P. 337 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 33601-33700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dfprm3 33601	The (positive) prime elements of the integer ring are the prime numbers. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ ℙ = (ℕ ∩ (RPrime‘ℤring))
Theorem	zringfrac 33602*	The field of fractions of the ring of integers is isomorphic to the field of the rational numbers. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 𝑄 = (ℂfld ↾s ℚ)    &   ⊢ ∼ = (ℤring ~RL (ℤ ∖ {0}))    &   ⊢ 𝐹 = (𝑞 ∈ ℚ ↦ [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ )    ⇒   ⊢ 𝐹 ∈ (𝑄 RingIso ( Frac ‘ℤring))
21.3.10.46  Associative Algebra
Theorem	assaassd 33603	Left-associative property of an associative algebra, deduction version. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ × = (.r‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ AssAlg)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))
Theorem	assaassrd 33604	Right-associative property of an associative algebra, deduction version. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ × = (.r‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ AssAlg)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))
21.3.10.47  Univariate Polynomials
Theorem	0ringmon1p 33605	There are no monic polynomials over a zero ring. (Contributed by Thierry Arnoux, 5-Feb-2025.)
⊢ 𝑀 = (Monic1p‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → (♯‘𝐵) = 1)    ⇒   ⊢ (𝜑 → 𝑀 = ∅)
Theorem	fply1 33606	Conditions for a function to be a univariate polynomial. (Contributed by Thierry Arnoux, 19-Aug-2023.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = (Base‘(Poly1‘𝑅))    &   ⊢ (𝜑 → 𝐹:(ℕ0 ↑m 1o)⟶𝐵)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → 𝐹 ∈ 𝑃)
Theorem	ply1lvec 33607	In a division ring, the univariate polynomials form a vector space. (Contributed by Thierry Arnoux, 19-Feb-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    ⇒   ⊢ (𝜑 → 𝑃 ∈ LVec)
Theorem	evls1fn 33608	Functionality of the subring polynomial evaluation. (Contributed by Thierry Arnoux, 9-Feb-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆))    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    ⇒   ⊢ (𝜑 → 𝑂 Fn 𝑈)
Theorem	evls1dm 33609	The domain of the subring polynomial evaluation function. (Contributed by Thierry Arnoux, 9-Feb-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆))    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    ⇒   ⊢ (𝜑 → dom 𝑂 = 𝑈)
Theorem	evls1fvf 33610	The subring evaluation function for a univariate polynomial as a function, with domain and codomain. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆))    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑂‘𝑄):𝐵⟶𝐵)
Theorem	evl1fvf 33611	The univariate polynomial evaluation function as a function, with domain and codomain. (Contributed by Thierry Arnoux, 8-Jun-2025.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑂‘𝑄):𝐵⟶𝐵)
Theorem	evl1fpws 33612*	Evaluation of a univariate polynomial as a function in a power series. (Contributed by Thierry Arnoux, 23-Jan-2025.)
⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑈)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑅))    &   ⊢ 𝐴 = (coe1‘𝑀)    ⇒   ⊢ (𝜑 → (𝑂‘𝑀) = (𝑥 ∈ 𝐵 ↦ (𝑅 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))))))
Theorem	ressply1evls1 33613	Subring evaluation of a univariate polynomial is the same as the subring evaluation in the bigger ring. (Contributed by Thierry Arnoux, 14-Nov-2025.)
⊢ 𝐺 = (𝐸 ↾s 𝑅)    &   ⊢ 𝑂 = (𝐸 evalSub1 𝑆)    &   ⊢ 𝑄 = (𝐺 evalSub1 𝑆)    &   ⊢ 𝑃 = (Poly1‘𝐾)    &   ⊢ 𝐾 = (𝐸 ↾s 𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝐸))    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝐺))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑄‘𝐹) = ((𝑂‘𝐹) ↾ 𝑅))
Theorem	ressdeg1 33614	The degree of a univariate polynomial in a structure restriction. (Contributed by Thierry Arnoux, 20-Jan-2025.)
⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑈 = (Poly1‘𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    ⇒   ⊢ (𝜑 → (𝐷‘𝑃) = ((deg1‘𝐻)‘𝑃))
Theorem	ressply10g 33615	A restricted polynomial algebra has the same group identity (zero polynomial). (Contributed by Thierry Arnoux, 20-Jan-2025.)
⊢ 𝑆 = (Poly1‘𝑅)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (Poly1‘𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝑍 = (0g‘𝑆)    ⇒   ⊢ (𝜑 → 𝑍 = (0g‘𝑈))
Theorem	ressply1mon1p 33616	The monic polynomials of a restricted polynomial algebra. (Contributed by Thierry Arnoux, 21-Jan-2025.)
⊢ 𝑆 = (Poly1‘𝑅)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (Poly1‘𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝑀 = (Monic1p‘𝑅)    &   ⊢ 𝑁 = (Monic1p‘𝐻)    ⇒   ⊢ (𝜑 → 𝑁 = (𝐵 ∩ 𝑀))
Theorem	ressply1invg 33617	An element of a restricted polynomial algebra has the same group inverse. (Contributed by Thierry Arnoux, 30-Jan-2025.)
⊢ 𝑆 = (Poly1‘𝑅)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (Poly1‘𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝑃 = (𝑆 ↾s 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((invg‘𝑈)‘𝑋) = ((invg‘𝑃)‘𝑋))
Theorem	ressply1sub 33618	A restricted polynomial algebra has the same subtraction operation. (Contributed by Thierry Arnoux, 30-Jan-2025.)
⊢ 𝑆 = (Poly1‘𝑅)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (Poly1‘𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝑃 = (𝑆 ↾s 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋(-g‘𝑈)𝑌) = (𝑋(-g‘𝑃)𝑌))
Theorem	ressasclcl 33619	Closure of the univariate polynomial evaluation for scalars. (Contributed by Thierry Arnoux, 22-Jun-2025.)
⊢ 𝑊 = (Poly1‘𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑅)    ⇒   ⊢ (𝜑 → (𝐴‘𝑋) ∈ 𝐵)
Theorem	evls1subd 33620	Univariate polynomial evaluation of a difference of polynomials. (Contributed by Thierry Arnoux, 25-Apr-2025.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑊 = (Poly1‘𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐷 = (-g‘𝑊)    &   ⊢ − = (-g‘𝑆)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑄‘(𝑀𝐷𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) − ((𝑄‘𝑁)‘𝐶)))
Theorem	deg1le0eq0 33621	A polynomial with nonpositive degree is the zero polynomial iff its constant term is zero. Biconditional version of deg1scl 26066. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑂 = (0g‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐷‘𝐹) ≤ 0)    ⇒   ⊢ (𝜑 → (𝐹 = 𝑂 ↔ ((coe1‘𝐹)‘0) = 0 ))
Theorem	ply1asclunit 33622	A nonzero scalar polynomial over a field 𝐹 is a unit. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 𝑃 = (Poly1‘𝐹)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 0 = (0g‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ Field)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝐴‘𝑌) ∈ (Unit‘𝑃))
Theorem	ply1unit 33623	In a field 𝐹, a polynomial 𝐶 is a unit iff it has degree 0. This corresponds to the nonzero scalars, see ply1asclunit 33622. (Contributed by Thierry Arnoux, 25-Apr-2025.)
⊢ 𝑃 = (Poly1‘𝐹)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 0 = (0g‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ Field)    &   ⊢ 𝐷 = (deg1‘𝐹)    &   ⊢ (𝜑 → 𝐶 ∈ (Base‘𝑃))    ⇒   ⊢ (𝜑 → (𝐶 ∈ (Unit‘𝑃) ↔ (𝐷‘𝐶) = 0))
Theorem	evl1deg1 33624	Evaluation of a univariate polynomial of degree 1. (Contributed by Thierry Arnoux, 8-Jun-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ · = (.r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝐶 = (coe1‘𝑀)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝐴 = (𝐶‘1)    &   ⊢ 𝐵 = (𝐶‘0)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑈)    &   ⊢ (𝜑 → (𝐷‘𝑀) = 1)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑋) = ((𝐴 · 𝑋) + 𝐵))
Theorem	evl1deg2 33625	Evaluation of a univariate polynomial of degree 2. (Contributed by Thierry Arnoux, 22-Jun-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ · = (.r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑅))    &   ⊢ 𝐹 = (coe1‘𝑀)    &   ⊢ 𝐸 = (deg1‘𝑅)    &   ⊢ 𝐴 = (𝐹‘2)    &   ⊢ 𝐵 = (𝐹‘1)    &   ⊢ 𝐶 = (𝐹‘0)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑈)    &   ⊢ (𝜑 → (𝐸‘𝑀) = 2)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑋) = ((𝐴 · (2 ↑ 𝑋)) + ((𝐵 · 𝑋) + 𝐶)))
Theorem	evl1deg3 33626	Evaluation of a univariate polynomial of degree 3. (Contributed by Thierry Arnoux, 14-Jun-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ · = (.r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑅))    &   ⊢ 𝐹 = (coe1‘𝑀)    &   ⊢ 𝐸 = (deg1‘𝑅)    &   ⊢ 𝐴 = (𝐹‘3)    &   ⊢ 𝐵 = (𝐹‘2)    &   ⊢ 𝐶 = (𝐹‘1)    &   ⊢ 𝐷 = (𝐹‘0)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑈)    &   ⊢ (𝜑 → (𝐸‘𝑀) = 3)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑋) = (((𝐴 · (3 ↑ 𝑋)) + (𝐵 · (2 ↑ 𝑋))) + ((𝐶 · 𝑋) + 𝐷)))
Theorem	evls1monply1 33627	Subring evaluation of a scaled monomial. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝑄 = (𝑆 evalSub1 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑊 = (Poly1‘𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝑋 = (var1‘𝑈)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑊))    &   ⊢ ∧ = (.g‘(mulGrp‘𝑆))    &   ⊢ ∗ = ( ·𝑠 ‘𝑊)    &   ⊢ · = (.r‘𝑆)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑄‘(𝐴 ∗ (𝑁 ↑ 𝑋)))‘𝑌) = (𝐴 · (𝑁 ∧ 𝑌)))
Theorem	ply1dg1rt 33628	Express the root − 𝐵 / 𝐴 of a polynomial 𝐴 · 𝑋 + 𝐵 of degree 1 over a field. (Contributed by Thierry Arnoux, 8-Jun-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Field)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑈)    &   ⊢ (𝜑 → (𝐷‘𝐺) = 1)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ 𝐶 = (coe1‘𝐺)    &   ⊢ 𝐴 = (𝐶‘1)    &   ⊢ 𝐵 = (𝐶‘0)    &   ⊢ 𝑍 = ((𝑁‘𝐵) / 𝐴)    ⇒   ⊢ (𝜑 → (◡(𝑂‘𝐺) “ { 0 }) = {𝑍})
Theorem	ply1dg1rtn0 33629	Polynomials of degree 1 over a field always have some roots. (Contributed by Thierry Arnoux, 8-Jun-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Field)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑈)    &   ⊢ (𝜑 → (𝐷‘𝐺) = 1)    ⇒   ⊢ (𝜑 → (◡(𝑂‘𝐺) “ { 0 }) ≠ ∅)
Theorem	ply1mulrtss 33630	The roots of a factor 𝐹 are also roots of the product of polynomials (𝐹 · 𝐺). (Contributed by Thierry Arnoux, 8-Jun-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑈)    &   ⊢ · = (.r‘𝑃)    ⇒   ⊢ (𝜑 → (◡(𝑂‘𝐹) “ { 0 }) ⊆ (◡(𝑂‘(𝐹 · 𝐺)) “ { 0 }))
Theorem	deg1prod 33631*	Degree of a product of polynomials. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑀 = (mulGrp‘𝑃)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝐹:𝐴⟶(𝐵 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐷‘(𝑀 Σg 𝐹)) = Σ𝑘 ∈ 𝐴 (𝐷‘(𝐹‘𝑘)))
Theorem	ply1dg3rt0irred 33632	If a cubic polynomial over a field has no roots, it is irreducible. (Proposed by Saveliy Skresanov, 5-Jun-2025.) (Contributed by Thierry Arnoux, 8-Jun-2025.)
⊢ 0 = (0g‘𝐹)    &   ⊢ 𝑂 = (eval1‘𝐹)    &   ⊢ 𝐷 = (deg1‘𝐹)    &   ⊢ 𝑃 = (Poly1‘𝐹)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ Field)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐵)    &   ⊢ (𝜑 → (◡(𝑂‘𝑄) “ { 0 }) = ∅)    &   ⊢ (𝜑 → (𝐷‘𝑄) = 3)    ⇒   ⊢ (𝜑 → 𝑄 ∈ (Irred‘𝑃))
Theorem	m1pmeq 33633	If two monic polynomials 𝐼 and 𝐽 differ by a unit factor 𝐾, then they are equal. (Contributed by Thierry Arnoux, 27-Apr-2025.)
⊢ 𝑃 = (Poly1‘𝐹)    &   ⊢ 𝑀 = (Monic1p‘𝐹)    &   ⊢ 𝑈 = (Unit‘𝑃)    &   ⊢ · = (.r‘𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ Field)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐼 = (𝐾 · 𝐽))    ⇒   ⊢ (𝜑 → 𝐼 = 𝐽)
Theorem	ply1fermltl 33634	Fermat's little theorem for polynomials. If 𝑃 is prime, Then (𝑋 + 𝐴)↑𝑃 = ((𝑋↑𝑃) + 𝐴) modulo 𝑃. (Contributed by Thierry Arnoux, 24-Jul-2024.)
⊢ 𝑍 = (ℤ/nℤ‘𝑃)    &   ⊢ 𝑊 = (Poly1‘𝑍)    &   ⊢ 𝑋 = (var1‘𝑍)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑁 = (mulGrp‘𝑊)    &   ⊢ ↑ = (.g‘𝑁)    &   ⊢ 𝐶 = (algSc‘𝑊)    &   ⊢ 𝐴 = (𝐶‘((ℤRHom‘𝑍)‘𝐸))    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝐸 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))
Theorem	coe1mon 33635*	Coefficient vector of a monomial. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝜑 → (coe1‘(𝑁 ↑ 𝑋)) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 𝑁, 1 , 0 )))
Theorem	ply1moneq 33636	Two monomials are equal iff their powers are equal. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝑀 ↑ 𝑋) = (𝑁 ↑ 𝑋) ↔ 𝑀 = 𝑁))
Theorem	ply1coedeg 33637*	Decompose a univariate polynomial 𝐾 as a sum of powers, up to its degree 𝐷. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝑀 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑀)    &   ⊢ 𝐴 = (coe1‘𝐾)    &   ⊢ 𝐷 = ((deg1‘𝑅)‘𝐾)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐾 = (𝑃 Σg (𝑘 ∈ (0...𝐷) ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))))
Theorem	coe1zfv 33638	The coefficients of the zero univariate polynomial. (Contributed by Thierry Arnoux, 22-Jun-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((coe1‘𝑍)‘𝑁) = 0 )
Theorem	coe1vr1 33639*	Polynomial coefficient of the variable. (Contributed by Thierry Arnoux, 22-Jun-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝜑 → (coe1‘𝑋) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 1, 1 , 0 )))
Theorem	deg1vr 33640	The degree of the variable polynomial is 1. (Contributed by Thierry Arnoux, 22-Jun-2025.)
⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    ⇒   ⊢ (𝜑 → (𝐷‘𝑋) = 1)
Theorem	vr1nz 33641	A univariate polynomial variable cannot be the zero polynomial. (Contributed by Thierry Arnoux, 14-Nov-2025.)
⊢ 𝑋 = (var1‘𝑈)    &   ⊢ 𝑍 = (0g‘𝑃)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑈)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ NzRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    ⇒   ⊢ (𝜑 → 𝑋 ≠ 𝑍)
Theorem	ply1degltel 33642	Characterize elementhood in the set 𝑆 of polynomials of degree less than 𝑁. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑆 = (◡𝐷 “ (-∞[,)𝑁))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹 ∈ 𝐵 ∧ (𝐷‘𝐹) ≤ (𝑁 − 1))))
Theorem	ply1degleel 33643	Characterize elementhood in the set 𝑆 of polynomials of degree less than 𝑁. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑆 = (◡𝐷 “ (-∞[,)𝑁))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹 ∈ 𝐵 ∧ (𝐷‘𝐹) < 𝑁)))
Theorem	ply1degltlss 33644	The space 𝑆 of the univariate polynomials of degree less than 𝑁 forms a vector subspace. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑆 = (◡𝐷 “ (-∞[,)𝑁))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → 𝑆 ∈ (LSubSp‘𝑃))
Theorem	gsummoncoe1fzo 33645*	A coefficient of the polynomial represented as a sum of scaled monomials is the coefficient of the corresponding scaled monomial. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ ∗ = ( ·𝑠 ‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑁)𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐿 ∈ (0..^𝑁))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝑘 = 𝐿 → 𝐴 = 𝐶)    ⇒   ⊢ (𝜑 → ((coe1‘(𝑃 Σg (𝑘 ∈ (0..^𝑁) ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = 𝐶)
Theorem	gsummoncoe1fz 33646*	A coefficient of the polynomial represented as a sum of scaled monomials is the coefficient of the corresponding scaled monomial. See gsummoncoe1fzo 33645. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ ∗ = ( ·𝑠 ‘𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    &   ⊢ (𝜑 → ∀𝑘 ∈ (0...𝐷)𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐿 ∈ (0...𝐷))    &   ⊢ (𝑘 = 𝐿 → 𝐴 = 𝐶)    ⇒   ⊢ (𝜑 → ((coe1‘(𝑃 Σg (𝑘 ∈ (0...𝐷) ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = 𝐶)
Theorem	ply1gsumz 33647*	If a polynomial given as a sum of scaled monomials by factors 𝐴 is the zero polynomial, then all factors 𝐴 are zero. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐹 = (𝑛 ∈ (0..^𝑁) ↦ (𝑛(.g‘(mulGrp‘𝑃))(var1‘𝑅)))    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑃)    &   ⊢ (𝜑 → 𝐴:(0..^𝑁)⟶𝐵)    &   ⊢ (𝜑 → (𝑃 Σg (𝐴 ∘f ( ·𝑠 ‘𝑃)𝐹)) = 𝑍)    ⇒   ⊢ (𝜑 → 𝐴 = ((0..^𝑁) × { 0 }))
Theorem	deg1addlt 33648	If both factors have degree bounded by 𝐿, then the sum of the polynomials also has degree bounded by 𝐿. See also deg1addle 26054. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 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	ig1pnunit 33649	The polynomial ideal generator is not a unit polynomial. (Contributed by Thierry Arnoux, 19-Mar-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐺 = (idlGen1p‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑃))    &   ⊢ (𝜑 → 𝐼 ≠ 𝑈)    ⇒   ⊢ (𝜑 → ¬ (𝐺‘𝐼) ∈ (Unit‘𝑃))
Theorem	ig1pmindeg 33650	The polynomial ideal generator is of minimum degree. (Contributed by Thierry Arnoux, 19-Mar-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐺 = (idlGen1p‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑃))    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐹 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝐷‘(𝐺‘𝐼)) ≤ (𝐷‘𝐹))
21.3.10.48  Polynomial quotient and polynomial remainder
Theorem	q1pdir 33651	Distribution of univariate polynomial quotient over addition. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝑁 = (Unic1p‘𝑅)    &   ⊢ / = (quot1p‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑁)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑈)    &   ⊢ + = (+g‘𝑃)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶)))
Theorem	q1pvsca 33652	Scalar multiplication property of the polynomial division. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝑁 = (Unic1p‘𝑅)    &   ⊢ / = (quot1p‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑁)    &   ⊢ × = ( ·𝑠 ‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝐵 × 𝐴) / 𝐶) = (𝐵 × (𝐴 / 𝐶)))
Theorem	r1pvsca 33653	Scalar multiplication property of the polynomial remainder operation. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝑁 = (Unic1p‘𝑅)    &   ⊢ 𝐸 = (rem1p‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑁)    &   ⊢ × = ( ·𝑠 ‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝐵 × 𝐴)𝐸𝐷) = (𝐵 × (𝐴𝐸𝐷)))
Theorem	r1p0 33654	Polynomial remainder operation of a division of the zero polynomial. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝑁 = (Unic1p‘𝑅)    &   ⊢ 𝐸 = (rem1p‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑁)    &   ⊢ 0 = (0g‘𝑃)    ⇒   ⊢ (𝜑 → ( 0 𝐸𝐷) = 0 )
Theorem	r1pcyc 33655	The polynomial remainder operation is periodic. See modcyc 13854. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝑁 = (Unic1p‘𝑅)    &   ⊢ 𝐸 = (rem1p‘𝑅)    &   ⊢ + = (+g‘𝑃)    &   ⊢ · = (.r‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑁)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ((𝐴 + (𝐶 · 𝐵))𝐸𝐵) = (𝐴𝐸𝐵))
Theorem	r1padd1 33656	Addition property of the polynomial remainder operation, similar to modadd1 13856. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝑁 = (Unic1p‘𝑅)    &   ⊢ 𝐸 = (rem1p‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑁)    &   ⊢ (𝜑 → (𝐴𝐸𝐷) = (𝐵𝐸𝐷))    &   ⊢ + = (+g‘𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐶)𝐸𝐷) = ((𝐵 + 𝐶)𝐸𝐷))
Theorem	r1pid2OLD 33657	Obsolete version of r1pid2 26115 as of 21-Jun-2025. (Contributed by Thierry Arnoux, 2-Apr-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝑁 = (Unic1p‘𝑅)    &   ⊢ 𝐸 = (rem1p‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑁)    ⇒   ⊢ (𝜑 → ((𝐴𝐸𝐵) = 𝐴 ↔ (𝐷‘𝐴) < (𝐷‘𝐵)))
Theorem	r1plmhm 33658*	The univariate polynomial remainder function 𝐹 is a module homomorphism. Its image (𝐹 “s 𝑃) is sometimes called the "ring of remainders". (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝐸 = (rem1p‘𝑅)    &   ⊢ 𝑁 = (Unic1p‘𝑅)    &   ⊢ 𝐹 = (𝑓 ∈ 𝑈 ↦ (𝑓𝐸𝑀))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑁)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑃 LMHom (𝐹 “s 𝑃)))
Theorem	r1pquslmic 33659*	The univariate polynomial remainder ring (𝐹 “s 𝑃) is module isomorphic with the quotient ring. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝐸 = (rem1p‘𝑅)    &   ⊢ 𝑁 = (Unic1p‘𝑅)    &   ⊢ 𝐹 = (𝑓 ∈ 𝑈 ↦ (𝑓𝐸𝑀))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑁)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝐾))    ⇒   ⊢ (𝜑 → 𝑄 ≃𝑚 (𝐹 “s 𝑃))
21.3.10.49  Multivariate Polynomials
Theorem	psrbasfsupp 33660	Rewrite a finite support for nonnegative integers : For functions mapping a set 𝐼 to the nonnegative integers, having finite support can also be written as having a finite preimage of the positive integers. The latter expression is used for example in psrbas 21902, but with the former expression, theorems about finite support can be used more directly. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ 𝑓 finSupp 0}    ⇒   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
Syntax	cextv 33661	Extend class notation with the "variable extension" function.
class extendVars
Definition	df-extv 33662*	Define the "variable extension" function. The function ((𝐼extendVars𝑅)‘𝐴) converts polynomials with variables indexed by (𝐼 ∖ {𝐴}) into polynomials indexed by 𝐼, and therefore maps elements of ((𝐼 ∖ {𝐴}) mPoly 𝑅) onto (𝐼 mPoly 𝑅). (Contributed by Thierry Arnoux, 20-Jan-2026.)
⊢ extendVars = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑎 ∈ 𝑖 ↦ (𝑓 ∈ (Base‘((𝑖 ∖ {𝑎}) mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝑖 ∖ {𝑎}))), (0g‘𝑟))))))
Theorem	extvval 33663*	Value of the "variable extension" function. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ 𝐽 = (𝐼 ∖ {𝑎})    &   ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))    ⇒   ⊢ (𝜑 → (𝐼extendVars𝑅) = (𝑎 ∈ 𝐼 ↦ (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 )))))
Theorem	extvfval 33664*	The "variable extension" function evaluated for adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐼)    &   ⊢ 𝐽 = (𝐼 ∖ {𝐴})    &   ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))    ⇒   ⊢ (𝜑 → ((𝐼extendVars𝑅)‘𝐴) = (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ))))
Theorem	extvfv 33665*	The "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐼)    &   ⊢ 𝐽 = (𝐼 ∖ {𝐴})    &   ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑀)    ⇒   ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))
Theorem	extvfvv 33666*	The "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐼)    &   ⊢ 𝐽 = (𝐼 ∖ {𝐴})    &   ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ((((𝐼extendVars𝑅)‘𝐴)‘𝐹)‘𝑋) = if((𝑋‘𝐴) = 0, (𝐹‘(𝑋 ↾ 𝐽)), 0 ))
Theorem	extvfvvcl 33667*	Closure for the "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐽 = (𝐼 ∖ {𝐴})    &   ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ((((𝐼extendVars𝑅)‘𝐴)‘𝐹)‘𝑋) ∈ 𝐵)
Theorem	extvfvcl 33668*	Closure for the "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐽 = (𝐼 ∖ {𝐴})    &   ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑀)    &   ⊢ 𝑁 = (Base‘(𝐼 mPoly 𝑅))    ⇒   ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ 𝑁)
Theorem	extvfvalf 33669*	The "variable extension" function maps polynomials with variables indexed in 𝐽 to polynomials with variables indexed in 𝐼. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐽 = (𝐼 ∖ {𝐴})    &   ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐼)    &   ⊢ 𝑁 = (Base‘(𝐼 mPoly 𝑅))    ⇒   ⊢ (𝜑 → ((𝐼extendVars𝑅)‘𝐴):𝑀⟶𝑁)
Theorem	mvrvalind 33670*	Value of the generating elements of the power series structure, expressed using the indicator function. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ 𝐴 = ((𝟭‘𝐼)‘{𝑋})    ⇒   ⊢ (𝜑 → ((𝑉‘𝑋)‘𝐹) = if(𝐹 = 𝐴, 1 , 0 ))
Theorem	mplmulmvr 33671*	Multiply a polynomial 𝐹 with a variable 𝑋 (i.e. with a monic monomial). (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑋 = ((𝐼 mVar 𝑅)‘𝑌)    &   ⊢ 𝑀 = (Base‘𝑃)    &   ⊢ · = (.r‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 𝐴 = ((𝟭‘𝐼)‘{𝑌})    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑀)    ⇒   ⊢ (𝜑 → (𝑋 · 𝐹) = (𝑏 ∈ 𝐷 ↦ if((𝑏‘𝑌) = 0, 0 , (𝐹‘(𝑏 ∘f − 𝐴)))))
Theorem	evlscaval 33672	Polynomial evaluation for scalars. See evlsscaval 42988. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑄 = (𝐼 eval 𝑅)    &   ⊢ 𝑊 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐿:𝐼⟶𝐵)    ⇒   ⊢ (𝜑 → ((𝑄‘(𝐴‘𝑋))‘𝐿) = 𝑋)
Theorem	evlvarval 33673	Polynomial evaluation builder for a variable. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑄 = (𝐼 eval 𝑆)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑆)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ ∙ = (.r‘𝑃)    &   ⊢ · = (.r‘𝑆)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑍)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    &   ⊢ 𝑉 = (𝐼 mVar 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    ⇒   ⊢ (𝜑 → ((𝑉‘𝑋) ∈ 𝐵 ∧ ((𝑄‘(𝑉‘𝑋))‘𝐴) = (𝐴‘𝑋)))
Theorem	evlextv 33674	Evaluating a variable-extended polynomial is the same as evaluating the polynomial in the original set of variables (in both cases, the additionial variable is ignored). (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑄 = (𝐼 eval 𝑅)    &   ⊢ 𝑂 = (𝐽 eval 𝑅)    &   ⊢ 𝐽 = (𝐼 ∖ {𝑌})    &   ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐸 = (𝐼extendVars𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐴:𝐼⟶𝐵)    ⇒   ⊢ (𝜑 → ((𝑄‘((𝐸‘𝑌)‘𝐹))‘𝐴) = ((𝑂‘𝐹)‘(𝐴 ↾ 𝐽)))
Theorem	mplvrpmlem 33675*	Lemma for mplvrpmga 33677 and others. (Contributed by Thierry Arnoux, 11-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼)    &   ⊢ 𝑃 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})    ⇒   ⊢ (𝜑 → (𝑋 ∘ 𝐷) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
Theorem	mplvrpmfgalem 33676*	Permuting variables in a multivariate polynomial conserves finite support. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼)    &   ⊢ 𝑃 = (Base‘𝑆)    &   ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑀)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝑄𝐴𝐹) finSupp 0 )
Theorem	mplvrpmga 33677*	The action of permuting variables in a multivariate polynomial is a group action. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼)    &   ⊢ 𝑃 = (Base‘𝑆)    &   ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝑆 GrpAct 𝑀))
Theorem	mplvrpmmhm 33678*	The action of permuting variables in a multivariate polynomial is a monoid homomorphism. (Contributed by Thierry Arnoux, 11-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼)    &   ⊢ 𝑃 = (Base‘𝑆)    &   ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ 𝐹 = (𝑓 ∈ 𝑀 ↦ (𝐷𝐴𝑓))    &   ⊢ 𝑊 = (𝐼 mPoly 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑊 MndHom 𝑊))
Theorem	mplvrpmrhm 33679*	The action of permuting variables in a multivariate polynomial is a ring homomorphism. (Contributed by Thierry Arnoux, 15-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼)    &   ⊢ 𝑃 = (Base‘𝑆)    &   ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ 𝐹 = (𝑓 ∈ 𝑀 ↦ (𝐷𝐴𝑓))    &   ⊢ 𝑊 = (𝐼 mPoly 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑊 RingHom 𝑊))
Theorem	psrgsum 33680*	Finite commutative sums of power series are taken componentwise. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝑆 Σg 𝐹) = (𝑦 ∈ 𝐷 ↦ (𝑅 Σg (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)‘𝑦)))))
Theorem	psrmon 33681*	A monomial is a power series. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐷)    ⇒   ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵)
Theorem	psrmonmul 33682*	The product of two power series monomials adds the exponent vectors together. For example, the product of (𝑥↑2)(𝑦↑2) with (𝑦↑1)(𝑧↑3) is (𝑥↑2)(𝑦↑3)(𝑧↑3), where the exponent vectors ⟨2, 2, 0⟩ and ⟨0, 1, 3⟩ are added to give ⟨2, 3, 3⟩. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐷)    &   ⊢ · = (.r‘𝑆)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), 1 , 0 )))
Theorem	psrmonmul2 33683*	The product of two power series monomials adds the exponent vectors together. Here, the function 𝐺 is a monomial builder, which maps a bag of variables with the monic monomial with only those variables. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐷)    &   ⊢ · = (.r‘𝑆)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐷)    &   ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 = 𝑦, 1 , 0 )))    ⇒   ⊢ (𝜑 → ((𝐺‘𝑋) · (𝐺‘𝑌)) = (𝐺‘(𝑋 ∘f + 𝑌)))
Theorem	psrmonprod 33684*	Finite product of bags of variables in a power series. Here the function 𝐺 maps a bag of variables to the corresponding monomial. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐷)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑆)    &   ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 = 𝑦, 1 , 0 )))    ⇒   ⊢ (𝜑 → (𝑀 Σg (𝐺 ∘ 𝐹)) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)‘𝑖))))))
Theorem	mplgsum 33685*	Finite commutative sums of polynomials are taken componentwise. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝑃 Σg 𝐹) = (𝑦 ∈ 𝐷 ↦ (𝑅 Σg (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)‘𝑦)))))
Theorem	mplmonprod 33686*	Finite product of monomials. Here the function 𝐺 maps a bag of variables to the corresponding monomial. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐷)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑃)    &   ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 = 𝑦, 1 , 0 )))    ⇒   ⊢ (𝜑 → (𝑀 Σg (𝐺 ∘ 𝐹)) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)‘𝑖))))))
21.3.10.50  The ring of symmetric polynomials
Syntax	csply 33687	Extend class notation with the symmetric polynomials.
class SymPoly
Syntax	cesply 33688	Extend class notation with the elementary symmetric polynomials.
class eSymPoly
Definition	df-sply 33689*	Define symmetric polynomials. See splyval 33691 for a more readable expression. (Contributed by Thierry Arnoux, 11-Jan-2026.)
⊢ SymPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((Base‘(𝑖 mPoly 𝑟))FixPts(𝑑 ∈ (Base‘(SymGrp‘𝑖)), 𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))))
Definition	df-esply 33690*	Define elementary symmetric polynomials. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ eSymPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑟) ∘ ((𝟭‘{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0})‘((𝟭‘𝑖) “ {𝑐 ∈ 𝒫 𝑖 ∣ (♯‘𝑐) = 𝑘})))))
Theorem	splyval 33691*	The symmetric polynomials for a given index 𝐼 of variables and base ring 𝑅. These are the fixed points of the action 𝐴 which permutes variables. (Contributed by Thierry Arnoux, 11-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼)    &   ⊢ 𝑃 = (Base‘𝑆)    &   ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐼SymPoly𝑅) = (𝑀FixPts𝐴))
Theorem	splysubrg 33692*	The symmetric polynomials form a subring of the ring of polynomials. (Contributed by Thierry Arnoux, 15-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼)    &   ⊢ 𝑃 = (Base‘𝑆)    &   ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → (𝐼SymPoly𝑅) ∈ (SubRing‘(𝐼 mPoly 𝑅)))
Theorem	issply 33693*	Conditions for being a symmetric polynomial. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼)    &   ⊢ 𝑃 = (Base‘𝑆)    &   ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑀)    &   ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑥 ∈ 𝐷) → (𝐹‘(𝑥 ∘ 𝑝)) = (𝐹‘𝑥))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐼SymPoly𝑅))
Theorem	esplyval 33694*	The elementary polynomials for a given index 𝐼 of variables and base ring 𝑅. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐼eSymPoly𝑅) = (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝑘})))))
Theorem	esplyfval 33695*	The 𝐾-th elementary polynomial for a given index 𝐼 of variables and base ring 𝑅. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))))
Theorem	esplyfval0 33696	The 0-th elementary symmetric polynomial is the constant 1. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝑈 = (1r‘(𝐼 mPoly 𝑅))    ⇒   ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘0) = 𝑈)
Theorem	esplyfval2 33697*	When 𝐾 is out-of-bounds, the 𝐾-th elementary symmetric polynomial is zero. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ (ℕ0 ∖ (0...(♯‘𝐼))))    &   ⊢ 𝑍 = (0g‘(𝐼 mPoly 𝑅))    ⇒   ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = 𝑍)
Theorem	esplylem 33698*	Lemma for esplyfv 33702 and others. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ⊆ 𝐷)
Theorem	esplympl 33699*	Elementary symmetric polynomials are polynomials. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    &   ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))    ⇒   ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) ∈ 𝑀)
Theorem	esplymhp 33700*	The 𝐾-th elementary symmetric polynomial is homogeneous of degree 𝐾. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    &   ⊢ 𝐻 = (𝐼 mHomP 𝑅)    ⇒   ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) ∈ (𝐻‘𝐾))