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Theorem List for Intuitionistic Logic Explorer - 13201-13300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremringdi 13201 Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( X  .x.  ( Y  .+  Z ) )  =  (
 ( X  .x.  Y )  .+  ( X  .x.  Z ) ) )
 
Theoremringdir 13202 Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .+  Y )  .x.  Z )  =  ( ( X  .x.  Z )  .+  ( Y  .x.  Z ) ) )
 
Theoremringidcl 13203 The unity element of a ring belongs to the base set of the ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  .1. 
 e.  B )
 
Theoremring0cl 13204 The zero element of a ring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  .0. 
 e.  B )
 
Theoremringidmlem 13205 Lemma for ringlidm 13206 and ringridm 13207. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  ( (  .1.  .x.  X )  =  X  /\  ( X  .x.  .1.  )  =  X ) )
 
Theoremringlidm 13206 The unity element of a ring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  (  .1.  .x.  X )  =  X )
 
Theoremringridm 13207 The unity element of a ring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  ( X  .x.  .1.  )  =  X )
 
Theoremisringid 13208* Properties showing that an element 
I is the unity element of a ring. (Contributed by NM, 7-Aug-2013.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  ( ( I  e.  B  /\  A. x  e.  B  ( ( I 
 .x.  x )  =  x  /\  ( x 
 .x.  I )  =  x ) )  <->  .1.  =  I ) )
 
Theoremringid 13209* The multiplication operation of a unital ring has (one or more) identity elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 24-Aug-2021.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  E. u  e.  B  ( ( u 
 .x.  X )  =  X  /\  ( X  .x.  u )  =  X )
 )
 
Theoremringadd2 13210* A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 24-Aug-2021.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  E. x  e.  B  ( X  .+  X )  =  ( ( x 
 .+  x )  .x.  X ) )
 
Theoremringo2times 13211 A ring element plus itself is two times the element. "Two" in an arbitrary unital ring is the sum of the unity element with itself. (Contributed by AV, 24-Aug-2021.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  A  e.  B ) 
 ->  ( A  .+  A )  =  ( (  .1.  .+  .1.  )  .x.  A ) )
 
Theoremringidss 13212 A subset of the multiplicative group has the multiplicative identity as its identity if the identity is in the subset. (Contributed by Mario Carneiro, 27-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  M  =  ( (mulGrp `  R )s  A )   &    |-  B  =  (
 Base `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  .1.  =  ( 0g `  M ) )
 
Theoremringacl 13213 Closure of the addition operation of a ring. (Contributed by Mario Carneiro, 14-Jan-2014.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  e.  B )
 
Theoremringcom 13214 Commutativity of the additive group of a ring. (Contributed by Gérard Lang, 4-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
 
Theoremringabl 13215 A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.)
 |-  ( R  e.  Ring  ->  R  e.  Abel )
 
Theoremringcmn 13216 A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( R  e.  Ring  ->  R  e. CMnd )
 
Theoremringpropd 13217* If two structures have the same group components (properties), one is a ring iff the other one is. (Contributed by Mario Carneiro, 6-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-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.  Ring  <->  L  e.  Ring )
 )
 
Theoremcrngpropd 13218* If two structures have the same group components (properties), one is a commutative ring 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.  CRing  <->  L  e.  CRing ) )
 
Theoremringprop 13219 If two structures have the same ring components (properties), one is a ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.)
 |-  ( Base `  K )  =  ( Base `  L )   &    |-  ( +g  `  K )  =  ( +g  `  L )   &    |-  ( .r `  K )  =  ( .r `  L )   =>    |-  ( K  e.  Ring  <->  L  e.  Ring )
 
Theoremisringd 13220* Properties that determine a ring. (Contributed by NM, 2-Aug-2013.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  R )
 )   &    |-  ( ph  ->  .x.  =  ( .r `  R ) )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  (
 ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .x.  y )  e.  B )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  ( ( x  .x.  y )  .x.  z )  =  ( x  .x.  ( y  .x.  z ) ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B )
 )  ->  ( x  .x.  ( y  .+  z
 ) )  =  ( ( x  .x.  y
 )  .+  ( x  .x.  z ) ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B )
 )  ->  ( ( x  .+  y )  .x.  z )  =  (
 ( x  .x.  z
 )  .+  ( y  .x.  z ) ) )   &    |-  ( ph  ->  .1.  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  (  .1.  .x.  x )  =  x )   &    |-  (
 ( ph  /\  x  e.  B )  ->  ( x  .x.  .1.  )  =  x )   =>    |-  ( ph  ->  R  e.  Ring )
 
Theoremiscrngd 13221* Properties that determine a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  R )
 )   &    |-  ( ph  ->  .x.  =  ( .r `  R ) )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  (
 ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .x.  y )  e.  B )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  ( ( x  .x.  y )  .x.  z )  =  ( x  .x.  ( y  .x.  z ) ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B )
 )  ->  ( x  .x.  ( y  .+  z
 ) )  =  ( ( x  .x.  y
 )  .+  ( x  .x.  z ) ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B )
 )  ->  ( ( x  .+  y )  .x.  z )  =  (
 ( x  .x.  z
 )  .+  ( y  .x.  z ) ) )   &    |-  ( ph  ->  .1.  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  (  .1.  .x.  x )  =  x )   &    |-  (
 ( ph  /\  x  e.  B )  ->  ( x  .x.  .1.  )  =  x )   &    |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .x.  y
 )  =  ( y 
 .x.  x ) )   =>    |-  ( ph  ->  R  e.  CRing
 )
 
Theoremringlz 13222 The zero of a unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  (  .0.  .x.  X )  =  .0.  )
 
Theoremringrz 13223 The zero of a unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  ( X  .x.  .0.  )  =  .0.  )
 
Theoremringsrg 13224 Any ring is also a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)
 |-  ( R  e.  Ring  ->  R  e. SRing )
 
Theoremring1eq0 13225 If one and zero are equal, then any two elements of a ring are equal. Alternately, every ring has one distinct from zero except the zero ring containing the single element  { 0 }. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  B  =  ( Base `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (  .1.  =  .0. 
 ->  X  =  Y ) )
 
Theoremringinvnz1ne0 13226* In a unital ring, a left invertible element is different from zero iff  .1.  =/=  .0.. (Contributed by FL, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |- 
 .0.  =  ( 0g `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  E. a  e.  B  ( a  .x.  X )  =  .1.  )   =>    |-  ( ph  ->  ( X  =/=  .0.  <->  .1.  =/=  .0.  )
 )
 
Theoremringinvnzdiv 13227* In a unital ring, a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.) (Revised by Jeff Madsen, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |- 
 .0.  =  ( 0g `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  E. a  e.  B  ( a  .x.  X )  =  .1.  )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( X  .x.  Y )  =  .0.  <->  Y  =  .0.  ) )
 
Theoremringnegl 13228 Negation in a ring is the same as left multiplication by -1. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  N  =  ( invg `  R )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  (
 ( N `  .1.  )  .x.  X )  =  ( N `  X ) )
 
Theoremringnegr 13229 Negation in a ring is the same as right multiplication by -1. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  N  =  ( invg `  R )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( X  .x.  ( N `  .1.  ) )  =  ( N `  X ) )
 
Theoremringmneg1 13230 Negation of a product in a ring. (mulneg1 8352 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  N  =  ( invg `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( N `  X )  .x.  Y )  =  ( N `  ( X  .x.  Y ) ) )
 
Theoremringmneg2 13231 Negation of a product in a ring. (mulneg2 8353 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  N  =  ( invg `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .x.  ( N `  Y ) )  =  ( N `  ( X  .x.  Y ) ) )
 
Theoremringm2neg 13232 Double negation of a product in a ring. (mul2neg 8355 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  N  =  ( invg `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( N `  X )  .x.  ( N `  Y ) )  =  ( X  .x.  Y ) )
 
Theoremringsubdi 13233 Ring multiplication distributes over subtraction. (subdi 8342 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 13234 Ring multiplication distributes over subtraction. (subdir 8343 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 13235 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 13236 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 13237 The class of rings is not empty (it is also inhabited, as shown at ring1 13236). (Contributed by AV, 29-Apr-2019.)
 |- 
 Ring  =/=  (/)
 
Theoremringressid 13238 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 12530. (Contributed by Jim Kingdon, 28-Feb-2025.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e.  Ring  ->  ( Gs  B )  e.  Ring )
 
7.3.5  Opposite ring
 
Syntaxcoppr 13239 The opposite ring operation.
 class oppr
 
Definitiondf-oppr 13240 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 13241 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 13242 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 13243 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 13244 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 13245 Existence of the opposite ring. If you know that  R is a ring, see opprring 13249. (Contributed by Jim Kingdon, 10-Jan-2025.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e.  V  ->  O  e.  _V )
 
Theoremopprsllem 13246 Lemma for opprbasg 13247 and oppraddg 13248. (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 13247 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 13248 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 ) )
 
Theoremopprring 13249 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 13250 Bidirectional form of opprring 13249. (Contributed by Mario Carneiro, 6-Dec-2014.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e.  V  ->  ( R  e.  Ring  <->  O  e.  Ring ) )
 
Theoremoppr0g 13251 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 13252 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 13253 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 ) )
 
Theoremmulgass3 13254 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 ) ) )
 
7.3.6  Divisibility
 
Syntaxcdsr 13255 Ring divisibility relation.
 class  ||r
 
Syntaxcui 13256 Units in a ring.
 class Unit
 
Syntaxcir 13257 Ring irreducibles.
 class Irred
 
Definitiondf-dvdsr 13258* 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 13259 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 13260* 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 13261 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 13262* 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 13263* 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 13264* 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 13265 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 13266 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 13267 Existence of the divisibility relation. (Contributed by Jim Kingdon, 28-Jan-2025.)
 |-  ( R  e. SRing  ->  (
 ||r `  R )  e.  _V )
 
Theoremdvdsrcl2 13268 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 13269 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 13270 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 13271 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 13272 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 13273 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 13274 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 13275 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 13276 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 13277 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 13278 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 13279 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 13280 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 13281 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 13282 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 13283 Reversal of unitmulcl 13282 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 13284 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 13285 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 13286 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 13287 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 13288 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 13289 Extend class notation with multiplicative inverse.
 class  invr
 
Definitiondf-invr 13290 Define multiplicative inverse. (Contributed by NM, 21-Sep-2011.)
 |- 
 invr  =  ( r  e.  _V  |->  ( invg `  ( (mulGrp `  r
 )s  (Unit `  r )
 ) ) )
 
Theoreminvrfvald 13291 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 13292 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 13293 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 13294 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 13295 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 13296 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 13297 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 13298 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 13299 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 13300 Extend class notation with ring division.
 class /r
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