morphism of G,H;
existence

proof end;

end;

theorem Th11: :: MOD_2:11

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

proof end;

theorem Th12: :: MOD_2:12

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

proof end;

theorem Th13: :: MOD_2:13

for R being Ring
for G being LeftMod of R holds id G is linear

proof end;

definition

let R be Ring;
let G be LeftMod of R;
func ID G -> strict Morphism of G,G equals :: MOD_2:def 12
LModMorphismStr(# G,G,(id G) #);
coherence

proof end;

end;

:: deftheorem defines ID MOD_2:def 12 :

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

proof end;

end;

theorem Th14: :: MOD_2:14

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 linear )

proof end;

theorem Th15: :: MOD_2:15

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 Th16: :: MOD_2:16

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_2:17

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 linear )

proof end;

theorem Th18: :: MOD_2:18

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 :Def13: :: MOD_2:def 13
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

proof end;

uniqueness

proof end;

end;

:: deftheorem Def13 defines * MOD_2:def 13 :

theorem :: MOD_2:19

canceled;

theorem Th20: :: MOD_2:20

for R being Ring
for G2, G3, G1 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 14
G * F;
coherence by Th20;

end;

:: deftheorem defines *' MOD_2:def 14 :

theorem Th21: :: MOD_2:21

for R being Ring
for G2, G3, G1 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 Th22: :: MOD_2:22

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_2:23

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 Th24: :: MOD_2:24

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_2:25

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_2:26

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;

registration

let x, y, z be set ;
cluster {x,y,z} -> non empty ;
coherence by ENUMSET1:def 1;

end;

theorem :: MOD_2:27

canceled;

theorem Th28: :: MOD_2:28

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

proof end;

theorem Th29: :: MOD_2:29

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

proof end;

theorem Th30: :: MOD_2:30

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

proof end;

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

proof end;

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

proof end;

Lm5: 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 :Def15: :: MOD_2:def 15
0 if a = 0
2 if a = 1
1 if a = 2
;
coherence by Lm4, Lm5;
consistency ;
let b be Element of {0,1,2};
func a + b -> Element of {0,1,2} equals :Def16: :: MOD_2:def 16
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 by Lm3, Lm4, Lm5;
consistency ;
func a * b -> Element of {0,1,2} equals :Def17: :: MOD_2:def 17
0 if b = 0
0 if a = 0
a if b = 1
b if a = 1
1 if ( a = 2 & b = 2 )
;
coherence by Lm4;
consistency ;

end;

:: deftheorem Def15 defines - MOD_2:def 15 :

:: deftheorem Def16 defines + MOD_2:def 16 :

:: deftheorem Def17 defines * MOD_2:def 17 :

definition

func add3 -> BinOp of {0,1,2} means :Def18: :: MOD_2:def 18
for a, b being Element of {0,1,2} holds it . a,b = a + b;
existence

proof end;

uniqueness

proof end;

func mult3 -> BinOp of {0,1,2} means :Def19: :: MOD_2:def 19
for a, b being Element of {0,1,2} holds it . a,b = a * b;
existence

proof end;

uniqueness

proof end;

func compl3 -> UnOp of {0,1,2} means :Def20: :: MOD_2:def 20
for a being Element of {0,1,2} holds it . a = - a;
existence

proof end;

uniqueness

proof end;

func unit3 -> Element of {0,1,2} equals :: MOD_2:def 21
1;
coherence by ENUMSET1:def 1;
func zero3 -> Element of {0,1,2} equals :: MOD_2:def 22
0;
coherence by ENUMSET1:def 1;

end;

:: deftheorem Def18 defines add3 MOD_2:def 18 :

:: deftheorem Def19 defines mult3 MOD_2:def 19 :

:: deftheorem Def20 defines compl3 MOD_2:def 20 :

:: deftheorem defines unit3 MOD_2:def 21 :

:: deftheorem defines zero3 MOD_2:def 22 :

definition

func Z3 -> strict doubleLoopStr equals :Def23: :: MOD_2:def 23
doubleLoopStr(# {0,1,2},add3 ,mult3 ,unit3 ,zero3 #);
coherence ;

end;

:: deftheorem Def23 defines Z3 MOD_2:def 23 :

registration

cluster Z3 -> non empty strict ;
coherence

proof end;

end;

Lm6: now

let h, e be Element of Z3 ; :: thesis:
assume A1: e = 1 ; :: thesis:
reconsider a = e, b = h as Element of {0,1,2} ;
thus h * e = mult3 . h,e
.= b * a by Def19
.= h by A1, Def17 ; :: thesis:
thus e * h = mult3 . e,h
.= a * b by Def19
.= h by A1, Def17 ; :: thesis:

end;

registration

cluster Z3 -> non empty unital strict ;
coherence

proof end;

end;

Lm7: 1. Z3 = 1

proof end;

theorem :: MOD_2:31

canceled;

theorem Th32: :: MOD_2:32

( 0. Z3 = 0 & 1. Z3 = 1 & 0. Z3 is Element of {0,1,2} & 1. Z3 is Element of {0,1,2} & the add of Z3 = add3 & the mult of Z3 = mult3 ) by Lm7;

Lm8: for x, y being Scalar of Z3
for X, Y being Element of {0,1,2} st X = x & Y = y holds
( x + y = X + Y & x * y = X * Y )

proof end;

Lm9: for x, y, z being Scalar of Z3
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;

Lm10: 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 Z3 -> non empty add-associative right_zeroed right_complementable unital strict ;
coherence

proof end;

end;

theorem Th33: :: MOD_2:33

for x, y being Scalar of Z3
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 Th34: :: MOD_2:34

for x, y, z being Scalar of Z3
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 Th35: :: MOD_2:35

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 Th36: :: MOD_2:36

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 <> 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 Th37: :: MOD_2:37

Z3 is Fanoian Field

proof end;

registration

cluster Z3 -> non empty Abelian add-associative right_zeroed right_complementable unital associative commutative strict distributive left_unital Field-like Fanoian ;
coherence by Th37;

end;

Lm11: for UN being Universe holds the carrier of Z3 in UN

proof end;

theorem Th38: :: MOD_2:38

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

proof end;

Lm12: 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_2:39

canceled;

theorem :: MOD_2:40

for UN being Universe holds
( the carrier of Z3 in UN & the add of Z3 is Element of UN & comp Z3 is Element of UN & the Zero of Z3 is Element of UN & the mult of Z3 is Element of UN & the unity of Z3 is Element of UN )

proof end;