P. 395 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 39401-39500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	submgmrcl 39401	Reverse closure for submagmas. (Contributed by AV, 24-Feb-2020.)
|- ( S e. (SubMgm ` M ) -> M e. Mgm )
Theorem	ismgmhm 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 ) ) ) ) )
Theorem	mgmhmf 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 )
Theorem	mgmhmpropd 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 ) )
Theorem	mgmhmlin 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 ) ) )
Theorem	mgmhmf1o 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 ) ) )
Theorem	idmgmhm 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 ) )
Theorem	issubmgm 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 ) ) )
Theorem	issubmgm2 39409	Submagmas are subsets that are also magmas. (Contributed by AV, 25-Feb-2020.)
|- B = ( Base ` M )   &    |- H = ( M ↾s S )   =>    |- ( M e. Mgm -> ( S e. (SubMgm ` M ) <-> ( S C_ B /\ H e. Mgm ) ) )
Theorem	rabsubmgmd 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 ) )
Theorem	submgmss 39411	Submagmas are subsets of the base set. (Contributed by AV, 26-Feb-2020.)
|- B = ( Base ` M )   =>    |- ( S e. (SubMgm ` M ) -> S C_ B )
Theorem	submgmid 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 ) )
Theorem	submgmcl 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 )
Theorem	submgmmgm 39414	Submagmas are themselves magmas under the given operation. (Contributed by AV, 26-Feb-2020.)
|- H = ( M ↾s S )   =>    |- ( S e. (SubMgm ` M ) -> H e. Mgm )
Theorem	submgmbas 39415	The base set of a submagma. (Contributed by AV, 26-Feb-2020.)
|- H = ( M ↾s S )   =>    |- ( S e. (SubMgm ` M ) -> S = ( Base ` H ) )
Theorem	subsubmgm 39416	A submagma of a submagma is a submagma. (Contributed by AV, 26-Feb-2020.)
|- H = ( G ↾s S )   =>    |- ( S e. (SubMgm ` G ) -> ( A e. (SubMgm ` H ) <-> ( A e. (SubMgm ` G ) /\ A C_ S ) ) )
Theorem	resmgmhm 39417	Restriction of a magma homomorphism to a submagma is a homomorphism. (Contributed by AV, 26-Feb-2020.)
|- U = ( S ↾s X )   =>    |- ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) -> ( F |` X ) e. ( U MgmHom T ) )
Theorem	resmgmhm2 39418	One direction of resmgmhm2b 39419. (Contributed by AV, 26-Feb-2020.)
|- U = ( T ↾s X )   =>    |- ( ( F e. ( S MgmHom U ) /\ X e. (SubMgm ` T ) ) -> F e. ( S MgmHom T ) )
Theorem	resmgmhm2b 39419	Restriction of the codomain of a homomorphism. (Contributed by AV, 26-Feb-2020.)
|- U = ( T ↾s X )   =>    |- ( ( X e. (SubMgm ` T ) /\ ran F C_ X ) -> ( F e. ( S MgmHom T ) <-> F e. ( S MgmHom U ) ) )
Theorem	mgmhmco 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 ) )
Theorem	mgmhmima 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 ) )
Theorem	mgmhmeql 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 ) )
Theorem	submgmacs 39423	Submagmas are an algebraic closure system. (Contributed by AV, 27-Feb-2020.)
|- B = ( Base ` G )   =>    |- ( G e. Mgm -> (SubMgm ` G ) e. (ACS ` B ) )
Theorem	ismhm0 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 ) ) )
Theorem	mhmismgmhm 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)
Theorem	opmpt2ismgm 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 )
Theorem	copissgrp 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 )
Theorem	copisnmnd 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 )
Theorem	0nodd 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
Theorem	1odd 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
Theorem	2nodd 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
Theorem	oddibas 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 = (ℂfld ↾s O )   =>    |- O = ( Base ` M )
Theorem	oddiadd 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 = (ℂfld ↾s O )   =>    |- + = ( +g ` M )
Theorem	oddinmgm 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 = (ℂfld ↾s O )   =>    |- M e/ Mgm
Theorem	nnsgrpmgm 39435	The structure of positive integers together with the addition of complex numbers is a magma. (Contributed by AV, 4-Feb-2020.)
|- M = (ℂfld ↾s NN )   =>    |- M e. Mgm
Theorem	nnsgrp 39436	The structure of positive integers together with the addition of complex numbers is a semigroup. (Contributed by AV, 4-Feb-2020.)
|- M = (ℂfld ↾s NN )   =>    |- M e. SGrp
Theorem	nnsgrpnmnd 39437	The structure of positive integers together with the addition of complex numbers is not a monoid. (Contributed by AV, 4-Feb-2020.)
|- M = (ℂfld ↾s 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.
Syntax	ccllaw 39438	Extend class notation for the closure law.
class clLaw
Syntax	casslaw 39439	Extend class notation for the associative law.
class assLaw
Syntax	ccomlaw 39440	Extend class notation for the commutative law.
class comLaw
Definition	df-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 }
Definition	df-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 ) }
Definition	df-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 ) ) }
Theorem	iscllaw 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 ) )
Theorem	iscomlaw 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 ) ) )
Theorem	clcllaw 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 )
Theorem	isasslaw 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 ) ) ) )
Theorem	asslawass 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 ) ) )
Theorem	mgmplusgiopALT 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 ) )
Theorem	sgrpplusgaopALT 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.
Syntax	cintop 39451	Extend class notation with class of internal (binary) operations for a set.
class intOp
Syntax	cclintop 39452	Extend class notation with class of closed operations for a set.
class clIntOp
Syntax	cassintop 39453	Extend class notation with class of associative operations for a set.
class assIntOp
Definition	df-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 ) ) )
Definition	df-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 ) )
Definition	df-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 } )
Theorem	intopval 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 ) ) )
Theorem	intop 39458	An internal (binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
|- ( .o. e. ( M intOp N ) -> .o. : ( M X. M ) --> N )
Theorem	clintopval 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 ) ) )
Theorem	assintopval 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 } )
Theorem	assintopmap 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 } )
Theorem	isclintop 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 ) )
Theorem	clintop 39463	A closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
|- ( .o. e. ( clIntOp ` M ) -> .o. : ( M X. M ) --> M )
Theorem	assintop 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 ) )
Theorem	isassintop 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 ) ) ) ) )
Theorem	clintopcllaw 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 )
Theorem	assintopcllaw 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 )
Theorem	assintopasslaw 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 )
Theorem	assintopass 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
Syntax	cmgm2 39470	Extend class notation with class of all magmas.
class MgmALT
Syntax	ccmgm2 39471	Extend class notation with class of all commutative magmas.
class CMgmALT
Syntax	csgrp2 39472	Extend class notation with class of all semigroups.
class SGrpALT
Syntax	ccsgrp2 39473	Extend class notation with class of all commutative semigroups.
class CSGrpALT
Definition	df-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 ) }
Definition	df-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 ) }
Definition	df-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 ) }
Definition	df-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 ) }
Theorem	ismgmALT 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 ) )
Theorem	iscmgmALT 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 ) )
Theorem	issgrpALT 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 ) )
Theorem	iscsgrpALT 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 ) )
Theorem	mgm2mgm 39482	Equivalence of the two definitions of a magma. (Contributed by AV, 16-Jan-2020.)
|- ( M e. MgmALT <-> M e. Mgm )
Theorem	sgrp2sgrp 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)
Theorem	idfusubc0 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 ) ) ) >. )
Theorem	idfusubc 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 ) ) ) >. )
Theorem	inclfusubc 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)
Theorem	lmod0rng 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 ) } )
Theorem	nzrneg1ne0 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 ) )
Theorem	0ringdif 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. } ) )
Theorem	0ringbas 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. } )
Theorem	0ring1eq0 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. )
Theorem	nrhmzr 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).
Syntax	crng 39493	Extend class notation with class of all non-unital rings.
class Rng
Definition	df-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 ) ) ) ) }
Theorem	isrng 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 ) ) ) ) )
Theorem	rngabl 39496	A non-unital ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)
|- ( R e. Rng -> R e. Abel )
Theorem	rngmgp 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 )
Theorem	ringrng 39498	A unital ring is a (non-unital) ring. (Contributed by AV, 6-Jan-2020.)
|- ( R e. Ring -> R e. Rng )
Theorem	ringssrng 39499	The unital rings are (non-unital) rings. (Contributed by AV, 20-Mar-2020.)
|- Ring C_ Rng
Theorem	isringrng 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 ) ) )