:: MODCAT_1 semantic presentation

definition

let R be Ring;
mode LeftMod_DOMAIN of R -> non empty set means :Def1: :: MODCAT_1:def 1
for x being Element of it holds x is strict LeftMod of R;
existence

proof end;

end;

:: deftheorem Def1 defines LeftMod_DOMAIN MODCAT_1:def 1 :

definition

let R be Ring;
let V be LeftMod_DOMAIN of R;
:: original: Element
redefine mode Element of V -> LeftMod of R;
coherence by Def1;

end;

registration

let R be Ring;
let V be LeftMod_DOMAIN of R;
cluster strict Element of V;
existence

proof end;

end;

definition

let R be Ring;
mode LModMorphism_DOMAIN of R -> non empty set means :Def2: :: MODCAT_1:def 2
for x being Element of it holds x is strict LModMorphism of R;
existence

proof end;

end;

:: deftheorem Def2 defines LModMorphism_DOMAIN MODCAT_1:def 2 :

definition

let R be Ring;
let M be LModMorphism_DOMAIN of R;
:: original: Element
redefine mode Element of M -> LModMorphism of R;
coherence by Def2;

end;

registration

let R be Ring;
let M be LModMorphism_DOMAIN of R;
cluster strict Element of M;
existence

proof end;

end;

theorem :: MODCAT_1:1

canceled;

theorem :: MODCAT_1:2

canceled;

theorem Th3: :: MODCAT_1:3

for R being Ring
for f being strict LModMorphism of R holds {f} is LModMorphism_DOMAIN of R

proof end;

definition

let R be Ring;
let G, H be LeftMod of R;
mode LModMorphism_DOMAIN of G,H -> LModMorphism_DOMAIN of R means :Def3: :: MODCAT_1:def 3
for x being Element of it holds x is strict Morphism of G,H;
existence

proof end;

end;

:: deftheorem Def3 defines LModMorphism_DOMAIN MODCAT_1:def 3 :

theorem Th4: :: MODCAT_1:4

for D being non empty set
for R being Ring
for G, H being LeftMod of R holds
( D is LModMorphism_DOMAIN of G,H iff for x being Element of D holds x is strict Morphism of G,H )

proof end;

theorem :: MODCAT_1:5

for R being Ring
for G, H being LeftMod of R
for f being strict Morphism of G,H holds {f} is LModMorphism_DOMAIN of G,H

proof end;

definition

let R be Ring;
let G, H be LeftMod of R;
func Morphs G,H -> LModMorphism_DOMAIN of G,H means :Def4: :: MODCAT_1:def 4
for x being set holds
( x in it iff x is strict Morphism of G,H );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def4 defines Morphs MODCAT_1:def 4 :

definition

let R be Ring;
let G, H be LeftMod of R;
let M be LModMorphism_DOMAIN of G,H;
:: original: Element
redefine mode Element of M -> Morphism of G,H;
coherence by Def3;

end;

definition

let x, y be set ;
let R be Ring;
pred GO x,y,R means :Def5: :: MODCAT_1:def 5
ex x1, x2 being set st
( x = [x1,x2] & ex G being strict LeftMod of R st
( y = G & x1 = LoopStr(# the carrier of G,the add of G,the Zero of G #) & x2 = the lmult of G ) );

end;

:: deftheorem Def5 defines GO MODCAT_1:def 5 :

theorem Th6: :: MODCAT_1:6

for R being Ring
for x, y1, y2 being set st GO x,y1,R & GO x,y2,R holds
y1 = y2

proof end;

theorem Th7: :: MODCAT_1:7

for R being Ring
for UN being Universe ex x being set st
( x in { [G,f] where G is Element of GroupObjects UN, f is Element of Funcs [:the carrier of R,{{} }:],{{} } : verum } & GO x, TrivialLMod R,R )

proof end;

definition

let UN be Universe;
let R be Ring;
func LModObjects UN,R -> set means :Def6: :: MODCAT_1:def 6
for y being set holds
( y in it iff ex x being set st
( x in { [G,f] where G is Element of GroupObjects UN, f is Element of Funcs [:the carrier of R,{{} }:],{{} } : verum } & GO x,y,R ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def6 defines LModObjects MODCAT_1:def 6 :

theorem Th8: :: MODCAT_1:8

for UN being Universe
for R being Ring holds TrivialLMod R in LModObjects UN,R

proof end;

registration

let UN be Universe;
let R be Ring;
cluster LModObjects UN,R -> non empty ;
coherence by Th8;

end;

theorem Th9: :: MODCAT_1:9

for UN being Universe
for R being Ring
for x being Element of LModObjects UN,R holds x is strict LeftMod of R

proof end;

definition

let UN be Universe;
let R be Ring;
:: original: LModObjects
redefine func LModObjects UN,R -> LeftMod_DOMAIN of R;
coherence

proof end;

end;

definition

let R be Ring;
let V be LeftMod_DOMAIN of R;
func Morphs V -> LModMorphism_DOMAIN of R means :Def7: :: MODCAT_1:def 7
for x being set holds
( x in it iff ex G, H being strict Element of V st x is strict Morphism of G,H );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def7 defines Morphs MODCAT_1:def 7 :

definition

let R be Ring;
let V be LeftMod_DOMAIN of R;
let F be Element of Morphs V;
func dom' F -> Element of V equals :: MODCAT_1:def 8
dom F;
coherence

proof end;

func cod' F -> Element of V equals :: MODCAT_1:def 9
cod F;
coherence

proof end;

end;

:: deftheorem defines dom' MODCAT_1:def 8 :

:: deftheorem defines cod' MODCAT_1:def 9 :

definition

let R be Ring;
let V be LeftMod_DOMAIN of R;
let G be Element of V;
func ID G -> strict Element of Morphs V equals :: MODCAT_1:def 10
ID G;
coherence

proof end;

end;

:: deftheorem defines ID MODCAT_1:def 10 :

definition

let R be Ring;
let V be LeftMod_DOMAIN of R;
func dom V -> Function of Morphs V,V means :Def11: :: MODCAT_1:def 11
for f being Element of Morphs V holds it . f = dom' f;
existence

proof end;

uniqueness

proof end;

func cod V -> Function of Morphs V,V means :Def12: :: MODCAT_1:def 12
for f being Element of Morphs V holds it . f = cod' f;
existence

proof end;

uniqueness

proof end;

func ID V -> Function of V, Morphs V means :Def13: :: MODCAT_1:def 13
for G being Element of V holds it . G = ID G;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def11 defines dom MODCAT_1:def 11 :

:: deftheorem Def12 defines cod MODCAT_1:def 12 :

:: deftheorem Def13 defines ID MODCAT_1:def 13 :

theorem Th10: :: MODCAT_1:10

for R being Ring
for V being LeftMod_DOMAIN of R
for g, f being Element of Morphs V st dom' g = cod' f holds
ex G1, G2, G3 being strict Element of V st
( g is Morphism of G2,G3 & f is Morphism of G1,G2 )

proof end;

theorem Th11: :: MODCAT_1:11

for R being Ring
for V being LeftMod_DOMAIN of R
for g, f being Element of Morphs V st dom' g = cod' f holds
g * f in Morphs V

proof end;

theorem Th12: :: MODCAT_1:12

for R being Ring
for V being LeftMod_DOMAIN of R
for g, f being Element of Morphs V st dom g = cod f holds
g * f in Morphs V

proof end;

definition

let R be Ring;
let V be LeftMod_DOMAIN of R;
func comp V -> PartFunc of [:(Morphs V),(Morphs V):], Morphs V means :Def14: :: MODCAT_1:def 14
( ( for g, f being Element of Morphs V holds
( [g,f] in dom it iff dom' g = cod' f ) ) & ( for g, f being Element of Morphs V st [g,f] in dom it holds
it . [g,f] = g * f ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def14 defines comp MODCAT_1:def 14 :

theorem Th13: :: MODCAT_1:13

for R being Ring
for V being LeftMod_DOMAIN of R
for g, f being Element of Morphs V holds
( [g,f] in dom (comp V) iff dom g = cod f )

proof end;

definition

let UN be Universe;
let R be Ring;
func LModCat UN,R -> strict CatStr equals :: MODCAT_1:def 15
CatStr(# (LModObjects UN,R),(Morphs (LModObjects UN,R)),(dom (LModObjects UN,R)),(cod (LModObjects UN,R)),(comp (LModObjects UN,R)),(ID (LModObjects UN,R)) #);
coherence ;

end;

:: deftheorem defines LModCat MODCAT_1:def 15 :

theorem Th14: :: MODCAT_1:14

for UN being Universe
for R being Ring
for f, g being Morphism of (LModCat UN,R) holds
( [g,f] in dom the Comp of (LModCat UN,R) iff dom g = cod f )

proof end;

theorem Th15: :: MODCAT_1:15

for UN being Universe
for R being Ring
for f being Morphism of (LModCat UN,R)
for f' being Element of Morphs (LModObjects UN,R)
for b being Object of (LModCat UN,R)
for b' being Element of LModObjects UN,R holds
( f is strict Element of Morphs (LModObjects UN,R) & f' is Morphism of (LModCat UN,R) & b is strict Element of LModObjects UN,R & b' is Object of (LModCat UN,R) )

proof end;

theorem Th16: :: MODCAT_1:16

for UN being Universe
for R being Ring
for b being Object of (LModCat UN,R)
for b' being Element of LModObjects UN,R st b = b' holds
id b = ID b'

proof end;

theorem Th17: :: MODCAT_1:17

for UN being Universe
for R being Ring
for f being Morphism of (LModCat UN,R)
for f' being Element of Morphs (LModObjects UN,R) st f = f' holds
( dom f = dom f' & cod f = cod f' )

proof end;

theorem Th18: :: MODCAT_1:18

for UN being Universe
for R being Ring
for f, g being Morphism of (LModCat UN,R)
for f', g' being Element of Morphs (LModObjects UN,R) st f = f' & g = g' holds
( ( dom g = cod f implies dom g' = cod f' ) & ( dom g' = cod f' implies dom g = cod f ) & ( dom g = cod f implies [g',f'] in dom (comp (LModObjects UN,R)) ) & ( [g',f'] in dom (comp (LModObjects UN,R)) implies dom g = cod f ) & ( dom g = cod f implies g * f = g' * f' ) & ( dom f = dom g implies dom f' = dom g' ) & ( dom f' = dom g' implies dom f = dom g ) & ( cod f = cod g implies cod f' = cod g' ) & ( cod f' = cod g' implies cod f = cod g ) )

proof end;

Lm1: for UN being Universe
for R being Ring
for f, g being Morphism of (LModCat UN,R) st dom g = cod f holds
( dom (g * f) = dom f & cod (g * f) = cod g )

proof end;

Lm2: for UN being Universe
for R being Ring
for f, g, h being Morphism of (LModCat UN,R) st dom h = cod g & dom g = cod f holds
h * (g * f) = (h * g) * f

proof end;

Lm3: for UN being Universe
for R being Ring
for b being Object of (LModCat UN,R) holds
( dom (id b) = b & cod (id b) = b & ( for f being Morphism of (LModCat UN,R) st cod f = b holds
(id b) * f = f ) & ( for g being Morphism of (LModCat UN,R) st dom g = b holds
g * (id b) = g ) )

proof end;

registration

let UN be Universe;
let R be Ring;
cluster LModCat UN,R -> strict Category-like ;
coherence

proof end;

end;