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Theorem List for Intuitionistic Logic Explorer - 13101-13200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiscmn 13101* The predicate "is a commutative monoid". (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( G  e. CMnd  <->  ( G  e.  Mnd  /\  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  (
 y  .+  x )
 ) )
 
Theoremisabl2 13102* The predicate "is an Abelian (commutative) group". (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( G  e.  Abel  <->  ( G  e.  Grp  /\  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
 
Theoremcmnpropd 13103* If two structures have the same group components (properties), one is a commutative monoid iff the other one is. (Contributed 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  ->  ( K  e. CMnd  <->  L  e. CMnd ) )
 
Theoremablpropd 13104* If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 6-Dec-2014.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  L ) y ) )   =>    |-  ( ph  ->  ( K  e.  Abel 
 <->  L  e.  Abel )
 )
 
Theoremablprop 13105 If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 11-Oct-2013.)
 |-  ( Base `  K )  =  ( Base `  L )   &    |-  ( +g  `  K )  =  ( +g  `  L )   =>    |-  ( K  e.  Abel  <->  L  e.  Abel )
 
Theoremiscmnd 13106* Properties that determine a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
 )  =  ( y 
 .+  x ) )   =>    |-  ( ph  ->  G  e. CMnd )
 
Theoremisabld 13107* Properties that determine an Abelian group. (Contributed by NM, 6-Aug-2013.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
 )  =  ( y 
 .+  x ) )   =>    |-  ( ph  ->  G  e.  Abel
 )
 
Theoremisabli 13108* Properties that determine an Abelian group. (Contributed by NM, 4-Sep-2011.)
 |-  G  e.  Grp   &    |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  (
 ( x  e.  B  /\  y  e.  B )  ->  ( x  .+  y )  =  (
 y  .+  x )
 )   =>    |-  G  e.  Abel
 
Theoremcmnmnd 13109 A commutative monoid is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  ( G  e. CMnd  ->  G  e.  Mnd )
 
Theoremcmncom 13110 A commutative monoid is commutative. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e. CMnd  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
 
Theoremablcom 13111 An Abelian group operation is commutative. (Contributed by NM, 26-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
 
Theoremcmn32 13112 Commutative/associative law for commutative monoids. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .+  Y ) 
 .+  Z )  =  ( ( X  .+  Z )  .+  Y ) )
 
Theoremcmn4 13113 Commutative/associative law for commutative monoids. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  ( ( X  .+  Y )  .+  ( Z 
 .+  W ) )  =  ( ( X 
 .+  Z )  .+  ( Y  .+  W ) ) )
 
Theoremcmn12 13114 Commutative/associative law for commutative monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( X 
 .+  ( Y  .+  Z ) )  =  ( Y  .+  ( X  .+  Z ) ) )
 
Theoremabl32 13115 Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 ( X  .+  Y )  .+  Z )  =  ( ( X  .+  Z )  .+  Y ) )
 
Theoremcmnmndd 13116 A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.)
 |-  ( ph  ->  G  e. CMnd )   =>    |-  ( ph  ->  G  e.  Mnd )
 
Theoremrinvmod 13117* Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6070. (Contributed by AV, 31-Dec-2023.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  E* w  e.  B  ( A  .+  w )  =  .0.  )
 
Theoremablinvadd 13118 The inverse of an Abelian group operation. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  N  =  ( invg `  G )   =>    |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .+  Y ) )  =  ( ( N `
  X )  .+  ( N `  Y ) ) )
 
Theoremablsub2inv 13119 Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  N  =  ( invg `  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( N `  X )  .-  ( N `  Y ) )  =  ( Y  .-  X ) )
 
Theoremablsubadd 13120 Relationship between Abelian group subtraction and addition. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .-  Y )  =  Z  <->  ( Y  .+  Z )  =  X ) )
 
Theoremablsub4 13121 Commutative/associative subtraction law for Abelian groups. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B ) 
 /\  ( Z  e.  B  /\  W  e.  B ) )  ->  ( ( X  .+  Y ) 
 .-  ( Z  .+  W ) )  =  ( ( X  .-  Z )  .+  ( Y 
 .-  W ) ) )
 
Theoremabladdsub4 13122 Abelian group addition/subtraction law. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B ) 
 /\  ( Z  e.  B  /\  W  e.  B ) )  ->  ( ( X  .+  Y )  =  ( Z  .+  W )  <->  ( X  .-  Z )  =  ( W  .-  Y ) ) )
 
Theoremabladdsub 13123 Associative-type law for group subtraction and addition. (Contributed by NM, 19-Apr-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .+  Y )  .-  Z )  =  (
 ( X  .-  Z )  .+  Y ) )
 
Theoremablpncan2 13124 Cancellation law for subtraction in an Abelian group. (Contributed by NM, 2-Oct-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .+  Y )  .-  X )  =  Y )
 
Theoremablpncan3 13125 A cancellation law for Abelian groups. (Contributed by NM, 23-Mar-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( X  .+  ( Y  .-  X ) )  =  Y )
 
Theoremablsubsub 13126 Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e.  Abel
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  ( X  .-  ( Y  .-  Z ) )  =  ( ( X  .-  Y )  .+  Z ) )
 
Theoremablsubsub4 13127 Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e.  Abel
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 ( X  .-  Y )  .-  Z )  =  ( X  .-  ( Y  .+  Z ) ) )
 
Theoremablpnpcan 13128 Cancellation law for mixed addition and subtraction. (pnpcan 8198 analog.) (Contributed by NM, 29-May-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e.  Abel
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 ( X  .+  Y )  .-  ( X  .+  Z ) )  =  ( Y  .-  Z ) )
 
Theoremablnncan 13129 Cancellation law for group subtraction. (nncan 8188 analog.) (Contributed by NM, 7-Apr-2015.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .-  ( X  .-  Y ) )  =  Y )
 
Theoremablsub32 13130 Swap the second and third terms in a double group subtraction. (Contributed by NM, 7-Apr-2015.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 ( X  .-  Y )  .-  Z )  =  ( ( X  .-  Z )  .-  Y ) )
 
Theoremablnnncan 13131 Cancellation law for group subtraction. (nnncan 8194 analog.) (Contributed by NM, 29-Feb-2008.) (Revised by AV, 27-Aug-2021.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 ( X  .-  ( Y  .-  Z ) ) 
 .-  Z )  =  ( X  .-  Y ) )
 
Theoremablnnncan1 13132 Cancellation law for group subtraction. (nnncan1 8195 analog.) (Contributed by NM, 7-Apr-2015.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 ( X  .-  Y )  .-  ( X  .-  Z ) )  =  ( Z  .-  Y ) )
 
Theoremablsubsub23 13133 Swap subtrahend and result of group subtraction. (Contributed by NM, 14-Dec-2007.) (Revised by AV, 7-Oct-2021.)
 |-  V  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  ->  ( ( A  .-  B )  =  C  <->  ( A  .-  C )  =  B ) )
 
Theoremsubcmnd 13134 A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  ( ph  ->  H  =  ( G ↾s  S ) )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  H  e.  Mnd )   &    |-  ( ph  ->  S  e.  V )   =>    |-  ( ph  ->  H  e. CMnd )
 
7.3  Rings
 
7.3.1  Multiplicative Group
 
Syntaxcmgp 13135 Multiplicative group.
 class mulGrp
 
Definitiondf-mgp 13136 Define a structure that puts the multiplication operation of a ring in the addition slot. Note that this will not actually be a group for the average ring, or even for a field, but it will be a monoid, and we get a group if we restrict to the elements that have inverses. This allows us to formalize such notions as "the multiplication operation of a ring is a monoid" or "the multiplicative identity" in terms of the identity of a monoid (df-ur 13148). (Contributed by Mario Carneiro, 21-Dec-2014.)
 |- mulGrp  =  ( w  e.  _V  |->  ( w sSet  <. ( +g  ` 
 ndx ) ,  ( .r `  w ) >. ) )
 
Theoremfnmgp 13137 The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |- mulGrp  Fn  _V
 
Theoremmgpvalg 13138 Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.)
 |-  M  =  (mulGrp `  R )   &    |- 
 .x.  =  ( .r `  R )   =>    |-  ( R  e.  V  ->  M  =  ( R sSet  <. ( +g  `  ndx ) ,  .x.  >. ) )
 
Theoremmgpplusgg 13139 Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.)
 |-  M  =  (mulGrp `  R )   &    |- 
 .x.  =  ( .r `  R )   =>    |-  ( R  e.  V  ->  .x.  =  ( +g  `  M ) )
 
Theoremmgpex 13140 Existence of the multiplication group. If  R is known to be a semiring, see srgmgp 13156. (Contributed by Jim Kingdon, 10-Jan-2025.)
 |-  M  =  (mulGrp `  R )   =>    |-  ( R  e.  V  ->  M  e.  _V )
 
Theoremmgpbasg 13141 Base set of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  B  =  ( Base `  R )   =>    |-  ( R  e.  V  ->  B  =  ( Base `  M ) )
 
Theoremmgpscag 13142 The multiplication monoid has the same (if any) scalars as the original ring. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  S  =  (Scalar `  R )   =>    |-  ( R  e.  V  ->  S  =  (Scalar `  M ) )
 
Theoremmgptsetg 13143 Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   =>    |-  ( R  e.  V  ->  (TopSet `  R )  =  (TopSet `  M )
 )
 
Theoremmgptopng 13144 Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  J  =  ( TopOpen `  R )   =>    |-  ( R  e.  V  ->  J  =  ( TopOpen `  M ) )
 
Theoremmgpdsg 13145 Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  B  =  ( dist `  R )   =>    |-  ( R  e.  V  ->  B  =  ( dist `  M ) )
 
Theoremmgpress 13146 Subgroup commutes with the multiplicative group operator. (Contributed by Mario Carneiro, 10-Jan-2015.) (Proof shortened by AV, 18-Oct-2024.)
 |-  S  =  ( R ↾s  A )   &    |-  M  =  (mulGrp `  R )   =>    |-  ( ( R  e.  V  /\  A  e.  W )  ->  ( M ↾s  A )  =  (mulGrp `  S ) )
 
7.3.2  Ring unity (multiplicative identity)

In Wikipedia "Identity element", see https://en.wikipedia.org/wiki/Identity_element (18-Jan-2025): "... an identity with respect to multiplication is called a multiplicative identity (often denoted as 1). ... The distinction between additive and multiplicative identity is used most often for sets that support both binary operations, such as rings, integral domains, and fields. The multiplicative identity is often called unity in the latter context (a ring with unity). This should not be confused with a unit in ring theory, which is any element having a multiplicative inverse. By its own definition, unity itself is necessarily a unit."

Calling the multiplicative identity of a ring a unity is taken from the definition of a ring with unity in section 17.3 of [BeauregardFraleigh] p. 135, "A ring ( R , + , . ) is a ring with unity if R is not the zero ring and ( R , . ) is a monoid. In this case, the identity element of ( R , . ) is denoted by 1 and is called the unity of R." This definition of a "ring with unity" corresponds to our definition of a unital ring (see df-ring 13186).

Some authors call the multiplicative identity "unit" or "unit element" (for example in section I, 2.2 of [BourbakiAlg1] p. 14, definition in section 1.3 of [Hall] p. 4, or in section I, 1 of [Lang] p. 3), whereas other authors use the term "unit" for an element having a multiplicative inverse (for example in section 17.3 of [BeauregardFraleigh] p. 135, in definition in [Roman] p. 26, or even in section II, 1 of [Lang] p. 84). Sometimes, the multiplicative identity is simply called "one" (see, for example, chapter 8 in [Schechter] p. 180).

To avoid this ambiguity of the term "unit", also mentioned in Wikipedia, we call the multiplicative identity of a structure with a multiplication (usually a ring) a "ring unity", or straightly "multiplicative identity".

The term "unit" will be used for an element having a multiplicative inverse (see https://us.metamath.org/mpeuni/df-unit.html 13186 in set.mm), and we have "the ring unity is a unit", see https://us.metamath.org/mpeuni/1unit.html 13186.

 
Syntaxcur 13147 Extend class notation with ring unity.
 class  1r
 
Definitiondf-ur 13148 Define the multiplicative identity, i.e., the monoid identity (df-0g 12712) of the multiplicative monoid (df-mgp 13136) of a ring-like structure. This multiplicative identity is also called "ring unity" or "unity element".

This definition works by transferring the multiplicative operation from the  .r slot to the  +g slot and then looking at the element which is then the  0g element, that is an identity with respect to the operation which started out in the  .r slot.

See also dfur2g 13150, which derives the "traditional" definition as the unique element of a ring which is left- and right-neutral under multiplication. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

 |- 
 1r  =  ( 0g 
 o. mulGrp )
 
Theoremringidvalg 13149 The value of the unity element of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  G  =  (mulGrp `  R )   &    |- 
 .1.  =  ( 1r `  R )   =>    |-  ( R  e.  V  ->  .1.  =  ( 0g
 `  G ) )
 
Theoremdfur2g 13150* The multiplicative identity is the unique element of the ring that is left- and right-neutral on all elements under multiplication. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  V  ->  .1.  =  ( iota
 e ( e  e.  B  /\  A. x  e.  B  ( ( e 
 .x.  x )  =  x  /\  ( x 
 .x.  e )  =  x ) ) ) )
 
7.3.3  Semirings
 
Syntaxcsrg 13151 Extend class notation with the class of all semirings.
 class SRing
 
Definitiondf-srg 13152* Define class of all semirings. A semiring is a set equipped with two everywhere-defined internal operations, whose first one is an additive commutative monoid structure and the second one is a multiplicative monoid structure, and where multiplication is (left- and right-) distributive over addition. Like with rings, the additive identity is an absorbing element of the multiplicative law, but in the case of semirings, this has to be part of the definition, as it cannot be deduced from distributivity alone. Definition of [Golan] p. 1. Note that our semirings are unital. Such semirings are sometimes called "rigs", being "rings without negatives". (Contributed by Thierry Arnoux, 21-Mar-2018.)
 |- SRing  =  { f  e. CMnd  |  ( (mulGrp `  f )  e.  Mnd  /\  [. ( Base `  f )  /  r ]. [. ( +g  `  f
 )  /  p ]. [. ( .r `  f )  /  t ]. [. ( 0g
 `  f )  /  n ]. A. x  e.  r  ( A. y  e.  r  A. z  e.  r  ( ( x t ( y p z ) )  =  ( ( x t y ) p ( x t z ) )  /\  ( ( x p y ) t z )  =  ( ( x t z ) p ( y t z ) ) )  /\  (
 ( n t x )  =  n  /\  ( x t n )  =  n ) ) ) }
 
Theoremissrg 13153* The predicate "is a semiring". (Contributed by Thierry Arnoux, 21-Mar-2018.)
 |-  B  =  ( Base `  R )   &    |-  G  =  (mulGrp `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e. SRing  <->  ( R  e. CMnd  /\  G  e.  Mnd  /\  A. x  e.  B  (
 A. y  e.  B  A. z  e.  B  ( ( x  .x.  (
 y  .+  z )
 )  =  ( ( x  .x.  y )  .+  ( x  .x.  z
 ) )  /\  (
 ( x  .+  y
 )  .x.  z )  =  ( ( x  .x.  z )  .+  ( y 
 .x.  z ) ) )  /\  ( (  .0.  .x.  x )  =  .0.  /\  ( x  .x.  .0.  )  =  .0.  ) ) ) )
 
Theoremsrgcmn 13154 A semiring is a commutative monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
 |-  ( R  e. SRing  ->  R  e. CMnd )
 
Theoremsrgmnd 13155 A semiring is a monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
 |-  ( R  e. SRing  ->  R  e.  Mnd )
 
Theoremsrgmgp 13156 A semiring is a monoid under multiplication. (Contributed by Thierry Arnoux, 21-Mar-2018.)
 |-  G  =  (mulGrp `  R )   =>    |-  ( R  e. SRing  ->  G  e.  Mnd )
 
Theoremsrgdilem 13157 Lemma for srgdi 13162 and srgdir 13163. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .x.  ( Y  .+  Z ) )  =  ( ( X  .x.  Y )  .+  ( X 
 .x.  Z ) )  /\  ( ( X  .+  Y )  .x.  Z )  =  ( ( X 
 .x.  Z )  .+  ( Y  .x.  Z ) ) ) )
 
Theoremsrgcl 13158 Closure of the multiplication operation of a semiring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e. SRing  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .x.  Y )  e.  B )
 
Theoremsrgass 13159 Associative law for the multiplication operation of a semiring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .x.  Y )  .x.  Z )  =  ( X 
 .x.  ( Y  .x.  Z ) ) )
 
Theoremsrgideu 13160* The unity element of a semiring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( R  e. SRing  ->  E! u  e.  B  A. x  e.  B  ( ( u  .x.  x )  =  x  /\  ( x  .x.  u )  =  x ) )
 
Theoremsrgfcl 13161 Functionality of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by AV, 24-Aug-2021.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e. SRing  /\  .x.  Fn  ( B  X.  B ) ) 
 ->  .x.  : ( B  X.  B ) --> B )
 
Theoremsrgdi 13162 Distributive law for the multiplication operation of a semiring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( X  .x.  ( Y  .+  Z ) )  =  (
 ( X  .x.  Y )  .+  ( X  .x.  Z ) ) )
 
Theoremsrgdir 13163 Distributive law for the multiplication operation of a semiring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e. SRing  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .+  Y )  .x.  Z )  =  ( ( X  .x.  Z )  .+  ( Y  .x.  Z ) ) )
 
Theoremsrgidcl 13164 The unity element of a semiring belongs to the base set of the semiring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e. SRing  ->  .1.  e.  B )
 
Theoremsrg0cl 13165 The zero element of a semiring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e. SRing  ->  .0.  e.  B )
 
Theoremsrgidmlem 13166 Lemma for srglidm 13167 and srgridm 13168. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e. SRing  /\  X  e.  B ) 
 ->  ( (  .1.  .x.  X )  =  X  /\  ( X  .x.  .1.  )  =  X ) )
 
Theoremsrglidm 13167 The unity element of a semiring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011.) (Revised by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e. SRing  /\  X  e.  B ) 
 ->  (  .1.  .x.  X )  =  X )
 
Theoremsrgridm 13168 The unity element of a semiring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.) (Revised by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e. SRing  /\  X  e.  B ) 
 ->  ( X  .x.  .1.  )  =  X )
 
Theoremissrgid 13169* Properties showing that an element 
I is the unity element of a semiring. (Contributed by NM, 7-Aug-2013.) (Revised by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e. SRing  ->  ( ( I  e.  B  /\  A. x  e.  B  ( ( I  .x.  x )  =  x  /\  ( x  .x.  I
 )  =  x ) )  <->  .1.  =  I ) )
 
Theoremsrgacl 13170 Closure of the addition operation of a semiring. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   =>    |-  ( ( R  e. SRing  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .+  Y )  e.  B )
 
Theoremsrgcom 13171 Commutativity of the additive group of a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   =>    |-  ( ( R  e. SRing  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .+  Y )  =  ( Y  .+  X ) )
 
Theoremsrgrz 13172 The zero of a semiring is a right-absorbing element. (Contributed by Thierry Arnoux, 1-Apr-2018.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. SRing  /\  X  e.  B ) 
 ->  ( X  .x.  .0.  )  =  .0.  )
 
Theoremsrglz 13173 The zero of a semiring is a left-absorbing element. (Contributed by AV, 23-Aug-2019.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e. SRing  /\  X  e.  B ) 
 ->  (  .0.  .x.  X )  =  .0.  )
 
Theoremsrgisid 13174* In a semiring, the only left-absorbing element is the additive identity. Remark in [Golan] p. 1. (Contributed by Thierry Arnoux, 1-May-2018.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  ( ph  ->  R  e. SRing )   &    |-  ( ph  ->  Z  e.  B )   &    |-  (
 ( ph  /\  x  e.  B )  ->  ( Z  .x.  x )  =  Z )   =>    |-  ( ph  ->  Z  =  .0.  )
 
Theoremsrg1zr 13175 The only semiring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .*  =  ( .r `  R )   =>    |-  ( ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B )  /\  .* 
 Fn  ( B  X.  B ) )  /\  Z  e.  B )  ->  ( B  =  { Z }  <->  (  .+  =  { <.
 <. Z ,  Z >. ,  Z >. }  /\  .*  =  { <. <. Z ,  Z >. ,  Z >. } )
 ) )
 
Theoremsrgen1zr 13176 The only semiring with one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 14-Feb-2010.) (Revised by AV, 25-Jan-2020.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .*  =  ( .r `  R )   &    |-  Z  =  ( 0g
 `  R )   =>    |-  ( ( R  e. SRing  /\  .+  Fn  ( B  X.  B )  /\  .* 
 Fn  ( B  X.  B ) )  ->  ( B  ~~  1o  <->  (  .+  =  { <.
 <. Z ,  Z >. ,  Z >. }  /\  .*  =  { <. <. Z ,  Z >. ,  Z >. } )
 ) )
 
Theoremsrgmulgass 13177 An associative property between group multiple and ring multiplication for semirings. (Contributed by AV, 23-Aug-2019.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  (.g `  R )   &    |-  .X.  =  ( .r `  R )   =>    |-  ( ( R  e. SRing  /\  ( N  e.  NN0  /\  X  e.  B  /\  Y  e.  B )
 )  ->  ( ( N  .x.  X )  .X.  Y )  =  ( N 
 .x.  ( X  .X.  Y ) ) )
 
Theoremsrgpcomp 13178 If two elements of a semiring commute, they also commute if one of the elements is raised to a higher power. (Contributed by AV, 23-Aug-2019.)
 |-  S  =  ( Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  G  =  (mulGrp `  R )   &    |-  .^  =  (.g `  G )   &    |-  ( ph  ->  R  e. SRing )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  K  e.  NN0 )   &    |-  ( ph  ->  ( A  .X.  B )  =  ( B  .X.  A ) )   =>    |-  ( ph  ->  (
 ( K  .^  B )  .X.  A )  =  ( A  .X.  ( K  .^  B ) ) )
 
Theoremsrgpcompp 13179 If two elements of a semiring commute, they also commute if the elements are raised to a higher power. (Contributed by AV, 23-Aug-2019.)
 |-  S  =  ( Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  G  =  (mulGrp `  R )   &    |-  .^  =  (.g `  G )   &    |-  ( ph  ->  R  e. SRing )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  K  e.  NN0 )   &    |-  ( ph  ->  ( A  .X.  B )  =  ( B  .X.  A ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( ( N  .^  A )  .X.  ( K 
 .^  B ) ) 
 .X.  A )  =  ( ( ( N  +  1 )  .^  A ) 
 .X.  ( K  .^  B ) ) )
 
Theoremsrgpcomppsc 13180 If two elements of a semiring commute, they also commute if the elements are raised to a higher power and a scalar multiplication is involved. (Contributed by AV, 23-Aug-2019.)
 |-  S  =  ( Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  G  =  (mulGrp `  R )   &    |-  .^  =  (.g `  G )   &    |-  ( ph  ->  R  e. SRing )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  K  e.  NN0 )   &    |-  ( ph  ->  ( A  .X.  B )  =  ( B  .X.  A ) )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  .x.  =  (.g `  R )   &    |-  ( ph  ->  C  e.  NN0 )   =>    |-  ( ph  ->  (
 ( C  .x.  (
 ( N  .^  A )  .X.  ( K  .^  B ) ) ) 
 .X.  A )  =  ( C  .x.  ( (
 ( N  +  1 )  .^  A )  .X.  ( K  .^  B ) ) ) )
 
Theoremsrglmhm 13181* Left-multiplication in a semiring by a fixed element of the ring is a monoid homomorphism. (Contributed by AV, 23-Aug-2019.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e. SRing  /\  X  e.  B )  ->  ( x  e.  B  |->  ( X  .x.  x ) )  e.  ( R MndHom  R )
 )
 
Theoremsrgrmhm 13182* Right-multiplication in a semiring by a fixed element of the ring is a monoid homomorphism. (Contributed by AV, 23-Aug-2019.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e. SRing  /\  X  e.  B )  ->  ( x  e.  B  |->  ( x  .x.  X ) )  e.  ( R MndHom  R ) )
 
Theoremsrg1expzeq1 13183 The exponentiation (by a nonnegative integer) of the multiplicative identity of a semiring, analogous to mulgnn0z 13015. (Contributed by AV, 25-Nov-2019.)
 |-  G  =  (mulGrp `  R )   &    |- 
 .x.  =  (.g `  G )   &    |- 
 .1.  =  ( 1r `  R )   =>    |-  ( ( R  e. SRing  /\  N  e.  NN0 )  ->  ( N  .x.  .1.  )  =  .1.  )
 
7.3.4  Definition and basic properties of unital rings
 
Syntaxcrg 13184 Extend class notation with class of all (unital) rings.
 class  Ring
 
Syntaxccrg 13185 Extend class notation with class of all (unital) commutative rings.
 class  CRing
 
Definitiondf-ring 13186* Define class of all (unital) rings. A unital ring is a set equipped with two everywhere-defined internal operations, whose first one is an additive group structure and the second one is a multiplicative monoid structure, and where the addition is left- and right-distributive for the multiplication. Definition 1 in [BourbakiAlg1] p. 92 or definition of a ring with identity in part Preliminaries of [Roman] p. 19. So that the additive structure must be abelian (see ringcom 13219), care must be taken that in the case of a non-unital ring, the commutativity of addition must be postulated and cannot be proved from the other conditions. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |- 
 Ring  =  { f  e.  Grp  |  ( (mulGrp `  f )  e.  Mnd  /\  [. ( Base `  f )  /  r ]. [. ( +g  `  f )  /  p ]. [. ( .r
 `  f )  /  t ]. A. x  e.  r  A. y  e.  r  A. z  e.  r  ( ( x t ( y p z ) )  =  ( ( x t y ) p ( x t z ) )  /\  ( ( x p y ) t z )  =  ( ( x t z ) p ( y t z ) ) ) ) }
 
Definitiondf-cring 13187 Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |- 
 CRing  =  { f  e.  Ring  |  (mulGrp `  f
 )  e. CMnd }
 
Theoremisring 13188* The predicate "is a (unital) ring". Definition of "ring with unit" in [Schechter] p. 187. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  R )   &    |-  G  =  (mulGrp `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( R  e.  Ring  <->  ( R  e.  Grp  /\  G  e.  Mnd  /\  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( ( x 
 .x.  ( y  .+  z ) )  =  ( ( x  .x.  y )  .+  ( x 
 .x.  z ) ) 
 /\  ( ( x 
 .+  y )  .x.  z )  =  (
 ( x  .x.  z
 )  .+  ( y  .x.  z ) ) ) ) )
 
Theoremringgrp 13189 A ring is a group. (Contributed by NM, 15-Sep-2011.)
 |-  ( R  e.  Ring  ->  R  e.  Grp )
 
Theoremringmgp 13190 A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  G  =  (mulGrp `  R )   =>    |-  ( R  e.  Ring  ->  G  e.  Mnd )
 
Theoremiscrng 13191 A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  G  =  (mulGrp `  R )   =>    |-  ( R  e.  CRing  <->  ( R  e.  Ring  /\  G  e. CMnd ) )
 
Theoremcrngmgp 13192 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  G  =  (mulGrp `  R )   =>    |-  ( R  e.  CRing  ->  G  e. CMnd )
 
Theoremringgrpd 13193 A ring is a group. (Contributed by SN, 16-May-2024.)
 |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  R  e.  Grp )
 
Theoremringmnd 13194 A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( R  e.  Ring  ->  R  e.  Mnd )
 
Theoremringmgm 13195 A ring is a magma. (Contributed by AV, 31-Jan-2020.)
 |-  ( R  e.  Ring  ->  R  e. Mgm )
 
Theoremcrngring 13196 A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( R  e.  CRing  ->  R  e.  Ring )
 
Theoremcrngringd 13197 A commutative ring is a ring. (Contributed by SN, 16-May-2024.)
 |-  ( ph  ->  R  e.  CRing )   =>    |-  ( ph  ->  R  e.  Ring )
 
Theoremcrnggrpd 13198 A commutative ring is a group. (Contributed by SN, 16-May-2024.)
 |-  ( ph  ->  R  e.  CRing )   =>    |-  ( ph  ->  R  e.  Grp )
 
Theoremmgpf 13199 Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (mulGrp  |`  Ring ) : Ring --> Mnd
 
Theoremringdilem 13200 Properties of a unital ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  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 ) )  /\  ( ( X  .+  Y )  .x.  Z )  =  ( ( X 
 .x.  Z )  .+  ( Y  .x.  Z ) ) ) )
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