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Theorem List for Metamath Proof Explorer - 33001-33100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlidlssbas 33001 The base set of the restriction of the ring to a (left) ideal is a subset of the base set of the ring. (Contributed by AV, 17-Feb-2020.)
 |-  L  =  (LIdeal `  R )   &    |-  I  =  ( Rs  U )   =>    |-  ( U  e.  L  ->  ( Base `  I )  C_  ( Base `  R )
 )
 
Theoremlidlbas 33002 A (left) ideal of a ring is the base set of the restriction of the ring to this ideal. (Contributed by AV, 17-Feb-2020.)
 |-  L  =  (LIdeal `  R )   &    |-  I  =  ( Rs  U )   =>    |-  ( U  e.  L  ->  ( Base `  I )  =  U )
 
Theoremlidlabl 33003 A (left) ideal of a ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)
 |-  L  =  (LIdeal `  R )   &    |-  I  =  ( Rs  U )   =>    |-  ( ( R  e.  Ring  /\  U  e.  L ) 
 ->  I  e.  Abel )
 
Theoremlidlmmgm 33004 The multiplicative group of a (left) ideal of a ring is a magma. (Contributed by AV, 17-Feb-2020.)
 |-  L  =  (LIdeal `  R )   &    |-  I  =  ( Rs  U )   =>    |-  ( ( R  e.  Ring  /\  U  e.  L ) 
 ->  (mulGrp `  I )  e. Mgm )
 
Theoremlidlmsgrp 33005 The multiplicative group of a (left) ideal of a ring is a semigroup. (Contributed by AV, 17-Feb-2020.)
 |-  L  =  (LIdeal `  R )   &    |-  I  =  ( Rs  U )   =>    |-  ( ( R  e.  Ring  /\  U  e.  L ) 
 ->  (mulGrp `  I )  e. SGrp )
 
Theoremlidlrng 33006 A (left) ideal of a ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.)
 |-  L  =  (LIdeal `  R )   &    |-  I  =  ( Rs  U )   =>    |-  ( ( R  e.  Ring  /\  U  e.  L ) 
 ->  I  e. Rng )
 
Theoremzlidlring 33007 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  =  ( Rs  U )   &    |-  B  =  (
 Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  U  =  {  .0.  } )  ->  I  e.  Ring
 )
 
Theoremuzlidlring 33008 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  =  ( Rs  U )   &    |-  B  =  (
 Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. Domn  /\  U  e.  L ) 
 ->  ( I  e.  Ring  <->  ( U  =  {  .0.  }  \/  U  =  B ) ) )
 
Theoremlidldomnnring 33009 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  =  ( Rs  U )   &    |-  B  =  (
 Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. Domn  /\  ( U  e.  L  /\  U  =/=  {  .0.  } 
 /\  U  =/=  B ) )  ->  I  e/  Ring
 )
 
21.24.10.6  The non-unital ring of even integers
 
Theorem0even 33010* 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
 
Theorem1neven 33011* 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
 
Theorem2even 33012* 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
 
Theorem2zlidl 33013* 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 ` ring )   =>    |-  E  e.  U
 
Theorem2zrng 33014* 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 32894. (Contributed by AV, 20-Feb-2020.)
 |-  E  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }   &    |-  U  =  (LIdeal ` ring )   &    |-  R  =  (rings  E )   =>    |-  R  e. Rng
 
Theorem2zrngbas 33015* 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  =  (flds  E )   =>    |-  E  =  ( Base `  R )
 
Theorem2zrngadd 33016* 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  =  (flds  E )   =>    |- 
 +  =  ( +g  `  R )
 
Theorem2zrng0 33017* 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  =  (flds  E )   =>    |-  0  =  ( 0g
 `  R )
 
Theorem2zrngamgm 33018* R is an (additive) magma. (Contributed by AV, 6-Jan-2020.)
 |-  E  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }   &    |-  R  =  (flds  E )   =>    |-  R  e. Mgm
 
Theorem2zrngasgrp 33019* R is an (additive) semigroup. (Contributed by AV, 4-Feb-2020.)
 |-  E  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }   &    |-  R  =  (flds  E )   =>    |-  R  e. SGrp
 
Theorem2zrngamnd 33020* R is an (additive) monoid. (Contributed by AV, 11-Feb-2020.)
 |-  E  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }   &    |-  R  =  (flds  E )   =>    |-  R  e.  Mnd
 
Theorem2zrngacmnd 33021* 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  =  (flds  E )   =>    |-  R  e. CMnd
 
Theorem2zrngagrp 33022* R is an (additive) group. (Contributed by AV, 6-Jan-2020.)
 |-  E  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }   &    |-  R  =  (flds  E )   =>    |-  R  e.  Grp
 
Theorem2zrngaabl 33023* 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  =  (flds  E )   =>    |-  R  e.  Abel
 
Theorem2zrngmul 33024* 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  =  (flds  E )   =>    |- 
 x.  =  ( .r
 `  R )
 
Theorem2zrngmmgm 33025* R is a (multiplicative) magma. (Contributed by AV, 11-Feb-2020.)
 |-  E  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }   &    |-  R  =  (flds  E )   &    |-  M  =  (mulGrp `  R )   =>    |-  M  e. Mgm
 
Theorem2zrngmsgrp 33026* R is a (multiplicative) semigroup. (Contributed by AV, 4-Feb-2020.)
 |-  E  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }   &    |-  R  =  (flds  E )   &    |-  M  =  (mulGrp `  R )   =>    |-  M  e. SGrp
 
Theorem2zrngALT 33027* The ring of integers restricted to the even integers is a (non-unital) ring, the "ring of even integers". Alternate version of 2zrng 33014, 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 33023) and a multiplicative semigroup (see 2zrngmsgrp 33026). (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  =  (flds  E )   &    |-  M  =  (mulGrp `  R )   =>    |-  R  e. Rng
 
Theorem2zrngnmlid 33028* 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  =  (flds  E )   &    |-  M  =  (mulGrp `  R )   =>    |- 
 A. b  e.  E  E. a  e.  E  ( b  x.  a
 )  =/=  a
 
Theorem2zrngnmrid 33029* 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  =  (flds  E )   &    |-  M  =  (mulGrp `  R )   =>    |- 
 A. a  e.  ( E  \  { 0 } ) A. b  e.  E  ( a  x.  b )  =/=  a
 
Theorem2zrngnmlid2 33030* 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  =  (flds  E )   &    |-  M  =  (mulGrp `  R )   =>    |- 
 A. a  e.  ( E  \  { 0 } ) A. b  e.  E  ( b  x.  a )  =/=  a
 
Theorem2zrngnring 33031* 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  =  (flds  E )   &    |-  M  =  (mulGrp `  R )   =>    |-  R  e/  Ring
 
21.24.10.7  A constructed not unital ring
 
Theoremplusgndxnmulrndx 33032 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 )
 
Theorembasendxnmulrndx 33033 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 )
 
Theoremcznrnglem 33034 Lemma for cznrng 33036: 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/n `  N )   &    |-  B  =  (
 Base `  Y )   &    |-  X  =  ( Y sSet  <. ( .r
 `  ndx ) ,  ( x  e.  B ,  y  e.  B  |->  C )
 >. )   =>    |-  B  =  ( Base `  X )
 
Theoremcznabel 33035 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/n `  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 )
 
Theoremcznrng 33036* 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/n `  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 )
 
Theoremcznnring 33037* 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/n `  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 )
 
21.24.10.8  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 33040. 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 33041 or dfrngc2 33053.

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 33046, 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 33047, whereas the composition is the ordinary composition of functions, see rngccofval 33051 and rngcco 33052.

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 33055, 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 33058. It follows that the category of non-unital rings RngCat is actually a category, see rngccat 33059 with the identity function as identity arrow, see rngcid 33060.

 
Syntaxcrngc 33038 Extend class notation to include the category Rng.
 class RngCat
 
SyntaxcrngcALTV 33039 Extend class notation to include the category Rng. (New usage is discouraged.)
 class RngCatALTV
 
Definitiondf-rngc 33040 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 ) ) ) ) )
 
Definitiondf-rngcALTV 33041* 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 ) ) ) >. } )
 
TheoremrngcvalALTV 33042* 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.  >. } )
 
Theoremrngcval 33043 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 ) )
 
Theoremrnghmresfn 33044 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 ) )
 
Theoremrnghmresel 33045 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 ) )
 
Theoremrngcbas 33046 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 )
 )
 
Theoremrngchomfval 33047 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 ) ) )
 
Theoremrngchom 33048 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 )
 )
 
Theoremelrngchom 33049 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 )
 ) )
 
Theoremrngchomfeqhom 33050 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 )
 )
 
Theoremrngccofval 33051 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 ) ) )
 
Theoremrngcco 33052 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 ) )
 
Theoremdfrngc2 33053 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.  >. } )
 
Theoremrnghmsscmap2 33054* 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 ) ) ) )
 
Theoremrnghmsscmap 33055* 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 ) ) ) )
 
Theoremrnghmsubcsetclem1 33056 Lemma 1 for rnghmsubcsetc 33058. (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 ) )
 
Theoremrnghmsubcsetclem2 33057* Lemma 2 for rnghmsubcsetc 33058. (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 ) )
 
Theoremrnghmsubcsetc 33058 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 )
 )
 
Theoremrngccat 33059 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 )
 
Theoremrngcid 33060 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 ) )
 
Theoremrngcsect 33061 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 ) ) ) )
 
Theoremrngcinv 33062 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 ) ) )
 
Theoremrngciso 33063 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 ) ) )
 
TheoremrngcbasALTV 33064 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 )
 )
 
TheoremrngchomfvalALTV 33065* 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 ) ) )
 
TheoremrngchomALTV 33066 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 )
 )
 
TheoremelrngchomALTV 33067 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 )
 ) )
 
TheoremrngccofvalALTV 33068* 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 ) ) ) )
 
TheoremrngccoALTV 33069 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 ) )
 
TheoremrngccatidALTV 33070* Lemma for rngccatALTV 33071. (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 ) ) ) ) )
 
TheoremrngccatALTV 33071 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 )
 
TheoremrngcidALTV 33072 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 ) )
 
TheoremrngcsectALTV 33073 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 ) ) ) )
 
TheoremrngcinvALTV 33074 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 ) ) )
 
TheoremrngcisoALTV 33075 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 ) ) )
 
TheoremrngchomffvalALTV 33076* 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 ) ) )
 
TheoremrngchomrnghmresALTV 33077 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 ) ) )
 
Theoremrngcifuestrc 33078* 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 )
 
Theoremfuncrngcsetc 33079* 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 33080, using cofuval2 15378 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 33078, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 15620. (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 )
 
TheoremfuncrngcsetcALT 33080* Alternate proof of funcrngcsetc 33079, using cofuval2 15378 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 33078, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 15620. Surprisingly, this proof is longer than the direct proof given in funcrngcsetc 33079. (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 )
 
Theoremzrinitorngc 33081 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 )
 )
 
Theoremzrtermorngc 33082 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 )
 )
 
Theoremzrzeroorngc 33083 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 )
 )
 
21.24.10.9  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 33086. 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 33099. 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 33092, and the morphisms/arrows are the ring homomorphisms restricted to this subset of the rings  ( RingHom  |`  ( B  X.  B ) ), see ringchomfval 33093, whereas the composition is the ordinary composition of functions, see ringccofval 33097 and ringcco 33098.

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 33101, it can be shown that the ring homomorphisms between rings are a subcategory (Subcat) of the category of extensible structures, see rhmsubcsetc 33104. It follows that the category of rings RingCat is actually a category, see ringccat 33105 with the identity function as identity arrow, see ringcid 33106.

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 33107, and that the ring homomorphisms between rings are a subcategory of the category of non-unital rings, see rhmsubcrngc 33110. 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 33111:  ( (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 33116.

 
Syntaxcringc 33084 Extend class notation to include the category Ring.
 class RingCat
 
SyntaxcringcALTV 33085 Extend class notation to include the category Ring. (New usage is discouraged.)
 class RingCatALTV
 
Definitiondf-ringc 33086 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 ) ) ) ) )
 
Definitiondf-ringcALTV 33087* 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 ) ) ) >. } )
 
TheoremringcvalALTV 33088* 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.  >. } )
 
Theoremringcval 33089 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 ) )
 
Theoremrhmresfn 33090 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 ) )
 
Theoremrhmresel 33091 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 ) )
 
Theoremringcbas 33092 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 ) )
 
Theoremringchomfval 33093 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 ) ) )
 
Theoremringchom 33094 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 )
 )
 
Theoremelringchom 33095 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 )
 ) )
 
Theoremringchomfeqhom 33096 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 )
 )
 
Theoremringccofval 33097 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 ) ) )
 
Theoremringcco 33098 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 ) )
 
Theoremdfringc2 33099 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.  >. } )
 
Theoremrhmsscmap2 33100* 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 ) ) ) )
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