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Theorem List for Intuitionistic Logic Explorer - 13601-13700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdvdsr02 13601 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.  ) )
 
Theoremisunitd 13602 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.)
 |-  ( ph  ->  U  =  (Unit `  R )
 )   &    |-  ( ph  ->  .1.  =  ( 1r `  R ) )   &    |-  ( ph  ->  .||  =  ( ||r
 `  R ) )   &    |-  ( ph  ->  S  =  (oppr `  R ) )   &    |-  ( ph  ->  E  =  (
 ||r `  S ) )   &    |-  ( ph  ->  R  e. SRing )   =>    |-  ( ph  ->  ( X  e.  U 
 <->  ( X  .||  .1.  /\  X E  .1.  )
 ) )
 
Theorem1unit 13603 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 )
 
Theoremunitcld 13604 A unit is an element of the base set. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  U  =  (Unit `  R )
 )   &    |-  ( ph  ->  R  e. SRing )   &    |-  ( ph  ->  X  e.  U )   =>    |-  ( ph  ->  X  e.  B )
 
Theoremunitssd 13605 The set of units is contained in the base set. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  U  =  (Unit `  R )
 )   &    |-  ( ph  ->  R  e. SRing )   =>    |-  ( ph  ->  U  C_  B )
 
Theoremopprunitd 13606 Being a unit is a symmetric property, so it transfers to the opposite ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  ( ph  ->  U  =  (Unit `  R )
 )   &    |-  ( ph  ->  S  =  (oppr `  R ) )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  U  =  (Unit `  S ) )
 
Theoremcrngunit 13607 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 13608 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 13609 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 13610 Reversal of unitmulcl 13609 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 ) ) )
 
Theoremunitgrpbasd 13611 The base set of the group of units. (Contributed by Mario Carneiro, 25-Dec-2014.)
 |-  ( ph  ->  U  =  (Unit `  R )
 )   &    |-  ( ph  ->  G  =  ( (mulGrp `  R )s  U ) )   &    |-  ( ph  ->  R  e. SRing )   =>    |-  ( ph  ->  U  =  (
 Base `  G ) )
 
Theoremunitgrp 13612 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 13613 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 13614 The identity of the group of units of a ring is the ring unity. (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 13615 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 13616 Extend class notation with multiplicative inverse.
 class  invr
 
Definitiondf-invr 13617 Define multiplicative inverse. (Contributed by NM, 21-Sep-2011.)
 |- 
 invr  =  ( r  e.  _V  |->  ( invg `  ( (mulGrp `  r
 )s  (Unit `  r )
 ) ) )
 
Theoreminvrfvald 13618 Multiplicative inverse function for a ring. (Contributed by NM, 21-Sep-2011.) (Revised by Mario Carneiro, 25-Dec-2014.)
 |-  ( ph  ->  U  =  (Unit `  R )
 )   &    |-  ( ph  ->  G  =  ( (mulGrp `  R )s  U ) )   &    |-  ( ph  ->  I  =  (
 invr `  R ) )   &    |-  ( ph  ->  R  e.  Ring
 )   =>    |-  ( ph  ->  I  =  ( invg `  G ) )
 
Theoremunitinvcl 13619 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 13620 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 )
 
Theoremringinvcl 13621 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 13622 A unit times its inverse is the ring unity. (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 13623 A unit times its inverse is the ring unity. (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 13624 The inverse of the ring unity is the ring unity. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  I  =  ( invr `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  ( I `  .1.  )  =  .1.  )
 
Theorem0unit 13625 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 13626 The negative of a unit is a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  N  =  ( invg `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U ) 
 ->  ( N `  X )  e.  U )
 
Syntaxcdvr 13627 Extend class notation with ring division.
 class /r
 
Definitiondf-dvr 13628* 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
 ) ) ) )
 
Theoremdvrfvald 13629* Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Proof shortened by AV, 2-Mar-2024.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  .x.  =  ( .r `  R ) )   &    |-  ( ph  ->  U  =  (Unit `  R ) )   &    |-  ( ph  ->  I  =  ( invr `  R ) )   &    |-  ( ph  ->  ./  =  (/r `  R ) )   &    |-  ( ph  ->  R  e. SRing )   =>    |-  ( ph  ->  ./  =  ( x  e.  B ,  y  e.  U  |->  ( x 
 .x.  ( I `  y ) ) ) )
 
Theoremdvrvald 13630 Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  .x.  =  ( .r `  R ) )   &    |-  ( ph  ->  U  =  (Unit `  R ) )   &    |-  ( ph  ->  I  =  ( invr `  R ) )   &    |-  ( ph  ->  ./  =  (/r `  R ) )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  U )   =>    |-  ( ph  ->  ( X  ./  Y )  =  ( X  .x.  ( I `  Y ) ) )
 
Theoremdvrcl 13631 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 13632 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 13633 A ring element divided by itself is the ring unity. (dividap 8720 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 13634 A ring element divided by the ring unity is itself. (div1 8722 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 13635 An associative law for division. (divassap 8709 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 13636 A cancellation law for division. (divcanap1 8700 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 13637 A cancellation law for division. (divcanap3 8717 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 13638 Equality in terms of ratio equal to ring unity. (diveqap1 8724 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 ) )
 
Theoremdvrdir 13639 Distributive law for the division operation of a ring. (Contributed by Thierry Arnoux, 30-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  ./  =  (/r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  U )
 )  ->  ( ( X  .+  Y )  ./  Z )  =  (
 ( X  ./  Z )  .+  ( Y  ./  Z ) ) )
 
Theoremrdivmuldivd 13640 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 30-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  ./  =  (/r `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  U )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  W  e.  U )   =>    |-  ( ph  ->  (
 ( X  ./  Y )  .x.  ( Z  ./  W ) )  =  ( ( X  .x.  Z )  ./  ( Y  .x.  W ) ) )
 
Theoremringinvdv 13641 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 ) )
 
Theoremrngidpropdg 13642* The ring unity 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  ->  K  e.  V )   &    |-  ( ph  ->  L  e.  W )   =>    |-  ( ph  ->  ( 1r `  K )  =  ( 1r `  L ) )
 
Theoremdvdsrpropdg 13643* 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  ->  K  e. SRing )   &    |-  ( ph  ->  L  e. SRing )   =>    |-  ( ph  ->  ( ||r `  K )  =  (
 ||r `  L ) )
 
Theoremunitpropdg 13644* 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  ->  K  e.  Ring )   &    |-  ( ph  ->  L  e.  Ring )   =>    |-  ( ph  ->  (Unit `  K )  =  (Unit `  L ) )
 
Theoreminvrpropdg 13645* 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  ->  K  e.  Ring )   &    |-  ( ph  ->  L  e.  Ring )   =>    |-  ( ph  ->  ( invr `  K )  =  ( invr `  L )
 )
 
7.3.8  Ring homomorphisms
 
Syntaxcrh 13646 Extend class notation with the ring homomorphisms.
 class RingHom
 
Syntaxcrs 13647 Extend class notation with the ring isomorphisms.
 class RingIso
 
Definitiondf-rhm 13648* 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-rim 13649* 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 13650* 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 13651 Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  ( F  e.  ( R RingHom  S )  ->  R  e.  Ring )
 
Theoremrhmrcl2 13652 Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  ( F  e.  ( R RingHom  S )  ->  S  e.  Ring )
 
Theoremrhmex 13653 Set existence for ring homomorphism. (Contributed by Jim Kingdon, 16-May-2025.)
 |-  ( ( R  e.  V  /\  S  e.  W )  ->  ( R RingHom  S )  e.  _V )
 
Theoremisrhm 13654 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 13655 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 )
 )
 
Theoremrimrcl 13656 Reverse closure for an isomorphism of rings. (Contributed by AV, 22-Oct-2019.)
 |-  ( F  e.  ( R RingIso  S )  ->  ( R  e.  _V  /\  S  e.  _V ) )
 
Theoremisrim0 13657 A ring isomorphism is a homomorphism whose converse is also a homomorphism. (Contributed by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 10-Jan-2025.)
 |-  ( F  e.  ( R RingIso  S )  <->  ( F  e.  ( R RingHom  S )  /\  `' F  e.  ( S RingHom  R ) ) )
 
Theoremrhmghm 13658 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 13659 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 13660 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 13661* 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 13662* 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 13663 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 )
 
Theoremrhmf1o 13664 A ring homomorphism is bijective iff its converse is also a ring homomorphism. (Contributed by AV, 22-Oct-2019.)
 |-  B  =  ( Base `  R )   &    |-  C  =  (
 Base `  S )   =>    |-  ( F  e.  ( R RingHom  S )  ->  ( F : B -1-1-onto-> C  <->  `' F  e.  ( S RingHom  R ) ) )
 
Theoremisrim 13665 An isomorphism of rings is a bijective homomorphism. (Contributed by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 12-Jan-2025.)
 |-  B  =  ( Base `  R )   &    |-  C  =  (
 Base `  S )   =>    |-  ( F  e.  ( R RingIso  S )  <->  ( F  e.  ( R RingHom  S )  /\  F : B -1-1-onto-> C ) )
 
Theoremrimf1o 13666 An isomorphism of rings is a bijection. (Contributed by AV, 22-Oct-2019.)
 |-  B  =  ( Base `  R )   &    |-  C  =  (
 Base `  S )   =>    |-  ( F  e.  ( R RingIso  S )  ->  F : B -1-1-onto-> C )
 
Theoremrimrhm 13667 A ring isomorphism is a homomorphism. (Contributed by AV, 22-Oct-2019.) Remove hypotheses. (Revised by SN, 10-Jan-2025.)
 |-  ( F  e.  ( R RingIso  S )  ->  F  e.  ( R RingHom  S )
 )
 
Theoremrhmfn 13668 The mapping of two rings to the ring homomorphisms between them is a function. (Contributed by AV, 1-Mar-2020.)
 |- RingHom  Fn  ( Ring  X.  Ring )
 
Theoremrhmval 13669 The ring homomorphisms between two rings. (Contributed by AV, 1-Mar-2020.)
 |-  ( ( R  e.  Ring  /\  S  e.  Ring )  ->  ( R RingHom  S )  =  ( ( R  GrpHom  S )  i^i  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) ) ) )
 
Theoremrhmco 13670 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 )
 )
 
Theoremrhmdvdsr 13671 A ring homomorphism preserves the divisibility relation. (Contributed by Thierry Arnoux, 22-Oct-2017.)
 |-  X  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   &    |-  ./  =  ( ||r `  S )   =>    |-  ( ( ( F  e.  ( R RingHom  S ) 
 /\  A  e.  X  /\  B  e.  X ) 
 /\  A  .||  B ) 
 ->  ( F `  A )  ./  ( F `  B ) )
 
Theoremrhmopp 13672 A ring homomorphism is also a ring homomorphism for the opposite rings. (Contributed by Thierry Arnoux, 27-Oct-2017.)
 |-  ( F  e.  ( R RingHom  S )  ->  F  e.  ( (oppr `  R ) RingHom  (oppr `  S ) ) )
 
Theoremelrhmunit 13673 Ring homomorphisms preserve unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
 |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R ) )  ->  ( F `
  A )  e.  (Unit `  S )
 )
 
Theoremrhmunitinv 13674 Ring homomorphisms preserve the inverse of unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
 |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R ) )  ->  ( F `
  ( ( invr `  R ) `  A ) )  =  (
 ( invr `  S ) `  ( F `  A ) ) )
 
7.3.9  Nonzero rings and zero rings
 
Syntaxcnzr 13675 The class of nonzero rings.
 class NzRing
 
Definitiondf-nzr 13676 A nonzero or nontrivial ring is a ring with at least two values, or equivalently where 1 and 0 are different. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |- NzRing  =  { r  e.  Ring  |  ( 1r `  r
 )  =/=  ( 0g `  r ) }
 
Theoremisnzr 13677 Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |- 
 .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e. NzRing  <->  ( R  e.  Ring  /\  .1.  =/=  .0.  )
 )
 
Theoremnzrnz 13678 One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |- 
 .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e. NzRing  ->  .1.  =/=  .0.  )
 
Theoremnzrring 13679 A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Proof shortened by SN, 23-Feb-2025.)
 |-  ( R  e. NzRing  ->  R  e.  Ring )
 
Theoremisnzr2 13680 Equivalent characterization of nonzero rings: they have at least two elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  R )   =>    |-  ( R  e. NzRing  <->  ( R  e.  Ring  /\  2o  ~<_  B ) )
 
Theoremopprnzrbg 13681 The opposite of a nonzero ring is nonzero, bidirectional form of opprnzr 13682. (Contributed by SN, 20-Jun-2025.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e.  V  ->  ( R  e. NzRing  <->  O  e. NzRing ) )
 
Theoremopprnzr 13682 The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e. NzRing  ->  O  e. NzRing )
 
Theoremringelnzr 13683 A ring is nonzero if it has a nonzero element. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by Mario Carneiro, 13-Jun-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  B  =  (
 Base `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  ( B  \  {  .0.  } ) )  ->  R  e. NzRing )
 
Theoremnzrunit 13684 A unit is nonzero in any nonzero ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  U  =  (Unit `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. NzRing  /\  A  e.  U ) 
 ->  A  =/=  .0.  )
 
Theorem01eq0ring 13685 If the zero and the identity element of a ring are the same, the ring is the zero ring. (Contributed by AV, 16-Apr-2019.) (Proof shortened by SN, 23-Feb-2025.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  .0.  =  .1.  )  ->  B  =  {  .0.  } )
 
7.3.10  Local rings
 
Syntaxclring 13686 Extend class notation with class of all local rings.
 class LRing
 
Definitiondf-lring 13687* A local ring is a nonzero ring where for any two elements summing to one, at least one is invertible. Any field is a local ring; the ring of integers is an example of a ring which is not a local ring. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.)
 |- LRing  =  { r  e. NzRing  |  A. x  e.  ( Base `  r ) A. y  e.  ( Base `  r )
 ( ( x (
 +g  `  r )
 y )  =  ( 1r `  r ) 
 ->  ( x  e.  (Unit `  r )  \/  y  e.  (Unit `  r )
 ) ) }
 
Theoremislring 13688* The predicate "is a local ring". (Contributed by SN, 23-Feb-2025.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( R  e. LRing  <->  ( R  e. NzRing  /\ 
 A. x  e.  B  A. y  e.  B  ( ( x  .+  y
 )  =  .1.  ->  ( x  e.  U  \/  y  e.  U )
 ) ) )
 
Theoremlringnzr 13689 A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.)
 |-  ( R  e. LRing  ->  R  e. NzRing )
 
Theoremlringring 13690 A local ring is a ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)
 |-  ( R  e. LRing  ->  R  e.  Ring )
 
Theoremlringnz 13691 A local ring is a nonzero ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)
 |- 
 .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e. LRing  ->  .1.  =/=  .0.  )
 
Theoremlringuplu 13692 If the sum of two elements of a local ring is invertible, then at least one of the summands must be invertible. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  U  =  (Unit `  R )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  R )
 )   &    |-  ( ph  ->  R  e. LRing )   &    |-  ( ph  ->  ( X  .+  Y )  e.  U )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  e.  U  \/  Y  e.  U )
 )
 
7.3.11  Subrings
 
7.3.11.1  Subrings of non-unital rings
 
Syntaxcsubrng 13693 Extend class notation with all subrings of a non-unital ring.
 class SubRng
 
Definitiondf-subrng 13694* Define a subring of a non-unital ring as a set of elements that is a non-unital ring in its own right. In this section, a subring of a non-unital ring is simply called "subring", unless it causes any ambiguity with SubRing. (Contributed by AV, 14-Feb-2025.)
 |- SubRng  =  ( w  e. Rng  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s
 )  e. Rng } )
 
Theoremissubrng 13695 The subring of non-unital ring predicate. (Contributed by AV, 14-Feb-2025.)
 |-  B  =  ( Base `  R )   =>    |-  ( A  e.  (SubRng `  R )  <->  ( R  e. Rng  /\  ( Rs  A )  e. Rng  /\  A  C_  B ) )
 
Theoremsubrngss 13696 A subring is a subset. (Contributed by AV, 14-Feb-2025.)
 |-  B  =  ( Base `  R )   =>    |-  ( A  e.  (SubRng `  R )  ->  A  C_  B )
 
Theoremsubrngid 13697 Every non-unital ring is a subring of itself. (Contributed by AV, 14-Feb-2025.)
 |-  B  =  ( Base `  R )   =>    |-  ( R  e. Rng  ->  B  e.  (SubRng `  R ) )
 
Theoremsubrngrng 13698 A subring is a non-unital ring. (Contributed by AV, 14-Feb-2025.)
 |-  S  =  ( Rs  A )   =>    |-  ( A  e.  (SubRng `  R )  ->  S  e. Rng )
 
Theoremsubrngrcl 13699 Reverse closure for a subring predicate. (Contributed by AV, 14-Feb-2025.)
 |-  ( A  e.  (SubRng `  R )  ->  R  e. Rng )
 
Theoremsubrngsubg 13700 A subring is a subgroup. (Contributed by AV, 14-Feb-2025.)
 |-  ( A  e.  (SubRng `  R )  ->  A  e.  (SubGrp `  R )
 )
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