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Type | Label | Description |
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Statement | ||
Theorem | evlslem1 21301* | Lemma for evlseu 21302, 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 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ 𝑇 = (mulGrp‘𝑆) & ⊢ ↑ = (.g‘𝑇) & ⊢ · = (.r‘𝑆) & ⊢ 𝑉 = (𝐼 mVar 𝑅) & ⊢ 𝐸 = (𝑝 ∈ 𝐵 ↦ (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺)))))) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) & ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) & ⊢ 𝐴 = (algSc‘𝑃) ⇒ ⊢ (𝜑 → (𝐸 ∈ (𝑃 RingHom 𝑆) ∧ (𝐸 ∘ 𝐴) = 𝐹 ∧ (𝐸 ∘ 𝑉) = 𝐺)) | ||
Theorem | evlseu 21302* | 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) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) & ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) ⇒ ⊢ (𝜑 → ∃!𝑚 ∈ (𝑃 RingHom 𝑆)((𝑚 ∘ 𝐴) = 𝐹 ∧ (𝑚 ∘ 𝑉) = 𝐺)) | ||
Theorem | reldmevls 21303 | Well-behaved binary operation property of evalSub. (Contributed by Stefan O'Rear, 19-Mar-2015.) |
⊢ Rel dom evalSub | ||
Theorem | mpfrcl 21304 | Reverse closure for the set of polynomial functions. (Contributed by Stefan O'Rear, 19-Mar-2015.) |
⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) ⇒ ⊢ (𝑋 ∈ 𝑄 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆))) | ||
Theorem | evlsval 21305* | 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 = (𝑆 ↾s 𝑅) & ⊢ 𝑇 = (𝑆 ↑s (𝐵 ↑m 𝐼)) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ 𝑋 = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) & ⊢ 𝑌 = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥))) ⇒ ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = (℩𝑓 ∈ (𝑊 RingHom 𝑇)((𝑓 ∘ 𝐴) = 𝑋 ∧ (𝑓 ∘ 𝑉) = 𝑌))) | ||
Theorem | evlsval2 21306* | 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 = (𝑆 ↾s 𝑅) & ⊢ 𝑇 = (𝑆 ↑s (𝐵 ↑m 𝐼)) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ 𝑋 = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) & ⊢ 𝑌 = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥))) ⇒ ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑄 ∈ (𝑊 RingHom 𝑇) ∧ ((𝑄 ∘ 𝐴) = 𝑋 ∧ (𝑄 ∘ 𝑉) = 𝑌))) | ||
Theorem | evlsrhm 21307 | 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 = (𝑆 ↾s 𝑅) & ⊢ 𝑇 = (𝑆 ↑s (𝐵 ↑m 𝐼)) & ⊢ 𝐵 = (Base‘𝑆) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇)) | ||
Theorem | evlssca 21308 | 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 = (𝑆 ↾s 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑋 ∈ 𝑅) ⇒ ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = ((𝐵 ↑m 𝐼) × {𝑋})) | ||
Theorem | evlsvar 21309* | 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 = (𝑆 ↾s 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑄‘(𝑉‘𝑋)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑋))) | ||
Theorem | evlsgsumadd 21310* | 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 = (0g‘𝑊) & ⊢ 𝑈 = (𝑆 ↾s 𝑅) & ⊢ 𝑃 = (𝑆 ↑s (𝐾 ↑m 𝐼)) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑁 ⊆ ℕ0) & ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 0 ) ⇒ ⊢ (𝜑 → (𝑄‘(𝑊 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝑃 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌)))) | ||
Theorem | evlsgsummul 21311* | 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 = (1r‘𝑊) & ⊢ 𝑈 = (𝑆 ↾s 𝑅) & ⊢ 𝑃 = (𝑆 ↑s (𝐾 ↑m 𝐼)) & ⊢ 𝐻 = (mulGrp‘𝑃) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑁 ⊆ ℕ0) & ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 1 ) ⇒ ⊢ (𝜑 → (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌)))) | ||
Theorem | evlspw 21312 | 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) & ⊢ ↑ = (.g‘𝐺) & ⊢ 𝑈 = (𝑆 ↾s 𝑅) & ⊢ 𝑃 = (𝑆 ↑s (𝐾 ↑m 𝐼)) & ⊢ 𝐻 = (mulGrp‘𝑃) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁(.g‘𝐻)(𝑄‘𝑋))) | ||
Theorem | evlsvarpw 21313 | 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) & ⊢ ↑ = (.g‘𝐺) & ⊢ 𝑋 = ((𝐼 mVar 𝑈)‘𝑌) & ⊢ 𝑈 = (𝑆 ↾s 𝑅) & ⊢ 𝑃 = (𝑆 ↑s (𝐵 ↑m 𝐼)) & ⊢ 𝐻 = (mulGrp‘𝑃) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁(.g‘𝐻)(𝑄‘𝑋))) | ||
Theorem | evlval 21314 | 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) | ||
Theorem | evlrhm 21315 | 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 = (𝑅 ↑s (𝐵 ↑m 𝐼)) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ CRing) → 𝑄 ∈ (𝑊 RingHom 𝑇)) | ||
Theorem | evlsscasrng 21316 | 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 = (𝑆 ↾s 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑆) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ 𝐶 = (algSc‘𝑃) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑋 ∈ 𝑅) ⇒ ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (𝑂‘(𝐶‘𝑋))) | ||
Theorem | evlsca 21317 | 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) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = ((𝐵 ↑m 𝐼) × {𝑋})) | ||
Theorem | evlsvarsrng 21318 | 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 = (𝑆 ↾s 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝐴) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑄‘(𝑉‘𝑋)) = (𝑂‘(𝑉‘𝑋))) | ||
Theorem | evlvar 21319* | Simple polynomial evaluation maps variables to projections. (Contributed by AV, 12-Sep-2019.) |
⊢ 𝑄 = (𝐼 eval 𝑆) & ⊢ 𝑉 = (𝐼 mVar 𝑆) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑄‘(𝑉‘𝑋)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑋))) | ||
Theorem | mpfconst 21320 | Constants are multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑋 ∈ 𝑅) ⇒ ⊢ (𝜑 → ((𝐵 ↑m 𝐼) × {𝑋}) ∈ 𝑄) | ||
Theorem | mpfproj 21321* | Projections are multivariate polynomial functions. (Contributed by Mario Carneiro, 20-Mar-2015.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑓 ∈ (𝐵 ↑m 𝐼) ↦ (𝑓‘𝐽)) ∈ 𝑄) | ||
Theorem | mpfsubrg 21322 | 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) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (SubRing‘(𝑆 ↑s ((Base‘𝑆) ↑m 𝐼)))) | ||
Theorem | mpff 21323 | Polynomial functions are functions. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ 𝐵 = (Base‘𝑆) ⇒ ⊢ (𝐹 ∈ 𝑄 → 𝐹:(𝐵 ↑m 𝐼)⟶𝐵) | ||
Theorem | mpfaddcl 21324 | The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ + = (+g‘𝑆) ⇒ ⊢ ((𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → (𝐹 ∘f + 𝐺) ∈ 𝑄) | ||
Theorem | mpfmulcl 21325 | The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ · = (.r‘𝑆) ⇒ ⊢ ((𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄) → (𝐹 ∘f · 𝐺) ∈ 𝑄) | ||
Theorem | mpfind 21326* | 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) & ⊢ + = (+g‘𝑆) & ⊢ · = (.r‘𝑆) & ⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜁) & ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜎) & ⊢ (𝑥 = ((𝐵 ↑m 𝐼) × {𝑓}) → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑓)) → (𝜓 ↔ 𝜃)) & ⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏)) & ⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂)) & ⊢ (𝑥 = (𝑓 ∘f + 𝑔) → (𝜓 ↔ 𝜁)) & ⊢ (𝑥 = (𝑓 ∘f · 𝑔) → (𝜓 ↔ 𝜎)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜌)) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑅) → 𝜒) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐼) → 𝜃) & ⊢ (𝜑 → 𝐴 ∈ 𝑄) ⇒ ⊢ (𝜑 → 𝜌) | ||
Syntax | cslv 21327 | Select a subset of variables in a multivariate polynomial. |
class selectVars | ||
Syntax | cmhp 21328 | Multivariate polynomials. |
class mHomP | ||
Syntax | cpsd 21329 | Power series partial derivative function. |
class mPSDer | ||
Syntax | cai 21330 | Algebraically independent. |
class AlgInd | ||
Definition | df-selv 21331* | Define the "variable selection" function. The function ((𝐼 selectVars 𝑅)‘𝐽) maps elements of (𝐼 mPoly 𝑅) bijectively onto (𝐽 mPoly ((𝐼 ∖ 𝐽) 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)) ↦ ⦋((𝑖 ∖ 𝑗) mPoly 𝑟) / 𝑢⦌⦋(𝑗 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝑖 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝑖 ↦ if(𝑥 ∈ 𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝑖 ∖ 𝑗) mVar 𝑟)‘𝑥)))))))) | ||
Definition | df-mhp 21332* | 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 (0g‘𝑟)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}})) | ||
Definition | df-psd 21333* | Define the differentiation operation on multivariate polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ mPSDer = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥 ∈ 𝑖 ↦ (𝑓 ∈ (Base‘(𝑖 mPwSer 𝑟)) ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑘‘𝑥) + 1)(.g‘𝑟)(𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0))))))))) | ||
Definition | df-algind 21334* | 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 (𝑤 ↾s 𝑘))) ↦ ((((𝑣 evalSub 𝑤)‘𝑘)‘𝑓)‘( I ↾ 𝑣)))}) | ||
Theorem | selvffval 21335* | Value of the "variable selection" function. (Contributed by SN, 4-Nov-2023.) |
⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐼 selectVars 𝑅) = (𝑗 ∈ 𝒫 𝐼 ↦ (𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 ∖ 𝑗) mPoly 𝑅) / 𝑢⦌⦋(𝑗 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥)))))))) | ||
Theorem | selvfval 21336* | Value of the "variable selection" function. (Contributed by SN, 4-Nov-2023.) |
⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) ⇒ ⊢ (𝜑 → ((𝐼 selectVars 𝑅)‘𝐽) = (𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 ∖ 𝐽) mPoly 𝑅) / 𝑢⦌⦋(𝐽 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))))) | ||
Theorem | selvval 21337* | Value of the "variable selection" function. (Contributed by SN, 4-Nov-2023.) |
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅) & ⊢ 𝑇 = (𝐽 mPoly 𝑈) & ⊢ 𝐶 = (algSc‘𝑇) & ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈)) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) = ((((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷 ∘ 𝐹))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))))) | ||
Theorem | mhpfval 21338* | 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 = (0g‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐻 = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}})) | ||
Theorem | mhpval 21339* | 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 = (0g‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐻‘𝑁) = {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}}) | ||
Theorem | ismhp 21340* | 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 = (0g‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝐻‘𝑁) ↔ (𝑋 ∈ 𝐵 ∧ (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}))) | ||
Theorem | ismhp2 21341* | Deduce a homogeneous polynomial from its properties. (Contributed by SN, 25-May-2024.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) ⇒ ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) | ||
Theorem | ismhp3 21342* | A polynomial is homogeneous iff the degree of every nonzero term is the same. (Contributed by SN, 22-Jul-2024.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝐻‘𝑁) ↔ ∀𝑑 ∈ 𝐷 ((𝑋‘𝑑) ≠ 0 → ((ℂfld ↾s ℕ0) Σg 𝑑) = 𝑁))) | ||
Theorem | mhpmpl 21343 | A homogeneous polynomial is a polynomial. (Contributed by Steven Nguyen, 25-Aug-2023.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝐵) | ||
Theorem | mhpdeg 21344* | All nonzero terms of a homogeneous polynomial have degree 𝑁. (Contributed by Steven Nguyen, 25-Aug-2023.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) ⇒ ⊢ (𝜑 → (𝑋 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) | ||
Theorem | mhp0cl 21345* | The zero polynomial is homogeneous. Under df-mhp 21332, it has any (nonnegative integer) degree which loosely corresponds to the value "undefined". The values -∞ and 0 are also used in Metamath (by df-mdeg 25226 and df-dgr 25361 respectively) and the literature: https://math.stackexchange.com/a/1796314/593843 25361. (Contributed by SN, 12-Sep-2023.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Grp) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐷 × { 0 }) ∈ (𝐻‘𝑁)) | ||
Theorem | mhpsclcl 21346 | A scalar (or constant) polynomial has degree 0. Compare deg1scl 25287. In other contexts, there may be an exception for the zero polynomial, but under df-mhp 21332 the zero polynomial can be any degree (see mhp0cl 21345) so there is no exception. (Contributed by SN, 25-May-2024.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐴 = (algSc‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐶 ∈ 𝐾) ⇒ ⊢ (𝜑 → (𝐴‘𝐶) ∈ (𝐻‘0)) | ||
Theorem | mhpvarcl 21347 | A power series variable is a polynomial of degree 1. (Contributed by SN, 25-May-2024.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑉 = (𝐼 mVar 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑉‘𝑋) ∈ (𝐻‘1)) | ||
Theorem | mhpmulcl 21348 | A product of homogeneous polynomials is a homogeneous polynomial whose degree is the sum of the degrees of the factors. Compare mdegmulle2 25253 (which shows less-than-or-equal instead of equal). (Contributed by SN, 22-Jul-2024.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑌 = (𝐼 mPoly 𝑅) & ⊢ · = (.r‘𝑌) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑃 ∈ (𝐻‘𝑀)) & ⊢ (𝜑 → 𝑄 ∈ (𝐻‘𝑁)) ⇒ ⊢ (𝜑 → (𝑃 · 𝑄) ∈ (𝐻‘(𝑀 + 𝑁))) | ||
Theorem | mhppwdeg 21349 | Degree of a homogeneous polynomial raised to a power. General version of deg1pw 25294. (Contributed by SN, 26-Jul-2024.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑇 = (mulGrp‘𝑃) & ⊢ ↑ = (.g‘𝑇) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑀)) ⇒ ⊢ (𝜑 → (𝑁 ↑ 𝑋) ∈ (𝐻‘(𝑀 · 𝑁))) | ||
Theorem | mhpaddcl 21350 | Homogeneous polynomials are closed under addition. (Contributed by SN, 26-Aug-2023.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ + = (+g‘𝑃) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Grp) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) & ⊢ (𝜑 → 𝑌 ∈ (𝐻‘𝑁)) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝐻‘𝑁)) | ||
Theorem | mhpinvcl 21351 | Homogeneous polynomials are closed under taking the opposite. (Contributed by SN, 12-Sep-2023.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑀 = (invg‘𝑃) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Grp) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) ⇒ ⊢ (𝜑 → (𝑀‘𝑋) ∈ (𝐻‘𝑁)) | ||
Theorem | mhpsubg 21352 | Homogeneous polynomials form a subgroup of the polynomials. (Contributed by SN, 25-Sep-2023.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Grp) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐻‘𝑁) ∈ (SubGrp‘𝑃)) | ||
Theorem | mhpvscacl 21353 | Homogeneous polynomials are closed under scalar multiplication. (Contributed by SN, 25-Sep-2023.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) & ⊢ (𝜑 → 𝐹 ∈ (𝐻‘𝑁)) ⇒ ⊢ (𝜑 → (𝑋 · 𝐹) ∈ (𝐻‘𝑁)) | ||
Theorem | mhplss 21354 | Homogeneous polynomials form a linear subspace of the polynomials. (Contributed by SN, 25-Sep-2023.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐻‘𝑁) ∈ (LSubSp‘𝑃)) | ||
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 21384. 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 21384). 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". | ||
Syntax | cps1 21355 | Univariate power series. |
class PwSer1 | ||
Syntax | cv1 21356 | The base variable of a univariate power series. |
class var1 | ||
Syntax | cpl1 21357 | Univariate polynomials. |
class Poly1 | ||
Syntax | cco1 21358 | Coefficient function for a univariate polynomial. |
class coe1 | ||
Syntax | ctp1 21359 | Convert a univariate polynomial representation to multivariate. |
class toPoly1 | ||
Definition | df-psr1 21360 | Define the algebra of univariate power series. (Contributed by Mario Carneiro, 29-Dec-2014.) |
⊢ PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅)) | ||
Definition | df-vr1 21361 | 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)‘∅)) | ||
Definition | df-ply1 21362 | Define the algebra of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ Poly1 = (𝑟 ∈ V ↦ ((PwSer1‘𝑟) ↾s (Base‘(1o mPoly 𝑟)))) | ||
Definition | df-coe1 21363* | Define the coefficient function for a univariate polynomial. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
⊢ coe1 = (𝑓 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (𝑓‘(1o × {𝑛})))) | ||
Definition | df-toply1 21364* | 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 ∈ (ℕ0 ↑m 1o) ↦ (𝑓‘(𝑛‘∅)))) | ||
Theorem | psr1baslem 21365 | The set of finite bags on 1o is just the set of all functions from 1o to ℕ0. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ (ℕ0 ↑m 1o) = {𝑓 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin} | ||
Theorem | psr1val 21366 | Value of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.) |
⊢ 𝑆 = (PwSer1‘𝑅) ⇒ ⊢ 𝑆 = ((1o ordPwSer 𝑅)‘∅) | ||
Theorem | psr1crng 21367 | The ring of univariate power series is a commutative ring. (Contributed by Mario Carneiro, 8-Feb-2015.) |
⊢ 𝑆 = (PwSer1‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → 𝑆 ∈ CRing) | ||
Theorem | psr1assa 21368 | The ring of univariate power series is an associative algebra. (Contributed by Mario Carneiro, 8-Feb-2015.) |
⊢ 𝑆 = (PwSer1‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → 𝑆 ∈ AssAlg) | ||
Theorem | psr1tos 21369 | The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 2-Jun-2015.) |
⊢ 𝑆 = (PwSer1‘𝑅) ⇒ ⊢ (𝑅 ∈ Toset → 𝑆 ∈ Toset) | ||
Theorem | psr1bas2 21370 | The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 3-Jul-2015.) |
⊢ 𝑆 = (PwSer1‘𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑂 = (1o mPwSer 𝑅) ⇒ ⊢ 𝐵 = (Base‘𝑂) | ||
Theorem | psr1bas 21371 | The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.) |
⊢ 𝑆 = (PwSer1‘𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐾 = (Base‘𝑅) ⇒ ⊢ 𝐵 = (𝐾 ↑m (ℕ0 ↑m 1o)) | ||
Theorem | vr1val 21372 | 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 = (var1‘𝑅) ⇒ ⊢ 𝑋 = ((1o mVar 𝑅)‘∅) | ||
Theorem | vr1cl2 21373 | The variable 𝑋 is a member of the power series algebra 𝑅[[𝑋]]. (Contributed by Mario Carneiro, 8-Feb-2015.) |
⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑆 = (PwSer1‘𝑅) & ⊢ 𝐵 = (Base‘𝑆) ⇒ ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) | ||
Theorem | ply1val 21374 | The value of the set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑆 = (PwSer1‘𝑅) ⇒ ⊢ 𝑃 = (𝑆 ↾s (Base‘(1o mPoly 𝑅))) | ||
Theorem | ply1bas 21375 | The value of the base set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑆 = (PwSer1‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) ⇒ ⊢ 𝑈 = (Base‘(1o mPoly 𝑅)) | ||
Theorem | ply1lss 21376 | Univariate polynomials form a linear subspace of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑆 = (PwSer1‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) ⇒ ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (LSubSp‘𝑆)) | ||
Theorem | ply1subrg 21377 | Univariate polynomials form a subring of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑆 = (PwSer1‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) ⇒ ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (SubRing‘𝑆)) | ||
Theorem | ply1crng 21378 | The ring of univariate polynomials is a commutative ring. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) | ||
Theorem | ply1assa 21379 | The ring of univariate polynomials is an associative algebra. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg) | ||
Theorem | psr1bascl 21380 | A univariate power series is a multivariate power series on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
⊢ 𝑃 = (PwSer1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (Base‘(1o mPwSer 𝑅))) | ||
Theorem | psr1basf 21381 | Univariate power series base set elements are functions. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
⊢ 𝑃 = (PwSer1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑m 1o)⟶𝐾) | ||
Theorem | ply1basf 21382 | Univariate polynomial base set elements are functions. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑m 1o)⟶𝐾) | ||
Theorem | ply1bascl 21383 | A univariate polynomial is a univariate power series. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (Base‘(PwSer1‘𝑅))) | ||
Theorem | ply1bascl2 21384 | A univariate polynomial is a multivariate polynomial on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (Base‘(1o mPoly 𝑅))) | ||
Theorem | coe1fval 21385* | Value of the univariate polynomial coefficient function. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ (𝐹 ∈ 𝑉 → 𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛})))) | ||
Theorem | coe1fv 21386 | Value of an evaluated coefficient in a polynomial coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴‘𝑁) = (𝐹‘(1o × {𝑁}))) | ||
Theorem | fvcoe1 21387 | Value of a multivariate coefficient in terms of the coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ (ℕ0 ↑m 1o)) → (𝐹‘𝑋) = (𝐴‘(𝑋‘∅))) | ||
Theorem | coe1fval3 21388* | Univariate power series coefficient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
⊢ 𝐴 = (coe1‘𝐹) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (PwSer1‘𝑅) & ⊢ 𝐺 = (𝑦 ∈ ℕ0 ↦ (1o × {𝑦})) ⇒ ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝐹 ∘ 𝐺)) | ||
Theorem | coe1f2 21389 | Functionality of univariate power series coefficient vectors. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
⊢ 𝐴 = (coe1‘𝐹) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (PwSer1‘𝑅) & ⊢ 𝐾 = (Base‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐵 → 𝐴:ℕ0⟶𝐾) | ||
Theorem | coe1fval2 21390* | Univariate polynomial coefficient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
⊢ 𝐴 = (coe1‘𝐹) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐺 = (𝑦 ∈ ℕ0 ↦ (1o × {𝑦})) ⇒ ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝐹 ∘ 𝐺)) | ||
Theorem | coe1f 21391 | Functionality of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
⊢ 𝐴 = (coe1‘𝐹) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐾 = (Base‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐵 → 𝐴:ℕ0⟶𝐾) | ||
Theorem | coe1fvalcl 21392 | A coefficient of a univariate polynomial over a class/ring is an element of this class/ring. (Contributed by AV, 9-Oct-2019.) |
⊢ 𝐴 = (coe1‘𝐹) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐾 = (Base‘𝑅) ⇒ ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0) → (𝐴‘𝑁) ∈ 𝐾) | ||
Theorem | coe1sfi 21393 | Finite support of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 19-Jul-2019.) |
⊢ 𝐴 = (coe1‘𝐹) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐵 → 𝐴 finSupp 0 ) | ||
Theorem | coe1fsupp 21394* | 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 = (coe1‘𝐹) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐾 = (Base‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐵 → 𝐴 ∈ {𝑔 ∈ (𝐾 ↑m ℕ0) ∣ 𝑔 finSupp 0 }) | ||
Theorem | mptcoe1fsupp 21395* | A mapping involving coefficients of polynomials is finitely supported. (Contributed by AV, 12-Oct-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((coe1‘𝑀)‘𝑘)) finSupp 0 ) | ||
Theorem | coe1ae0 21396* | The coefficient vector of a univariate polynomial is 0 almost everywhere. (Contributed by AV, 19-Oct-2019.) |
⊢ 𝐴 = (coe1‘𝐹) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐵 → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝐴‘𝑛) = 0 )) | ||
Theorem | vr1cl 21397 | The generator of a univariate polynomial algebra is contained in the base set. (Contributed by Stefan O'Rear, 19-Mar-2015.) |
⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) | ||
Theorem | opsr0 21398 | 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) & ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) ⇒ ⊢ (𝜑 → (0g‘𝑆) = (0g‘𝑂)) | ||
Theorem | opsr1 21399 | 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) & ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) ⇒ ⊢ (𝜑 → (1r‘𝑆) = (1r‘𝑂)) | ||
Theorem | mplplusg 21400 | 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) & ⊢ + = (+g‘𝑌) ⇒ ⊢ + = (+g‘𝑆) |
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