HomeTheorem List (p. 343 of 510)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | esplyindfv 34201* | 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 34202 | 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 34203* | 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 34204* | Lemma for vieta 34205: 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 34205* | 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 26628 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 34206 | 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 34207 | 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 34208 | 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 34209 | 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 34210 | 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 34211 | 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 34212 | 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 34213 | 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 34214 | 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 34215 | 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 34216 | 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 34217 | 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 34218 | 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 34219 | 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 34220* | 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 34221 | Finite version of lvecdim 21428 which does not require the axiom of choice. The axiom of choice is used in acsmapd 18721, 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 34222* | 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 34223 | 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 34224 | Extend class notation with the dimension of a vector space. |
| class dim | ||
| Definition | df-dim 34225 | Define the dimension of a vector space as the cardinality of its bases. Note that by lvecdim 21428, all bases are equinumerous. (Contributed by Thierry Arnoux, 6-May-2023.) |
| ⊢ dim = (𝑓 ∈ V ↦ ∪ (♯ “ (LBasis‘𝑓))) | ||
| Theorem | dimval 34226 | 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 34227 | The dimension of a vector space 𝐹 is the cardinality of one of its bases. This version of dimval 34226 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 34228 | Closure of the vector space dimension. (Contributed by Thierry Arnoux, 18-May-2023.) |
| ⊢ (𝑉 ∈ LVec → (dim‘𝑉) ∈ ℕ0*) | ||
| Theorem | lmimdim 34229 | 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 34230 | 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 34231 | 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 34232 | 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 34233 | 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 34234* | 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 34235 | 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 | frlmdim 34236 | 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 34237 | 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 34238 | 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 34239 | Dimension of the generalized Euclidean space. (Contributed by Thierry Arnoux, 20-May-2023.) |
| ⊢ 𝐻 = (ℝ^‘𝐼) ⇒ ⊢ (𝐼 ∈ 𝑉 → (dim‘𝐻) = (♯‘𝐼)) | ||
| Theorem | matdim 34240 | 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 34241 | 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 40013 and for example lsatlspsn 40030. (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 34242 | 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 34243 | 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 34244 | 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 34245 | 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 34246 | 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 34247* | Lemma for ply1degltdim 34248. (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 34248 | 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 34249 | Lemma for lindsun 34250. (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 34250 | 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 34251 | The linear spans of two disjunct independent sets only have a trivial intersection. This can be seen as the opposite direction of lindsun 34250. (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 34252 | 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 34253 | 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 34254* | Lemma for fedgmul 34256. (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 34255* | Lemma for fedgmul 34256. (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 34256 | 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 34257 | 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 34258 | 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 34259* | In an associative algebra 𝐴, left-multiplication by a fixed element of the algebra is a module homomorphism, analogous to ringlghm 20536. (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 34260* | In an associative algebra 𝐴, left-multiplication by a fixed element of the algebra is bijective. See also lactlmhm 34259. (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 34261 | 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 34262 | 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 34263 | Syntax for the field extension relation. |
| class /FldExt | ||
| Syntax | cfinext 34264 | Syntax for the finite field extension relation. |
| class /FinExt | ||
| Syntax | cextdg 34265 | Syntax for the field extension degree operation. |
| class [:] | ||
| Definition | df-fldext 34266* | 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 34267* | 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 34268* | Definition of the finite field extension relation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ /FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0)} | ||
| Theorem | relfldext 34269 | The field extension is a relation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ Rel /FldExt | ||
| Theorem | brfldext 34270 | The field extension relation explicited. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))) | ||
| Theorem | ccfldextrr 34271 | The field of the complex numbers is an extension of the field of the real numbers. (Contributed by Thierry Arnoux, 20-Jul-2023.) |
| ⊢ ℂfld/FldExtℝfld | ||
| Theorem | fldextfld1 34272 | A field extension is only defined if the extension is a field. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field) | ||
| Theorem | fldextfld2 34273 | A field extension is only defined if the subfield is a field. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field) | ||
| Theorem | fldextsubrg 34274 | Field extension implies a subring relation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ 𝑈 = (Base‘𝐹) ⇒ ⊢ (𝐸/FldExt𝐹 → 𝑈 ∈ (SubRing‘𝐸)) | ||
| Theorem | sdrgfldext 34275 | A field 𝐸 and any sub-division-ring 𝐹 of 𝐸 form a field extension. (Contributed by Thierry Arnoux, 26-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝐸) & ⊢ (𝜑 → 𝐸 ∈ Field) & ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) ⇒ ⊢ (𝜑 → 𝐸/FldExt(𝐸 ↾s 𝐹)) | ||
| Theorem | fldextress 34276 | Field extension implies a structure restriction relation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ (𝐸/FldExt𝐹 → 𝐹 = (𝐸 ↾s (Base‘𝐹))) | ||
| Theorem | brfinext 34277 | The finite field extension relation explicited. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸[:]𝐹) ∈ ℕ0)) | ||
| Theorem | extdgval 34278 | Value of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ (𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) | ||
| Theorem | fldextsdrg 34279 | Deduce sub-division-ring from field extension. (Contributed by Thierry Arnoux, 26-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐸/FldExt𝐹) ⇒ ⊢ (𝜑 → 𝐵 ∈ (SubDRing‘𝐸)) | ||
| Theorem | fldextsralvec 34280 | The subring algebra associated with a field extension is a vector space. (Contributed by Thierry Arnoux, 3-Aug-2023.) |
| ⊢ (𝐸/FldExt𝐹 → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec) | ||
| Theorem | extdgcl 34281 | Closure of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ (𝐸/FldExt𝐹 → (𝐸[:]𝐹) ∈ ℕ0*) | ||
| Theorem | extdggt0 34282 | Degrees of field extension are greater than zero. (Contributed by Thierry Arnoux, 30-Jul-2023.) |
| ⊢ (𝐸/FldExt𝐹 → 0 < (𝐸[:]𝐹)) | ||
| Theorem | fldexttr 34283 | Field extension is a transitive relation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐾) | ||
| Theorem | fldextid 34284 | The field extension relation is reflexive. (Contributed by Thierry Arnoux, 30-Jul-2023.) |
| ⊢ (𝐹 ∈ Field → 𝐹/FldExt𝐹) | ||
| Theorem | extdgid 34285 | A trivial field extension has degree one. (Contributed by Thierry Arnoux, 4-Aug-2023.) |
| ⊢ (𝐸 ∈ Field → (𝐸[:]𝐸) = 1) | ||
| Theorem | fldsdrgfldext 34286 | A sub-division-ring of a field forms a field extension. (Contributed by Thierry Arnoux, 19-Oct-2025.) |
| ⊢ 𝐺 = (𝐹 ↾s 𝐴) & ⊢ (𝜑 → 𝐹 ∈ Field) & ⊢ (𝜑 → 𝐴 ∈ (SubDRing‘𝐹)) ⇒ ⊢ (𝜑 → 𝐹/FldExt𝐺) | ||
| Theorem | fldsdrgfldext2 34287 | A sub-sub-division-ring of a field forms a field extension. (Contributed by Thierry Arnoux, 19-Oct-2025.) |
| ⊢ 𝐺 = (𝐹 ↾s 𝐴) & ⊢ (𝜑 → 𝐹 ∈ Field) & ⊢ (𝜑 → 𝐴 ∈ (SubDRing‘𝐹)) & ⊢ (𝜑 → 𝐵 ∈ (SubDRing‘𝐺)) & ⊢ 𝐻 = (𝐹 ↾s 𝐵) ⇒ ⊢ (𝜑 → 𝐺/FldExt𝐻) | ||
| Theorem | extdgmul 34288 | The multiplicativity formula for degrees of field extensions. Given 𝐸 a field extension of 𝐹, itself a field extension of 𝐾, the degree of the extension 𝐸/FldExt𝐾 is the product of the degrees of the extensions 𝐸/FldExt𝐹 and 𝐹/FldExt𝐾. Proposition 1.2 of [Lang], p. 224. (Contributed by Thierry Arnoux, 30-Jul-2023.) |
| ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐹) ·e (𝐹[:]𝐾))) | ||
| Theorem | finextfldext 34289 | A finite field extension is a field extension. (Contributed by Thierry Arnoux, 10-Jan-2026.) |
| ⊢ (𝜑 → 𝐸/FinExt𝐹) ⇒ ⊢ (𝜑 → 𝐸/FldExt𝐹) | ||
| Theorem | finexttrb 34290 | The extension 𝐸 of 𝐾 is finite if and only if 𝐸 is finite over 𝐹 and 𝐹 is finite over 𝐾. Corollary 1.3 of [Lang] , p. 225. (Contributed by Thierry Arnoux, 30-Jul-2023.) |
| ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸/FinExt𝐾 ↔ (𝐸/FinExt𝐹 ∧ 𝐹/FinExt𝐾))) | ||
| Theorem | extdg1id 34291 | If the degree of the extension 𝐸/FldExt𝐹 is 1, then 𝐸 and 𝐹 are identical. (Contributed by Thierry Arnoux, 6-Aug-2023.) |
| ⊢ ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → 𝐸 = 𝐹) | ||
| Theorem | extdg1b 34292 | The degree of the extension 𝐸/FldExt𝐹 is 1 iff 𝐸 and 𝐹 are the same structure. (Contributed by Thierry Arnoux, 6-Aug-2023.) |
| ⊢ (𝐸/FldExt𝐹 → ((𝐸[:]𝐹) = 1 ↔ 𝐸 = 𝐹)) | ||
| Theorem | fldgenfldext 34293 | A subfield 𝐹 extended with a set 𝐴 forms a field extension. (Contributed by Thierry Arnoux, 22-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝐸) & ⊢ 𝐾 = (𝐸 ↾s 𝐹) & ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴))) & ⊢ (𝜑 → 𝐸 ∈ Field) & ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝐿/FldExt𝐾) | ||
| Theorem | fldextchr 34294 | The characteristic of a subfield is the same as the characteristic of the larger field. (Contributed by Thierry Arnoux, 20-Aug-2023.) |
| ⊢ (𝐸/FldExt𝐹 → (chr‘𝐹) = (chr‘𝐸)) | ||
| Theorem | evls1fldgencl 34295 | Closure of the subring polynomial evaluation in the field extention. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝐵 = (Base‘𝐸) & ⊢ 𝑂 = (𝐸 evalSub1 𝐹) & ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹)) & ⊢ 𝑈 = (Base‘𝑃) & ⊢ (𝜑 → 𝐸 ∈ Field) & ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝑈) ⇒ ⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴}))) | ||
| Theorem | ccfldsrarelvec 34296 | The subring algebra of the complex numbers over the real numbers is a left vector space. (Contributed by Thierry Arnoux, 20-Aug-2023.) |
| ⊢ ((subringAlg ‘ℂfld)‘ℝ) ∈ LVec | ||
| Theorem | ccfldextdgrr 34297 | The degree of the field extension of the complex numbers over the real numbers is 2. (Suggested by GL, 4-Aug-2023.) (Contributed by Thierry Arnoux, 20-Aug-2023.) |
| ⊢ (ℂfld[:]ℝfld) = 2 | ||
| Theorem | fldextrspunlsplem 34298* | Lemma for fldextrspunlsp 34299: First direction. Part of the proof of Proposition 5, Chapter 5, of [BourbakiAlg2] p. 116. (Contributed by Thierry Arnoux, 13-Oct-2025.) |
| ⊢ 𝐾 = (𝐿 ↾s 𝐹) & ⊢ 𝐼 = (𝐿 ↾s 𝐺) & ⊢ 𝐽 = (𝐿 ↾s 𝐻) & ⊢ (𝜑 → 𝐿 ∈ Field) & ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼)) & ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿)) & ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿)) & ⊢ 𝑁 = (RingSpan‘𝐿) & ⊢ 𝐶 = (𝑁‘(𝐺 ∪ 𝐻)) & ⊢ 𝐸 = (𝐿 ↾s 𝐶) & ⊢ (𝜑 → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝑃:𝐻⟶𝐺) & ⊢ (𝜑 → 𝑃 finSupp (0g‘𝐿)) & ⊢ (𝜑 → 𝑋 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)𝑓)))) ⇒ ⊢ (𝜑 → ∃𝑎 ∈ (𝐺 ↑m 𝐵)(𝑎 finSupp (0g‘𝐿) ∧ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑎‘𝑏)(.r‘𝐿)𝑏))))) | ||
| Theorem | fldextrspunlsp 34299 | Lemma for fldextrspunfld 34301. The subring generated by the union of two field extensions 𝐺 and 𝐻 is the vector sub- 𝐺 space generated by a basis 𝐵 of 𝐻. Part of the proof of Proposition 5, Chapter 5, of [BourbakiAlg2] p. 116. (Contributed by Thierry Arnoux, 13-Oct-2025.) |
| ⊢ 𝐾 = (𝐿 ↾s 𝐹) & ⊢ 𝐼 = (𝐿 ↾s 𝐺) & ⊢ 𝐽 = (𝐿 ↾s 𝐻) & ⊢ (𝜑 → 𝐿 ∈ Field) & ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼)) & ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿)) & ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿)) & ⊢ 𝑁 = (RingSpan‘𝐿) & ⊢ 𝐶 = (𝑁‘(𝐺 ∪ 𝐻)) & ⊢ 𝐸 = (𝐿 ↾s 𝐶) & ⊢ (𝜑 → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) & ⊢ (𝜑 → 𝐵 ∈ Fin) ⇒ ⊢ (𝜑 → 𝐶 = ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝐵)) | ||
| Theorem | fldextrspunlem1 34300 | Lemma for fldextrspunfld 34301. Part of the proof of Proposition 5, Chapter 5, of [BourbakiAlg2] p. 116. (Contributed by Thierry Arnoux, 13-Oct-2025.) |
| ⊢ 𝐾 = (𝐿 ↾s 𝐹) & ⊢ 𝐼 = (𝐿 ↾s 𝐺) & ⊢ 𝐽 = (𝐿 ↾s 𝐻) & ⊢ (𝜑 → 𝐿 ∈ Field) & ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼)) & ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿)) & ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿)) & ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ0) & ⊢ 𝑁 = (RingSpan‘𝐿) & ⊢ 𝐶 = (𝑁‘(𝐺 ∪ 𝐻)) & ⊢ 𝐸 = (𝐿 ↾s 𝐶) ⇒ ⊢ (𝜑 → (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (𝐽[:]𝐾)) | ||
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