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Theorem List for Metamath Proof Explorer - 21301-21400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmpfsubrg 21301 Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) (Revised by AV, 19-Sep-2021.)
𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)       ((𝐼𝑉𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (SubRing‘(𝑆s ((Base‘𝑆) ↑m 𝐼))))
 
Theoremmpff 21302 Polynomial functions are functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)    &   𝐵 = (Base‘𝑆)       (𝐹𝑄𝐹:(𝐵m 𝐼)⟶𝐵)
 
Theoremmpfaddcl 21303 The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)    &    + = (+g𝑆)       ((𝐹𝑄𝐺𝑄) → (𝐹f + 𝐺) ∈ 𝑄)
 
Theoremmpfmulcl 21304 The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)    &    · = (.r𝑆)       ((𝐹𝑄𝐺𝑄) → (𝐹f · 𝐺) ∈ 𝑄)
 
Theoremmpfind 21305* Prove a property of polynomials by "structural" induction, under a simplified model of structure which loses the sum of products structure. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝐵 = (Base‘𝑆)    &    + = (+g𝑆)    &    · = (.r𝑆)    &   𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)    &   ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜁)    &   ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜎)    &   (𝑥 = ((𝐵m 𝐼) × {𝑓}) → (𝜓𝜒))    &   (𝑥 = (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑓)) → (𝜓𝜃))    &   (𝑥 = 𝑓 → (𝜓𝜏))    &   (𝑥 = 𝑔 → (𝜓𝜂))    &   (𝑥 = (𝑓f + 𝑔) → (𝜓𝜁))    &   (𝑥 = (𝑓f · 𝑔) → (𝜓𝜎))    &   (𝑥 = 𝐴 → (𝜓𝜌))    &   ((𝜑𝑓𝑅) → 𝜒)    &   ((𝜑𝑓𝐼) → 𝜃)    &   (𝜑𝐴𝑄)       (𝜑𝜌)
 
11.3.3  Additional definitions for (multivariate) polynomials
 
Syntaxcslv 21306 Select a subset of variables in a multivariate polynomial.
class selectVars
 
Syntaxcmhp 21307 Multivariate polynomials.
class mHomP
 
Syntaxcpsd 21308 Power series partial derivative function.
class mPSDer
 
Syntaxcai 21309 Algebraically independent.
class AlgInd
 
Definitiondf-selv 21310* Define the "variable selection" function. The function ((𝐼 selectVars 𝑅)‘𝐽) maps elements of (𝐼 mPoly 𝑅) bijectively onto (𝐽 mPoly ((𝐼𝐽) mPoly 𝑅)) in the natural way, for example if 𝐼 = {𝑥, 𝑦} and 𝐽 = {𝑦} it would map 1 + 𝑥 + 𝑦 + 𝑥𝑦 ∈ ({𝑥, 𝑦} mPoly ℤ) to (1 + 𝑥) + (1 + 𝑥)𝑦 ∈ ({𝑦} mPoly ({𝑥} mPoly ℤ)). This, for example, allows one to treat a multivariate polynomial as a univariate polynomial with coefficients in a polynomial ring with one less variable. (Contributed by Mario Carneiro, 21-Mar-2015.)
selectVars = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ ((𝑖𝑗) mPoly 𝑟) / 𝑢(𝑗 mPoly 𝑢) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝑖 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝑓))‘(𝑥𝑖 ↦ if(𝑥𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝑖𝑗) mVar 𝑟)‘𝑥))))))))
 
Definitiondf-mhp 21311* Define the subspaces of order- 𝑛 homogeneous polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
mHomP = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ∣ (𝑓 supp (0g𝑟)) ⊆ {𝑔 ∈ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑔) = 𝑛}}))
 
Definitiondf-psd 21312* Define the differentiation operation on multivariate polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
mPSDer = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥𝑖 ↦ (𝑓 ∈ (Base‘(𝑖 mPwSer 𝑟)) ↦ (𝑘 ∈ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ↦ (((𝑘𝑥) + 1)(.g𝑟)(𝑓‘(𝑘f + (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)))))))))
 
Definitiondf-algind 21313* Define the predicate "the set 𝑣 is algebraically independent in the algebra 𝑤". A collection of vectors is algebraically independent if no nontrivial polynomial with elements from the subset evaluates to zero. (Contributed by Mario Carneiro, 21-Mar-2015.)
AlgInd = (𝑤 ∈ V, 𝑘 ∈ 𝒫 (Base‘𝑤) ↦ {𝑣 ∈ 𝒫 (Base‘𝑤) ∣ Fun (𝑓 ∈ (Base‘(𝑣 mPoly (𝑤s 𝑘))) ↦ ((((𝑣 evalSub 𝑤)‘𝑘)‘𝑓)‘( I ↾ 𝑣)))})
 
Theoremselvffval 21314* Value of the "variable selection" function. (Contributed by SN, 4-Nov-2023.)
(𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)       (𝜑 → (𝐼 selectVars 𝑅) = (𝑗 ∈ 𝒫 𝐼 ↦ (𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ ((𝐼𝑗) mPoly 𝑅) / 𝑢(𝑗 mPoly 𝑢) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝑓))‘(𝑥𝐼 ↦ if(𝑥𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝑗) mVar 𝑅)‘𝑥))))))))
 
Theoremselvfval 21315* Value of the "variable selection" function. (Contributed by SN, 4-Nov-2023.)
(𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)    &   (𝜑𝐽𝐼)       (𝜑 → ((𝐼 selectVars 𝑅)‘𝐽) = (𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ ((𝐼𝐽) mPoly 𝑅) / 𝑢(𝐽 mPoly 𝑢) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝑓))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥)))))))
 
Theoremselvval 21316* Value of the "variable selection" function. (Contributed by SN, 4-Nov-2023.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑈 = ((𝐼𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   (𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)    &   (𝜑𝐽𝐼)    &   (𝜑𝐹𝐵)       (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) = ((((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
 
Theoremmhpfval 21317* Value of the "homogeneous polynomial" function. (Contributed by Steven Nguyen, 25-Aug-2023.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)       (𝜑𝐻 = (𝑛 ∈ ℕ0 ↦ {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛}}))
 
Theoremmhpval 21318* Value of the "homogeneous polynomial" function. (Contributed by Steven Nguyen, 25-Aug-2023.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐻𝑁) = {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑁}})
 
Theoremismhp 21319* Property of being a homogeneous polynomial. (Contributed by Steven Nguyen, 25-Aug-2023.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝑋 ∈ (𝐻𝑁) ↔ (𝑋𝐵 ∧ (𝑋 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑁})))
 
Theoremismhp2 21320* Deduce a homogeneous polynomial from its properties. (Contributed by SN, 25-May-2024.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋𝐵)    &   (𝜑 → (𝑋 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑁})       (𝜑𝑋 ∈ (𝐻𝑁))
 
Theoremismhp3 21321* A polynomial is homogeneous iff the degree of every nonzero term is the same. (Contributed by SN, 22-Jul-2024.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑋 ∈ (𝐻𝑁) ↔ ∀𝑑𝐷 ((𝑋𝑑) ≠ 0 → ((ℂflds0) Σg 𝑑) = 𝑁)))
 
Theoremmhpmpl 21322 A homogeneous polynomial is a polynomial. (Contributed by Steven Nguyen, 25-Aug-2023.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋 ∈ (𝐻𝑁))       (𝜑𝑋𝐵)
 
Theoremmhpdeg 21323* All nonzero terms of a homogeneous polynomial have degree 𝑁. (Contributed by Steven Nguyen, 25-Aug-2023.)
𝐻 = (𝐼 mHomP 𝑅)    &    0 = (0g𝑅)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋 ∈ (𝐻𝑁))       (𝜑 → (𝑋 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑁})
 
Theoremmhp0cl 21324* The zero polynomial is homogeneous. Under df-mhp 21311, it has any (nonnegative integer) degree which loosely corresponds to the value "undefined". The values -∞ and 0 are also used in Metamath (by df-mdeg 25205 and df-dgr 25340 respectively) and the literature: https://math.stackexchange.com/a/1796314/593843 25340. (Contributed by SN, 12-Sep-2023.)
𝐻 = (𝐼 mHomP 𝑅)    &    0 = (0g𝑅)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Grp)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐷 × { 0 }) ∈ (𝐻𝑁))
 
Theoremmhpsclcl 21325 A scalar (or constant) polynomial has degree 0. Compare deg1scl 25266. In other contexts, there may be an exception for the zero polynomial, but under df-mhp 21311 the zero polynomial can be any degree (see mhp0cl 21324) so there is no exception. (Contributed by SN, 25-May-2024.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝐾 = (Base‘𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐶𝐾)       (𝜑 → (𝐴𝐶) ∈ (𝐻‘0))
 
Theoremmhpvarcl 21326 A power series variable is a polynomial of degree 1. (Contributed by SN, 25-May-2024.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑉 = (𝐼 mVar 𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐼)       (𝜑 → (𝑉𝑋) ∈ (𝐻‘1))
 
Theoremmhpmulcl 21327 A product of homogeneous polynomials is a homogeneous polynomial whose degree is the sum of the degrees of the factors. Compare mdegmulle2 25232 (which shows less-than-or-equal instead of equal). (Contributed by SN, 22-Jul-2024.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑌 = (𝐼 mPoly 𝑅)    &    · = (.r𝑌)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑃 ∈ (𝐻𝑀))    &   (𝜑𝑄 ∈ (𝐻𝑁))       (𝜑 → (𝑃 · 𝑄) ∈ (𝐻‘(𝑀 + 𝑁)))
 
Theoremmhppwdeg 21328 Degree of a homogeneous polynomial raised to a power. General version of deg1pw 25273. (Contributed by SN, 26-Jul-2024.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝑇 = (mulGrp‘𝑃)    &    = (.g𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋 ∈ (𝐻𝑀))       (𝜑 → (𝑁 𝑋) ∈ (𝐻‘(𝑀 · 𝑁)))
 
Theoremmhpaddcl 21329 Homogeneous polynomials are closed under addition. (Contributed by SN, 26-Aug-2023.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &    + = (+g𝑃)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Grp)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋 ∈ (𝐻𝑁))    &   (𝜑𝑌 ∈ (𝐻𝑁))       (𝜑 → (𝑋 + 𝑌) ∈ (𝐻𝑁))
 
Theoremmhpinvcl 21330 Homogeneous polynomials are closed under taking the opposite. (Contributed by SN, 12-Sep-2023.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝑀 = (invg𝑃)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Grp)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋 ∈ (𝐻𝑁))       (𝜑 → (𝑀𝑋) ∈ (𝐻𝑁))
 
Theoremmhpsubg 21331 Homogeneous polynomials form a subgroup of the polynomials. (Contributed by SN, 25-Sep-2023.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Grp)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐻𝑁) ∈ (SubGrp‘𝑃))
 
Theoremmhpvscacl 21332 Homogeneous polynomials are closed under scalar multiplication. (Contributed by SN, 25-Sep-2023.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &    · = ( ·𝑠𝑃)    &   𝐾 = (Base‘𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋𝐾)    &   (𝜑𝐹 ∈ (𝐻𝑁))       (𝜑 → (𝑋 · 𝐹) ∈ (𝐻𝑁))
 
Theoremmhplss 21333 Homogeneous polynomials form a linear subspace of the polynomials. (Contributed by SN, 25-Sep-2023.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐻𝑁) ∈ (LSubSp‘𝑃))
 
11.3.4  Univariate polynomials

According to Wikipedia ("Polynomial", 23-Dec-2019, https://en.wikipedia.org/wiki/Polynomial) "A polynomial in one indeterminate is called a univariate polynomial, a polynomial in more than one indeterminate is called a multivariate polynomial." In this sense univariate polynomials are defined as multivariate polynomials restricted to one indeterminate/polynomial variable in the following, see ply1bascl2 21363.

According to the definition in Wikipedia "a polynomial can either be zero or can be written as the sum of a finite number of nonzero terms. Each term consists of the product of a number - called the coefficient of the term - and a finite number of indeterminates, raised to nonnegative integer powers.". By this, a term of a univariate polynomial (often also called "polynomial term") is the product of a coefficient (usually a member of the underlying ring) and the variable, raised to a nonnegative integer power.

A (univariate) polynomial which has only one term is called (univariate) monomial - therefore, the notions "term" and "monomial" are often used synonymously, see also the definition in [Lang] p. 102. Sometimes, however, a monomial is defined as power product, "a product of powers of variables with nonnegative integer exponents", see Wikipedia ("Monomial", 23-Dec-2019, https://en.wikipedia.org/wiki/Mononomial 21363). In [Lang] p. 101, such terms are called "primitive monomials". To avoid any ambiguity, the notion "primitive monomial" is used for such power products ("x^i") in the following, whereas the synonym for "term" ("ai x^i") will be "scaled monomial".

 
Syntaxcps1 21334 Univariate power series.
class PwSer1
 
Syntaxcv1 21335 The base variable of a univariate power series.
class var1
 
Syntaxcpl1 21336 Univariate polynomials.
class Poly1
 
Syntaxcco1 21337 Coefficient function for a univariate polynomial.
class coe1
 
Syntaxctp1 21338 Convert a univariate polynomial representation to multivariate.
class toPoly1
 
Definitiondf-psr1 21339 Define the algebra of univariate power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅))
 
Definitiondf-vr1 21340 Define the base element of a univariate power series (the 𝑋 element of the set 𝑅[𝑋] of polynomials and also the 𝑋 in the set 𝑅[[𝑋]] of power series). (Contributed by Mario Carneiro, 8-Feb-2015.)
var1 = (𝑟 ∈ V ↦ ((1o mVar 𝑟)‘∅))
 
Definitiondf-ply1 21341 Define the algebra of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1 = (𝑟 ∈ V ↦ ((PwSer1𝑟) ↾s (Base‘(1o mPoly 𝑟))))
 
Definitiondf-coe1 21342* Define the coefficient function for a univariate polynomial. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1 = (𝑓 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (𝑓‘(1o × {𝑛}))))
 
Definitiondf-toply1 21343* Define a function which maps a coefficient function for a univariate polynomial to the corresponding polynomial object. (Contributed by Mario Carneiro, 12-Jun-2015.)
toPoly1 = (𝑓 ∈ V ↦ (𝑛 ∈ (ℕ0m 1o) ↦ (𝑓‘(𝑛‘∅))))
 
Theorempsr1baslem 21344 The set of finite bags on 1o is just the set of all functions from 1o to 0. (Contributed by Mario Carneiro, 9-Feb-2015.)
(ℕ0m 1o) = {𝑓 ∈ (ℕ0m 1o) ∣ (𝑓 “ ℕ) ∈ Fin}
 
Theorempsr1val 21345 Value of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑆 = (PwSer1𝑅)       𝑆 = ((1o ordPwSer 𝑅)‘∅)
 
Theorempsr1crng 21346 The ring of univariate power series is a commutative ring. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑆 = (PwSer1𝑅)       (𝑅 ∈ CRing → 𝑆 ∈ CRing)
 
Theorempsr1assa 21347 The ring of univariate power series is an associative algebra. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑆 = (PwSer1𝑅)       (𝑅 ∈ CRing → 𝑆 ∈ AssAlg)
 
Theorempsr1tos 21348 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 2-Jun-2015.)
𝑆 = (PwSer1𝑅)       (𝑅 ∈ Toset → 𝑆 ∈ Toset)
 
Theorempsr1bas2 21349 The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (PwSer1𝑅)    &   𝐵 = (Base‘𝑆)    &   𝑂 = (1o mPwSer 𝑅)       𝐵 = (Base‘𝑂)
 
Theorempsr1bas 21350 The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑆 = (PwSer1𝑅)    &   𝐵 = (Base‘𝑆)    &   𝐾 = (Base‘𝑅)       𝐵 = (𝐾m (ℕ0m 1o))
 
Theoremvr1val 21351 The value of the generator of the power series algebra (the 𝑋 in 𝑅[[𝑋]]). Since all univariate polynomial rings over a fixed base ring 𝑅 are isomorphic, we don't bother to pass this in as a parameter; internally we are actually using the empty set as this generator and 1o = {∅} is the index set (but for most purposes this choice should not be visible anyway). (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 12-Jun-2015.)
𝑋 = (var1𝑅)       𝑋 = ((1o mVar 𝑅)‘∅)
 
Theoremvr1cl2 21352 The variable 𝑋 is a member of the power series algebra 𝑅[[𝑋]]. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑋 = (var1𝑅)    &   𝑆 = (PwSer1𝑅)    &   𝐵 = (Base‘𝑆)       (𝑅 ∈ Ring → 𝑋𝐵)
 
Theoremply1val 21353 The value of the set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)    &   𝑆 = (PwSer1𝑅)       𝑃 = (𝑆s (Base‘(1o mPoly 𝑅)))
 
Theoremply1bas 21354 The value of the base set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)    &   𝑆 = (PwSer1𝑅)    &   𝑈 = (Base‘𝑃)       𝑈 = (Base‘(1o mPoly 𝑅))
 
Theoremply1lss 21355 Univariate polynomials form a linear subspace of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)    &   𝑆 = (PwSer1𝑅)    &   𝑈 = (Base‘𝑃)       (𝑅 ∈ Ring → 𝑈 ∈ (LSubSp‘𝑆))
 
Theoremply1subrg 21356 Univariate polynomials form a subring of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)    &   𝑆 = (PwSer1𝑅)    &   𝑈 = (Base‘𝑃)       (𝑅 ∈ Ring → 𝑈 ∈ (SubRing‘𝑆))
 
Theoremply1crng 21357 The ring of univariate polynomials is a commutative ring. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ CRing → 𝑃 ∈ CRing)
 
Theoremply1assa 21358 The ring of univariate polynomials is an associative algebra. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
 
Theorempsr1bascl 21359 A univariate power series is a multivariate power series on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝑃 = (PwSer1𝑅)    &   𝐵 = (Base‘𝑃)       (𝐹𝐵𝐹 ∈ (Base‘(1o mPwSer 𝑅)))
 
Theorempsr1basf 21360 Univariate power series base set elements are functions. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝑃 = (PwSer1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)       (𝐹𝐵𝐹:(ℕ0m 1o)⟶𝐾)
 
Theoremply1basf 21361 Univariate polynomial base set elements are functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)       (𝐹𝐵𝐹:(ℕ0m 1o)⟶𝐾)
 
Theoremply1bascl 21362 A univariate polynomial is a univariate power series. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)       (𝐹𝐵𝐹 ∈ (Base‘(PwSer1𝑅)))
 
Theoremply1bascl2 21363 A univariate polynomial is a multivariate polynomial on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)       (𝐹𝐵𝐹 ∈ (Base‘(1o mPoly 𝑅)))
 
Theoremcoe1fval 21364* Value of the univariate polynomial coefficient function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐴 = (coe1𝐹)       (𝐹𝑉𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))
 
Theoremcoe1fv 21365 Value of an evaluated coefficient in a polynomial coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐴 = (coe1𝐹)       ((𝐹𝑉𝑁 ∈ ℕ0) → (𝐴𝑁) = (𝐹‘(1o × {𝑁})))
 
Theoremfvcoe1 21366 Value of a multivariate coefficient in terms of the coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐴 = (coe1𝐹)       ((𝐹𝑉𝑋 ∈ (ℕ0m 1o)) → (𝐹𝑋) = (𝐴‘(𝑋‘∅)))
 
Theoremcoe1fval3 21367* Univariate power series coefficient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (PwSer1𝑅)    &   𝐺 = (𝑦 ∈ ℕ0 ↦ (1o × {𝑦}))       (𝐹𝐵𝐴 = (𝐹𝐺))
 
Theoremcoe1f2 21368 Functionality of univariate power series coefficient vectors. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (PwSer1𝑅)    &   𝐾 = (Base‘𝑅)       (𝐹𝐵𝐴:ℕ0𝐾)
 
Theoremcoe1fval2 21369* Univariate polynomial coefficient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (Poly1𝑅)    &   𝐺 = (𝑦 ∈ ℕ0 ↦ (1o × {𝑦}))       (𝐹𝐵𝐴 = (𝐹𝐺))
 
Theoremcoe1f 21370 Functionality of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (Poly1𝑅)    &   𝐾 = (Base‘𝑅)       (𝐹𝐵𝐴:ℕ0𝐾)
 
Theoremcoe1fvalcl 21371 A coefficient of a univariate polynomial over a class/ring is an element of this class/ring. (Contributed by AV, 9-Oct-2019.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (Poly1𝑅)    &   𝐾 = (Base‘𝑅)       ((𝐹𝐵𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ 𝐾)
 
Theoremcoe1sfi 21372 Finite support of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 19-Jul-2019.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑅)       (𝐹𝐵𝐴 finSupp 0 )
 
Theoremcoe1fsupp 21373* The coefficient vector of a univariate polynomial is a finitely supported mapping from the nonnegative integers to the elements of the coefficient class/ring for the polynomial. (Contributed by AV, 3-Oct-2019.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑅)    &   𝐾 = (Base‘𝑅)       (𝐹𝐵𝐴 ∈ {𝑔 ∈ (𝐾m0) ∣ 𝑔 finSupp 0 })
 
Theoremmptcoe1fsupp 21374* A mapping involving coefficients of polynomials is finitely supported. (Contributed by AV, 12-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑘 ∈ ℕ0 ↦ ((coe1𝑀)‘𝑘)) finSupp 0 )
 
Theoremcoe1ae0 21375* The coefficient vector of a univariate polynomial is 0 almost everywhere. (Contributed by AV, 19-Oct-2019.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑅)       (𝐹𝐵 → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝐴𝑛) = 0 ))
 
Theoremvr1cl 21376 The generator of a univariate polynomial algebra is contained in the base set. (Contributed by Stefan O'Rear, 19-Mar-2015.)
𝑋 = (var1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)       (𝑅 ∈ Ring → 𝑋𝐵)
 
Theoremopsr0 21377 Zero in the ordered power series ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑 → (0g𝑆) = (0g𝑂))
 
Theoremopsr1 21378 One in the ordered power series ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑 → (1r𝑆) = (1r𝑂))
 
Theoremmplplusg 21379 Value of addition in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑌 = (𝐼 mPoly 𝑅)    &   𝑆 = (𝐼 mPwSer 𝑅)    &    + = (+g𝑌)        + = (+g𝑆)
 
Theoremmplmulr 21380 Value of multiplication in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑌 = (𝐼 mPoly 𝑅)    &   𝑆 = (𝐼 mPwSer 𝑅)    &    · = (.r𝑌)        · = (.r𝑆)
 
Theorempsr1plusg 21381 Value of addition in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑌 = (PwSer1𝑅)    &   𝑆 = (1o mPwSer 𝑅)    &    + = (+g𝑌)        + = (+g𝑆)
 
Theorempsr1vsca 21382 Value of scalar multiplication in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑌 = (PwSer1𝑅)    &   𝑆 = (1o mPwSer 𝑅)    &    · = ( ·𝑠𝑌)        · = ( ·𝑠𝑆)
 
Theorempsr1mulr 21383 Value of multiplication in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑌 = (PwSer1𝑅)    &   𝑆 = (1o mPwSer 𝑅)    &    · = (.r𝑌)        · = (.r𝑆)
 
Theoremply1plusg 21384 Value of addition in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
𝑌 = (Poly1𝑅)    &   𝑆 = (1o mPoly 𝑅)    &    + = (+g𝑌)        + = (+g𝑆)
 
Theoremply1vsca 21385 Value of scalar multiplication in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
𝑌 = (Poly1𝑅)    &   𝑆 = (1o mPoly 𝑅)    &    · = ( ·𝑠𝑌)        · = ( ·𝑠𝑆)
 
Theoremply1mulr 21386 Value of multiplication in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
𝑌 = (Poly1𝑅)    &   𝑆 = (1o mPoly 𝑅)    &    · = (.r𝑌)        · = (.r𝑆)
 
Theoremressply1bas2 21387 The base set of a restricted polynomial algebra consists of power series in the subring which are also polynomials (in the parent ring). (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (Poly1𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (Poly1𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑊 = (PwSer1𝐻)    &   𝐶 = (Base‘𝑊)    &   𝐾 = (Base‘𝑆)       (𝜑𝐵 = (𝐶𝐾))
 
Theoremressply1bas 21388 A restricted polynomial algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (Poly1𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (Poly1𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)       (𝜑𝐵 = (Base‘𝑃))
 
Theoremressply1add 21389 A restricted polynomial algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (Poly1𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (Poly1𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(+g𝑈)𝑌) = (𝑋(+g𝑃)𝑌))
 
Theoremressply1mul 21390 A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (Poly1𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (Poly1𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑈)𝑌) = (𝑋(.r𝑃)𝑌))
 
Theoremressply1vsca 21391 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (Poly1𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (Poly1𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)       ((𝜑 ∧ (𝑋𝑇𝑌𝐵)) → (𝑋( ·𝑠𝑈)𝑌) = (𝑋( ·𝑠𝑃)𝑌))
 
Theoremsubrgply1 21392 A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (Poly1𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (Poly1𝐻)    &   𝐵 = (Base‘𝑈)       (𝑇 ∈ (SubRing‘𝑅) → 𝐵 ∈ (SubRing‘𝑆))
 
Theoremgsumply1subr 21393 Evaluate a group sum in a polynomial ring over a subring. (Contributed by AV, 22-Sep-2019.) (Proof shortened by AV, 31-Jan-2020.)
𝑆 = (Poly1𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (Poly1𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → (𝑆 Σg 𝐹) = (𝑈 Σg 𝐹))
 
Theorempsrbaspropd 21394 Property deduction for power series base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(𝜑 → (Base‘𝑅) = (Base‘𝑆))       (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆)))
 
Theorempsrplusgpropd 21395* Property deduction for power series addition. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑𝐵 = (Base‘𝑆))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑅)𝑦) = (𝑥(+g𝑆)𝑦))       (𝜑 → (+g‘(𝐼 mPwSer 𝑅)) = (+g‘(𝐼 mPwSer 𝑆)))
 
Theoremmplbaspropd 21396* Property deduction for polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Jul-2019.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑𝐵 = (Base‘𝑆))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑅)𝑦) = (𝑥(+g𝑆)𝑦))       (𝜑 → (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑆)))
 
Theorempsropprmul 21397 Reversing multiplication in a ring reverses multiplication in the power series ring. (Contributed by Stefan O'Rear, 27-Mar-2015.)
𝑌 = (𝐼 mPwSer 𝑅)    &   𝑆 = (oppr𝑅)    &   𝑍 = (𝐼 mPwSer 𝑆)    &    · = (.r𝑌)    &    = (.r𝑍)    &   𝐵 = (Base‘𝑌)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) = (𝐺 · 𝐹))
 
Theoremply1opprmul 21398 Reversing multiplication in a ring reverses multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 27-Mar-2015.)
𝑌 = (Poly1𝑅)    &   𝑆 = (oppr𝑅)    &   𝑍 = (Poly1𝑆)    &    · = (.r𝑌)    &    = (.r𝑍)    &   𝐵 = (Base‘𝑌)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) = (𝐺 · 𝐹))
 
Theorem00ply1bas 21399 Lemma for ply1basfvi 21400 and deg1fvi 25238. (Contributed by Stefan O'Rear, 28-Mar-2015.)
∅ = (Base‘(Poly1‘∅))
 
Theoremply1basfvi 21400 Protection compatibility of the univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(Base‘(Poly1𝑅)) = (Base‘(Poly1‘( I ‘𝑅)))
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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 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