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Theorem List for Metamath Proof Explorer - 17901-18000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
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 17902, 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 17904. 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 17901 Extend class notation with class of all monoids.
class Mnd
 
Definitiondf-mnd 17902* 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 17909), whose operation is associative (see mndass 17910) and has a two-sided neutral element (see mndid 17911), see also ismnd 17904. (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.)
Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑒𝑏𝑥𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)}
 
Theoremismnddef 17903* 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 ∈ Smgrp ∧ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
 
Theoremismnd 17904* 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 17909), whose operation is associative (so, a semigroup, see also mndass 17910) and has a two-sided neutral element (see mndid 17911). (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 17905* 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)
 
Theoremsgrpidmnd 17906* A semigroup with an identity element which is not the empty set is a monoid. Of course there could be monoids with the empty set as identity element (see, for example, the monoid of the power set of a class under union, pwmnd 18042 and pwmndid 18041), but these cannot be proven to be monoids with this theorem. (Contributed by AV, 29-Jan-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Smgrp ∧ ∃𝑒𝐵 (𝑒 ≠ ∅ ∧ 𝑒 = 0 )) → 𝐺 ∈ Mnd)
 
Theoremmndsgrp 17907 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 ∈ Smgrp)
 
Theoremmndmgm 17908 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 17909 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 17910 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 17911* 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))
 
Theoremmndideu 17912* 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.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ Mnd → ∃!𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))
 
Theoremmnd32g 17913 Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌))       (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))
 
Theoremmnd12g 17914 Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))       (𝜑 → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))
 
Theoremmnd4g 17915 Commutative/associative law for commutative monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑊𝐵)    &   (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌))       (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))
 
Theoremmndidcl 17916 The identity element of a monoid belongs to the monoid. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)       (𝐺 ∈ Mnd → 0𝐵)
 
Theoremmndbn0 17917 The base set of a monoid is not empty. Statement in [Lang] p. 3. (Contributed by AV, 29-Dec-2023.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ Mnd → 𝐵 ≠ ∅)
 
Theoremhashfinmndnn 17918 A finite monoid has positive integer size. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐵 ∈ Fin)       (𝜑 → (♯‘𝐵) ∈ ℕ)
 
Theoremmndplusf 17919 The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 3-Feb-2020.)
𝐵 = (Base‘𝐺)    &    = (+𝑓𝐺)       (𝐺 ∈ Mnd → :(𝐵 × 𝐵)⟶𝐵)
 
Theoremmndlrid 17920 A monoid's identity element is a two-sided identity. (Contributed by NM, 18-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑋𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))
 
Theoremmndlid 17921 The identity element of a monoid is a left identity. (Contributed by NM, 18-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑋𝐵) → ( 0 + 𝑋) = 𝑋)
 
Theoremmndrid 17922 The identity element of a monoid is a right identity. (Contributed by NM, 18-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑋𝐵) → (𝑋 + 0 ) = 𝑋)
 
Theoremismndd 17923* Deduce a monoid from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑+ = (+g𝐺))    &   ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑0𝐵)    &   ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)    &   ((𝜑𝑥𝐵) → (𝑥 + 0 ) = 𝑥)       (𝜑𝐺 ∈ Mnd)
 
Theoremmndpfo 17924 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.)
𝐵 = (Base‘𝐺)    &    = (+𝑓𝐺)       (𝐺 ∈ Mnd → :(𝐵 × 𝐵)–onto𝐵)
 
Theoremmndfo 17925 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.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)–onto𝐵)
 
Theoremmndpropd 17926* 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.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))       (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
 
Theoremmndprop 17927 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.)
(Base‘𝐾) = (Base‘𝐿)    &   (+g𝐾) = (+g𝐿)       (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd)
 
Theoremissubmnd 17928* Characterize a submonoid by closure properties. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)    &   𝐻 = (𝐺s 𝑆)       ((𝐺 ∈ Mnd ∧ 𝑆𝐵0𝑆) → (𝐻 ∈ Mnd ↔ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆))
 
Theoremress0g 17929 0g is unaffected by restriction. This is a bit more generic than submnd0 17930. (Contributed by Thierry Arnoux, 23-Oct-2017.)
𝑆 = (𝑅s 𝐴)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Mnd ∧ 0𝐴𝐴𝐵) → 0 = (0g𝑆))
 
Theoremsubmnd0 17930 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. See, for example, smndex1mnd 43980 and smndex1n0mnd 43982) (Contributed by Mario Carneiro, 10-Jan-2015.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝐻 = (𝐺s 𝑆)       (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆𝐵0𝑆)) → 0 = (0g𝐻))
 
Theoremmndinvmod 17931* Uniqueness of an inverse element in a monoid, if it exists. (Contributed by AV, 20-Jan-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝐵)       (𝜑 → ∃*𝑤𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))
 
Theoremprdsplusgcl 17932 Structure product pointwise sums are closed when the factors are monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    + = (+g𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅:𝐼⟶Mnd)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐹 + 𝐺) ∈ 𝐵)
 
Theoremprdsidlem 17933* Characterization of identity in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    + = (+g𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅:𝐼⟶Mnd)    &    0 = (0g𝑅)       (𝜑 → ( 0𝐵 ∧ ∀𝑥𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
 
Theoremprdsmndd 17934 The product of a family of monoids is a monoid. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Mnd)       (𝜑𝑌 ∈ Mnd)
 
Theoremprds0g 17935 Zero in a product of monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Mnd)       (𝜑 → (0g𝑅) = (0g𝑌))
 
Theorempwsmnd 17936 The structure power of a monoid is a monoid. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ Mnd ∧ 𝐼𝑉) → 𝑌 ∈ Mnd)
 
Theorempws0g 17937 Zero in a product of monoids. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &    0 = (0g𝑅)       ((𝑅 ∈ Mnd ∧ 𝐼𝑉) → (𝐼 × { 0 }) = (0g𝑌))
 
Theoremimasmnd2 17938* The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &    + = (+g𝑅)    &   (𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   (𝜑𝑅𝑊)    &   ((𝜑𝑥𝑉𝑦𝑉) → (𝑥 + 𝑦) ∈ 𝑉)    &   ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))    &   (𝜑0𝑉)    &   ((𝜑𝑥𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹𝑥))    &   ((𝜑𝑥𝑉) → (𝐹‘(𝑥 + 0 )) = (𝐹𝑥))       (𝜑 → (𝑈 ∈ Mnd ∧ (𝐹0 ) = (0g𝑈)))
 
Theoremimasmnd 17939* The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &    + = (+g𝑅)    &   (𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   (𝜑𝑅 ∈ Mnd)    &    0 = (0g𝑅)       (𝜑 → (𝑈 ∈ Mnd ∧ (𝐹0 ) = (0g𝑈)))
 
Theoremimasmndf1 17940 The image of a monoid under an injection is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
𝑈 = (𝐹s 𝑅)    &   𝑉 = (Base‘𝑅)       ((𝐹:𝑉1-1𝐵𝑅 ∈ Mnd) → 𝑈 ∈ Mnd)
 
Theoremxpsmnd 17941 The binary product of monoids is a monoid. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)       ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → 𝑇 ∈ Mnd)
 
Theoremmnd1 17942 The (smallest) structure representing a trivial monoid consists of one element. (Contributed by AV, 28-Apr-2019.) (Proof shortened by AV, 11-Feb-2020.)
𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}       (𝐼𝑉𝑀 ∈ Mnd)
 
Theoremmnd1id 17943 The singleton element of a trivial monoid is its identity element. (Contributed by AV, 23-Jan-2020.)
𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}       (𝐼𝑉 → (0g𝑀) = 𝐼)
 
10.1.6  Monoid homomorphisms and submonoids
 
Syntaxcmhm 17944 Hom-set generator class for monoids.
class MndHom
 
Syntaxcsubmnd 17945 Class function taking a monoid to its lattice of submonoids.
class SubMnd
 
Definitiondf-mhm 17946* 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 = (𝑠 ∈ Mnd, 𝑡 ∈ Mnd ↦ {𝑓 ∈ ((Base‘𝑡) ↑m (Base‘𝑠)) ∣ (∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ∧ (𝑓‘(0g𝑠)) = (0g𝑡))})
 
Definitiondf-submnd 17947* 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 = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0g𝑠) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑠)𝑦) ∈ 𝑡)})
 
Theoremismhm 17948* Property of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)    &    + = (+g𝑆)    &    = (+g𝑇)    &    0 = (0g𝑆)    &   𝑌 = (0g𝑇)       (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))
 
Theoremmhmrcl1 17949 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
(𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
 
Theoremmhmrcl2 17950 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
(𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑇 ∈ Mnd)
 
Theoremmhmf 17951 A monoid homomorphism is a function. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)       (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:𝐵𝐶)
 
Theoremmhmpropd 17952* Monoid homomorphism depends only on the monoidal attributes of structures. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 7-Nov-2015.)
(𝜑𝐵 = (Base‘𝐽))    &   (𝜑𝐶 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑𝐶 = (Base‘𝑀))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐽)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦))       (𝜑 → (𝐽 MndHom 𝐾) = (𝐿 MndHom 𝑀))
 
Theoremmhmlin 17953 A monoid homomorphism commutes with composition. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐵 = (Base‘𝑆)    &    + = (+g𝑆)    &    = (+g𝑇)       ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))
 
Theoremmhm0 17954 A monoid homomorphism preserves zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
0 = (0g𝑆)    &   𝑌 = (0g𝑇)       (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹0 ) = 𝑌)
 
Theoremidmhm 17955 The identity homomorphism on a monoid. (Contributed by AV, 14-Feb-2020.)
𝐵 = (Base‘𝑀)       (𝑀 ∈ Mnd → ( I ↾ 𝐵) ∈ (𝑀 MndHom 𝑀))
 
Theoremmhmf1o 17956 A monoid homomorphism is bijective iff its converse is also a monoid homomorphism. (Contributed by AV, 22-Oct-2019.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹:𝐵1-1-onto𝐶𝐹 ∈ (𝑆 MndHom 𝑅)))
 
Theoremsubmrcl 17957 Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.)
(𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd)
 
Theoremissubm 17958* Expand definition of a submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐵 = (Base‘𝑀)    &    0 = (0g𝑀)    &    + = (+g𝑀)       (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆𝐵0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
 
Theoremissubm2 17959 Submonoids are subsets that are also monoids with the same zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐵 = (Base‘𝑀)    &    0 = (0g𝑀)    &   𝐻 = (𝑀s 𝑆)       (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆𝐵0𝑆𝐻 ∈ Mnd)))
 
Theoremissubmndb 17960 The submonoid predicate. Analogous to issubg 18219. (Contributed by AV, 1-Feb-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)       (𝑆 ∈ (SubMnd‘𝐺) ↔ ((𝐺 ∈ Mnd ∧ (𝐺s 𝑆) ∈ Mnd) ∧ (𝑆𝐵0𝑆)))
 
Theoremissubmd 17961* Deduction for proving a submonoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &    0 = (0g𝑀)    &   (𝜑𝑀 ∈ Mnd)    &   (𝜑𝜒)    &   ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝜃𝜏))) → 𝜂)    &   (𝑧 = 0 → (𝜓𝜒))    &   (𝑧 = 𝑥 → (𝜓𝜃))    &   (𝑧 = 𝑦 → (𝜓𝜏))    &   (𝑧 = (𝑥 + 𝑦) → (𝜓𝜂))       (𝜑 → {𝑧𝐵𝜓} ∈ (SubMnd‘𝑀))
 
Theoremmndissubm 17962 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. Analogous to grpissubg 18239. (Contributed by AV, 17-Feb-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = (Base‘𝐻)    &    0 = (0g𝐺)       ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → ((𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ∈ (SubMnd‘𝐺)))
 
Theoremresmndismnd 17963 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 other monoid restricted to the base set of the monoid is a monoid. Analogous to resgrpisgrp 18240. (Contributed by AV, 17-Feb-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = (Base‘𝐻)    &    0 = (0g𝐺)       ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → ((𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) → (𝐺s 𝑆) ∈ Mnd))
 
Theoremsubmss 17964 Submonoids are subsets of the base set. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐵 = (Base‘𝑀)       (𝑆 ∈ (SubMnd‘𝑀) → 𝑆𝐵)
 
Theoremsubmid 17965 Every monoid is trivially a submonoid of itself. (Contributed by Stefan O'Rear, 15-Aug-2015.)
𝐵 = (Base‘𝑀)       (𝑀 ∈ Mnd → 𝐵 ∈ (SubMnd‘𝑀))
 
Theoremsubm0cl 17966 Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
0 = (0g𝑀)       (𝑆 ∈ (SubMnd‘𝑀) → 0𝑆)
 
Theoremsubmcl 17967 Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.)
+ = (+g𝑀)       ((𝑆 ∈ (SubMnd‘𝑀) ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
 
Theoremsubmmnd 17968 Submonoids are themselves monoids under the given operation. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐻 = (𝑀s 𝑆)       (𝑆 ∈ (SubMnd‘𝑀) → 𝐻 ∈ Mnd)
 
Theoremsubmbas 17969 The base set of a submonoid. (Contributed by Stefan O'Rear, 15-Jun-2015.)
𝐻 = (𝑀s 𝑆)       (𝑆 ∈ (SubMnd‘𝑀) → 𝑆 = (Base‘𝐻))
 
Theoremsubm0 17970 Submonoids have the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐻 = (𝑀s 𝑆)    &    0 = (0g𝑀)       (𝑆 ∈ (SubMnd‘𝑀) → 0 = (0g𝐻))
 
Theoremsubsubm 17971 A submonoid of a submonoid is a submonoid. (Contributed by Mario Carneiro, 21-Jun-2015.)
𝐻 = (𝐺s 𝑆)       (𝑆 ∈ (SubMnd‘𝐺) → (𝐴 ∈ (SubMnd‘𝐻) ↔ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)))
 
Theorem0subm 17972 The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.)
0 = (0g𝐺)       (𝐺 ∈ Mnd → { 0 } ∈ (SubMnd‘𝐺))
 
Theoreminsubm 17973 The intersection of two submonoids is a submonoid. (Contributed by AV, 25-Feb-2024.)
((𝐴 ∈ (SubMnd‘𝑀) ∧ 𝐵 ∈ (SubMnd‘𝑀)) → (𝐴𝐵) ∈ (SubMnd‘𝑀))
 
Theorem0mhm 17974 The constant zero linear function between two monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
0 = (0g𝑁)    &   𝐵 = (Base‘𝑀)       ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁))
 
Theoremresmhm 17975 Restriction of a monoid homomorphism to a submonoid is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
𝑈 = (𝑆s 𝑋)       ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹𝑋) ∈ (𝑈 MndHom 𝑇))
 
Theoremresmhm2 17976 One direction of resmhm2b 17977. (Contributed by Mario Carneiro, 18-Jun-2015.)
𝑈 = (𝑇s 𝑋)       ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
 
Theoremresmhm2b 17977 Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.)
𝑈 = (𝑇s 𝑋)       ((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))
 
Theoremmhmco 17978 The composition of monoid homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 MndHom 𝑈))
 
Theoremmhmima 17979 The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.)
((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (𝐹𝑋) ∈ (SubMnd‘𝑁))
 
Theoremmhmeql 17980 The equalizer of two monoid homomorphisms is a submonoid. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → dom (𝐹𝐺) ∈ (SubMnd‘𝑆))
 
Theoremsubmacs 17981 Submonoids are an algebraic closure system. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
 
Theoremmndind 17982* Induction in a monoid. In this theorem, 𝜓(𝑥) is the "generic" proposition to be be proved (the first four hypotheses tell its values at y, y+z, 0, A respectively). The two induction hypotheses mndind.i1 and mndind.i2 tell that it is true at 0, that if it is true at y then it is true at y+z (provided z is in 𝐺). The hypothesis mndind.k tells that 𝐺 is generating. (Contributed by SO, 14-Jul-2018.)
(𝑥 = 𝑦 → (𝜓𝜒))    &   (𝑥 = (𝑦 + 𝑧) → (𝜓𝜃))    &   (𝑥 = 0 → (𝜓𝜏))    &   (𝑥 = 𝐴 → (𝜓𝜂))    &    0 = (0g𝑀)    &    + = (+g𝑀)    &   𝐵 = (Base‘𝑀)    &   (𝜑𝑀 ∈ Mnd)    &   (𝜑𝐺𝐵)    &   (𝜑𝐵 = ((mrCls‘(SubMnd‘𝑀))‘𝐺))    &   (𝜑𝜏)    &   (((𝜑𝑦𝐵𝑧𝐺) ∧ 𝜒) → 𝜃)    &   (𝜑𝐴𝐵)       (𝜑𝜂)
 
Theoremprdspjmhm 17983* A projection from a product of monoids to one of the factors is a monoid homomorphism. (Contributed by Mario Carneiro, 6-May-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆𝑋)    &   (𝜑𝑅:𝐼⟶Mnd)    &   (𝜑𝐴𝐼)       (𝜑 → (𝑥𝐵 ↦ (𝑥𝐴)) ∈ (𝑌 MndHom (𝑅𝐴)))
 
Theorempwspjmhm 17984* A projection from a product of monoids to one of the factors is a monoid homomorphism. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)       ((𝑅 ∈ Mnd ∧ 𝐼𝑉𝐴𝐼) → (𝑥𝐵 ↦ (𝑥𝐴)) ∈ (𝑌 MndHom 𝑅))
 
Theorempwsdiagmhm 17985* Diagonal monoid homomorphism into a structure power. (Contributed by Stefan O'Rear, 12-Mar-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑅)    &   𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))       ((𝑅 ∈ Mnd ∧ 𝐼𝑊) → 𝐹 ∈ (𝑅 MndHom 𝑌))
 
Theorempwsco1mhm 17986* Right composition with a function on the index sets yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑌 = (𝑅s 𝐴)    &   𝑍 = (𝑅s 𝐵)    &   𝐶 = (Base‘𝑍)    &   (𝜑𝑅 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → (𝑔𝐶 ↦ (𝑔𝐹)) ∈ (𝑍 MndHom 𝑌))
 
Theorempwsco2mhm 17987* Left composition with a monoid homomorphism yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑌 = (𝑅s 𝐴)    &   𝑍 = (𝑆s 𝐴)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹 ∈ (𝑅 MndHom 𝑆))       (𝜑 → (𝑔𝐵 ↦ (𝐹𝑔)) ∈ (𝑌 MndHom 𝑍))
 
10.1.7  Iterated sums in a monoid

One important use of words is as formal composites in cases where order is significant, using the general sum operator df-gsum 16706. If order is not significant, it is simpler to use families instead.

 
Theoremgsumvallem2 17988* Lemma for properties of the set of identities of 𝐺. The set of identities of a monoid is exactly the unique identity element. (Contributed by Mario Carneiro, 7-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}       (𝐺 ∈ Mnd → 𝑂 = { 0 })
 
Theoremgsumsubm 17989 Evaluate a group sum in a submonoid. (Contributed by Mario Carneiro, 19-Dec-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝑆 ∈ (SubMnd‘𝐺))    &   (𝜑𝐹:𝐴𝑆)    &   𝐻 = (𝐺s 𝑆)       (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
 
Theoremgsumz 17990* Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.)
0 = (0g𝐺)       ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
 
Theoremgsumwsubmcl 17991 Closure of the composite in any submonoid. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)
((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑊 ∈ Word 𝑆) → (𝐺 Σg 𝑊) ∈ 𝑆)
 
Theoremgsumws1 17992 A singleton composite recovers the initial symbol. (Contributed by Stefan O'Rear, 16-Aug-2015.)
𝐵 = (Base‘𝐺)       (𝑆𝐵 → (𝐺 Σg ⟨“𝑆”⟩) = 𝑆)
 
Theoremgsumwcl 17993 Closure of the composite of a word in a structure 𝐺. (Contributed by Stefan O'Rear, 15-Aug-2015.)
𝐵 = (Base‘𝐺)       ((𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵) → (𝐺 Σg 𝑊) ∈ 𝐵)
 
Theoremgsumsgrpccat 17994 Homomorphic property of not empty composites of a group sum over a semigroup. Formerly part of proof for gsumccat 17996. (Contributed by AV, 26-Dec-2023.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Smgrp ∧ (𝑊 ∈ Word 𝐵𝑋 ∈ Word 𝐵) ∧ (𝑊 ≠ ∅ ∧ 𝑋 ≠ ∅)) → (𝐺 Σg (𝑊 ++ 𝑋)) = ((𝐺 Σg 𝑊) + (𝐺 Σg 𝑋)))
 
TheoremgsumccatOLD 17995 Obsolete version of gsumccat 17996 as of 13-Jan-2024. Homomorphic property of composites. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵𝑋 ∈ Word 𝐵) → (𝐺 Σg (𝑊 ++ 𝑋)) = ((𝐺 Σg 𝑊) + (𝐺 Σg 𝑋)))
 
Theoremgsumccat 17996 Homomorphic property of composites. Second formula in [Lang] p. 4. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 26-Dec-2023.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵𝑋 ∈ Word 𝐵) → (𝐺 Σg (𝑊 ++ 𝑋)) = ((𝐺 Σg 𝑊) + (𝐺 Σg 𝑋)))
 
Theoremgsumws2 17997 Valuation of a pair in a monoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑆𝐵𝑇𝐵) → (𝐺 Σg ⟨“𝑆𝑇”⟩) = (𝑆 + 𝑇))
 
Theoremgsumccatsn 17998 Homomorphic property of composites with a singleton. (Contributed by AV, 20-Jan-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵𝑍𝐵) → (𝐺 Σg (𝑊 ++ ⟨“𝑍”⟩)) = ((𝐺 Σg 𝑊) + 𝑍))
 
Theoremgsumspl 17999 The primary purpose of the splice construction is to enable local rewrites. Thus, in any monoidal valuation, if a splice does not cause a local change it does not cause a global change. (Contributed by Stefan O'Rear, 23-Aug-2015.)
𝐵 = (Base‘𝑀)    &   (𝜑𝑀 ∈ Mnd)    &   (𝜑𝑆 ∈ Word 𝐵)    &   (𝜑𝐹 ∈ (0...𝑇))    &   (𝜑𝑇 ∈ (0...(♯‘𝑆)))    &   (𝜑𝑋 ∈ Word 𝐵)    &   (𝜑𝑌 ∈ Word 𝐵)    &   (𝜑 → (𝑀 Σg 𝑋) = (𝑀 Σg 𝑌))       (𝜑 → (𝑀 Σg (𝑆 splice ⟨𝐹, 𝑇, 𝑋⟩)) = (𝑀 Σg (𝑆 splice ⟨𝐹, 𝑇, 𝑌⟩)))
 
Theoremgsumwmhm 18000 Behavior of homomorphisms on finite monoidal sums. (Contributed by Stefan O'Rear, 27-Aug-2015.)
𝐵 = (Base‘𝑀)       ((𝐻 ∈ (𝑀 MndHom 𝑁) ∧ 𝑊 ∈ Word 𝐵) → (𝐻‘(𝑀 Σg 𝑊)) = (𝑁 Σg (𝐻𝑊)))
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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 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