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Theorem List for Metamath Proof Explorer - 17601-17700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempsref2 17601 A poset is antisymmetric and reflexive. (Contributed by FL, 3-Aug-2009.)
(𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))

Theorempstr2 17602 A poset is transitive. (Contributed by FL, 3-Aug-2009.)
(𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)

Theorempslem 17603 Lemma for psref 17605 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 17604 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 17605 A poset is reflexive. (Contributed by NM, 13-May-2008.)
𝑋 = dom 𝑅       ((𝑅 ∈ PosetRel ∧ 𝐴𝑋) → 𝐴𝑅𝐴)

Theorempsrn 17606 The range of a poset equals it domain. (Contributed by NM, 7-Jul-2008.)
𝑋 = dom 𝑅       (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)

Theorempsasym 17607 A poset is antisymmetric. (Contributed by NM, 12-May-2008.)
((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵𝐵𝑅𝐴) → 𝐴 = 𝐵)

Theorempstr 17608 A poset is transitive. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.)
((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Theoremcnvps 17609 The converse of a poset is a poset. In the general case (𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel) is not true. See cnvpsb 17610 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 17610 The converse of a poset is a poset. (Contributed by FL, 5-Jan-2009.)
(Rel 𝑅 → (𝑅 ∈ PosetRel ↔ 𝑅 ∈ PosetRel))

Theorempsss 17611 Any subset of a partially ordered set is partially ordered. (Contributed by FL, 24-Jan-2010.)
(𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)

Theorempsssdm2 17612 Field of a subposet. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝑋 = dom 𝑅       (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋𝐴))

Theorempsssdm 17613 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 17614 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 17615* 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 17616 A toset is a linear order. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵𝐵𝑅𝐴))

Theoremtsrlemax 17617 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 17618 A toset is a poset. (Contributed by Mario Carneiro, 9-Sep-2015.)
(𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)

Theoremcnvtsr 17619 The converse of a toset is a toset. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑅 ∈ TosetRel → 𝑅 ∈ TosetRel )

Theoremtsrss 17620 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 17621 The domain of is *. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)
* = dom ≤

Theoremlern 17622 The range of is *. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
* = ran ≤

Theoremlefld 17623 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 17624 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 17625 Extend class notation with the class of directed sets.
class DirRel

Syntaxctail 17626 Extend class notation with the tail function for directed sets.
class tail

Definitiondf-dir 17627 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 17628* Define the tail function for directed sets. (Contributed by Jeff Hankins, 25-Nov-2009.)
tail = (𝑟 ∈ DirRel ↦ (𝑥 𝑟 ↦ (𝑟 “ {𝑥})))

Theoremisdir 17629 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 17630 A direction is a relation. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
(𝑅 ∈ DirRel → Rel 𝑅)

Theoremdirdm 17631 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 17632 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 17633 A direction is transitive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
(((𝑅 ∈ DirRel ∧ 𝐶𝑉) ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴𝑅𝐶)

Theoremdirge 17634* 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 17635 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-mpt2 6929, binary operations are defined by a rule, and with df-ov 6927, 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 × 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 ).

The definition of magmas (Mgm, see df-mgm 17639) 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 17636 Extend class notation with group addition as a function.
class +𝑓

Syntaxcmgm 17637 Extend class notation with class of all magmas.
class Mgm

Definitiondf-plusf 17638* 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 17648), while +g only has closure (mgmcl 17642). (Contributed by Mario Carneiro, 14-Aug-2015.)
+𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)𝑦)))

Definitiondf-mgm 17639* 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 17640* 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 17641* 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 17642 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 17643 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)

Theoremplusffval 17644* The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (+𝑓𝐺)        = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦))

Theoremplusfval 17645 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 17646 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 17647 The group addition operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)
𝐵 = (Base‘𝐺)    &    = (+𝑓𝐺)        Fn (𝐵 × 𝐵)

Theoremmgmplusf 17648 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 17649* 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 17650 The internal operation for a set is the trivial operation iff the set is a singleton. Formerly part of proof of ring1zr 19683. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 23-Jan-2020.)
(( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 = {𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))

Theoremmgmb1mgm1 17651 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 17652 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 17653 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 17654 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 17655* 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 17656) is an important property of monoids (see mndid 17700), and therefore also for groups (see grpid 17855), 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 16499. Related theorems which are already valid for magmas are provided in the following.

Theoremmgmidmo 17656* 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 17657* 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 17658* 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 17659 The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.)
0g Fn V

Theorem0g0 17660 The identity element function evaluates to the empty set on an empty structure. (Contributed by Stefan O'Rear, 2-Oct-2015.)
∅ = (0g‘∅)

Theoremismgmid 17661* 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 17662* 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 17663* 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 17664* 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 )

Theoremgrpidd 17665* 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 17666* 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 })

10.1.3  Ordered sums in a magma

The symbol Σg is mostly used in the context of abelian groups. Therefore, it is called "group sum". It can be defined, however, in arbitrary magmas. 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 16500.

Theoremgsumvalx 17667* Expand out the substitutions in df-gsum 16500. (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 17668* Expand out the substitutions in df-gsum 16500. (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 17669 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 17713 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 17670* Lemma for gsumpropd2 17671. (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 17671* A stronger version of gsumpropd 17669, working for magma, where only the closure of the addition operation on a common base is required, see gsummgmpropd 17672. (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 17672* A stronger version of gsumpropd 17669 if at least one of the involved structures is a magma, see gsumpropd2 17671. (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 17673* 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 17674* 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 17675 Value of the empty group sum. (Contributed by Mario Carneiro, 7-Dec-2014.)
0 = (0g𝐺)       (𝐺 Σg ∅) = 0

Theoremgsumval2a 17676* 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 17677 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))

Theoremgsumprval 17678 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 17679 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 (SGrp, see df-sgrp 17681) is a set together with an associative binary operation (see Wikipedia, Semigroup, 8-Jan-2020, https://en.wikipedia.org/wiki/Semigroup). 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 17680 Extend class notation with class of all semigroups.
class SGrp

Definitiondf-sgrp 17681* A semigroup is a set equipped with an everywhere defined internal operation (so, a magma, see df-mgm 17639), 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.)
SGrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}

Theoremissgrp 17682* The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (Revised by by AV, 6-Jan-2020.)
𝐵 = (Base‘𝑀)    &    = (+g𝑀)       (𝑀 ∈ SGrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))

Theoremissgrpv 17683* The predicate "is a semigroup" for a structure which is a set. (Contributed by AV, 1-Feb-2020.)
𝐵 = (Base‘𝑀)    &    = (+g𝑀)       (𝑀𝑉 → (𝑀 ∈ SGrp ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))))

Theoremissgrpn0 17684* 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 ∈ SGrp ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))))

Theoremisnsgrp 17685 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 ∉ SGrp))

Theoremsgrpmgm 17686 A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
(𝑀 ∈ SGrp → 𝑀 ∈ Mgm)

Theoremsgrpass 17687 A semigroup operation is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 30-Jan-2020.)
𝐵 = (Base‘𝐺)    &    = (+g𝐺)       ((𝐺 ∈ SGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))

Theoremsgrp0 17688 Any set with an empty base set and any group operation is a semigroup. (Contributed by AV, 28-Aug-2021.)
((𝑀𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ SGrp)

Theoremsgrp0b 17689 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⟩} ∈ SGrp

Theoremsgrp1 17690 The structure with one element and the only closed internal operation for a singleton is a semigroup. (Contributed by AV, 10-Feb-2020.)
𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}       (𝐼𝑉𝑀 ∈ SGrp)

10.1.5  Definition and basic properties of monoids

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 17692, 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 17694. 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".

Syntaxcmnd 17691 Extend class notation with class of all monoids.
class Mnd

Definitiondf-mnd 17692* 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 17698), whose operation is associative (see mndass 17699) and has a two-sided neutral element (see mndid 17700), see also ismnd 17694. (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.)
Mnd = {𝑔 ∈ SGrp ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑒𝑏𝑥𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)}

Theoremismnddef 17693* The predicate "is a monoid", corresponding 1-to-1 to the definition. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 1-Feb-2020.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ Mnd ↔ (𝐺 ∈ SGrp ∧ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))

Theoremismnd 17694* The predicate "is a monoid". This is the definig 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 17698), whose operation is associative (so, a semigroup, see also mndass 17699) and has a two-sided neutral element (see mndid 17700). (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ Mnd ↔ (∀𝑎𝐵𝑏𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐))) ∧ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))

Theoremisnmnd 17695* A condition for a structure not to be a monoid: every element of the base set is not a left identity for at least one element of the base set. (Contributed by AV, 4-Feb-2020.)
𝐵 = (Base‘𝑀)    &    = (+g𝑀)       (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥𝑀 ∉ Mnd)

Theoremmndsgrp 17696 A monoid is a semigroup. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) (Proof shortened by AV, 6-Feb-2020.)
(𝐺 ∈ Mnd → 𝐺 ∈ SGrp)

Theoremmndmgm 17697 A monoid is a magma. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) (Proof shortened by AV, 6-Feb-2020.)
(𝑀 ∈ Mnd → 𝑀 ∈ Mgm)

Theoremmndcl 17698 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.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)

Theoremmndass 17699 A monoid operation is associative. (Contributed by NM, 14-Aug-2011.) (Proof shortened by AV, 8-Feb-2020.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))

Theoremmndid 17700* A monoid has a two-sided identity element. (Contributed by NM, 16-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ Mnd → ∃𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 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 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43671
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