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Theorem List for Metamath Proof Explorer - 19901-20000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcevl 19901 Evaluation of a multivariate polynomial.
class eval

Definitiondf-evls 19902* 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 19903* 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 19904* 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 19905* 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 19906* 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 19907* 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)    &    0 = (0g𝑇)    &   (𝜑𝑇 ∈ CMnd)    &   (𝜑𝐵𝐷)    &   (𝜑𝐺:𝐼𝐶)    &   (𝜑𝐼 ∈ V)       (𝜑 → (𝑇 Σg (𝐵𝑓 · 𝐺)) ∈ 𝐶)

Theoremevlslem2 19908* 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)))

Theoremevlslem6 19909* Lemma for evlseu 19912. 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)    &   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)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑏𝐷 ↦ ((𝐹‘(𝑌𝑏)) · (𝑇 Σg (𝑏𝑓 𝐺)))):𝐷𝐶 ∧ (𝑏𝐷 ↦ ((𝐹‘(𝑌𝑏)) · (𝑇 Σg (𝑏𝑓 𝐺)))) finSupp (0g𝑆)))

Theoremevlslem3 19910* Lemma for evlseu 19912. 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))))

Theoremevlslem1 19911* Lemma for evlseu 19912, 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)    &   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)    &   𝐴 = (algSc‘𝑃)       (𝜑 → (𝐸 ∈ (𝑃 RingHom 𝑆) ∧ (𝐸𝐴) = 𝐹 ∧ (𝐸𝑉) = 𝐺))

Theoremevlseu 19912* 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 19913 Well-behaved binary operation property of evalSub. (Contributed by Stefan O'Rear, 19-Mar-2015.)
Rel dom evalSub

Theoremmpfrcl 19914 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 19915* 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 19916* 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 19917 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 19918 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 19919* 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)))

Theoremevlval 19920 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 19921 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 19922 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 19923 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 19924 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 19925* 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 19926 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 19927* 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 19928 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 19929 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 19930 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 19931 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 19932* 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)

10.10.3  Additional definitions for (multivariate) polynomials

Remark: There are no theorems using these definitions yet!

Syntaxcmhp 19933 Multivariate polynomials.
class mHomP

Syntaxcpsd 19934 Power series partial derivative function.
class mPSDer

Syntaxcslv 19935 Select a subset of variables in a multivariate polynomial.
class selectVars

Syntaxcai 19936 Algebraically independent.
class AlgInd

Definitiondf-mhp 19937* 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 19938* 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-selv 19939* 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 ∈ (𝑖 mPoly 𝑟) ↦ ((𝑖𝑗) mPoly 𝑟) / 𝑠(𝑥 ∈ (Scalar‘𝑠) ↦ (𝑥( ·𝑠𝑠)(1r𝑠))) / 𝑐((((𝑖 evalSub 𝑠)‘(𝑐s 𝑟))‘(𝑐𝑓))‘(𝑥𝑖 ↦ if(𝑥𝑗, ((𝑗 mVar ((𝑖𝑗) mPoly 𝑟))‘𝑥), (𝑐 ∘ (((𝑖𝑗) mVar 𝑟)‘𝑥))))))))

Definitiondf-algind 19940* 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)))})

10.10.4  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 19970.

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). 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 19941 Univariate power series.
class PwSer1

Syntaxcv1 19942 The base variable of a univariate power series.
class var1

Syntaxcpl1 19943 Univariate polynomials.
class Poly1

Syntaxcco1 19944 Coefficient function for a univariate polynomial.
class coe1

Syntaxctp1 19945 Convert a univariate polynomial representation to multivariate.
class toPoly1

Definitiondf-psr1 19946 Define the algebra of univariate power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅))

Definitiondf-vr1 19947 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 19948 Define the algebra of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1 = (𝑟 ∈ V ↦ ((PwSer1𝑟) ↾s (Base‘(1o mPoly 𝑟))))

Definitiondf-coe1 19949* 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 19950* 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 19951 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 19952 Value of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑆 = (PwSer1𝑅)       𝑆 = ((1o ordPwSer 𝑅)‘∅)

Theorempsr1crng 19953 The ring of univariate power series is a commutative ring. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑆 = (PwSer1𝑅)       (𝑅 ∈ CRing → 𝑆 ∈ CRing)

Theorempsr1assa 19954 The ring of univariate power series is an associative algebra. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑆 = (PwSer1𝑅)       (𝑅 ∈ CRing → 𝑆 ∈ AssAlg)

Theorempsr1tos 19955 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 2-Jun-2015.)
𝑆 = (PwSer1𝑅)       (𝑅 ∈ Toset → 𝑆 ∈ Toset)

Theorempsr1bas2 19956 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 19957 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 19958 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 19959 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 19960 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 19961 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 19962 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 19963 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 19964 The ring of univariate polynomials is a commutative ring. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ CRing → 𝑃 ∈ CRing)

Theoremply1assa 19965 The ring of univariate polynomials is an associative algebra. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)

Theorempsr1bascl 19966 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 19967 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 19968 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 19969 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 19970 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 19971* Value of the univariate polynomial coefficient function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐴 = (coe1𝐹)       (𝐹𝑉𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))

Theoremcoe1fv 19972 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 19973 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 19974* 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 19975 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 19976* 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 19977 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 19978 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 19979 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 19980* 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 19981* 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 19982* 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 19983 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 19984 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 19985 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 19986 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 19987 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 19988 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 19989 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 19990 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 19991 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 19992 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 19993 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 19994 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 19995 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 19996 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 19997 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 19998 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 19999 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 20000 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))

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