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Theorem List for Metamath Proof Explorer - 39401-39500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsubmgmrcl 39401 Reverse closure for submagmas. (Contributed by AV, 24-Feb-2020.)
 |-  ( S  e.  (SubMgm `  M )  ->  M  e. Mgm )
 
Theoremismgmhm 39402* Property of a magma homomorphism. (Contributed by AV, 25-Feb-2020.)
 |-  B  =  ( Base `  S )   &    |-  C  =  ( Base `  T )   &    |-  .+  =  ( +g  `  S )   &    |-  .+^  =  (
 +g  `  T )   =>    |-  ( F  e.  ( S MgmHom  T )  <->  ( ( S  e. Mgm  /\  T  e. Mgm ) 
 /\  ( F : B
 --> C  /\  A. x  e.  B  A. y  e.  B  ( F `  ( x  .+  y ) )  =  ( ( F `  x )  .+^  ( F `  y
 ) ) ) ) )
 
Theoremmgmhmf 39403 A magma homomorphism is a function. (Contributed by AV, 25-Feb-2020.)
 |-  B  =  ( Base `  S )   &    |-  C  =  ( Base `  T )   =>    |-  ( F  e.  ( S MgmHom  T )  ->  F : B
 --> C )
 
Theoremmgmhmpropd 39404* Magma homomorphism depends only on the operation of structures. (Contributed by AV, 25-Feb-2020.)
 |-  ( ph  ->  B  =  (
 Base `  J ) )   &    |-  ( ph  ->  C  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ph  ->  C  =  ( Base `  M )
 )   &    |-  ( ph  ->  B  =/= 
 (/) )   &    |-  ( ph  ->  C  =/=  (/) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  ( x ( +g  `  J ) y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  C ) )  ->  ( x ( +g  `  K ) y )  =  ( x ( +g  `  M ) y ) )   =>    |-  ( ph  ->  ( J MgmHom  K )  =  ( L MgmHom  M ) )
 
Theoremmgmhmlin 39405 A magma homomorphism preserves the binary operation. (Contributed by AV, 25-Feb-2020.)
 |-  B  =  ( Base `  S )   &    |-  .+  =  ( +g  `  S )   &    |-  .+^  =  (
 +g  `  T )   =>    |-  (
 ( F  e.  ( S MgmHom  T )  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `
  ( X  .+  Y ) )  =  ( ( F `  X )  .+^  ( F `
  Y ) ) )
 
Theoremmgmhmf1o 39406 A magma homomorphism is bijective iff its converse is also a magma homomorphism. (Contributed by AV, 25-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   =>    |-  ( F  e.  ( R MgmHom  S )  ->  ( F : B -1-1-onto-> C  <->  `' F  e.  ( S MgmHom  R ) ) )
 
Theoremidmgmhm 39407 The identity homomorphism on a magma. (Contributed by AV, 27-Feb-2020.)
 |-  B  =  ( Base `  M )   =>    |-  ( M  e. Mgm  ->  (  _I  |`  B )  e.  ( M MgmHom  M ) )
 
Theoremissubmgm 39408* Expand definition of a submagma. (Contributed by AV, 25-Feb-2020.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   =>    |-  ( M  e. Mgm  ->  ( S  e.  (SubMgm `  M ) 
 <->  ( S  C_  B  /\  A. x  e.  S  A. y  e.  S  ( x  .+  y )  e.  S ) ) )
 
Theoremissubmgm2 39409 Submagmas are subsets that are also magmas. (Contributed by AV, 25-Feb-2020.)
 |-  B  =  ( Base `  M )   &    |-  H  =  ( Ms  S )   =>    |-  ( M  e. Mgm  ->  ( S  e.  (SubMgm `  M ) 
 <->  ( S  C_  B  /\  H  e. Mgm ) ) )
 
Theoremrabsubmgmd 39410* Deduction for proving that a restricted class abstraction is a submagma. (Contributed by AV, 26-Feb-2020.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  ( ph  ->  M  e. Mgm )   &    |-  (
 ( ph  /\  ( ( x  e.  B  /\  y  e.  B )  /\  ( th  /\  ta ) ) )  ->  et )   &    |-  ( z  =  x  ->  ( ps  <->  th ) )   &    |-  ( z  =  y  ->  ( ps  <->  ta ) )   &    |-  ( z  =  ( x  .+  y
 )  ->  ( ps  <->  et ) )   =>    |-  ( ph  ->  { z  e.  B  |  ps }  e.  (SubMgm `  M )
 )
 
Theoremsubmgmss 39411 Submagmas are subsets of the base set. (Contributed by AV, 26-Feb-2020.)
 |-  B  =  ( Base `  M )   =>    |-  ( S  e.  (SubMgm `  M )  ->  S  C_  B )
 
Theoremsubmgmid 39412 Every magma is trivially a submagma of itself. (Contributed by AV, 26-Feb-2020.)
 |-  B  =  ( Base `  M )   =>    |-  ( M  e. Mgm  ->  B  e.  (SubMgm `  M ) )
 
Theoremsubmgmcl 39413 Submagmas are closed under the monoid operation. (Contributed by AV, 26-Feb-2020.)
 |-  .+  =  ( +g  `  M )   =>    |-  (
 ( S  e.  (SubMgm `  M )  /\  X  e.  S  /\  Y  e.  S )  ->  ( X 
 .+  Y )  e.  S )
 
Theoremsubmgmmgm 39414 Submagmas are themselves magmas under the given operation. (Contributed by AV, 26-Feb-2020.)
 |-  H  =  ( Ms  S )   =>    |-  ( S  e.  (SubMgm `  M )  ->  H  e. Mgm )
 
Theoremsubmgmbas 39415 The base set of a submagma. (Contributed by AV, 26-Feb-2020.)
 |-  H  =  ( Ms  S )   =>    |-  ( S  e.  (SubMgm `  M )  ->  S  =  ( Base `  H )
 )
 
Theoremsubsubmgm 39416 A submagma of a submagma is a submagma. (Contributed by AV, 26-Feb-2020.)
 |-  H  =  ( Gs  S )   =>    |-  ( S  e.  (SubMgm `  G )  ->  ( A  e.  (SubMgm `  H ) 
 <->  ( A  e.  (SubMgm `  G )  /\  A  C_  S ) ) )
 
Theoremresmgmhm 39417 Restriction of a magma homomorphism to a submagma is a homomorphism. (Contributed by AV, 26-Feb-2020.)
 |-  U  =  ( Ss  X )   =>    |-  ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S ) )  ->  ( F  |`  X )  e.  ( U MgmHom  T ) )
 
Theoremresmgmhm2 39418 One direction of resmgmhm2b 39419. (Contributed by AV, 26-Feb-2020.)
 |-  U  =  ( Ts  X )   =>    |-  ( ( F  e.  ( S MgmHom  U )  /\  X  e.  (SubMgm `  T ) )  ->  F  e.  ( S MgmHom  T ) )
 
Theoremresmgmhm2b 39419 Restriction of the codomain of a homomorphism. (Contributed by AV, 26-Feb-2020.)
 |-  U  =  ( Ts  X )   =>    |-  ( ( X  e.  (SubMgm `  T )  /\  ran 
 F  C_  X )  ->  ( F  e.  ( S MgmHom  T )  <->  F  e.  ( S MgmHom  U ) ) )
 
Theoremmgmhmco 39420 The composition of magma homomorphisms is a homomorphism. (Contributed by AV, 27-Feb-2020.)
 |-  (
 ( F  e.  ( T MgmHom  U )  /\  G  e.  ( S MgmHom  T )
 )  ->  ( F  o.  G )  e.  ( S MgmHom  U ) )
 
Theoremmgmhmima 39421 The homomorphic image of a submagma is a submagma. (Contributed by AV, 27-Feb-2020.)
 |-  (
 ( F  e.  ( M MgmHom  N )  /\  X  e.  (SubMgm `  M )
 )  ->  ( F " X )  e.  (SubMgm `  N ) )
 
Theoremmgmhmeql 39422 The equalizer of two magma homomorphisms is a submagma. (Contributed by AV, 27-Feb-2020.)
 |-  (
 ( F  e.  ( S MgmHom  T )  /\  G  e.  ( S MgmHom  T )
 )  ->  dom  ( F  i^i  G )  e.  (SubMgm `  S )
 )
 
Theoremsubmgmacs 39423 Submagmas are an algebraic closure system. (Contributed by AV, 27-Feb-2020.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e. Mgm  ->  (SubMgm `  G )  e.  (ACS `  B ) )
 
Theoremismhm0 39424 Property of a monoid homomorphism, expressed by a magma homomorphism. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  S )   &    |-  C  =  ( Base `  T )   &    |-  .+  =  ( +g  `  S )   &    |-  .+^  =  (
 +g  `  T )   &    |-  .0.  =  ( 0g `  S )   &    |-  Y  =  ( 0g
 `  T )   =>    |-  ( F  e.  ( S MndHom  T )  <->  ( ( S  e.  Mnd  /\  T  e.  Mnd )  /\  ( F  e.  ( S MgmHom  T )  /\  ( F `  .0.  )  =  Y ) ) )
 
Theoremmhmismgmhm 39425 Each monoid homomorphism is a magma homomorphism. (Contributed by AV, 29-Feb-2020.)
 |-  ( F  e.  ( R MndHom  S )  ->  F  e.  ( R MgmHom  S ) )
 
21.33.10.4  Examples and counterexamples for magmas, semigroups and monoids (extension)
 
Theoremopmpt2ismgm 39426* A structure with a group addition operation in maps-to notation is a magma if the operation value is contained in the base set. (Contributed by AV, 16-Feb-2020.)
 |-  B  =  ( Base `  M )   &    |-  ( +g  `  M )  =  ( x  e.  B ,  y  e.  B  |->  C )   &    |-  ( ph  ->  B  =/=  (/) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  C  e.  B )   =>    |-  ( ph  ->  M  e. Mgm )
 
Theoremcopissgrp 39427* A structure with a constant group addition operation is a semigroup if the constant is contained in the base set. (Contributed by AV, 16-Feb-2020.)
 |-  B  =  ( Base `  M )   &    |-  ( +g  `  M )  =  ( x  e.  B ,  y  e.  B  |->  C )   &    |-  ( ph  ->  B  =/=  (/) )   &    |-  ( ph  ->  C  e.  B )   =>    |-  ( ph  ->  M  e. SGrp )
 
Theoremcopisnmnd 39428* A structure with a constant group addition operation and at least two elements is not a monoid. (Contributed by AV, 16-Feb-2020.)
 |-  B  =  ( Base `  M )   &    |-  ( +g  `  M )  =  ( x  e.  B ,  y  e.  B  |->  C )   &    |-  ( ph  ->  C  e.  B )   &    |-  ( ph  ->  1  <  ( # `
  B ) )   =>    |-  ( ph  ->  M  e/  Mnd )
 
Theorem0nodd 39429* 0 is not an odd integer. (Contributed by AV, 3-Feb-2020.)
 |-  O  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( (
 2  x.  x )  +  1 ) }   =>    |-  0  e/  O
 
Theorem1odd 39430* 1 is an odd integer. (Contributed by AV, 3-Feb-2020.)
 |-  O  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( (
 2  x.  x )  +  1 ) }   =>    |-  1  e.  O
 
Theorem2nodd 39431* 2 is not an odd integer. (Contributed by AV, 3-Feb-2020.)
 |-  O  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( (
 2  x.  x )  +  1 ) }   =>    |-  2  e/  O
 
Theoremoddibas 39432* Lemma 1 for oddinmgm 39434: The base set of M is the set of all odd integers. (Contributed by AV, 3-Feb-2020.)
 |-  O  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( (
 2  x.  x )  +  1 ) }   &    |-  M  =  (flds  O )   =>    |-  O  =  ( Base `  M )
 
Theoremoddiadd 39433* Lemma 2 for oddinmgm 39434: The group addition operation of M is the addition of complex numbers. (Contributed by AV, 3-Feb-2020.)
 |-  O  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( (
 2  x.  x )  +  1 ) }   &    |-  M  =  (flds  O )   =>    |- 
 +  =  ( +g  `  M )
 
Theoremoddinmgm 39434* The structure of all odd integers together with the addition of complex numbers is not a magma. Remark: the structure of the complementary subset of the set of integers, the even integers, is a magma, actually an abelian group, see 2zrngaabl 39563, and even a non-unital ring, see 2zrng 39554. (Contributed by AV, 3-Feb-2020.)
 |-  O  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( (
 2  x.  x )  +  1 ) }   &    |-  M  =  (flds  O )   =>    |-  M  e/ Mgm
 
Theoremnnsgrpmgm 39435 The structure of positive integers together with the addition of complex numbers is a magma. (Contributed by AV, 4-Feb-2020.)
 |-  M  =  (flds  NN )   =>    |-  M  e. Mgm
 
Theoremnnsgrp 39436 The structure of positive integers together with the addition of complex numbers is a semigroup. (Contributed by AV, 4-Feb-2020.)
 |-  M  =  (flds  NN )   =>    |-  M  e. SGrp
 
Theoremnnsgrpnmnd 39437 The structure of positive integers together with the addition of complex numbers is not a monoid. (Contributed by AV, 4-Feb-2020.)
 |-  M  =  (flds  NN )   =>    |-  M  e/  Mnd
 
21.33.11  Magmas and internal binary operations (alternate approach)

With df-mpt2 6310, binary operations are defined by a rule, and with df-ov 6308, the value of a binary operation applied to two operands can be expressed. In both cases, the two operands can belong to different sets, and the result can be an element of a third set. However, according to Wikipedia "Binary operation", see https://en.wikipedia.org/wiki/Binary_operation (19-Jan-2020), "... a binary operation on a set  S is a mapping of the elements of the Cartesian product  S  X.  S to S:  f : ( S  X.  S
--> S ). Because the result of performing the operation on a pair of elements of S is again an element of S, the operation is called a closed binary operation on S (or sometimes expressed as having the property of closure).". To distinguish this more restrictive definition (in Wikipedia and most of the literature) from the general case, we call binary operations mapping the elements of the Cartesian product  S  X.  S internal binary operations, see df-intop 39454. If, in addition, the result is also contained in the set  S, the operation is called closed internal binary operation, see df-clintop 39455. Therefore, a "binary operation on a set  S" according to Wikipedia is a "closed internal binary operation" in our terminology. If the sets are different, the operation is explicitly called external binary operation (see Wikipedia https://en.wikipedia.org/wiki/Binary_operation#External_binary_operations ).

Taking a step back, we define "laws" applicable for "binary operations" (which even need not to be functions), according to the definition in [Hall] p. 1 and [BourbakiAlg1] p. 1, p. 4 and p. 7. These laws are used, on the one hand, to specialize internal binary operations (see df-clintop 39455 and df-assintop 39456), and on the other hand to define the common algebraic structures like magmas, groups, rings, etc. Internal binary operations, which obeys these laws, are defined afterwards. Notice that in [BourbakiAlg1] p. 1, p. 4 and p. 7, these operations are called "laws" by themselves.

In the following, an alternative definition df-cllaw 39441 for an internal binary operation is provided, which does not require to be a function, but only closure. Therefore, this definition could be used as binary operation (slot 2) defined for a magma as extensible structure, see mgmplusgiopALT 39449, or for an alternative definition df-mgm2 39474 for a magma as extensible structure. Similar results are obtained for an associative operation resp. semigroups.

 
21.33.11.1  Laws for internal binary operations

In this subsection, the "laws" applicable for "binary operations" according to the definition in [Hall] p. 1 and [BourbakiAlg1] p. 1, p. 4 and p. 7 are defined. These laws are called "internal laws" in [BourbakiAlg1] p. xxi.

 
Syntaxccllaw 39438 Extend class notation for the closure law.
 class clLaw
 
Syntaxcasslaw 39439 Extend class notation for the associative law.
 class assLaw
 
Syntaxccomlaw 39440 Extend class notation for the commutative law.
 class comLaw
 
Definitiondf-cllaw 39441* The closure law for binary operations, see definitions of laws A0. and M0. in section 1.1 of [Hall] p. 1, or definition 1 in [BourbakiAlg1] p. 1: the value of a binary operation applied to two operands of a given sets is an element of this set. By this definition, the closure law is expressed as binary relation: a binary operation is related to a set by clLaw if the closure law holds for this binary operation regarding this set. Note that the binary operation needs not to be a function. (Contributed by AV, 7-Jan-2020.)
 |- clLaw  =  { <. o ,  m >.  | 
 A. x  e.  m  A. y  e.  m  ( x o y )  e.  m }
 
Definitiondf-comlaw 39442* The commutative law for binary operations, see definitions of laws A2. and M2. in section 1.1 of [Hall] p. 1, or definition 8 in [BourbakiAlg1] p. 7: the value of a binary operation applied to two operands equals the value of a binary operation applied to the two operands in reversed order. By this definition, the commutative law is expressed as binary relation: a binary operation is related to a set by comLaw if the commutative law holds for this binary operation regarding this set. Note that the binary operation needs neither to be closed nor to be a function. (Contributed by AV, 7-Jan-2020.)
 |- comLaw  =  { <. o ,  m >.  | 
 A. x  e.  m  A. y  e.  m  ( x o y )  =  ( y o x ) }
 
Definitiondf-asslaw 39443* The associative law for binary operations, see definitions of laws A1. and M1. in section 1.1 of [Hall] p. 1, or definition 5 in [BourbakiAlg1] p. 4: the value of a binary operation applied the value of the binary operation applied to two operands and a third operand equals the value of the binary operation applied to the first operand and the value of the binary operation applied to the second and third operand. By this definition, the associative law is expressed as binary relation: a binary operation is related to a set by assLaw if the associative law holds for this binary operation regarding this set. Note that the binary operation needs neither to be closed nor to be a function. (Contributed by FL, 1-Nov-2009.) (Revised by AV, 13-Jan-2020.)
 |- assLaw  =  { <. o ,  m >.  | 
 A. x  e.  m  A. y  e.  m  A. z  e.  m  (
 ( x o y ) o z )  =  ( x o ( y o z ) ) }
 
Theoremiscllaw 39444* The predicate "is a closed operation". (Contributed by AV, 13-Jan-2020.)
 |-  (
 (  .o.  e.  V  /\  M  e.  W ) 
 ->  (  .o. clLaw  M  <->  A. x  e.  M  A. y  e.  M  ( x  .o.  y )  e.  M ) )
 
Theoremiscomlaw 39445* The predicate "is a commutative operation". (Contributed by AV, 20-Jan-2020.)
 |-  (
 (  .o.  e.  V  /\  M  e.  W ) 
 ->  (  .o. comLaw  M  <->  A. x  e.  M  A. y  e.  M  ( x  .o.  y )  =  ( y  .o. 
 x ) ) )
 
Theoremclcllaw 39446 Closure of a closed operation. (Contributed by FL, 14-Sep-2010.) (Revised by AV, 21-Jan-2020.)
 |-  (
 (  .o. clLaw  M  /\  X  e.  M  /\  Y  e.  M )  ->  ( X  .o.  Y )  e.  M )
 
Theoremisasslaw 39447* The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (Revised by AV, 13-Jan-2020.)
 |-  (
 (  .o.  e.  V  /\  M  e.  W ) 
 ->  (  .o. assLaw  M  <->  A. x  e.  M  A. y  e.  M  A. z  e.  M  (
 ( x  .o.  y
 )  .o.  z )  =  ( x  .o.  (
 y  .o.  z )
 ) ) )
 
Theoremasslawass 39448* Associativity of an associative operation. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 21-Jan-2020.)
 |-  (  .o. assLaw  M  ->  A. x  e.  M  A. y  e.  M  A. z  e.  M  ( ( x  .o.  y )  .o.  z )  =  ( x  .o.  ( y  .o.  z ) ) )
 
TheoremmgmplusgiopALT 39449 Slot 2 (group operation) of a magma as extensible structure is a closed operation on the base set. (Contributed by AV, 13-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( M  e. Mgm  ->  ( +g  `  M ) clLaw  ( Base `  M ) )
 
TheoremsgrpplusgaopALT 39450 Slot 2 (group operation) of a semigroup as extensible structure is an associative operation on the base set. (Contributed by AV, 13-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( G  e. SGrp  ->  ( +g  `  G ) assLaw  ( Base `  G ) )
 
21.33.11.2  Internal binary operations

In this subsection, "internal binary operations" obeying different laws are defined.

 
Syntaxcintop 39451 Extend class notation with class of internal (binary) operations for a set.
 class intOp
 
Syntaxcclintop 39452 Extend class notation with class of closed operations for a set.
 class clIntOp
 
Syntaxcassintop 39453 Extend class notation with class of associative operations for a set.
 class assIntOp
 
Definitiondf-intop 39454* Function mapping a set to the class of all internal (binary) operations for this set. (Contributed by AV, 20-Jan-2020.)
 |- intOp  =  ( m  e.  _V ,  n  e.  _V  |->  ( n 
 ^m  ( m  X.  m ) ) )
 
Definitiondf-clintop 39455 Function mapping a set to the class of all closed (internal binary) operations for this set, see definition in section 1.2 of [Hall] p. 2, definition in section I.1 of [Bruck] p. 1, or definition 1 in [BourbakiAlg1] p. 1, where it is called "a law of composition". (Contributed by AV, 20-Jan-2020.)
 |- clIntOp  =  ( m  e.  _V  |->  ( m intOp  m ) )
 
Definitiondf-assintop 39456* Function mapping a set to the class of all associative (closed internal binary) operations for this set, see definition 5 in [BourbakiAlg1] p. 4, where it is called "an associative law of composition". (Contributed by AV, 20-Jan-2020.)
 |- assIntOp  =  ( m  e.  _V  |->  { o  e.  ( clIntOp  `  m )  |  o assLaw  m }
 )
 
Theoremintopval 39457 The internal (binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
 |-  (
 ( M  e.  V  /\  N  e.  W ) 
 ->  ( M intOp  N )  =  ( N  ^m  ( M  X.  M ) ) )
 
Theoremintop 39458 An internal (binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
 |-  (  .o.  e.  ( M intOp  N )  ->  .o.  : ( M  X.  M ) --> N )
 
Theoremclintopval 39459 The closed (internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
 |-  ( M  e.  V  ->  ( clIntOp  `  M )  =  ( M  ^m  ( M  X.  M ) ) )
 
Theoremassintopval 39460* The associative (closed internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
 |-  ( M  e.  V  ->  ( assIntOp  `  M )  =  {
 o  e.  ( clIntOp  `  M )  |  o assLaw  M }
 )
 
Theoremassintopmap 39461* The associative (closed internal binary) operations for a set, expressed with set exponentiation. (Contributed by AV, 20-Jan-2020.)
 |-  ( M  e.  V  ->  ( assIntOp  `  M )  =  {
 o  e.  ( M 
 ^m  ( M  X.  M ) )  |  o assLaw  M } )
 
Theoremisclintop 39462 The predicate "is a closed (internal binary) operations for a set". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
 |-  ( M  e.  V  ->  (  .o.  e.  ( clIntOp  `  M ) 
 <->  .o.  : ( M  X.  M ) --> M ) )
 
Theoremclintop 39463 A closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
 |-  (  .o.  e.  ( clIntOp  `  M ) 
 ->  .o.  : ( M  X.  M ) --> M )
 
Theoremassintop 39464 An associative (closed internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
 |-  (  .o.  e.  ( assIntOp  `  M ) 
 ->  (  .o.  : ( M  X.  M ) --> M  /\  .o. assLaw  M ) )
 
Theoremisassintop 39465* The predicate "is an associative (closed internal binary) operations for a set". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
 |-  ( M  e.  V  ->  (  .o.  e.  ( assIntOp  `  M ) 
 <->  (  .o.  : ( M  X.  M ) --> M  /\  A. x  e.  M  A. y  e.  M  A. z  e.  M  ( ( x  .o.  y )  .o.  z )  =  ( x  .o.  ( y  .o.  z ) ) ) ) )
 
Theoremclintopcllaw 39466 The closure law holds for a closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
 |-  (  .o.  e.  ( clIntOp  `  M ) 
 ->  .o. clLaw  M )
 
Theoremassintopcllaw 39467 The closure low holds for an associative (closed internal binary) operation for a set. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
 |-  (  .o.  e.  ( assIntOp  `  M ) 
 ->  .o. clLaw  M )
 
Theoremassintopasslaw 39468 The associative low holds for a associative (closed internal binary) operation for a set. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
 |-  (  .o.  e.  ( assIntOp  `  M ) 
 ->  .o. assLaw  M )
 
Theoremassintopass 39469* An associative (closed internal binary) operation for a set is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
 |-  (  .o.  e.  ( assIntOp  `  M ) 
 ->  A. x  e.  M  A. y  e.  M  A. z  e.  M  (
 ( x  .o.  y
 )  .o.  z )  =  ( x  .o.  (
 y  .o.  z )
 ) )
 
21.33.11.3  Alternative definitions for Magmas and Semigroups
 
Syntaxcmgm2 39470 Extend class notation with class of all magmas.
 class MgmALT
 
Syntaxccmgm2 39471 Extend class notation with class of all commutative magmas.
 class CMgmALT
 
Syntaxcsgrp2 39472 Extend class notation with class of all semigroups.
 class SGrpALT
 
Syntaxccsgrp2 39473 Extend class notation with class of all commutative semigroups.
 class CSGrpALT
 
Definitiondf-mgm2 39474 A magma is a set equipped with a closed operation. Definition 1 of [BourbakiAlg1] p. 1, or definition of a groupoid in section I.1 of [Bruck] p. 1. Note: The term "groupoid" is now widely used to refer to other objects: (small) categories all of whose morphisms are invertible, or groups with a partial function replacing the binary operation. Therefore, we will only use the term "magma" for the present notion in set.mm. (Contributed by AV, 6-Jan-2020.)
 |- MgmALT  =  { m  |  ( +g  `  m ) clLaw  ( Base `  m ) }
 
Definitiondf-cmgm2 39475 A commutative magma is a magma with a commutative operation. Definition 8 of [BourbakiAlg1] p. 7. (Contributed by AV, 20-Jan-2020.)
 |- CMgmALT  =  { m  e. MgmALT  |  ( +g  `  m ) comLaw  ( Base `  m ) }
 
Definitiondf-sgrp2 39476 A semigroup is a magma with an associative operation. Definition in section II.1 of [Bruck] p. 23, or of an "associative magma" in definition 5 of [BourbakiAlg1] p. 4, or of a semi-group in section 1.3 of [Hall] p. 7. (Contributed by AV, 6-Jan-2020.)
 |- SGrpALT  =  {
 g  e. MgmALT  |  ( +g  `  g ) assLaw  ( Base `  g ) }
 
Definitiondf-csgrp2 39477 A commutative semigroup is a semigroup with a commutative operation. (Contributed by AV, 20-Jan-2020.)
 |- CSGrpALT  =  {
 g  e. SGrpALT  |  ( +g  `  g ) comLaw  ( Base `  g ) }
 
TheoremismgmALT 39478 The predicate "is a magma." (Contributed by AV, 16-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  ( +g  `  M )   =>    |-  ( M  e.  V  ->  ( M  e. MgmALT  <->  .o. clLaw  B ) )
 
TheoremiscmgmALT 39479 The predicate "is a commutative magma." (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  ( +g  `  M )   =>    |-  ( M  e. CMgmALT  <->  ( M  e. MgmALT  /\ 
 .o. comLaw  B ) )
 
TheoremissgrpALT 39480 The predicate "is a semigroup." (Contributed by AV, 16-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  ( +g  `  M )   =>    |-  ( M  e. SGrpALT  <->  ( M  e. MgmALT  /\ 
 .o. assLaw  B ) )
 
TheoremiscsgrpALT 39481 The predicate "is a commutative semigroup." (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  ( +g  `  M )   =>    |-  ( M  e. CSGrpALT  <->  ( M  e. SGrpALT  /\ 
 .o. comLaw  B ) )
 
Theoremmgm2mgm 39482 Equivalence of the two definitions of a magma. (Contributed by AV, 16-Jan-2020.)
 |-  ( M  e. MgmALT  <->  M  e. Mgm )
 
Theoremsgrp2sgrp 39483 Equivalence of the two definitions of a semigroup. (Contributed by AV, 16-Jan-2020.)
 |-  ( M  e. SGrpALT  <->  M  e. SGrp )
 
21.33.12  Categories (extension)
 
21.33.12.1  Subcategories (extension)
 
Theoremidfusubc0 39484* The identity functor for a subcategory is an "inclusion functor" from the subcategory into its supercategory. (Contributed by AV, 29-Mar-2020.)
 |-  S  =  ( C  |`cat  J )   &    |-  I  =  (idfunc `  S )   &    |-  B  =  (
 Base `  S )   =>    |-  ( J  e.  (Subcat `  C )  ->  I  =  <. (  _I  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x ( Hom  `  S ) y ) ) ) >. )
 
Theoremidfusubc 39485* The identity functor for a subcategory is an "inclusion functor" from the subcategory into its supercategory. (Contributed by AV, 29-Mar-2020.)
 |-  S  =  ( C  |`cat  J )   &    |-  I  =  (idfunc `  S )   &    |-  B  =  (
 Base `  S )   =>    |-  ( J  e.  (Subcat `  C )  ->  I  =  <. (  _I  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x J y ) ) ) >. )
 
Theoreminclfusubc 39486* The "inclusion functor" from a subcategory of a category into the category itself. (Contributed by AV, 30-Mar-2020.)
 |-  ( ph  ->  J  e.  (Subcat `  C ) )   &    |-  S  =  ( C  |`cat  J )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  F  =  (  _I  |`  B )
 )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x J y ) ) ) )   =>    |-  ( ph  ->  F ( S  Func  C ) G )
 
21.33.13  Rings (extension)
 
21.33.13.1  Nonzero rings (extension)
 
Theoremlmod0rng 39487 If the scalar ring of a module is the zero ring, the module is the zero module, i.e. the base set of the module is the singleton consisting of the identity element only. (Contributed by AV, 17-Apr-2019.)
 |-  (
 ( M  e.  LMod  /\ 
 -.  (Scalar `  M )  e. NzRing )  ->  ( Base `  M )  =  {
 ( 0g `  M ) } )
 
Theoremnzrneg1ne0 39488 The additive inverse of the 1 in a nonzero ring is not zero ( -1 =/= 0 ). (Contributed by AV, 29-Apr-2019.)
 |-  ( R  e. NzRing  ->  ( ( invg `  R ) `  ( 1r `  R ) )  =/=  ( 0g `  R ) )
 
Theorem0ringdif 39489 A zero ring is a ring which is not a nonzero ring. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e.  ( Ring  \ NzRing )  <->  ( R  e.  Ring  /\  B  =  {  .0.  } ) )
 
Theorem0ringbas 39490 The base set of a zero ring, a ring which is not a nonzero ring, is the singleton of the zero element. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e.  ( Ring  \ NzRing )  ->  B  =  {  .0.  } )
 
Theorem0ring1eq0 39491 In a zero ring, a ring which is not a nonzero ring, the unit equals the zero element. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .1.  =  ( 1r `  R )   =>    |-  ( R  e.  ( Ring  \ NzRing )  ->  .1.  =  .0.  )
 
Theoremnrhmzr 39492 There is no ring homomorphism from the zero ring into a nonzero ring. (Contributed by AV, 18-Apr-2020.)
 |-  (
 ( Z  e.  ( Ring  \ NzRing )  /\  R  e. NzRing )  ->  ( Z RingHom  R )  =  (/) )
 
21.33.13.2  Non-unital rings ("rngs")

According to Wikipedia, "... in abstract algebra, a rng (or pseudo-ring or non-unital ring) is an algebraic structure satisfying the same properties as a [unital] ring, without assuming the existence of a multiplicative identity. The term "rng" (pronounced rung) is meant to suggest that it is a "ring" without "i", i.e. without the requirement for an "identity element"." (see https://en.wikipedia.org/wiki/Rng_(algebra), 6-Jan-2020).

 
Syntaxcrng 39493 Extend class notation with class of all non-unital rings.
 class Rng
 
Definitiondf-rng0 39494* Define class of all (non-unital) rings. A non-unital ring (or rng, or pseudoring) is a set equipped with two everywhere-defined internal operations, whose first one is an additive abelian group operation and the second one is a multiplicative semigroup operation, and where the addition is left- and right-distributive for the multiplication. Definition of a pseudo-ring in section I.8.1 of [BourbakiAlg1] p. 93 or the definition of a ring in part Preliminaries of [Roman] p. 18. As almost always in mathematics, "non-unital" means "not necessarily unital". Therefore, by talking about a ring (in general) or a non-unital ring the "unital" case is always included. In contrast to a unital ring, the commutativity of addition must be postulated and cannot be proven from the other conditions. (Contributed by AV, 6-Jan-2020.)
 |- Rng  =  {
 f  e.  Abel  |  ( (mulGrp `  f )  e. SGrp  /\  [. ( Base `  f
 )  /  b ]. [. ( +g  `  f
 )  /  p ]. [. ( .r `  f )  /  t ]. A. x  e.  b  A. y  e.  b  A. z  e.  b  ( ( x t ( y p z ) )  =  ( ( x t y ) p ( x t z ) )  /\  ( ( x p y ) t z )  =  ( ( x t z ) p ( y t z ) ) ) ) }
 
Theoremisrng 39495* The predicate "is a non-unital ring." (Contributed by AV, 6-Jan-2020.)
 |-  B  =  ( Base `  R )   &    |-  G  =  (mulGrp `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( R  e. Rng  <->  ( R  e.  Abel  /\  G  e. SGrp  /\  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 ) ) ) ) )
 
Theoremrngabl 39496 A non-unital ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)
 |-  ( R  e. Rng  ->  R  e.  Abel
 )
 
Theoremrngmgp 39497 A non-unital ring is a semigroup under multiplication. (Contributed by AV, 17-Feb-2020.)
 |-  G  =  (mulGrp `  R )   =>    |-  ( R  e. Rng  ->  G  e. SGrp )
 
Theoremringrng 39498 A unital ring is a (non-unital) ring. (Contributed by AV, 6-Jan-2020.)
 |-  ( R  e.  Ring  ->  R  e. Rng )
 
Theoremringssrng 39499 The unital rings are (non-unital) rings. (Contributed by AV, 20-Mar-2020.)
 |-  Ring  C_ Rng
 
Theoremisringrng 39500* The predicate "is a unital ring" as extension of the predicate "is a non-unital ring". (Contributed by AV, 17-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( R  e.  Ring  <->  ( R  e. Rng  /\ 
 E. x  e.  B  A. y  e.  B  ( ( x  .x.  y
 )  =  y  /\  ( y  .x.  x )  =  y ) ) )
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