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Theorem List for Metamath Proof Explorer - 17801-17900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-ps 17801 Define the class of all posets (partially ordered sets) with weak ordering (e.g., "less than or equal to" instead of "less than"). A poset is a relation which is transitive, reflexive, and antisymmetric. (Contributed by NM, 11-May-2008.)
PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟𝑟) ⊆ 𝑟 ∧ (𝑟𝑟) = ( I ↾ 𝑟))}
 
Definitiondf-tsr 17802 Define the class of all totally ordered sets. (Contributed by FL, 1-Nov-2009.)
TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟𝑟)}
 
Theoremisps 17803 The predicate "is a poset" i.e. a transitive, reflexive, antisymmetric relation. (Contributed by NM, 11-May-2008.)
(𝑅𝐴 → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))
 
Theorempsrel 17804 A poset is a relation. (Contributed by NM, 12-May-2008.)
(𝐴 ∈ PosetRel → Rel 𝐴)
 
Theorempsref2 17805 A poset is antisymmetric and reflexive. (Contributed by FL, 3-Aug-2009.)
(𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))
 
Theorempstr2 17806 A poset is transitive. (Contributed by FL, 3-Aug-2009.)
(𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
 
Theorempslem 17807 Lemma for psref 17809 and others. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.)
(𝑅 ∈ PosetRel → (((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶) ∧ (𝐴 𝑅𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)))
 
Theorempsdmrn 17808 The domain and range of a poset equal its field. (Contributed by NM, 13-May-2008.)
(𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))
 
Theorempsref 17809 A poset is reflexive. (Contributed by NM, 13-May-2008.)
𝑋 = dom 𝑅       ((𝑅 ∈ PosetRel ∧ 𝐴𝑋) → 𝐴𝑅𝐴)
 
Theorempsrn 17810 The range of a poset equals it domain. (Contributed by NM, 7-Jul-2008.)
𝑋 = dom 𝑅       (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
 
Theorempsasym 17811 A poset is antisymmetric. (Contributed by NM, 12-May-2008.)
((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)
 
Theorempstr 17812 A poset is transitive. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.)
((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
 
Theoremcnvps 17813 The converse of a poset is a poset. In the general case (𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel) is not true. See cnvpsb 17814 for a special case where the property holds. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)
(𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)
 
Theoremcnvpsb 17814 The converse of a poset is a poset. (Contributed by FL, 5-Jan-2009.)
(Rel 𝑅 → (𝑅 ∈ PosetRel ↔ 𝑅 ∈ PosetRel))
 
Theorempsss 17815 Any subset of a partially ordered set is partially ordered. (Contributed by FL, 24-Jan-2010.)
(𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
 
Theorempsssdm2 17816 Field of a subposet. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝑋 = dom 𝑅       (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋𝐴))
 
Theorempsssdm 17817 Field of a subposet. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 9-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅 ∈ PosetRel ∧ 𝐴𝑋) → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)
 
Theoremistsr 17818 The predicate is a toset. (Contributed by FL, 1-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
𝑋 = dom 𝑅       (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅𝑅)))
 
Theoremistsr2 17819* The predicate is a toset. (Contributed by FL, 1-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
𝑋 = dom 𝑅       (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥)))
 
Theoremtsrlin 17820 A toset is a linear order. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵𝐵𝑅𝐴))
 
Theoremtsrlemax 17821 Two ways of saying a number is less than or equal to the maximum of two others. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵𝐴𝑅𝐶)))
 
Theoremtsrps 17822 A toset is a poset. (Contributed by Mario Carneiro, 9-Sep-2015.)
(𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
 
Theoremcnvtsr 17823 The converse of a toset is a toset. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑅 ∈ TosetRel → 𝑅 ∈ TosetRel )
 
Theoremtsrss 17824 Any subset of a totally ordered set is totally ordered. (Contributed by FL, 24-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Nov-2013.)
(𝑅 ∈ TosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ TosetRel )
 
Theoremledm 17825 The domain of is *. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)
* = dom ≤
 
Theoremlern 17826 The range of is *. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
* = ran ≤
 
Theoremlefld 17827 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 17828 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.)
≤ ∈ TosetRel
 
9.2.7  Directed sets, nets
 
Syntaxcdir 17829 Extend class notation with the class of directed sets.
class DirRel
 
Syntaxctail 17830 Extend class notation with the tail function for directed sets.
class tail
 
Definitiondf-dir 17831 Define the class of directed sets (the order relation itself is sometimes called a direction, and a directed set is a set equipped with a direction). (Contributed by Jeff Hankins, 25-Nov-2009.)
DirRel = {𝑟 ∣ ((Rel 𝑟 ∧ ( I ↾ 𝑟) ⊆ 𝑟) ∧ ((𝑟𝑟) ⊆ 𝑟 ∧ ( 𝑟 × 𝑟) ⊆ (𝑟𝑟)))}
 
Definitiondf-tail 17832* Define the tail function for directed sets. (Contributed by Jeff Hankins, 25-Nov-2009.)
tail = (𝑟 ∈ DirRel ↦ (𝑥 𝑟 ↦ (𝑟 “ {𝑥})))
 
Theoremisdir 17833 A condition for a relation to be a direction. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
𝐴 = 𝑅       (𝑅𝑉 → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝐴) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ (𝐴 × 𝐴) ⊆ (𝑅𝑅)))))
 
Theoremreldir 17834 A direction is a relation. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
(𝑅 ∈ DirRel → Rel 𝑅)
 
Theoremdirdm 17835 A direction's domain is equal to its field. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
(𝑅 ∈ DirRel → dom 𝑅 = 𝑅)
 
Theoremdirref 17836 A direction is reflexive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
𝑋 = dom 𝑅       ((𝑅 ∈ DirRel ∧ 𝐴𝑋) → 𝐴𝑅𝐴)
 
Theoremdirtr 17837 A direction is transitive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
(((𝑅 ∈ DirRel ∧ 𝐶𝑉) ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴𝑅𝐶)
 
Theoremdirge 17838* 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.)
𝑋 = dom 𝑅       ((𝑅 ∈ DirRel ∧ 𝐴𝑋𝐵𝑋) → ∃𝑥𝑋 (𝐴𝑅𝑥𝐵𝑅𝑥))
 
Theoremtsrdir 17839 A totally ordered set is a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
(𝐴 ∈ TosetRel → 𝐴 ∈ DirRel)
 
PART 10  BASIC ALGEBRAIC STRUCTURES
 
10.1  Monoids
 
10.1.1  Magmas

According to Wikipedia ("Magma (algebra)", 08-Jan-2020, https://en.wikipedia.org/wiki/magma_(algebra)) "In abstract algebra, a magma [...] is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation. The binary operation must be closed by definition but no other properties are imposed.".

Since the concept of a "binary operation" is used in different variants, these differences are explained in more detail in the following:

With df-mpo 7145, binary operations are defined by a rule, and with df-ov 7143, 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 7143 (19-Jan-2020), "... a binary operation on a set 𝑆 is a mapping of the elements of the Cartesian product 𝑆 × 𝑆 to S: 𝑓:𝑆 × 𝑆𝑆. 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, binary operations mapping the elements of the Cartesian product 𝑆 × 𝑆 are more precisely called internal binary operations. If, in addition, the result is also contained in the set 𝑆, the operation should be called closed internal binary operation. Therefore, a "binary operation on a set 𝑆" according to Wikipedia is a "closed internal binary operation" in a more precise 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 7143).

The definition of magmas (Mgm, see df-mgm 17843) concentrates on the closure property of the associated operation, and poses no additional restrictions on it. In this way, it is most general and flexible.

 
Syntaxcplusf 17840 Extend class notation with group addition as a function.
class +𝑓
 
Syntaxcmgm 17841 Extend class notation with class of all magmas.
class Mgm
 
Definitiondf-plusf 17842* Define group addition function. Usually we will use +g directly instead of +𝑓, and they have the same behavior in most cases. The main advantage of +𝑓 for any magma is that it is a guaranteed function (mgmplusf 17853), while +g only has closure (mgmcl 17846). (Contributed by Mario Carneiro, 14-Aug-2015.)
+𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)𝑦)))
 
Definitiondf-mgm 17843* A magma is a set equipped with an everywhere defined internal operation. Definition 1 in [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 FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
Mgm = {𝑔[(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏}
 
Theoremismgm 17844* The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
𝐵 = (Base‘𝑀)    &    = (+g𝑀)       (𝑀𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
 
Theoremismgmn0 17845* The predicate "is a magma" for a structure with a nonempty base set. (Contributed by AV, 29-Jan-2020.)
𝐵 = (Base‘𝑀)    &    = (+g𝑀)       (𝐴𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
 
Theoremmgmcl 17846 Closure of the operation of a magma. (Contributed by FL, 14-Sep-2010.) (Revised by AV, 13-Jan-2020.)
𝐵 = (Base‘𝑀)    &    = (+g𝑀)       ((𝑀 ∈ Mgm ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
 
Theoremisnmgm 17847 A condition for a structure not to be a magma. (Contributed by AV, 30-Jan-2020.) (Proof shortened by NM, 5-Feb-2020.)
𝐵 = (Base‘𝑀)    &    = (+g𝑀)       ((𝑋𝐵𝑌𝐵 ∧ (𝑋 𝑌) ∉ 𝐵) → 𝑀 ∉ Mgm)
 
Theoremmgmsscl 17848 If the base set of a magma is contained in the base set of another magma, and the group operation of the magma is the restriction of the group operation of the other magma to its base set, then the base set of the magma is closed under the group operation of the other magma. Formerly part of proof of grpissubg 18290. (Contributed by AV, 17-Feb-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = (Base‘𝐻)       (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋𝑆𝑌𝑆)) → (𝑋(+g𝐺)𝑌) ∈ 𝑆)
 
Theoremplusffval 17849* The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 2-Mar-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (+𝑓𝐺)        = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦))
 
Theoremplusfval 17850 The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (+𝑓𝐺)       ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + 𝑌))
 
Theoremplusfeq 17851 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.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (+𝑓𝐺)       ( + Fn (𝐵 × 𝐵) → = + )
 
Theoremplusffn 17852 The group addition operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)
𝐵 = (Base‘𝐺)    &    = (+𝑓𝐺)        Fn (𝐵 × 𝐵)
 
Theoremmgmplusf 17853 The group addition function of a magma is a function into its base set. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revisd by AV, 28-Jan-2020.)
𝐵 = (Base‘𝑀)    &    = (+𝑓𝑀)       (𝑀 ∈ Mgm → :(𝐵 × 𝐵)⟶𝐵)
 
Theoremissstrmgm 17854* Characterize a substructure as submagma by closure properties. (Contributed by AV, 30-Aug-2021.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐻 = (𝐺s 𝑆)       ((𝐻𝑉𝑆𝐵) → (𝐻 ∈ Mgm ↔ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆))
 
Theoremintopsn 17855 The internal operation for a set is the trivial operation iff the set is a singleton. Formerly part of proof of ring1zr 20039. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 23-Jan-2020.)
(( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 = {𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
 
Theoremmgmb1mgm1 17856 The only magma with a base set consisting of one element is the trivial magma (at least if its operation is an internal binary operation). (Contributed by AV, 23-Jan-2020.) (Revised by AV, 7-Feb-2020.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)       ((𝑀 ∈ Mgm ∧ 𝑍𝐵+ Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
 
Theoremmgm0 17857 Any set with an empty base set and any group operation is a magma. (Contributed by AV, 28-Aug-2021.)
((𝑀𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Mgm)
 
Theoremmgm0b 17858 The structure with an empty base set and any group operation is a magma. (Contributed by AV, 28-Aug-2021.)
{⟨(Base‘ndx), ∅⟩, ⟨(+g‘ndx), 𝑂⟩} ∈ Mgm
 
Theoremmgm1 17859 The structure with one element and the only closed internal operation for a singleton is a magma. (Contributed by AV, 10-Feb-2020.)
𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}       (𝐼𝑉𝑀 ∈ Mgm)
 
Theoremopifismgm 17860* A structure with a group addition operation expressed by a conditional operator is a magma if both values of the conditional operator are contained in the base set. (Contributed by AV, 9-Feb-2020.)
𝐵 = (Base‘𝑀)    &   (+g𝑀) = (𝑥𝐵, 𝑦𝐵 ↦ if(𝜓, 𝐶, 𝐷))    &   (𝜑𝐵 ≠ ∅)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐶𝐵)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐷𝐵)       (𝜑𝑀 ∈ Mgm)
 
10.1.2  Identity elements

According to Wikipedia ("Identity element", 7-Feb-2020, https://en.wikipedia.org/wiki/Identity_element): "In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it.". Or in more detail "... an element e of S is called a left identity if e * a = a for all a in S, and a right identity if a * e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity." We concentrate on two-sided identities in the following. The existence of an identity (an identity is unique if it exists, see mgmidmo 17861) is an important property of monoids (see mndid 17912), and therefore also for groups (see grpid 18130), but also for magmas not required to be associative. Magmas with an identity element are called "unital magmas" (see Definition 2 in [BourbakiAlg1] p. 12) or, if the magmas are cancellative, "loops" (see definition in [Bruck] p. 15).

In the context of extensible structures, the identity element (of any magma 𝑀) is defined as "group identity element" (0g𝑀), see df-0g 16706. Related theorems which are already valid for magmas are provided in the following.

 
Theoremmgmidmo 17861* 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.)
∃*𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)
 
Theoremgrpidval 17862* The value of the identity element of a group. (Contributed by NM, 20-Aug-2011.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)        0 = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
 
Theoremgrpidpropd 17863* 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, they have the same identity element. (Contributed by Mario Carneiro, 27-Nov-2014.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))       (𝜑 → (0g𝐾) = (0g𝐿))
 
Theoremfn0g 17864 The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.)
0g Fn V
 
Theorem0g0 17865 The identity element function evaluates to the empty set on an empty structure. (Contributed by Stefan O'Rear, 2-Oct-2015.)
∅ = (0g‘∅)
 
Theoremismgmid 17866* The identity element of a magma, if it exists, belongs to the base set. (Contributed by Mario Carneiro, 27-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑 → ∃𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))       (𝜑 → ((𝑈𝐵 ∧ ∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ 0 = 𝑈))
 
Theoremmgmidcl 17867* The identity element of a magma, if it exists, belongs to the base set. (Contributed by Mario Carneiro, 27-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑 → ∃𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))       (𝜑0𝐵)
 
Theoremmgmlrid 17868* The identity element of a magma, if it exists, is a left and right identity. (Contributed by Mario Carneiro, 27-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑 → ∃𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))       ((𝜑𝑋𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))
 
Theoremismgmid2 17869* Show that a given element is the identity element of a magma. (Contributed by Mario Carneiro, 27-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑𝑈𝐵)    &   ((𝜑𝑥𝐵) → (𝑈 + 𝑥) = 𝑥)    &   ((𝜑𝑥𝐵) → (𝑥 + 𝑈) = 𝑥)       (𝜑𝑈 = 0 )
 
Theoremlidrideqd 17870* If there is a left and right identity element for any binary operation (group operation) +, both identity elements are equal. Generalization of statement in [Lang] p. 3: it is sufficient that "e" is a left identity element and "e`" is a right identity element instead of both being (two-sided) identity elements. (Contributed by AV, 26-Dec-2023.)
(𝜑𝐿𝐵)    &   (𝜑𝑅𝐵)    &   (𝜑 → ∀𝑥𝐵 (𝐿 + 𝑥) = 𝑥)    &   (𝜑 → ∀𝑥𝐵 (𝑥 + 𝑅) = 𝑥)       (𝜑𝐿 = 𝑅)
 
Theoremlidrididd 17871* If there is a left and right identity element for any binary operation (group operation) +, the left identity element (and therefore also the right identity element according to lidrideqd 17870) is equal to the two-sided identity element. (Contributed by AV, 26-Dec-2023.)
(𝜑𝐿𝐵)    &   (𝜑𝑅𝐵)    &   (𝜑 → ∀𝑥𝐵 (𝐿 + 𝑥) = 𝑥)    &   (𝜑 → ∀𝑥𝐵 (𝑥 + 𝑅) = 𝑥)    &   𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       (𝜑𝐿 = 0 )
 
Theoremgrpidd 17872* Deduce the identity element of a magma from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑+ = (+g𝐺))    &   (𝜑0𝐵)    &   ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)    &   ((𝜑𝑥𝐵) → (𝑥 + 0 ) = 𝑥)       (𝜑0 = (0g𝐺))
 
Theoremmgmidsssn0 17873* Property of the set of identities of 𝐺. Either 𝐺 has no identities, and 𝑂 = ∅, or it has one and this identity is unique and identified by the 0g function. (Contributed by Mario Carneiro, 7-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}       (𝐺𝑉𝑂 ⊆ { 0 })
 
Theoremgrprinvlem 17874* Lemma for grprinvd 17875. (Contributed by NM, 9-Aug-2013.)
((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   (𝜑𝑂𝐵)    &   ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = 𝑥)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂)    &   ((𝜑𝜓) → 𝑋𝐵)    &   ((𝜑𝜓) → (𝑋 + 𝑋) = 𝑋)       ((𝜑𝜓) → 𝑋 = 𝑂)
 
Theoremgrprinvd 17875* 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.)
((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   (𝜑𝑂𝐵)    &   ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = 𝑥)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂)    &   ((𝜑𝜓) → 𝑋𝐵)    &   ((𝜑𝜓) → 𝑁𝐵)    &   ((𝜑𝜓) → (𝑁 + 𝑋) = 𝑂)       ((𝜑𝜓) → (𝑋 + 𝑁) = 𝑂)
 
Theoremgrpridd 17876* 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.)
((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   (𝜑𝑂𝐵)    &   ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = 𝑥)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂)       ((𝜑𝑥𝐵) → (𝑥 + 𝑂) = 𝑥)
 
10.1.3  Iterated sums in a magma

The symbol Σg is mostly used in the context of abelian groups. Therefore, it is usually called "group sum". It can be defined, however, in arbitrary magmas (then it should be called "iterated sum"). If the magma is not required to be commutative or associative, then the order of the summands and the order in which summations are done become important. If the magma is not unital, then one cannot define a meaningful empty sum. See Remark 2. in the comment for df-gsum 16707.

 
Theoremgsumvalx 17877* Expand out the substitutions in df-gsum 16707. (Contributed by Mario Carneiro, 18-Sep-2015.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝑂 = {𝑠𝐵 ∣ ∀𝑡𝐵 ((𝑠 + 𝑡) = 𝑡 ∧ (𝑡 + 𝑠) = 𝑡)}    &   (𝜑𝑊 = (𝐹 “ (V ∖ 𝑂)))    &   (𝜑𝐺𝑉)    &   (𝜑𝐹𝑋)    &   (𝜑 → dom 𝐹 = 𝐴)       (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹𝑂, 0 , if(𝐴 ∈ ran ..., (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)))))))
 
Theoremgsumval 17878* Expand out the substitutions in df-gsum 16707. (Contributed by Mario Carneiro, 7-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝑂 = {𝑠𝐵 ∣ ∀𝑡𝐵 ((𝑠 + 𝑡) = 𝑡 ∧ (𝑡 + 𝑠) = 𝑡)}    &   (𝜑𝑊 = (𝐹 “ (V ∖ 𝑂)))    &   (𝜑𝐺𝑉)    &   (𝜑𝐴𝑋)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹𝑂, 0 , if(𝐴 ∈ ran ..., (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)))))))
 
Theoremgsumpropd 17879 The group sum depends only on the base set and additive operation. Note that for entirely unrestricted functions, there can be dependency on out-of-domain values of the operation, so this is somewhat weaker than mndpropd 17927 etc. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by Mario Carneiro, 18-Sep-2015.)
(𝜑𝐹𝑉)    &   (𝜑𝐺𝑊)    &   (𝜑𝐻𝑋)    &   (𝜑 → (Base‘𝐺) = (Base‘𝐻))    &   (𝜑 → (+g𝐺) = (+g𝐻))       (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
 
Theoremgsumpropd2lem 17880* Lemma for gsumpropd2 17881. (Contributed by Thierry Arnoux, 28-Jun-2017.)
(𝜑𝐹𝑉)    &   (𝜑𝐺𝑊)    &   (𝜑𝐻𝑋)    &   (𝜑 → (Base‘𝐺) = (Base‘𝐻))    &   ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) ∈ (Base‘𝐺))    &   ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) = (𝑠(+g𝐻)𝑡))    &   (𝜑 → Fun 𝐹)    &   (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))    &   𝐴 = (𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g𝐺)𝑡) = 𝑡 ∧ (𝑡(+g𝐺)𝑠) = 𝑡)}))    &   𝐵 = (𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g𝐻)𝑡) = 𝑡 ∧ (𝑡(+g𝐻)𝑠) = 𝑡)}))       (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
 
Theoremgsumpropd2 17881* A stronger version of gsumpropd 17879, working for magma, where only the closure of the addition operation on a common base is required, see gsummgmpropd 17882. (Contributed by Thierry Arnoux, 28-Jun-2017.)
(𝜑𝐹𝑉)    &   (𝜑𝐺𝑊)    &   (𝜑𝐻𝑋)    &   (𝜑 → (Base‘𝐺) = (Base‘𝐻))    &   ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) ∈ (Base‘𝐺))    &   ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) = (𝑠(+g𝐻)𝑡))    &   (𝜑 → Fun 𝐹)    &   (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))       (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
 
Theoremgsummgmpropd 17882* A stronger version of gsumpropd 17879 if at least one of the involved structures is a magma, see gsumpropd2 17881. (Contributed by AV, 31-Jan-2020.)
(𝜑𝐹𝑉)    &   (𝜑𝐺𝑊)    &   (𝜑𝐻𝑋)    &   (𝜑 → (Base‘𝐺) = (Base‘𝐻))    &   (𝜑𝐺 ∈ Mgm)    &   ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) = (𝑠(+g𝐻)𝑡))    &   (𝜑 → Fun 𝐹)    &   (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))       (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
 
Theoremgsumress 17883* The group sum in a substructure is the same as the group sum in the original structure. The only requirement on the substructure is that it contain the identity element; neither 𝐺 nor 𝐻 need be groups. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐻 = (𝐺s 𝑆)    &   (𝜑𝐺𝑉)    &   (𝜑𝐴𝑋)    &   (𝜑𝑆𝐵)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑0𝑆)    &   ((𝜑𝑥𝐵) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))       (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
 
Theoremgsumval1 17884* Value of the group sum operation when every element being summed is an identity of 𝐺. (Contributed by Mario Carneiro, 7-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}    &   (𝜑𝐺𝑉)    &   (𝜑𝐴𝑊)    &   (𝜑𝐹:𝐴𝑂)       (𝜑 → (𝐺 Σg 𝐹) = 0 )
 
Theoremgsum0 17885 Value of the empty group sum. (Contributed by Mario Carneiro, 7-Dec-2014.)
0 = (0g𝐺)       (𝐺 Σg ∅) = 0
 
Theoremgsumval2a 17886* Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐹:(𝑀...𝑁)⟶𝐵)    &   𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}    &   (𝜑 → ¬ ran 𝐹𝑂)       (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
 
Theoremgsumval2 17887 Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐹:(𝑀...𝑁)⟶𝐵)       (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
 
Theoremgsumsplit1r 17888 Splitting off the rightmost summand of a group sum. This corresponds to the (inductive) definition of a (finite) product in [Lang] p. 4, first formula. (Contributed by AV, 26-Dec-2023.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐹:(𝑀...(𝑁 + 1))⟶𝐵)       (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ (𝑀...𝑁))) + (𝐹‘(𝑁 + 1))))
 
Theoremgsumprval 17889 Value of the group sum operation over a pair of sequential integers. (Contributed by AV, 14-Dec-2018.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 = (𝑀 + 1))    &   (𝜑𝐹:{𝑀, 𝑁}⟶𝐵)       (𝜑 → (𝐺 Σg 𝐹) = ((𝐹𝑀) + (𝐹𝑁)))
 
Theoremgsumpr12val 17890 Value of the group sum operation over the pair {1, 2}. (Contributed by AV, 14-Dec-2018.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝐹:{1, 2}⟶𝐵)       (𝜑 → (𝐺 Σg 𝐹) = ((𝐹‘1) + (𝐹‘2)))
 
10.1.4  Semigroups

A semigroup (Smgrp, see df-sgrp 17892) is a set together with an associative binary operation (see Wikipedia, Semigroup, 8-Jan-2020, https://en.wikipedia.org/wiki/Semigroup 17892). 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.

 
Syntaxcsgrp 17891 Extend class notation with class of all semigroups.
class Smgrp
 
Definitiondf-sgrp 17892* A semigroup is a set equipped with an everywhere defined internal operation (so, a magma, see df-mgm 17843), 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.)
Smgrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
 
Theoremissgrp 17893* The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
𝐵 = (Base‘𝑀)    &    = (+g𝑀)       (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
 
Theoremissgrpv 17894* The predicate "is a semigroup" for a structure which is a set. (Contributed by AV, 1-Feb-2020.)
𝐵 = (Base‘𝑀)    &    = (+g𝑀)       (𝑀𝑉 → (𝑀 ∈ Smgrp ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))))
 
Theoremissgrpn0 17895* The predicate "is a semigroup" for a structure with a nonempty base set. (Contributed by AV, 1-Feb-2020.)
𝐵 = (Base‘𝑀)    &    = (+g𝑀)       (𝐴𝐵 → (𝑀 ∈ Smgrp ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))))
 
Theoremisnsgrp 17896 A condition for a structure not to be a semigroup. (Contributed by AV, 30-Jan-2020.)
𝐵 = (Base‘𝑀)    &    = (+g𝑀)       ((𝑋𝐵𝑌𝐵𝑍𝐵) → (((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍)) → 𝑀 ∉ Smgrp))
 
Theoremsgrpmgm 17897 A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
(𝑀 ∈ Smgrp → 𝑀 ∈ Mgm)
 
Theoremsgrpass 17898 A semigroup operation is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 30-Jan-2020.)
𝐵 = (Base‘𝐺)    &    = (+g𝐺)       ((𝐺 ∈ Smgrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))
 
Theoremsgrp0 17899 Any set with an empty base set and any group operation is a semigroup. (Contributed by AV, 28-Aug-2021.)
((𝑀𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Smgrp)
 
Theoremsgrp0b 17900 The structure with an empty base set and any group operation is a semigroup. (Contributed by AV, 28-Aug-2021.)
{⟨(Base‘ndx), ∅⟩, ⟨(+g‘ndx), 𝑂⟩} ∈ Smgrp
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