HomeTheorem List (p. 338 of 501)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | extvfvcl 33701* | 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 33702* | 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 33703* | 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 33704* | 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 33705 | Polynomial evaluation for scalars. See evlsscaval 42806. (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 33706 | 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 33707 | 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 33708* | Lemma for mplvrpmga 33710 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 33709* | 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 33710* | 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 33711* | 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 33712* | 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)) | ||
| Syntax | csply 33713 | Extend class notation with the symmetric polynomials. |
| class SymPoly | ||
| Syntax | cesply 33714 | Extend class notation with the elementary symmetric polynomials. |
| class eSymPoly | ||
| Definition | df-sply 33715* | Define symmetric polynomials. See splyval 33717 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 33716* | 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 33717* | 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 33718* | 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 33719* | 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 33720* | 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 33721* | 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 33722 | 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 33723* | 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 33724* | Lemma for esplyfv 33728 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 33725* | 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 33726* | 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 33727* | 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 33728* | 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 33729* | 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 33730* | 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 | esplyind 33731* | 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 33732* | 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 33733 | 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 33734* | 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 33735* | Lemma for vieta 33736: 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 33736* | 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 26276 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))) | ||
| Theorem | sra1r 33737 | 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 33738 | 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 33739 | 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 33740 | 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 33741 | 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 33742 | 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 33743 | 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 33744 | 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 33745 | 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))) | ||
| Theorem | drgext0g 33746 | 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 33747 | 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 33748 | 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 33749 | 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 33750 | 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 33751* | 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 33752 | Finite version of lvecdim 21112 which does not require the axiom of choice. The axiom of choice is used in acsmapd 18477, 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 33753* | 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 33754 | 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 33755 | Extend class notation with the dimension of a vector space. |
| class dim | ||
| Definition | df-dim 33756 | Define the dimension of a vector space as the cardinality of its bases. Note that by lvecdim 21112, all bases are equinumerous. (Contributed by Thierry Arnoux, 6-May-2023.) |
| ⊢ dim = (𝑓 ∈ V ↦ ∪ (♯ “ (LBasis‘𝑓))) | ||
| Theorem | dimval 33757 | 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 33758 | The dimension of a vector space 𝐹 is the cardinality of one of its bases. This version of dimval 33757 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 33759 | Closure of the vector space dimension. (Contributed by Thierry Arnoux, 18-May-2023.) |
| ⊢ (𝑉 ∈ LVec → (dim‘𝑉) ∈ ℕ0*) | ||
| Theorem | lmimdim 33760 | 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 33761 | 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)) | ||
| Theorem | lvecdim0i 33762 | A vector space of dimension zero is reduced to its identity element. (Contributed by Thierry Arnoux, 31-Jul-2023.) |
| ⊢ 0 = (0g‘𝑉) ⇒ ⊢ ((𝑉 ∈ LVec ∧ (dim‘𝑉) = 0) → (Base‘𝑉) = { 0 }) | ||
| Theorem | lvecdim0 33763 | A vector space of dimension zero is reduced to its identity element, biconditional version. (Contributed by Thierry Arnoux, 31-Jul-2023.) |
| ⊢ 0 = (0g‘𝑉) ⇒ ⊢ (𝑉 ∈ LVec → ((dim‘𝑉) = 0 ↔ (Base‘𝑉) = { 0 })) | ||
| Theorem | lssdimle 33764 | The dimension of a linear subspace is less than or equal to the dimension of the parent vector space. This is corollary 5.4 of [Lang] p. 141. (Contributed by Thierry Arnoux, 20-May-2023.) |
| ⊢ 𝑋 = (𝑊 ↾s 𝑈) ⇒ ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (dim‘𝑋) ≤ (dim‘𝑊)) | ||
| Theorem | dimpropd 33765* | If two structures have the same components (properties), they have the same dimension. (Contributed by Thierry Arnoux, 18-May-2023.) |
| ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → 𝐵 ⊆ 𝑊) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) & ⊢ 𝐹 = (Scalar‘𝐾) & ⊢ 𝐺 = (Scalar‘𝐿) & ⊢ (𝜑 → 𝑃 = (Base‘𝐹)) & ⊢ (𝜑 → 𝑃 = (Base‘𝐺)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘𝐹)𝑦) = (𝑥(+g‘𝐺)𝑦)) & ⊢ (𝜑 → 𝐾 ∈ LVec) & ⊢ (𝜑 → 𝐿 ∈ LVec) ⇒ ⊢ (𝜑 → (dim‘𝐾) = (dim‘𝐿)) | ||
| Theorem | rlmdim 33766 | The left vector space induced by a ring over itself has dimension 1. (Contributed by Thierry Arnoux, 5-Aug-2023.) Generalize to division rings. (Revised by SN, 22-Mar-2025.) |
| ⊢ 𝑉 = (ringLMod‘𝐹) ⇒ ⊢ (𝐹 ∈ DivRing → (dim‘𝑉) = 1) | ||
| Theorem | rgmoddimOLD 33767 | Obsolete version of rlmdim 33766 as of 21-Mar-2025. (Contributed by Thierry Arnoux, 5-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝑉 = (ringLMod‘𝐹) ⇒ ⊢ (𝐹 ∈ Field → (dim‘𝑉) = 1) | ||
| Theorem | frlmdim 33768 | Dimension of a free left module. (Contributed by Thierry Arnoux, 20-May-2023.) |
| ⊢ 𝐹 = (𝑅 freeLMod 𝐼) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (dim‘𝐹) = (♯‘𝐼)) | ||
| Theorem | tnglvec 33769 | Augmenting a structure with a norm conserves left vector spaces. (Contributed by Thierry Arnoux, 20-May-2023.) |
| ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) ⇒ ⊢ (𝑁 ∈ 𝑉 → (𝐺 ∈ LVec ↔ 𝑇 ∈ LVec)) | ||
| Theorem | tngdim 33770 | Dimension of a left vector space augmented with a norm. (Contributed by Thierry Arnoux, 20-May-2023.) |
| ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) ⇒ ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (dim‘𝐺) = (dim‘𝑇)) | ||
| Theorem | rrxdim 33771 | Dimension of the generalized Euclidean space. (Contributed by Thierry Arnoux, 20-May-2023.) |
| ⊢ 𝐻 = (ℝ^‘𝐼) ⇒ ⊢ (𝐼 ∈ 𝑉 → (dim‘𝐻) = (♯‘𝐼)) | ||
| Theorem | matdim 33772 | Dimension of the space of square matrices. (Contributed by Thierry Arnoux, 20-May-2023.) |
| ⊢ 𝐴 = (𝐼 Mat 𝑅) & ⊢ 𝑁 = (♯‘𝐼) ⇒ ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ DivRing) → (dim‘𝐴) = (𝑁 · 𝑁)) | ||
| Theorem | lbslsat 33773 | A nonzero vector 𝑋 is a basis of a line spanned by the singleton 𝑋. Spans of nonzero singletons are sometimes called "atoms", see df-lsatoms 39232 and for example lsatlspsn 39249. (Contributed by Thierry Arnoux, 20-May-2023.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑌 = (𝑊 ↾s (𝑁‘{𝑋})) ⇒ ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → {𝑋} ∈ (LBasis‘𝑌)) | ||
| Theorem | lsatdim 33774 | A line, spanned by a nonzero singleton, has dimension 1. (Contributed by Thierry Arnoux, 20-May-2023.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑌 = (𝑊 ↾s (𝑁‘{𝑋})) ⇒ ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (dim‘𝑌) = 1) | ||
| Theorem | drngdimgt0 33775 | The dimension of a vector space that is also a division ring is greater than zero. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → 0 < (dim‘𝐹)) | ||
| Theorem | lmhmlvec2 33776 | A homomorphism of left vector spaces has a left vector space as codomain. (Contributed by Thierry Arnoux, 7-May-2023.) |
| ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LVec) | ||
| Theorem | kerlmhm 33777 | The kernel of a vector space homomorphism is a vector space itself. (Contributed by Thierry Arnoux, 7-May-2023.) |
| ⊢ 0 = (0g‘𝑈) & ⊢ 𝐾 = (𝑉 ↾s (◡𝐹 “ { 0 })) ⇒ ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐾 ∈ LVec) | ||
| Theorem | imlmhm 33778 | The image of a vector space homomorphism is a vector space itself. (Contributed by Thierry Arnoux, 7-May-2023.) |
| ⊢ 𝐼 = (𝑈 ↾s ran 𝐹) ⇒ ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐼 ∈ LVec) | ||
| Theorem | ply1degltdimlem 33779* | Lemma for ply1degltdim 33780. (Contributed by Thierry Arnoux, 20-Feb-2025.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑆 = (◡𝐷 “ (-∞[,)𝑁)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ 𝐸 = (𝑃 ↾s 𝑆) & ⊢ 𝐹 = (𝑛 ∈ (0..^𝑁) ↦ (𝑛(.g‘(mulGrp‘𝑃))(var1‘𝑅))) ⇒ ⊢ (𝜑 → ran 𝐹 ∈ (LBasis‘𝐸)) | ||
| Theorem | ply1degltdim 33780 | The space 𝑆 of the univariate polynomials of degree less than 𝑁 has dimension 𝑁. (Contributed by Thierry Arnoux, 20-Feb-2025.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑆 = (◡𝐷 “ (-∞[,)𝑁)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ 𝐸 = (𝑃 ↾s 𝑆) ⇒ ⊢ (𝜑 → (dim‘𝐸) = 𝑁) | ||
| Theorem | lindsunlem 33781 | Lemma for lindsun 33782. (Contributed by Thierry Arnoux, 9-May-2023.) |
| ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ (LIndS‘𝑊)) & ⊢ (𝜑 → 𝑉 ∈ (LIndS‘𝑊)) & ⊢ (𝜑 → ((𝑁‘𝑈) ∩ (𝑁‘𝑉)) = { 0 }) & ⊢ 𝑂 = (0g‘(Scalar‘𝑊)) & ⊢ 𝐹 = (Base‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝐶 ∈ 𝑈) & ⊢ (𝜑 → 𝐾 ∈ (𝐹 ∖ {𝑂})) & ⊢ (𝜑 → (𝐾( ·𝑠 ‘𝑊)𝐶) ∈ (𝑁‘((𝑈 ∪ 𝑉) ∖ {𝐶}))) ⇒ ⊢ (𝜑 → ⊥) | ||
| Theorem | lindsun 33782 | Condition for the union of two independent sets to be an independent set. (Contributed by Thierry Arnoux, 9-May-2023.) |
| ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ (LIndS‘𝑊)) & ⊢ (𝜑 → 𝑉 ∈ (LIndS‘𝑊)) & ⊢ (𝜑 → ((𝑁‘𝑈) ∩ (𝑁‘𝑉)) = { 0 }) ⇒ ⊢ (𝜑 → (𝑈 ∪ 𝑉) ∈ (LIndS‘𝑊)) | ||
| Theorem | lbsdiflsp0 33783 | The linear spans of two disjunct independent sets only have a trivial intersection. This can be seen as the opposite direction of lindsun 33782. (Contributed by Thierry Arnoux, 17-May-2023.) |
| ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽 ∧ 𝑉 ⊆ 𝐵) → ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉)) = { 0 }) | ||
| Theorem | dimkerim 33784 | Given a linear map 𝐹 between vector spaces 𝑉 and 𝑈, the dimension of the vector space 𝑉 is the sum of the dimension of 𝐹 's kernel and the dimension of 𝐹's image. Second part of theorem 5.3 in [Lang] p. 141 This can also be described as the Rank-nullity theorem, (dim‘𝐼) being the rank of 𝐹 (the dimension of its image), and (dim‘𝐾) its nullity (the dimension of its kernel). (Contributed by Thierry Arnoux, 17-May-2023.) |
| ⊢ 0 = (0g‘𝑈) & ⊢ 𝐾 = (𝑉 ↾s (◡𝐹 “ { 0 })) & ⊢ 𝐼 = (𝑈 ↾s ran 𝐹) ⇒ ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (dim‘𝑉) = ((dim‘𝐾) +𝑒 (dim‘𝐼))) | ||
| Theorem | qusdimsum 33785 | Let 𝑊 be a vector space, and let 𝑋 be a subspace. Then the dimension of 𝑊 is the sum of the dimension of 𝑋 and the dimension of the quotient space of 𝑋. First part of theorem 5.3 in [Lang] p. 141. (Contributed by Thierry Arnoux, 20-May-2023.) |
| ⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑌 = (𝑊 /s (𝑊 ~QG 𝑈)) ⇒ ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (dim‘𝑊) = ((dim‘𝑋) +𝑒 (dim‘𝑌))) | ||
| Theorem | fedgmullem1 33786* | Lemma for fedgmul 33788. (Contributed by Thierry Arnoux, 20-Jul-2023.) |
| ⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑉) & ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈) & ⊢ 𝐶 = ((subringAlg ‘𝐹)‘𝑉) & ⊢ 𝐹 = (𝐸 ↾s 𝑈) & ⊢ 𝐾 = (𝐸 ↾s 𝑉) & ⊢ (𝜑 → 𝐸 ∈ DivRing) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝐾 ∈ DivRing) & ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸)) & ⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐹)) & ⊢ 𝐷 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑖(.r‘𝐸)𝑗)) & ⊢ 𝐻 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖)) & ⊢ (𝜑 → 𝑋 ∈ (LBasis‘𝐶)) & ⊢ (𝜑 → 𝑌 ∈ (LBasis‘𝐵)) & ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐴)) & ⊢ (𝜑 → 𝐿:𝑌⟶(Base‘(Scalar‘𝐵))) & ⊢ (𝜑 → 𝐿 finSupp (0g‘(Scalar‘𝐵))) & ⊢ (𝜑 → 𝑍 = (𝐵 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗)))) & ⊢ (𝜑 → 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋)) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶))) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐿‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))) ⇒ ⊢ (𝜑 → (𝐻 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑍 = (𝐴 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷)))) | ||
| Theorem | fedgmullem2 33787* | Lemma for fedgmul 33788. (Contributed by Thierry Arnoux, 20-Jul-2023.) |
| ⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑉) & ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈) & ⊢ 𝐶 = ((subringAlg ‘𝐹)‘𝑉) & ⊢ 𝐹 = (𝐸 ↾s 𝑈) & ⊢ 𝐾 = (𝐸 ↾s 𝑉) & ⊢ (𝜑 → 𝐸 ∈ DivRing) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝐾 ∈ DivRing) & ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸)) & ⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐹)) & ⊢ 𝐷 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑖(.r‘𝐸)𝑗)) & ⊢ 𝐻 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖)) & ⊢ (𝜑 → 𝑋 ∈ (LBasis‘𝐶)) & ⊢ (𝜑 → 𝑌 ∈ (LBasis‘𝐵)) & ⊢ (𝜑 → 𝑊 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋)))) & ⊢ (𝜑 → (𝐴 Σg (𝑊 ∘f ( ·𝑠 ‘𝐴)𝐷)) = (0g‘𝐴)) ⇒ ⊢ (𝜑 → 𝑊 = ((𝑌 × 𝑋) × {(0g‘(Scalar‘𝐴))})) | ||
| Theorem | fedgmul 33788 | The multiplicativity formula for degrees of field extensions. Given 𝐸 a field extension of 𝐹, itself a field extension of 𝐾, we have [𝐸:𝐾] = [𝐸:𝐹][𝐹:𝐾]. Proposition 1.2 of [Lang], p. 224. Here (dim‘𝐴) is the degree of the extension 𝐸 of 𝐾, (dim‘𝐵) is the degree of the extension 𝐸 of 𝐹, and (dim‘𝐶) is the degree of the extension 𝐹 of 𝐾. This proof is valid for infinite dimensions, and is actually valid for division ring extensions, not just field extensions. (Contributed by Thierry Arnoux, 25-Jul-2023.) |
| ⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑉) & ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈) & ⊢ 𝐶 = ((subringAlg ‘𝐹)‘𝑉) & ⊢ 𝐹 = (𝐸 ↾s 𝑈) & ⊢ 𝐾 = (𝐸 ↾s 𝑉) & ⊢ (𝜑 → 𝐸 ∈ DivRing) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝐾 ∈ DivRing) & ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸)) & ⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐹)) ⇒ ⊢ (𝜑 → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶))) | ||
| Theorem | dimlssid 33789 | If the dimension of a linear subspace 𝐿 is the dimension of the whole vector space 𝐸, then 𝐿 is the whole space. (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| ⊢ 𝐵 = (Base‘𝐸) & ⊢ (𝜑 → 𝐸 ∈ LVec) & ⊢ (𝜑 → (dim‘𝐸) ∈ ℕ0) & ⊢ (𝜑 → 𝐿 ∈ (LSubSp‘𝐸)) & ⊢ (𝜑 → (dim‘(𝐸 ↾s 𝐿)) = (dim‘𝐸)) ⇒ ⊢ (𝜑 → 𝐿 = 𝐵) | ||
| Theorem | lvecendof1f1o 33790 | If an endomorphism 𝑈 of a vector space 𝐸 of finite dimension is injective, then it is bijective. Item (b) of Corollary of Proposition 9 in [BourbakiAlg1] p. 298 . (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| ⊢ 𝐵 = (Base‘𝐸) & ⊢ (𝜑 → 𝐸 ∈ LVec) & ⊢ (𝜑 → (dim‘𝐸) ∈ ℕ0) & ⊢ (𝜑 → 𝑈 ∈ (𝐸 LMHom 𝐸)) & ⊢ (𝜑 → 𝑈:𝐵–1-1→𝐵) ⇒ ⊢ (𝜑 → 𝑈:𝐵–1-1-onto→𝐵) | ||
| Theorem | lactlmhm 33791* | In an associative algebra 𝐴, left-multiplication by a fixed element of the algebra is a module homomorphism, analogous to ringlghm 20247. (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = (.r‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐶 · 𝑥)) & ⊢ (𝜑 → 𝐴 ∈ AssAlg) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐴 LMHom 𝐴)) | ||
| Theorem | assalactf1o 33792* | In an associative algebra 𝐴, left-multiplication by a fixed element of the algebra is bijective. See also lactlmhm 33791. (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = (.r‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐶 · 𝑥)) & ⊢ (𝜑 → 𝐴 ∈ AssAlg) & ⊢ 𝐸 = (RLReg‘𝐴) & ⊢ 𝐾 = (Scalar‘𝐴) & ⊢ (𝜑 → 𝐾 ∈ DivRing) & ⊢ (𝜑 → (dim‘𝐴) ∈ ℕ0) & ⊢ (𝜑 → 𝐶 ∈ 𝐸) ⇒ ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐵) | ||
| Theorem | assarrginv 33793 | If an element 𝑋 of an associative algebra 𝐴 over a division ring 𝐾 is regular, then it is a unit. Proposition 2. in Chapter 5. of [BourbakiAlg2] p. 113. (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| ⊢ 𝐸 = (RLReg‘𝐴) & ⊢ 𝑈 = (Unit‘𝐴) & ⊢ 𝐾 = (Scalar‘𝐴) & ⊢ (𝜑 → 𝐴 ∈ AssAlg) & ⊢ (𝜑 → 𝐾 ∈ DivRing) & ⊢ (𝜑 → (dim‘𝐴) ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ 𝐸) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝑈) | ||
| Theorem | assafld 33794 | If an algebra 𝐴 of finite degree over a division ring 𝐾 is an integral domain, then it is a field. Corollary of Proposition 2. in Chapter 5. of [BourbakiAlg2] p. 113. (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| ⊢ 𝐾 = (Scalar‘𝐴) & ⊢ (𝜑 → 𝐴 ∈ AssAlg) & ⊢ (𝜑 → 𝐴 ∈ IDomn) & ⊢ (𝜑 → 𝐾 ∈ DivRing) & ⊢ (𝜑 → (dim‘𝐴) ∈ ℕ0) ⇒ ⊢ (𝜑 → 𝐴 ∈ Field) | ||
| Syntax | cfldext 33795 | Syntax for the field extension relation. |
| class /FldExt | ||
| Syntax | cfinext 33796 | Syntax for the finite field extension relation. |
| class /FinExt | ||
| Syntax | cextdg 33797 | Syntax for the field extension degree operation. |
| class [:] | ||
| Definition | df-fldext 33798* | Definition of the field extension relation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))} | ||
| Definition | df-extdg 33799* | Definition of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ [:] = (𝑒 ∈ V, 𝑓 ∈ (/FldExt “ {𝑒}) ↦ (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓)))) | ||
| Definition | df-finext 33800* | Definition of the finite field extension relation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ /FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0)} | ||
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