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**Theorem List for Metamath Proof Explorer -** 25801-25900 *Has distinct variable group(s)
Type	Label	Description
Statement

PART 14 BASIC REAL AND COMPLEX FUNCTIONS

14.1 Polynomials

14.1.1 Polynomial degrees

Syntax	cmdg 25801	Multivariate polynomial degree.
class mDeg

Syntax	cdg1 25802	Univariate polynomial degree.
class deg₁

Definition	df-mdeg 25803*	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 25938. (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 (0_g‘𝑟)) ↦ (ℂ_fld Σ_g ℎ)), ℝ^*, < )))

Definition	df-deg1 25804	Define the degree of a univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ deg₁ = (𝑟 ∈ V ↦ (1_o mDeg 𝑟))

Theorem	reldmmdeg 25805	Multivariate degree is a binary operation. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ Rel dom mDeg

Theorem	tdeglem1 25806*	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 ↦ (ℂ_fld Σ_g ℎ)) ⇒ ⊢ 𝐻:𝐴⟶ℕ₀

Theorem	tdeglem1OLD 25807*	Obsolete version of tdeglem1 25806 as of 7-Aug-2024. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐴 = {𝑚 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂ_fld Σ_g ℎ)) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐻:𝐴⟶ℕ₀)

Theorem	tdeglem3 25808*	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 ↦ (ℂ_fld Σ_g ℎ)) ⇒ ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐻‘(𝑋 ∘_f + 𝑌)) = ((𝐻‘𝑋) + (𝐻‘𝑌)))

Theorem	tdeglem3OLD 25809*	Obsolete version of tdeglem3 25808 as of 7-Aug-2024. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐴 = {𝑚 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂ_fld Σ_g ℎ)) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐻‘(𝑋 ∘_f + 𝑌)) = ((𝐻‘𝑋) + (𝐻‘𝑌)))

Theorem	tdeglem4 25810*	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 ↦ (ℂ_fld Σ_g ℎ)) ⇒ ⊢ (𝑋 ∈ 𝐴 → ((𝐻‘𝑋) = 0 ↔ 𝑋 = (𝐼 × {0})))

Theorem	tdeglem4OLD 25811*	Obsolete version of tdeglem4 25810 as of 7-Aug-2024. (Contributed by Stefan O'Rear, 29-Mar-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐴 = {𝑚 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂ_fld Σ_g ℎ)) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ((𝐻‘𝑋) = 0 ↔ 𝑋 = (𝐼 × {0})))

Theorem	tdeglem2 25812	Simplification of total degree for the univariate case. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ (ℎ ∈ (ℕ₀ ↑_m 1_o) ↦ (ℎ‘∅)) = (ℎ ∈ (ℕ₀ ↑_m 1_o) ↦ (ℂ_fld Σ_g ℎ))

Theorem	mdegfval 25813*	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 = (0_g‘𝑅) & ⊢ 𝐴 = {𝑚 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂ_fld Σ_g ℎ)) ⇒ ⊢ 𝐷 = (𝑓 ∈ 𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ^*, < ))

Theorem	mdegval 25814*	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 = (0_g‘𝑅) & ⊢ 𝐴 = {𝑚 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂ_fld Σ_g ℎ)) ⇒ ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ^*, < ))

Theorem	mdegleb 25815*	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 = (0_g‘𝑅) & ⊢ 𝐴 = {𝑚 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂ_fld Σ_g ℎ)) ⇒ ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ^*) → ((𝐷‘𝐹) ≤ 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐺 < (𝐻‘𝑥) → (𝐹‘𝑥) = 0 )))

Theorem	mdeglt 25816*	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 = (0_g‘𝑅) & ⊢ 𝐴 = {𝑚 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂ_fld Σ_g ℎ)) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → (𝐷‘𝐹) < (𝐻‘𝑋)) ⇒ ⊢ (𝜑 → (𝐹‘𝑋) = 0 )

Theorem	mdegldg 25817*	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 = (0_g‘𝑅) & ⊢ 𝐴 = {𝑚 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂ_fld Σ_g ℎ)) & ⊢ 𝑌 = (0_g‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → ∃𝑥 ∈ 𝐴 ((𝐹‘𝑥) ≠ 0 ∧ (𝐻‘𝑥) = (𝐷‘𝐹)))

Theorem	mdegxrcl 25818	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 25819	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 25820	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 25821	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 = (0_g‘𝑃) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (𝐷‘ 0 ) = -∞)

Theorem	mdegnn0cl 25822	Degree of a nonzero polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 0 = (0_g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ₀)

Theorem	degltlem1 25823	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 25824	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 25825	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 25826	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 25827	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 25828	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 25829*	Lemma for mdegmulle2 25830. (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 ↦ (ℂ_fld Σ_g 𝑏)) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐽 + 𝐾))

Theorem	mdegmulle2 25830	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 25831	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 = (1_o mDeg 𝑅)

Theorem	deg1xrf 25832	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 25833	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 25834	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 25835*	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 25836	Univariate polynomial degree respects protection. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ ( deg₁ ‘𝑅) = ( deg₁ ‘( I ‘𝑅))

Theorem	deg1propd 25837*	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 25838	Degree of the zero univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ 𝐷 = ( deg₁ ‘𝑅) & ⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 0 = (0_g‘𝑃) ⇒ ⊢ (𝑅 ∈ Ring → (𝐷‘ 0 ) = -∞)

Theorem	deg1nn0cl 25839	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 = (0_g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ₀)

Theorem	deg1n0ima 25840	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 = (0_g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ (𝑅 ∈ Ring → (𝐷 “ (𝐵 ∖ { 0 })) ⊆ ℕ₀)

Theorem	deg1nn0clb 25841	A polynomial is nonzero iff it has definite degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ 𝐷 = ( deg₁ ‘𝑅) & ⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 0 = (0_g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → (𝐹 ≠ 0 ↔ (𝐷‘𝐹) ∈ ℕ₀))

Theorem	deg1lt0 25842	A polynomial is zero iff it has negative degree. (Contributed by Stefan O'Rear, 1-Apr-2015.)
⊢ 𝐷 = ( deg₁ ‘𝑅) & ⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 0 = (0_g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((𝐷‘𝐹) < 0 ↔ 𝐹 = 0 ))

Theorem	deg1ldg 25843	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 = (0_g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑌 = (0_g‘𝑅) & ⊢ 𝐴 = (coe₁‘𝐹) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐴‘(𝐷‘𝐹)) ≠ 𝑌)

Theorem	deg1ldgn 25844	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 = (0_g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑌 = (0_g‘𝑅) & ⊢ 𝐴 = (coe₁‘𝐹) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ ℕ₀) & ⊢ (𝜑 → (𝐴‘𝑋) = 𝑌) ⇒ ⊢ (𝜑 → (𝐷‘𝐹) ≠ 𝑋)

Theorem	deg1ldgdomn 25845	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 = (0_g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝐴 = (coe₁‘𝐹) ⇒ ⊢ ((𝑅 ∈ Domn ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐴‘(𝐷‘𝐹)) ∈ 𝐸)

Theorem	deg1leb 25846*	Property of being of limited degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ 𝐷 = ( deg₁ ‘𝑅) & ⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0_g‘𝑅) & ⊢ 𝐴 = (coe₁‘𝐹) ⇒ ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ^*) → ((𝐷‘𝐹) ≤ 𝐺 ↔ ∀𝑥 ∈ ℕ₀ (𝐺 < 𝑥 → (𝐴‘𝑥) = 0 )))

Theorem	deg1val 25847	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 = (0_g‘𝑅) & ⊢ 𝐴 = (coe₁‘𝐹) ⇒ ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐴 supp 0 ), ℝ^*, < ))

Theorem	deg1lt 25848	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 = (0_g‘𝑅) & ⊢ 𝐴 = (coe₁‘𝐹) ⇒ ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ₀ ∧ (𝐷‘𝐹) < 𝐺) → (𝐴‘𝐺) = 0 )

Theorem	deg1ge 25849	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 = (0_g‘𝑅) & ⊢ 𝐴 = (coe₁‘𝐹) ⇒ ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ₀ ∧ (𝐴‘𝐺) ≠ 0 ) → 𝐺 ≤ (𝐷‘𝐹))

Theorem	coe1mul3 25850	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 25851	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 = (0_g‘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 25852	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 25853	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 25854	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 25855	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 25856	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 25857	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 = (inv_g‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷‘(𝑁‘𝐹)) = (𝐷‘𝐹))

Theorem	deg1suble 25858	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 25859	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 25860	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 25861	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 25862	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 25863	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 25864	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 = (0_g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐹 ≠ 0 ) → (𝐷‘(𝐴‘𝐹)) = 0)

Theorem	deg1mul2 25865	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 = (0_g‘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	deg1mul3 25866	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 25867	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 25868	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 25869	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 = (0_g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ₀) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) = 𝐹)

Theorem	deg1pwle 25870	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 25871	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 25872	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 25873	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 25874	Corollary of deg1mul2 25865: 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 25875	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)

14.1.2 The division algorithm for univariate polynomials

Syntax	cmn1 25876	Monic polynomials.
class Monic_1p

Syntax	cuc1p 25877	Unitic polynomials.
class Unic_1p

Syntax	cq1p 25878	Univariate polynomial quotient.
class quot_1p

Syntax	cr1p 25879	Univariate polynomial remainder.
class rem_1p

Syntax	cig1p 25880	Univariate polynomial ideal generator.
class idlGen_1p

Definition	df-mon1 25881*	Define the set of monic univariate polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ Monic_1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly₁‘𝑟)) ∣ (𝑓 ≠ (0_g‘(Poly₁‘𝑟)) ∧ ((coe₁‘𝑓)‘(( deg₁ ‘𝑟)‘𝑓)) = (1_r‘𝑟))})

Definition	df-uc1p 25882*	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 25888. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ Unic_1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly₁‘𝑟)) ∣ (𝑓 ≠ (0_g‘(Poly₁‘𝑟)) ∧ ((coe₁‘𝑓)‘(( deg₁ ‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))})

Definition	df-q1p 25883*	Define the quotient of two univariate polynomials, which is guaranteed to exist and be unique by ply1divalg 25888. We actually use the reversed version for better harmony with our divisibility df-dvdsr 20249. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ quot_1p = (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 25884*	Define the remainder after dividing two univariate polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ rem_1p = (𝑟 ∈ V ↦ ⦋(Base‘(Poly₁‘𝑟)) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓(-_g‘(Poly₁‘𝑟))((𝑓(quot_1p‘𝑟)𝑔)(._r‘(Poly₁‘𝑟))𝑔))))

Definition	df-ig1p 25885*	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.)
⊢ idlGen_1p = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly₁‘𝑟)) ↦ if(𝑖 = {(0_g‘(Poly₁‘𝑟))}, (0_g‘(Poly₁‘𝑟)), (℩𝑔 ∈ (𝑖 ∩ (Monic_1p‘𝑟))(( deg₁ ‘𝑟)‘𝑔) = inf((( deg₁ ‘𝑟) “ (𝑖 ∖ {(0_g‘(Poly₁‘𝑟))})), ℝ, < )))))

Theorem	ply1divmo 25886*	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 = (0_g‘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 25887*	Lemma for ply1divalg 25888: existence part. (Contributed by Stefan O'Rear, 27-Mar-2015.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 𝐷 = ( deg₁ ‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ − = (-_g‘𝑃) & ⊢ 0 = (0_g‘𝑃) & ⊢ ∙ = (._r‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ≠ 0 ) & ⊢ 1 = (1_r‘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 25888*	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 = (0_g‘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 25889*	Reverse the order of multiplication in ply1divalg 25888 via the opposite ring. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 𝐷 = ( deg₁ ‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ − = (-_g‘𝑃) & ⊢ 0 = (0_g‘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 25890*	Value of the set of unitic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0_g‘𝑃) & ⊢ 𝐷 = ( deg₁ ‘𝑅) & ⊢ 𝐶 = (Unic_1p‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ 𝐶 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe₁‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)}

Theorem	isuc1p 25891	Being a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0_g‘𝑃) & ⊢ 𝐷 = ( deg₁ ‘𝑅) & ⊢ 𝐶 = (Unic_1p‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe₁‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈))

Theorem	mon1pval 25892*	Value of the set of monic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0_g‘𝑃) & ⊢ 𝐷 = ( deg₁ ‘𝑅) & ⊢ 𝑀 = (Monic_1p‘𝑅) & ⊢ 1 = (1_r‘𝑅) ⇒ ⊢ 𝑀 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe₁‘𝑓)‘(𝐷‘𝑓)) = 1 )}

Theorem	ismon1p 25893	Being a monic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0_g‘𝑃) & ⊢ 𝐷 = ( deg₁ ‘𝑅) & ⊢ 𝑀 = (Monic_1p‘𝑅) & ⊢ 1 = (1_r‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝑀 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe₁‘𝐹)‘(𝐷‘𝐹)) = 1 ))

Theorem	uc1pcl 25894	Unitic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐶 = (Unic_1p‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐶 → 𝐹 ∈ 𝐵)

Theorem	mon1pcl 25895	Monic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑀 = (Monic_1p‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝑀 → 𝐹 ∈ 𝐵)

Theorem	uc1pn0 25896	Unitic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 0 = (0_g‘𝑃) & ⊢ 𝐶 = (Unic_1p‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐶 → 𝐹 ≠ 0 )

Theorem	mon1pn0 25897	Monic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 0 = (0_g‘𝑃) & ⊢ 𝑀 = (Monic_1p‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝑀 → 𝐹 ≠ 0 )

Theorem	uc1pdeg 25898	Unitic polynomials have nonnegative degrees. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝐷 = ( deg₁ ‘𝑅) & ⊢ 𝐶 = (Unic_1p‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐶) → (𝐷‘𝐹) ∈ ℕ₀)

Theorem	uc1pldg 25899	Unitic polynomials have unit leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝐷 = ( deg₁ ‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐶 = (Unic_1p‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐶 → ((coe₁‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)

Theorem	mon1pldg 25900	Unitic polynomials have one leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝐷 = ( deg₁ ‘𝑅) & ⊢ 1 = (1_r‘𝑅) & ⊢ 𝑀 = (Monic_1p‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝑀 → ((coe₁‘𝐹)‘(𝐷‘𝐹)) = 1 )

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