P. 336 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 33501-33600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rprmasso 33501	In an integral domain, the associate of a prime is a prime. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∥ 𝑌)    &   ⊢ (𝜑 → 𝑌 ∥ 𝑋)    ⇒   ⊢ (𝜑 → 𝑌 ∈ 𝑃)
Theorem	rprmasso2 33502	In an integral domain, if a prime element divides another, they are associates. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∥ 𝑌)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    ⇒   ⊢ (𝜑 → 𝑌 ∥ 𝑋)
Theorem	rprmasso3 33503*	In an integral domain, if a prime element divides another, they are associates. (Contributed by Thierry Arnoux, 27-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∥ 𝑌)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝜑 → ∃𝑡 ∈ 𝑈 (𝑡 · 𝑋) = 𝑌)
Theorem	unitmulrprm 33504	A ring unit multiplied by a ring prime is a ring prime. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝐼 · 𝑄) ∈ 𝑃)
Theorem	rprmndvdsru 33505	A ring prime element does not divide any ring unit. (Contributed by Thierry Arnoux, 27-May-2025.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ¬ 𝑄 ∥ 𝑇)
Theorem	rprmirredlem 33506	Lemma for rprmirred 33507. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝑄 ≠ 0 )    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑈))    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑄 = (𝑋 · 𝑌))    &   ⊢ (𝜑 → 𝑄 ∥ 𝑋)    ⇒   ⊢ (𝜑 → 𝑌 ∈ 𝑈)
Theorem	rprmirred 33507	In an integral domain, ring primes are irreducible. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 𝐼 = (Irred‘𝑅)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    ⇒   ⊢ (𝜑 → 𝑄 ∈ 𝐼)
Theorem	rprmirredb 33508	In a principal ideal domain, the converse of rprmirred 33507 holds, i.e. irreducible elements are prime. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 𝐼 = (Irred‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ PID)    ⇒   ⊢ (𝜑 → 𝐼 = 𝑃)
Theorem	rprmdvdspow 33509	If a prime element divides a ring "power", it divides the term. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ ↑ = (.g‘𝑀)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑄 ∥ (𝑁 ↑ 𝑋))    ⇒   ⊢ (𝜑 → 𝑄 ∥ 𝑋)
Theorem	rprmdvdsprod 33510*	If a prime element 𝑄 divides a product, then it divides one term. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 finSupp 1 )    &   ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)    &   ⊢ (𝜑 → 𝑄 ∥ (𝑀 Σg 𝐹))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (𝐹 supp 1 )𝑄 ∥ (𝐹‘𝑥))
Theorem	1arithidomlem1 33511*	Lemma for 1arithidom 33513. (Contributed by Thierry Arnoux, 30-May-2025.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐽 = (0..^(♯‘𝐹))    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝐹 ∈ Word 𝑃)    &   ⊢ (𝜑 → 𝐺 ∈ Word 𝑃)    &   ⊢ (𝜑 → (𝑀 Σg 𝐹) = (𝑀 Σg 𝐺))    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    &   ⊢ (𝜑 → ∀𝑔 ∈ Word 𝑃(∃𝑘 ∈ 𝑈 (𝑀 Σg 𝐹) = (𝑘 · (𝑀 Σg 𝑔)) → ∃𝑤∃𝑢 ∈ (𝑈 ↑m (0..^(♯‘𝐹)))(𝑤:(0..^(♯‘𝐹))–1-1-onto→(0..^(♯‘𝐹)) ∧ 𝑔 = (𝑢 ∘f · (𝐹 ∘ 𝑤)))))    &   ⊢ (𝜑 → 𝐻 ∈ Word 𝑃)    &   ⊢ (𝜑 → ∃𝑘 ∈ 𝑈 (𝑀 Σg (𝐹 ++ ⟨“𝑄”⟩)) = (𝑘 · (𝑀 Σg 𝐻)))    &   ⊢ (𝜑 → 𝐾 ∈ (0..^(♯‘𝐻)))    &   ⊢ (𝜑 → 𝑄(∥r‘𝑅)(𝐻‘𝐾))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑈)    &   ⊢ (𝜑 → (𝑇 · 𝑄) = (𝐻‘𝐾))    &   ⊢ (𝜑 → 𝑆:(0..^(♯‘𝐻))–1-1-onto→(0..^(♯‘𝐻)))    &   ⊢ (𝜑 → (𝐻 ∘ 𝑆) = (((𝐻 ∘ 𝑆) prefix ((♯‘𝐻) − 1)) ++ ⟨“(𝐻‘𝐾)”⟩))    &   ⊢ (𝜑 → 𝑁 ∈ 𝑈)    &   ⊢ (𝜑 → (𝑀 Σg (𝐹 ++ ⟨“𝑄”⟩)) = (𝑁 · (𝑀 Σg 𝐻)))    ⇒   ⊢ (𝜑 → ∃𝑐∃𝑑 ∈ (𝑈 ↑m (0..^(♯‘𝐹)))(𝑐:(0..^(♯‘𝐹))–1-1-onto→(0..^(♯‘𝐹)) ∧ ((𝐻 ∘ 𝑆) prefix ((♯‘𝐻) − 1)) = (𝑑 ∘f · (𝐹 ∘ 𝑐))))
Theorem	1arithidomlem2 33512*	Lemma for 1arithidom 33513: induction step. (Contributed by Thierry Arnoux, 27-May-2025.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐽 = (0..^(♯‘𝐹))    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝐹 ∈ Word 𝑃)    &   ⊢ (𝜑 → 𝐺 ∈ Word 𝑃)    &   ⊢ (𝜑 → (𝑀 Σg 𝐹) = (𝑀 Σg 𝐺))    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    &   ⊢ (𝜑 → ∀𝑔 ∈ Word 𝑃(∃𝑘 ∈ 𝑈 (𝑀 Σg 𝐹) = (𝑘 · (𝑀 Σg 𝑔)) → ∃𝑤∃𝑢 ∈ (𝑈 ↑m (0..^(♯‘𝐹)))(𝑤:(0..^(♯‘𝐹))–1-1-onto→(0..^(♯‘𝐹)) ∧ 𝑔 = (𝑢 ∘f · (𝐹 ∘ 𝑤)))))    &   ⊢ (𝜑 → 𝐻 ∈ Word 𝑃)    &   ⊢ (𝜑 → ∃𝑘 ∈ 𝑈 (𝑀 Σg (𝐹 ++ ⟨“𝑄”⟩)) = (𝑘 · (𝑀 Σg 𝐻)))    &   ⊢ (𝜑 → 𝐾 ∈ (0..^(♯‘𝐻)))    &   ⊢ (𝜑 → 𝑄(∥r‘𝑅)(𝐻‘𝐾))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑈)    &   ⊢ (𝜑 → (𝑇 · 𝑄) = (𝐻‘𝐾))    &   ⊢ (𝜑 → 𝑆:(0..^(♯‘𝐻))–1-1-onto→(0..^(♯‘𝐻)))    &   ⊢ (𝜑 → (𝐻 ∘ 𝑆) = (((𝐻 ∘ 𝑆) prefix ((♯‘𝐻) − 1)) ++ ⟨“(𝐻‘𝐾)”⟩))    &   ⊢ (𝜑 → 𝑁 ∈ 𝑈)    &   ⊢ (𝜑 → (𝑀 Σg (𝐹 ++ ⟨“𝑄”⟩)) = (𝑁 · (𝑀 Σg 𝐻)))    &   ⊢ (𝜑 → 𝐷 ∈ (𝑈 ↑m (0..^(♯‘𝐹))))    &   ⊢ (𝜑 → 𝐶:(0..^(♯‘𝐹))–1-1-onto→(0..^(♯‘𝐹)))    &   ⊢ (𝜑 → ((𝐻 ∘ 𝑆) prefix ((♯‘𝐻) − 1)) = (𝐷 ∘f · (𝐹 ∘ 𝐶)))    ⇒   ⊢ (𝜑 → (((𝐶 ++ ⟨“(♯‘𝐹)”⟩) ∘ ◡𝑆):(0..^(♯‘(𝐹 ++ ⟨“𝑄”⟩)))–1-1-onto→(0..^(♯‘(𝐹 ++ ⟨“𝑄”⟩))) ∧ 𝐻 = (((𝐷 ++ ⟨“𝑇”⟩) ∘ ◡𝑆) ∘f · ((𝐹 ++ ⟨“𝑄”⟩) ∘ ((𝐶 ++ ⟨“(♯‘𝐹)”⟩) ∘ ◡𝑆)))))
Theorem	1arithidom 33513*	Uniqueness of prime factorizations in an integral domain 𝑅. Given two equal products 𝐹 and 𝐺 of prime elements, 𝐹 and 𝐺 are equal up to a renumbering 𝑤 and a multiplication by units 𝑢. See also 1arith 16849. Chapter VII, Paragraph 3, Section 3, Proposition 2 of [BourbakiCAlg2], p. 228. (Contributed by Thierry Arnoux, 27-May-2025.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐽 = (0..^(♯‘𝐹))    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝐹 ∈ Word 𝑃)    &   ⊢ (𝜑 → 𝐺 ∈ Word 𝑃)    &   ⊢ (𝜑 → (𝑀 Σg 𝐹) = (𝑀 Σg 𝐺))    ⇒   ⊢ (𝜑 → ∃𝑤∃𝑢 ∈ (𝑈 ↑m 𝐽)(𝑤:𝐽–1-1-onto→𝐽 ∧ 𝐺 = (𝑢 ∘f · (𝐹 ∘ 𝑤))))
21.3.10.43  Unique factorization domains
Syntax	cufd 33514	Class of unique factorization domains.
class UFD
Definition	df-ufd 33515*	Define the class of unique factorization domains. A unique factorization domain (UFD for short), is an integral domain such that every nonzero prime ideal contains a prime element (this is a characterization due to Irving Kaplansky). A UFD is sometimes also called a "factorial ring" following the terminology of Bourbaki. (Contributed by Mario Carneiro, 17-Feb-2015.) Exclude the 0 prime ideal. (Revised by Thierry Arnoux, 9-May-2025.) Exclude the 0 ring. (Revised by Thierry Arnoux, 14-Jun-2025.)
⊢ UFD = {𝑟 ∈ IDomn ∣ ∀𝑖 ∈ ((PrmIdeal‘𝑟) ∖ {{(0g‘𝑟)}})(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅}
Theorem	isufd 33516*	The property of being a Unique Factorization Domain. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝐼 = (PrmIdeal‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑖 ∈ (𝐼 ∖ {{ 0 }})(𝑖 ∩ 𝑃) ≠ ∅))
Theorem	ufdprmidl 33517*	In a unique factorization domain 𝑅, a nonzero prime ideal 𝐽 contains a prime element 𝑝. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ 𝐼 = (PrmIdeal‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ UFD)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐽 ≠ { 0 })    ⇒   ⊢ (𝜑 → ∃𝑝 ∈ 𝑃 𝑝 ∈ 𝐽)
Theorem	ufdidom 33518	A nonzero unique factorization domain is an integral domain. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ (𝜑 → 𝑅 ∈ UFD)    ⇒   ⊢ (𝜑 → 𝑅 ∈ IDomn)
Theorem	pidufd 33519	Every principal ideal domain is a unique factorization domain. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ (𝜑 → 𝑅 ∈ PID)    ⇒   ⊢ (𝜑 → 𝑅 ∈ UFD)
Theorem	1arithufdlem1 33520*	Lemma for 1arithufd 33524. The set 𝑆 of elements which can be written as a product of primes is not empty. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ UFD)    &   ⊢ (𝜑 → ¬ 𝑅 ∈ DivRing)    &   ⊢ 𝑆 = {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓)}    ⇒   ⊢ (𝜑 → 𝑆 ≠ ∅)
Theorem	1arithufdlem2 33521*	Lemma for 1arithufd 33524. The set 𝑆 of elements which can be written as a product of primes is multiplicatively closed. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ UFD)    &   ⊢ (𝜑 → ¬ 𝑅 ∈ DivRing)    &   ⊢ 𝑆 = {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓)}    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝑆)
Theorem	1arithufdlem3 33522*	Lemma for 1arithufd 33524. If a product (𝑌 · 𝑋) can be written as a product of primes, with 𝑋 non-unit, nonzero, so can 𝑋. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ UFD)    &   ⊢ (𝜑 → ¬ 𝑅 ∈ DivRing)    &   ⊢ 𝑆 = {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓)}    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑌 · 𝑋) ∈ 𝑆)    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝑆)
Theorem	1arithufdlem4 33523*	Lemma for 1arithufd 33524. Nonzero ring, non-field case. Those trivial cases are handled in the final proof. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ UFD)    &   ⊢ (𝜑 → ¬ 𝑅 ∈ DivRing)    &   ⊢ 𝑆 = {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓)}    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝑆)
Theorem	1arithufd 33524*	Existence of a factorization into irreducible elements in a unique factorization domain. Any non-zero, non-unit element 𝑋 of a UFD 𝑅 can be written as a product of primes 𝑓. As shown in 1arithidom 33513, that factorization is unique, up to the order of the factors and multiplication by units. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ UFD)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    ⇒   ⊢ (𝜑 → ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σg 𝑓))
Theorem	dfufd2lem 33525	Lemma for dfufd2 33526. (Contributed by Thierry Arnoux, 6-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝐼 ∈ (PrmIdeal‘𝑅))    &   ⊢ (𝜑 → 𝐹 ∈ Word 𝑃)    &   ⊢ (𝜑 → (𝑀 Σg 𝐹) ∈ 𝐼)    &   ⊢ (𝜑 → (𝑀 Σg 𝐹) ≠ 0 )    ⇒   ⊢ (𝜑 → (𝐼 ∩ 𝑃) ≠ ∅)
Theorem	dfufd2 33526*	Alternative definition of unique factorization domain (UFD). This is often the textbook definition. Chapter VII, Paragraph 3, Section 3, Proposition 2 of [BourbakiCAlg2], p. 228. (Contributed by Thierry Arnoux, 6-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑥 ∈ ((𝐵 ∖ 𝑈) ∖ { 0 })∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓)))
21.3.10.44  The ring of integers
Theorem	zringidom 33527	The ring of integers is an integral domain. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ ℤring ∈ IDomn
Theorem	zringpid 33528	The ring of integers is a principal ideal domain. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ ℤring ∈ PID
Theorem	dfprm3 33529	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 33530*	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))
21.3.10.45  Univariate Polynomials
Theorem	0ringmon1p 33531	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 33532	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 33533	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 33534	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 33535	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 33536	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 33537	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 33538*	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 33539	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 33540	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 33541	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 33542	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 33543	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 33544	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 33545	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 33546	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 33547	A polynomial with nonpositive degree is the zero polynomial iff its constant term is zero. Biconditional version of deg1scl 26055. (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 33548	A non-zero 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 33549	In a field 𝐹, a polynomial 𝐶 is a unit iff it has degree 0. This corresponds to the nonzero scalars, see ply1asclunit 33548. (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 33550	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 33551	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 33552	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 33553	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 33554	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 33555	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 33556	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	ply1dg3rt0irred 33557	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 33558	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 33559	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 33560*	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 33561	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	coe1zfv 33562	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 33563*	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 33564	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 33565	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 33566	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 33567	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 33568	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 33569*	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	ply1gsumz 33570*	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 33571	If both factors have degree bounded by 𝐿, then the sum of the polynomials also has degree bounded by 𝐿. See also deg1addle 26043. (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 33572	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 33573	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))
21.3.10.46  Polynomial quotient and polynomial remainder
Theorem	q1pdir 33574	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 33575	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 33576	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 33577	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 33578	The polynomial remainder operation is periodic. See modcyc 13820. (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 33579	Addition property of the polynomial remainder operation, similar to modadd1 13822. (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 33580	Obsolete version of r1pid2 26104 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 33581*	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 33582*	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))
21.3.10.47  Multivariate Polynomials
Theorem	psrbasfsupp 33583	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 21880, 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	mplvrpmlem 33584*	Lemma for mplvrpmga 33586 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 33585*	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 33586*	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 33587*	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 33588*	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))
21.3.10.48  The ring of symmetric polynomials
Syntax	csply 33589	Extend class notation with the symmetric polynomials.
class SymPoly
Syntax	cesply 33590	Extend class notation with the elementary symmetric polynomials.
class eSymPoly
Definition	df-sply 33591*	Define symmetric polynomials. See splyval 33593 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 33592*	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 33593*	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 33594*	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 33595*	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 33596*	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 33597*	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	esplylem 33598*	Lemma for esplyfv 33602 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 33599*	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 33600*	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))