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Theorem List for Metamath Proof Explorer - 15701-15800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-dvdsr 15701* Define the (right) divisibility relation in a ring. Access to the left divisibility relation is available through  ( ||r `
 (oppr
`  R ) ). (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  ||r  =  ( w  e.  _V  |->  {
 <. x ,  y >.  |  ( x  e.  ( Base `  w )  /\  E. z  e.  ( Base `  w ) ( z ( .r `  w ) x )  =  y ) } )
 
Definitiondf-unit 15702 Define the set of units in a ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |- Unit  =  ( w  e.  _V  |->  ( `' ( ( ||r
 `  w )  i^i  ( ||r
 `  (oppr `  w ) ) )
 " { ( 1r
 `  w ) }
 ) )
 
Definitiondf-irred 15703* Define the set of irreducible elements in a ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |- Irred  =  ( w  e.  _V  |->  [_ ( ( Base `  w )  \  (Unit `  w ) )  /  b ]_ { z  e.  b  |  A. x  e.  b  A. y  e.  b  ( x ( .r `  w ) y )  =/=  z } )
 
Theoremreldvdsr 15704 The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  .||  =  ( ||r
 `  R )   =>    |-  Rel  .||
 
Theoremdvdsrval 15705* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   &    |- 
 .x.  =  ( .r `  R )   =>    |-  .||  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  (
 z  .x.  x )  =  y ) }
 
Theoremdvdsr 15706* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   &    |- 
 .x.  =  ( .r `  R )   =>    |-  ( X  .||  Y  <->  ( X  e.  B  /\  E. z  e.  B  ( z  .x.  X )  =  Y ) )
 
Theoremdvdsr2 15707* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   &    |- 
 .x.  =  ( .r `  R )   =>    |-  ( X  e.  B  ->  ( X  .||  Y  <->  E. z  e.  B  ( z  .x.  X )  =  Y ) )
 
Theoremdvdsrmul 15708 A left-multiple of  X is divisible by  X. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   &    |- 
 .x.  =  ( .r `  R )   =>    |-  ( ( X  e.  B  /\  Y  e.  B )  ->  X  .||  ( Y 
 .x.  X ) )
 
Theoremdvdsrcl 15709 Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   =>    |-  ( X  .||  Y  ->  X  e.  B )
 
Theoremdvdsrcl2 15710 Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  .||  Y )  ->  Y  e.  B )
 
Theoremdvdsrid 15711 An element in a (unital) ring divides itself. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  X  .||  X )
 
Theoremdvdsrtr 15712 Divisibility is transitive. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   =>    |-  ( ( R  e.  Ring  /\  Y  .||  Z  /\  Z  .||  X )  ->  Y  .||  X )
 
Theoremdvdsrmul1 15713 The divisibility relation is preserved under right-multiplication. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   &    |- 
 .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  Z  e.  B  /\  X  .||  Y )  ->  ( X  .x.  Z )  .||  ( Y  .x.  Z ) )
 
Theoremdvdsrneg 15714 An element divides its negative. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   &    |-  N  =  ( inv
 g `  R )   =>    |-  (
 ( R  e.  Ring  /\  X  e.  B ) 
 ->  X  .||  ( N `  X ) )
 
Theoremdvdsr01 15715 In a ring, zero is divisible by all elements. ("Zero divisor" as a term has a somewhat different meaning, see df-rlreg 16298.) (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  X  .||  .0.  )
 
Theoremdvdsr02 15716 Only zero is divisible by zero. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  (  .0.  .||  X  <->  X  =  .0.  ) )
 
Theoremisunit 15717 Property of being a unit of a ring. A unit is an element that left- and right-divides one. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 8-Dec-2015.)
 |-  U  =  (Unit `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  .|| 
 =  ( ||r
 `  R )   &    |-  S  =  (oppr `  R )   &    |-  E  =  (
 ||r `  S )   =>    |-  ( X  e.  U  <->  ( X  .||  .1.  /\  X E  .1.  ) )
 
Theorem1unit 15718 The multiplicative identity is a unit. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  .1. 
 e.  U )
 
Theoremunitcl 15719 A unit is an element of the base set. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( X  e.  U  ->  X  e.  B )
 
Theoremunitss 15720 The set of units is contained in the base set. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   =>    |-  U  C_  B
 
Theoremopprunit 15721 Being a unit is a symmetric property, so it transfers to the opposite ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  S  =  (oppr `  R )   =>    |-  U  =  (Unit `  S )
 
Theoremcrngunit 15722 Property of being a unit in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  U  =  (Unit `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  .|| 
 =  ( ||r
 `  R )   =>    |-  ( R  e.  CRing  ->  ( X  e.  U  <->  X  .||  .1.  ) )
 
Theoremdvdsunit 15723 A divisor of a unit is a unit. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  U  =  (Unit `  R )   &    |-  .||  =  ( ||r `  R )   =>    |-  ( ( R  e.  CRing  /\  Y  .||  X  /\  X  e.  U )  ->  Y  e.  U )
 
Theoremunitmulcl 15724 The product of units is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y )  e.  U )
 
Theoremunitmulclb 15725 Reversal of unitmulcl 15724 in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  U  =  (Unit `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  B  =  ( Base `  R )   =>    |-  (
 ( R  e.  CRing  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .x.  Y )  e.  U  <->  ( X  e.  U  /\  Y  e.  U ) ) )
 
Theoremunitgrpbas 15726 The base set of the group of units. (Contributed by Mario Carneiro, 25-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  G  =  ( (mulGrp `  R )s  U )   =>    |-  U  =  ( Base `  G )
 
Theoremunitgrp 15727 The group of units is a group under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  G  =  ( (mulGrp `  R )s  U )   =>    |-  ( R  e.  Ring  ->  G  e.  Grp )
 
Theoremunitabl 15728 The group of units of a commutative ring is abelian. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  U  =  (Unit `  R )   &    |-  G  =  ( (mulGrp `  R )s  U )   =>    |-  ( R  e.  CRing  ->  G  e.  Abel )
 
Theoremunitgrpid 15729 The identity of the multiplicative group is  1r. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  G  =  ( (mulGrp `  R )s  U )   &    |- 
 .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  .1.  =  ( 0g `  G ) )
 
Theoremunitsubm 15730 The group of units is a submonoid of the multiplicative monoid of the ring. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  U  =  (Unit `  R )   &    |-  M  =  (mulGrp `  R )   =>    |-  ( R  e.  Ring  ->  U  e.  (SubMnd `  M ) )
 
Syntaxcinvr 15731 Extend class notation with multiplicative inverse.
 class  invr
 
Definitiondf-invr 15732 Define multiplicative inverse. (Contributed by NM, 21-Sep-2011.)
 |- 
 invr  =  ( r  e.  _V  |->  ( inv g `  ( (mulGrp `  r
 )s  (Unit `  r )
 ) ) )
 
Theoreminvrfval 15733 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 15734 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 15735 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 15736 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 15737 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 15738 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 15739 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 15740 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 15741 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 15742 Extend class notation with ring division.
 class /r
 
Definitiondf-dvr 15743* 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 15744* 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 15745 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 15746 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 15747 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 15748 A cancellation law for division. (divid 9661 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 15749 A cancellation law for division. (div1 9663 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 15750 An associative law for division. (divass 9652 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 15751 A cancellation law for division. (divcan1 9643 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 15752 A cancellation law for division. (divcan3 9658 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 15753 A cancellation law for division. (diveq1 9664 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 15754 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 15755* 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 15756* 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 15757* 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 15758* 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 15759* 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 15760* 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 15761* Expand out the set differences from isirred 15759. (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 15762 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 15763 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 15764 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 15765 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 15766 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 15767 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 15768 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 15769 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 15770 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 15771 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 15772 Ring homomorphisms.
 class RingHom
 
Syntaxcrs 15773 Ring isomorphisms.
 class RingIso
 
Definitiondf-rnghom 15774* 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 15775* 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 15776* 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 15777 Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  ( F  e.  ( R RingHom  S )  ->  R  e.  Ring )
 
Theoremrhmrcl2 15778 Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  ( F  e.  ( R RingHom  S )  ->  S  e.  Ring )
 
Theoremisrhm 15779 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 15780 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 15781 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 15782 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 15783 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 15784* 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 15785* 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 15786 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 15787 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 15788* 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 15789* 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 15790 Extend class notation with class of all division rings.
 class  DivRing
 
Syntaxcfield 15791 Class of fields.
 class Field
 
Definitiondf-drng 15792 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 15793 A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
 |- Field  =  ( DivRing  i^i  CRing )
 
Theoremisdrng 15794 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 15795 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 15796 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 15797 A division ring is a ring. (Contributed by NM, 8-Sep-2011.)
 |-  ( R  e.  DivRing  ->  R  e.  Ring )
 
Theoremdrnggrp 15798 A division ring is a group. (Contributed by NM, 8-Sep-2011.)
 |-  ( R  e.  DivRing  ->  R  e.  Grp )
 
Theoremisfld 15799 A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
 |-  ( R  e. Field  <->  ( R  e.  DivRing  /\  R  e.  CRing ) )
 
Theoremisdrng2 15800 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 ) )
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