HomeTheorem List (p. 336 of 498)
GIF version.
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List
| Home |
Metamath
Proof Explorer Theorem List (p. 336 of 498) | < Previous Next > |
| Bad symbols? Try the
GIF version. |
||
|
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
||
| Color key: | Metamath Proof Explorer
(1-30847) |
Hilbert Space Explorer
(30848-32370) |
Users' Mathboxes
(32371-49794) |
| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | rprmirredlem 33501 | Lemma for rprmirred 33502. (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 33502 | 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 33503 | In a principal ideal domain, the converse of rprmirred 33502 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 33504 | 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 33505* | 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 33506* | Lemma for 1arithidom 33508. (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 33507* | Lemma for 1arithidom 33508: 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 33508* | 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 16898. 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)))) | ||
| Syntax | cufd 33509 | Class of unique factorization domains. |
| class UFD | ||
| Definition | df-ufd 33510* | 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 33511* | 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 33512* | 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 33513 | A nonzero unique factorization domain is an integral domain. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ (𝜑 → 𝑅 ∈ UFD) ⇒ ⊢ (𝜑 → 𝑅 ∈ IDomn) | ||
| Theorem | pidufd 33514 | Every principal ideal domain is a unique factorization domain. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ (𝜑 → 𝑅 ∈ PID) ⇒ ⊢ (𝜑 → 𝑅 ∈ UFD) | ||
| Theorem | 1arithufdlem1 33515* | Lemma for 1arithufd 33519. 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 33516* | Lemma for 1arithufd 33519. 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 33517* | Lemma for 1arithufd 33519. 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 33518* | Lemma for 1arithufd 33519. 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 33519* | 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 33508, 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 33520 | Lemma for dfufd2 33521. (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 33521* | 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))) | ||
| Theorem | zringidom 33522 | The ring of integers is an integral domain. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ ℤring ∈ IDomn | ||
| Theorem | zringpid 33523 | The ring of integers is a principal ideal domain. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ ℤring ∈ PID | ||
| Theorem | dfprm3 33524 | 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 33525* | 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)) | ||
| Theorem | 0ringmon1p 33526 | 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 33527 | 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 33528 | 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 33529 | 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 33530 | 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 33531 | 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 33532 | 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 33533* | 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 33534 | 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 33535 | 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 33536 | 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 33537 | 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 33538 | 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 33539 | 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 33540 | 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 33541 | 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 33542 | A polynomial with nonpositive degree is the zero polynomial iff its constant term is zero. Biconditional version of deg1scl 26018. (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 33543 | 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 33544 | In a field 𝐹, a polynomial 𝐶 is a unit iff it has degree 0. This corresponds to the nonzero scalars, see ply1asclunit 33543. (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 33545 | 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 33546 | 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 33547 | 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 | ply1dg1rt 33548 | 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 33549 | 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 33550 | 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 33551 | 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 33552 | 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 33553 | 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 33554* | 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 33555 | 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 33556 | 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 33557* | 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 33558 | 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 33559 | 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 33560 | 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 33561 | 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 33562 | 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 33563* | 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 33564* | 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 33565 | If both factors have degree bounded by 𝐿, then the sum of the polynomials also has degree bounded by 𝐿. See also deg1addle 26006. (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 33566 | 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 33567 | 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)) | ||
| Theorem | q1pdir 33568 | 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 33569 | 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 33570 | 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 33571 | 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 33572 | The polynomial remainder operation is periodic. See modcyc 13868. (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 33573 | Addition property of the polynomial remainder operation, similar to modadd1 13870. (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 33574 | Obsolete version of r1pid2 26067 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 33575* | 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 33576* | 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)) | ||
| Theorem | sra1r 33577 | 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 33578 | 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 33579 | 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 33580 | 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 33581 | 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 33582 | 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 33583 | 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 33584 | 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 | drgext0g 33585 | 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 33586 | 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 33587 | 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 33588 | 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)) | ||
| Theorem | drgextlsp 33589 | The scalar field is a subspace of a subring algebra. (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 ∈ (LSubSp‘𝐵)) | ||
| Theorem | drgextgsum 33590* | Group sum in 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 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐸 Σg (𝑖 ∈ 𝑋 ↦ 𝑌)) = (𝐵 Σg (𝑖 ∈ 𝑋 ↦ 𝑌))) | ||
| Theorem | lvecdimfi 33591 | Finite version of lvecdim 21067 which does not require the axiom of choice. The axiom of choice is used in acsmapd 18513, which is required in the infinite case. Suggested by Gérard Lang. (Contributed by Thierry Arnoux, 24-May-2023.) |
| ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑆 ∈ 𝐽) & ⊢ (𝜑 → 𝑇 ∈ 𝐽) & ⊢ (𝜑 → 𝑆 ∈ Fin) ⇒ ⊢ (𝜑 → 𝑆 ≈ 𝑇) | ||
| Theorem | exsslsb 33592* | Any finite generating set 𝑆 of a vector space 𝑊 contains a basis. (Contributed by Thierry Arnoux, 13-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝐾 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑆 ∈ Fin) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → (𝐾‘𝑆) = 𝐵) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ 𝐽 𝑠 ⊆ 𝑆) | ||
| Theorem | lbslelsp 33593 | The size of a basis 𝑋 of a vector space 𝑊 is less than the size of a generating set 𝑌. (Contributed by Thierry Arnoux, 13-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝐾 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝐽) & ⊢ (𝜑 → 𝑌 ⊆ 𝐵) & ⊢ (𝜑 → (𝐾‘𝑌) = 𝐵) ⇒ ⊢ (𝜑 → (♯‘𝑋) ≤ (♯‘𝑌)) | ||
| Syntax | cldim 33594 | Extend class notation with the dimension of a vector space. |
| class dim | ||
| Definition | df-dim 33595 | Define the dimension of a vector space as the cardinality of its bases. Note that by lvecdim 21067, all bases are equinumerous. (Contributed by Thierry Arnoux, 6-May-2023.) |
| ⊢ dim = (𝑓 ∈ V ↦ ∪ (♯ “ (LBasis‘𝑓))) | ||
| Theorem | dimval 33596 | The dimension of a vector space 𝐹 is the cardinality of one of its bases. (Contributed by Thierry Arnoux, 6-May-2023.) |
| ⊢ 𝐽 = (LBasis‘𝐹) ⇒ ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → (dim‘𝐹) = (♯‘𝑆)) | ||
| Theorem | dimvalfi 33597 | The dimension of a vector space 𝐹 is the cardinality of one of its bases. This version of dimval 33596 does not depend on the axiom of choice, but it is limited to the case where the base 𝑆 is finite. (Contributed by Thierry Arnoux, 24-May-2023.) |
| ⊢ 𝐽 = (LBasis‘𝐹) ⇒ ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → (dim‘𝐹) = (♯‘𝑆)) | ||
| Theorem | dimcl 33598 | Closure of the vector space dimension. (Contributed by Thierry Arnoux, 18-May-2023.) |
| ⊢ (𝑉 ∈ LVec → (dim‘𝑉) ∈ ℕ0*) | ||
| Theorem | lmimdim 33599 | Module isomorphisms preserve vector space dimensions. (Contributed by Thierry Arnoux, 25-Feb-2025.) |
| ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMIso 𝑇)) & ⊢ (𝜑 → 𝑆 ∈ LVec) ⇒ ⊢ (𝜑 → (dim‘𝑆) = (dim‘𝑇)) | ||
| Theorem | lmicdim 33600 | Module isomorphisms preserve vector space dimensions. (Contributed by Thierry Arnoux, 25-Mar-2025.) |
| ⊢ (𝜑 → 𝑆 ≃𝑚 𝑇) & ⊢ (𝜑 → 𝑆 ∈ LVec) ⇒ ⊢ (𝜑 → (dim‘𝑆) = (dim‘𝑇)) | ||
| < Previous Next > |
| Copyright terms: Public domain | < Previous Next > |