P. 339 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 33801-33900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ply1degltel 33801	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 33802	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 33803	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 33804*	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 33805*	A coefficient of the polynomial represented as a sum of scaled monomials is the coefficient of the corresponding scaled monomial. See gsummoncoe1fzo 33804. (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 33806*	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 33807	If both factors have degree bounded by 𝐿, then the sum of the polynomials also has degree bounded by 𝐿. See also deg1addle 26219. (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 33808	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 33809	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 33810	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 33811	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 33812	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 33813	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 33814	The polynomial remainder operation is periodic. See modcyc 13930. (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 33815	Addition property of the polynomial remainder operation, similar to modadd1 13932. (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	r1plmhm 33816*	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 33817*	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 33818	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 22044, 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}
Theorem	psrnzr 33819	The ring of power series over a nonzero ring form a nonzero ring. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    ⇒   ⊢ (𝜑 → 𝑆 ∈ NzRing)
Theorem	mplnzr 33820	The multivariate polynomials over a nonzero ring form a nonzero ring. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    ⇒   ⊢ (𝜑 → 𝑃 ∈ NzRing)
Theorem	0mplrim 33821*	Build a ring isomorphism between multivariate polynomials with no variables and the underlying ring. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (∅ mPoly 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝑝‘∅))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑃 RingIso 𝑅))
Theorem	0mplric 33822	Multivariate polynomials with no variables are isomorphic with the underlying ring. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (∅ mPoly 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → 𝑃 ≃𝑟 𝑅)
Theorem	mplasclco 33823*	Case where composing an algebra scalar lifting functions with a scalar leads to a scalar. This is useful when working with selectVars. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝑆 = (Base‘𝑅)    &   ⊢ 𝑂 = (𝐽 mPoly 𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑄 = (𝐼 mPoly 𝑂)    &   ⊢ 𝐴 = (algSc‘𝑂)    &   ⊢ 𝐵 = (algSc‘𝑃)    &   ⊢ 𝐶 = (algSc‘𝑄)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝐸 = {𝑗 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑗 “ ℕ) ∈ Fin}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ⊆ 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐴 ∘ (𝐵‘𝑋)) = (𝐶‘(𝐴‘𝑋)))
Theorem	selvascl 33824	The "variable selection" function evaluated at a scalar. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐶 = (algSc‘𝑇)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)    &   ⊢ 𝑇 = (𝐽 mPoly 𝑈)    &   ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐽 ⊆ 𝐼)    ⇒   ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘(𝐴‘𝑋)) = (𝐷‘𝑋))
Theorem	selvply1rhmlema 33825*	Lemma for selvply1rhm 33832. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = ({𝑋} mPoly 𝑅)    &   ⊢ · = (.r‘𝑃)    &   ⊢ × = (.r‘𝑄)    &   ⊢ 𝑄 = (Poly1‘𝑅)    &   ⊢ 𝑀 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑓‘{⟨𝑋, (𝑛‘∅)⟩})))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑀‘𝐹) ∈ (Base‘𝑄))
Theorem	selvply1rhmlemb 33826*	Lemma for selvply1rhm 33832. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = ({𝑋} mPoly 𝑅)    &   ⊢ · = (.r‘𝑃)    &   ⊢ × = (.r‘𝑄)    &   ⊢ 𝑄 = (Poly1‘𝑅)    &   ⊢ 𝑀 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑓‘{⟨𝑋, (𝑛‘∅)⟩})))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑀‘(𝐹 · 𝐺)) = ((𝑀‘𝐹) × (𝑀‘𝐺)))
Theorem	selvply1rhmlem1 33827*	Lemma for selvply1rhm 33832. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝑈)    &   ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩})))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    ⇒   ⊢ (𝜑 → 𝐻:𝐵⟶(Base‘𝑄))
Theorem	selvply1rhmlem2 33828*	Lemma for selvply1rhm 33832: Image of the ring unit by the mapping 𝐻 (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝑈)    &   ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩})))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    ⇒   ⊢ (𝜑 → (𝐻‘(1r‘𝑃)) = (1r‘𝑄))
Theorem	selvply1rhmlem3 33829*	Lemma for selvply1rhm 33832. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝑈)    &   ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩})))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ (ℕ0 ↑m 1o))    ⇒   ⊢ (𝜑 → ((𝐻‘𝐹)‘𝑁) = ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝐹)‘{⟨𝑋, (𝑁‘∅)⟩}))
Theorem	selvply1rhmlem4 33830*	Lemma for selvply1rhm 33832: The mapping 𝐻 is linear. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝑈)    &   ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩})))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻‘(𝐹(+g‘𝑃)𝐺)) = ((𝐻‘𝐹)(+g‘𝑄)(𝐻‘𝐺)))
Theorem	selvply1rhmlem5 33831*	Lemma for selvply1rhm 33832. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝑈)    &   ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩})))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ 𝑀 = (𝑞 ∈ (Base‘({𝑋} mPoly 𝑈)) ↦ (𝑠 ∈ (ℕ0 ↑m 1o) ↦ (𝑞‘{⟨𝑋, (𝑠‘∅)⟩})))    ⇒   ⊢ (𝜑 → (𝐻‘𝐹) = (𝑀‘(((𝐼 selectVars 𝑅)‘{𝑋})‘𝐹)))
Theorem	selvply1rhm 33832*	Build a ring homomorphism 𝐻 between the multivariate polynomials 𝑃 with variables in 𝐼 and the univariate polynomials 𝑄 in a single variable 𝑋 element of 𝐼. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝑈)    &   ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩})))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    ⇒   ⊢ (𝜑 → 𝐻 ∈ (𝑃 RingHom 𝑄))
Theorem	selvply1rhm0 33833*	The ring homomorphism 𝐻 built in selvply1rhm 33832 is injective. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝑈)    &   ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩})))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ 0 = (0g‘𝑄)    &   ⊢ 𝑍 = (0g‘𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐻‘𝐹) = 0 )    ⇒   ⊢ (𝜑 → 𝐹 = 𝑍)
Theorem	mplidomlem 33834*	Lemma for mplidom 33835. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ 𝐻 = (𝑓 ∈ 𝐶 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑛‘∅)⟩})))    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ 𝑆 = ((𝑗 ∪ {𝑥}) mPoly 𝑅)    &   ⊢ 𝑈 = (((𝑗 ∪ {𝑥}) ∖ {𝑥}) mPoly 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝑈)    ⇒   ⊢ (𝜑 → 𝑃 ∈ IDomn)
Theorem	mplidom 33835	The multivariate polynomials over an integral domain form an integral domain. See ply1idom 26243. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    ⇒   ⊢ (𝜑 → 𝑃 ∈ IDomn)
Syntax	cextv 33836	Extend class notation with the "variable extension" function.
class extendVars
Definition	df-extv 33837*	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 33838*	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 33839*	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 33840*	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 33841*	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 33842*	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 33843*	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 33844*	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 33845*	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 33846*	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 33847	Polynomial evaluation for scalars. See evlsscaval 22237. (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 33848	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 33849	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 33850*	Lemma for mplvrpmga 33852 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 33851*	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 33852*	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 33853*	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 33854*	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 33855*	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 33856*	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 33857*	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 33858*	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 33859*	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 33860*	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 33861*	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 33862	Extend class notation with the symmetric polynomials.
class SymPoly
Syntax	cesply 33863	Extend class notation with the elementary symmetric polynomials.
class eSymPoly
Definition	df-sply 33864*	Define symmetric polynomials. See splyval 33866 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 33865*	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 33866*	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 33867*	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 33868*	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 33869*	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 33870*	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 33871	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 33872*	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 33873*	Lemma for esplyfv 33877 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 33874*	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 33875*	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))
Theorem	esplyfv1 33876*	Coefficient for the 𝐾-th elementary symmetric polynomial and a bag of variables 𝐹 where variables are not raised to a power. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ (0...(♯‘𝐼)))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝜑 → ran 𝐹 ⊆ {0, 1})    ⇒   ⊢ (𝜑 → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if((♯‘(𝐹 supp 0)) = 𝐾, 1 , 0 ))
Theorem	esplyfv 33877*	Coefficient for the 𝐾-th elementary symmetric polynomial and a bag of variables 𝐹: the coefficient is 1 for the bags of exactly 𝐾 variables, having exponent at most 1. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ (0...(♯‘𝐼)))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝜑 → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if((ran 𝐹 ⊆ {0, 1} ∧ (♯‘(𝐹 supp 0)) = 𝐾), 1 , 0 ))
Theorem	esplysply 33878*	The 𝐾-th elementary symmetric polynomial is symmetric. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ (0...(♯‘𝐼)))    ⇒   ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) ∈ (𝐼SymPoly𝑅))
Theorem	esplyfval3 33879*	Alternate expression for the value of the 𝐾-th elementary symmetric polynomial. (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 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = (𝑓 ∈ 𝐷 ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 )))
Theorem	esplyfval1 33880	The first elementary symmetric polynomial is the sum of all variables. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑊 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ 𝐸 = (𝐼eSymPoly𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → (𝐸‘1) = (𝑊 Σg 𝑉))
Theorem	esplyfvaln 33881	The last elementary symmetric polynomial is the product of all variables. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑊 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ 𝐸 = (𝐼eSymPoly𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ 𝑁 = (♯‘𝐼)    &   ⊢ 𝑀 = (mulGrp‘𝑊)    ⇒   ⊢ (𝜑 → (𝐸‘𝑁) = (𝑀 Σg 𝑉))
Theorem	esplyind 33882*	A recursive formula for the elementary symmetric polynomials. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝑊 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ + = (+g‘𝑊)    &   ⊢ · = (.r‘𝑊)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 𝐺 = ((𝐼extendVars𝑅)‘𝑌)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐼)    &   ⊢ 𝐽 = (𝐼 ∖ {𝑌})    &   ⊢ 𝐸 = (𝐽eSymPoly𝑅)    &   ⊢ (𝜑 → 𝐾 ∈ (1...(♯‘𝐼)))    &   ⊢ 𝐶 = {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}    ⇒   ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = (((𝑉‘𝑌) · (𝐺‘(𝐸‘(𝐾 − 1)))) + (𝐺‘(𝐸‘𝐾))))
Theorem	esplyindfv 33883*	A recursive formula for the elementary symmetric polynomials, evaluated at a given set of points 𝑍. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐼)    &   ⊢ 𝐽 = (𝐼 ∖ {𝑌})    &   ⊢ 𝐸 = (𝐽eSymPoly𝑅)    &   ⊢ (𝜑 → 𝐾 ∈ (0...(♯‘𝐽)))    &   ⊢ 𝐶 = {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}    &   ⊢ 𝐹 = ((𝐼eSymPoly𝑅)‘(𝐾 + 1))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑄 = (𝐼 eval 𝑅)    &   ⊢ 𝑂 = (𝐽 eval 𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ (𝜑 → 𝑍:𝐼⟶𝐵)    ⇒   ⊢ (𝜑 → ((𝑄‘𝐹)‘𝑍) = (((𝑍‘𝑌) · ((𝑂‘(𝐸‘𝐾))‘(𝑍 ↾ 𝐽))) + ((𝑂‘(𝐸‘(𝐾 + 1)))‘(𝑍 ↾ 𝐽))))
Theorem	esplyfvn 33884	Express the last elementary symmetric polynomial, evaluated at a given set of points 𝑍, in terms of the last elementary symmetric polynomial with one less variable. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑄 = (𝐼 eval 𝑅)    &   ⊢ 𝑂 = (𝐽 eval 𝑅)    &   ⊢ 𝐸 = (𝐼eSymPoly𝑅)    &   ⊢ 𝐹 = (𝐽eSymPoly𝑅)    &   ⊢ 𝐻 = (♯‘𝐼)    &   ⊢ 𝐾 = (♯‘𝐽)    &   ⊢ 𝐽 = (𝐼 ∖ {𝑌})    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑍:𝐼⟶𝐵)    ⇒   ⊢ (𝜑 → ((𝑄‘(𝐸‘𝐻))‘𝑍) = ((𝑍‘𝑌) · ((𝑂‘(𝐹‘𝐾))‘(𝑍 ↾ 𝐽))))
Theorem	vietadeg1 33885*	The degree of a product of 𝐻 of linear polynomials of the form 𝑋 − 𝑍 is 𝐻. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ − = (-g‘𝑊)    &   ⊢ 𝑀 = (mulGrp‘𝑊)    &   ⊢ 𝑄 = (𝐼 eval 𝑅)    &   ⊢ 𝐸 = (𝐼eSymPoly𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑅))    &   ⊢ 𝐻 = (♯‘𝐼)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝑍:𝐼⟶𝐵)    &   ⊢ 𝐹 = (𝑀 Σg (𝑛 ∈ 𝐼 ↦ (𝑋 − (𝐴‘(𝑍‘𝑛)))))    &   ⊢ 𝐷 = (deg1‘𝑅)    ⇒   ⊢ (𝜑 → (𝐷‘𝐹) = 𝐻)
Theorem	vietalem 33886*	Lemma for vieta 33887: induction step. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ − = (-g‘𝑊)    &   ⊢ 𝑀 = (mulGrp‘𝑊)    &   ⊢ 𝑄 = (𝐼 eval 𝑅)    &   ⊢ 𝐸 = (𝐼eSymPoly𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑅))    &   ⊢ 𝐻 = (♯‘𝐼)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝑍:𝐼⟶𝐵)    &   ⊢ 𝐹 = (𝑀 Σg (𝑛 ∈ 𝐼 ↦ (𝑋 − (𝐴‘(𝑍‘𝑛)))))    &   ⊢ (𝜑 → 𝐾 ∈ (0...𝐻))    &   ⊢ (𝜑 → 𝑌 ∈ 𝐼)    &   ⊢ 𝐽 = (𝐼 ∖ {𝑌})    &   ⊢ (𝜑 → ∀𝑧 ∈ (𝐵 ↑m 𝐽)∀𝑘 ∈ (0...(♯‘𝐽))((coe1‘(𝑀 Σg (𝑛 ∈ 𝐽 ↦ (𝑋 − (𝐴‘(𝑧‘𝑛))))))‘((♯‘𝐽) − 𝑘)) = ((𝑘 ↑ (𝑁‘ 1 )) · (((𝐽 eval 𝑅)‘((𝐽eSymPoly𝑅)‘𝑘))‘𝑧)))    &   ⊢ (𝜑 → ((deg1‘𝑅)‘(𝑀 Σg (𝑛 ∈ 𝐽 ↦ (𝑋 − (𝐴‘((𝑍 ↾ 𝐽)‘𝑛)))))) = (♯‘𝐽))    ⇒   ⊢ (𝜑 → ((coe1‘𝐹)‘𝐾) = (((𝐻 − 𝐾) ↑ (𝑁‘ 1 )) · ((𝑄‘(𝐸‘(𝐻 − 𝐾)))‘𝑍)))
Theorem	vieta 33887*	Vieta's Formulas: Coefficients of a monic polynomial 𝐹 expressed as a product of linear polynomials of the form 𝑋 − 𝑍 can be expressed in terms of elementary symmetric polynomials. The formulas appear in Chapter 6 of [Lang], p. 190. Theorem vieta1 26434 is a special case for the complex numbers, for the case 𝐾 = 1. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ − = (-g‘𝑊)    &   ⊢ 𝑀 = (mulGrp‘𝑊)    &   ⊢ 𝑄 = (𝐼 eval 𝑅)    &   ⊢ 𝐸 = (𝐼eSymPoly𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑅))    &   ⊢ 𝐻 = (♯‘𝐼)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝑍:𝐼⟶𝐵)    &   ⊢ 𝐹 = (𝑀 Σg (𝑛 ∈ 𝐼 ↦ (𝑋 − (𝐴‘(𝑍‘𝑛)))))    &   ⊢ (𝜑 → 𝐾 ∈ (0...𝐻))    &   ⊢ 𝐶 = (coe1‘𝐹)    ⇒   ⊢ (𝜑 → (𝐶‘(𝐻 − 𝐾)) = ((𝐾 ↑ (𝑁‘ 1 )) · ((𝑄‘(𝐸‘𝐾))‘𝑍)))
21.3.10.51  The subring algebra
Theorem	sra1r 33888	The unity element of a subring algebra. (Contributed by Thierry Arnoux, 24-Jul-2023.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 1 = (1r‘𝑊))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   ⊢ (𝜑 → 1 = (1r‘𝐴))
Theorem	sradrng 33889	Condition for a subring algebra to be a division ring. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ 𝐴 = ((subringAlg ‘𝑅)‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝑉 ⊆ 𝐵) → 𝐴 ∈ DivRing)
Theorem	sraidom 33890	Condition for a subring algebra to be an integral domain. (Contributed by Thierry Arnoux, 13-Oct-2025.)
⊢ 𝐴 = ((subringAlg ‘𝑅)‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝑉 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ IDomn)
Theorem	srasubrg 33891	A subring of the original structure is also a subring of the constructed subring algebra. (Contributed by Thierry Arnoux, 24-Jul-2023.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝑊))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐴))
Theorem	sralvec 33892	Given a sub division ring 𝐹 of a division ring 𝐸, 𝐸 may be considered as a vector space over 𝐹, which becomes the field of scalars. (Contributed by Thierry Arnoux, 24-May-2023.)
⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑈)    &   ⊢ 𝐹 = (𝐸 ↾s 𝑈)    ⇒   ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec)
Theorem	srafldlvec 33893	Given a subfield 𝐹 of a field 𝐸, 𝐸 may be considered as a vector space over 𝐹, which becomes the field of scalars. (Contributed by Thierry Arnoux, 24-May-2023.)
⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑈)    &   ⊢ 𝐹 = (𝐸 ↾s 𝑈)    ⇒   ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec)
Theorem	resssra 33894	The subring algebra of a restricted structure is the restriction of the subring algebra. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝐴 = (Base‘𝑅)    &   ⊢ 𝑆 = (𝑅 ↾s 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((subringAlg ‘𝑆)‘𝐶) = (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵))
Theorem	lsssra 33895	A subring is a subspace of the subring algebra. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝑊 = ((subringAlg ‘𝑅)‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑅)    &   ⊢ 𝑆 = (𝑅 ↾s 𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝐶 ∈ (SubRing‘𝑆))    ⇒   ⊢ (𝜑 → 𝐵 ∈ (LSubSp‘𝑊))
Theorem	srapwov 33896	The "power" operation on a subring algebra. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆)    &   ⊢ (𝜑 → 𝑊 ∈ Ring)    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   ⊢ (𝜑 → (.g‘(mulGrp‘𝑊)) = (.g‘(mulGrp‘𝐴)))
21.3.10.52  Division Ring Extensions
Theorem	drgext0g 33897	The additive neutral element of a division ring extension. (Contributed by Thierry Arnoux, 17-Jul-2023.)
⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ DivRing)    &   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸))    ⇒   ⊢ (𝜑 → (0g‘𝐸) = (0g‘𝐵))
Theorem	drgextvsca 33898	The scalar multiplication operation of a division ring extension. (Contributed by Thierry Arnoux, 17-Jul-2023.)
⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ DivRing)    &   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸))    ⇒   ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘𝐵))
Theorem	drgext0gsca 33899	The additive neutral element of the scalar field of a division ring extension. (Contributed by Thierry Arnoux, 17-Jul-2023.)
⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ DivRing)    &   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸))    ⇒   ⊢ (𝜑 → (0g‘𝐵) = (0g‘(Scalar‘𝐵)))
Theorem	drgextsubrg 33900	The scalar field is a subring of a division ring extension. (Contributed by Thierry Arnoux, 17-Jul-2023.)
⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ DivRing)    &   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸))    &   ⊢ 𝐹 = (𝐸 ↾s 𝑈)    &   ⊢ (𝜑 → 𝐹 ∈ DivRing)    ⇒   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐵))