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Theorem List for Metamath Proof Explorer - 17401-17500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsubmmnd 17401 Submonoids are themselves monoids under the given operation. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐻 = (𝑀s 𝑆)       (𝑆 ∈ (SubMnd‘𝑀) → 𝐻 ∈ Mnd)

Theoremsubmbas 17402 The base set of a submonoid. (Contributed by Stefan O'Rear, 15-Jun-2015.)
𝐻 = (𝑀s 𝑆)       (𝑆 ∈ (SubMnd‘𝑀) → 𝑆 = (Base‘𝐻))

Theoremsubm0 17403 Submonoids have the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐻 = (𝑀s 𝑆)    &    0 = (0g𝑀)       (𝑆 ∈ (SubMnd‘𝑀) → 0 = (0g𝐻))

Theoremsubsubm 17404 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)))

Theorem0mhm 17405 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 17406 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 17407 One direction of resmhm2b 17408. (Contributed by Mario Carneiro, 18-Jun-2015.)
𝑈 = (𝑇s 𝑋)       ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑇))

Theoremresmhm2b 17408 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 17409 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 17410 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 17411 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 17412 Submonoids are an algebraic closure system. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))

Theoremmrcmndind 17413* (( From SO's determinants branch )). TODO: Appropriate description to be added! (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 17414* 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 17415* 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 17416* 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 17417* 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 17418* 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  Ordered 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 16150. If order is not significant, it is simpler to use families instead.

Theoremgsumvallem2 17419* 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 17420 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 17421* 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 17422 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 17423 A singleton composite recovers the initial symbol. (Contributed by Stefan O'Rear, 16-Aug-2015.)
𝐵 = (Base‘𝐺)       (𝑆𝐵 → (𝐺 Σg ⟨“𝑆”⟩) = 𝑆)

Theoremgsumwcl 17424 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)

Theoremgsumccat 17425 Homomorphic property of composites. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵𝑋 ∈ Word 𝐵) → (𝐺 Σg (𝑊 ++ 𝑋)) = ((𝐺 Σg 𝑊) + (𝐺 Σg 𝑋)))

Theoremgsumws2 17426 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 17427 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 17428 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 17429 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 17430* 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 17431 Extend class definition with the free monoid construction.
class freeMnd

Syntaxcvrmd 17432 Extend class notation with free monoid injection.
class varFMnd

Definitiondf-frmd 17433 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 17434* 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 17435 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 17436 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 17437 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 17438 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 17439 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 17440* 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 17441 The value of the generating elements of a free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
𝑈 = (varFMnd𝐼)       ((𝐼𝑉𝐴𝐼) → (𝑈𝐴) = ⟨“𝐴”⟩)

Theoremvrmdf 17442 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 17443 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 17444 The identity of the free monoid is the empty word. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑀 = (freeMnd‘𝐼)       ∅ = (0g𝑀)

Theoremfrmdsssubm 17445 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 17446 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 17447 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 17448* 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 17449* 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 17450* Lemma for frmdup3 17451. (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 17451* 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)

10.1.9  Examples and counterexamples for magmas, semigroups and monoids

Theoremmgm2nsgrplem1 17452* Lemma 1 for mgm2nsgrp 17456: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 17301). (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 17453* Lemma 2 for mgm2nsgrp 17456. (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 17454* Lemma 3 for mgm2nsgrp 17456. (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 17455* Lemma 4 for mgm2nsgrp 17456: 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 ∉ SGrp)

Theoremmgm2nsgrp 17456* 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 ∉ SGrp))

Theoremsgrp2nmndlem1 17457* Lemma 1 for sgrp2nmnd 17464: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 17301). (Contributed by AV, 29-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))       ((𝐴𝑉𝐵𝑊) → 𝑀 ∈ Mgm)

Theoremsgrp2nmndlem2 17458* Lemma 2 for sgrp2nmnd 17464. (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 17459* Lemma 3 for sgrp2nmnd 17464. (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 17460* 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 17461* 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 17462* Lemma 4 for sgrp2nmnd 17464: 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 ∈ SGrp)

Theoremsgrp2nmndlem5 17463* Lemma 5 for sgrp2nmnd 17464: 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 17464* 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 ∈ SGrp ∧ 𝑀 ∉ Mnd))

Theoremmgmnsgrpex 17465 There is a magma which is not a semigroup. (Contributed by AV, 29-Jan-2020.)
𝑚 ∈ Mgm 𝑚 ∉ SGrp

Theoremsgrpnmndex 17466 There is a semigroup which is not a monoid. (Contributed by AV, 29-Jan-2020.)
𝑚 ∈ SGrp 𝑚 ∉ Mnd

Theoremsgrpssmgm 17467 The class of all semigroups is a proper subclass of the class of all magmas. (Contributed by AV, 29-Jan-2020.)
SGrp ⊊ Mgm

Theoremmndsssgrp 17468 The class of all monoids is a proper subclass of the class of all semigroups. (Contributed by AV, 29-Jan-2020.)
Mnd ⊊ SGrp

10.2  Groups

10.2.1  Definition and basic properties

Syntaxcgrp 17469 Extend class notation with class of all groups.
class Grp

Syntaxcminusg 17470 Extend class notation with inverse of group element.
class invg

Syntaxcsg 17471 Extend class notation with group subtraction (or division) operation.
class -g

Definitiondf-grp 17472* Define class of all groups. A group is a monoid (df-mnd 17342) 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 15910) and an internal group operation (notated (+g𝐺) per df-plusg 16001). 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 17477), associativity (so ((𝑎+g𝑏)+g𝑐) = (𝑎+g(𝑏+g𝑐)) for any a, b, c, see grpass 17478), 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 17480). Groups need not be commutative; a commutative group is an Abelian group (see df-abl 18242). Subgroups can often be formed from groups, see df-subg 17638. An example of an (Abelian) group is the set of complex numbers over the group operation + (addition), as proven in cnaddablx 18317; an Abelian group is a group as proven in ablgrp 18244. Other structures include groups, including unital rings (df-ring 18595) and fields (df-field 18798). (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 17473* 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 17474* 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 17475* 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 ))

Theoremgrpmnd 17476 A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
(𝐺 ∈ Grp → 𝐺 ∈ Mnd)

Theoremgrpcl 17477 Closure of the operation of a group. (Contributed by NM, 14-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)

Theoremgrpass 17478 A group operation is associative. (Contributed by NM, 14-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))

Theoremgrpinvex 17479* Every member of a group has a left inverse. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 0 )

Theoremgrpideu 17480* The two-sided identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 8-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       (𝐺 ∈ Grp → ∃!𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))

Theoremgrpplusf 17481 The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
𝐵 = (Base‘𝐺)    &   𝐹 = (+𝑓𝐺)       (𝐺 ∈ Grp → 𝐹:(𝐵 × 𝐵)⟶𝐵)

Theoremgrpplusfo 17482 The group addition operation is a function onto the base set/set of group elements. (Contributed by NM, 30-Oct-2006.) (Revised by AV, 30-Aug-2021.)
𝐵 = (Base‘𝐺)    &   𝐹 = (+𝑓𝐺)       (𝐺 ∈ Grp → 𝐹:(𝐵 × 𝐵)–onto𝐵)

Theoremresgrpplusfrn 17483 The underlying set of a group operation which is a restriction of a structure. (Contributed by Paul Chapman, 25-Mar-2008.) (Revised by AV, 30-Aug-2021.)
𝐵 = (Base‘𝐺)    &   𝐻 = (𝐺s 𝑆)    &   𝐹 = (+𝑓𝐻)       ((𝐻 ∈ Grp ∧ 𝑆𝐵) → 𝑆 = ran 𝐹)

Theoremgrppropd 17484* If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))       (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))

Theoremgrpprop 17485 If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by NM, 11-Oct-2013.)
(Base‘𝐾) = (Base‘𝐿)    &   (+g𝐾) = (+g𝐿)       (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)

Theoremgrppropstr 17486 Generalize a specific 2-element group 𝐿 to show that any set 𝐾 with the same (relevant) properties is also a group. (Contributed by NM, 28-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
(Base‘𝐾) = 𝐵    &   (+g𝐾) = +    &   𝐿 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}       (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)

Theoremgrpss 17487 Show that a structure extending a constructed group (e.g., a ring) is also a group. This allows us to prove that a constructed potential ring 𝑅 is a group before we know that it is also a ring. (Theorem ringgrp 18598, on the other hand, requires that we know in advance that 𝑅 is a ring.) (Contributed by NM, 11-Oct-2013.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    &   𝑅 ∈ V    &   𝐺𝑅    &   Fun 𝑅       (𝐺 ∈ Grp ↔ 𝑅 ∈ Grp)

Theoremisgrpd2e 17488* Deduce a group from its properties. In this version of isgrpd2 17489, we don't assume there is an expression for the inverse of 𝑥. (Contributed by NM, 10-Aug-2013.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑+ = (+g𝐺))    &   (𝜑0 = (0g𝐺))    &   (𝜑𝐺 ∈ Mnd)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 0 )       (𝜑𝐺 ∈ Grp)

Theoremisgrpd2 17489* Deduce a group from its properties. 𝑁 (negative) is normally dependent on 𝑥 i.e. read it as 𝑁(𝑥). Note: normally we don't use a 𝜑 antecedent on hypotheses that name structure components, since they can be eliminated with eqid 2651, but we make an exception for theorems such as isgrpd2 17489, ismndd 17360, and islmodd 18917 since theorems using them often rewrite the structure components. (Contributed by NM, 10-Aug-2013.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑+ = (+g𝐺))    &   (𝜑0 = (0g𝐺))    &   (𝜑𝐺 ∈ Mnd)    &   ((𝜑𝑥𝐵) → 𝑁𝐵)    &   ((𝜑𝑥𝐵) → (𝑁 + 𝑥) = 0 )       (𝜑𝐺 ∈ Grp)

Theoremisgrpde 17490* Deduce a group from its properties. In this version of isgrpd 17491, we don't assume there is an expression for the inverse of 𝑥. (Contributed by NM, 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) → ∃𝑦𝐵 (𝑦 + 𝑥) = 0 )       (𝜑𝐺 ∈ Grp)

Theoremisgrpd 17491* Deduce a group from its properties. Unlike isgrpd2 17489, this one goes straight from the base properties rather than going through Mnd. 𝑁 (negative) is normally dependent on 𝑥 i.e. read it as 𝑁(𝑥). (Contributed by NM, 6-Jun-2013.) (Revised 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) → 𝑁𝐵)    &   ((𝜑𝑥𝐵) → (𝑁 + 𝑥) = 0 )       (𝜑𝐺 ∈ Grp)

Theoremisgrpi 17492* Properties that determine a group. 𝑁 (negative) is normally dependent on 𝑥 i.e. read it as 𝑁(𝑥). (Contributed by NM, 3-Sep-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ((𝑥𝐵𝑦𝐵𝑧𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &    0𝐵    &   (𝑥𝐵 → ( 0 + 𝑥) = 𝑥)    &   (𝑥𝐵𝑁𝐵)    &   (𝑥𝐵 → (𝑁 + 𝑥) = 0 )       𝐺 ∈ Grp

Theoremgrpsgrp 17493 A group is a semigroup. (Contributed by AV, 28-Aug-2021.)
(𝐺 ∈ Grp → 𝐺 ∈ SGrp)

Theoremdfgrp2 17494* Alternate definition of a group as semigroup with a left identity and a left inverse for each element. This "definition" is weaker than df-grp 17472, based on the definition of a monoid which provides a left and a right identity. (Contributed by AV, 28-Aug-2021.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ Grp ↔ (𝐺 ∈ SGrp ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)))

Theoremdfgrp2e 17495* Alternate definition of a group as a set with a closed, associative operation, a left identity and a left inverse for each element. Alternate definition in [Lang] p. 7. (Contributed by NM, 10-Oct-2006.) (Revised by AV, 28-Aug-2021.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ Grp ↔ (∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)))

Theoremisgrpix 17496* Properties that determine a group. Read 𝑁 as 𝑁(𝑥). Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use. (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.)
𝐵 ∈ V    &    + ∈ V    &   𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}    &   ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ((𝑥𝐵𝑦𝐵𝑧𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &    0𝐵    &   (𝑥𝐵 → ( 0 + 𝑥) = 𝑥)    &   (𝑥𝐵𝑁𝐵)    &   (𝑥𝐵 → (𝑁 + 𝑥) = 0 )       𝐺 ∈ Grp

Theoremgrpidcl 17497 The identity element of a group belongs to the group. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)       (𝐺 ∈ Grp → 0𝐵)

Theoremgrpbn0 17498 The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ Grp → 𝐵 ≠ ∅)

Theoremgrplid 17499 The identity element of a group is a left identity. (Contributed by NM, 18-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ( 0 + 𝑋) = 𝑋)

Theoremgrprid 17500 The identity element of a group is a right identity. (Contributed by NM, 18-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + 0 ) = 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 42801-42879
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