HomeHome Metamath Proof Explorer
Theorem List (p. 389 of 394)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-26406)
  Hilbert Space Explorer  Hilbert Space Explorer
(26407-27929)
  Users' Mathboxes  Users' Mathboxes
(27930-39310)
 

Theorem List for Metamath Proof Explorer - 38801-38900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremringcinv 38801 An inverse in the category of unital rings is the converse operation. (Contributed by AV, 14-Feb-2020.)
 |-  C  =  (RingCat `  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 RingIso  Y )  /\  G  =  `' F ) ) )
 
Theoremringciso 38802 An isomorphism in the category of unital rings is a bijection. (Contributed by AV, 14-Feb-2020.)
 |-  C  =  (RingCat `  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 RingIso  Y ) ) )
 
Theoremringcbasbas 38803 An element of the base set of the base set of the category of unital rings (i.e. the base set of a ring) belongs to the considered weak universe. (Contributed by AV, 15-Feb-2020.)
 |-  C  =  (RingCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  U  e. WUni )   =>    |-  (
 ( ph  /\  R  e.  B )  ->  ( Base `  R )  e.  U )
 
Theoremfuncringcsetc 38804* The "natural forgetful functor" from the category of unital rings into the category of sets which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets. (Contributed by AV, 26-Mar-2020.)
 |-  R  =  (RingCat `  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 RingHom  y ) ) ) )   =>    |-  ( ph  ->  F ( R  Func  S ) G )
 
TheoremfuncringcsetcALTV2lem1 38805* Lemma 1 for funcringcsetcALTV2 38814. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
 |-  R  =  (RingCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   =>    |-  ( ( ph  /\  X  e.  B )  ->  ( F `  X )  =  ( Base `  X )
 )
 
TheoremfuncringcsetcALTV2lem2 38806* Lemma 2 for funcringcsetcALTV2 38814. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
 |-  R  =  (RingCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   =>    |-  ( ( ph  /\  X  e.  B )  ->  ( F `  X )  e.  U )
 
TheoremfuncringcsetcALTV2lem3 38807* Lemma 3 for funcringcsetcALTV2 38814. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
 |-  R  =  (RingCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   =>    |-  ( ph  ->  F : B --> C )
 
TheoremfuncringcsetcALTV2lem4 38808* Lemma 4 for funcringcsetcALTV2 38814. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
 |-  R  =  (RingCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x RingHom  y ) ) ) )   =>    |-  ( ph  ->  G  Fn  ( B  X.  B ) )
 
TheoremfuncringcsetcALTV2lem5 38809* Lemma 5 for funcringcsetcALTV2 38814. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
 |-  R  =  (RingCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x RingHom  y ) ) ) )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X G Y )  =  (  _I  |`  ( X RingHom  Y ) ) )
 
TheoremfuncringcsetcALTV2lem6 38810* Lemma 6 for funcringcsetcALTV2 38814. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
 |-  R  =  (RingCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x RingHom  y ) ) ) )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )  /\  H  e.  ( X RingHom  Y ) )  ->  ( ( X G Y ) `  H )  =  H )
 
TheoremfuncringcsetcALTV2lem7 38811* Lemma 7 for funcringcsetcALTV2 38814. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
 |-  R  =  (RingCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x RingHom  y ) ) ) )   =>    |-  ( ( ph  /\  X  e.  B )  ->  (
 ( X G X ) `  ( ( Id
 `  R ) `  X ) )  =  ( ( Id `  S ) `  ( F `  X ) ) )
 
TheoremfuncringcsetcALTV2lem8 38812* Lemma 8 for funcringcsetcALTV2 38814. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
 |-  R  =  (RingCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x RingHom  y ) ) ) )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X G Y ) : ( X ( Hom  `  R ) Y ) --> ( ( F `  X ) ( Hom  `  S ) ( F `  Y ) ) )
 
TheoremfuncringcsetcALTV2lem9 38813* Lemma 9 for funcringcsetcALTV2 38814. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
 |-  R  =  (RingCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x RingHom  y ) ) ) )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( H  e.  ( X ( Hom  `  R ) Y )  /\  K  e.  ( Y ( Hom  `  R ) Z ) ) )  ->  (
 ( X G Z ) `  ( K (
 <. X ,  Y >. (comp `  R ) Z ) H ) )  =  ( ( ( Y G Z ) `  K ) ( <. ( F `  X ) ,  ( F `  Y ) >. (comp `  S ) ( F `
  Z ) ) ( ( X G Y ) `  H ) ) )
 
TheoremfuncringcsetcALTV2 38814* The "natural forgetful functor" from the category of unital rings into the category of sets which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets. (Contributed by AV, 16-Feb-2020.) (New usage is discouraged.)
 |-  R  =  (RingCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x RingHom  y ) ) ) )   =>    |-  ( ph  ->  F ( R  Func  S ) G )
 
TheoremringcbasALTV 38815 Set of objects of the category of rings (in a universe). (Contributed by AV, 13-Feb-2020.) (New usage is discouraged.)
 |-  C  =  (RingCatALTV `  U )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  U  e.  V )   =>    |-  ( ph  ->  B  =  ( U  i^i  Ring ) )
 
TheoremringchomfvalALTV 38816* Set of arrows of the category of rings (in a universe). (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
 |-  C  =  (RingCatALTV `  U )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  U  e.  V )   &    |-  H  =  ( Hom  `  C )   =>    |-  ( ph  ->  H  =  ( x  e.  B ,  y  e.  B  |->  ( x RingHom  y ) ) )
 
TheoremringchomALTV 38817 Set of arrows of the category of rings (in a universe). (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
 |-  C  =  (RingCatALTV `  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 )
 )
 
TheoremelringchomALTV 38818 A morphism of rings is a function. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
 |-  C  =  (RingCatALTV `  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 )
 ) )
 
TheoremringccofvalALTV 38819* Composition in the category of rings. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
 |-  C  =  (RingCatALTV `  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 ) RingHom  z ) ,  f  e.  ( ( 1st `  v
 ) RingHom  ( 2nd `  v
 ) )  |->  ( g  o.  f ) ) ) )
 
TheoremringccoALTV 38820 Composition in the category of rings. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
 |-  C  =  (RingCatALTV `  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 RingHom  Y )
 )   &    |-  ( ph  ->  G  e.  ( Y RingHom  Z )
 )   =>    |-  ( ph  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  =  ( G  o.  F ) )
 
TheoremringccatidALTV 38821* Lemma for ringccatALTV 38822. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
 |-  C  =  (RingCatALTV `  U )   &    |-  B  =  ( Base `  C )   =>    |-  ( U  e.  V  ->  ( C  e.  Cat  /\  ( Id `  C )  =  ( x  e.  B  |->  (  _I  |`  ( Base `  x ) ) ) ) )
 
TheoremringccatALTV 38822 The category of rings is a category. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
 |-  C  =  (RingCatALTV `  U )   =>    |-  ( U  e.  V  ->  C  e.  Cat )
 
TheoremringcidALTV 38823 The identity arrow in the category of rings is the identity function. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
 |-  C  =  (RingCatALTV `  U )   &    |-  B  =  ( Base `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  X  e.  B )   &    |-  S  =  ( Base `  X )   =>    |-  ( ph  ->  (  .1.  `  X )  =  (  _I  |`  S ) )
 
TheoremringcsectALTV 38824 A section in the category of rings, written out. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
 |-  C  =  (RingCatALTV `  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 RingHom  Y )  /\  G  e.  ( Y RingHom  X )  /\  ( G  o.  F )  =  (  _I  |`  E ) ) ) )
 
TheoremringcinvALTV 38825 An inverse in the category of rings is the converse operation. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
 |-  C  =  (RingCatALTV `  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 RingIso  Y )  /\  G  =  `' F ) ) )
 
TheoremringcisoALTV 38826 An isomorphism in the category of rings is a bijection. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
 |-  C  =  (RingCatALTV `  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 RingIso  Y ) ) )
 
TheoremringcbasbasALTV 38827 An element of the base set of the base set of the category of rings (i.e. the base set of a ring) belongs to the considered weak universe. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
 |-  C  =  (RingCatALTV `  U )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  U  e. WUni )   =>    |-  (
 ( ph  /\  R  e.  B )  ->  ( Base `  R )  e.  U )
 
Theoremfuncringcsetclem1ALTV 38828* Lemma 1 for funcringcsetcALTV 38837. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
 |-  R  =  (RingCatALTV `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   =>    |-  ( ( ph  /\  X  e.  B )  ->  ( F `  X )  =  ( Base `  X )
 )
 
Theoremfuncringcsetclem2ALTV 38829* Lemma 2 for funcringcsetcALTV 38837. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
 |-  R  =  (RingCatALTV `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   =>    |-  ( ( ph  /\  X  e.  B )  ->  ( F `  X )  e.  U )
 
Theoremfuncringcsetclem3ALTV 38830* Lemma 3 for funcringcsetcALTV 38837. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
 |-  R  =  (RingCatALTV `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   =>    |-  ( ph  ->  F : B --> C )
 
Theoremfuncringcsetclem4ALTV 38831* Lemma 4 for funcringcsetcALTV 38837. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
 |-  R  =  (RingCatALTV `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x RingHom  y ) ) ) )   =>    |-  ( ph  ->  G  Fn  ( B  X.  B ) )
 
Theoremfuncringcsetclem5ALTV 38832* Lemma 5 for funcringcsetcALTV 38837. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
 |-  R  =  (RingCatALTV `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x RingHom  y ) ) ) )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X G Y )  =  (  _I  |`  ( X RingHom  Y ) ) )
 
Theoremfuncringcsetclem6ALTV 38833* Lemma 6 for funcringcsetcALTV 38837. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
 |-  R  =  (RingCatALTV `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x RingHom  y ) ) ) )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )  /\  H  e.  ( X RingHom  Y ) )  ->  ( ( X G Y ) `  H )  =  H )
 
Theoremfuncringcsetclem7ALTV 38834* Lemma 7 for funcringcsetcALTV 38837. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
 |-  R  =  (RingCatALTV `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x RingHom  y ) ) ) )   =>    |-  ( ( ph  /\  X  e.  B )  ->  (
 ( X G X ) `  ( ( Id
 `  R ) `  X ) )  =  ( ( Id `  S ) `  ( F `  X ) ) )
 
Theoremfuncringcsetclem8ALTV 38835* Lemma 8 for funcringcsetcALTV 38837. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
 |-  R  =  (RingCatALTV `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x RingHom  y ) ) ) )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X G Y ) : ( X ( Hom  `  R ) Y ) --> ( ( F `  X ) ( Hom  `  S ) ( F `  Y ) ) )
 
Theoremfuncringcsetclem9ALTV 38836* Lemma 9 for funcringcsetcALTV 38837. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
 |-  R  =  (RingCatALTV `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x RingHom  y ) ) ) )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( H  e.  ( X ( Hom  `  R ) Y )  /\  K  e.  ( Y ( Hom  `  R ) Z ) ) )  ->  (
 ( X G Z ) `  ( K (
 <. X ,  Y >. (comp `  R ) Z ) H ) )  =  ( ( ( Y G Z ) `  K ) ( <. ( F `  X ) ,  ( F `  Y ) >. (comp `  S ) ( F `
  Z ) ) ( ( X G Y ) `  H ) ) )
 
TheoremfuncringcsetcALTV 38837* The "natural forgetful functor" from the category of rings into the category of sets which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets. (Contributed by AV, 16-Feb-2020.) (New usage is discouraged.)
 |-  R  =  (RingCatALTV `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x RingHom  y ) ) ) )   =>    |-  ( ph  ->  F ( R  Func  S ) G )
 
Theoremirinitoringc 38838 The ring of integers is an initial object in the category of unital rings (within a universe containing the ring of integers). Example 7.2 (6) of [Adamek] p. 101 , and example in [Lang] p. 58. (Contributed by AV, 3-Apr-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->ring  e.  U )   &    |-  C  =  (RingCat `  U )   =>    |-  ( ph  ->ring  e.  (InitO `  C )
 )
 
Theoremzrtermoringc 38839 The zero ring is a terminal object in the category of unital rings. (Contributed by AV, 17-Apr-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RingCat `  U )   &    |-  ( ph  ->  Z  e.  ( Ring  \ NzRing ) )   &    |-  ( ph  ->  Z  e.  U )   =>    |-  ( ph  ->  Z  e.  (TermO `  C )
 )
 
Theoremzrninitoringc 38840* The zero ring is not an initial object in the category of unital rings (if the universe contains at least one unital ring different from the zero ring). (Contributed by AV, 18-Apr-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RingCat `  U )   &    |-  ( ph  ->  Z  e.  ( Ring  \ NzRing ) )   &    |-  ( ph  ->  Z  e.  U )   &    |-  ( ph  ->  E. r  e.  ( Base `  C ) r  e. NzRing )   =>    |-  ( ph  ->  Z  e/  (InitO `  C )
 )
 
Theoremnzerooringczr 38841 There is no zero object in the category of unital rings (at least in a universe which contains the zero ring and the ring of integers). Example 7.9 (3) in [Adamek] p. 103. (Contributed by AV, 18-Apr-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RingCat `  U )   &    |-  ( ph  ->  Z  e.  ( Ring  \ NzRing ) )   &    |-  ( ph  ->  Z  e.  U )   &    |-  ( ph  ->ring  e.  U )   =>    |-  ( ph  ->  (ZeroO `  C )  =  (/) )
 
21.33.12.10  Subcategories of the category of rings
 
Theoremsrhmsubclem1 38842* Lemma 1 for srhmsubc 38845. (Contributed by AV, 19-Feb-2020.)
 |-  A. r  e.  S  r  e.  Ring   &    |-  C  =  ( U  i^i  S )   =>    |-  ( X  e.  C  ->  X  e.  ( U  i^i  Ring ) )
 
Theoremsrhmsubclem2 38843* Lemma 2 for srhmsubc 38845. (Contributed by AV, 19-Feb-2020.)
 |-  A. r  e.  S  r  e.  Ring   &    |-  C  =  ( U  i^i  S )   =>    |-  ( ( U  e.  V  /\  X  e.  C )  ->  X  e.  ( Base `  (RingCat `  U ) ) )
 
Theoremsrhmsubclem3 38844* Lemma 3 for srhmsubc 38845. (Contributed by AV, 19-Feb-2020.)
 |-  A. r  e.  S  r  e.  Ring   &    |-  C  =  ( U  i^i  S )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
 ) )   =>    |-  ( ( U  e.  V  /\  ( X  e.  C  /\  Y  e.  C ) )  ->  ( X J Y )  =  ( X ( Hom  `  (RingCat `  U )
 ) Y ) )
 
Theoremsrhmsubc 38845* According to df-subc 15668, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15696 and subcss2 15699). Therefore, the set of special ring homomorphisms (i.e. ring homomorphisms from a special ring to another ring of that kind) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.)
 |-  A. r  e.  S  r  e.  Ring   &    |-  C  =  ( U  i^i  S )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
 ) )   =>    |-  ( U  e.  V  ->  J  e.  (Subcat `  (RingCat `  U ) ) )
 
Theoremsringcat 38846* The restriction of the category of (unital) rings to the set of special ring homomorphisms is a category. (Contributed by AV, 19-Feb-2020.)
 |-  A. r  e.  S  r  e.  Ring   &    |-  C  =  ( U  i^i  S )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
 ) )   =>    |-  ( U  e.  V  ->  ( (RingCat `  U )  |`cat  J )  e.  Cat )
 
Theoremcrhmsubc 38847* According to df-subc 15668, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15696 and subcss2 15699). Therefore, the set of commutative ring homomorphisms (i.e. ring homomorphisms from a commutative ring to a commutative ring) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.)
 |-  C  =  ( U  i^i  CRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
 ) )   =>    |-  ( U  e.  V  ->  J  e.  (Subcat `  (RingCat `  U ) ) )
 
Theoremcringcat 38848* The restriction of the category of (unital) rings to the set of commutative ring homomorphisms is a category, the "category of commutative rings". (Contributed by AV, 19-Feb-2020.)
 |-  C  =  ( U  i^i  CRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
 ) )   =>    |-  ( U  e.  V  ->  ( (RingCat `  U )  |`cat  J )  e.  Cat )
 
Theoremdrhmsubc 38849* According to df-subc 15668, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15696 and subcss2 15699). Therefore, the set of division ring homomorphisms is a "subcategory" of the category of (unital) rings. (Contributed by AV, 20-Feb-2020.)
 |-  C  =  ( U  i^i  DivRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )   =>    |-  ( U  e.  V  ->  J  e.  (Subcat `  (RingCat `  U ) ) )
 
Theoremdrngcat 38850* The restriction of the category of (unital) rings to the set of division ring homomorphisms is a category, the "category of division rings". (Contributed by AV, 20-Feb-2020.)
 |-  C  =  ( U  i^i  DivRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )   =>    |-  ( U  e.  V  ->  ( (RingCat `  U )  |`cat  J )  e.  Cat )
 
Theoremfldcat 38851* The restriction of the category of (unital) rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.)
 |-  C  =  ( U  i^i  DivRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )   &    |-  D  =  ( U  i^i Field )   &    |-  F  =  ( r  e.  D ,  s  e.  D  |->  ( r RingHom  s ) )   =>    |-  ( U  e.  V  ->  ( (RingCat `  U )  |`cat  F )  e.  Cat )
 
Theoremfldc 38852* The restriction of the category of division rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.)
 |-  C  =  ( U  i^i  DivRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )   &    |-  D  =  ( U  i^i Field )   &    |-  F  =  ( r  e.  D ,  s  e.  D  |->  ( r RingHom  s ) )   =>    |-  ( U  e.  V  ->  ( ( (RingCat `  U )  |`cat  J )  |`cat  F )  e.  Cat )
 
Theoremfldhmsubc 38853* According to df-subc 15668, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15696 and subcss2 15699). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.)
 |-  C  =  ( U  i^i  DivRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )   &    |-  D  =  ( U  i^i Field )   &    |-  F  =  ( r  e.  D ,  s  e.  D  |->  ( r RingHom  s ) )   =>    |-  ( U  e.  V  ->  F  e.  (Subcat `  ( (RingCat `  U )  |`cat  J ) ) )
 
Theoremrngcrescrhm 38854 The category of non-unital rings (in a universe) restricted to the ring homomorphisms between unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCat `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ph  ->  ( C  |`cat  H )  =  ( ( Cs  R ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
 
Theoremrhmsubclem1 38855 Lemma 1 for rhmsubc 38859. (Contributed by AV, 2-Mar-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCat `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ph  ->  H  Fn  ( R  X.  R ) )
 
Theoremrhmsubclem2 38856 Lemma 2 for rhmsubc 38859. (Contributed by AV, 2-Mar-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCat `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ( ph  /\  X  e.  R  /\  Y  e.  R )  ->  ( X H Y )  =  ( X RingHom  Y )
 )
 
Theoremrhmsubclem3 38857* Lemma 3 for rhmsubc 38859. (Contributed by AV, 2-Mar-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCat `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ( ph  /\  x  e.  R )  ->  (
 ( Id `  (RngCat `  U ) ) `  x )  e.  ( x H x ) )
 
Theoremrhmsubclem4 38858* Lemma 4 for rhmsubc 38859. (Contributed by AV, 2-Mar-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCat `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ( ( (
 ph  /\  x  e.  R )  /\  ( y  e.  R  /\  z  e.  R ) )  /\  ( f  e.  ( x H y )  /\  g  e.  ( y H z ) ) )  ->  ( g
 ( <. x ,  y >. (comp `  (RngCat `  U ) ) z ) f )  e.  ( x H z ) )
 
Theoremrhmsubc 38859 According to df-subc 15668, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15696 and subcss2 15699). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCat `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ph  ->  H  e.  (Subcat `  (RngCat `  U ) ) )
 
Theoremrhmsubccat 38860 The restriction of the category of non-unital rings to the set of unital ring homomorphisms is a category. (Contributed by AV, 4-Mar-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCat `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ph  ->  (
 (RngCat `  U )  |`cat  H )  e.  Cat )
 
TheoremsrhmsubcALTVlem1 38861* Lemma 1 for srhmsubcALTV 38864. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
 |-  A. r  e.  S  r  e.  Ring   &    |-  C  =  ( U  i^i  S )   =>    |-  ( X  e.  C  ->  X  e.  ( U  i^i  Ring ) )
 
TheoremsrhmsubcALTVlem2 38862* Lemma 2 for srhmsubcALTV 38864. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
 |-  A. r  e.  S  r  e.  Ring   &    |-  C  =  ( U  i^i  S )   =>    |-  ( ( U  e.  V  /\  X  e.  C )  ->  X  e.  ( Base `  (RingCatALTV `  U ) ) )
 
TheoremsrhmsubcALTVlem3 38863* Lemma 3 for srhmsubcALTV 38864. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
 |-  A. r  e.  S  r  e.  Ring   &    |-  C  =  ( U  i^i  S )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
 ) )   =>    |-  ( ( U  e.  V  /\  ( X  e.  C  /\  Y  e.  C ) )  ->  ( X J Y )  =  ( X ( Hom  `  (RingCatALTV `  U ) ) Y ) )
 
TheoremsrhmsubcALTV 38864* According to df-subc 15668, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15696 and subcss2 15699). Therefore, the set of special ring homomorphisms (i.e. ring homomorphisms from a special ring to another ring of that kind) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
 |-  A. r  e.  S  r  e.  Ring   &    |-  C  =  ( U  i^i  S )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
 ) )   =>    |-  ( U  e.  V  ->  J  e.  (Subcat `  (RingCatALTV `  U ) ) )
 
TheoremsringcatALTV 38865* The restriction of the category of (unital) rings to the set of special ring homomorphisms is a category. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
 |-  A. r  e.  S  r  e.  Ring   &    |-  C  =  ( U  i^i  S )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
 ) )   =>    |-  ( U  e.  V  ->  ( (RingCatALTV `  U )  |`cat  J )  e.  Cat )
 
TheoremcrhmsubcALTV 38866* According to df-subc 15668, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15696 and subcss2 15699). Therefore, the set of commutative ring homomorphisms (i.e. ring homomorphisms from a commutative ring to a commutative ring) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
 |-  C  =  ( U  i^i  CRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
 ) )   =>    |-  ( U  e.  V  ->  J  e.  (Subcat `  (RingCatALTV `  U ) ) )
 
TheoremcringcatALTV 38867* The restriction of the category of (unital) rings to the set of commutative ring homomorphisms is a category, the "category of commutative rings". (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
 |-  C  =  ( U  i^i  CRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
 ) )   =>    |-  ( U  e.  V  ->  ( (RingCatALTV `  U )  |`cat  J )  e.  Cat )
 
TheoremdrhmsubcALTV 38868* According to df-subc 15668, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15696 and subcss2 15699). Therefore, the set of division ring homomorphisms is a "subcategory" of the category of (unital) rings. (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
 |-  C  =  ( U  i^i  DivRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )   =>    |-  ( U  e.  V  ->  J  e.  (Subcat `  (RingCatALTV `  U ) ) )
 
TheoremdrngcatALTV 38869* The restriction of the category of (unital) rings to the set of division ring homomorphisms is a category, the "category of division rings". (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
 |-  C  =  ( U  i^i  DivRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )   =>    |-  ( U  e.  V  ->  ( (RingCatALTV `  U )  |`cat  J )  e.  Cat )
 
TheoremfldcatALTV 38870* The restriction of the category of (unital) rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
 |-  C  =  ( U  i^i  DivRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )   &    |-  D  =  ( U  i^i Field )   &    |-  F  =  ( r  e.  D ,  s  e.  D  |->  ( r RingHom  s ) )   =>    |-  ( U  e.  V  ->  ( (RingCatALTV `  U )  |`cat  F )  e.  Cat )
 
TheoremfldcALTV 38871* The restriction of the category of division rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
 |-  C  =  ( U  i^i  DivRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )   &    |-  D  =  ( U  i^i Field )   &    |-  F  =  ( r  e.  D ,  s  e.  D  |->  ( r RingHom  s ) )   =>    |-  ( U  e.  V  ->  ( ( (RingCatALTV `  U )  |`cat  J )  |`cat  F )  e.  Cat )
 
TheoremfldhmsubcALTV 38872* According to df-subc 15668, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15696 and subcss2 15699). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
 |-  C  =  ( U  i^i  DivRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )   &    |-  D  =  ( U  i^i Field )   &    |-  F  =  ( r  e.  D ,  s  e.  D  |->  ( r RingHom  s ) )   =>    |-  ( U  e.  V  ->  F  e.  (Subcat `  ( (RingCatALTV `  U )  |`cat  J ) ) )
 
TheoremrngcrescrhmALTV 38873 The category of non-unital rings (in a universe) restricted to the ring homomorphisms between unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.) (New usage is discouraged.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCatALTV `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ph  ->  ( C  |`cat  H )  =  ( ( Cs  R ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
 
TheoremrhmsubcALTVlem1 38874 Lemma 1 for rhmsubcALTV 38878. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCatALTV `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ph  ->  H  Fn  ( R  X.  R ) )
 
TheoremrhmsubcALTVlem2 38875 Lemma 2 for rhmsubcALTV 38878. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCatALTV `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ( ph  /\  X  e.  R  /\  Y  e.  R )  ->  ( X H Y )  =  ( X RingHom  Y )
 )
 
TheoremrhmsubcALTVlem3 38876* Lemma 3 for rhmsubcALTV 38878. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCatALTV `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ( ph  /\  x  e.  R )  ->  (
 ( Id `  (RngCatALTV `  U ) ) `  x )  e.  ( x H x ) )
 
TheoremrhmsubcALTVlem4 38877* Lemma 4 for rhmsubcALTV 38878. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCatALTV `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ( ( (
 ph  /\  x  e.  R )  /\  ( y  e.  R  /\  z  e.  R ) )  /\  ( f  e.  ( x H y )  /\  g  e.  ( y H z ) ) )  ->  ( g
 ( <. x ,  y >. (comp `  (RngCatALTV `  U ) ) z ) f )  e.  ( x H z ) )
 
TheoremrhmsubcALTV 38878 According to df-subc 15668, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15696 and subcss2 15699). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCatALTV `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ph  ->  H  e.  (Subcat `  (RngCatALTV `  U ) ) )
 
TheoremrhmsubcALTVcat 38879 The restriction of the category of non-unital rings to the set of unital ring homomorphisms is a category. (Contributed by AV, 4-Mar-2020.) (New usage is discouraged.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCatALTV `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ph  ->  (
 (RngCatALTV `  U )  |`cat  H )  e.  Cat )
 
21.33.13  Basic algebraic structures (extension)
 
21.33.13.1  Auxiliary theorems
 
Theoremrabeqsn 38880* Conditions for a restricted class abstraction to be a singleton. (Contributed by AV, 18-Apr-2019.)
 |-  ( { x  e.  V  |  ph }  =  { X }  <->  A. x ( ( x  e.  V  /\  ph )  <->  x  =  X ) )
 
Theoremrabsssn 38881* Conditions for a restricted class abstraction to be a subset of a singleton, i.e. to be a singleton or the empty set. (Contributed by AV, 18-Apr-2019.)
 |-  ( { x  e.  V  |  ph }  C_  { X } 
 <-> 
 A. x  e.  V  ( ph  ->  x  =  X ) )
 
Theoremxpprsng 38882 The Cartesian product of an unordered pair and a singleton. (Contributed by AV, 20-May-2019.)
 |-  (
 ( A  e.  V  /\  B  e.  W  /\  C  e.  U )  ->  ( { A ,  B }  X.  { C } )  =  { <. A ,  C >. , 
 <. B ,  C >. } )
 
Theoremopeliun2xp 38883 Membership of an ordered pair in a union of Cartesian products over its second component, analogous to opeliunxp 4906. (Contributed by AV, 30-Mar-2019.)
 |-  ( <. C ,  y >.  e.  U_ y  e.  B  ( A  X.  { y } )  <->  ( y  e.  B  /\  C  e.  A ) )
 
Theoremeliunxp2 38884* Membership in a union of Cartesian products over its second component, analogous to eliunxp 4992. (Contributed by AV, 30-Mar-2019.)
 |-  ( C  e.  U_ y  e.  B  ( A  X.  { y } )  <->  E. x E. y
 ( C  =  <. x ,  y >.  /\  ( x  e.  A  /\  y  e.  B )
 ) )
 
Theoremmpt2mptx2 38885* Express a two-argument function as a one-argument function, or vice-versa. In this version 
A ( y ) is not assumed to be constant w.r.t  y, analogous to mpt2mptx 6401. (Contributed by AV, 30-Mar-2019.)
 |-  (
 z  =  <. x ,  y >.  ->  C  =  D )   =>    |-  ( z  e.  U_ y  e.  B  ( A  X.  { y }
 )  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
 
Theoremcbvmpt2x2 38886* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6384 allows  A to be a function of  y, analogous to cbvmpt2x 6383. (Contributed by AV, 30-Mar-2019.)
 |-  F/_ z A   &    |-  F/_ y D   &    |-  F/_ z C   &    |-  F/_ w C   &    |-  F/_ x E   &    |-  F/_ y E   &    |-  (
 y  =  z  ->  A  =  D )   &    |-  (
 ( y  =  z 
 /\  x  =  w )  ->  C  =  E )   =>    |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( w  e.  D ,  z  e.  B  |->  E )
 
Theoremdmmpt2ssx2 38887* The domain of a mapping is a subset of its base classes expressed as union of Cartesian products over its second component, analogous to dmmpt2ssx 6872. (Contributed by AV, 30-Mar-2019.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |- 
 dom  F  C_  U_ y  e.  B  ( A  X.  { y } )
 
Theoremmpt2exxg2 38888* Existence of an operation class abstraction (version for dependent domains, i.e. the first base class may depend on the second base class), analogous to mpt2exxg 6881. (Contributed by AV, 30-Mar-2019.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( ( B  e.  R  /\  A. y  e.  B  A  e.  S )  ->  F  e.  _V )
 
Theoremovmpt2rdxf 38889* Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 6436. (Contributed by AV, 30-Mar-2019.)
 |-  ( ph  ->  F  =  ( x  e.  C ,  y  e.  D  |->  R ) )   &    |-  ( ( ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )   &    |-  ( ( ph  /\  y  =  B ) 
 ->  C  =  L )   &    |-  ( ph  ->  A  e.  L )   &    |-  ( ph  ->  B  e.  D )   &    |-  ( ph  ->  S  e.  X )   &    |- 
 F/ x ph   &    |-  F/ y ph   &    |-  F/_ y A   &    |-  F/_ x B   &    |-  F/_ x S   &    |-  F/_ y S   =>    |-  ( ph  ->  ( A F B )  =  S )
 
Theoremovmpt2rdx 38890* Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 6436. (Contributed by AV, 30-Mar-2019.)
 |-  ( ph  ->  F  =  ( x  e.  C ,  y  e.  D  |->  R ) )   &    |-  ( ( ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )   &    |-  ( ( ph  /\  y  =  B ) 
 ->  C  =  L )   &    |-  ( ph  ->  A  e.  L )   &    |-  ( ph  ->  B  e.  D )   &    |-  ( ph  ->  S  e.  X )   =>    |-  ( ph  ->  ( A F B )  =  S )
 
Theoremovmpt2x2 38891* The value of an operation class abstraction. Variant of ovmpt2ga 6440 which does not require  D and  x to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
 |-  (
 ( x  =  A  /\  y  =  B )  ->  R  =  S )   &    |-  ( y  =  B  ->  C  =  L )   &    |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )   =>    |-  ( ( A  e.  L  /\  B  e.  D  /\  S  e.  H )  ->  ( A F B )  =  S )
 
Theoremfdmdifeqresdif 38892* The restriction of a conditional mapping to function values of a function having a domain which is a difference with a singleton equals this function. (Contributed by AV, 23-Apr-2019.)
 |-  F  =  ( x  e.  D  |->  if ( x  =  Y ,  X ,  ( G `
  x ) ) )   =>    |-  ( G : ( D  \  { Y } ) --> R  ->  G  =  ( F  |`  ( D 
 \  { Y }
 ) ) )
 
Theoremoffvalfv 38893* The function operation expressed as a mapping with function values. (Contributed by AV, 6-Apr-2019.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  A )   =>    |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  A  |->  ( ( F `  x ) R ( G `  x ) ) ) )
 
Theoremofaddmndmap 38894 The function operation applied to the addition for functions (with the same domain) into a monoid is a function (with the same domain) into the monoid. (Contributed by AV, 6-Apr-2019.)
 |-  R  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   =>    |-  (
 ( M  e.  Mnd  /\  V  e.  Y  /\  ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) ) ) 
 ->  ( A  oF  .+  B )  e.  ( R  ^m  V ) )
 
Theoremmapsnop 38895 A singleton of an ordered pair as an element of the mapping operation. (Contributed by AV, 12-Apr-2019.)
 |-  F  =  { <. X ,  Y >. }   =>    |-  ( ( X  e.  V  /\  Y  e.  R  /\  R  e.  W ) 
 ->  F  e.  ( R 
 ^m  { X } )
 )
 
Theoremmapprop 38896 An unordered pair containing two ordered pairs as an element of the mapping operation. (Contributed by AV, 16-Apr-2019.)
 |-  F  =  { <. X ,  A >. ,  <. Y ,  B >. }   =>    |-  ( ( ( X  e.  V  /\  A  e.  R )  /\  ( Y  e.  V  /\  B  e.  R )  /\  ( X  =/=  Y  /\  R  e.  W ) )  ->  F  e.  ( R  ^m  { X ,  Y } ) )
 
Theoremztprmneprm 38897 A prime is not an integer multiple of another prime. (Contributed by AV, 23-May-2019.)
 |-  (
 ( Z  e.  ZZ  /\  A  e.  Prime  /\  B  e.  Prime )  ->  (
 ( Z  x.  A )  =  B  ->  A  =  B ) )
 
Theorem2t6m3t4e0 38898 2 times 6 minus 3 times 4 equals 0. (Contributed by AV, 24-May-2019.)
 |-  (
 ( 2  x.  6
 )  -  ( 3  x.  4 ) )  =  0
 
Theoremssnn0ssfz 38899* For any finite subset of  NN0, find a superset in the form of a set of sequential integers, analogous to ssnnssfz 28203. (Contributed by AV, 30-Sep-2019.)
 |-  ( A  e.  ( ~P NN0 
 i^i  Fin )  ->  E. n  e.  NN0  A  C_  (
 0 ... n ) )
 
Theoremnn0sumltlt 38900 If the sum of two nonnegative integers is less than a third integer, then one of the summands is already less than this third integer. (Contributed by AV, 19-Oct-2019.)
 |-  (
 ( a  e.  NN0  /\  b  e.  NN0  /\  c  e.  NN0 )  ->  (
 ( a  +  b
 )  <  c  ->  b  <  c ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 392 39101-39200 393 39201-39300 394 39301-39310
  Copyright terms: Public domain < Previous  Next >