:: Category of Rings
:: by Micha{\l} Muzalewski
::
:: Received December 5, 1991
:: Copyright (c) 1991-2011 Association of Mizar Users

begin

definition

let G, H be non empty doubleLoopStr ;
let IT be Function of G,H;
canceled;
attr IT is linear means :Def2: :: RINGCAT1:def 2
( IT is additive & IT is multiplicative & IT is unity-preserving );

end;

:: deftheorem RINGCAT1:def 1 :
canceled;

:: deftheorem Def2 defines linear RINGCAT1:def 2 :
for G, H being non empty doubleLoopStr
for IT being Function of G,H holds
( IT is linear iff ( IT is additive & IT is multiplicative & IT is unity-preserving ) );

registration

let G, H be non empty doubleLoopStr ;
cluster V6() V25( the carrier of G, the carrier of H) linear -> additive unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of H:];
coherence
for b₁ being Function of G,H st b₁ is linear holds
( b₁ is additive & b₁ is multiplicative & b₁ is unity-preserving ) by Def2;
cluster V6() V25( the carrier of G, the carrier of H) additive unity-preserving multiplicative -> linear Element of bool [: the carrier of G, the carrier of H:];
coherence
for b₁ being Function of G,H st b₁ is additive & b₁ is multiplicative & b₁ is unity-preserving holds
b₁ is linear by Def2;

end;

theorem :: RINGCAT1:1

canceled;

theorem :: RINGCAT1:2

canceled;

theorem Th3: :: RINGCAT1:3

for G1, G2, G3 being non empty doubleLoopStr
for f being Function of G1,G2
for g being Function of G2,G3 st f is linear & g is linear holds
g * f is linear

proof end;

definition

attr c₁ is strict ;
struct RingMorphismStr -> ;
aggr RingMorphismStr(# Dom, Cod, Fun #) -> RingMorphismStr ;
sel Dom c₁ -> Ring;
sel Cod c₁ -> Ring;
sel Fun c₁ -> Function of the Dom of c₁, the Cod of c₁;

end;

definition

let f be RingMorphismStr ;
func dom f -> Ring equals :: RINGCAT1:def 3
the Dom of f;
coherence
the Dom of f is Ring ;
func cod f -> Ring equals :: RINGCAT1:def 4
the Cod of f;
coherence
the Cod of f is Ring ;
func fun f -> Function of the Dom of f, the Cod of f equals :: RINGCAT1:def 5
the Fun of f;
coherence
the Fun of f is Function of the Dom of f, the Cod of f ;

end;

:: deftheorem defines dom RINGCAT1:def 3 :
for f being RingMorphismStr holds dom f = the Dom of f;

:: deftheorem defines cod RINGCAT1:def 4 :
for f being RingMorphismStr holds cod f = the Cod of f;

:: deftheorem defines fun RINGCAT1:def 5 :
for f being RingMorphismStr holds fun f = the Fun of f;

Lm1: for G being non empty doubleLoopStr holds id G is linear

proof end;

registration

let G be non empty doubleLoopStr ;
cluster id G -> linear ;
coherence
id G is linear by Lm1;

end;

definition

let IT be RingMorphismStr ;
attr IT is RingMorphism-like means :Def6: :: RINGCAT1:def 6
fun IT is linear ;

end;

:: deftheorem Def6 defines RingMorphism-like RINGCAT1:def 6 :
for IT being RingMorphismStr holds
( IT is RingMorphism-like iff fun IT is linear );

registration

cluster strict RingMorphism-like RingMorphismStr ;
existence
ex b₁ being RingMorphismStr st
( b₁ is strict & b₁ is RingMorphism-like )

proof end;

end;

definition

mode RingMorphism is RingMorphism-like RingMorphismStr ;

end;

definition

let G be Ring;
func ID G -> RingMorphism equals :: RINGCAT1:def 7
RingMorphismStr(# G,G,(id G) #);
coherence
RingMorphismStr(# G,G,(id G) #) is RingMorphism

proof end;

end;

:: deftheorem defines ID RINGCAT1:def 7 :
for G being Ring holds ID G = RingMorphismStr(# G,G,(id G) #);

registration

let G be Ring;
cluster ID G -> strict ;
coherence
ID G is strict ;

end;

definition

let G, H be Ring;
pred G <= H means :Def8: :: RINGCAT1:def 8
ex F being RingMorphism st
( dom F = G & cod F = H );
reflexivity
for G being Ring ex F being RingMorphism st
( dom F = G & cod F = G )

proof end;

end;

:: deftheorem Def8 defines <= RINGCAT1:def 8 :
for G, H being Ring holds
( G <= H iff ex F being RingMorphism st
( dom F = G & cod F = H ) );

Lm2: for F being RingMorphism holds RingMorphismStr(# the Dom of F, the Cod of F, the Fun of F #) is RingMorphism-like

proof end;

definition

let G, H be Ring;
assume A1: G <= H ;
mode Morphism of G,H -> strict RingMorphism means :Def9: :: RINGCAT1:def 9
( dom it = G & cod it = H );
existence
ex b₁ being strict RingMorphism st
( dom b₁ = G & cod b₁ = H )

proof end;

end;

:: deftheorem Def9 defines Morphism RINGCAT1:def 9 :
for G, H being Ring st G <= H holds
for b₃ being strict RingMorphism holds
( b₃ is Morphism of G,H iff ( dom b₃ = G & cod b₃ = H ) );

registration

let G, H be Ring;
cluster strict RingMorphism-like Morphism of G,H;
existence
ex b₁ being Morphism of G,H st b₁ is strict

proof end;

end;

definition

let G be Ring;
:: original: ID
redefine func ID G -> strict Morphism of G,G;
coherence
ID G is strict Morphism of G,G

proof end;

end;

Lm3: for F being RingMorphism holds the Fun of F is linear

proof end;

Lm4: for G, H being Ring
for f being strict RingMorphismStr st dom f = G & cod f = H & fun f is linear holds
f is Morphism of G,H

proof end;

Lm5: for G, H being Ring
for f being Function of G,H st f is linear holds
RingMorphismStr(# G,H,f #) is Morphism of G,H

proof end;

Lm6: for F being RingMorphism ex G, H being Ring st
( G <= H & dom F = G & cod F = H & RingMorphismStr(# the Dom of F, the Cod of F, the Fun of F #) is Morphism of G,H )

proof end;

theorem :: RINGCAT1:4

canceled;

theorem Th5: :: RINGCAT1:5

for g, f being RingMorphism st dom g = cod f holds
ex G1, G2, G3 being Ring st
( G1 <= G2 & G2 <= G3 & RingMorphismStr(# the Dom of g, the Cod of g, the Fun of g #) is Morphism of G2,G3 & RingMorphismStr(# the Dom of f, the Cod of f, the Fun of f #) is Morphism of G1,G2 )

proof end;

theorem Th6: :: RINGCAT1:6

for F being strict RingMorphism holds
( F is Morphism of dom F, cod F & dom F <= cod F )

proof end;

Lm7: for G, H being Ring
for F being Morphism of G,H st G <= H holds
ex f being Function of G,H st
( F = RingMorphismStr(# G,H,f #) & f is linear )

proof end;

Lm8: for G, H being Ring
for F being Morphism of G,H st G <= H holds
ex f being Function of G,H st F = RingMorphismStr(# G,H,f #)

proof end;

theorem :: RINGCAT1:7

for F being strict RingMorphism ex G, H being Ring ex f being Function of G,H st
( F is Morphism of G,H & F = RingMorphismStr(# G,H,f #) & f is linear )

proof end;

definition

let G, F be RingMorphism;
assume A1: dom G = cod F ;
func G * F -> strict RingMorphism means :Def10: :: RINGCAT1:def 10
for G1, G2, G3 being Ring
for g being Function of G2,G3
for f being Function of G1,G2 st RingMorphismStr(# the Dom of G, the Cod of G, the Fun of G #) = RingMorphismStr(# G2,G3,g #) & RingMorphismStr(# the Dom of F, the Cod of F, the Fun of F #) = RingMorphismStr(# G1,G2,f #) holds
it = RingMorphismStr(# G1,G3,(g * f) #);
existence
ex b₁ being strict RingMorphism st
for G1, G2, G3 being Ring
for g being Function of G2,G3
for f being Function of G1,G2 st RingMorphismStr(# the Dom of G, the Cod of G, the Fun of G #) = RingMorphismStr(# G2,G3,g #) & RingMorphismStr(# the Dom of F, the Cod of F, the Fun of F #) = RingMorphismStr(# G1,G2,f #) holds
b₁ = RingMorphismStr(# G1,G3,(g * f) #)

proof end;

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

proof end;

end;

:: deftheorem Def10 defines * RINGCAT1:def 10 :
for G, F being RingMorphism st dom G = cod F holds
for b₃ being strict RingMorphism holds
( b₃ = G * F iff for G1, G2, G3 being Ring
for g being Function of G2,G3
for f being Function of G1,G2 st RingMorphismStr(# the Dom of G, the Cod of G, the Fun of G #) = RingMorphismStr(# G2,G3,g #) & RingMorphismStr(# the Dom of F, the Cod of F, the Fun of F #) = RingMorphismStr(# G1,G2,f #) holds
b₃ = RingMorphismStr(# G1,G3,(g * f) #) );

theorem Th8: :: RINGCAT1:8

for G1, G2, G3 being Ring st G1 <= G2 & G2 <= G3 holds
G1 <= G3

proof end;

theorem Th9: :: RINGCAT1:9

for G2, G3, G1 being Ring
for G being Morphism of G2,G3
for F being Morphism of G1,G2 st G1 <= G2 & G2 <= G3 holds
G * F is Morphism of G1,G3

proof end;

definition

let G1, G2, G3 be Ring;
let G be Morphism of G2,G3;
let F be Morphism of G1,G2;
assume A1: ( G1 <= G2 & G2 <= G3 ) ;
func G *' F -> strict Morphism of G1,G3 equals :Def11: :: RINGCAT1:def 11
G * F;
coherence
G * F is strict Morphism of G1,G3 by A1, Th9;

end;

:: deftheorem Def11 defines *' RINGCAT1:def 11 :
for G1, G2, G3 being Ring
for G being Morphism of G2,G3
for F being Morphism of G1,G2 st G1 <= G2 & G2 <= G3 holds
G *' F = G * F;

theorem Th10: :: RINGCAT1:10

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

proof end;

theorem Th11: :: RINGCAT1:11

for f, g being strict RingMorphism st dom g = cod f holds
( dom (g * f) = dom f & cod (g * f) = cod g )

proof end;

theorem Th12: :: RINGCAT1:12

for G1, G2, G3, G4 being Ring
for f being Morphism of G1,G2
for g being Morphism of G2,G3
for h being Morphism of G3,G4 st G1 <= G2 & G2 <= G3 & G3 <= G4 holds
h * (g * f) = (h * g) * f

proof end;

theorem Th13: :: RINGCAT1:13

for f, g, h being strict RingMorphism st dom h = cod g & dom g = cod f holds
h * (g * f) = (h * g) * f

proof end;

theorem Th14: :: RINGCAT1:14

for G being Ring holds
( dom (ID G) = G & cod (ID G) = G & ( for f being strict RingMorphism st cod f = G holds
(ID G) * f = f ) & ( for g being strict RingMorphism st dom g = G holds
g * (ID G) = g ) )

proof end;

definition

let IT be set ;
attr IT is Ring_DOMAIN-like means :Def12: :: RINGCAT1:def 12
for x being Element of IT holds x is strict Ring;

end;

:: deftheorem Def12 defines Ring_DOMAIN-like RINGCAT1:def 12 :
for IT being set holds
( IT is Ring_DOMAIN-like iff for x being Element of IT holds x is strict Ring );

registration

cluster non empty Ring_DOMAIN-like set ;
existence
ex b₁ being set st
( b₁ is Ring_DOMAIN-like & not b₁ is empty )

proof end;

end;

definition

mode Ring_DOMAIN is non empty Ring_DOMAIN-like set ;

end;

definition

let V be Ring_DOMAIN;
:: original: Element
redefine mode Element of V -> Ring;
coherence
for b₁ being Element of V holds b₁ is Ring by Def12;

end;

registration

let V be Ring_DOMAIN;
cluster non empty right_complementable strict Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital Element of V;
existence
ex b₁ being Element of V st b₁ is strict

proof end;

end;

definition

let IT be set ;
attr IT is RingMorphism_DOMAIN-like means :Def13: :: RINGCAT1:def 13
for x being set st x in IT holds
x is strict RingMorphism;

end;

:: deftheorem Def13 defines RingMorphism_DOMAIN-like RINGCAT1:def 13 :
for IT being set holds
( IT is RingMorphism_DOMAIN-like iff for x being set st x in IT holds
x is strict RingMorphism );

registration

cluster non empty RingMorphism_DOMAIN-like set ;
existence
ex b₁ being non empty set st b₁ is RingMorphism_DOMAIN-like

proof end;

end;

definition

mode RingMorphism_DOMAIN is non empty RingMorphism_DOMAIN-like set ;

end;

definition

let M be RingMorphism_DOMAIN;
:: original: Element
redefine mode Element of M -> RingMorphism;
coherence
for b₁ being Element of M holds b₁ is RingMorphism by Def13;

end;

registration

let M be RingMorphism_DOMAIN;
cluster strict RingMorphism-like Element of M;
existence
ex b₁ being Element of M st b₁ is strict

proof end;

end;

theorem :: RINGCAT1:15

canceled;

theorem :: RINGCAT1:16

canceled;

theorem Th17: :: RINGCAT1:17

for f being strict RingMorphism holds {f} is RingMorphism_DOMAIN

proof end;

definition

let G, H be Ring;
mode RingMorphism_DOMAIN of G,H -> RingMorphism_DOMAIN means :Def14: :: RINGCAT1:def 14
for x being Element of it holds x is Morphism of G,H;
existence
ex b₁ being RingMorphism_DOMAIN st
for x being Element of b₁ holds x is Morphism of G,H

proof end;

end;

:: deftheorem Def14 defines RingMorphism_DOMAIN RINGCAT1:def 14 :
for G, H being Ring
for b₃ being RingMorphism_DOMAIN holds
( b₃ is RingMorphism_DOMAIN of G,H iff for x being Element of b₃ holds x is Morphism of G,H );

theorem Th18: :: RINGCAT1:18

for D being non empty set
for G, H being Ring holds
( D is RingMorphism_DOMAIN of G,H iff for x being Element of D holds x is Morphism of G,H )

proof end;

theorem :: RINGCAT1:19

for G, H being Ring
for f being Morphism of G,H holds {f} is RingMorphism_DOMAIN of G,H

proof end;

definition

let G, H be Ring;
assume A1: G <= H ;
func Morphs (G,H) -> RingMorphism_DOMAIN of G,H means :Def15: :: RINGCAT1:def 15
for x being set holds
( x in it iff x is Morphism of G,H );
existence
ex b₁ being RingMorphism_DOMAIN of G,H st
for x being set holds
( x in b₁ iff x is Morphism of G,H )

proof end;

uniqueness
for b₁, b₂ being RingMorphism_DOMAIN of G,H st ( for x being set holds
( x in b₁ iff x is Morphism of G,H ) ) & ( for x being set holds
( x in b₂ iff x is Morphism of G,H ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def15 defines Morphs RINGCAT1:def 15 :
for G, H being Ring st G <= H holds
for b₃ being RingMorphism_DOMAIN of G,H holds
( b₃ = Morphs (G,H) iff for x being set holds
( x in b₃ iff x is Morphism of G,H ) );

definition

let G, H be Ring;
let M be RingMorphism_DOMAIN of G,H;
:: original: Element
redefine mode Element of M -> Morphism of G,H;
coherence
for b₁ being Element of M holds b₁ is Morphism of G,H by Def14;

end;

registration

let G, H be Ring;
let M be RingMorphism_DOMAIN of G,H;
cluster strict RingMorphism-like Element of M;
existence
ex b₁ being Element of M st b₁ is strict

proof end;

end;

definition

let x, y be set ;
pred GO x,y means :Def16: :: RINGCAT1:def 16
ex x1, x2, x3, x4, x5, x6 being set st
( x = [[x1,x2,x3,x4],x5,x6] & ex G being strict Ring st
( y = G & x1 = the carrier of G & x2 = the addF of G & x3 = comp G & x4 = 0. G & x5 = the multF of G & x6 = 1. G ) );

end;

:: deftheorem Def16 defines GO RINGCAT1:def 16 :
for x, y being set holds
( GO x,y iff ex x1, x2, x3, x4, x5, x6 being set st
( x = [[x1,x2,x3,x4],x5,x6] & ex G being strict Ring st
( y = G & x1 = the carrier of G & x2 = the addF of G & x3 = comp G & x4 = 0. G & x5 = the multF of G & x6 = 1. G ) ) );

theorem Th20: :: RINGCAT1:20

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

proof end;

theorem Th21: :: RINGCAT1:21

for UN being Universe ex x being set st
( x in UN & GO x, Z3 )

proof end;

definition

let UN be Universe;
func RingObjects UN -> set means :Def17: :: RINGCAT1:def 17
for y being set holds
( y in it iff ex x being set st
( x in UN & GO x,y ) );
existence
ex b₁ being set st
for y being set holds
( y in b₁ iff ex x being set st
( x in UN & GO x,y ) )

proof end;

uniqueness
for b₁, b₂ being set st ( for y being set holds
( y in b₁ iff ex x being set st
( x in UN & GO x,y ) ) ) & ( for y being set holds
( y in b₂ iff ex x being set st
( x in UN & GO x,y ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def17 defines RingObjects RINGCAT1:def 17 :
for UN being Universe
for b₂ being set holds
( b₂ = RingObjects UN iff for y being set holds
( y in b₂ iff ex x being set st
( x in UN & GO x,y ) ) );

theorem Th22: :: RINGCAT1:22

for UN being Universe holds Z3 in RingObjects UN

proof end;

registration

let UN be Universe;
cluster RingObjects UN -> non empty ;
coherence
not RingObjects UN is empty by Th22;

end;

theorem Th23: :: RINGCAT1:23

for UN being Universe
for x being Element of RingObjects UN holds x is strict Ring

proof end;

registration

let UN be Universe;
cluster RingObjects UN -> Ring_DOMAIN-like ;
coherence
RingObjects UN is Ring_DOMAIN-like

proof end;

end;

definition

let V be Ring_DOMAIN;
func Morphs V -> RingMorphism_DOMAIN means :Def18: :: RINGCAT1:def 18
for x being set holds
( x in it iff ex G, H being Element of V st
( G <= H & x is Morphism of G,H ) );
existence
ex b₁ being RingMorphism_DOMAIN st
for x being set holds
( x in b₁ iff ex G, H being Element of V st
( G <= H & x is Morphism of G,H ) )

proof end;

uniqueness
for b₁, b₂ being RingMorphism_DOMAIN st ( for x being set holds
( x in b₁ iff ex G, H being Element of V st
( G <= H & x is Morphism of G,H ) ) ) & ( for x being set holds
( x in b₂ iff ex G, H being Element of V st
( G <= H & x is Morphism of G,H ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def18 defines Morphs RINGCAT1:def 18 :
for V being Ring_DOMAIN
for b₂ being RingMorphism_DOMAIN holds
( b₂ = Morphs V iff for x being set holds
( x in b₂ iff ex G, H being Element of V st
( G <= H & x is Morphism of G,H ) ) );

definition

let V be Ring_DOMAIN;
let F be Element of Morphs V;
:: original: dom
redefine func dom F -> Element of V;
coherence
dom F is Element of V

proof end;

:: original: cod
redefine func cod F -> Element of V;
coherence
cod F is Element of V

proof end;

end;

definition

let V be Ring_DOMAIN;
let G be Element of V;
func ID G -> strict Element of Morphs V equals :: RINGCAT1:def 19
ID G;
coherence
ID G is strict Element of Morphs V by Def18;

end;

:: deftheorem defines ID RINGCAT1:def 19 :
for V being Ring_DOMAIN
for G being Element of V holds ID G = ID G;

definition

let V be Ring_DOMAIN;
func dom V -> Function of (Morphs V),V means :Def20: :: RINGCAT1:def 20
for f being Element of Morphs V holds it . f = dom f;
existence
ex b₁ being Function of (Morphs V),V st
for f being Element of Morphs V holds b₁ . f = dom f

proof end;

uniqueness
for b₁, b₂ being Function of (Morphs V),V st ( for f being Element of Morphs V holds b₁ . f = dom f ) & ( for f being Element of Morphs V holds b₂ . f = dom f ) holds
b₁ = b₂

proof end;

func cod V -> Function of (Morphs V),V means :Def21: :: RINGCAT1:def 21
for f being Element of Morphs V holds it . f = cod f;
existence
ex b₁ being Function of (Morphs V),V st
for f being Element of Morphs V holds b₁ . f = cod f

proof end;

uniqueness
for b₁, b₂ being Function of (Morphs V),V st ( for f being Element of Morphs V holds b₁ . f = cod f ) & ( for f being Element of Morphs V holds b₂ . f = cod f ) holds
b₁ = b₂

proof end;

func ID V -> Function of V,(Morphs V) means :Def22: :: RINGCAT1:def 22
for G being Element of V holds it . G = ID G;
existence
ex b₁ being Function of V,(Morphs V) st
for G being Element of V holds b₁ . G = ID G

proof end;

uniqueness
for b₁, b₂ being Function of V,(Morphs V) st ( for G being Element of V holds b₁ . G = ID G ) & ( for G being Element of V holds b₂ . G = ID G ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def20 defines dom RINGCAT1:def 20 :
for V being Ring_DOMAIN
for b₂ being Function of (Morphs V),V holds
( b₂ = dom V iff for f being Element of Morphs V holds b₂ . f = dom f );

:: deftheorem Def21 defines cod RINGCAT1:def 21 :
for V being Ring_DOMAIN
for b₂ being Function of (Morphs V),V holds
( b₂ = cod V iff for f being Element of Morphs V holds b₂ . f = cod f );

:: deftheorem Def22 defines ID RINGCAT1:def 22 :
for V being Ring_DOMAIN
for b₂ being Function of V,(Morphs V) holds
( b₂ = ID V iff for G being Element of V holds b₂ . G = ID G );

theorem Th24: :: RINGCAT1:24

for V being Ring_DOMAIN
for g, f being Element of Morphs V st dom g = cod f holds
ex G1, G2, G3 being Element of V st
( G1 <= G2 & G2 <= G3 & g is Morphism of G2,G3 & f is Morphism of G1,G2 )

proof end;

theorem Th25: :: RINGCAT1:25

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

proof end;

definition

let V be Ring_DOMAIN;
func comp V -> PartFunc of [:(Morphs V),(Morphs V):],(Morphs V) means :Def23: :: RINGCAT1:def 23
( ( 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
ex b₁ being PartFunc of [:(Morphs V),(Morphs V):],(Morphs V) st
( ( for g, f being Element of Morphs V holds
( [g,f] in dom b₁ iff dom g = cod f ) ) & ( for g, f being Element of Morphs V st [g,f] in dom b₁ holds
b₁ . (g,f) = g * f ) )

proof end;

uniqueness
for b₁, b₂ being PartFunc of [:(Morphs V),(Morphs V):],(Morphs V) st ( for g, f being Element of Morphs V holds
( [g,f] in dom b₁ iff dom g = cod f ) ) & ( for g, f being Element of Morphs V st [g,f] in dom b₁ holds
b₁ . (g,f) = g * f ) & ( for g, f being Element of Morphs V holds
( [g,f] in dom b₂ iff dom g = cod f ) ) & ( for g, f being Element of Morphs V st [g,f] in dom b₂ holds
b₂ . (g,f) = g * f ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def23 defines comp RINGCAT1:def 23 :
for V being Ring_DOMAIN
for b₂ being PartFunc of [:(Morphs V),(Morphs V):],(Morphs V) holds
( b₂ = comp V iff ( ( for g, f being Element of Morphs V holds
( [g,f] in dom b₂ iff dom g = cod f ) ) & ( for g, f being Element of Morphs V st [g,f] in dom b₂ holds
b₂ . (g,f) = g * f ) ) );

definition

let UN be Universe;
func RingCat UN -> CatStr equals :: RINGCAT1:def 24
CatStr(# (RingObjects UN),(Morphs (RingObjects UN)),(dom (RingObjects UN)),(cod (RingObjects UN)),(comp (RingObjects UN)),(ID (RingObjects UN)) #);
coherence
CatStr(# (RingObjects UN),(Morphs (RingObjects UN)),(dom (RingObjects UN)),(cod (RingObjects UN)),(comp (RingObjects UN)),(ID (RingObjects UN)) #) is CatStr ;

end;

:: deftheorem defines RingCat RINGCAT1:def 24 :
for UN being Universe holds RingCat UN = CatStr(# (RingObjects UN),(Morphs (RingObjects UN)),(dom (RingObjects UN)),(cod (RingObjects UN)),(comp (RingObjects UN)),(ID (RingObjects UN)) #);

registration

let UN be Universe;
cluster RingCat UN -> non empty non void strict ;
coherence
( RingCat UN is strict & not RingCat UN is void & not RingCat UN is empty ) ;

end;

theorem Th26: :: RINGCAT1:26

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

proof end;

theorem Th27: :: RINGCAT1:27

for UN being Universe
for f being Morphism of (RingCat UN)
for f9 being Element of Morphs (RingObjects UN)
for b being Object of (RingCat UN)
for b9 being Element of RingObjects UN holds
( f is strict Element of Morphs (RingObjects UN) & f9 is Morphism of (RingCat UN) & b is strict Element of RingObjects UN & b9 is Object of (RingCat UN) )

proof end;

theorem Th28: :: RINGCAT1:28

for UN being Universe
for b being Object of (RingCat UN)
for b9 being Element of RingObjects UN st b = b9 holds
id b = ID b9

proof end;

theorem :: RINGCAT1:29

canceled;

theorem Th30: :: RINGCAT1:30

for UN being Universe
for f, g being Morphism of (RingCat UN)
for f9, g9 being Element of Morphs (RingObjects UN) st f = f9 & g = g9 holds
( ( dom g = cod f implies dom g9 = cod f9 ) & ( dom g9 = cod f9 implies dom g = cod f ) & ( dom g = cod f implies [g9,f9] in dom (comp (RingObjects UN)) ) & ( [g9,f9] in dom (comp (RingObjects UN)) implies dom g = cod f ) & ( dom g = cod f implies g * f = g9 * f9 ) & ( dom f = dom g implies dom f9 = dom g9 ) & ( dom f9 = dom g9 implies dom f = dom g ) & ( cod f = cod g implies cod f9 = cod g9 ) & ( cod f9 = cod g9 implies cod f = cod g ) )

proof end;

Lm9: for UN being Universe
for f, g being Morphism of (RingCat UN) st dom g = cod f holds
( dom (g * f) = dom f & cod (g * f) = cod g )

proof end;

Lm10: for UN being Universe
for f, g, h being Morphism of (RingCat UN) st dom h = cod g & dom g = cod f holds
h * (g * f) = (h * g) * f

proof end;

Lm11: for UN being Universe
for b being Object of (RingCat UN) holds
( dom (id b) = b & cod (id b) = b & ( for f being Morphism of (RingCat UN) st cod f = b holds
(id b) * f = f ) & ( for g being Morphism of (RingCat UN) st dom g = b holds
g * (id b) = g ) )

proof end;

registration

let UN be Universe;
cluster RingCat UN -> Category-like ;
coherence
RingCat UN is Category-like

proof end;

end;