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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.)
SubMgm Mgm

Theoremismgmhm 39402* Property of a magma homomorphism. (Contributed by AV, 25-Feb-2020.)
MgmHom Mgm Mgm

Theoremmgmhmf 39403 A magma homomorphism is a function. (Contributed by AV, 25-Feb-2020.)
MgmHom

Theoremmgmhmpropd 39404* Magma homomorphism depends only on the operation of structures. (Contributed by AV, 25-Feb-2020.)
MgmHom MgmHom

Theoremmgmhmlin 39405 A magma homomorphism preserves the binary operation. (Contributed by AV, 25-Feb-2020.)
MgmHom

Theoremmgmhmf1o 39406 A magma homomorphism is bijective iff its converse is also a magma homomorphism. (Contributed by AV, 25-Feb-2020.)
MgmHom MgmHom

Theoremidmgmhm 39407 The identity homomorphism on a magma. (Contributed by AV, 27-Feb-2020.)
Mgm MgmHom

Theoremissubmgm 39408* Expand definition of a submagma. (Contributed by AV, 25-Feb-2020.)
Mgm SubMgm

Theoremissubmgm2 39409 Submagmas are subsets that are also magmas. (Contributed by AV, 25-Feb-2020.)
s        Mgm SubMgm Mgm

Theoremrabsubmgmd 39410* Deduction for proving that a restricted class abstraction is a submagma. (Contributed by AV, 26-Feb-2020.)
Mgm                                   SubMgm

Theoremsubmgmss 39411 Submagmas are subsets of the base set. (Contributed by AV, 26-Feb-2020.)
SubMgm

Theoremsubmgmid 39412 Every magma is trivially a submagma of itself. (Contributed by AV, 26-Feb-2020.)
Mgm SubMgm

Theoremsubmgmcl 39413 Submagmas are closed under the monoid operation. (Contributed by AV, 26-Feb-2020.)
SubMgm

Theoremsubmgmmgm 39414 Submagmas are themselves magmas under the given operation. (Contributed by AV, 26-Feb-2020.)
s        SubMgm Mgm

Theoremsubmgmbas 39415 The base set of a submagma. (Contributed by AV, 26-Feb-2020.)
s        SubMgm

Theoremsubsubmgm 39416 A submagma of a submagma is a submagma. (Contributed by AV, 26-Feb-2020.)
s        SubMgm SubMgm SubMgm

Theoremresmgmhm 39417 Restriction of a magma homomorphism to a submagma is a homomorphism. (Contributed by AV, 26-Feb-2020.)
s        MgmHom SubMgm MgmHom

Theoremresmgmhm2 39418 One direction of resmgmhm2b 39419. (Contributed by AV, 26-Feb-2020.)
s        MgmHom SubMgm MgmHom

Theoremresmgmhm2b 39419 Restriction of the codomain of a homomorphism. (Contributed by AV, 26-Feb-2020.)
s        SubMgm MgmHom MgmHom

Theoremmgmhmco 39420 The composition of magma homomorphisms is a homomorphism. (Contributed by AV, 27-Feb-2020.)
MgmHom MgmHom MgmHom

Theoremmgmhmima 39421 The homomorphic image of a submagma is a submagma. (Contributed by AV, 27-Feb-2020.)
MgmHom SubMgm SubMgm

Theoremmgmhmeql 39422 The equalizer of two magma homomorphisms is a submagma. (Contributed by AV, 27-Feb-2020.)
MgmHom MgmHom SubMgm

Theoremsubmgmacs 39423 Submagmas are an algebraic closure system. (Contributed by AV, 27-Feb-2020.)
Mgm SubMgm ACS

Theoremismhm0 39424 Property of a monoid homomorphism, expressed by a magma homomorphism. (Contributed by AV, 17-Apr-2020.)
MndHom MgmHom

Theoremmhmismgmhm 39425 Each monoid homomorphism is a magma homomorphism. (Contributed by AV, 29-Feb-2020.)
MndHom MgmHom

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.)
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.)
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.)

Theorem0nodd 39429* 0 is not an odd integer. (Contributed by AV, 3-Feb-2020.)

Theorem1odd 39430* 1 is an odd integer. (Contributed by AV, 3-Feb-2020.)

Theorem2nodd 39431* 2 is not an odd integer. (Contributed by AV, 3-Feb-2020.)

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.)
flds

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.)
flds

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.)
flds        Mgm

Theoremnnsgrpmgm 39435 The structure of positive integers together with the addition of complex numbers is a magma. (Contributed by AV, 4-Feb-2020.)
flds        Mgm

Theoremnnsgrp 39436 The structure of positive integers together with the addition of complex numbers is a semigroup. (Contributed by AV, 4-Feb-2020.)
flds        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.)
flds

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 is a mapping of the elements of the Cartesian product to 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 internal binary operations, see df-intop 39454. If, in addition, the result is also contained in the set , the operation is called closed internal binary operation, see df-clintop 39455. Therefore, a "binary operation on a set " 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.
clLaw

Syntaxcasslaw 39439 Extend class notation for the associative law.
assLaw

Syntaxccomlaw 39440 Extend class notation for the commutative law.
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

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

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

Theoremiscllaw 39444* The predicate "is a closed operation". (Contributed by AV, 13-Jan-2020.)
clLaw

Theoremiscomlaw 39445* The predicate "is a commutative operation". (Contributed by AV, 20-Jan-2020.)
comLaw

Theoremclcllaw 39446 Closure of a closed operation. (Contributed by FL, 14-Sep-2010.) (Revised by AV, 21-Jan-2020.)
clLaw

Theoremisasslaw 39447* The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (Revised by AV, 13-Jan-2020.)
assLaw

Theoremasslawass 39448* Associativity of an associative operation. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 21-Jan-2020.)
assLaw

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.)
Mgm clLaw

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.)
SGrp assLaw

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.
intOp

Syntaxcclintop 39452 Extend class notation with class of closed operations for a set.
clIntOp

Syntaxcassintop 39453 Extend class notation with class of associative operations for a set.
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

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 intOp

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 clIntOp assLaw

Theoremintopval 39457 The internal (binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
intOp

Theoremintop 39458 An internal (binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
intOp

Theoremclintopval 39459 The closed (internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
clIntOp

Theoremassintopval 39460* The associative (closed internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
assIntOp clIntOp assLaw

Theoremassintopmap 39461* The associative (closed internal binary) operations for a set, expressed with set exponentiation. (Contributed by AV, 20-Jan-2020.)
assIntOp assLaw

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.)
clIntOp

Theoremclintop 39463 A closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
clIntOp

Theoremassintop 39464 An associative (closed internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
assIntOp assLaw

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.)
assIntOp

Theoremclintopcllaw 39466 The closure law holds for a closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
clIntOp clLaw

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.)
assIntOp clLaw

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.)
assIntOp assLaw

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.)
assIntOp

21.33.11.3  Alternative definitions for Magmas and Semigroups

Syntaxcmgm2 39470 Extend class notation with class of all magmas.
MgmALT

Syntaxccmgm2 39471 Extend class notation with class of all commutative magmas.
CMgmALT

Syntaxcsgrp2 39472 Extend class notation with class of all semigroups.
SGrpALT

Syntaxccsgrp2 39473 Extend class notation with class of all commutative semigroups.
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 clLaw

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 MgmALT comLaw

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 MgmALT assLaw

Definitiondf-csgrp2 39477 A commutative semigroup is a semigroup with a commutative operation. (Contributed by AV, 20-Jan-2020.)
CSGrpALT SGrpALT comLaw

TheoremismgmALT 39478 The predicate "is a magma." (Contributed by AV, 16-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
MgmALT clLaw

TheoremiscmgmALT 39479 The predicate "is a commutative magma." (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
CMgmALT MgmALT comLaw

TheoremissgrpALT 39480 The predicate "is a semigroup." (Contributed by AV, 16-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
SGrpALT MgmALT assLaw

TheoremiscsgrpALT 39481 The predicate "is a commutative semigroup." (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
CSGrpALT SGrpALT comLaw

Theoremmgm2mgm 39482 Equivalence of the two definitions of a magma. (Contributed by AV, 16-Jan-2020.)
MgmALT Mgm

Theoremsgrp2sgrp 39483 Equivalence of the two definitions of a semigroup. (Contributed by AV, 16-Jan-2020.)
SGrpALT 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.)
cat        idfunc              Subcat

Theoremidfusubc 39485* The identity functor for a subcategory is an "inclusion functor" from the subcategory into its supercategory. (Contributed by AV, 29-Mar-2020.)
cat        idfunc              Subcat

Theoreminclfusubc 39486* The "inclusion functor" from a subcategory of a category into the category itself. (Contributed by AV, 30-Mar-2020.)
Subcat       cat

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.)
Scalar NzRing

Theoremnzrneg1ne0 39488 The additive inverse of the 1 in a nonzero ring is not zero ( -1 =/= 0 ). (Contributed by AV, 29-Apr-2019.)
NzRing

Theorem0ringdif 39489 A zero ring is a ring which is not a nonzero ring. (Contributed by AV, 17-Apr-2020.)
NzRing

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.)
NzRing

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.)
NzRing

Theoremnrhmzr 39492 There is no ring homomorphism from the zero ring into a nonzero ring. (Contributed by AV, 18-Apr-2020.)
NzRing NzRing RingHom

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.
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 mulGrp SGrp

Theoremisrng 39495* The predicate "is a non-unital ring." (Contributed by AV, 6-Jan-2020.)
mulGrp                     Rng SGrp

Theoremrngabl 39496 A non-unital ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)
Rng

Theoremrngmgp 39497 A non-unital ring is a semigroup under multiplication. (Contributed by AV, 17-Feb-2020.)
mulGrp       Rng SGrp

Theoremringrng 39498 A unital ring is a (non-unital) ring. (Contributed by AV, 6-Jan-2020.)
Rng

Theoremringssrng 39499 The unital rings are (non-unital) rings. (Contributed by AV, 20-Mar-2020.)
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.)
Rng

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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