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Theorem List for Metamath Proof Explorer - 18001-18100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremgsumwmhm 18001 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)))

Theoremgsumwspan 18002* The submonoid generated by a set of elements is precisely the set of elements which can be expressed as finite products of the generator. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝐵 = (Base‘𝑀)    &   𝐾 = (mrCls‘(SubMnd‘𝑀))       ((𝑀 ∈ Mnd ∧ 𝐺𝐵) → (𝐾𝐺) = ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))

10.1.8  Free monoids

Syntaxcfrmd 18003 Extend class definition with the free monoid construction.
class freeMnd

Syntaxcvrmd 18004 Extend class notation with free monoid injection.
class varFMnd

Definitiondf-frmd 18005 Define a free monoid over a set 𝑖 of generators, defined as the set of finite strings on 𝐼 with the operation of concatenation. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd = (𝑖 ∈ V ↦ {⟨(Base‘ndx), Word 𝑖⟩, ⟨(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))⟩})

Definitiondf-vrmd 18006* Define a free monoid over a set 𝑖 of generators, defined as the set of finite strings on 𝐼 with the operation of concatenation. (Contributed by Mario Carneiro, 27-Sep-2015.)
varFMnd = (𝑖 ∈ V ↦ (𝑗𝑖 ↦ ⟨“𝑗”⟩))

Theoremfrmdval 18007 Value of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑀 = (freeMnd‘𝐼)    &   (𝐼𝑉𝐵 = Word 𝐼)    &    + = ( ++ ↾ (𝐵 × 𝐵))       (𝐼𝑉𝑀 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩})

Theoremfrmdbas 18008 The base set of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝑀 = (freeMnd‘𝐼)    &   𝐵 = (Base‘𝑀)       (𝐼𝑉𝐵 = Word 𝐼)

Theoremfrmdelbas 18009 An element of the base set of a free monoid is a string on the generators. (Contributed by Mario Carneiro, 27-Feb-2016.)
𝑀 = (freeMnd‘𝐼)    &   𝐵 = (Base‘𝑀)       (𝑋𝐵𝑋 ∈ Word 𝐼)

Theoremfrmdplusg 18010 The monoid operation of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝑀 = (freeMnd‘𝐼)    &   𝐵 = (Base‘𝑀)    &    + = (+g𝑀)        + = ( ++ ↾ (𝐵 × 𝐵))

Theoremfrmdadd 18011 Value of the monoid operation of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑀 = (freeMnd‘𝐼)    &   𝐵 = (Base‘𝑀)    &    + = (+g𝑀)       ((𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑋 ++ 𝑌))

Theoremvrmdfval 18012* The canonical injection from the generating set 𝐼 to the base set of the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
𝑈 = (varFMnd𝐼)       (𝐼𝑉𝑈 = (𝑗𝐼 ↦ ⟨“𝑗”⟩))

Theoremvrmdval 18013 The value of the generating elements of a free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
𝑈 = (varFMnd𝐼)       ((𝐼𝑉𝐴𝐼) → (𝑈𝐴) = ⟨“𝐴”⟩)

Theoremvrmdf 18014 The mapping from the index set to the generators is a function into the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
𝑈 = (varFMnd𝐼)       (𝐼𝑉𝑈:𝐼⟶Word 𝐼)

Theoremfrmdmnd 18015 A free monoid is a monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝑀 = (freeMnd‘𝐼)       (𝐼𝑉𝑀 ∈ Mnd)

Theoremfrmd0 18016 The identity of the free monoid is the empty word. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑀 = (freeMnd‘𝐼)       ∅ = (0g𝑀)

Theoremfrmdsssubm 18017 The set of words taking values in a subset is a (free) submonoid of the free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝑀 = (freeMnd‘𝐼)       ((𝐼𝑉𝐽𝐼) → Word 𝐽 ∈ (SubMnd‘𝑀))

Theoremfrmdgsum 18018 Any word in a free monoid can be expressed as the sum of the singletons composing it. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑀 = (freeMnd‘𝐼)    &   𝑈 = (varFMnd𝐼)       ((𝐼𝑉𝑊 ∈ Word 𝐼) → (𝑀 Σg (𝑈𝑊)) = 𝑊)

Theoremfrmdss2 18019 A subset of generators is contained in a submonoid iff the set of words on the generators is in the submonoid. This can be viewed as an elementary way of saying "the monoidal closure of 𝐽 is Word 𝐽". (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑀 = (freeMnd‘𝐼)    &   𝑈 = (varFMnd𝐼)       ((𝐼𝑉𝐽𝐼𝐴 ∈ (SubMnd‘𝑀)) → ((𝑈𝐽) ⊆ 𝐴 ↔ Word 𝐽𝐴))

Theoremfrmdup1 18020* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑀 = (freeMnd‘𝐼)    &   𝐵 = (Base‘𝐺)    &   𝐸 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴𝑥)))    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐼𝑋)    &   (𝜑𝐴:𝐼𝐵)       (𝜑𝐸 ∈ (𝑀 MndHom 𝐺))

Theoremfrmdup2 18021* The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑀 = (freeMnd‘𝐼)    &   𝐵 = (Base‘𝐺)    &   𝐸 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴𝑥)))    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐼𝑋)    &   (𝜑𝐴:𝐼𝐵)    &   𝑈 = (varFMnd𝐼)    &   (𝜑𝑌𝐼)       (𝜑 → (𝐸‘(𝑈𝑌)) = (𝐴𝑌))

Theoremfrmdup3lem 18022* Lemma for frmdup3 18023. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑀 = (freeMnd‘𝐼)    &   𝐵 = (Base‘𝐺)    &   𝑈 = (varFMnd𝐼)       (((𝐺 ∈ Mnd ∧ 𝐼𝑉𝐴:𝐼𝐵) ∧ (𝐹 ∈ (𝑀 MndHom 𝐺) ∧ (𝐹𝑈) = 𝐴)) → 𝐹 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴𝑥))))

Theoremfrmdup3 18023* Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 18-Jul-2016.)
𝑀 = (freeMnd‘𝐼)    &   𝐵 = (Base‘𝐺)    &   𝑈 = (varFMnd𝐼)       ((𝐺 ∈ Mnd ∧ 𝐼𝑉𝐴:𝐼𝐵) → ∃!𝑚 ∈ (𝑀 MndHom 𝐺)(𝑚𝑈) = 𝐴)

10.1.8.1  Monoid of endofunctions

According to Wikipedia ("Endomorphism", 25-Jan-2024, https://en.wikipedia.org/wiki/Endomorphism) "An endofunction is a function whose domain is equal to its codomain.". An endofunction is sometimes also called "self-mapping" (see https://www.wikidata.org/wiki/Q1691962) or "self-map" (see https://mathworld.wolfram.com/Self-Map.html), in German "Selbstabbildung" (see https://de.wikipedia.org/wiki/Selbstabbildung).

Syntaxcefmnd 18024 Extend class notation to include the class of monoids of endofunctions.
class EndoFMnd

Definitiondf-efmnd 18025* Define the monoid of endofunctions on set 𝑥. We represent the monoid as the set of functions from 𝑥 to itself ((𝑥m 𝑥)) under function composition, and topologize it as a function space assuming the set is discrete. Analogous to the former definition of SymGrp, see df-symg 18487 and symgvalstruct 18516. (Contributed by AV, 25-Jan-2024.)
EndoFMnd = (𝑥 ∈ V ↦ (𝑥m 𝑥) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑥 × {𝒫 𝑥}))⟩})

Theoremefmnd 18026* The monoid of endofunctions on set 𝐴. (Contributed by AV, 25-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (𝐴m 𝐴)    &    + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔))    &   𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))       (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})

Theoremefmndbas 18027 The base set of the monoid of endofunctions on class 𝐴. (Contributed by AV, 25-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)       𝐵 = (𝐴m 𝐴)

Theoremefmndbasabf 18028* The base set of the monoid of endofunctions on class 𝐴 is the set of functions from 𝐴 into itself. (Contributed by AV, 29-Mar-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)       𝐵 = {𝑓𝑓:𝐴𝐴}

Theoremelefmndbas 18029 Two ways of saying a function is a mapping of 𝐴 to itself. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴𝑉 → (𝐹𝐵𝐹:𝐴𝐴))

Theoremelefmndbas2 18030 Two ways of saying a function is a mapping of 𝐴 to itself. (Contributed by AV, 27-Jan-2024.) (Proof shortened by AV, 29-Mar-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐹𝑉 → (𝐹𝐵𝐹:𝐴𝐴))

Theoremefmndbasf 18031 Elements in the monoid of endofunctions on 𝐴 are functions from 𝐴 into itself. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐹𝐵𝐹:𝐴𝐴)

Theoremefmndhash 18032 The monoid of endofunctions on 𝑛 objects has cardinality 𝑛𝑛. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴 ∈ Fin → (♯‘𝐵) = ((♯‘𝐴)↑(♯‘𝐴)))

Theoremefmndbasfi 18033 The monoid of endofunctions on a finite set 𝐴 is finite. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴 ∈ Fin → 𝐵 ∈ Fin)

Theoremefmndfv 18034 The function value of an endofunction. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)       ((𝐹𝐵𝑋𝐴) → (𝐹𝑋) ∈ 𝐴)

Theoremefmndtset 18035 The topology of the monoid of endofunctions on 𝐴. This component is defined on a larger set than the true base - the product topology is defined on the set of all functions, not just endofunctions - but the definition of TopOpen ensures that it is trimmed down before it gets use. (Contributed by AV, 25-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)       (𝐴𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) = (TopSet‘𝐺))

Theoremefmndplusg 18036* The group operation of a monoid of endofunctions is the function composition. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    &    + = (+g𝐺)        + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔))

Theoremefmndov 18037 The value of the group operation of the monoid of endofunctions on 𝐴. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑋𝑌))

Theoremefmndcl 18038 The group operation of the monoid of endofunctions on 𝐴 is closed. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)

Theoremefmndtopn 18039 The topology of the monoid of endofunctions on 𝐴. (Contributed by AV, 31-Jan-2024.)
𝐺 = (EndoFMnd‘𝑋)    &   𝐵 = (Base‘𝐺)       (𝑋𝑉 → ((∏t‘(𝑋 × {𝒫 𝑋})) ↾t 𝐵) = (TopOpen‘𝐺))

Theoremsymggrplem 18040* Lemma for symggrp 18519 and efmndsgrp 18042. Conditions for an operation to be associative. Formerly part of proof for symggrp 18519. (Contributed by AV, 28-Jan-2024.)
((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑥𝑦))       ((𝑋𝐵𝑌𝐵𝑍𝐵) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))

Theoremefmndmgm 18041 The monoid of endofunctions on a class 𝐴 is a magma. (Contributed by AV, 28-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)       𝐺 ∈ Mgm

Theoremefmndsgrp 18042 The monoid of endofunctions on a class 𝐴 is a semigroup. (Contributed by AV, 28-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)       𝐺 ∈ Smgrp

Theoremielefmnd 18043 The identity function restricted to a set 𝐴 is an element of the base set of the monoid of endofunctions on 𝐴. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)       (𝐴𝑉 → ( I ↾ 𝐴) ∈ (Base‘𝐺))

Theoremefmndid 18044 The identity function restricted to a set 𝐴 is the identity element of the monoid of endofunctions on 𝐴. (Contributed by AV, 25-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)       (𝐴𝑉 → ( I ↾ 𝐴) = (0g𝐺))

Theoremefmndmnd 18045 The monoid of endofunctions on a set 𝐴 is actually a monoid. (Contributed by AV, 31-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)       (𝐴𝑉𝐺 ∈ Mnd)

Theoremefmnd0nmnd 18046 Even the monoid of endofunctions on the empty set is actually a monoid. (Contributed by AV, 31-Jan-2024.)
(EndoFMnd‘∅) ∈ Mnd

Theoremefmndbas0 18047 The base set of the monoid of endofunctions on the empty set is the singleton containing the empty set. (Contributed by AV, 27-Jan-2024.) (Proof shortened by AV, 31-Mar-2024.)
(Base‘(EndoFMnd‘∅)) = {∅}

Theoremefmnd1hash 18048 The monoid of endofunctions on a singleton has cardinality 1. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    &   𝐴 = {𝐼}       (𝐼𝑉 → (♯‘𝐵) = 1)

Theoremefmnd1bas 18049 The monoid of endofunctions on a singleton consists of the identity only. (Contributed by AV, 31-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    &   𝐴 = {𝐼}       (𝐼𝑉𝐵 = {{⟨𝐼, 𝐼⟩}})

Theoremefmnd2hash 18050 The monoid of endofunctions on a (proper) pair has cardinality 4. (Contributed by AV, 18-Feb-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    &   𝐴 = {𝐼, 𝐽}       ((𝐼𝑉𝐽𝑊𝐼𝐽) → (♯‘𝐵) = 4)

Theoremsubmefmnd 18051* If the base set of a monoid is contained in the base set of the monoid of endofunctions on a set 𝐴, contains the identity function and has the function composition as group operation, then its base set is a submonoid of the monoid of endofunctions on set 𝐴. Analogous to pgrpsubgsymg 18528. (Contributed by AV, 17-Feb-2024.)
𝑀 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝑀)    &    0 = (0g𝑀)    &   𝐹 = (Base‘𝑆)       (𝐴𝑉 → (((𝑆 ∈ Mnd ∧ 𝐹𝐵0𝐹) ∧ (+g𝑆) = (𝑓𝐹, 𝑔𝐹 ↦ (𝑓𝑔))) → 𝐹 ∈ (SubMnd‘𝑀)))

Theoremsursubmefmnd 18052* The set of surjective endofunctions on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. (Contributed by AV, 25-Feb-2024.)
𝑀 = (EndoFMnd‘𝐴)       (𝐴𝑉 → {:𝐴onto𝐴} ∈ (SubMnd‘𝑀))

Theoreminjsubmefmnd 18053* The set of injective endofunctions on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. (Contributed by AV, 25-Feb-2024.)
𝑀 = (EndoFMnd‘𝐴)       (𝐴𝑉 → {:𝐴1-1𝐴} ∈ (SubMnd‘𝑀))

Theoremidressubmefmnd 18054 The singleton containing only the identity function restricted to a set is a submonoid of the monoid of endofunctions on this set. (Contributed by AV, 17-Feb-2024.)
𝐺 = (EndoFMnd‘𝐴)       (𝐴𝑉 → {( I ↾ 𝐴)} ∈ (SubMnd‘𝐺))

Theoremidresefmnd 18055 The structure with the singleton containing only the identity function restricted to a set 𝐴 as base set and the function composition as group operation, constructed by (structure) restricting the monoid of endofunctions on 𝐴 to that singleton, is a monoid whose base set is a subset of the base set of the monoid of endofunctions on 𝐴. (Contributed by AV, 17-Feb-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐸 = (𝐺s {( I ↾ 𝐴)})       (𝐴𝑉 → (𝐸 ∈ Mnd ∧ (Base‘𝐸) ⊆ (Base‘𝐺)))

Theoremsmndex1ibas 18056 The modulo function 𝐼 is an endofunction on 0. (Contributed by AV, 12-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))       𝐼 ∈ (Base‘𝑀)

Theoremsmndex1iidm 18057* The modulo function 𝐼 is idempotent. (Contributed by AV, 12-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))       (𝐼𝐼) = 𝐼

Theoremsmndex1gbas 18058* The constant functions (𝐺𝐾) are endofunctions on 0. (Contributed by AV, 12-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))       (𝐾 ∈ (0..^𝑁) → (𝐺𝐾) ∈ (Base‘𝑀))

Theoremsmndex1gid 18059* The composition of a constant function (𝐺𝐾) with another endofunction on 0 results in (𝐺𝐾) itself. (Contributed by AV, 14-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))       ((𝐹 ∈ (Base‘𝑀) ∧ 𝐾 ∈ (0..^𝑁)) → ((𝐺𝐾) ∘ 𝐹) = (𝐺𝐾))

Theoremsmndex1igid 18060* The composition of the modulo function 𝐼 and a constant function (𝐺𝐾) results in (𝐺𝐾) itself. (Contributed by AV, 14-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))       (𝐾 ∈ (0..^𝑁) → (𝐼 ∘ (𝐺𝐾)) = (𝐺𝐾))

Theoremsmndex1basss 18061* The modulo function 𝐼 and the constant functions (𝐺𝐾) are endofunctions on 0. (Contributed by AV, 12-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))    &   𝐵 = ({𝐼} ∪ 𝑛 ∈ (0..^𝑁){(𝐺𝑛)})       𝐵 ⊆ (Base‘𝑀)

Theoremsmndex1bas 18062* The base set of the monoid of endofunctions on 0 restricted to the modulo function 𝐼 and the constant functions (𝐺𝐾). (Contributed by AV, 12-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))    &   𝐵 = ({𝐼} ∪ 𝑛 ∈ (0..^𝑁){(𝐺𝑛)})    &   𝑆 = (𝑀s 𝐵)       (Base‘𝑆) = 𝐵

Theoremsmndex1mgm 18063* The monoid of endofunctions on 0 restricted to the modulo function 𝐼 and the constant functions (𝐺𝐾) is a magma. (Contributed by AV, 14-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))    &   𝐵 = ({𝐼} ∪ 𝑛 ∈ (0..^𝑁){(𝐺𝑛)})    &   𝑆 = (𝑀s 𝐵)       𝑆 ∈ Mgm

Theoremsmndex1sgrp 18064* The monoid of endofunctions on 0 restricted to the modulo function 𝐼 and the constant functions (𝐺𝐾) is a semigroup. (Contributed by AV, 14-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))    &   𝐵 = ({𝐼} ∪ 𝑛 ∈ (0..^𝑁){(𝐺𝑛)})    &   𝑆 = (𝑀s 𝐵)       𝑆 ∈ Smgrp

Theoremsmndex1mndlem 18065* Lemma for smndex1mnd 18066 and smndex1id 18067. (Contributed by AV, 16-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))    &   𝐵 = ({𝐼} ∪ 𝑛 ∈ (0..^𝑁){(𝐺𝑛)})    &   𝑆 = (𝑀s 𝐵)       (𝑋𝐵 → ((𝐼𝑋) = 𝑋 ∧ (𝑋𝐼) = 𝑋))

Theoremsmndex1mnd 18066* The monoid of endofunctions on 0 restricted to the modulo function 𝐼 and the constant functions (𝐺𝐾) is a monoid. (Contributed by AV, 16-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))    &   𝐵 = ({𝐼} ∪ 𝑛 ∈ (0..^𝑁){(𝐺𝑛)})    &   𝑆 = (𝑀s 𝐵)       𝑆 ∈ Mnd

Theoremsmndex1id 18067* The modulo function 𝐼 is the identity of the monoid of endofunctions on 0 restricted to the modulo function 𝐼 and the constant functions (𝐺𝐾). (Contributed by AV, 16-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))    &   𝐵 = ({𝐼} ∪ 𝑛 ∈ (0..^𝑁){(𝐺𝑛)})    &   𝑆 = (𝑀s 𝐵)       𝐼 = (0g𝑆)

Theoremsmndex1n0mnd 18068* The identity of the monoid 𝑀 of endofunctions on set 0 is not contained in the base set of the constructed monoid 𝑆. (Contributed by AV, 17-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))    &   𝐵 = ({𝐼} ∪ 𝑛 ∈ (0..^𝑁){(𝐺𝑛)})    &   𝑆 = (𝑀s 𝐵)       (0g𝑀) ∉ 𝐵

Theoremnsmndex1 18069* The base set 𝐵 of the constructed monoid 𝑆 is not a submonoid of the monoid 𝑀 of endofunctions on set 0, although 𝑀 ∈ Mnd and 𝑆 ∈ Mnd and 𝐵 ⊆ (Base‘𝑀) hold. (Contributed by AV, 17-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))    &   𝐵 = ({𝐼} ∪ 𝑛 ∈ (0..^𝑁){(𝐺𝑛)})    &   𝑆 = (𝑀s 𝐵)       𝐵 ∉ (SubMnd‘𝑀)

Theoremsmndex2dbas 18070 The doubling function 𝐷 is an endofunction on 0. (Contributed by AV, 18-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝐵 = (Base‘𝑀)    &    0 = (0g𝑀)    &   𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥))       𝐷𝐵

Theoremsmndex2dnrinv 18071 The doubling function 𝐷 has no right inverse in the monoid of endofunctions on 0. (Contributed by AV, 18-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝐵 = (Base‘𝑀)    &    0 = (0g𝑀)    &   𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥))       𝑓𝐵 (𝐷𝑓) ≠ 0

Theoremsmndex2hbas 18072 The halving functions 𝐻 are endofunctions on 0. (Contributed by AV, 18-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝐵 = (Base‘𝑀)    &    0 = (0g𝑀)    &   𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥))    &   𝑁 ∈ ℕ0    &   𝐻 = (𝑥 ∈ ℕ0 ↦ if(2 ∥ 𝑥, (𝑥 / 2), 𝑁))       𝐻𝐵

Theoremsmndex2dlinvh 18073* The halving functions 𝐻 are left inverses of the doubling function 𝐷. (Contributed by AV, 18-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝐵 = (Base‘𝑀)    &    0 = (0g𝑀)    &   𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥))    &   𝑁 ∈ ℕ0    &   𝐻 = (𝑥 ∈ ℕ0 ↦ if(2 ∥ 𝑥, (𝑥 / 2), 𝑁))       (𝐻𝐷) = 0

10.1.9  Examples and counterexamples for magmas, semigroups and monoids

Theoremmgm2nsgrplem1 18074* Lemma 1 for mgm2nsgrp 18078: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 17856). (Contributed by AV, 27-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))       ((𝐴𝑉𝐵𝑊) → 𝑀 ∈ Mgm)

Theoremmgm2nsgrplem2 18075* Lemma 2 for mgm2nsgrp 18078. (Contributed by AV, 27-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))    &    = (+g𝑀)       ((𝐴𝑉𝐵𝑊) → ((𝐴 𝐴) 𝐵) = 𝐴)

Theoremmgm2nsgrplem3 18076* Lemma 3 for mgm2nsgrp 18078. (Contributed by AV, 28-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))    &    = (+g𝑀)       ((𝐴𝑉𝐵𝑊) → (𝐴 (𝐴 𝐵)) = 𝐵)

Theoremmgm2nsgrplem4 18077* Lemma 4 for mgm2nsgrp 18078: M is not a semigroup. (Contributed by AV, 28-Jan-2020.) (Proof shortened by AV, 31-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))       ((♯‘𝑆) = 2 → 𝑀 ∉ Smgrp)

Theoremmgm2nsgrp 18078* A small magma (with two elements) which is not a semigroup. (Contributed by AV, 28-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))       ((♯‘𝑆) = 2 → (𝑀 ∈ Mgm ∧ 𝑀 ∉ Smgrp))

Theoremsgrp2nmndlem1 18079* Lemma 1 for sgrp2nmnd 18086: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 17856). (Contributed by AV, 29-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))       ((𝐴𝑉𝐵𝑊) → 𝑀 ∈ Mgm)

Theoremsgrp2nmndlem2 18080* Lemma 2 for sgrp2nmnd 18086. (Contributed by AV, 29-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    &    = (+g𝑀)       ((𝐴𝑆𝐶𝑆) → (𝐴 𝐶) = 𝐴)

Theoremsgrp2nmndlem3 18081* Lemma 3 for sgrp2nmnd 18086. (Contributed by AV, 29-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    &    = (+g𝑀)       ((𝐶𝑆𝐵𝑆𝐴𝐵) → (𝐵 𝐶) = 𝐵)

Theoremsgrp2rid2 18082* A small semigroup (with two elements) with two right identities which are different if 𝐴𝐵. (Contributed by AV, 10-Feb-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    &    = (+g𝑀)       ((𝐴𝑉𝐵𝑊) → ∀𝑥𝑆𝑦𝑆 (𝑦 𝑥) = 𝑦)

Theoremsgrp2rid2ex 18083* A small semigroup (with two elements) with two right identities which are different. (Contributed by AV, 10-Feb-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    &    = (+g𝑀)       ((♯‘𝑆) = 2 → ∃𝑥𝑆𝑧𝑆𝑦𝑆 (𝑥𝑧 ∧ (𝑦 𝑥) = 𝑦 ∧ (𝑦 𝑧) = 𝑦))

Theoremsgrp2nmndlem4 18084* Lemma 4 for sgrp2nmnd 18086: M is a semigroup. (Contributed by AV, 29-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))       ((♯‘𝑆) = 2 → 𝑀 ∈ Smgrp)

Theoremsgrp2nmndlem5 18085* Lemma 5 for sgrp2nmnd 18086: M is not a monoid. (Contributed by AV, 29-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))       ((♯‘𝑆) = 2 → 𝑀 ∉ Mnd)

Theoremsgrp2nmnd 18086* A small semigroup (with two elements) which is not a monoid. (Contributed by AV, 26-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))       ((♯‘𝑆) = 2 → (𝑀 ∈ Smgrp ∧ 𝑀 ∉ Mnd))

Theoremmgmnsgrpex 18087 There is a magma which is not a semigroup. (Contributed by AV, 29-Jan-2020.)
𝑚 ∈ Mgm 𝑚 ∉ Smgrp

Theoremsgrpnmndex 18088 There is a semigroup which is not a monoid. (Contributed by AV, 29-Jan-2020.)
𝑚 ∈ Smgrp 𝑚 ∉ Mnd

Theoremsgrpssmgm 18089 The class of all semigroups is a proper subclass of the class of all magmas. (Contributed by AV, 29-Jan-2020.)
Smgrp ⊊ Mgm

Theoremmndsssgrp 18090 The class of all monoids is a proper subclass of the class of all semigroups. (Contributed by AV, 29-Jan-2020.)
Mnd ⊊ Smgrp

Theorempwmndgplus 18091* The operation of the monoid of the power set of a class 𝐴 under union. (Contributed by AV, 27-Feb-2024.)
(Base‘𝑀) = 𝒫 𝐴    &   (+g𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥𝑦))       ((𝑋 ∈ 𝒫 𝐴𝑌 ∈ 𝒫 𝐴) → (𝑋(+g𝑀)𝑌) = (𝑋𝑌))

Theorempwmndid 18092* The identity of the monoid of the power set of a class 𝐴 under union is the empty set. (Contributed by AV, 27-Feb-2024.)
(Base‘𝑀) = 𝒫 𝐴    &   (+g𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥𝑦))       (0g𝑀) = ∅

Theorempwmnd 18093* The power set of a class 𝐴 is a monoid under union. (Contributed by AV, 27-Feb-2024.)
(Base‘𝑀) = 𝒫 𝐴    &   (+g𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥𝑦))       𝑀 ∈ Mnd

10.2  Groups

10.2.1  Definition and basic properties

Syntaxcgrp 18094 Extend class notation with class of all groups.
class Grp

Syntaxcminusg 18095 Extend class notation with inverse of group element.
class invg

Syntaxcsg 18096 Extend class notation with group subtraction (or division) operation.
class -g

Definitiondf-grp 18097* Define class of all groups. A group is a monoid (df-mnd 17903) whose internal operation is such that every element admits a left inverse (which can be proven to be a two-sided inverse). Thus, a group 𝐺 is an algebraic structure formed from a base set of elements (notated (Base‘𝐺) per df-base 16480) and an internal group operation (notated (+g𝐺) per df-plusg 16569). The operation combines any two elements of the group base set and must satisfy the 4 group axioms: closure (the result of the group operation must always be a member of the base set, see grpcl 18102), associativity (so ((𝑎+g𝑏)+g𝑐) = (𝑎+g(𝑏+g𝑐)) for any a, b, c, see grpass 18103), identity (there must be an element 𝑒 = (0g𝐺) such that 𝑒+g𝑎 = 𝑎+g𝑒 = 𝑎 for any a), and inverse (for each element a in the base set, there must be an element 𝑏 = invg𝑎 in the base set such that 𝑎+g𝑏 = 𝑏+g𝑎 = 𝑒). It can be proven that the identity element is unique (grpideu 18105). Groups need not be commutative; a commutative group is an Abelian group (see df-abl 18900). Subgroups can often be formed from groups, see df-subg 18267. An example of an (Abelian) group is the set of complex numbers over the group operation + (addition), as proven in cnaddablx 18979; an Abelian group is a group as proven in ablgrp 18902. Other structures include groups, including unital rings (df-ring 19290) and fields (df-field 19496). (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔)}

Definitiondf-minusg 18098* Define inverse of group element. (Contributed by NM, 24-Aug-2011.)
invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (𝑤 ∈ (Base‘𝑔)(𝑤(+g𝑔)𝑥) = (0g𝑔))))

Definitiondf-sbg 18099* Define group subtraction (also called division for multiplicative groups). (Contributed by NM, 31-Mar-2014.)
-g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦))))

Theoremisgrp 18100* The predicate "is a group." (This theorem demonstrates the use of symbols as variable names, first proposed by FL in 2010.) (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑎𝐵𝑚𝐵 (𝑚 + 𝑎) = 0 ))

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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 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45272
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