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Theorem List for Intuitionistic Logic Explorer - 13301-13400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremringsubdi 13301 Ring multiplication distributes over subtraction. (subdi 8355 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .-  =  ( -g `  R )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  ( X  .x.  ( Y  .-  Z ) )  =  ( ( X  .x.  Y )  .-  ( X  .x.  Z ) ) )
 
Theoremringsubdir 13302 Ring multiplication distributes over subtraction. (subdir 8356 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .-  =  ( -g `  R )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 ( X  .-  Y )  .x.  Z )  =  ( ( X  .x.  Z )  .-  ( Y  .x.  Z ) ) )
 
Theoremmulgass2 13303 An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  (.g `  R )   &    |-  .X.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  ( ( N  .x.  X )  .X.  Y )  =  ( N  .x.  ( X  .X.  Y ) ) )
 
Theoremring1 13304 The (smallest) structure representing a zero ring. (Contributed by AV, 28-Apr-2019.)
 |-  M  =  { <. (
 Base `  ndx ) ,  { Z } >. , 
 <. ( +g  `  ndx ) ,  { <. <. Z ,  Z >. ,  Z >. }
 >. ,  <. ( .r `  ndx ) ,  { <. <. Z ,  Z >. ,  Z >. } >. }   =>    |-  ( Z  e.  V  ->  M  e.  Ring )
 
Theoremringn0 13305 The class of rings is not empty (it is also inhabited, as shown at ring1 13304). (Contributed by AV, 29-Apr-2019.)
 |- 
 Ring  =/=  (/)
 
Theoremringressid 13306 A ring restricted to its base set is a ring. It will usually be the original ring exactly, of course, but to show that needs additional conditions such as those in strressid 12544. (Contributed by Jim Kingdon, 28-Feb-2025.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e.  Ring  ->  ( Gs  B )  e.  Ring )
 
Theoremimasring 13307* The image structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |- 
 .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ( ph  /\  ( a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `
  a )  =  ( F `  p )  /\  ( F `  b )  =  ( F `  q ) ) 
 ->  ( F `  (
 a  .+  b )
 )  =  ( F `
  ( p  .+  q ) ) ) )   &    |-  ( ( ph  /\  ( a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `
  a )  =  ( F `  p )  /\  ( F `  b )  =  ( F `  q ) ) 
 ->  ( F `  (
 a  .x.  b )
 )  =  ( F `
  ( p  .x.  q ) ) ) )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  ( U  e.  Ring  /\  ( F `  .1.  )  =  ( 1r `  U ) ) )
 
Theoremimasringf1 13308 The image of a ring under an injection is a ring. (Contributed by AV, 27-Feb-2025.)
 |-  U  =  ( F 
 "s 
 R )   &    |-  V  =  (
 Base `  R )   =>    |-  ( ( F : V -1-1-> B  /\  R  e.  Ring )  ->  U  e.  Ring )
 
Theoremqusring2 13309* The quotient structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  ( ph  ->  U  =  ( R  /.s  .~  ) )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |- 
 .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  .~  Er  V )   &    |-  ( ph  ->  ( ( a  .~  p  /\  b  .~  q ) 
 ->  ( a  .+  b
 )  .~  ( p  .+  q ) ) )   &    |-  ( ph  ->  ( (
 a  .~  p  /\  b  .~  q )  ->  ( a  .x.  b ) 
 .~  ( p  .x.  q ) ) )   &    |-  ( ph  ->  R  e.  Ring
 )   =>    |-  ( ph  ->  ( U  e.  Ring  /\  [  .1.  ]  .~  =  ( 1r `  U ) ) )
 
7.3.6  Opposite ring
 
Syntaxcoppr 13310 The opposite ring operation.
 class oppr
 
Definitiondf-oppr 13311 Define an opposite ring, which is the same as the original ring but with multiplication written the other way around. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |- oppr  =  ( f  e.  _V  |->  ( f sSet  <. ( .r
 `  ndx ) , tpos  ( .r `  f ) >. ) )
 
Theoremopprvalg 13312 Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  O  =  (oppr `  R )   =>    |-  ( R  e.  V  ->  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. ) )
 
Theoremopprmulfvalg 13313 Value of the multiplication operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  O  =  (oppr `  R )   &    |-  .xb  =  ( .r `  O )   =>    |-  ( R  e.  V  ->  .xb  = tpos  .x.  )
 
Theoremopprmulg 13314 Value of the multiplication operation of an opposite ring. Hypotheses eliminated by a suggestion of Stefan O'Rear, 30-Aug-2015. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  O  =  (oppr `  R )   &    |-  .xb  =  ( .r `  O )   =>    |-  ( ( R  e.  V  /\  X  e.  W  /\  Y  e.  U )  ->  ( X 
 .xb  Y )  =  ( Y  .x.  X )
 )
 
Theoremcrngoppr 13315 In a commutative ring, the opposite ring is equivalent to the original ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  O  =  (oppr `  R )   &    |-  .xb  =  ( .r `  O )   =>    |-  ( ( R  e.  CRing  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  Y )  =  ( X 
 .xb  Y ) )
 
Theoremopprex 13316 Existence of the opposite ring. If you know that  R is a ring, see opprring 13322. (Contributed by Jim Kingdon, 10-Jan-2025.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e.  V  ->  O  e.  _V )
 
Theoremopprsllem 13317 Lemma for opprbasg 13318 and oppraddg 13319. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.)
 |-  O  =  (oppr `  R )   &    |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )   &    |-  ( E `  ndx )  =/=  ( .r `  ndx )   =>    |-  ( R  e.  V  ->  ( E `  R )  =  ( E `  O ) )
 
Theoremopprbasg 13318 Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
 |-  O  =  (oppr `  R )   &    |-  B  =  ( Base `  R )   =>    |-  ( R  e.  V  ->  B  =  ( Base `  O ) )
 
Theoremoppraddg 13319 Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
 |-  O  =  (oppr `  R )   &    |- 
 .+  =  ( +g  `  R )   =>    |-  ( R  e.  V  ->  .+  =  ( +g  `  O ) )
 
Theoremopprrng 13320 An opposite non-unital ring is a non-unital ring. (Contributed by AV, 15-Feb-2025.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e. Rng  ->  O  e. Rng )
 
Theoremopprrngbg 13321 A set is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 13320. (Contributed by AV, 15-Feb-2025.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e.  V  ->  ( R  e. Rng  <->  O  e. Rng ) )
 
Theoremopprring 13322 An opposite ring is a ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e.  Ring  ->  O  e.  Ring )
 
Theoremopprringbg 13323 Bidirectional form of opprring 13322. (Contributed by Mario Carneiro, 6-Dec-2014.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e.  V  ->  ( R  e.  Ring  <->  O  e.  Ring ) )
 
Theoremoppr0g 13324 Additive identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  O  =  (oppr `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( R  e.  V  ->  .0.  =  ( 0g
 `  O ) )
 
Theoremoppr1g 13325 Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  O  =  (oppr `  R )   &    |- 
 .1.  =  ( 1r `  R )   =>    |-  ( R  e.  V  ->  .1.  =  ( 1r
 `  O ) )
 
Theoremopprnegg 13326 The negative function in an opposite ring. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  O  =  (oppr `  R )   &    |-  N  =  ( invg `  R )   =>    |-  ( R  e.  V  ->  N  =  ( invg `  O ) )
 
Theoremopprsubgg 13327 Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e.  V  ->  (SubGrp `  R )  =  (SubGrp `  O )
 )
 
Theoremmulgass3 13328 An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  (.g `  R )   &    |-  .X.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  ( X  .X.  ( N 
 .x.  Y ) )  =  ( N  .x.  ( X  .X.  Y ) ) )
 
7.3.7  Divisibility
 
Syntaxcdsr 13329 Ring divisibility relation.
 class  ||r
 
Syntaxcui 13330 Units in a ring.
 class Unit
 
Syntaxcir 13331 Ring irreducibles.
 class Irred
 
Definitiondf-dvdsr 13332* 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 13333 Define the set of units in a ring, that is, all elements with a left and right multiplicative inverse. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |- Unit  =  ( w  e.  _V  |->  ( `' ( ( ||r
 `  w )  i^i  ( ||r
 `  (oppr `  w ) ) )
 " { ( 1r
 `  w ) }
 ) )
 
Definitiondf-irred 13334* 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 } )
 
Theoremreldvdsrsrg 13335 The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2025.)
 |-  ( R  e. SRing  ->  Rel  ( ||r
 `  R ) )
 
Theoremdvdsrvald 13336* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  .||  =  (
 ||r `  R ) )   &    |-  ( ph  ->  R  e. SRing )   &    |-  ( ph  ->  .x.  =  ( .r `  R ) )   =>    |-  ( ph  ->  .||  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }
 )
 
Theoremdvdsrd 13337* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  .||  =  (
 ||r `  R ) )   &    |-  ( ph  ->  R  e. SRing )   &    |-  ( ph  ->  .x.  =  ( .r `  R ) )   =>    |-  ( ph  ->  ( X  .|| 
 Y 
 <->  ( X  e.  B  /\  E. z  e.  B  ( z  .x.  X )  =  Y ) ) )
 
Theoremdvdsr2d 13338* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  .||  =  (
 ||r `  R ) )   &    |-  ( ph  ->  R  e. SRing )   &    |-  ( ph  ->  .x.  =  ( .r `  R ) )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( X  .||  Y  <->  E. z  e.  B  ( z  .x.  X )  =  Y ) )
 
Theoremdvdsrmuld 13339 A left-multiple of  X is divisible by  X. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  .||  =  (
 ||r `  R ) )   &    |-  ( ph  ->  R  e. SRing )   &    |-  ( ph  ->  .x.  =  ( .r `  R ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  X  .||  ( Y  .x.  X ) )
 
Theoremdvdsrcld 13340 Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  .||  =  (
 ||r `  R ) )   &    |-  ( ph  ->  R  e. SRing )   &    |-  ( ph  ->  X  .||  Y )   =>    |-  ( ph  ->  X  e.  B )
 
Theoremdvdsrex 13341 Existence of the divisibility relation. (Contributed by Jim Kingdon, 28-Jan-2025.)
 |-  ( R  e. SRing  ->  (
 ||r `  R )  e.  _V )
 
Theoremdvdsrcl2 13342 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 13343 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 13344 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 13345 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 13346 An element divides its negative. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   &    |-  N  =  ( invg `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  X  .||  ( N `  X ) )
 
Theoremdvdsr01 13347 In a ring, zero is divisible by all elements. ("Zero divisor" as a term has a somewhat different meaning.) (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 13348 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 13349 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 13350 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 13351 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 13352 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 13353 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 13354 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 13355 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 13356 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 13357 Reversal of unitmulcl 13356 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 13358 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 13359 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 13360 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 13361 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 13362 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 13363 Extend class notation with multiplicative inverse.
 class  invr
 
Definitiondf-invr 13364 Define multiplicative inverse. (Contributed by NM, 21-Sep-2011.)
 |- 
 invr  =  ( r  e.  _V  |->  ( invg `  ( (mulGrp `  r
 )s  (Unit `  r )
 ) ) )
 
Theoreminvrfvald 13365 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 13366 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 13367 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 13368 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 13369 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 13370 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 13371 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 13372 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 13373 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 13374 Extend class notation with ring division.
 class /r
 
Definitiondf-dvr 13375* 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 13376* 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 13377 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 13378 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 13379 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 13380 A ring element divided by itself is the ring unity. (dividap 8671 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 13381 A ring element divided by the ring unity is itself. (div1 8673 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 13382 An associative law for division. (divassap 8660 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 13383 A cancellation law for division. (divcanap1 8651 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 13384 A cancellation law for division. (divcanap3 8668 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 13385 Equality in terms of ratio equal to ring unity. (diveqap1 8675 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 13386 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 13387 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 13388 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 13389* 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 13390* 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 13391* 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 13392* 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 13393 Extend class notation with the ring homomorphisms.
 class RingHom
 
Syntaxcrs 13394 Extend class notation with the ring isomorphisms.
 class RingIso
 
Definitiondf-rnghom 13395* 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 13396* 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 ) } )
 
7.3.9  Nonzero rings and zero rings
 
Syntaxcnzr 13397 The class of nonzero rings.
 class NzRing
 
Definitiondf-nzr 13398 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 13399 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 13400 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.  )
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