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Theorem List for Metamath Proof Explorer - 39501-39600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrngdir 39501 Distributive law for the multiplication operation of a nonunital ring (right-distributivity). (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e. Rng  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .+  Y )  .x.  Z )  =  ( ( X  .x.  Z )  .+  ( Y  .x.  Z ) ) )
 
Theoremrngcl 39502 Closure of the multiplication operation of a nonunital ring. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e. Rng  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  Y )  e.  B )
 
Theoremrnglz 39503 The zero of a nonunital ring is a left-absorbing element. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. Rng  /\  X  e.  B ) 
 ->  (  .0.  .x.  X )  =  .0.  )
 
21.33.13.3  Rng homomorphisms
 
Syntaxcrngh 39504 non-unital ring homomorphisms.
 class RngHomo
 
Syntaxcrngs 39505 non-unital ring isomorphisms.
 class RngIsom
 
Definitiondf-rnghomo 39506* Define the set of non-unital ring homomorphisms from  r to  s. (Contributed by AV, 20-Feb-2020.)
 |- RngHomo  =  ( r  e. Rng ,  s  e. Rng 
 |->  [_ ( Base `  r
 )  /  v ]_ [_ ( Base `  s )  /  w ]_ { f  e.  ( w  ^m  v
 )  |  A. x  e.  v  A. y  e.  v  ( ( f `
  ( x (
 +g  `  r )
 y ) )  =  ( ( f `  x ) ( +g  `  s ) ( f `
  y ) ) 
 /\  ( f `  ( x ( .r `  r ) y ) )  =  ( ( f `  x ) ( .r `  s
 ) ( f `  y ) ) ) } )
 
Definitiondf-rngisom 39507* Define the set of non-unital ring isomorphisms from  r to  s. (Contributed by AV, 20-Feb-2020.)
 |- RngIsom  =  ( r  e.  _V ,  s  e.  _V  |->  { f  e.  ( r RngHomo  s )  |  `' f  e.  ( s RngHomo  r ) } )
 
Theoremrnghmrcl 39508 Reverse closure of a non-unital ring homomorphism. (Contributed by AV, 22-Feb-2020.)
 |-  ( F  e.  ( R RngHomo  S )  ->  ( R  e. Rng  /\  S  e. Rng )
 )
 
Theoremrnghmfn 39509 The mapping of two non-unital rings to the non-unital ring homomorphisms between them is a function. (Contributed by AV, 1-Mar-2020.)
 |- RngHomo  Fn  (Rng  X. Rng
 )
 
Theoremrnghmval 39510* The set of the non-unital ring homomorphisms between two non-unital rings. (Contributed by AV, 22-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .*  =  ( .r `  S )   &    |-  C  =  (
 Base `  S )   &    |-  .+  =  ( +g  `  R )   &    |-  .+b  =  ( +g  `  S )   =>    |-  (
 ( R  e. Rng  /\  S  e. Rng )  ->  ( R RngHomo  S )  =  {
 f  e.  ( C 
 ^m  B )  | 
 A. x  e.  B  A. y  e.  B  ( ( f `  ( x  .+  y ) )  =  ( ( f `
  x )  .+b  ( f `  y
 ) )  /\  (
 f `  ( x  .x.  y ) )  =  ( ( f `  x )  .*  (
 f `  y )
 ) ) } )
 
Theoremisrnghm 39511* A function is a non-unital ring homomorphism iff it is a group homomorphism and preserves multiplication. (Contributed by AV, 22-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .*  =  ( .r `  S )   =>    |-  ( F  e.  ( R RngHomo  S )  <->  ( ( R  e. Rng  /\  S  e. Rng ) 
 /\  ( F  e.  ( R  GrpHom  S ) 
 /\  A. x  e.  B  A. y  e.  B  ( F `  ( x 
 .x.  y ) )  =  ( ( F `
  x )  .*  ( F `  y ) ) ) ) )
 
Theoremisrnghmmul 39512 A function is a non-unital ring homomorphism iff it preserves both addition and multiplication. (Contributed by AV, 27-Feb-2020.)
 |-  M  =  (mulGrp `  R )   &    |-  N  =  (mulGrp `  S )   =>    |-  ( F  e.  ( R RngHomo  S )  <->  ( ( R  e. Rng  /\  S  e. Rng ) 
 /\  ( F  e.  ( R  GrpHom  S ) 
 /\  F  e.  ( M MgmHom  N ) ) ) )
 
Theoremrnghmmgmhm 39513 A non-unital ring homomorphism is a homomorphism of multiplicative magmas. (Contributed by AV, 27-Feb-2020.)
 |-  M  =  (mulGrp `  R )   &    |-  N  =  (mulGrp `  S )   =>    |-  ( F  e.  ( R RngHomo  S )  ->  F  e.  ( M MgmHom  N ) )
 
Theoremrnghmval2 39514 The non-unital ring homomorphisms between two non-unital rings. (Contributed by AV, 1-Mar-2020.)
 |-  (
 ( R  e. Rng  /\  S  e. Rng )  ->  ( R RngHomo  S )  =  ( ( R  GrpHom  S )  i^i  ( (mulGrp `  R ) MgmHom  (mulGrp `  S )
 ) ) )
 
Theoremisrngisom 39515 An isomorphism of non-unital rings is a homomorphism whose converse is also a homomorphism. (Contributed by AV, 22-Feb-2020.)
 |-  (
 ( R  e.  V  /\  S  e.  W ) 
 ->  ( F  e.  ( R RngIsom  S )  <->  ( F  e.  ( R RngHomo  S )  /\  `' F  e.  ( S RngHomo  R ) ) ) )
 
Theoremrngimrcl 39516 Reverse closure for an isomorphism of non-unital rings. (Contributed by AV, 22-Feb-2020.)
 |-  ( F  e.  ( R RngIsom  S )  ->  ( R  e.  _V  /\  S  e.  _V ) )
 
Theoremrnghmghm 39517 A non-unital ring homomorphism is an additive group homomorphism. (Contributed by AV, 23-Feb-2020.)
 |-  ( F  e.  ( R RngHomo  S )  ->  F  e.  ( R  GrpHom  S ) )
 
Theoremrnghmf 39518 A ring homomorphism is a function. (Contributed by AV, 23-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   =>    |-  ( F  e.  ( R RngHomo  S )  ->  F : B
 --> C )
 
Theoremrnghmmul 39519 A homomorphism of non-unital rings preserves multiplication. (Contributed by AV, 23-Feb-2020.)
 |-  X  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .X. 
 =  ( .r `  S )   =>    |-  ( ( F  e.  ( R RngHomo  S )  /\  A  e.  X  /\  B  e.  X )  ->  ( F `  ( A  .x.  B ) )  =  ( ( F `
  A )  .X.  ( F `  B ) ) )
 
Theoremisrnghm2d 39520* Demonstration of non-unital ring homomorphism. (Contributed by AV, 23-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .X. 
 =  ( .r `  S )   &    |-  ( ph  ->  R  e. Rng )   &    |-  ( ph  ->  S  e. Rng )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  ( F `
  ( x  .x.  y ) )  =  ( ( F `  x )  .X.  ( F `
  y ) ) )   &    |-  ( ph  ->  F  e.  ( R  GrpHom  S ) )   =>    |-  ( ph  ->  F  e.  ( R RngHomo  S )
 )
 
Theoremisrnghmd 39521* Demonstration of non-unital ring homomorphism. (Contributed by AV, 23-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .X. 
 =  ( .r `  S )   &    |-  ( ph  ->  R  e. Rng )   &    |-  ( ph  ->  S  e. Rng )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  ( F `
  ( x  .x.  y ) )  =  ( ( F `  x )  .X.  ( F `
  y ) ) )   &    |-  C  =  (
 Base `  S )   &    |-  .+  =  ( +g  `  R )   &    |-  .+^  =  (
 +g  `  S )   &    |-  ( ph  ->  F : B --> C )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  ( F `
  ( x  .+  y ) )  =  ( ( F `  x )  .+^  ( F `
  y ) ) )   =>    |-  ( ph  ->  F  e.  ( R RngHomo  S )
 )
 
Theoremrnghmf1o 39522 A non-unital ring homomorphism is bijective iff its converse is also a non-unital ring homomorphism. (Contributed by AV, 27-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   =>    |-  ( F  e.  ( R RngHomo  S )  ->  ( F : B -1-1-onto-> C  <->  `' F  e.  ( S RngHomo  R ) ) )
 
Theoremisrngim 39523 An isomorphism of non-unital rings is a bijective homomorphism. (Contributed by AV, 23-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   =>    |-  (
 ( R  e.  V  /\  S  e.  W ) 
 ->  ( F  e.  ( R RngIsom  S )  <->  ( F  e.  ( R RngHomo  S )  /\  F : B -1-1-onto-> C ) ) )
 
Theoremrngimf1o 39524 An isomorphism of non-unital rings is a bijection. (Contributed by AV, 23-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   =>    |-  ( F  e.  ( R RngIsom  S )  ->  F : B
 -1-1-onto-> C )
 
Theoremrngimrnghm 39525 An isomorphism of non-unital rings is a homomorphism. (Contributed by AV, 23-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   =>    |-  ( F  e.  ( R RngIsom  S )  ->  F  e.  ( R RngHomo  S ) )
 
Theoremrnghmco 39526 The composition of non-unital ring homomorphisms is a homomorphism. (Contributed by AV, 27-Feb-2020.)
 |-  (
 ( F  e.  ( T RngHomo  U )  /\  G  e.  ( S RngHomo  T )
 )  ->  ( F  o.  G )  e.  ( S RngHomo  U ) )
 
Theoremidrnghm 39527 The identity homomorphism on a non-unital ring. (Contributed by AV, 27-Feb-2020.)
 |-  B  =  ( Base `  R )   =>    |-  ( R  e. Rng  ->  (  _I  |`  B )  e.  ( R RngHomo  R ) )
 
Theoremc0mgm 39528* The constant mapping to zero is a magma homomorphism into a monoid. Remark: Instead of the assumption that T is a monoid, it would be sufficient that T is a magma with a right or left identity. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  T )   &    |-  H  =  ( x  e.  B  |->  .0.  )   =>    |-  (
 ( S  e. Mgm  /\  T  e.  Mnd )  ->  H  e.  ( S MgmHom  T ) )
 
Theoremc0mhm 39529* The constant mapping to zero is a monoid homomorphism. (Contributed by AV, 16-Apr-2020.)
 |-  B  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  T )   &    |-  H  =  ( x  e.  B  |->  .0.  )   =>    |-  (
 ( S  e.  Mnd  /\  T  e.  Mnd )  ->  H  e.  ( S MndHom  T ) )
 
Theoremc0ghm 39530* The constant mapping to zero is a group homomorphism. (Contributed by AV, 16-Apr-2020.)
 |-  B  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  T )   &    |-  H  =  ( x  e.  B  |->  .0.  )   =>    |-  (
 ( S  e.  Grp  /\  T  e.  Grp )  ->  H  e.  ( S 
 GrpHom  T ) )
 
Theoremc0rhm 39531* The constant mapping to zero is a ring homomorphism from any ring to the zero ring. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  T )   &    |-  H  =  ( x  e.  B  |->  .0.  )   =>    |-  (
 ( S  e.  Ring  /\  T  e.  ( Ring  \ NzRing
 ) )  ->  H  e.  ( S RingHom  T )
 )
 
Theoremc0rnghm 39532* The constant mapping to zero is a nonunital ring homomorphism from any nonunital ring to the zero ring. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  T )   &    |-  H  =  ( x  e.  B  |->  .0.  )   =>    |-  (
 ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing
 ) )  ->  H  e.  ( S RngHomo  T )
 )
 
Theoremc0snmgmhm 39533* The constant mapping to zero is a magma homomorphism from a magma with one element to any monoid. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  T )   &    |-  .0.  =  ( 0g `  S )   &    |-  H  =  ( x  e.  B  |->  .0.  )   =>    |-  (
 ( S  e.  Mnd  /\  T  e. Mgm  /\  ( # `
  B )  =  1 )  ->  H  e.  ( T MgmHom  S )
 )
 
Theoremc0snmhm 39534* The constant mapping to zero is a monoid homomorphism from the trivial monoid (consisting of the zero only) to any monoid. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  T )   &    |-  .0.  =  ( 0g `  S )   &    |-  H  =  ( x  e.  B  |->  .0.  )   &    |-  Z  =  ( 0g `  T )   =>    |-  ( ( S  e.  Mnd  /\  T  e.  Mnd  /\  B  =  { Z } )  ->  H  e.  ( T MndHom  S ) )
 
Theoremc0snghm 39535* The constant mapping to zero is a group homomorphism from the trivial group (consisting of the zero only) to any group. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  T )   &    |-  .0.  =  ( 0g `  S )   &    |-  H  =  ( x  e.  B  |->  .0.  )   &    |-  Z  =  ( 0g `  T )   =>    |-  ( ( S  e.  Grp  /\  T  e.  Grp  /\  B  =  { Z } )  ->  H  e.  ( T  GrpHom  S ) )
 
Theoremzrrnghm 39536* The constant mapping to zero is a nonunital ring homomorphism from the zero ring to any nonunital ring. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  T )   &    |-  .0.  =  ( 0g `  S )   &    |-  H  =  ( x  e.  B  |->  .0.  )   =>    |-  (
 ( S  e. Rng  /\  T  e.  ( Ring  \ NzRing
 ) )  ->  H  e.  ( T RngHomo  S )
 )
 
21.33.13.4  Ring homomorphisms (extension)
 
Theoremrhmfn 39537 The mapping of two rings to the ring homomorphisms between them is a function. (Contributed by AV, 1-Mar-2020.)
 |- RingHom  Fn  ( Ring  X.  Ring )
 
Theoremrhmval 39538 The ring homomorphisms between two rings. (Contributed by AV, 1-Mar-2020.)
 |-  (
 ( R  e.  Ring  /\  S  e.  Ring )  ->  ( R RingHom  S )  =  ( ( R  GrpHom  S )  i^i  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) ) ) )
 
Theoremrhmisrnghm 39539 Each unital ring homomorphism is a non-unital ring homomorphism. (Contributed by AV, 29-Feb-2020.)
 |-  ( F  e.  ( R RingHom  S )  ->  F  e.  ( R RngHomo  S ) )
 
21.33.13.5  Ideals as non-unital rings
 
Theoremlidldomn1 39540* If a (left) ideal (which is not the zero ideal) of a domain has a multiplicative identity element, the identity element is the identity of the domain. (Contributed by AV, 17-Feb-2020.)
 |-  L  =  (LIdeal `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. Domn  /\  ( U  e.  L  /\  U  =/=  {  .0.  } )  /\  I  e.  U )  ->  ( A. x  e.  U  ( ( I  .x.  x )  =  x  /\  ( x  .x.  I
 )  =  x ) 
 ->  I  =  .1.  ) )
 
Theoremlidlssbas 39541 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 39542 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 39543 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 39544 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 39545 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 39546 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 39547 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 39548 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 39549 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
 )
 
21.33.13.6  The non-unital ring of even integers
 
Theorem0even 39550* 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 39551* 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 39552* 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 39553* 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 39554* 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 39434. (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 39555* 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 39556* 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 39557* 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 39558* 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 39559* 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 39560* 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 39561* 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 39562* 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 39563* 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 39564* 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 39565* 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 39566* 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 39567* The ring of integers restricted to the even integers is a (non-unital) ring, the "ring of even integers". Alternate version of 2zrng 39554, 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 39563) and a multiplicative semigroup (see 2zrngmsgrp 39566). (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 39568* 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 39569* 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 39570* 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 39571* 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
 
21.33.13.7  A constructed not unital ring
 
Theoremplusgndxnmulrndx 39572 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 39573 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 39574 Lemma for cznrng 39576: 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 39575 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 39576* 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 39577* 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 )
 
21.33.13.8  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 39580. 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 39581 or dfrngc2 39593.

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 39586, 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 39587, whereas the composition is the ordinary composition of functions, see rngccofval 39591 and rngcco 39592.

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 39595, 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 39598. It follows that the category of non-unital rings RngCat is actually a category, see rngccat 39599 with the identity function as identity arrow, see rngcid 39600.

 
Syntaxcrngc 39578 Extend class notation to include the category Rng.
 class RngCat
 
SyntaxcrngcALTV 39579 Extend class notation to include the category Rng. (New usage is discouraged.)
 class RngCatALTV
 
Definitiondf-rngc 39580 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 39581* 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 39582* 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 39583 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 39584 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 39585 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 39586 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 39587 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 39588 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 39589 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 39590 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 39591 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 39592 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 39593 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 39594* 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 39595* 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 39596 Lemma 1 for rnghmsubcsetc 39598. (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 39597* Lemma 2 for rnghmsubcsetc 39598. (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 39598 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 39599 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 39600 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 ) )
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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