P. 403 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 40201-40300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	zlidlring 40201	The zero (left) ideal of a non-unital ring is a unital ring (the zero ring). (Contributed by AV, 16-Feb-2020.)
|- L = (LIdeal ` R )   &    |- I = ( R ↾s U )   &    |- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Ring /\ U = { .0. } ) -> I e. Ring )
Theorem	uzlidlring 40202	Only the zero (left) ideal or the unit (left) ideal of a domain is a unital ring. (Contributed by AV, 18-Feb-2020.)
|- L = (LIdeal ` R )   &    |- I = ( R ↾s U )   &    |- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Domn /\ U e. L ) -> ( I e. Ring <-> ( U = { .0. } \/ U = B ) ) )
Theorem	lidldomnnring 40203	A (left) ideal of a domain which is neither the zero ideal nor the unit ideal is not a unital ring. (Contributed by AV, 18-Feb-2020.)
|- L = (LIdeal ` R )   &    |- I = ( R ↾s U )   &    |- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } /\ U =/= B ) ) -> I e/ Ring )
21.33.13.6  The non-unital ring of even integers
Theorem	0even 40204*	0 is an even integer. (Contributed by AV, 11-Feb-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   =>    |- 0 e. E
Theorem	1neven 40205*	1 is not an even integer. (Contributed by AV, 12-Feb-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   =>    |- 1 e/ E
Theorem	2even 40206*	2 is an even integer. (Contributed by AV, 12-Feb-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   =>    |- 2 e. E
Theorem	2zlidl 40207*	The even integers are a (left) ideal of the ring of integers. (Contributed by AV, 20-Feb-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   &    |- U = (LIdeal ` ℤring )   =>    |- E e. U
Theorem	2zrng 40208*	The ring of integers restricted to the even integers is a (non-unital) ring, the "ring of even integers". Remark: the structure of the complementary subset of the set of integers, the odd integers, is not even a magma, see oddinmgm 40088. (Contributed by AV, 20-Feb-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   &    |- U = (LIdeal ` ℤring )   &    |- R = (ℤring ↾s E )   =>    |- R e. Rng
Theorem	2zrngbas 40209*	The base set of R is the set of all even integers. (Contributed by AV, 31-Jan-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   &    |- R = (ℂfld ↾s E )   =>    |- E = ( Base ` R )
Theorem	2zrngadd 40210*	The group addition operation of R is the addition of complex numbers. (Contributed by AV, 31-Jan-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   &    |- R = (ℂfld ↾s E )   =>    |- + = ( +g ` R )
Theorem	2zrng0 40211*	The additive identity of R is the complex number 0. (Contributed by AV, 11-Feb-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   &    |- R = (ℂfld ↾s E )   =>    |- 0 = ( 0g ` R )
Theorem	2zrngamgm 40212*	R is an (additive) magma. (Contributed by AV, 6-Jan-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   &    |- R = (ℂfld ↾s E )   =>    |- R e. Mgm
Theorem	2zrngasgrp 40213*	R is an (additive) semigroup. (Contributed by AV, 4-Feb-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   &    |- R = (ℂfld ↾s E )   =>    |- R e. SGrp
Theorem	2zrngamnd 40214*	R is an (additive) monoid. (Contributed by AV, 11-Feb-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   &    |- R = (ℂfld ↾s E )   =>    |- R e. Mnd
Theorem	2zrngacmnd 40215*	R is a commutative (additive) monoid. (Contributed by AV, 11-Feb-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   &    |- R = (ℂfld ↾s E )   =>    |- R e. CMnd
Theorem	2zrngagrp 40216*	R is an (additive) group. (Contributed by AV, 6-Jan-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   &    |- R = (ℂfld ↾s E )   =>    |- R e. Grp
Theorem	2zrngaabl 40217*	R is an (additive) abelian group. (Contributed by AV, 11-Feb-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   &    |- R = (ℂfld ↾s E )   =>    |- R e. Abel
Theorem	2zrngmul 40218*	The ring multiplication operation of R is the multiplication on complex numbers. (Contributed by AV, 31-Jan-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   &    |- R = (ℂfld ↾s E )   =>    |- x. = ( .r ` R )
Theorem	2zrngmmgm 40219*	R is a (multiplicative) magma. (Contributed by AV, 11-Feb-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   &    |- R = (ℂfld ↾s E )   &    |- M = (mulGrp ` R )   =>    |- M e. Mgm
Theorem	2zrngmsgrp 40220*	R is a (multiplicative) semigroup. (Contributed by AV, 4-Feb-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   &    |- R = (ℂfld ↾s E )   &    |- M = (mulGrp ` R )   =>    |- M e. SGrp
Theorem	2zrngALT 40221*	The ring of integers restricted to the even integers is a (non-unital) ring, the "ring of even integers". Alternate version of 2zrng 40208, based on a restriction of the field of the complex numbers. The proof is based on the facts that the ring of even integers is an additive abelian group (see 2zrngaabl 40217) and a multiplicative semigroup (see 2zrngmsgrp 40220). (Contributed by AV, 11-Feb-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   &    |- R = (ℂfld ↾s E )   &    |- M = (mulGrp ` R )   =>    |- R e. Rng
Theorem	2zrngnmlid 40222*	R has no multiplicative (left) identity. (Contributed by AV, 12-Feb-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   &    |- R = (ℂfld ↾s E )   &    |- M = (mulGrp ` R )   =>    |- A. b e. E E. a e. E ( b x. a ) =/= a
Theorem	2zrngnmrid 40223*	R has no multiplicative (right) identity. (Contributed by AV, 12-Feb-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   &    |- R = (ℂfld ↾s E )   &    |- M = (mulGrp ` R )   =>    |- A. a e. ( E \ { 0 } ) A. b e. E ( a x. b ) =/= a
Theorem	2zrngnmlid2 40224*	R has no multiplicative (left) identity. (Contributed by AV, 12-Feb-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   &    |- R = (ℂfld ↾s E )   &    |- M = (mulGrp ` R )   =>    |- A. a e. ( E \ { 0 } ) A. b e. E ( b x. a ) =/= a
Theorem	2zrngnring 40225*	R is not a unital ring. (Contributed by AV, 6-Jan-2020.)
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }   &    |- R = (ℂfld ↾s E )   &    |- M = (mulGrp ` R )   =>    |- R e/ Ring
21.33.13.7  A constructed not unital ring
Theorem	plusgndxnmulrndx 40226	The slot for the group (addition) operation is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.)
|- ( +g ` ndx ) =/= ( .r ` ndx )
Theorem	basendxnmulrndx 40227	The slot for the base set is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.)
|- ( Base ` ndx ) =/= ( .r ` ndx )
Theorem	cznrnglem 40228	Lemma for cznrng 40230: The base set of the ring constructed from a ℤ/nℤ structure by replacing the (multiplicative) ring operation by a constant operation is the base set of the ℤ/nℤ structure. (Contributed by AV, 16-Feb-2020.)
|- Y = (ℤ/nℤ ` N )   &    |- B = ( Base ` Y )   &    |- X = ( Y sSet <. ( .r ` ndx ) , ( x e. B , y e. B |-> C ) >. )   =>    |- B = ( Base ` X )
Theorem	cznabel 40229	The ring constructed from a ℤ/nℤ structure by replacing the (multiplicative) ring operation by a constant operation is an abelian group. (Contributed by AV, 16-Feb-2020.)
|- Y = (ℤ/nℤ ` N )   &    |- B = ( Base ` Y )   &    |- X = ( Y sSet <. ( .r ` ndx ) , ( x e. B , y e. B |-> C ) >. )   =>    |- ( ( N e. NN /\ C e. B ) -> X e. Abel )
Theorem	cznrng 40230*	The ring constructed from a ℤ/nℤ structure by replacing the (multiplicative) ring operation by a constant operation is a non-unital ring. (Contributed by AV, 17-Feb-2020.)
|- Y = (ℤ/nℤ ` N )   &    |- B = ( Base ` Y )   &    |- X = ( Y sSet <. ( .r ` ndx ) , ( x e. B , y e. B |-> C ) >. )   &    |- .0. = ( 0g ` Y )   =>    |- ( ( N e. NN /\ C = .0. ) -> X e. Rng )
Theorem	cznnring 40231*	The ring constructed from a ℤ/nℤ structure with 1 < n by replacing the (multiplicative) ring operation by a constant operation is not a unital ring. (Contributed by AV, 17-Feb-2020.)
|- Y = (ℤ/nℤ ` N )   &    |- B = ( Base ` Y )   &    |- X = ( Y sSet <. ( .r ` ndx ) , ( x e. B , y e. B |-> C ) >. )   &    |- .0. = ( 0g ` Y )   =>    |- ( ( N e. ( ZZ>= ` 2 ) /\ C e. B ) -> X e/ Ring )
21.33.13.8  The category of non-unital rings The "category of non-unital rings" RngCat is the category of all non-unital rings Rng in a universe and non-unital ring homomorphisms RngHomo between these rings. This category is defined as "category restriction" of the category of extensible structures ExtStrCat, which restricts the objects to non-unital rings and the morphisms to the non-unital ring homomorphisms, while the composition of morphisms is preserved, see df-rngc 40234. Alternatively, the category of non-unital rings could have been defined as extensible structure consisting of three components/slots for the objects, morphisms and composition, see df-rngcALTV 40235 or dfrngc2 40247. Since we consider only "small categories" (i.e. categories whose objects and morphisms are actually sets and not proper classes), the objects of the category (i.e. the base set of the category regarded as extensible structure) are a subset of the non-unital rings (relativized to a subset or "universe" u) ( u i^i Rng ), see rngcbas 40240, and the morphisms/arrows are the non-unital ring homomorphisms restricted to this subset of the non-unital rings ( RngHomo |` ( B X. B ) ), see rngchomfval 40241, whereas the composition is the ordinary composition of functions, see rngccofval 40245 and rngcco 40246. By showing that the non-unital ring homomorphisms between non-unital rings are a subcategory subset ( C_cat) of the mappings between base sets of extensible structures, see rnghmsscmap 40249, it can be shown that the non-unital ring homomorphisms between non-unital rings are a subcategory (Subcat) of the category of extensible structures, see rnghmsubcsetc 40252. It follows that the category of non-unital rings RngCat is actually a category, see rngccat 40253 with the identity function as identity arrow, see rngcid 40254.
Syntax	crngc 40232	Extend class notation to include the category Rng.
class RngCat
Syntax	crngcALTV 40233	Extend class notation to include the category Rng. (New usage is discouraged.)
class RngCatALTV
Definition	df-rngc 40234	Definition of the category Rng, relativized to a subset u. This is the category of all non-unital rings in u and homomorphisms between these rings. Generally, we will take u to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
|- RngCat = ( u e. _V |-> ( (ExtStrCat ` u ) |`cat ( RngHomo |` ( ( u i^i Rng ) X. ( u i^i Rng ) ) ) ) )
Definition	df-rngcALTV 40235*	Definition of the category Rng, relativized to a subset u. This is the category of all non-unital rings in u and homomorphisms between these rings. Generally, we will take u to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
|- RngCatALTV = ( u e. _V |-> [_ ( u i^i Rng ) / b ]_ { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , ( x e. b , y e. b |-> ( x RngHomo y ) ) >. , <. (comp ` ndx ) , ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) RngHomo z ) , f e. ( ( 1st ` v ) RngHomo ( 2nd ` v ) ) |-> ( g o. f ) ) ) >. } )
Theorem	rngcvalALTV 40236*	Value of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
|- C = (RngCatALTV ` U )   &    |- ( ph -> U e. V )   &    |- ( ph -> B = ( U i^i Rng ) )   &    |- ( ph -> H = ( x e. B , y e. B |-> ( x RngHomo y ) ) )   &    |- ( ph -> .x. = ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) RngHomo z ) , f e. ( ( 1st ` v ) RngHomo ( 2nd ` v ) ) |-> ( g o. f ) ) ) )   =>    |- ( ph -> C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
Theorem	rngcval 40237	Value of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
|- C = (RngCat ` U )   &    |- ( ph -> U e. V )   &    |- ( ph -> B = ( U i^i Rng ) )   &    |- ( ph -> H = ( RngHomo |` ( B X. B ) ) )   =>    |- ( ph -> C = ( (ExtStrCat ` U ) |`cat H ) )
Theorem	rnghmresfn 40238	The class of non-unital ring homomorphisms restricted to subsets of non-unital rings is a function. (Contributed by AV, 4-Mar-2020.)
|- ( ph -> B = ( U i^i Rng ) )   &    |- ( ph -> H = ( RngHomo |` ( B X. B ) ) )   =>    |- ( ph -> H Fn ( B X. B ) )
Theorem	rnghmresel 40239	An element of the non-unital ring homomorphisms restricted to a subset of non-unital rings is a non-unital ring homomorphisms. (Contributed by AV, 9-Mar-2020.)
|- ( ph -> H = ( RngHomo |` ( B X. B ) ) )   =>    |- ( ( ph /\ ( X e. B /\ Y e. B ) /\ F e. ( X H Y ) ) -> F e. ( X RngHomo Y ) )
Theorem	rngcbas 40240	Set of objects of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
|- C = (RngCat ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   =>    |- ( ph -> B = ( U i^i Rng ) )
Theorem	rngchomfval 40241	Set of arrows of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
|- C = (RngCat ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- H = ( Hom ` C )   =>    |- ( ph -> H = ( RngHomo |` ( B X. B ) ) )
Theorem	rngchom 40242	Set of arrows of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.)
|- C = (RngCat ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- H = ( Hom ` C )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X H Y ) = ( X RngHomo Y ) )
Theorem	elrngchom 40243	A morphism of non-unital rings is a function. (Contributed by AV, 27-Feb-2020.)
|- C = (RngCat ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- H = ( Hom ` C )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( F e. ( X H Y ) -> F : ( Base ` X ) --> ( Base ` Y ) ) )
Theorem	rngchomfeqhom 40244	The functionalized Hom-set operation equals the Hom-set operation in the category of non-unital rings (in a universe). (Contributed by AV, 9-Mar-2020.)
|- C = (RngCat ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   =>    |- ( ph -> ( Hom f ` C ) = ( Hom ` C ) )
Theorem	rngccofval 40245	Composition in the category of non-unital rings. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
|- C = (RngCat ` U )   &    |- ( ph -> U e. V )   &    |- .x. = (comp ` C )   =>    |- ( ph -> .x. = (comp ` (ExtStrCat ` U ) ) )
Theorem	rngcco 40246	Composition in the category of non-unital rings. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
|- C = (RngCat ` U )   &    |- ( ph -> U e. V )   &    |- .x. = (comp ` C )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. U )   &    |- ( ph -> Z e. U )   &    |- ( ph -> F : ( Base ` X ) --> ( Base ` Y ) )   &    |- ( ph -> G : ( Base ` Y ) --> ( Base ` Z ) )   =>    |- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) = ( G o. F ) )
Theorem	dfrngc2 40247	Alternate definition of the category of non-unital rings (in a universe). (Contributed by AV, 16-Mar-2020.)
|- C = (RngCat ` U )   &    |- ( ph -> U e. V )   &    |- ( ph -> B = ( U i^i Rng ) )   &    |- ( ph -> H = ( RngHomo |` ( B X. B ) ) )   &    |- ( ph -> .x. = (comp ` (ExtStrCat ` U ) ) )   =>    |- ( ph -> C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
Theorem	rnghmsscmap2 40248*	The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory subset of the mappings between base sets of non-unital rings (in the same universe). (Contributed by AV, 6-Mar-2020.)
|- ( ph -> U e. V )   &    |- ( ph -> R = (Rng i^i U ) )   =>    |- ( ph -> ( RngHomo |` ( R X. R ) ) C_cat ( x e. R , y e. R |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) )
Theorem	rnghmsscmap 40249*	The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory subset of the mappings between base sets of extensible structures (in the same universe). (Contributed by AV, 9-Mar-2020.)
|- ( ph -> U e. V )   &    |- ( ph -> R = (Rng i^i U ) )   =>    |- ( ph -> ( RngHomo |` ( R X. R ) ) C_cat ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) )
Theorem	rnghmsubcsetclem1 40250	Lemma 1 for rnghmsubcsetc 40252. (Contributed by AV, 9-Mar-2020.)
|- C = (ExtStrCat ` U )   &    |- ( ph -> U e. V )   &    |- ( ph -> B = (Rng i^i U ) )   &    |- ( ph -> H = ( RngHomo |` ( B X. B ) ) )   =>    |- ( ( ph /\ x e. B ) -> ( ( Id ` C ) ` x ) e. ( x H x ) )
Theorem	rnghmsubcsetclem2 40251*	Lemma 2 for rnghmsubcsetc 40252. (Contributed by AV, 9-Mar-2020.)
|- C = (ExtStrCat ` U )   &    |- ( ph -> U e. V )   &    |- ( ph -> B = (Rng i^i U ) )   &    |- ( ph -> H = ( RngHomo |` ( B X. B ) ) )   =>    |- ( ( ph /\ x e. B ) -> A. y e. B A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. (comp ` C ) z ) f ) e. ( x H z ) )
Theorem	rnghmsubcsetc 40252	The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory of the category of extensible structures. (Contributed by AV, 9-Mar-2020.)
|- C = (ExtStrCat ` U )   &    |- ( ph -> U e. V )   &    |- ( ph -> B = (Rng i^i U ) )   &    |- ( ph -> H = ( RngHomo |` ( B X. B ) ) )   =>    |- ( ph -> H e. (Subcat ` C ) )
Theorem	rngccat 40253	The category of non-unital rings is a category. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 9-Mar-2020.)
|- C = (RngCat ` U )   =>    |- ( U e. V -> C e. Cat )
Theorem	rngcid 40254	The identity arrow in the category of non-unital rings is the identity function. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 10-Mar-2020.)
|- C = (RngCat ` U )   &    |- B = ( Base ` C )   &    |- .1. = ( Id ` C )   &    |- ( ph -> U e. V )   &    |- ( ph -> X e. B )   &    |- S = ( Base ` X )   =>    |- ( ph -> ( .1. ` X ) = ( _I |` S ) )
Theorem	rngcsect 40255	A section in the category of non-unital rings, written out. (Contributed by AV, 28-Feb-2020.)
|- C = (RngCat ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- E = ( Base ` X )   &    |- S = (Sect ` C )   =>    |- ( ph -> ( F ( X S Y ) G <-> ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) /\ ( G o. F ) = ( _I |` E ) ) ) )
Theorem	rngcinv 40256	An inverse in the category of non-unital rings is the converse operation. (Contributed by AV, 28-Feb-2020.)
|- C = (RngCat ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- N = (Inv ` C )   =>    |- ( ph -> ( F ( X N Y ) G <-> ( F e. ( X RngIsom Y ) /\ G = `' F ) ) )
Theorem	rngciso 40257	An isomorphism in the category of non-unital rings is a bijection. (Contributed by AV, 28-Feb-2020.)
|- C = (RngCat ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- I = ( Iso ` C )   =>    |- ( ph -> ( F e. ( X I Y ) <-> F e. ( X RngIsom Y ) ) )
Theorem	rngcbasALTV 40258	Set of objects of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
|- C = (RngCatALTV ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   =>    |- ( ph -> B = ( U i^i Rng ) )
Theorem	rngchomfvalALTV 40259*	Set of arrows of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
|- C = (RngCatALTV ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- H = ( Hom ` C )   =>    |- ( ph -> H = ( x e. B , y e. B |-> ( x RngHomo y ) ) )
Theorem	rngchomALTV 40260	Set of arrows of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
|- C = (RngCatALTV ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- H = ( Hom ` C )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X H Y ) = ( X RngHomo Y ) )
Theorem	elrngchomALTV 40261	A morphism of non-unital rings is a function. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
|- C = (RngCatALTV ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- H = ( Hom ` C )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( F e. ( X H Y ) -> F : ( Base ` X ) --> ( Base ` Y ) ) )
Theorem	rngccofvalALTV 40262*	Composition in the category of non-unital rings. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
|- C = (RngCatALTV ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- .x. = (comp ` C )   =>    |- ( ph -> .x. = ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) RngHomo z ) , f e. ( ( 1st ` v ) RngHomo ( 2nd ` v ) ) |-> ( g o. f ) ) ) )
Theorem	rngccoALTV 40263	Composition in the category of non-unital rings. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
|- C = (RngCatALTV ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- .x. = (comp ` C )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   &    |- ( ph -> F e. ( X RngHomo Y ) )   &    |- ( ph -> G e. ( Y RngHomo Z ) )   =>    |- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) = ( G o. F ) )
Theorem	rngccatidALTV 40264*	Lemma for rngccatALTV 40265. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
|- C = (RngCatALTV ` U )   &    |- B = ( Base ` C )   =>    |- ( U e. V -> ( C e. Cat /\ ( Id ` C ) = ( x e. B |-> ( _I |` ( Base ` x ) ) ) ) )
Theorem	rngccatALTV 40265	The category of non-unital rings is a category. (Contributed by AV, 27-Feb-2020.) (New usage is discouraged.)
|- C = (RngCatALTV ` U )   =>    |- ( U e. V -> C e. Cat )
Theorem	rngcidALTV 40266	The identity arrow in the category of non-unital rings is the identity function. (Contributed by AV, 27-Feb-2020.) (New usage is discouraged.)
|- C = (RngCatALTV ` U )   &    |- B = ( Base ` C )   &    |- .1. = ( Id ` C )   &    |- ( ph -> U e. V )   &    |- ( ph -> X e. B )   &    |- S = ( Base ` X )   =>    |- ( ph -> ( .1. ` X ) = ( _I |` S ) )
Theorem	rngcsectALTV 40267	A section in the category of non-unital rings, written out. (Contributed by AV, 28-Feb-2020.) (New usage is discouraged.)
|- C = (RngCatALTV ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- E = ( Base ` X )   &    |- S = (Sect ` C )   =>    |- ( ph -> ( F ( X S Y ) G <-> ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) /\ ( G o. F ) = ( _I |` E ) ) ) )
Theorem	rngcinvALTV 40268	An inverse in the category of non-unital rings is the converse operation. (Contributed by AV, 28-Feb-2020.) (New usage is discouraged.)
|- C = (RngCatALTV ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- N = (Inv ` C )   =>    |- ( ph -> ( F ( X N Y ) G <-> ( F e. ( X RngIsom Y ) /\ G = `' F ) ) )
Theorem	rngcisoALTV 40269	An isomorphism in the category of non-unital rings is a bijection. (Contributed by AV, 28-Feb-2020.) (New usage is discouraged.)
|- C = (RngCatALTV ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- I = ( Iso ` C )   =>    |- ( ph -> ( F e. ( X I Y ) <-> F e. ( X RngIsom Y ) ) )
Theorem	rngchomffvalALTV 40270*	The value of the functionalized Hom-set operation in the category of non-unital rings (in a universe) in maps-to notation for an operation. (Contributed by AV, 1-Mar-2020.) (New usage is discouraged.)
|- C = (RngCatALTV ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- F = ( Hom f ` C )   =>    |- ( ph -> F = ( x e. B , y e. B |-> ( x RngHomo y ) ) )
Theorem	rngchomrnghmresALTV 40271	The value of the functionalized Hom-set operation in the category of non-unital rings (in a universe) as restriction of the non-unital ring homomorphisms. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
|- C = (RngCatALTV ` U )   &    |- B = (Rng i^i U )   &    |- ( ph -> U e. V )   &    |- F = ( Hom f ` C )   =>    |- ( ph -> F = ( RngHomo |` ( B X. B ) ) )
Theorem	rngcifuestrc 40272*	The "inclusion functor" from the category of non-unital rings into the category of extensible structures. (Contributed by AV, 30-Mar-2020.)
|- R = (RngCat ` U )   &    |- E = (ExtStrCat ` U )   &    |- B = ( Base ` R )   &    |- ( ph -> U e. V )   &    |- ( ph -> F = ( _I |` B ) )   &    |- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( x RngHomo y ) ) ) )   =>    |- ( ph -> F ( R Func E ) G )
Theorem	funcrngcsetc 40273*	The "natural forgetful functor" from the category of non-unital rings into the category of sets which sends each non-unital ring to its underlying set (base set) and the morphisms (non-unital ring homomorphisms) to mappings of the corresponding base sets. An alternate proof is provided in funcrngcsetcALT 40274, using cofuval2 15841 to construct the "natural forgetful functor" from the category of non-unital rings into the category of sets by composing the "inclusion functor" from the category of non-unital rings into the category of extensible structures, see rngcifuestrc 40272, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 16083. (Contributed by AV, 26-Mar-2020.)
|- R = (RngCat ` U )   &    |- S = ( SetCat ` U )   &    |- B = ( Base ` R )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )   &    |- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( x RngHomo y ) ) ) )   =>    |- ( ph -> F ( R Func S ) G )
Theorem	funcrngcsetcALT 40274*	Alternate proof of funcrngcsetc 40273, using cofuval2 15841 to construct the "natural forgetful functor" from the category of non-unital rings into the category of sets by composing the "inclusion functor" from the category of non-unital rings into the category of extensible structures, see rngcifuestrc 40272, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 16083. Surprisingly, this proof is longer than the direct proof given in funcrngcsetc 40273. (Contributed by AV, 30-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
|- R = (RngCat ` U )   &    |- S = ( SetCat ` U )   &    |- B = ( Base ` R )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )   &    |- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( x RngHomo y ) ) ) )   =>    |- ( ph -> F ( R Func S ) G )
Theorem	zrinitorngc 40275	The zero ring is an initial object in the category of nonunital rings. (Contributed by AV, 18-Apr-2020.)
|- ( ph -> U e. V )   &    |- C = (RngCat ` U )   &    |- ( ph -> Z e. ( Ring \ NzRing ) )   &    |- ( ph -> Z e. U )   =>    |- ( ph -> Z e. (InitO ` C ) )
Theorem	zrtermorngc 40276	The zero ring is a terminal object in the category of nonunital rings. (Contributed by AV, 17-Apr-2020.)
|- ( ph -> U e. V )   &    |- C = (RngCat ` U )   &    |- ( ph -> Z e. ( Ring \ NzRing ) )   &    |- ( ph -> Z e. U )   =>    |- ( ph -> Z e. (TermO ` C ) )
Theorem	zrzeroorngc 40277	The zero ring is a zero object in the category of non-unital rings. (Contributed by AV, 18-Apr-2020.)
|- ( ph -> U e. V )   &    |- C = (RngCat ` U )   &    |- ( ph -> Z e. ( Ring \ NzRing ) )   &    |- ( ph -> Z e. U )   =>    |- ( ph -> Z e. (ZeroO ` C ) )
21.33.13.9  The category of (unital) rings The "category of unital rings" RingCat is the category of all (unital) rings Ring in a universe and (unital) ring homomorphisms RingHom between these rings. This category is defined as "category restriction" of the category of extensible structures ExtStrCat, which restricts the objects to (unital) rings and the morphisms to the (unital) ring homomorphisms, while the composition of morphisms is preserved, see df-ringc 40280. Alternatively, the category of unital rings could have been defined as extensible structure consisting of three components/slots for the objects, morphisms and composition, see dfringc2 40293. In the following, we omit the predicate "unital", so that "ring" and "ring homomorphism" (without predicate) always mean "unital ring" and "unital ring homomorphism". Since we consider only "small categories" (i.e. categories whose objects and morphisms are actually sets and not proper classes), the objects of the category (i.e. the base set of the category regarded as extensible structure) are a subset of the rings (relativized to a subset or "universe" u) ( u i^i Ring ), see ringcbas 40286, and the morphisms/arrows are the ring homomorphisms restricted to this subset of the rings ( RingHom |` ( B X. B ) ), see ringchomfval 40287, whereas the composition is the ordinary composition of functions, see ringccofval 40291 and ringcco 40292. By showing that the ring homomorphisms between rings are a subcategory subset ( C_cat) of the mappings between base sets of extensible structures, see rhmsscmap 40295, it can be shown that the ring homomorphisms between rings are a subcategory (Subcat) of the category of extensible structures, see rhmsubcsetc 40298. It follows that the category of rings RingCat is actually a category, see ringccat 40299 with the identity function as identity arrow, see ringcid 40300. Furthermore, it is shown that the ring homomorphisms between rings are a subcategory subset of the non-unital ring homomorphisms between non-unital rings, see rhmsscrnghm 40301, and that the ring homomorphisms between rings are a subcategory of the category of non-unital rings, see rhmsubcrngc 40304. By this, the restriction of the category of non-unital rings to the set of unital ring homomorphisms is the category of unital rings, see rngcresringcat 40305: ( (RngCat ` U ) |`cat ( RingHom |` ( B X. B ) ) ) = (RingCat ` U ) ). Finally, it is shown that the "natural forgetful functor" from the category of rings into the category of sets is the function which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets, see funcringcsetc 40310.
Syntax	cringc 40278	Extend class notation to include the category Ring.
class RingCat
Syntax	cringcALTV 40279	Extend class notation to include the category Ring. (New usage is discouraged.)
class RingCatALTV
Definition	df-ringc 40280	Definition of the category Ring, relativized to a subset u. See also the note in [Lang] p. 91, and the item Rng in [Adamek] p. 478. This is the category of all unital rings in u and homomorphisms between these rings. Generally, we will take u to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. (Contributed by AV, 13-Feb-2020.) (Revised by AV, 8-Mar-2020.)
|- RingCat = ( u e. _V |-> ( (ExtStrCat ` u ) |`cat ( RingHom |` ( ( u i^i Ring ) X. ( u i^i Ring ) ) ) ) )
Definition	df-ringcALTV 40281*	Definition of the category Ring, relativized to a subset u. This is the category of all rings in u and homomorphisms between these rings. Generally, we will take u to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. (Contributed by AV, 13-Feb-2020.) (New usage is discouraged.)
|- RingCatALTV = ( u e. _V |-> [_ ( u i^i Ring ) / b ]_ { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , ( x e. b , y e. b |-> ( x RingHom y ) ) >. , <. (comp ` ndx ) , ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) RingHom z ) , f e. ( ( 1st ` v ) RingHom ( 2nd ` v ) ) |-> ( g o. f ) ) ) >. } )
Theorem	ringcvalALTV 40282*	Value of the category of rings (in a universe). (Contributed by AV, 13-Feb-2020.) (New usage is discouraged.)
|- C = (RingCatALTV ` U )   &    |- ( ph -> U e. V )   &    |- ( ph -> B = ( U i^i Ring ) )   &    |- ( ph -> H = ( x e. B , y e. B |-> ( x RingHom y ) ) )   &    |- ( ph -> .x. = ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) RingHom z ) , f e. ( ( 1st ` v ) RingHom ( 2nd ` v ) ) |-> ( g o. f ) ) ) )   =>    |- ( ph -> C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
Theorem	ringcval 40283	Value of the category of unital rings (in a universe). (Contributed by AV, 13-Feb-2020.) (Revised by AV, 8-Mar-2020.)
|- C = (RingCat ` U )   &    |- ( ph -> U e. V )   &    |- ( ph -> B = ( U i^i Ring ) )   &    |- ( ph -> H = ( RingHom |` ( B X. B ) ) )   =>    |- ( ph -> C = ( (ExtStrCat ` U ) |`cat H ) )
Theorem	rhmresfn 40284	The class of unital ring homomorphisms restricted to subsets of unital rings is a function. (Contributed by AV, 10-Mar-2020.)
|- ( ph -> B = ( U i^i Ring ) )   &    |- ( ph -> H = ( RingHom |` ( B X. B ) ) )   =>    |- ( ph -> H Fn ( B X. B ) )
Theorem	rhmresel 40285	An element of the unital ring homomorphisms restricted to a subset of unital rings is a unital ring homomorphism. (Contributed by AV, 10-Mar-2020.)
|- ( ph -> H = ( RingHom |` ( B X. B ) ) )   =>    |- ( ( ph /\ ( X e. B /\ Y e. B ) /\ F e. ( X H Y ) ) -> F e. ( X RingHom Y ) )
Theorem	ringcbas 40286	Set of objects of the category of unital rings (in a universe). (Contributed by AV, 13-Feb-2020.) (Revised by AV, 8-Mar-2020.)
|- C = (RingCat ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   =>    |- ( ph -> B = ( U i^i Ring ) )
Theorem	ringchomfval 40287	Set of arrows of the category of unital rings (in a universe). (Contributed by AV, 14-Feb-2020.) (Revised by AV, 8-Mar-2020.)
|- C = (RingCat ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- H = ( Hom ` C )   =>    |- ( ph -> H = ( RingHom |` ( B X. B ) ) )
Theorem	ringchom 40288	Set of arrows of the category of unital rings (in a universe). (Contributed by AV, 14-Feb-2020.)
|- C = (RingCat ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- H = ( Hom ` C )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X H Y ) = ( X RingHom Y ) )
Theorem	elringchom 40289	A morphism of unital rings is a function. (Contributed by AV, 14-Feb-2020.)
|- C = (RingCat ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- H = ( Hom ` C )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( F e. ( X H Y ) -> F : ( Base ` X ) --> ( Base ` Y ) ) )
Theorem	ringchomfeqhom 40290	The functionalized Hom-set operation equals the Hom-set operation in the category of unital rings (in a universe). (Contributed by AV, 9-Mar-2020.)
|- C = (RingCat ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   =>    |- ( ph -> ( Hom f ` C ) = ( Hom ` C ) )
Theorem	ringccofval 40291	Composition in the category of unital rings. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 8-Mar-2020.)
|- C = (RingCat ` U )   &    |- ( ph -> U e. V )   &    |- .x. = (comp ` C )   =>    |- ( ph -> .x. = (comp ` (ExtStrCat ` U ) ) )
Theorem	ringcco 40292	Composition in the category of unital rings. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 8-Mar-2020.)
|- C = (RingCat ` U )   &    |- ( ph -> U e. V )   &    |- .x. = (comp ` C )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. U )   &    |- ( ph -> Z e. U )   &    |- ( ph -> F : ( Base ` X ) --> ( Base ` Y ) )   &    |- ( ph -> G : ( Base ` Y ) --> ( Base ` Z ) )   =>    |- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) = ( G o. F ) )
Theorem	dfringc2 40293	Alternate definition of the category of unital rings (in a universe). (Contributed by AV, 16-Mar-2020.)
|- C = (RingCat ` U )   &    |- ( ph -> U e. V )   &    |- ( ph -> B = ( U i^i Ring ) )   &    |- ( ph -> H = ( RingHom |` ( B X. B ) ) )   &    |- ( ph -> .x. = (comp ` (ExtStrCat ` U ) ) )   =>    |- ( ph -> C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
Theorem	rhmsscmap2 40294*	The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the mappings between base sets of unital rings (in the same universe). (Contributed by AV, 6-Mar-2020.)
|- ( ph -> U e. V )   &    |- ( ph -> R = ( Ring i^i U ) )   =>    |- ( ph -> ( RingHom |` ( R X. R ) ) C_cat ( x e. R , y e. R |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) )
Theorem	rhmsscmap 40295*	The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the mappings between base sets of extensible structures (in the same universe). (Contributed by AV, 9-Mar-2020.)
|- ( ph -> U e. V )   &    |- ( ph -> R = ( Ring i^i U ) )   =>    |- ( ph -> ( RingHom |` ( R X. R ) ) C_cat ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) )
Theorem	rhmsubcsetclem1 40296	Lemma 1 for rhmsubcsetc 40298. (Contributed by AV, 9-Mar-2020.)
|- C = (ExtStrCat ` U )   &    |- ( ph -> U e. V )   &    |- ( ph -> B = ( Ring i^i U ) )   &    |- ( ph -> H = ( RingHom |` ( B X. B ) ) )   =>    |- ( ( ph /\ x e. B ) -> ( ( Id ` C ) ` x ) e. ( x H x ) )
Theorem	rhmsubcsetclem2 40297*	Lemma 2 for rhmsubcsetc 40298. (Contributed by AV, 9-Mar-2020.)
|- C = (ExtStrCat ` U )   &    |- ( ph -> U e. V )   &    |- ( ph -> B = ( Ring i^i U ) )   &    |- ( ph -> H = ( RingHom |` ( B X. B ) ) )   =>    |- ( ( ph /\ x e. B ) -> A. y e. B A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. (comp ` C ) z ) f ) e. ( x H z ) )
Theorem	rhmsubcsetc 40298	The unital ring homomorphisms between unital rings (in a universe) are a subcategory of the category of extensible structures. (Contributed by AV, 9-Mar-2020.)
|- C = (ExtStrCat ` U )   &    |- ( ph -> U e. V )   &    |- ( ph -> B = ( Ring i^i U ) )   &    |- ( ph -> H = ( RingHom |` ( B X. B ) ) )   =>    |- ( ph -> H e. (Subcat ` C ) )
Theorem	ringccat 40299	The category of unital rings is a category. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 9-Mar-2020.)
|- C = (RingCat ` U )   =>    |- ( U e. V -> C e. Cat )
Theorem	ringcid 40300	The identity arrow in the category of unital rings is the identity function. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 10-Mar-2020.)
|- C = (RingCat ` U )   &    |- B = ( Base ` C )   &    |- .1. = ( Id ` C )   &    |- ( ph -> U e. V )   &    |- ( ph -> X e. B )   &    |- S = ( Base ` X )   =>    |- ( ph -> ( .1. ` X ) = ( _I |` S ) )