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Theorem List for Metamath Proof Explorer - 20201-20300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmplsca 20201 The scalar field of a multivariate polynomial structure. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)       (𝜑𝑅 = (Scalar‘𝑃))

Theoremmplvsca2 20202 The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝑆 = (𝐼 mPwSer 𝑅)    &    · = ( ·𝑠𝑃)        · = ( ·𝑠𝑆)

Theoremmplvsca 20203* The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &    = ( ·𝑠𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝑃)    &    · = (.r𝑅)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝑋𝐾)    &   (𝜑𝐹𝐵)       (𝜑 → (𝑋 𝐹) = ((𝐷 × {𝑋}) ∘f · 𝐹))

Theoremmplvscaval 20204* The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &    = ( ·𝑠𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝑃)    &    · = (.r𝑅)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝑋𝐾)    &   (𝜑𝐹𝐵)    &   (𝜑𝑌𝐷)       (𝜑 → ((𝑋 𝐹)‘𝑌) = (𝑋 · (𝐹𝑌)))

Theoremmvrcl 20205 A power series variable is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝑉 = (𝐼 mVar 𝑅)    &   𝐵 = (Base‘𝑃)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐼)       (𝜑 → (𝑉𝑋) ∈ 𝐵)

Theoremmplgrp 20206 The polynomial ring is a group. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)       ((𝐼𝑉𝑅 ∈ Grp) → 𝑃 ∈ Grp)

Theoremmpllmod 20207 The polynomial ring is a left module. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)       ((𝐼𝑉𝑅 ∈ Ring) → 𝑃 ∈ LMod)

Theoremmplring 20208 The polynomial ring is a ring. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)       ((𝐼𝑉𝑅 ∈ Ring) → 𝑃 ∈ Ring)

Theoremmpllvec 20209 The polynomial ring is a vector space. (Contributed by SN, 29-Feb-2024.)
𝑃 = (𝐼 mPoly 𝑅)       ((𝐼𝑉𝑅 ∈ DivRing) → 𝑃 ∈ LVec)

Theoremmplcrng 20210 The polynomial ring is a commutative ring. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)       ((𝐼𝑉𝑅 ∈ CRing) → 𝑃 ∈ CRing)

Theoremmplassa 20211 The polynomial ring is an associative algebra. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)       ((𝐼𝑉𝑅 ∈ CRing) → 𝑃 ∈ AssAlg)

Theoremressmplbas2 20212 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 = (𝐼 mPoly 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝐼𝑉)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑊 = (𝐼 mPwSer 𝐻)    &   𝐶 = (Base‘𝑊)    &   𝐾 = (Base‘𝑆)       (𝜑𝐵 = (𝐶𝐾))

Theoremressmplbas 20213 A restricted polynomial algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPoly 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝐼𝑉)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)       (𝜑𝐵 = (Base‘𝑃))

Theoremressmpladd 20214 A restricted polynomial algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPoly 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝐼𝑉)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(+g𝑈)𝑌) = (𝑋(+g𝑃)𝑌))

Theoremressmplmul 20215 A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPoly 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝐼𝑉)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑈)𝑌) = (𝑋(.r𝑃)𝑌))

Theoremressmplvsca 20216 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPoly 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝐼𝑉)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)       ((𝜑 ∧ (𝑋𝑇𝑌𝐵)) → (𝑋( ·𝑠𝑈)𝑌) = (𝑋( ·𝑠𝑃)𝑌))

Theoremsubrgmpl 20217 A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPoly 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)       ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ∈ (SubRing‘𝑆))

Theoremsubrgmvr 20218 The variables in a subring polynomial algebra are the same as the original ring. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑉 = (𝐼 mVar 𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝐻 = (𝑅s 𝑇)       (𝜑𝑉 = (𝐼 mVar 𝐻))

Theoremsubrgmvrf 20219 The variables in a polynomial algebra are contained in every subring algebra. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑉 = (𝐼 mVar 𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)       (𝜑𝑉:𝐼𝐵)

Theoremmplmon 20220* A monomial is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐷)       (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵)

Theoremmplmonmul 20221* The product of two monomials adds the exponent vectors together. For example, the product of (𝑥↑2)(𝑦↑2) with (𝑦↑1)(𝑧↑3) is (𝑥↑2)(𝑦↑3)(𝑧↑3), where the exponent vectors ⟨2, 2, 0⟩ and ⟨0, 1, 3⟩ are added to give ⟨2, 3, 3⟩. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐷)    &    · = (.r𝑃)    &   (𝜑𝑌𝐷)       (𝜑 → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) · (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))) = (𝑦𝐷 ↦ if(𝑦 = (𝑋f + 𝑌), 1 , 0 )))

Theoremmplcoe1 20222* Decompose a polynomial into a finite sum of monomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   (𝜑𝐼𝑊)    &   𝐵 = (Base‘𝑃)    &    · = ( ·𝑠𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑𝑋 = (𝑃 Σg (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))

Theoremmplcoe3 20223* Decompose a monomial in one variable into a power of a variable. (Contributed by Mario Carneiro, 7-Jan-2015.) (Proof shortened by AV, 18-Jul-2019.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   (𝜑𝐼𝑊)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝑉 = (𝐼 mVar 𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐼)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑘𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 )) = (𝑁 (𝑉𝑋)))

Theoremmplcoe5lem 20224* Lemma for mplcoe4 20259. (Contributed by AV, 7-Oct-2019.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   (𝜑𝐼𝑊)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝑉 = (𝐼 mVar 𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑌𝐷)    &   (𝜑 → ∀𝑥𝐼𝑦𝐼 ((𝑉𝑦)(+g𝐺)(𝑉𝑥)) = ((𝑉𝑥)(+g𝐺)(𝑉𝑦)))    &   (𝜑𝑆𝐼)       (𝜑 → ran (𝑘𝑆 ↦ ((𝑌𝑘) (𝑉𝑘))) ⊆ ((Cntz‘𝐺)‘ran (𝑘𝑆 ↦ ((𝑌𝑘) (𝑉𝑘)))))

Theoremmplcoe5 20225* Decompose a monomial into a finite product of powers of variables. Instead of assuming that 𝑅 is a commutative ring (as in mplcoe2 20226), it is sufficient that 𝑅 is a ring and all the variables of the multivariate polynomial commute. (Contributed by AV, 7-Oct-2019.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   (𝜑𝐼𝑊)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝑉 = (𝐼 mVar 𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑌𝐷)    &   (𝜑 → ∀𝑥𝐼𝑦𝐼 ((𝑉𝑦)(+g𝐺)(𝑉𝑥)) = ((𝑉𝑥)(+g𝐺)(𝑉𝑦)))       (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝐺 Σg (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘)))))

Theoremmplcoe2 20226* Decompose a monomial into a finite product of powers of variables. (The assumption that 𝑅 is a commutative ring is not strictly necessary, because the submonoid of monomials is in the center of the multiplicative monoid of polynomials, but it simplifies the proof.) (Contributed by Mario Carneiro, 10-Jan-2015.) (Proof shortened by AV, 18-Oct-2019.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   (𝜑𝐼𝑊)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝑉 = (𝐼 mVar 𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐷)       (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝐺 Σg (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘)))))

Theoremmplbas2 20227 An alternative expression for the set of polynomials, as the smallest subalgebra of the set of power series that contains all the variable generators. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝑆 = (𝐼 mPwSer 𝑅)    &   𝑉 = (𝐼 mVar 𝑅)    &   𝐴 = (AlgSpan‘𝑆)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ CRing)       (𝜑 → (𝐴‘ran 𝑉) = (Base‘𝑃))

Theoremltbval 20228* Value of the well-order on finite bags. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝐶 = (𝑇 <bag 𝐼)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑉)    &   (𝜑𝑇𝑊)       (𝜑𝐶 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐷 ∧ ∃𝑧𝐼 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐼 (𝑧𝑇𝑤 → (𝑥𝑤) = (𝑦𝑤))))})

Theoremltbwe 20229* The finite bag order is a well-order, given a well-order of the index set. (Contributed by Mario Carneiro, 2-Jun-2015.)
𝐶 = (𝑇 <bag 𝐼)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑉)    &   (𝜑𝑇𝑊)    &   (𝜑𝑇 We 𝐼)       (𝜑𝐶 We 𝐷)

Theoremreldmopsr 20230 Lemma for ordered power series. (Contributed by Stefan O'Rear, 2-Oct-2015.)
Rel dom ordPwSer

Theoremopsrval 20231* The value of the "ordered power series" function. This is the same as mPwSer psrval 20118, but with the addition of a well-order on 𝐼 we can turn a strict order on 𝑅 into a strict order on the power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   𝐵 = (Base‘𝑆)    &    < = (lt‘𝑅)    &   𝐶 = (𝑇 <bag 𝐼)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}    &   (𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑𝑂 = (𝑆 sSet ⟨(le‘ndx), ⟩))

Theoremopsrle 20232* An alternative expression for the set of polynomials, as the smallest subalgebra of the set of power series that contains all the variable generators. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   𝐵 = (Base‘𝑆)    &    < = (lt‘𝑅)    &   𝐶 = (𝑇 <bag 𝐼)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &    = (le‘𝑂)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))})

Theoremopsrval2 20233 Self-referential expression for the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &    = (le‘𝑂)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑𝑂 = (𝑆 sSet ⟨(le‘ndx), ⟩))

Theoremopsrbaslem 20234 Get a component of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 9-Sep-2021.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 < 10       (𝜑 → (𝐸𝑆) = (𝐸𝑂))

Theoremopsrbas 20235 The base set of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑 → (Base‘𝑆) = (Base‘𝑂))

Theoremopsrplusg 20236 The addition operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑 → (+g𝑆) = (+g𝑂))

Theoremopsrmulr 20237 The multiplication operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑 → (.r𝑆) = (.r𝑂))

Theoremopsrvsca 20238 The scalar product operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑 → ( ·𝑠𝑆) = ( ·𝑠𝑂))

Theoremopsrsca 20239 The scalar ring of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))    &   (𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)       (𝜑𝑅 = (Scalar‘𝑂))

Theoremopsrtoslem1 20240* Lemma for opsrtos 20242. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Toset)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))    &   (𝜑𝑇 We 𝐼)    &   𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    < = (lt‘𝑅)    &   𝐶 = (𝑇 <bag 𝐼)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜓 ↔ ∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))))    &    = (le‘𝑂)       (𝜑 = (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∩ (𝐵 × 𝐵)) ∪ ( I ↾ 𝐵)))

Theoremopsrtoslem2 20241* Lemma for opsrtos 20242. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Toset)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))    &   (𝜑𝑇 We 𝐼)    &   𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    < = (lt‘𝑅)    &   𝐶 = (𝑇 <bag 𝐼)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜓 ↔ ∃𝑧𝐷 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝐷 (𝑤𝐶𝑧 → (𝑥𝑤) = (𝑦𝑤))))    &    = (le‘𝑂)       (𝜑𝑂 ∈ Toset)

Theoremopsrtos 20242 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Toset)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))    &   (𝜑𝑇 We 𝐼)       (𝜑𝑂 ∈ Toset)

Theoremopsrso 20243 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Toset)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))    &   (𝜑𝑇 We 𝐼)    &    = (lt‘𝑂)    &   𝐵 = (Base‘𝑂)       (𝜑 Or 𝐵)

Theoremopsrcrng 20244 The ring of ordered power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑𝑂 ∈ CRing)

Theoremopsrassa 20245 The ring of ordered power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑𝑂 ∈ AssAlg)

Theoremmplrcl 20246 Reverse closure for the polynomial index set. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)       (𝑋𝐵𝐼 ∈ V)

Theoremmplelsfi 20247 A polynomial treated as a coefficient function has finitely many nonzero terms. (Contributed by Stefan O'Rear, 22-Mar-2015.) (Revised by AV, 25-Jun-2019.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &   (𝜑𝐹𝐵)    &   (𝜑𝑅𝑉)       (𝜑𝐹 finSupp 0 )

Theoremmvrf2 20248 The power series/polynomial variable function maps indices to polynomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝑉 = (𝐼 mVar 𝑅)    &   𝐵 = (Base‘𝑃)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)       (𝜑𝑉:𝐼𝐵)

Theoremmplmon2 20249* Express a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &    · = ( ·𝑠𝑃)    &   𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐵 = (Base‘𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐾𝐷)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑋 · (𝑦𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 )))

Theorempsrbag0 20250* The empty bag is a bag. (Contributed by Stefan O'Rear, 9-Mar-2015.)
𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}       (𝐼𝑉 → (𝐼 × {0}) ∈ 𝐷)

Theorempsrbagsn 20251* A singleton bag is a bag. (Contributed by Stefan O'Rear, 9-Mar-2015.)
𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}       (𝐼𝑉 → (𝑥𝐼 ↦ if(𝑥 = 𝐾, 1, 0)) ∈ 𝐷)

Theoremmplascl 20252* Value of the scalar injection into the polynomial algebra. (Contributed by Stefan O'Rear, 9-Mar-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &   𝐵 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐴𝑋) = (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 𝑋, 0 )))

Theoremmplasclf 20253 The scalar injection is a function into the polynomial algebra. (Contributed by Stefan O'Rear, 9-Mar-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)       (𝜑𝐴:𝐾𝐵)

Theoremsubrgascl 20254 The scalar injection function in a subring algebra is the same up to a restriction to the subring. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   (𝜑𝐼𝑊)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝐶 = (algSc‘𝑈)       (𝜑𝐶 = (𝐴𝑇))

Theoremsubrgasclcl 20255 The scalars in a polynomial algebra are in the subring algebra iff the scalar value is in the subring. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   (𝜑𝐼𝑊)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝐵 = (Base‘𝑈)    &   𝐾 = (Base‘𝑅)    &   (𝜑𝑋𝐾)       (𝜑 → ((𝐴𝑋) ∈ 𝐵𝑋𝑇))

Theoremmplmon2cl 20256* A scaled monomial is a polynomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &   𝐶 = (Base‘𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑃)    &   (𝜑𝑋𝐶)    &   (𝜑𝐾𝐷)       (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 )) ∈ 𝐵)

Theoremmplmon2mul 20257* Product of scaled monomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &   𝐶 = (Base‘𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ CRing)    &    = (.r𝑃)    &    · = (.r𝑅)    &   (𝜑𝑋𝐷)    &   (𝜑𝑌𝐷)    &   (𝜑𝐹𝐶)    &   (𝜑𝐺𝐶)       (𝜑 → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 𝐹, 0 )) (𝑦𝐷 ↦ if(𝑦 = 𝑌, 𝐺, 0 ))) = (𝑦𝐷 ↦ if(𝑦 = (𝑋f + 𝑌), (𝐹 · 𝐺), 0 )))

Theoremmplind 20258* Prove a property of polynomials by "structural" induction, under a simplified model of structure which loses the sum of products structure. The commutativity condition is stronger than strictly needed. (Contributed by Stefan O'Rear, 11-Mar-2015.)
𝐾 = (Base‘𝑅)    &   𝑉 = (𝐼 mVar 𝑅)    &   𝑌 = (𝐼 mPoly 𝑅)    &    + = (+g𝑌)    &    · = (.r𝑌)    &   𝐶 = (algSc‘𝑌)    &   𝐵 = (Base‘𝑌)    &   ((𝜑 ∧ (𝑥𝐻𝑦𝐻)) → (𝑥 + 𝑦) ∈ 𝐻)    &   ((𝜑 ∧ (𝑥𝐻𝑦𝐻)) → (𝑥 · 𝑦) ∈ 𝐻)    &   ((𝜑𝑥𝐾) → (𝐶𝑥) ∈ 𝐻)    &   ((𝜑𝑥𝐼) → (𝑉𝑥) ∈ 𝐻)    &   (𝜑𝑋𝐵)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ CRing)       (𝜑𝑋𝐻)

Theoremmplcoe4 20259* Decompose a polynomial into a finite sum of scaled monomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &   𝐵 = (Base‘𝑃)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑𝑋 = (𝑃 Σg (𝑘𝐷 ↦ (𝑦𝐷 ↦ if(𝑦 = 𝑘, (𝑋𝑘), 0 )))))

10.9.2  Polynomial evaluation

Syntaxces 20260 Evaluation of a multivariate polynomial in a subring.
class evalSub

Syntaxcevl 20261 Evaluation of a multivariate polynomial.
class eval

Definitiondf-evls 20262* Define the evaluation map for the polynomial algebra. The function ((𝐼 evalSub 𝑆)‘𝑅):𝑉⟶(𝑆m (𝑆m 𝐼)) makes sense when 𝐼 is an index set, 𝑆 is a ring, 𝑅 is a subring of 𝑆, and where 𝑉 is the set of polynomials in (𝐼 mPoly 𝑅). This function maps an element of the formal polynomial algebra (with coefficients in 𝑅) to a function from assignments 𝐼𝑆 of the variables to elements of 𝑆 formed by evaluating the polynomial with the given assignments. (Contributed by Stefan O'Rear, 11-Mar-2015.)
evalSub = (𝑖 ∈ V, 𝑠 ∈ CRing ↦ (Base‘𝑠) / 𝑏(𝑟 ∈ (SubRing‘𝑠) ↦ (𝑖 mPoly (𝑠s 𝑟)) / 𝑤(𝑓 ∈ (𝑤 RingHom (𝑠s (𝑏m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥)))))))

Definitiondf-evl 20263* A simplification of evalSub when the evaluation ring is the same as the coefficient ring. (Contributed by Stefan O'Rear, 19-Mar-2015.)
eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))

Theoremevlslem4 20264* The support of a tensor product of ring element families is contained in the product of the supports. (Contributed by Stefan O'Rear, 8-Mar-2015.) (Revised by AV, 18-Jul-2019.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   ((𝜑𝑥𝐼) → 𝑋𝐵)    &   ((𝜑𝑦𝐽) → 𝑌𝐵)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)       (𝜑 → ((𝑥𝐼, 𝑦𝐽 ↦ (𝑋 · 𝑌)) supp 0 ) ⊆ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 )))

Theorempsrbagfsupp 20265* Finite bags have finite nonzero-support. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.)
𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}       ((𝑋𝐷𝐼𝑉) → 𝑋 finSupp 0)

Theorempsrbagev1 20266* A bag of multipliers provides the conditions for a valid sum. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) (Revised by AV, 11-Apr-2024.)
𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   𝐶 = (Base‘𝑇)    &    · = (.g𝑇)    &    0 = (0g𝑇)    &   (𝜑𝑇 ∈ CMnd)    &   (𝜑𝐵𝐷)    &   (𝜑𝐺:𝐼𝐶)    &   (𝜑𝐼𝑊)       (𝜑 → ((𝐵f · 𝐺):𝐼𝐶 ∧ (𝐵f · 𝐺) finSupp 0 ))

Theorempsrbagev2 20267* Closure of a sum using a bag of multipliers. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Proof shortened by AV, 18-Jul-2019.) (Revised by AV, 11-Apr-2024.)
𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   𝐶 = (Base‘𝑇)    &    · = (.g𝑇)    &   (𝜑𝑇 ∈ CMnd)    &   (𝜑𝐵𝐷)    &   (𝜑𝐺:𝐼𝐶)    &   (𝜑𝐼𝑊)       (𝜑 → (𝑇 Σg (𝐵f · 𝐺)) ∈ 𝐶)

Theoremevlslem2 20268* A linear function on the polynomial ring which is multiplicative on scaled monomials is generally multiplicative. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 11-Apr-2024.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &    · = (.r𝑆)    &    0 = (0g𝑅)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝐸 ∈ (𝑃 GrpHom 𝑆))    &   ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑗𝐷𝑖𝐷))) → (𝐸‘(𝑘𝐷 ↦ if(𝑘 = (𝑗f + 𝑖), ((𝑥𝑗)(.r𝑅)(𝑦𝑖)), 0 ))) = ((𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) · (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))       ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸‘(𝑥(.r𝑃)𝑦)) = ((𝐸𝑥) · (𝐸𝑦)))

Theoremevlslem3 20269* Lemma for evlseu 20272. Polynomial evaluation of a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.) (Revised by AV, 11-Apr-2024.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐶 = (Base‘𝑆)    &   𝐾 = (Base‘𝑅)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   𝑇 = (mulGrp‘𝑆)    &    = (.g𝑇)    &    · = (.r𝑆)    &   𝑉 = (𝐼 mVar 𝑅)    &   𝐸 = (𝑝𝐵 ↦ (𝑆 Σg (𝑏𝐷 ↦ ((𝐹‘(𝑝𝑏)) · (𝑇 Σg (𝑏f 𝐺))))))    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝐹 ∈ (𝑅 RingHom 𝑆))    &   (𝜑𝐺:𝐼𝐶)    &    0 = (0g𝑅)    &   (𝜑𝐴𝐷)    &   (𝜑𝐻𝐾)       (𝜑 → (𝐸‘(𝑥𝐷 ↦ if(𝑥 = 𝐴, 𝐻, 0 ))) = ((𝐹𝐻) · (𝑇 Σg (𝐴f 𝐺))))

Theoremevlslem6 20270* Lemma for evlseu 20272. Finiteness and consistency of the top-level sum. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 26-Jul-2019.) (Revised by AV, 11-Apr-2024.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐶 = (Base‘𝑆)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   𝑇 = (mulGrp‘𝑆)    &    = (.g𝑇)    &    · = (.r𝑆)    &   𝑉 = (𝐼 mVar 𝑅)    &   𝐸 = (𝑝𝐵 ↦ (𝑆 Σg (𝑏𝐷 ↦ ((𝐹‘(𝑝𝑏)) · (𝑇 Σg (𝑏f 𝐺))))))    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝐹 ∈ (𝑅 RingHom 𝑆))    &   (𝜑𝐺:𝐼𝐶)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑏𝐷 ↦ ((𝐹‘(𝑌𝑏)) · (𝑇 Σg (𝑏f 𝐺)))):𝐷𝐶 ∧ (𝑏𝐷 ↦ ((𝐹‘(𝑌𝑏)) · (𝑇 Σg (𝑏f 𝐺)))) finSupp (0g𝑆)))

Theoremevlslem1 20271* Lemma for evlseu 20272, give a formula for (the unique) polynomial evaluation homomorphism. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Proof shortened by AV, 26-Jul-2019.) (Revised by AV, 11-Apr-2024.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐶 = (Base‘𝑆)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   𝑇 = (mulGrp‘𝑆)    &    = (.g𝑇)    &    · = (.r𝑆)    &   𝑉 = (𝐼 mVar 𝑅)    &   𝐸 = (𝑝𝐵 ↦ (𝑆 Σg (𝑏𝐷 ↦ ((𝐹‘(𝑝𝑏)) · (𝑇 Σg (𝑏f 𝐺))))))    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝐹 ∈ (𝑅 RingHom 𝑆))    &   (𝜑𝐺:𝐼𝐶)    &   𝐴 = (algSc‘𝑃)       (𝜑 → (𝐸 ∈ (𝑃 RingHom 𝑆) ∧ (𝐸𝐴) = 𝐹 ∧ (𝐸𝑉) = 𝐺))

Theoremevlseu 20272* For a given interpretation of the variables 𝐺 and of the scalars 𝐹, this extends to a homomorphic interpretation of the polynomial ring in exactly one way. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 11-Apr-2024.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐶 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑃)    &   𝑉 = (𝐼 mVar 𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝐹 ∈ (𝑅 RingHom 𝑆))    &   (𝜑𝐺:𝐼𝐶)       (𝜑 → ∃!𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺))

Theoremreldmevls 20273 Well-behaved binary operation property of evalSub. (Contributed by Stefan O'Rear, 19-Mar-2015.)
Rel dom evalSub

Theoremmpfrcl 20274 Reverse closure for the set of polynomial functions. (Contributed by Stefan O'Rear, 19-Mar-2015.)
𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)       (𝑋𝑄 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))

Theoremevlsval 20275* Value of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 11-Mar-2015.) (Revised by AV, 18-Sep-2021.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑈)    &   𝑉 = (𝐼 mVar 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝑇 = (𝑆s (𝐵m 𝐼))    &   𝐵 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑊)    &   𝑋 = (𝑥𝑅 ↦ ((𝐵m 𝐼) × {𝑥}))    &   𝑌 = (𝑥𝐼 ↦ (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)))       ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = (𝑓 ∈ (𝑊 RingHom 𝑇)((𝑓𝐴) = 𝑋 ∧ (𝑓𝑉) = 𝑌)))

Theoremevlsval2 20276* Characterizing properties of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 12-Mar-2015.) (Revised by AV, 18-Sep-2021.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑈)    &   𝑉 = (𝐼 mVar 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝑇 = (𝑆s (𝐵m 𝐼))    &   𝐵 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑊)    &   𝑋 = (𝑥𝑅 ↦ ((𝐵m 𝐼) × {𝑥}))    &   𝑌 = (𝑥𝐼 ↦ (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)))       ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑄 ∈ (𝑊 RingHom 𝑇) ∧ ((𝑄𝐴) = 𝑋 ∧ (𝑄𝑉) = 𝑌)))

Theoremevlsrhm 20277 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Stefan O'Rear, 12-Mar-2015.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝑇 = (𝑆s (𝐵m 𝐼))    &   𝐵 = (Base‘𝑆)       ((𝐼𝑉𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇))

Theoremevlssca 20278 Polynomial evaluation maps scalars to constant functions. (Contributed by Stefan O'Rear, 13-Mar-2015.) (Proof shortened by AV, 18-Sep-2021.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑊)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝑅)       (𝜑 → (𝑄‘(𝐴𝑋)) = ((𝐵m 𝐼) × {𝑋}))

Theoremevlsvar 20279* Polynomial evaluation maps variables to projections. (Contributed by Stefan O'Rear, 12-Mar-2015.) (Proof shortened by AV, 18-Sep-2021.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑉 = (𝐼 mVar 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝐼)       (𝜑 → (𝑄‘(𝑉𝑋)) = (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑋)))

Theoremevlsgsumadd 20280* Polynomial evaluation maps (additive) group sums to group sums. (Contributed by SN, 13-Feb-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑈)    &    0 = (0g𝑊)    &   𝑈 = (𝑆s 𝑅)    &   𝑃 = (𝑆s (𝐾m 𝐼))    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   ((𝜑𝑥𝑁) → 𝑌𝐵)    &   (𝜑𝑁 ⊆ ℕ0)    &   (𝜑 → (𝑥𝑁𝑌) finSupp 0 )       (𝜑 → (𝑄‘(𝑊 Σg (𝑥𝑁𝑌))) = (𝑃 Σg (𝑥𝑁 ↦ (𝑄𝑌))))

Theoremevlsgsummul 20281* Polynomial evaluation maps (multiplicative) group sums to group sums. (Contributed by SN, 13-Feb-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑈)    &   𝐺 = (mulGrp‘𝑊)    &    1 = (1r𝑊)    &   𝑈 = (𝑆s 𝑅)    &   𝑃 = (𝑆s (𝐾m 𝐼))    &   𝐻 = (mulGrp‘𝑃)    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   ((𝜑𝑥𝑁) → 𝑌𝐵)    &   (𝜑𝑁 ⊆ ℕ0)    &   (𝜑 → (𝑥𝑁𝑌) finSupp 1 )       (𝜑 → (𝑄‘(𝐺 Σg (𝑥𝑁𝑌))) = (𝐻 Σg (𝑥𝑁 ↦ (𝑄𝑌))))

Theoremevlspw 20282 Polynomial evaluation for subrings maps the exponentiation of a polynomial to the exponentiation of the evaluated polynomial. (Contributed by SN, 29-Feb-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑈)    &   𝐺 = (mulGrp‘𝑊)    &    = (.g𝐺)    &   𝑈 = (𝑆s 𝑅)    &   𝑃 = (𝑆s (𝐾m 𝐼))    &   𝐻 = (mulGrp‘𝑃)    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑄‘(𝑁 𝑋)) = (𝑁(.g𝐻)(𝑄𝑋)))

Theoremevlsvarpw 20283 Polynomial evaluation for subrings maps the exponentiation of a variable to the exponentiation of the evaluated variable. (Contributed by SN, 21-Feb-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑈)    &   𝐺 = (mulGrp‘𝑊)    &    = (.g𝐺)    &   𝑋 = ((𝐼 mVar 𝑈)‘𝑌)    &   𝑈 = (𝑆s 𝑅)    &   𝑃 = (𝑆s (𝐵m 𝐼))    &   𝐻 = (mulGrp‘𝑃)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐼𝑉)    &   (𝜑𝑌𝐼)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝑄‘(𝑁 𝑋)) = (𝑁(.g𝐻)(𝑄𝑋)))

Theoremevlval 20284 Value of the simple/same ring evaluation map. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 12-Jun-2015.)
𝑄 = (𝐼 eval 𝑅)    &   𝐵 = (Base‘𝑅)       𝑄 = ((𝐼 evalSub 𝑅)‘𝐵)

Theoremevlrhm 20285 The simple evaluation map is a ring homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = (𝐼 eval 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑅)    &   𝑇 = (𝑅s (𝐵m 𝐼))       ((𝐼𝑉𝑅 ∈ CRing) → 𝑄 ∈ (𝑊 RingHom 𝑇))

Theoremevlsscasrng 20286 The evaluation of a scalar of a subring yields the same result as evaluated as a scalar over the ring itself. (Contributed by AV, 12-Sep-2019.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑂 = (𝐼 eval 𝑆)    &   𝑊 = (𝐼 mPoly 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝑃 = (𝐼 mPoly 𝑆)    &   𝐵 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑊)    &   𝐶 = (algSc‘𝑃)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝑅)       (𝜑 → (𝑄‘(𝐴𝑋)) = (𝑂‘(𝐶𝑋)))

Theoremevlsca 20287 Simple polynomial evaluation maps scalars to constant functions. (Contributed by AV, 12-Sep-2019.)
𝑄 = (𝐼 eval 𝑆)    &   𝑊 = (𝐼 mPoly 𝑆)    &   𝐵 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑊)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑄‘(𝐴𝑋)) = ((𝐵m 𝐼) × {𝑋}))

Theoremevlsvarsrng 20288 The evaluation of the variable of polynomials over subring yields the same result as evaluated as variable of the polynomials over the ring itself. (Contributed by AV, 12-Sep-2019.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑂 = (𝐼 eval 𝑆)    &   𝑉 = (𝐼 mVar 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐼𝐴)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝐼)       (𝜑 → (𝑄‘(𝑉𝑋)) = (𝑂‘(𝑉𝑋)))

Theoremevlvar 20289* Simple polynomial evaluation maps variables to projections. (Contributed by AV, 12-Sep-2019.)
𝑄 = (𝐼 eval 𝑆)    &   𝑉 = (𝐼 mVar 𝑆)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑋𝐼)       (𝜑 → (𝑄‘(𝑉𝑋)) = (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑋)))

Theoremmpfconst 20290 Constants are multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝐵 = (Base‘𝑆)    &   𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝑅)       (𝜑 → ((𝐵m 𝐼) × {𝑋}) ∈ 𝑄)

Theoremmpfproj 20291* Projections are multivariate polynomial functions. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝐵 = (Base‘𝑆)    &   𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝐽𝐼)       (𝜑 → (𝑓 ∈ (𝐵m 𝐼) ↦ (𝑓𝐽)) ∈ 𝑄)

Theoremmpfsubrg 20292 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 20293 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 20294 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 20295 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 20296* 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)

10.9.3  Additional definitions for (multivariate) polynomials

Syntaxcslv 20297 Select a subset of variables in a multivariate polynomial.
class selectVars

Syntaxcmhp 20298 Multivariate polynomials.
class mHomP

Syntaxcpsd 20299 Power series partial derivative function.
class mPSDer

Syntaxcai 20300 Algebraically independent.
class AlgInd

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 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 45101-45144
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