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Theorem List for Metamath Proof Explorer - 23801-23900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremftc2 23801* The Fundamental Theorem of Calculus, part two. If 𝐹 is a function continuous on [𝐴, 𝐵] and continuously differentiable on (𝐴, 𝐵), then the integral of the derivative of 𝐹 is equal to 𝐹(𝐵) − 𝐹(𝐴). This is part of Metamath 100 proof #15. (Contributed by Mario Carneiro, 2-Sep-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑 → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   (𝜑 → (ℝ D 𝐹) ∈ 𝐿1)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))       (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹𝐵) − (𝐹𝐴)))

Theoremftc2ditglem 23802* Lemma for ftc2ditg 23803. (Contributed by Mario Carneiro, 3-Sep-2014.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝐴 ∈ (𝑋[,]𝑌))    &   (𝜑𝐵 ∈ (𝑋[,]𝑌))    &   (𝜑 → (ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ))    &   (𝜑 → (ℝ D 𝐹) ∈ 𝐿1)    &   (𝜑𝐹 ∈ ((𝑋[,]𝑌)–cn→ℂ))       ((𝜑𝐴𝐵) → ⨜[𝐴𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹𝐵) − (𝐹𝐴)))

Theoremftc2ditg 23803* Directed integral analogue of ftc2 23801. (Contributed by Mario Carneiro, 3-Sep-2014.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝐴 ∈ (𝑋[,]𝑌))    &   (𝜑𝐵 ∈ (𝑋[,]𝑌))    &   (𝜑 → (ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ))    &   (𝜑 → (ℝ D 𝐹) ∈ 𝐿1)    &   (𝜑𝐹 ∈ ((𝑋[,]𝑌)–cn→ℂ))       (𝜑 → ⨜[𝐴𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹𝐵) − (𝐹𝐴)))

Theoremitgparts 23804* 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 𝐹𝐸 minus the integral of 𝐵 · 𝐶. (Contributed by Mario Carneiro, 3-Sep-2014.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑋𝑌)    &   (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))    &   ((𝜑𝑥 = 𝑋) → (𝐴 · 𝐶) = 𝐸)    &   ((𝜑𝑥 = 𝑌) → (𝐴 · 𝐶) = 𝐹)       (𝜑 → ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥 = ((𝐹𝐸) − ∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥))

Theoremitgsubstlem 23805* Lemma for itgsubst 23806. (Contributed by Mario Carneiro, 12-Sep-2014.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑋𝑌)    &   (𝜑𝑍 ∈ ℝ*)    &   (𝜑𝑊 ∈ 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 = 𝐴𝐶 = 𝐸)    &   (𝑥 = 𝑋𝐴 = 𝐾)    &   (𝑥 = 𝑌𝐴 = 𝐿)    &   (𝜑𝑀 ∈ (𝑍(,)𝑊))    &   (𝜑𝑁 ∈ (𝑍(,)𝑊))    &   ((𝜑𝑥 ∈ (𝑋[,]𝑌)) → 𝐴 ∈ (𝑀(,)𝑁))       (𝜑 → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)

Theoremitgsubst 23806* 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 23805, 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.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑋𝑌)    &   (𝜑𝑍 ∈ ℝ*)    &   (𝜑𝑊 ∈ 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 = 𝐴𝐶 = 𝐸)    &   (𝑥 = 𝑋𝐴 = 𝐾)    &   (𝑥 = 𝑌𝐴 = 𝐿)       (𝜑 → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)

PART 14  BASIC REAL AND COMPLEX FUNCTIONS

14.1  Polynomials

14.1.1  Polynomial degrees

Syntaxcmdg 23807 Multivariate polynomial degree.
class mDeg

Syntaxcdg1 23808 Univariate polynomial degree.
class deg1

Definitiondf-mdeg 23809* 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 23941. (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 ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg )), ℝ*, < )))

Definitiondf-deg1 23810 Define the degree of a univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1 = (𝑟 ∈ V ↦ (1𝑜 mDeg 𝑟))

Theoremreldmmdeg 23811 Multivariate degree is a binary operation. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Rel dom mDeg

Theoremtdeglem1 23812* Functionality of the total degree helper function. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
𝐴 = {𝑚 ∈ (ℕ0𝑚 𝐼) ∣ (𝑚 “ ℕ) ∈ Fin}    &   𝐻 = (𝐴 ↦ (ℂfld Σg ))       (𝐼𝑉𝐻:𝐴⟶ℕ0)

Theoremtdeglem3 23813* Additivity of the total degree helper function. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
𝐴 = {𝑚 ∈ (ℕ0𝑚 𝐼) ∣ (𝑚 “ ℕ) ∈ Fin}    &   𝐻 = (𝐴 ↦ (ℂfld Σg ))       ((𝐼𝑉𝑋𝐴𝑌𝐴) → (𝐻‘(𝑋𝑓 + 𝑌)) = ((𝐻𝑋) + (𝐻𝑌)))

Theoremtdeglem4 23814* There is only one multi-index with total degree 0. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝐴 = {𝑚 ∈ (ℕ0𝑚 𝐼) ∣ (𝑚 “ ℕ) ∈ Fin}    &   𝐻 = (𝐴 ↦ (ℂfld Σg ))       ((𝐼𝑉𝑋𝐴) → ((𝐻𝑋) = 0 ↔ 𝑋 = (𝐼 × {0})))

Theoremtdeglem2 23815 Simplification of total degree for the univariate case. (Contributed by Stefan O'Rear, 23-Mar-2015.)
( ∈ (ℕ0𝑚 1𝑜) ↦ (‘∅)) = ( ∈ (ℕ0𝑚 1𝑜) ↦ (ℂfld Σg ))

Theoremmdegfval 23816* 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 = (0g𝑅)    &   𝐴 = {𝑚 ∈ (ℕ0𝑚 𝐼) ∣ (𝑚 “ ℕ) ∈ Fin}    &   𝐻 = (𝐴 ↦ (ℂfld Σg ))       𝐷 = (𝑓𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < ))

Theoremmdegval 23817* 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 = (0g𝑅)    &   𝐴 = {𝑚 ∈ (ℕ0𝑚 𝐼) ∣ (𝑚 “ ℕ) ∈ Fin}    &   𝐻 = (𝐴 ↦ (ℂfld Σg ))       (𝐹𝐵 → (𝐷𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ*, < ))

Theoremmdegleb 23818* 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 = (0g𝑅)    &   𝐴 = {𝑚 ∈ (ℕ0𝑚 𝐼) ∣ (𝑚 “ ℕ) ∈ Fin}    &   𝐻 = (𝐴 ↦ (ℂfld Σg ))       ((𝐹𝐵𝐺 ∈ ℝ*) → ((𝐷𝐹) ≤ 𝐺 ↔ ∀𝑥𝐴 (𝐺 < (𝐻𝑥) → (𝐹𝑥) = 0 )))

Theoremmdeglt 23819* 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 = (0g𝑅)    &   𝐴 = {𝑚 ∈ (ℕ0𝑚 𝐼) ∣ (𝑚 “ ℕ) ∈ Fin}    &   𝐻 = (𝐴 ↦ (ℂfld Σg ))    &   (𝜑𝐹𝐵)    &   (𝜑𝑋𝐴)    &   (𝜑 → (𝐷𝐹) < (𝐻𝑋))       (𝜑 → (𝐹𝑋) = 0 )

Theoremmdegldg 23820* 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 = (0g𝑅)    &   𝐴 = {𝑚 ∈ (ℕ0𝑚 𝐼) ∣ (𝑚 “ ℕ) ∈ Fin}    &   𝐻 = (𝐴 ↦ (ℂfld Σg ))    &   𝑌 = (0g𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐹𝑌) → ∃𝑥𝐴 ((𝐹𝑥) ≠ 0 ∧ (𝐻𝑥) = (𝐷𝐹)))

Theoremmdegxrcl 23821 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)       (𝐹𝐵 → (𝐷𝐹) ∈ ℝ*)

Theoremmdegxrf 23822 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*

Theoremmdegcl 23823 Sharp closure for multivariate polynomials. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = (𝐼 mDeg 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)       (𝐹𝐵 → (𝐷𝐹) ∈ (ℕ0 ∪ {-∞}))

Theoremmdeg0 23824 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 = (0g𝑃)       ((𝐼𝑉𝑅 ∈ Ring) → (𝐷0 ) = -∞)

Theoremmdegnn0cl 23825 Degree of a nonzero polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = (𝐼 mDeg 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &    0 = (0g𝑃)    &   𝐵 = (Base‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐹0 ) → (𝐷𝐹) ∈ ℕ0)

Theoremdegltlem1 23826 Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 23-Mar-2015.)
((𝑋 ∈ (ℕ0 ∪ {-∞}) ∧ 𝑌 ∈ ℤ) → (𝑋 < 𝑌𝑋 ≤ (𝑌 − 1)))

Theoremdegltp1le 23827 Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 1-Apr-2015.)
((𝑋 ∈ (ℕ0 ∪ {-∞}) ∧ 𝑌 ∈ ℤ) → (𝑋 < (𝑌 + 1) ↔ 𝑋𝑌))

Theoremmdegaddle 23828 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)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    + = (+g𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ if((𝐷𝐹) ≤ (𝐷𝐺), (𝐷𝐺), (𝐷𝐹)))

Theoremmdegvscale 23829 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)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝑌)    &   (𝜑𝐹𝐾)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐷𝐺))

Theoremmdegvsca 23830 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)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &   𝐸 = (RLReg‘𝑅)    &    · = ( ·𝑠𝑌)    &   (𝜑𝐹𝐸)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐷‘(𝐹 · 𝐺)) = (𝐷𝐺))

Theoremmdegle0 23831 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)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &   𝐴 = (algSc‘𝑌)    &   (𝜑𝐹𝐵)       (𝜑 → ((𝐷𝐹) ≤ 0 ↔ 𝐹 = (𝐴‘(𝐹‘(𝐼 × {0})))))

Theoremmdegmullem 23832* Lemma for mdegmulle2 23833. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑌 = (𝐼 mPoly 𝑅)    &   𝐷 = (𝐼 mDeg 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    · = (.r𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑 → (𝐷𝐹) ≤ 𝐽)    &   (𝜑 → (𝐷𝐺) ≤ 𝐾)    &   𝐴 = {𝑎 ∈ (ℕ0𝑚 𝐼) ∣ (𝑎 “ ℕ) ∈ Fin}    &   𝐻 = (𝑏𝐴 ↦ (ℂfld Σg 𝑏))       (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐽 + 𝐾))

Theoremmdegmulle2 23833 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)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    · = (.r𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑 → (𝐷𝐹) ≤ 𝐽)    &   (𝜑 → (𝐷𝐺) ≤ 𝐾)       (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐽 + 𝐾))

Theoremdeg1fval 23834 Relate univariate polynomial degree to multivariate. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐷 = ( deg1𝑅)       𝐷 = (1𝑜 mDeg 𝑅)

Theoremdeg1xrf 23835 Functionality of univariate polynomial degree, weak range. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)       𝐷:𝐵⟶ℝ*

Theoremdeg1xrcl 23836 Closure of univariate polynomial degree in extended reals. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)       (𝐹𝐵 → (𝐷𝐹) ∈ ℝ*)

Theoremdeg1cl 23837 Sharp closure of univariate polynomial degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)       (𝐹𝐵 → (𝐷𝐹) ∈ (ℕ0 ∪ {-∞}))

Theoremmdegpropd 23838* Property deduction for polynomial degree. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑𝐵 = (Base‘𝑆))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑅)𝑦) = (𝑥(+g𝑆)𝑦))       (𝜑 → (𝐼 mDeg 𝑅) = (𝐼 mDeg 𝑆))

Theoremdeg1fvi 23839 Univariate polynomial degree respects protection. (Contributed by Stefan O'Rear, 28-Mar-2015.)
( deg1𝑅) = ( deg1 ‘( I ‘𝑅))

Theoremdeg1propd 23840* Property deduction for polynomial degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑𝐵 = (Base‘𝑆))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑅)𝑦) = (𝑥(+g𝑆)𝑦))       (𝜑 → ( deg1𝑅) = ( deg1𝑆))

Theoremdeg1z 23841 Degree of the zero univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)       (𝑅 ∈ Ring → (𝐷0 ) = -∞)

Theoremdeg1nn0cl 23842 Degree of a nonzero univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝐵 = (Base‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐹0 ) → (𝐷𝐹) ∈ ℕ0)

Theoremdeg1n0ima 23843 Degree image of a set of polynomials which does not include zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝐵 = (Base‘𝑃)       (𝑅 ∈ Ring → (𝐷 “ (𝐵 ∖ { 0 })) ⊆ ℕ0)

Theoremdeg1nn0clb 23844 A polynomial is nonzero iff it has definite degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝐵 = (Base‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐵) → (𝐹0 ↔ (𝐷𝐹) ∈ ℕ0))

Theoremdeg1lt0 23845 A polynomial is zero iff it has negative degree. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝐵 = (Base‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐵) → ((𝐷𝐹) < 0 ↔ 𝐹 = 0 ))

Theoremdeg1ldg 23846 A nonzero univariate polynomial always has a nonzero leading coefficient. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝐵 = (Base‘𝑃)    &   𝑌 = (0g𝑅)    &   𝐴 = (coe1𝐹)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐹0 ) → (𝐴‘(𝐷𝐹)) ≠ 𝑌)

Theoremdeg1ldgn 23847 An index at which a polynomial is zero, cannot be its degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝐵 = (Base‘𝑃)    &   𝑌 = (0g𝑅)    &   𝐴 = (coe1𝐹)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑𝑋 ∈ ℕ0)    &   (𝜑 → (𝐴𝑋) = 𝑌)       (𝜑 → (𝐷𝐹) ≠ 𝑋)

Theoremdeg1ldgdomn 23848 A nonzero univariate polynomial over a domain always has a nonzero-divisor leading coefficient. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝐵 = (Base‘𝑃)    &   𝐸 = (RLReg‘𝑅)    &   𝐴 = (coe1𝐹)       ((𝑅 ∈ Domn ∧ 𝐹𝐵𝐹0 ) → (𝐴‘(𝐷𝐹)) ∈ 𝐸)

Theoremdeg1leb 23849* Property of being of limited degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &   𝐴 = (coe1𝐹)       ((𝐹𝐵𝐺 ∈ ℝ*) → ((𝐷𝐹) ≤ 𝐺 ↔ ∀𝑥 ∈ ℕ0 (𝐺 < 𝑥 → (𝐴𝑥) = 0 )))

Theoremdeg1val 23850 Value of the univariate degree as a supremum. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Jul-2019.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &   𝐴 = (coe1𝐹)       (𝐹𝐵 → (𝐷𝐹) = sup((𝐴 supp 0 ), ℝ*, < ))

Theoremdeg1lt 23851 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 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &   𝐴 = (coe1𝐹)       ((𝐹𝐵𝐺 ∈ ℕ0 ∧ (𝐷𝐹) < 𝐺) → (𝐴𝐺) = 0 )

Theoremdeg1ge 23852 Conversely, a nonzero coefficient sets a lower bound on the degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &   𝐴 = (coe1𝐹)       ((𝐹𝐵𝐺 ∈ ℕ0 ∧ (𝐴𝐺) ≠ 0 ) → 𝐺 ≤ (𝐷𝐹))

Theoremcoe1mul3 23853 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 = (Poly1𝑅)    &    = (.r𝑌)    &    · = (.r𝑅)    &   𝐵 = (Base‘𝑌)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑𝐼 ∈ ℕ0)    &   (𝜑 → (𝐷𝐹) ≤ 𝐼)    &   (𝜑𝐺𝐵)    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑 → (𝐷𝐺) ≤ 𝐽)       (𝜑 → ((coe1‘(𝐹 𝐺))‘(𝐼 + 𝐽)) = (((coe1𝐹)‘𝐼) · ((coe1𝐺)‘𝐽)))

Theoremcoe1mul4 23854 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 = (Poly1𝑅)    &    = (.r𝑌)    &    · = (.r𝑅)    &   𝐵 = (Base‘𝑌)    &   𝐷 = ( deg1𝑅)    &    0 = (0g𝑌)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑𝐹0 )    &   (𝜑𝐺𝐵)    &   (𝜑𝐺0 )       (𝜑 → ((coe1‘(𝐹 𝐺))‘((𝐷𝐹) + (𝐷𝐺))) = (((coe1𝐹)‘(𝐷𝐹)) · ((coe1𝐺)‘(𝐷𝐺))))

Theoremdeg1addle 23855 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 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    + = (+g𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ if((𝐷𝐹) ≤ (𝐷𝐺), (𝐷𝐺), (𝐷𝐹)))

Theoremdeg1addle2 23856 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 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    + = (+g𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑𝐿 ∈ ℝ*)    &   (𝜑 → (𝐷𝐹) ≤ 𝐿)    &   (𝜑 → (𝐷𝐺) ≤ 𝐿)       (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ 𝐿)

Theoremdeg1add 23857 Exact degree of a sum of two polynomials of unequal degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑌 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    + = (+g𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑 → (𝐷𝐺) < (𝐷𝐹))       (𝜑 → (𝐷‘(𝐹 + 𝐺)) = (𝐷𝐹))

Theoremdeg1vscale 23858 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 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝑌)    &   (𝜑𝐹𝐾)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐷𝐺))

Theoremdeg1vsca 23859 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 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &   𝐸 = (RLReg‘𝑅)    &    · = ( ·𝑠𝑌)    &   (𝜑𝐹𝐸)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐷‘(𝐹 · 𝐺)) = (𝐷𝐺))

Theoremdeg1invg 23860 The degree of the negated polynomial is the same as the original. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑌 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &   𝑁 = (invg𝑌)    &   (𝜑𝐹𝐵)       (𝜑 → (𝐷‘(𝑁𝐹)) = (𝐷𝐹))

Theoremdeg1suble 23861 The degree of a difference of polynomials is bounded by the maximum of degrees. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑌 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    = (-g𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐷‘(𝐹 𝐺)) ≤ if((𝐷𝐹) ≤ (𝐷𝐺), (𝐷𝐺), (𝐷𝐹)))

Theoremdeg1sub 23862 Exact degree of a difference of two polynomials of unequal degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑌 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    = (-g𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑 → (𝐷𝐺) < (𝐷𝐹))       (𝜑 → (𝐷‘(𝐹 𝐺)) = (𝐷𝐹))

Theoremdeg1mulle2 23863 Produce a bound on the product of two univariate polynomials given bounds on the factors. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑌 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    · = (.r𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑 → (𝐷𝐹) ≤ 𝐽)    &   (𝜑 → (𝐷𝐺) ≤ 𝐾)       (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐽 + 𝐾))

Theoremdeg1sublt 23864 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 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    = (-g𝑃)    &   (𝜑𝐿 ∈ ℕ0)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑 → (𝐷𝐹) ≤ 𝐿)    &   (𝜑𝐺𝐵)    &   (𝜑 → (𝐷𝐺) ≤ 𝐿)    &   𝐴 = (coe1𝐹)    &   𝐶 = (coe1𝐺)    &   (𝜑 → ((coe1𝐹)‘𝐿) = ((coe1𝐺)‘𝐿))       (𝜑 → (𝐷‘(𝐹 𝐺)) < 𝐿)

Theoremdeg1le0 23865 A polynomial has nonpositive degree iff it is a constant. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (algSc‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐵) → ((𝐷𝐹) ≤ 0 ↔ 𝐹 = (𝐴‘((coe1𝐹)‘0))))

Theoremdeg1sclle 23866 A scalar polynomial has nonpositive degree. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐾) → (𝐷‘(𝐴𝐹)) ≤ 0)

Theoremdeg1scl 23867 A nonzero scalar polynomial has zero degree. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝐹𝐾𝐹0 ) → (𝐷‘(𝐴𝐹)) = 0)

Theoremdeg1mul2 23868 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 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐸 = (RLReg‘𝑅)    &   𝐵 = (Base‘𝑃)    &    · = (.r𝑃)    &    0 = (0g𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑𝐹0 )    &   (𝜑 → ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝐸)    &   (𝜑𝐺𝐵)    &   (𝜑𝐺0 )       (𝜑 → (𝐷‘(𝐹 · 𝐺)) = ((𝐷𝐹) + (𝐷𝐺)))

Theoremdeg1mul3 23869 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 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐸 = (RLReg‘𝑅)    &   𝐵 = (Base‘𝑃)    &    · = (.r𝑃)    &   𝐴 = (algSc‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐸𝐺𝐵) → (𝐷‘((𝐴𝐹) · 𝐺)) = (𝐷𝐺))

Theoremdeg1mul3le 23870 Degree of multiplication of a polynomial on the left by a scalar. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝑃)    &    · = (.r𝑃)    &   𝐴 = (algSc‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐾𝐺𝐵) → (𝐷‘((𝐴𝐹) · 𝐺)) ≤ (𝐷𝐺))

Theoremdeg1tmle 23871 Limiting degree of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)       ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 𝑋))) ≤ 𝐹)

Theoremdeg1tm 23872 Exact degree of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ (𝐶𝐾𝐶0 ) ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 𝑋))) = 𝐹)

Theoremdeg1pwle 23873 Limiting degree of a variable power. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)       ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐹 𝑋)) ≤ 𝐹)

Theoremdeg1pw 23874 Exact degree of a variable power over a nontrivial ring. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)       ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐹 𝑋)) = 𝐹)

Theoremply1nz 23875 Univariate polynomials over a nonzero ring are a nonzero ring. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ NzRing → 𝑃 ∈ NzRing)

Theoremply1nzb 23876 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 = (Poly1𝑅)       (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ 𝑃 ∈ NzRing))

Theoremply1domn 23877 Corollary of deg1mul2 23868: 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 = (Poly1𝑅)       (𝑅 ∈ Domn → 𝑃 ∈ Domn)

Theoremply1idom 23878 The ring of univariate polynomials over an integral domain is itself an integral domain. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ IDomn → 𝑃 ∈ IDomn)

14.1.2  The division algorithm for univariate polynomials

Syntaxcmn1 23879 Monic polynomials.
class Monic1p

Syntaxcuc1p 23880 Unitic polynomials.
class Unic1p

Syntaxcq1p 23881 Univariate polynomial quotient.
class quot1p

Syntaxcr1p 23882 Univariate polynomial remainder.
class rem1p

Syntaxcig1p 23883 Univariate polynomial ideal generator.
class idlGen1p

Definitiondf-mon1 23884* Define the set of monic univariate polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Monic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) = (1r𝑟))})

Definitiondf-uc1p 23885* 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 23891. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟))})

Definitiondf-q1p 23886* Define the quotient of two univariate polynomials, which is guaranteed to exist and be unique by ply1divalg 23891. We actually use the reversed version for better harmony with our divisibility df-dvdsr 18635. (Contributed by Stefan O'Rear, 28-Mar-2015.)
quot1p = (𝑟 ∈ V ↦ (Poly1𝑟) / 𝑝(Base‘𝑝) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔))))

Definitiondf-r1p 23887* Define the remainder after dividing two univariate polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
rem1p = (𝑟 ∈ V ↦ (Base‘(Poly1𝑟)) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))))

Definitiondf-ig1p 23888* 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‘(Poly1𝑟)) ↦ if(𝑖 = {(0g‘(Poly1𝑟))}, (0g‘(Poly1𝑟)), (𝑔 ∈ (𝑖 ∩ (Monic1p𝑟))(( deg1𝑟)‘𝑔) = inf((( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})), ℝ, < )))))

Theoremply1divmo 23889* 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 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   𝐵 = (Base‘𝑃)    &    = (-g𝑃)    &    0 = (0g𝑃)    &    = (.r𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑𝐺0 )    &   (𝜑 → ((coe1𝐺)‘(𝐷𝐺)) ∈ 𝐸)    &   𝐸 = (RLReg‘𝑅)       (𝜑 → ∃*𝑞𝐵 (𝐷‘(𝐹 (𝐺 𝑞))) < (𝐷𝐺))

Theoremply1divex 23890* Lemma for ply1divalg 23891: existence part. (Contributed by Stefan O'Rear, 27-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   𝐵 = (Base‘𝑃)    &    = (-g𝑃)    &    0 = (0g𝑃)    &    = (.r𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑𝐺0 )    &    1 = (1r𝑅)    &   𝐾 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝐼𝐾)    &   (𝜑 → (((coe1𝐺)‘(𝐷𝐺)) · 𝐼) = 1 )       (𝜑 → ∃𝑞𝐵 (𝐷‘(𝐹 (𝐺 𝑞))) < (𝐷𝐺))

Theoremply1divalg 23891* 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 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   𝐵 = (Base‘𝑃)    &    = (-g𝑃)    &    0 = (0g𝑃)    &    = (.r𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑𝐺0 )    &   (𝜑 → ((coe1𝐺)‘(𝐷𝐺)) ∈ 𝑈)    &   𝑈 = (Unit‘𝑅)       (𝜑 → ∃!𝑞𝐵 (𝐷‘(𝐹 (𝐺 𝑞))) < (𝐷𝐺))

Theoremply1divalg2 23892* Reverse the order of multiplication in ply1divalg 23891 via the opposite ring. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   𝐵 = (Base‘𝑃)    &    = (-g𝑃)    &    0 = (0g𝑃)    &    = (.r𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑𝐺0 )    &   (𝜑 → ((coe1𝐺)‘(𝐷𝐺)) ∈ 𝑈)    &   𝑈 = (Unit‘𝑅)       (𝜑 → ∃!𝑞𝐵 (𝐷‘(𝐹 (𝑞 𝐺))) < (𝐷𝐺))

Theoremuc1pval 23893* Value of the set of unitic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑃)    &   𝐷 = ( deg1𝑅)    &   𝐶 = (Unic1p𝑅)    &   𝑈 = (Unit‘𝑅)       𝐶 = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)}

Theoremisuc1p 23894 Being a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑃)    &   𝐷 = ( deg1𝑅)    &   𝐶 = (Unic1p𝑅)    &   𝑈 = (Unit‘𝑅)       (𝐹𝐶 ↔ (𝐹𝐵𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈))

Theoremmon1pval 23895* Value of the set of monic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑃)    &   𝐷 = ( deg1𝑅)    &   𝑀 = (Monic1p𝑅)    &    1 = (1r𝑅)       𝑀 = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )}

Theoremismon1p 23896 Being a monic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑃)    &   𝐷 = ( deg1𝑅)    &   𝑀 = (Monic1p𝑅)    &    1 = (1r𝑅)       (𝐹𝑀 ↔ (𝐹𝐵𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) = 1 ))

Theoremuc1pcl 23897 Unitic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐶 = (Unic1p𝑅)       (𝐹𝐶𝐹𝐵)

Theoremmon1pcl 23898 Monic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑀 = (Monic1p𝑅)       (𝐹𝑀𝐹𝐵)

Theoremuc1pn0 23899 Unitic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝐶 = (Unic1p𝑅)       (𝐹𝐶𝐹0 )

Theoremmon1pn0 23900 Monic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝑀 = (Monic1p𝑅)       (𝐹𝑀𝐹0 )

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