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Type | Label | Description |
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Statement | ||
Theorem | ftc2ditg 26101* | Directed integral analogue of ftc2 26099. (Contributed by Mario Carneiro, 3-Sep-2014.) |
⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ (𝑋[,]𝑌)) & ⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌)) & ⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ)) & ⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿1) & ⊢ (𝜑 → 𝐹 ∈ ((𝑋[,]𝑌)–cn→ℂ)) ⇒ ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴))) | ||
Theorem | itgparts 26102* | Integration by parts. If 𝐵(𝑥) is the derivative of 𝐴(𝑥) and 𝐷(𝑥) is the derivative of 𝐶(𝑥), and 𝐸 = (𝐴 · 𝐵)(𝑋) and 𝐹 = (𝐴 · 𝐵)(𝑌), then under suitable integrability and differentiability assumptions, the integral of 𝐴 · 𝐷 from 𝑋 to 𝑌 is equal to 𝐹 − 𝐸 minus the integral of 𝐵 · 𝐶. (Contributed by Mario Carneiro, 3-Sep-2014.) |
⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶) ∈ ((𝑋[,]𝑌)–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ ((𝑋(,)𝑌)–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷) ∈ ((𝑋(,)𝑌)–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐷)) ∈ 𝐿1) & ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐵 · 𝐶)) ∈ 𝐿1) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵)) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷)) & ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐴 · 𝐶) = 𝐸) & ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝐴 · 𝐶) = 𝐹) ⇒ ⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥 = ((𝐹 − 𝐸) − ∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥)) | ||
Theorem | itgsubstlem 26103* | Lemma for itgsubst 26104. (Contributed by Mario Carneiro, 12-Sep-2014.) |
⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → 𝑍 ∈ ℝ*) & ⊢ (𝜑 → 𝑊 ∈ ℝ*) & ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊))) & ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿1)) & ⊢ (𝜑 → (𝑢 ∈ (𝑍(,)𝑊) ↦ 𝐶) ∈ ((𝑍(,)𝑊)–cn→ℂ)) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵)) & ⊢ (𝑢 = 𝐴 → 𝐶 = 𝐸) & ⊢ (𝑥 = 𝑋 → 𝐴 = 𝐾) & ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐿) & ⊢ (𝜑 → 𝑀 ∈ (𝑍(,)𝑊)) & ⊢ (𝜑 → 𝑁 ∈ (𝑍(,)𝑊)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → 𝐴 ∈ (𝑀(,)𝑁)) ⇒ ⊢ (𝜑 → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥) | ||
Theorem | itgsubst 26104* | Integration by 𝑢-substitution. If 𝐴(𝑥) is a continuous, differentiable function from [𝑋, 𝑌] to (𝑍, 𝑊), whose derivative is continuous and integrable, and 𝐶(𝑢) is a continuous function on (𝑍, 𝑊), then the integral of 𝐶(𝑢) from 𝐾 = 𝐴(𝑋) to 𝐿 = 𝐴(𝑌) is equal to the integral of 𝐶(𝐴(𝑥)) D 𝐴(𝑥) from 𝑋 to 𝑌. In this part of the proof we discharge the assumptions in itgsubstlem 26103, which use the fact that (𝑍, 𝑊) is open to shrink the interval a little to (𝑀, 𝑁) where 𝑍 < 𝑀 < 𝑁 < 𝑊- this is possible because 𝐴(𝑥) is a continuous function on a closed interval, so its range is in fact a closed interval, and we have some wiggle room on the edges. (Contributed by Mario Carneiro, 7-Sep-2014.) |
⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → 𝑍 ∈ ℝ*) & ⊢ (𝜑 → 𝑊 ∈ ℝ*) & ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊))) & ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿1)) & ⊢ (𝜑 → (𝑢 ∈ (𝑍(,)𝑊) ↦ 𝐶) ∈ ((𝑍(,)𝑊)–cn→ℂ)) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵)) & ⊢ (𝑢 = 𝐴 → 𝐶 = 𝐸) & ⊢ (𝑥 = 𝑋 → 𝐴 = 𝐾) & ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐿) ⇒ ⊢ (𝜑 → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥) | ||
Theorem | itgpowd 26105* | The integral of a monomial on a closed bounded interval of the real line. Co-authors TA and MC. (Contributed by Jon Pennant, 31-May-2019.) (Revised by Thierry Arnoux, 14-Jun-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → ∫(𝐴[,]𝐵)(𝑥↑𝑁) d𝑥 = (((𝐵↑(𝑁 + 1)) − (𝐴↑(𝑁 + 1))) / (𝑁 + 1))) | ||
Syntax | cmdg 26106 | Multivariate polynomial degree. |
class mDeg | ||
Syntax | cdg1 26107 | Univariate polynomial degree. |
class deg1 | ||
Definition | df-mdeg 26108* | Define the degree of a polynomial. Note (SO): as an experiment I am using a definition which makes the degree of the zero polynomial -∞, contrary to the convention used in df-dgr 26244. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.) |
⊢ mDeg = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ sup(ran (ℎ ∈ (𝑓 supp (0g‘𝑟)) ↦ (ℂfld Σg ℎ)), ℝ*, < ))) | ||
Definition | df-deg1 26109 | Define the degree of a univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ deg1 = (𝑟 ∈ V ↦ (1o mDeg 𝑟)) | ||
Theorem | reldmmdeg 26110 | Multivariate degree is a binary operation. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ Rel dom mDeg | ||
Theorem | tdeglem1 26111* | Functionality of the total degree helper function. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) Remove sethood antecedent. (Revised by SN, 7-Aug-2024.) |
⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) ⇒ ⊢ 𝐻:𝐴⟶ℕ0 | ||
Theorem | tdeglem3 26112* | Additivity of the total degree helper function. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) Remove a sethood antecedent. (Revised by SN, 7-Aug-2024.) |
⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) ⇒ ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐻‘(𝑋 ∘f + 𝑌)) = ((𝐻‘𝑋) + (𝐻‘𝑌))) | ||
Theorem | tdeglem4 26113* | There is only one multi-index with total degree 0. (Contributed by Stefan O'Rear, 29-Mar-2015.) Remove a sethood antecedent. (Revised by SN, 7-Aug-2024.) |
⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) ⇒ ⊢ (𝑋 ∈ 𝐴 → ((𝐻‘𝑋) = 0 ↔ 𝑋 = (𝐼 × {0}))) | ||
Theorem | tdeglem2 26114 | Simplification of total degree for the univariate case. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ (ℎ ∈ (ℕ0 ↑m 1o) ↦ (ℎ‘∅)) = (ℎ ∈ (ℕ0 ↑m 1o) ↦ (ℂfld Σg ℎ)) | ||
Theorem | mdegfval 26115* | Value of the multivariate degree function. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.) |
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) ⇒ ⊢ 𝐷 = (𝑓 ∈ 𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )) | ||
Theorem | mdegval 26116* | Value of the multivariate degree function at some particular polynomial. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.) |
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) ⇒ ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ*, < )) | ||
Theorem | mdegleb 26117* | Property of being of limited degree. (Contributed by Stefan O'Rear, 19-Mar-2015.) |
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) ⇒ ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ*) → ((𝐷‘𝐹) ≤ 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐺 < (𝐻‘𝑥) → (𝐹‘𝑥) = 0 ))) | ||
Theorem | mdeglt 26118* | If there is an upper limit on the degree of a polynomial that is lower than the degree of some exponent bag, then that exponent bag is unrepresented in the polynomial. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → (𝐷‘𝐹) < (𝐻‘𝑋)) ⇒ ⊢ (𝜑 → (𝐹‘𝑋) = 0 ) | ||
Theorem | mdegldg 26119* | A nonzero polynomial has some coefficient which witnesses its degree. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) & ⊢ 𝑌 = (0g‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → ∃𝑥 ∈ 𝐴 ((𝐹‘𝑥) ≠ 0 ∧ (𝐻‘𝑥) = (𝐷‘𝐹))) | ||
Theorem | mdegxrcl 26120 | Closure of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ ℝ*) | ||
Theorem | mdegxrf 26121 | Functionality of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ 𝐷:𝐵⟶ℝ* | ||
Theorem | mdegcl 26122 | Sharp closure for multivariate polynomials. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ (ℕ0 ∪ {-∞})) | ||
Theorem | mdeg0 26123 | Degree of the zero polynomial. (Contributed by Stefan O'Rear, 20-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 0 = (0g‘𝑃) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (𝐷‘ 0 ) = -∞) | ||
Theorem | mdegnn0cl 26124 | Degree of a nonzero polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0) | ||
Theorem | degltlem1 26125 | Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ ((𝑋 ∈ (ℕ0 ∪ {-∞}) ∧ 𝑌 ∈ ℤ) → (𝑋 < 𝑌 ↔ 𝑋 ≤ (𝑌 − 1))) | ||
Theorem | degltp1le 26126 | Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
⊢ ((𝑋 ∈ (ℕ0 ∪ {-∞}) ∧ 𝑌 ∈ ℤ) → (𝑋 < (𝑌 + 1) ↔ 𝑋 ≤ 𝑌)) | ||
Theorem | mdegaddle 26127 | The degree of a sum is at most the maximum of the degrees of the factors. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
⊢ 𝑌 = (𝐼 mPoly 𝑅) & ⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ + = (+g‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹))) | ||
Theorem | mdegvscale 26128 | The degree of a scalar multiple of a polynomial is at most the degree of the original polynomial. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
⊢ 𝑌 = (𝐼 mPoly 𝑅) & ⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐾) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐷‘𝐺)) | ||
Theorem | mdegvsca 26129 | The degree of a scalar multiple of a polynomial is exactly the degree of the original polynomial when the multiple is a nonzero-divisor. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
⊢ 𝑌 = (𝐼 mPoly 𝑅) & ⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐸) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = (𝐷‘𝐺)) | ||
Theorem | mdegle0 26130 | A polynomial has nonpositive degree iff it is a constant. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
⊢ 𝑌 = (𝐼 mPoly 𝑅) & ⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐴 = (algSc‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝐷‘𝐹) ≤ 0 ↔ 𝐹 = (𝐴‘(𝐹‘(𝐼 × {0}))))) | ||
Theorem | mdegmullem 26131* | Lemma for mdegmulle2 26132. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
⊢ 𝑌 = (𝐼 mPoly 𝑅) & ⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ · = (.r‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐽) & ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐾) & ⊢ 𝐴 = {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (𝑏 ∈ 𝐴 ↦ (ℂfld Σg 𝑏)) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐽 + 𝐾)) | ||
Theorem | mdegmulle2 26132 | The multivariate degree of a product of polynomials is at most the sum of the degrees of the polynomials. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
⊢ 𝑌 = (𝐼 mPoly 𝑅) & ⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ · = (.r‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐽) & ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐾) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐽 + 𝐾)) | ||
Theorem | deg1fval 26133 | Relate univariate polynomial degree to multivariate. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝐷 = (deg1‘𝑅) ⇒ ⊢ 𝐷 = (1o mDeg 𝑅) | ||
Theorem | deg1xrf 26134 | Functionality of univariate polynomial degree, weak range. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ 𝐷:𝐵⟶ℝ* | ||
Theorem | deg1xrcl 26135 | Closure of univariate polynomial degree in extended reals. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ ℝ*) | ||
Theorem | deg1cl 26136 | Sharp closure of univariate polynomial degree. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ (ℕ0 ∪ {-∞})) | ||
Theorem | mdegpropd 26137* | Property deduction for polynomial degree. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐵 = (Base‘𝑆)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦)) ⇒ ⊢ (𝜑 → (𝐼 mDeg 𝑅) = (𝐼 mDeg 𝑆)) | ||
Theorem | deg1fvi 26138 | Univariate polynomial degree respects protection. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ (deg1‘𝑅) = (deg1‘( I ‘𝑅)) | ||
Theorem | deg1propd 26139* | Property deduction for polynomial degree. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐵 = (Base‘𝑆)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦)) ⇒ ⊢ (𝜑 → (deg1‘𝑅) = (deg1‘𝑆)) | ||
Theorem | deg1z 26140 | Degree of the zero univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) ⇒ ⊢ (𝑅 ∈ Ring → (𝐷‘ 0 ) = -∞) | ||
Theorem | deg1nn0cl 26141 | Degree of a nonzero univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0) | ||
Theorem | deg1n0ima 26142 | Degree image of a set of polynomials which does not include zero. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ (𝑅 ∈ Ring → (𝐷 “ (𝐵 ∖ { 0 })) ⊆ ℕ0) | ||
Theorem | deg1nn0clb 26143 | A polynomial is nonzero iff it has definite degree. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → (𝐹 ≠ 0 ↔ (𝐷‘𝐹) ∈ ℕ0)) | ||
Theorem | deg1lt0 26144 | A polynomial is zero iff it has negative degree. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((𝐷‘𝐹) < 0 ↔ 𝐹 = 0 )) | ||
Theorem | deg1ldg 26145 | A nonzero univariate polynomial always has a nonzero leading coefficient. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑌 = (0g‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐴‘(𝐷‘𝐹)) ≠ 𝑌) | ||
Theorem | deg1ldgn 26146 | An index at which a polynomial is zero, cannot be its degree. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑌 = (0g‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ ℕ0) & ⊢ (𝜑 → (𝐴‘𝑋) = 𝑌) ⇒ ⊢ (𝜑 → (𝐷‘𝐹) ≠ 𝑋) | ||
Theorem | deg1ldgdomn 26147 | A nonzero univariate polynomial over a domain always has a nonzero-divisor leading coefficient. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ ((𝑅 ∈ Domn ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐴‘(𝐷‘𝐹)) ∈ 𝐸) | ||
Theorem | deg1leb 26148* | Property of being of limited degree. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ*) → ((𝐷‘𝐹) ≤ 𝐺 ↔ ∀𝑥 ∈ ℕ0 (𝐺 < 𝑥 → (𝐴‘𝑥) = 0 ))) | ||
Theorem | deg1val 26149 | Value of the univariate degree as a supremum. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Jul-2019.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐴 supp 0 ), ℝ*, < )) | ||
Theorem | deg1lt 26150 | If the degree of a univariate polynomial is less than some index, then that coefficient must be zero. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐷‘𝐹) < 𝐺) → (𝐴‘𝐺) = 0 ) | ||
Theorem | deg1ge 26151 | Conversely, a nonzero coefficient sets a lower bound on the degree. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐴‘𝐺) ≠ 0 ) → 𝐺 ≤ (𝐷‘𝐹)) | ||
Theorem | coe1mul3 26152 | The coefficient vector of multiplication in the univariate polynomial ring, at indices high enough that at most one component can be active in the sum. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
⊢ 𝑌 = (Poly1‘𝑅) & ⊢ ∙ = (.r‘𝑌) & ⊢ · = (.r‘𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐼 ∈ ℕ0) & ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐼) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐽) ⇒ ⊢ (𝜑 → ((coe1‘(𝐹 ∙ 𝐺))‘(𝐼 + 𝐽)) = (((coe1‘𝐹)‘𝐼) · ((coe1‘𝐺)‘𝐽))) | ||
Theorem | coe1mul4 26153 | Value of the "leading" coefficient of a product of two nonzero polynomials. This will fail to actually be the leading coefficient only if it is zero (requiring the basic ring to contain zero divisors). (Contributed by Stefan O'Rear, 25-Mar-2015.) |
⊢ 𝑌 = (Poly1‘𝑅) & ⊢ ∙ = (.r‘𝑌) & ⊢ · = (.r‘𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 0 = (0g‘𝑌) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ≠ 0 ) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ≠ 0 ) ⇒ ⊢ (𝜑 → ((coe1‘(𝐹 ∙ 𝐺))‘((𝐷‘𝐹) + (𝐷‘𝐺))) = (((coe1‘𝐹)‘(𝐷‘𝐹)) · ((coe1‘𝐺)‘(𝐷‘𝐺)))) | ||
Theorem | deg1addle 26154 | The degree of a sum is at most the maximum of the degrees of the factors. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
⊢ 𝑌 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ + = (+g‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹))) | ||
Theorem | deg1addle2 26155 | If both factors have degree bounded by 𝐿, then the sum of the polynomials also has degree bounded by 𝐿. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
⊢ 𝑌 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ + = (+g‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐿 ∈ ℝ*) & ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐿) & ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐿) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ 𝐿) | ||
Theorem | deg1add 26156 | Exact degree of a sum of two polynomials of unequal degree. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑌 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ + = (+g‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → (𝐷‘𝐺) < (𝐷‘𝐹)) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 + 𝐺)) = (𝐷‘𝐹)) | ||
Theorem | deg1vscale 26157 | The degree of a scalar times a polynomial is at most the degree of the original polynomial. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
⊢ 𝑌 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐾) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐷‘𝐺)) | ||
Theorem | deg1vsca 26158 | The degree of a scalar times a polynomial is exactly the degree of the original polynomial when the scalar is not a zero divisor. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑌 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐸) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = (𝐷‘𝐺)) | ||
Theorem | deg1invg 26159 | The degree of the negated polynomial is the same as the original. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑌 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑁 = (invg‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷‘(𝑁‘𝐹)) = (𝐷‘𝐹)) | ||
Theorem | deg1suble 26160 | The degree of a difference of polynomials is bounded by the maximum of degrees. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
⊢ 𝑌 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ≤ if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹))) | ||
Theorem | deg1sub 26161 | Exact degree of a difference of two polynomials of unequal degree. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑌 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → (𝐷‘𝐺) < (𝐷‘𝐹)) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) = (𝐷‘𝐹)) | ||
Theorem | deg1mulle2 26162 | Produce a bound on the product of two univariate polynomials given bounds on the factors. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
⊢ 𝑌 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ · = (.r‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐽) & ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐾) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐽 + 𝐾)) | ||
Theorem | deg1sublt 26163 | Subtraction of two polynomials limited to the same degree with the same leading coefficient gives a polynomial with a smaller degree. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ − = (-g‘𝑃) & ⊢ (𝜑 → 𝐿 ∈ ℕ0) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐿) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐿) & ⊢ 𝐴 = (coe1‘𝐹) & ⊢ 𝐶 = (coe1‘𝐺) & ⊢ (𝜑 → ((coe1‘𝐹)‘𝐿) = ((coe1‘𝐺)‘𝐿)) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) < 𝐿) | ||
Theorem | deg1le0 26164 | A polynomial has nonpositive degree iff it is a constant. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐴 = (algSc‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((𝐷‘𝐹) ≤ 0 ↔ 𝐹 = (𝐴‘((coe1‘𝐹)‘0)))) | ||
Theorem | deg1sclle 26165 | A scalar polynomial has nonpositive degree. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (algSc‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾) → (𝐷‘(𝐴‘𝐹)) ≤ 0) | ||
Theorem | deg1scl 26166 | A nonzero scalar polynomial has zero degree. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (algSc‘𝑃) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐹 ≠ 0 ) → (𝐷‘(𝐴‘𝐹)) = 0) | ||
Theorem | deg1mul2 26167 | Degree of multiplication of two nonzero polynomials when the first leads with a nonzero-divisor coefficient. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ · = (.r‘𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ≠ 0 ) & ⊢ (𝜑 → ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝐸) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ≠ 0 ) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = ((𝐷‘𝐹) + (𝐷‘𝐺))) | ||
Theorem | deg1mul 26168 | Degree of multiplication of two nonzero polynomials in a domain. (Contributed by metakunt, 6-May-2025.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ · = (.r‘𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ Domn) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ≠ 0 ) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ≠ 0 ) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = ((𝐷‘𝐹) + (𝐷‘𝐺))) | ||
Theorem | deg1mul3 26169 | Degree of multiplication of a polynomial on the left by a nonzero-dividing scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Proof shortened by AV, 25-Jul-2019.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ · = (.r‘𝑃) & ⊢ 𝐴 = (algSc‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (𝐷‘((𝐴‘𝐹) · 𝐺)) = (𝐷‘𝐺)) | ||
Theorem | deg1mul3le 26170 | Degree of multiplication of a polynomial on the left by a scalar. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ · = (.r‘𝑃) & ⊢ 𝐴 = (algSc‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐵) → (𝐷‘((𝐴‘𝐹) · 𝐺)) ≤ (𝐷‘𝐺)) | ||
Theorem | deg1tmle 26171 | Limiting degree of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ 𝑁 = (mulGrp‘𝑃) & ⊢ ↑ = (.g‘𝑁) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) ≤ 𝐹) | ||
Theorem | deg1tm 26172 | Exact degree of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ 𝑁 = (mulGrp‘𝑃) & ⊢ ↑ = (.g‘𝑁) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) = 𝐹) | ||
Theorem | deg1pwle 26173 | Limiting degree of a variable power. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑁 = (mulGrp‘𝑃) & ⊢ ↑ = (.g‘𝑁) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐹 ↑ 𝑋)) ≤ 𝐹) | ||
Theorem | deg1pw 26174 | Exact degree of a variable power over a nontrivial ring. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑁 = (mulGrp‘𝑃) & ⊢ ↑ = (.g‘𝑁) ⇒ ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐹 ↑ 𝑋)) = 𝐹) | ||
Theorem | ply1nz 26175 | Univariate polynomials over a nonzero ring are a nonzero ring. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (𝑅 ∈ NzRing → 𝑃 ∈ NzRing) | ||
Theorem | ply1nzb 26176 | Univariate polynomials are nonzero iff the base is nonzero. Or in contraposition, the univariate polynomials over the zero ring are also zero. (Contributed by Mario Carneiro, 13-Jun-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ 𝑃 ∈ NzRing)) | ||
Theorem | ply1domn 26177 | Corollary of deg1mul2 26167: the univariate polynomials over a domain are a domain. This is true for multivariate but with a much more complicated proof. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (𝑅 ∈ Domn → 𝑃 ∈ Domn) | ||
Theorem | ply1idom 26178 | The ring of univariate polynomials over an integral domain is itself an integral domain. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (𝑅 ∈ IDomn → 𝑃 ∈ IDomn) | ||
Syntax | cmn1 26179 | Monic polynomials. |
class Monic1p | ||
Syntax | cuc1p 26180 | Unitic polynomials. |
class Unic1p | ||
Syntax | cq1p 26181 | Univariate polynomial quotient. |
class quot1p | ||
Syntax | cr1p 26182 | Univariate polynomial remainder. |
class rem1p | ||
Syntax | cig1p 26183 | Univariate polynomial ideal generator. |
class idlGen1p | ||
Definition | df-mon1 26184* | Define the set of monic univariate polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ Monic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) = (1r‘𝑟))}) | ||
Definition | df-uc1p 26185* | Define the set of unitic univariate polynomials, as the polynomials with an invertible leading coefficient. This is not a standard concept but is useful to us as the set of polynomials which can be used as the divisor in the polynomial division theorem ply1divalg 26191. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))}) | ||
Definition | df-q1p 26186* | Define the quotient of two univariate polynomials, which is guaranteed to exist and be unique by ply1divalg 26191. We actually use the reversed version for better harmony with our divisibility df-dvdsr 20373. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ quot1p = (𝑟 ∈ V ↦ ⦋(Poly1‘𝑟) / 𝑝⦌⦋(Base‘𝑝) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (℩𝑞 ∈ 𝑏 ((deg1‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) < ((deg1‘𝑟)‘𝑔)))) | ||
Definition | df-r1p 26187* | Define the remainder after dividing two univariate polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ rem1p = (𝑟 ∈ V ↦ ⦋(Base‘(Poly1‘𝑟)) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔)))) | ||
Definition | df-ig1p 26188* | Define a choice function for generators of ideals over a division ring; this is the unique monic polynomial of minimal degree in the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Sep-2020.) |
⊢ idlGen1p = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1‘𝑟)) ↦ if(𝑖 = {(0g‘(Poly1‘𝑟))}, (0g‘(Poly1‘𝑟)), (℩𝑔 ∈ (𝑖 ∩ (Monic1p‘𝑟))((deg1‘𝑟)‘𝑔) = inf(((deg1‘𝑟) “ (𝑖 ∖ {(0g‘(Poly1‘𝑟))})), ℝ, < ))))) | ||
Theorem | ply1divmo 26189* | Uniqueness of a quotient in a polynomial division. For polynomials 𝐹, 𝐺 such that 𝐺 ≠ 0 and the leading coefficient of 𝐺 is not a zero divisor, there is at most one polynomial 𝑞 which satisfies 𝐹 = (𝐺 · 𝑞) + 𝑟 where the degree of 𝑟 is less than the degree of 𝐺. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Revised by NM, 17-Jun-2017.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ − = (-g‘𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ ∙ = (.r‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ≠ 0 ) & ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ 𝐸) & ⊢ 𝐸 = (RLReg‘𝑅) ⇒ ⊢ (𝜑 → ∃*𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) | ||
Theorem | ply1divex 26190* | Lemma for ply1divalg 26191: existence part. (Contributed by Stefan O'Rear, 27-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ − = (-g‘𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ ∙ = (.r‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ≠ 0 ) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝐾) & ⊢ (𝜑 → (((coe1‘𝐺)‘(𝐷‘𝐺)) · 𝐼) = 1 ) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) | ||
Theorem | ply1divalg 26191* | The division algorithm for univariate polynomials over a ring. For polynomials 𝐹, 𝐺 such that 𝐺 ≠ 0 and the leading coefficient of 𝐺 is a unit, there are unique polynomials 𝑞 and 𝑟 = 𝐹 − (𝐺 · 𝑞) such that the degree of 𝑟 is less than the degree of 𝐺. (Contributed by Stefan O'Rear, 27-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ − = (-g‘𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ ∙ = (.r‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ≠ 0 ) & ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ 𝑈) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) | ||
Theorem | ply1divalg2 26192* | Reverse the order of multiplication in ply1divalg 26191 via the opposite ring. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ − = (-g‘𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ ∙ = (.r‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ≠ 0 ) & ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ 𝑈) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) < (𝐷‘𝐺)) | ||
Theorem | uc1pval 26193* | Value of the set of unitic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝐶 = (Unic1p‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ 𝐶 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)} | ||
Theorem | isuc1p 26194 | Being a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝐶 = (Unic1p‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)) | ||
Theorem | mon1pval 26195* | Value of the set of monic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑀 = (Monic1p‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ 𝑀 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} | ||
Theorem | ismon1p 26196 | Being a monic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑀 = (Monic1p‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝑀 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 )) | ||
Theorem | uc1pcl 26197 | Unitic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐶 = (Unic1p‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐶 → 𝐹 ∈ 𝐵) | ||
Theorem | mon1pcl 26198 | Monic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑀 = (Monic1p‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝑀 → 𝐹 ∈ 𝐵) | ||
Theorem | uc1pn0 26199 | Unitic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐶 = (Unic1p‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐶 → 𝐹 ≠ 0 ) | ||
Theorem | mon1pn0 26200 | Monic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝑀 = (Monic1p‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝑀 → 𝐹 ≠ 0 ) |
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