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Theorem List for Metamath Proof Explorer - 20201-20300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmplelsfi 20201 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 20202 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 20203* 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 20204* 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 20205* 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 20206* 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 20207 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 20208 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 20209 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 20210* 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 20211* 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 20212* 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)    &   (𝜑𝐼 ∈ V)    &   (𝜑𝑅 ∈ CRing)       (𝜑𝑋𝐻)
 
Theoremmplcoe4 20213* 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 20214 Evaluation of a multivariate polynomial in a subring.
class evalSub
 
Syntaxcevl 20215 Evaluation of a multivariate polynomial.
class eval
 
Definitiondf-evls 20216* 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 20217* 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 20218* 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 20219* 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 20220* 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.)
𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   𝐶 = (Base‘𝑇)    &    · = (.g𝑇)    &    0 = (0g𝑇)    &   (𝜑𝑇 ∈ CMnd)    &   (𝜑𝐵𝐷)    &   (𝜑𝐺:𝐼𝐶)    &   (𝜑𝐼 ∈ V)       (𝜑 → ((𝐵f · 𝐺):𝐼𝐶 ∧ (𝐵f · 𝐺) finSupp 0 ))
 
Theorempsrbagev2 20221* Closure of a sum using a bag of multipliers. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Proof shortened by AV, 18-Jul-2019.)
𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   𝐶 = (Base‘𝑇)    &    · = (.g𝑇)    &   (𝜑𝑇 ∈ CMnd)    &   (𝜑𝐵𝐷)    &   (𝜑𝐺:𝐼𝐶)    &   (𝜑𝐼 ∈ V)       (𝜑 → (𝑇 Σg (𝐵f · 𝐺)) ∈ 𝐶)
 
Theoremevlslem2 20222* A linear function on the polynomial ring which is multiplicative on scaled monomials is generally multiplicative. (Contributed by Stefan O'Rear, 9-Mar-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &    · = (.r𝑆)    &    0 = (0g𝑅)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝐼 ∈ V)    &   (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 20223* Lemma for evlseu 20226. Polynomial evaluation of a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
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))))))    &   (𝜑𝐼 ∈ V)    &   (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 20224* Lemma for evlseu 20226. Finiteness and consistency of the top-level sum. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 26-Jul-2019.)
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))))))    &   (𝜑𝐼 ∈ V)    &   (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 20225* Lemma for evlseu 20226, give a formula for (the unique) polynomial evaluation homomorphism. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Proof shortened by AV, 26-Jul-2019.)
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))))))    &   (𝜑𝐼 ∈ V)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝐹 ∈ (𝑅 RingHom 𝑆))    &   (𝜑𝐺:𝐼𝐶)    &   𝐴 = (algSc‘𝑃)       (𝜑 → (𝐸 ∈ (𝑃 RingHom 𝑆) ∧ (𝐸𝐴) = 𝐹 ∧ (𝐸𝑉) = 𝐺))
 
Theoremevlseu 20226* 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.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐶 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑃)    &   𝑉 = (𝐼 mVar 𝑅)    &   (𝜑𝐼 ∈ V)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝐹 ∈ (𝑅 RingHom 𝑆))    &   (𝜑𝐺:𝐼𝐶)       (𝜑 → ∃!𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚𝐴) = 𝐹 ∧ (𝑚𝑉) = 𝐺))
 
Theoremreldmevls 20227 Well-behaved binary operation property of evalSub. (Contributed by Stefan O'Rear, 19-Mar-2015.)
Rel dom evalSub
 
Theoremmpfrcl 20228 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 20229* 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 20230* 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 20231 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 20232 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 20233* 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 20234* 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 20235* 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 20236 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 20237 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 20238 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 20239 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 20240 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 20241 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 20242 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 20243* 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 20244 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 20245* 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 20246 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 20247 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 20248 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 20249 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 20250* 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 20251 Select a subset of variables in a multivariate polynomial.
class selectVars
 
Syntaxcmhp 20252 Multivariate polynomials.
class mHomP
 
Syntaxcpsd 20253 Power series partial derivative function.
class mPSDer
 
Syntaxcai 20254 Algebraically independent.
class AlgInd
 
Definitiondf-selv 20255* 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 20256* 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} ∣ Σ𝑗 ∈ ℕ0 (𝑔𝑗) = 𝑛}}))
 
Definitiondf-psd 20257* 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 20258* 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 20259* 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 20260* 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 20261* 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 20262* 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 ∣ Σ𝑗 ∈ ℕ0 (𝑔𝑗) = 𝑛}}))
 
Theoremmhpval 20263* 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 ∣ Σ𝑗 ∈ ℕ0 (𝑔𝑗) = 𝑁}})
 
Theoremismhp 20264* 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 ∣ Σ𝑗 ∈ ℕ0 (𝑔𝑗) = 𝑁})))
 
Theoremmhpmpl 20265 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 20266* 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 ∣ Σ𝑗 ∈ ℕ0 (𝑔𝑗) = 𝑁})
 
Theoremmhp0cl 20267* The zero polynomial is homogeneous. (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))
 
Theoremmhpaddcl 20268 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 20269 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 20270 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 20271 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 20272 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))
 
10.9.4  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 20302.

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 20302). 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 20273 Univariate power series.
class PwSer1
 
Syntaxcv1 20274 The base variable of a univariate power series.
class var1
 
Syntaxcpl1 20275 Univariate polynomials.
class Poly1
 
Syntaxcco1 20276 Coefficient function for a univariate polynomial.
class coe1
 
Syntaxctp1 20277 Convert a univariate polynomial representation to multivariate.
class toPoly1
 
Definitiondf-psr1 20278 Define the algebra of univariate power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅))
 
Definitiondf-vr1 20279 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 20280 Define the algebra of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1 = (𝑟 ∈ V ↦ ((PwSer1𝑟) ↾s (Base‘(1o mPoly 𝑟))))
 
Definitiondf-coe1 20281* 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 20282* 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 20283 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 20284 Value of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑆 = (PwSer1𝑅)       𝑆 = ((1o ordPwSer 𝑅)‘∅)
 
Theorempsr1crng 20285 The ring of univariate power series is a commutative ring. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑆 = (PwSer1𝑅)       (𝑅 ∈ CRing → 𝑆 ∈ CRing)
 
Theorempsr1assa 20286 The ring of univariate power series is an associative algebra. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑆 = (PwSer1𝑅)       (𝑅 ∈ CRing → 𝑆 ∈ AssAlg)
 
Theorempsr1tos 20287 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 2-Jun-2015.)
𝑆 = (PwSer1𝑅)       (𝑅 ∈ Toset → 𝑆 ∈ Toset)
 
Theorempsr1bas2 20288 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 20289 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 20290 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 20291 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 20292 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 20293 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 20294 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 20295 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 20296 The ring of univariate polynomials is a commutative ring. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ CRing → 𝑃 ∈ CRing)
 
Theoremply1assa 20297 The ring of univariate polynomials is an associative algebra. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
 
Theorempsr1bascl 20298 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 20299 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 20300 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)
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