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Statement | ||
Theorem | lidrididd 12801* |
If there is a left and right identity element for any binary operation
(group operation) ![]() |
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Theorem | grpidd 12802* | Deduce the identity element of a magma from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.) |
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Theorem | mgmidsssn0 12803* |
Property of the set of identities of ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | grprinvlem 12804* | Lemma for grprinvd 12805. (Contributed by NM, 9-Aug-2013.) |
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Theorem | grprinvd 12805* | Deduce right inverse from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.) |
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Theorem | grpridd 12806* | Deduce right identity from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.) |
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A semigroup (Smgrp, see df-sgrp 12808) is a set together with an associative binary operation (see Wikipedia, Semigroup, 8-Jan-2020, https://en.wikipedia.org/wiki/Semigroup 12808). In other words, a semigroup is an associative magma. The notion of semigroup is a generalization of that of group where the existence of an identity or inverses is not required. | ||
Syntax | csgrp 12807 | Extend class notation with class of all semigroups. |
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Definition | df-sgrp 12808* | A semigroup is a set equipped with an everywhere defined internal operation (so, a magma, see df-mgm 12775), whose operation is associative. Definition in section II.1 of [Bruck] p. 23, or of an "associative magma" in definition 5 of [BourbakiAlg1] p. 4 . (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) |
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Theorem | issgrp 12809* | The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) |
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Theorem | issgrpv 12810* | The predicate "is a semigroup" for a structure which is a set. (Contributed by AV, 1-Feb-2020.) |
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Theorem | issgrpn0 12811* | The predicate "is a semigroup" for a structure with a nonempty base set. (Contributed by AV, 1-Feb-2020.) |
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Theorem | isnsgrp 12812 | A condition for a structure not to be a semigroup. (Contributed by AV, 30-Jan-2020.) |
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Theorem | sgrpmgm 12813 | A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) |
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Theorem | sgrpass 12814 | A semigroup operation is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 30-Jan-2020.) |
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Theorem | sgrp0 12815 | Any set with an empty base set and any group operation is a semigroup. (Contributed by AV, 28-Aug-2021.) |
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Theorem | sgrp1 12816 | The structure with one element and the only closed internal operation for a singleton is a semigroup. (Contributed by AV, 10-Feb-2020.) |
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According to Wikipedia ("Monoid", https://en.wikipedia.org/wiki/Monoid, 6-Feb-2020,) "In abstract algebra [...] a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are semigroups with identity.". In the following, monoids are defined in the second way (as semigroups with identity), see df-mnd 12818, whereas many authors define magmas in the first way (as algebraic structure with a single associative binary operation and an identity element, i.e. without the need of a definition for/knowledge about semigroups), see ismnd 12820. See, for example, the definition in [Lang] p. 3: "A monoid is a set G, with a law of composition which is associative, and having a unit element". | ||
Syntax | cmnd 12817 | Extend class notation with class of all monoids. |
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Definition | df-mnd 12818* | A monoid is a semigroup, which has a two-sided neutral element. Definition 2 in [BourbakiAlg1] p. 12. In other words (according to the definition in [Lang] p. 3), a monoid is a set equipped with an everywhere defined internal operation (see mndcl 12824), whose operation is associative (see mndass 12825) and has a two-sided neutral element (see mndid 12826), see also ismnd 12820. (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.) |
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Theorem | ismnddef 12819* | The predicate "is a monoid", corresponding 1-to-1 to the definition. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 1-Feb-2020.) |
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Theorem | ismnd 12820* | The predicate "is a monoid". This is the defining theorem of a monoid by showing that a set is a monoid if and only if it is a set equipped with a closed, everywhere defined internal operation (so, a magma, see mndcl 12824), whose operation is associative (so, a semigroup, see also mndass 12825) and has a two-sided neutral element (see mndid 12826). (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.) |
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Theorem | sgrpidmndm 12821* | A semigroup with an identity element which is inhabited is a monoid. Of course there could be monoids with the empty set as identity element, but these cannot be proven to be monoids with this theorem. (Contributed by AV, 29-Jan-2024.) |
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Theorem | mndsgrp 12822 | A monoid is a semigroup. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) (Proof shortened by AV, 6-Feb-2020.) |
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Theorem | mndmgm 12823 | A monoid is a magma. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) (Proof shortened by AV, 6-Feb-2020.) |
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Theorem | mndcl 12824 | Closure of the operation of a monoid. (Contributed by NM, 14-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Proof shortened by AV, 8-Feb-2020.) |
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Theorem | mndass 12825 | A monoid operation is associative. (Contributed by NM, 14-Aug-2011.) (Proof shortened by AV, 8-Feb-2020.) |
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Theorem | mndid 12826* | A monoid has a two-sided identity element. (Contributed by NM, 16-Aug-2011.) |
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Theorem | mndideu 12827* | The two-sided identity element of a monoid is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by Mario Carneiro, 8-Dec-2014.) |
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Theorem | mnd32g 12828 | Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.) |
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Theorem | mnd12g 12829 | Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.) |
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Theorem | mnd4g 12830 | Commutative/associative law for commutative monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.) |
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Theorem | mndidcl 12831 | The identity element of a monoid belongs to the monoid. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
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Theorem | mndbn0 12832 | The base set of a monoid is not empty. (It is also inhabited, as seen at mndidcl 12831). Statement in [Lang] p. 3. (Contributed by AV, 29-Dec-2023.) |
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Theorem | hashfinmndnn 12833 | A finite monoid has positive integer size. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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Theorem | mndplusf 12834 | The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 3-Feb-2020.) |
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Theorem | mndlrid 12835 | A monoid's identity element is a two-sided identity. (Contributed by NM, 18-Aug-2011.) |
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Theorem | mndlid 12836 | The identity element of a monoid is a left identity. (Contributed by NM, 18-Aug-2011.) |
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Theorem | mndrid 12837 | The identity element of a monoid is a right identity. (Contributed by NM, 18-Aug-2011.) |
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Theorem | ismndd 12838* | Deduce a monoid from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.) |
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Theorem | mndpfo 12839 | The addition operation of a monoid as a function is an onto function. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 11-Oct-2013.) (Revised by AV, 23-Jan-2020.) |
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Theorem | mndfo 12840 | The addition operation of a monoid is an onto function (assuming it is a function). (Contributed by Mario Carneiro, 11-Oct-2013.) (Proof shortened by AV, 23-Jan-2020.) |
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Theorem | mndpropd 12841* | If two structures have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.) |
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Theorem | mndprop 12842 | If two structures have the same group components (properties), one is a monoid iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) |
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Theorem | issubmnd 12843* | Characterize a submonoid by closure properties. (Contributed by Mario Carneiro, 10-Jan-2015.) |
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Theorem | ress0g 12844 |
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Theorem | submnd0 12845 | The zero of a submonoid is the same as the zero in the parent monoid. (Note that we must add the condition that the zero of the parent monoid is actually contained in the submonoid, because it is possible to have "subsets that are monoids" which are not submonoids because they have a different identity element. (Contributed by Mario Carneiro, 10-Jan-2015.) |
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Theorem | mndinvmod 12846* | Uniqueness of an inverse element in a monoid, if it exists. (Contributed by AV, 20-Jan-2024.) |
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Theorem | mnd1 12847 | The (smallest) structure representing a trivial monoid consists of one element. (Contributed by AV, 28-Apr-2019.) (Proof shortened by AV, 11-Feb-2020.) |
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Theorem | mnd1id 12848 | The singleton element of a trivial monoid is its identity element. (Contributed by AV, 23-Jan-2020.) |
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Syntax | cmhm 12849 | Hom-set generator class for monoids. |
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Syntax | csubmnd 12850 | Class function taking a monoid to its lattice of submonoids. |
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Definition | df-mhm 12851* | A monoid homomorphism is a function on the base sets which preserves the binary operation and the identity. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Definition | df-submnd 12852* | A submonoid is a subset of a monoid which contains the identity and is closed under the operation. Such subsets are themselves monoids with the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | ismhm 12853* | Property of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | mhmrcl1 12854 | Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | mhmrcl2 12855 | Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | mhmf 12856 | A monoid homomorphism is a function. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | mhmpropd 12857* | Monoid homomorphism depends only on the monoidal attributes of structures. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 7-Nov-2015.) |
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Theorem | mhmlin 12858 | A monoid homomorphism commutes with composition. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | mhm0 12859 | A monoid homomorphism preserves zero. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | idmhm 12860 | The identity homomorphism on a monoid. (Contributed by AV, 14-Feb-2020.) |
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Theorem | mhmf1o 12861 | A monoid homomorphism is bijective iff its converse is also a monoid homomorphism. (Contributed by AV, 22-Oct-2019.) |
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Theorem | submrcl 12862 | Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | issubm 12863* | Expand definition of a submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | issubm2 12864 | Submonoids are subsets that are also monoids with the same zero. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | issubmd 12865* | Deduction for proving a submonoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.) |
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Theorem | mndissubm 12866 | If the base set of a monoid is contained in the base set of another monoid, and the group operation of the monoid is the restriction of the group operation of the other monoid to its base set, and the identity element of the the other monoid is contained in the base set of the monoid, then the (base set of the) monoid is a submonoid of the other monoid. (Contributed by AV, 17-Feb-2024.) |
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Theorem | submss 12867 | Submonoids are subsets of the base set. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | submid 12868 | Every monoid is trivially a submonoid of itself. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
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Theorem | subm0cl 12869 | Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.) |
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Theorem | submcl 12870 | Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.) |
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Theorem | 0subm 12871 | The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.) |
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Theorem | insubm 12872 | The intersection of two submonoids is a submonoid. (Contributed by AV, 25-Feb-2024.) |
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Theorem | 0mhm 12873 | The constant zero linear function between two monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
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Theorem | mhmco 12874 | The composition of monoid homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) |
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Theorem | mhmima 12875 | The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.) |
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Theorem | mhmeql 12876 | The equalizer of two monoid homomorphisms is a submonoid. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
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Syntax | cgrp 12877 | Extend class notation with class of all groups. |
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Syntax | cminusg 12878 | Extend class notation with inverse of group element. |
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Syntax | csg 12879 | Extend class notation with group subtraction (or division) operation. |
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Definition | df-grp 12880* |
Define class of all groups. A group is a monoid (df-mnd 12818) whose
internal operation is such that every element admits a left inverse
(which can be proven to be a two-sided inverse). Thus, a group ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Definition | df-minusg 12881* | Define inverse of group element. (Contributed by NM, 24-Aug-2011.) |
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Definition | df-sbg 12882* | Define group subtraction (also called division for multiplicative groups). (Contributed by NM, 31-Mar-2014.) |
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Theorem | isgrp 12883* | The predicate "is a group". (This theorem demonstrates the use of symbols as variable names, first proposed by FL in 2010.) (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.) |
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Theorem | grpmnd 12884 | A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.) |
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Theorem | grpcl 12885 | Closure of the operation of a group. (Contributed by NM, 14-Aug-2011.) |
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Theorem | grpass 12886 | A group operation is associative. (Contributed by NM, 14-Aug-2011.) |
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Theorem | grpinvex 12887* | Every member of a group has a left inverse. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) |
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Theorem | grpideu 12888* | The two-sided identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 8-Dec-2014.) |
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Theorem | grpmndd 12889 | A group is a monoid. (Contributed by SN, 1-Jun-2024.) |
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Theorem | grpcld 12890 | Closure of the operation of a group. (Contributed by SN, 29-Jul-2024.) |
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Theorem | grpplusf 12891 | The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.) |
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Theorem | grpplusfo 12892 | The group addition operation is a function onto the base set/set of group elements. (Contributed by NM, 30-Oct-2006.) (Revised by AV, 30-Aug-2021.) |
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Theorem | grppropd 12893* | If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) |
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Theorem | grpprop 12894 | If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by NM, 11-Oct-2013.) |
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Theorem | grppropstrg 12895 |
Generalize a specific 2-element group ![]() ![]() |
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Theorem | isgrpd2e 12896* |
Deduce a group from its properties. In this version of isgrpd2 12897, we
don't assume there is an expression for the inverse of ![]() |
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Theorem | isgrpd2 12897* |
Deduce a group from its properties. ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | isgrpde 12898* |
Deduce a group from its properties. In this version of isgrpd 12899, we
don't assume there is an expression for the inverse of ![]() |
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Theorem | isgrpd 12899* |
Deduce a group from its properties. Unlike isgrpd2 12897, this one goes
straight from the base properties rather than going through ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | isgrpi 12900* |
Properties that determine a group. ![]() ![]() ![]() ![]() ![]() ![]() |
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