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Theorem List for Metamath Proof Explorer - 14701-14800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlern 14701 The range of is . (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)

Theoremlefld 14702 The field of the 'less or equal to' relationship on the extended real. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)

Theoremletsr 14703 The "less than or equal to" relationship on the extended reals is a toset. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)

9.2.7  Directed sets, nets

Syntaxcdir 14704 Extend class notation with the class of all directed sets.

Syntaxctail 14705 Extend class notation with the tail function.

Definitiondf-dir 14706 Define the class of all directed sets/directions. (Contributed by Jeff Hankins, 25-Nov-2009.)

Definitiondf-tail 14707* Define the tail function for directed sets. (Contributed by Jeff Hankins, 25-Nov-2009.)

Theoremisdir 14708 A condition for a relation to be a direction. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Theoremreldir 14709 A direction is a relation. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Theoremdirdm 14710 A direction's domain is equal to its field. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Theoremdirref 14711 A direction is reflexive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Theoremdirtr 14712 A direction is transitive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Theoremdirge 14713* For any two elements of a directed set, there exists a third element greater than or equal to both. (Note that this does not say that the two elements have a least upper bound.) (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Theoremtsrdir 14714 A totally ordered set is a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

PART 10  BASIC ALGEBRAIC STRUCTURES

10.1  Monoids

10.1.1  Definition and basic properties

Syntaxcmnd 14715 Extend class notation with class of all monoids.

Syntaxcgrp 14716 Extend class notation with class of all groups.

Syntaxcminusg 14717 Extend class notation with inverse of group element.

Syntaxcplusf 14718 Extend class notation with group addition as a function.

Syntaxcsg 14719 Extend class notation with group subtraction (or division) operation.

Syntaxcmg 14720 Extend class notation with a function mapping a group operation to the power operation for the group.
.g

Definitiondf-mnd 14721* Definition of a monoid. A monoid is a set equipped with an everywhere defined internal operation (so, a magma, see mndcl 14726), whose operation is associative (so, a semigroup, see mndass 14727) and has a two-sided neutral element (see mndid 14728). (Contributed by Mario Carneiro, 6-Jan-2015.)

Definitiondf-plusf 14722* Define group addition function. Usually we will use directly instead of , and they have the same behavior in most cases. The main advantage of is that it is a guaranteed function (mndplusf 14737), while only has closure (mndcl 14726). (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremismnd 14723* The predicate "is a monoid." (Contributed by Mario Carneiro, 6-Jan-2015.)

Theoremmgmidmo 14724* A two-sided identity element is unique (if it exists) in any magma. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by NM, 17-Jun-2017.)

Theoremmndlem1 14725 Lemma for monoid properties. (Contributed by NM, 14-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremmndcl 14726 Closure of the operation of a monoid. (Contributed by NM, 14-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremmndass 14727 A monoid operation is associative. (Contributed by NM, 14-Aug-2011.)

Theoremmndid 14728* A monoid has a two-sided identity element. (Contributed by NM, 16-Aug-2011.)

Theoremmndideu 14729* 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.)

Theoremmnd32g 14730 Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)

Theoremmnd12g 14731 Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)

Theoremmnd4g 14732 Commutative/associative law for commutative monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)

Theoremplusffval 14733* The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremplusfval 14734 The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremplusfeq 14735 If the addition operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremplusffn 14736 The group addition operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)

Theoremmndplusf 14737 The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremgrpidval 14738* The value of the identity element of a group. (Contributed by NM, 20-Aug-2011.) (Revised by Mario Carneiro, 2-Oct-2015.)

Theoremfn0g 14739 The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.)

Theorem0g0 14740 The identity element function evaluates to the empty set on an empty structure. (Contributed by Stefan O'Rear, 2-Oct-2015.)

Theoremismgmid 14741* The identity element of a magma, if it exists, belongs to the base set. (Contributed by Mario Carneiro, 27-Dec-2014.)

Theoremmgmidcl 14742* The identity element of a magma, if it exists, belongs to the base set. (Contributed by Mario Carneiro, 27-Dec-2014.)

Theoremmgmlrid 14743* The identity element of a magma, if it exists, is a left and right identity. (Contributed by Mario Carneiro, 27-Dec-2014.)

Theoremismgmid2 14744* Show that a given element is the identity element of a magma. (Contributed by Mario Carneiro, 27-Dec-2014.)

Theoremmndidcl 14745 The identity element of a monoid belongs to the monoid. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremmndlrid 14746 A monoid's identity element is a two-sided identity. (Contributed by NM, 18-Aug-2011.)

Theoremmndlid 14747 The identity element of a monoid is a left identity. (Contributed by NM, 18-Aug-2011.)

Theoremmndrid 14748 The identity element of a monoid is a right identity. (Contributed by NM, 18-Aug-2011.)

Theoremgrpidd 14749* Deduce the identity element of a monoid from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)

Theoremismndd 14750* Deduce a monoid from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)

Theoremmndfo 14751 The addition operation of a monoid is an onto function (assuming it is a function). (Contributed by Mario Carneiro, 11-Oct-2013.)

Theoremmndpropd 14752* If two structures have the same group components (properties), one is a monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)

Theoremgrpidpropd 14753* If two structures have the same group components (properties), they have the same identity element. (Contributed by Mario Carneiro, 27-Nov-2014.)

Theoremmndprop 14754 If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.)

Theoremissubmnd 14755* Characterize a submonoid by closure properties. (Contributed by Mario Carneiro, 10-Jan-2015.)
s

Theoremsubmnd0 14756 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.)
s

Theoremprdsplusgcl 14757 Structure product pointwise sums are closed when the factors are monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s

Theoremprdsidlem 14758* Characterization of identity in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
s

Theoremprdsmndd 14759 The product of a family of monoids is a monoid. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s

Theoremprds0g 14760 Zero in a product of monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s

Theorempwsmnd 14761 The structure power of a monoid is a monoid. (Contributed by Mario Carneiro, 11-Jan-2015.)
s

Theorempws0g 14762 Zero in a product of monoids. (Contributed by Mario Carneiro, 11-Jan-2015.)
s

Theoremimasmnd2 14763* The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremimasmnd 14764* The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremimasmndf1 14765 The image of a monoid under an injection is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremxpsmnd 14766 The binary product of monoids is a monoid. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

10.1.2  Monoid homomorphisms and submonoids

Syntaxcmhm 14767 Hom-set generator class for monoids.
MndHom

Syntaxcsubmnd 14768 Class function taking a monoid to its lattice of submonoids.
SubMnd

Definitiondf-mhm 14769* 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.)
MndHom

Definitiondf-submnd 14770* 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.)
SubMnd

Theoremismhm 14771* Property of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremmhmrcl1 14772 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremmhmrcl2 14773 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremmhmf 14774 A monoid homomorphism is a function. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremmhmpropd 14775* Monoid homomorphism depends only on the monoidal attributes of structures. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 7-Nov-2015.)
MndHom MndHom

Theoremmhmlin 14776 A monoid homomorphism commutes with composition. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremmhm0 14777 A monoid homorphism preserves zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremsubmrcl 14778 Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremissubm 14779* Expand definition of a submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremissubm2 14780 Submonoids are subsets that are also monoids with the same zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
s        SubMnd

Theoremsubmss 14781 Submonoids are subsets of the base set. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremsubmid 14782 Every monoid is trivially a submonoid of itself. (Contributed by Stefan O'Rear, 15-Aug-2015.)
SubMnd

Theoremsubm0cl 14783 Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremsubmcl 14784 Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.)
SubMnd

Theoremsubmmnd 14785 Submonoids are themselves monoids under the given operation. (Contributed by Mario Carneiro, 7-Mar-2015.)
s        SubMnd

Theoremsubmbas 14786 The base set of a submonoid. (Contributed by Stefan O'Rear, 15-Jun-2015.)
s        SubMnd

Theoremsubm0 14787 Submonoids have the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.)
s               SubMnd

Theoremsubsubm 14788 A submonoid of a submonoid is a submonoid. (Contributed by Mario Carneiro, 21-Jun-2015.)
s        SubMnd SubMnd SubMnd

Theorem0mhm 14789 The constant zero linear function between two monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
MndHom

Theoremresmhm 14790 Restriction of a monoid homomorphism to a submonoid is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
s        MndHom SubMnd MndHom

Theoremresmhm2 14791 One direction of resmhm2b 14792. (Contributed by Mario Carneiro, 18-Jun-2015.)
s        MndHom SubMnd MndHom

Theoremresmhm2b 14792 Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.)
s        SubMnd MndHom MndHom

Theoremmhmco 14793 The composition of monoid homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
MndHom MndHom MndHom

Theoremmhmima 14794 The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.)
MndHom SubMnd SubMnd

Theoremmhmeql 14795 The equalizer of two monoid homomorphisms is a submonoid. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
MndHom MndHom SubMnd

Theoremsubmacs 14796 Submonoids are an algebraic closure system. (Contributed by Stefan O'Rear, 22-Aug-2015.)
SubMnd ACS

Theoremprdspjmhm 14797* A projection from a product of monoids to one of the factors is a monoid homomorphism. (Contributed by Mario Carneiro, 6-May-2015.)
s                                          MndHom

Theorempwspjmhm 14798* A projection from a product of monoids to one of the factors is a monoid homomorphism. (Contributed by Mario Carneiro, 15-Jun-2015.)
s               MndHom

Theorempwsdiagmhm 14799* Diagonal monoid homomorphism into a structure power. (Contributed by Stefan O'Rear, 12-Mar-2015.)
s                      MndHom

Theorempwsco1mhm 14800* Right composition with a function on the index sets yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
s        s                                           MndHom

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