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Theorem List for Metamath Proof Explorer - 15401-15500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcinvr 15401 Extend class notation with multiplicative inverse.
 class  invr
 
Definitiondf-invr 15402 Define multiplicative inverse. (Contributed by NM, 21-Sep-2011.)
 |- 
 invr  =  ( r  e.  _V  |->  ( inv g `  ( (mulGrp `  r
 )s  (Unit `  r )
 ) ) )
 
Theoreminvrfval 15403 Multiplicative inverse function for a division ring. (Contributed by NM, 21-Sep-2011.) (Revised by Mario Carneiro, 25-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  G  =  ( (mulGrp `  R )s  U )   &    |-  I  =  ( invr `  R )   =>    |-  I  =  ( inv
 g `  G )
 
Theoremunitinvcl 15404 The inverse of a unit exists and is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  I  =  (
 invr `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( I `
  X )  e.  U )
 
Theoremunitinvinv 15405 The inverse of the inverse of a unit is the same element. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  I  =  (
 invr `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( I `
  ( I `  X ) )  =  X )
 
Theoremrnginvcl 15406 The inverse of a unit is an element of the ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  I  =  (
 invr `  R )   &    |-  B  =  ( Base `  R )   =>    |-  (
 ( R  e.  Ring  /\  X  e.  U ) 
 ->  ( I `  X )  e.  B )
 
Theoremunitlinv 15407 A unit times its inverse is the identity. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  I  =  (
 invr `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U ) 
 ->  ( ( I `  X )  .x.  X )  =  .1.  )
 
Theoremunitrinv 15408 A unit times its inverse is the identity. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  I  =  (
 invr `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U ) 
 ->  ( X  .x.  ( I `  X ) )  =  .1.  )
 
Theorem1rinv 15409 The inverse of the identity is the identity. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  I  =  ( invr `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  ( I `  .1.  )  =  .1.  )
 
Theorem0unit 15410 The additive identity is a unit if and only if  1  =  0, i.e. we are in the zero ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  (  .0.  e.  U  <->  .1.  =  .0.  )
 )
 
Theoremunitnegcl 15411 The negative of a unit is a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  N  =  ( inv g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U ) 
 ->  ( N `  X )  e.  U )
 
Syntaxcdvr 15412 Extend class notation with ring division.
 class /r
 
Definitiondf-dvr 15413* Define ring division. (Contributed by Mario Carneiro, 2-Jul-2014.)
 |- /r  =  ( r  e.  _V  |->  ( x  e.  ( Base `  r ) ,  y  e.  (Unit `  r )  |->  ( x ( .r `  r
 ) ( ( invr `  r ) `  y
 ) ) ) )
 
Theoremdvrfval 15414* Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  U  =  (Unit `  R )   &    |-  I  =  ( invr `  R )   &    |-  ./  =  (/r `  R )   =>    |-  ./  =  ( x  e.  B ,  y  e.  U  |->  ( x  .x.  ( I `  y ) ) )
 
Theoremdvrval 15415 Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  U  =  (Unit `  R )   &    |-  I  =  ( invr `  R )   &    |-  ./  =  (/r `  R )   =>    |-  ( ( X  e.  B  /\  Y  e.  U )  ->  ( X  ./  Y )  =  ( X  .x.  ( I `  Y ) ) )
 
Theoremdvrcl 15416 Closure of division operation. (Contributed by Mario Carneiro, 2-Jul-2014.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  ./  =  (/r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  U )  ->  ( X  ./  Y )  e.  B )
 
Theoremunitdvcl 15417 The units are closed under division. (Contributed by Mario Carneiro, 2-Jul-2014.)
 |-  U  =  (Unit `  R )   &    |-  ./  =  (/r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  ./  Y )  e.  U )
 
Theoremdvrid 15418 An cancellation law for division. (divid 9405 analog.) (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  U  =  (Unit `  R )   &    |-  ./  =  (/r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U ) 
 ->  ( X  ./  X )  =  .1.  )
 
Theoremdvr1 15419 An cancellation law for division. (div1 9407 analog.) (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  B  =  ( Base `  R )   &    |-  ./  =  (/r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  ( X  ./  .1.  )  =  X )
 
Theoremdvrass 15420 An associative law for division. (divass 9396 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  ./  =  (/r `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  U ) )  ->  ( ( X  .x.  Y )  ./  Z )  =  ( X  .x.  ( Y  ./  Z ) ) )
 
Theoremdvrcan1 15421 A cancellation law for division. (divcan1 9387 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  ./  =  (/r `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  U )  ->  ( ( X 
 ./  Y )  .x.  Y )  =  X )
 
Theoremdvrcan3 15422 A cancellation law for division. (divcan3 9402 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 18-Jun-2015.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  ./  =  (/r `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  U )  ->  ( ( X 
 .x.  Y )  ./  Y )  =  X )
 
Theoremdvreq1 15423 A cancellation law for division. (diveq1 9408 analog.) (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  ./  =  (/r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  U )  ->  ( ( X  ./  Y )  =  .1.  <->  X  =  Y ) )
 
Theoremrnginvdv 15424 Write the inverse function in terms of division. (Contributed by Mario Carneiro, 2-Jul-2014.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  ./  =  (/r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  I  =  ( invr `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U ) 
 ->  ( I `  X )  =  (  .1.  ./  X ) )
 
Theoremrngidpropd 15425* The ring identity depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )   =>    |-  ( ph  ->  ( 1r `  K )  =  ( 1r `  L ) )
 
Theoremdvdsrpropd 15426* The divisibility relation depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )   =>    |-  ( ph  ->  ( ||r `  K )  =  (
 ||r `  L ) )
 
Theoremunitpropd 15427* The set of units depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )   =>    |-  ( ph  ->  (Unit `  K )  =  (Unit `  L ) )
 
Theoreminvrpropd 15428* The ring inverse function depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )   =>    |-  ( ph  ->  ( invr `  K )  =  ( invr `  L )
 )
 
Theoremisirred 15429* An irreducible element of a ring is a non-unit that is not the product of two non-units. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  I  =  (Irred `  R )   &    |-  N  =  ( B  \  U )   &    |-  .x. 
 =  ( .r `  R )   =>    |-  ( X  e.  I  <->  ( X  e.  N  /\  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/= 
 X ) )
 
Theoremisnirred 15430* The property of being a non-irreducible (reducible) element in a ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  I  =  (Irred `  R )   &    |-  N  =  ( B  \  U )   &    |-  .x. 
 =  ( .r `  R )   =>    |-  ( X  e.  B  ->  ( -.  X  e.  I 
 <->  ( X  e.  U  \/  E. x  e.  N  E. y  e.  N  ( x  .x.  y )  =  X ) ) )
 
Theoremisirred2 15431* Expand out the set differences from isirred 15429. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  I  =  (Irred `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( X  e.  I 
 <->  ( X  e.  B  /\  -.  X  e.  U  /\  A. x  e.  B  A. y  e.  B  ( ( x  .x.  y
 )  =  X  ->  ( x  e.  U  \/  y  e.  U )
 ) ) )
 
Theoremopprirred 15432 Irreducibility is symmetric, so the irreducible elements of the opposite ring are the same as the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  S  =  (oppr `  R )   &    |-  I  =  (Irred `  R )   =>    |-  I  =  (Irred `  S )
 
Theoremirredn0 15433 The additive identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  I  =  (Irred `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  I ) 
 ->  X  =/=  .0.  )
 
Theoremirredcl 15434 An irreducible element is in the ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  I  =  (Irred `  R )   &    |-  B  =  (
 Base `  R )   =>    |-  ( X  e.  I  ->  X  e.  B )
 
Theoremirrednu 15435 An irreducible element is not a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  I  =  (Irred `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( X  e.  I  ->  -.  X  e.  U )
 
Theoremirredn1 15436 The multiplicative identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  I  =  (Irred `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  I ) 
 ->  X  =/=  .1.  )
 
Theoremirredrmul 15437 The product of an irreducible element and a unit is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  I  =  (Irred `  R )   &    |-  U  =  (Unit `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  I  /\  Y  e.  U )  ->  ( X  .x.  Y )  e.  I )
 
Theoremirredlmul 15438 The product of a unit and an irreducible element is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  I  =  (Irred `  R )   &    |-  U  =  (Unit `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  I
 )  ->  ( X  .x.  Y )  e.  I
 )
 
Theoremirredmul 15439 If product of two elements is irreducible, then one of the elements must be a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  I  =  (Irred `  R )   &    |-  B  =  (
 Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( X  e.  B  /\  Y  e.  B  /\  ( X  .x.  Y )  e.  I )  ->  ( X  e.  U  \/  Y  e.  U ) )
 
Theoremirredneg 15440 The negative of an irreducible element is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  I  =  (Irred `  R )   &    |-  N  =  ( inv g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  I ) 
 ->  ( N `  X )  e.  I )
 
Theoremirrednegb 15441 An element is irreducible iff its negative is. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  I  =  (Irred `  R )   &    |-  N  =  ( inv g `  R )   &    |-  B  =  ( Base `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  ( X  e.  I  <->  ( N `  X )  e.  I ) )
 
10.4.5  Ring homomorphisms
 
Syntaxcrh 15442 Ring homomorphisms.
 class RingHom
 
Syntaxcrs 15443 Ring isomorphisms.
 class RingIso
 
Definitiondf-rnghom 15444* Define the set of ring homomorphisms from  r to  s. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |- RingHom  =  ( r  e.  Ring ,  s  e.  Ring  |->  [_ ( Base `  r )  /  v ]_ [_ ( Base `  s )  /  w ]_
 { f  e.  ( w  ^m  v )  |  ( ( f `  ( 1r `  r ) )  =  ( 1r
 `  s )  /\  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-rngiso 15445* Define the set of ring isomorphisms from  r to  s. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |- RingIso  =  ( r  e.  _V ,  s  e.  _V  |->  { f  e.  ( r RingHom  s )  |  `' f  e.  ( s RingHom  r ) } )
 
Theoremdfrhm2 15446* The property of a ring homomorphism can be decomposed into separate homomorphic conditions for addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |- RingHom  =  ( r  e.  Ring ,  s  e.  Ring  |->  ( ( r  GrpHom  s )  i^i  ( (mulGrp `  r
 ) MndHom  (mulGrp `  s )
 ) ) )
 
Theoremrhmrcl1 15447 Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  ( F  e.  ( R RingHom  S )  ->  R  e.  Ring )
 
Theoremrhmrcl2 15448 Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  ( F  e.  ( R RingHom  S )  ->  S  e.  Ring )
 
Theoremisrhm 15449 A function is a ring homomorphism iff it preserves both addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  N  =  (mulGrp `  S )   =>    |-  ( F  e.  ( R RingHom  S )  <->  ( ( R  e.  Ring  /\  S  e.  Ring
 )  /\  ( F  e.  ( R  GrpHom  S ) 
 /\  F  e.  ( M MndHom  N ) ) ) )
 
Theoremrhmmhm 15450 A ring homomorphism is a homomorphism of multiplicative monoids. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  N  =  (mulGrp `  S )   =>    |-  ( F  e.  ( R RingHom  S )  ->  F  e.  ( M MndHom  N )
 )
 
Theoremrhmghm 15451 A ring homomorphism is an additive group homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  ( F  e.  ( R RingHom  S )  ->  F  e.  ( R  GrpHom  S ) )
 
Theoremrhmf 15452 A ring homomorphism is a function. (Contributed by Stefan O'Rear, 8-Mar-2015.)
 |-  B  =  ( Base `  R )   &    |-  C  =  (
 Base `  S )   =>    |-  ( F  e.  ( R RingHom  S )  ->  F : B --> C )
 
Theoremrhmmul 15453 A homomorphism of rings preserves multiplication. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  X  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .X.  =  ( .r `  S )   =>    |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  X  /\  B  e.  X )  ->  ( F `  ( A  .x.  B ) )  =  ( ( F `
  A )  .X.  ( F `  B ) ) )
 
Theoremisrhm2d 15454* Demonstration of ring homomorphism. (Contributed by Mario Carneiro, 13-Jun-2015.)
 |-  B  =  ( Base `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  N  =  ( 1r `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  .X.  =  ( .r `  S )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  S  e.  Ring )   &    |-  ( ph  ->  ( F `  .1.  )  =  N )   &    |-  ( ( 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 RingHom  S )
 )
 
Theoremisrhmd 15455* Demonstration of ring homomorphism. (Contributed by Stefan O'Rear, 8-Mar-2015.)
 |-  B  =  ( Base `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  N  =  ( 1r `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  .X.  =  ( .r `  S )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  S  e.  Ring )   &    |-  ( ph  ->  ( F `  .1.  )  =  N )   &    |-  ( ( 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 RingHom  S )
 )
 
Theoremrhm1 15456 Ring homomorphisms are required to fix 1. (Contributed by Stefan O'Rear, 8-Mar-2015.)
 |- 
 .1.  =  ( 1r `  R )   &    |-  N  =  ( 1r `  S )   =>    |-  ( F  e.  ( R RingHom  S )  ->  ( F `  .1.  )  =  N )
 
Theoremrhmco 15457 The composition of ring homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  ( ( F  e.  ( T RingHom  U )  /\  G  e.  ( S RingHom  T ) )  ->  ( F  o.  G )  e.  ( S RingHom  U )
 )
 
Theorempwsco1rhm 15458* Right composition with a function on the index sets yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Y  =  ( R 
 ^s 
 A )   &    |-  Z  =  ( R  ^s  B )   &    |-  C  =  (
 Base `  Z )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  (
 g  e.  C  |->  ( g  o.  F ) )  e.  ( Z RingHom  Y ) )
 
Theorempwsco2rhm 15459* Left composition with a ring homomorphism yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Y  =  ( R 
 ^s 
 A )   &    |-  Z  =  ( S  ^s  A )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F  e.  ( R RingHom  S )
 )   =>    |-  ( ph  ->  (
 g  e.  B  |->  ( F  o.  g ) )  e.  ( Y RingHom  Z ) )
 
10.5  Division rings and Fields
 
10.5.1  Definition and basic properties
 
Syntaxcdr 15460 Extend class notation with class of all division rings.
 class  DivRing
 
Syntaxcfield 15461 Class of fields.
 class Field
 
Definitiondf-drng 15462 Define class of all division rings. A division ring is a ring in which the set of units is exactly the nonzero elements of the ring. (Contributed by NM, 18-Oct-2012.)
 |-  DivRing  =  { r  e. 
 Ring  |  (Unit `  r
 )  =  ( (
 Base `  r )  \  { ( 0g `  r ) } ) }
 
Definitiondf-field 15463 A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
 |- Field  =  ( DivRing  i^i  CRing )
 
Theoremisdrng 15464 The predicate "is a division ring". (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e.  DivRing  <->  ( R  e.  Ring  /\  U  =  ( B 
 \  {  .0.  }
 ) ) )
 
Theoremdrngunit 15465 Elementhood in the set of units when  R is a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e.  DivRing  ->  ( X  e.  U  <->  ( X  e.  B  /\  X  =/=  .0.  ) ) )
 
Theoremdrngui 15466 The set of units of a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  R  e.  DivRing   =>    |-  ( B  \  {  .0.  } )  =  (Unit `  R )
 
Theoremdrngrng 15467 A division ring is a ring. (Contributed by NM, 8-Sep-2011.)
 |-  ( R  e.  DivRing  ->  R  e.  Ring )
 
Theoremdrnggrp 15468 A division ring is a group. (Contributed by NM, 8-Sep-2011.)
 |-  ( R  e.  DivRing  ->  R  e.  Grp )
 
Theoremisfld 15469 A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
 |-  ( R  e. Field  <->  ( R  e.  DivRing  /\  R  e.  CRing ) )
 
Theoremisdrng2 15470 A division ring can equivalently be defined as a ring such that the nonzero elements form a group under multiplication (from which it follows that this is the same group as the group of units). (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  G  =  ( (mulGrp `  R )s  ( B  \  {  .0.  } ) )   =>    |-  ( R  e.  DivRing  <->  ( R  e.  Ring  /\  G  e.  Grp ) )
 
Theoremdrngprop 15471 If two structures have the same ring components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 28-Dec-2014.)
 |-  ( Base `  K )  =  ( Base `  L )   &    |-  ( +g  `  K )  =  ( +g  `  L )   &    |-  ( .r `  K )  =  ( .r `  L )   =>    |-  ( K  e.  DivRing  <->  L  e.  DivRing )
 
Theoremdrngmgp 15472 A division ring contains a multiplicative group. (Contributed by NM, 8-Sep-2011.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  G  =  ( (mulGrp `  R )s  ( B  \  {  .0.  } ) )   =>    |-  ( R  e.  DivRing  ->  G  e.  Grp )
 
Theoremdrngmcl 15473 The product of two nonzero elements of a division ring is nonzero. (Contributed by NM, 7-Sep-2011.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  DivRing  /\  X  e.  ( B 
 \  {  .0.  }
 )  /\  Y  e.  ( B  \  {  .0.  } ) )  ->  ( X  .x.  Y )  e.  ( B  \  {  .0.  } ) )
 
Theoremdrngid 15474 A division ring's unit is the identity element of its multiplicative group. (Contributed by NM, 7-Sep-2011.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  G  =  ( (mulGrp `  R )s  ( B  \  {  .0.  }
 ) )   =>    |-  ( R  e.  DivRing  ->  .1.  =  ( 0g `  G ) )
 
Theoremdrngunz 15475 A division ring's unit is different from its zero. (Contributed by NM, 8-Sep-2011.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  DivRing  ->  .1.  =/=  .0.  )
 
Theoremdrngid2 15476 Properties showing that an element 
I is the identity element of a division ring. (Contributed by Mario Carneiro, 11-Oct-2013.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .1.  =  ( 1r `  R )   =>    |-  ( R  e.  DivRing  ->  ( ( I  e.  B  /\  I  =/= 
 .0.  /\  ( I  .x.  I )  =  I
 ) 
 <->  .1.  =  I ) )
 
Theoremdrnginvrcl 15477 Closure of the multiplicative inverse in a division ring. (reccl 9385 analog.) (Contributed by NM, 19-Apr-2014.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  I  =  ( invr `  R )   =>    |-  ( ( R  e.  DivRing  /\  X  e.  B  /\  X  =/=  .0.  )  ->  ( I `  X )  e.  B )
 
Theoremdrnginvrn0 15478 The multiplicative inverse in a division ring is nonzero. (recne0 9391 analog.) (Contributed by NM, 19-Apr-2014.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  I  =  ( invr `  R )   =>    |-  ( ( R  e.  DivRing  /\  X  e.  B  /\  X  =/=  .0.  )  ->  ( I `  X )  =/=  .0.  )
 
Theoremdrnginvrl 15479 Property of the multiplicative inverse in a division ring. (recid2 9393 analog.) (Contributed by NM, 19-Apr-2014.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  .x. 
 =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  I  =  ( invr `  R )   =>    |-  ( ( R  e.  DivRing  /\  X  e.  B  /\  X  =/=  .0.  )  ->  ( ( I `  X )  .x.  X )  =  .1.  )
 
Theoremdrnginvrr 15480 Property of the multiplicative inverse in a division ring. (recid 9392 analog.) (Contributed by NM, 19-Apr-2014.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  .x. 
 =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  I  =  ( invr `  R )   =>    |-  ( ( R  e.  DivRing  /\  X  e.  B  /\  X  =/=  .0.  )  ->  ( X  .x.  ( I `
  X ) )  =  .1.  )
 
Theoremdrngmul0or 15481 A product is zero iff one of its factors is zero. (Contributed by NM, 8-Oct-2014.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  .x. 
 =  ( .r `  R )   &    |-  ( ph  ->  R  e.  DivRing )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( X  .x.  Y )  =  .0.  <->  ( X  =  .0.  \/  Y  =  .0.  ) ) )
 
Theoremdrngmulne0 15482 A product is nonzero iff both its factors are nonzero. (Contributed by NM, 18-Oct-2014.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  .x. 
 =  ( .r `  R )   &    |-  ( ph  ->  R  e.  DivRing )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( X  .x.  Y )  =/=  .0.  <->  ( X  =/=  .0.  /\  Y  =/=  .0.  )
 ) )
 
Theoremdrngmuleq0 15483 An element is zero iff its product with a nonzero element is zero. (Contributed by NM, 8-Oct-2014.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  .x. 
 =  ( .r `  R )   &    |-  ( ph  ->  R  e.  DivRing )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Y  =/=  .0.  )   =>    |-  ( ph  ->  (
 ( X  .x.  Y )  =  .0.  <->  X  =  .0.  ) )
 
Theoremopprdrng 15484 The opposite of a division ring is also a division ring. (Contributed by NM, 18-Oct-2014.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e.  DivRing  <->  O  e.  DivRing )
 
Theoremisdrngd 15485* Properties that determine a division ring.  I (reciprocal) is normally dependent on  x i.e. read it as  I ( x )." (Contributed by NM, 2-Aug-2013.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  .x.  =  ( .r `  R ) )   &    |-  ( ph  ->  .0. 
 =  ( 0g `  R ) )   &    |-  ( ph  ->  .1.  =  ( 1r `  R ) )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  ( y  e.  B  /\  y  =/=  .0.  )
 )  ->  ( x  .x.  y )  =/=  .0.  )   &    |-  ( ph  ->  .1.  =/=  .0.  )   &    |-  ( ( ph  /\  ( x  e.  B  /\  x  =/=  .0.  )
 )  ->  I  e.  B )   &    |-  ( ( ph  /\  ( x  e.  B  /\  x  =/=  .0.  )
 )  ->  I  =/=  .0.  )   &    |-  ( ( ph  /\  ( x  e.  B  /\  x  =/=  .0.  )
 )  ->  ( I  .x.  x )  =  .1.  )   =>    |-  ( ph  ->  R  e. 
 DivRing )
 
Theoremisdrngrd 15486* Properties that determine a division ring.  I (reciprocal) is normally dependent on  x i.e. read it as  I ( x )." This version of isdrngd 15485 requires a right reciprocal instead of left. (Contributed by NM, 10-Aug-2013.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  .x.  =  ( .r `  R ) )   &    |-  ( ph  ->  .0. 
 =  ( 0g `  R ) )   &    |-  ( ph  ->  .1.  =  ( 1r `  R ) )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  ( y  e.  B  /\  y  =/=  .0.  )
 )  ->  ( x  .x.  y )  =/=  .0.  )   &    |-  ( ph  ->  .1.  =/=  .0.  )   &    |-  ( ( ph  /\  ( x  e.  B  /\  x  =/=  .0.  )
 )  ->  I  e.  B )   &    |-  ( ( ph  /\  ( x  e.  B  /\  x  =/=  .0.  )
 )  ->  I  =/=  .0.  )   &    |-  ( ( ph  /\  ( x  e.  B  /\  x  =/=  .0.  )
 )  ->  ( x  .x.  I )  =  .1.  )   =>    |-  ( ph  ->  R  e. 
 DivRing )
 
Theoremdrngpropd 15487* If two structures have the same group components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 27-Jun-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )   =>    |-  ( ph  ->  ( K  e.  DivRing  <->  L  e.  DivRing ) )
 
Theoremfldpropd 15488* If two structures have the same group components (properties), one is a field iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )   =>    |-  ( ph  ->  ( K  e. Field  <->  L  e. Field ) )
 
10.5.2  Subrings of a ring
 
Syntaxcsubrg 15489 Extend class notation with all subrings of a ring.
 class SubRing
 
Syntaxcrgspn 15490 Extend class notation with span of a set of elements over a ring.
 class RingSpan
 
Definitiondf-subrg 15491* Define a subring of a ring as a set of elements that is a ring in its own right and contains the multiplicative identity.

The additional constraint is necessary because the multiplicative identity of a ring, unlike the additive identity of a ring/group or the multiplicative identity of a field, cannot be identified by a local property. Thus it is possible for a subset of a ring to be a ring while not containing the true identity if it contains a false identity. For instance, the subset  ( ZZ  X.  {
0 } ) of  ( ZZ  X.  ZZ ) (where multiplication is component-wise) contains the false identity  <. 1 ,  0 >. which preserves every element of the subset and thus appears to be the identity of the subset, but is not the identity of the larger ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)

 |- SubRing  =  ( w  e.  Ring  |->  { s  e.  ~P ( Base `  w )  |  ( ( ws  s )  e.  Ring  /\  ( 1r
 `  w )  e.  s ) } )
 
Definitiondf-rgspn 15492* The ring-span of a set of elements in a ring is the smallest subring which contains all of them. (Contributed by Stefan O'Rear, 7-Dec-2014.)
 |- RingSpan  =  ( w  e.  _V  |->  ( s  e.  ~P ( Base `  w )  |-> 
 |^| { t  e.  (SubRing `  w )  |  s 
 C_  t } )
 )
 
Theoremissubrg 15493 The subring predicate. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  B  =  ( Base `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( A  e.  (SubRing `  R )  <->  ( ( R  e.  Ring  /\  ( Rs  A )  e.  Ring )  /\  ( A  C_  B  /\  .1.  e.  A ) ) )
 
Theoremsubrgss 15494 A subring is a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  B  =  ( Base `  R )   =>    |-  ( A  e.  (SubRing `  R )  ->  A  C_  B )
 
Theoremsubrgid 15495 Every ring is a subring of itself. (Contributed by Stefan O'Rear, 30-Nov-2014.)
 |-  B  =  ( Base `  R )   =>    |-  ( R  e.  Ring  ->  B  e.  (SubRing `  R ) )
 
Theoremsubrgrng 15496 A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  S  =  ( Rs  A )   =>    |-  ( A  e.  (SubRing `  R )  ->  S  e.  Ring )
 
Theoremsubrgcrng 15497 A subring of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  S  =  ( Rs  A )   =>    |-  ( ( R  e.  CRing  /\  A  e.  (SubRing `  R ) )  ->  S  e.  CRing
 )
 
Theoremsubrgrcl 15498 Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.)
 |-  ( A  e.  (SubRing `  R )  ->  R  e.  Ring )
 
Theoremsubrgsubg 15499 A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.)
 |-  ( A  e.  (SubRing `  R )  ->  A  e.  (SubGrp `  R )
 )
 
Theoremsubrg0 15500 A subring always has the same additive identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  S  =  ( Rs  A )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( A  e.  (SubRing `  R )  ->  .0.  =  ( 0g `  S ) )
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