HomeTheorem List (p. 334 of 497)
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. 334 of 497) | < 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
(1-30845) |
Hilbert Space Explorer
(30846-32368) |
Users' Mathboxes
(32369-49617) |
| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | resv1r 33301 | 1r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ 𝐻 = (𝐺 ↾v 𝐴) & ⊢ 1 = (1r‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → 1 = (1r‘𝐻)) | ||
| Theorem | resvcmn 33302 | Scalar restriction preserves commutative monoids. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ 𝐻 = (𝐺 ↾v 𝐴) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐺 ∈ CMnd ↔ 𝐻 ∈ CMnd)) | ||
| Theorem | gzcrng 33303 | The gaussian integers form a commutative ring. (Contributed by Thierry Arnoux, 18-Mar-2018.) |
| ⊢ (ℂfld ↾s ℤ[i]) ∈ CRing | ||
| Theorem | cnfldfld 33304 | The complex numbers form a field. (Contributed by Thierry Arnoux, 6-Jul-2025.) |
| ⊢ ℂfld ∈ Field | ||
| Theorem | reofld 33305 | The real numbers form an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
| ⊢ ℝfld ∈ oField | ||
| Theorem | nn0omnd 33306 | The nonnegative integers form an ordered monoid. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
| ⊢ (ℂfld ↾s ℕ0) ∈ oMnd | ||
| Theorem | rearchi 33307 | The field of the real numbers is Archimedean. See also arch 12496. (Contributed by Thierry Arnoux, 9-Apr-2018.) |
| ⊢ ℝfld ∈ Archi | ||
| Theorem | nn0archi 33308 | The monoid of the nonnegative integers is Archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.) |
| ⊢ (ℂfld ↾s ℕ0) ∈ Archi | ||
| Theorem | xrge0slmod 33309 | The extended nonnegative real numbers form a semiring left module. One could also have used subringAlg to get the same structure. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞)) & ⊢ 𝑊 = (𝐺 ↾v (0[,)+∞)) ⇒ ⊢ 𝑊 ∈ SLMod | ||
| Theorem | qusker 33310* | The kernel of a quotient map. (Contributed by Thierry Arnoux, 20-May-2023.) |
| ⊢ 𝑉 = (Base‘𝑀) & ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺)) & ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺)) & ⊢ 0 = (0g‘𝑁) ⇒ ⊢ (𝐺 ∈ (NrmSGrp‘𝑀) → (◡𝐹 “ { 0 }) = 𝐺) | ||
| Theorem | eqgvscpbl 33311 | The left coset equivalence relation is compatible with the scalar multiplication operation. (Contributed by Thierry Arnoux, 18-May-2023.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ ∼ = (𝑀 ~QG 𝐺) & ⊢ 𝑆 = (Base‘(Scalar‘𝑀)) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀)) & ⊢ (𝜑 → 𝐾 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑋 ∼ 𝑌 → (𝐾 · 𝑋) ∼ (𝐾 · 𝑌))) | ||
| Theorem | qusvscpbl 33312* | The quotient map distributes over the scalar multiplication. (Contributed by Thierry Arnoux, 18-May-2023.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ ∼ = (𝑀 ~QG 𝐺) & ⊢ 𝑆 = (Base‘(Scalar‘𝑀)) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀)) & ⊢ (𝜑 → 𝐾 ∈ 𝑆) & ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺)) & ⊢ ∙ = ( ·𝑠 ‘𝑁) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)) & ⊢ (𝜑 → 𝑈 ∈ 𝐵) & ⊢ (𝜑 → 𝑉 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝐹‘𝑈) = (𝐹‘𝑉) → (𝐹‘(𝐾 · 𝑈)) = (𝐹‘(𝐾 · 𝑉)))) | ||
| Theorem | qusvsval 33313 | Value of the scalar multiplication operation on the quotient structure. (Contributed by Thierry Arnoux, 18-May-2023.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ ∼ = (𝑀 ~QG 𝐺) & ⊢ 𝑆 = (Base‘(Scalar‘𝑀)) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀)) & ⊢ (𝜑 → 𝐾 ∈ 𝑆) & ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺)) & ⊢ ∙ = ( ·𝑠 ‘𝑁) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐾 ∙ [𝑋](𝑀 ~QG 𝐺)) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺)) | ||
| Theorem | imaslmod 33314* | The image structure of a left module is a left module. (Contributed by Thierry Arnoux, 15-May-2023.) |
| ⊢ (𝜑 → 𝑁 = (𝐹 “s 𝑀)) & ⊢ 𝑉 = (Base‘𝑀) & ⊢ 𝑆 = (Base‘(Scalar‘𝑀)) & ⊢ + = (+g‘𝑀) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ 0 = (0g‘𝑀) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) & ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → (𝐹‘(𝑘 · 𝑎)) = (𝐹‘(𝑘 · 𝑏)))) & ⊢ (𝜑 → 𝑀 ∈ LMod) ⇒ ⊢ (𝜑 → 𝑁 ∈ LMod) | ||
| Theorem | imasmhm 33315* | Given a function 𝐹 with homomorphic properties, build the image of a monoid. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) & ⊢ + = (+g‘𝑊) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) & ⊢ (𝜑 → 𝑊 ∈ Mnd) ⇒ ⊢ (𝜑 → ((𝐹 “s 𝑊) ∈ Mnd ∧ 𝐹 ∈ (𝑊 MndHom (𝐹 “s 𝑊)))) | ||
| Theorem | imasghm 33316* | Given a function 𝐹 with homomorphic properties, build the image of a group. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) & ⊢ + = (+g‘𝑊) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) & ⊢ (𝜑 → 𝑊 ∈ Grp) ⇒ ⊢ (𝜑 → ((𝐹 “s 𝑊) ∈ Grp ∧ 𝐹 ∈ (𝑊 GrpHom (𝐹 “s 𝑊)))) | ||
| Theorem | imasrhm 33317* | Given a function 𝐹 with homomorphic properties, build the image of a ring. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) & ⊢ + = (+g‘𝑊) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) & ⊢ · = (.r‘𝑊) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)))) & ⊢ (𝜑 → 𝑊 ∈ Ring) ⇒ ⊢ (𝜑 → ((𝐹 “s 𝑊) ∈ Ring ∧ 𝐹 ∈ (𝑊 RingHom (𝐹 “s 𝑊)))) | ||
| Theorem | imaslmhm 33318* | Given a function 𝐹 with homomorphic properties, build the image of a left module. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) & ⊢ + = (+g‘𝑊) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐷) & ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐾 ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → (𝐹‘(𝑘 × 𝑎)) = (𝐹‘(𝑘 × 𝑏)))) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ × = ( ·𝑠 ‘𝑊) ⇒ ⊢ (𝜑 → ((𝐹 “s 𝑊) ∈ LMod ∧ 𝐹 ∈ (𝑊 LMHom (𝐹 “s 𝑊)))) | ||
| Theorem | quslmod 33319 | If 𝐺 is a submodule in 𝑀, then 𝑁 = 𝑀 / 𝐺 is a left module, called the quotient module of 𝑀 by 𝐺. (Contributed by Thierry Arnoux, 18-May-2023.) |
| ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺)) & ⊢ 𝑉 = (Base‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀)) ⇒ ⊢ (𝜑 → 𝑁 ∈ LMod) | ||
| Theorem | quslmhm 33320* | If 𝐺 is a submodule of 𝑀, then the "natural map" from elements to their cosets is a left module homomorphism from 𝑀 to 𝑀 / 𝐺. (Contributed by Thierry Arnoux, 18-May-2023.) |
| ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺)) & ⊢ 𝑉 = (Base‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀)) & ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑀 LMHom 𝑁)) | ||
| Theorem | quslvec 33321 | If 𝑆 is a vector subspace in 𝑊, then 𝑄 = 𝑊 / 𝑆 is a vector space, called the quotient space of 𝑊 by 𝑆. (Contributed by Thierry Arnoux, 18-May-2023.) |
| ⊢ 𝑄 = (𝑊 /s (𝑊 ~QG 𝑆)) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑆 ∈ (LSubSp‘𝑊)) ⇒ ⊢ (𝜑 → 𝑄 ∈ LVec) | ||
| Theorem | ecxpid 33322 | The equivalence class of a cartesian product is the whole set. (Contributed by Thierry Arnoux, 15-Jan-2024.) |
| ⊢ (𝑋 ∈ 𝐴 → [𝑋](𝐴 × 𝐴) = 𝐴) | ||
| Theorem | qsxpid 33323 | The quotient set of a cartesian product is trivial. (Contributed by Thierry Arnoux, 16-Jan-2024.) |
| ⊢ (𝐴 ≠ ∅ → (𝐴 / (𝐴 × 𝐴)) = {𝐴}) | ||
| Theorem | qusxpid 33324 | The Group quotient equivalence relation for the whole group is the cartesian product, i.e. all elements are in the same equivalence class. (Contributed by Thierry Arnoux, 16-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → (𝐺 ~QG 𝐵) = (𝐵 × 𝐵)) | ||
| Theorem | qustriv 33325 | The quotient of a group 𝐺 by itself is trivial. (Contributed by Thierry Arnoux, 15-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐵)) ⇒ ⊢ (𝐺 ∈ Grp → (Base‘𝑄) = {𝐵}) | ||
| Theorem | qustrivr 33326 | Converse of qustriv 33325. (Contributed by Thierry Arnoux, 15-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐻)) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ (Base‘𝑄) = {𝐻}) → 𝐻 = 𝐵) | ||
| Theorem | znfermltl 33327 | Fermat's little theorem in ℤ/nℤ. (Contributed by Thierry Arnoux, 24-Jul-2024.) |
| ⊢ 𝑍 = (ℤ/nℤ‘𝑃) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ ↑ = (.g‘(mulGrp‘𝑍)) ⇒ ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ 𝐵) → (𝑃 ↑ 𝐴) = 𝐴) | ||
| Theorem | islinds5 33328* | A set is linearly independent if and only if it has no non-trivial representations of zero. (Contributed by Thierry Arnoux, 18-May-2023.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑂 = (0g‘𝑊) & ⊢ 0 = (0g‘𝐹) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑉 ⊆ 𝐵) → (𝑉 ∈ (LIndS‘𝑊) ↔ ∀𝑎 ∈ (𝐾 ↑m 𝑉)((𝑎 finSupp 0 ∧ (𝑊 Σg (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣))) = 𝑂) → 𝑎 = (𝑉 × { 0 })))) | ||
| Theorem | ellspds 33329* | Variation on ellspd 21760. (Contributed by Thierry Arnoux, 18-May-2023.) |
| ⊢ 𝑁 = (LSpan‘𝑀) & ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 0 = (0g‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ LMod) & ⊢ (𝜑 → 𝑉 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝑁‘𝑉) ↔ ∃𝑎 ∈ (𝐾 ↑m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣)))))) | ||
| Theorem | 0ellsp 33330 | Zero is in all spans. (Contributed by Thierry Arnoux, 8-May-2023.) |
| ⊢ 0 = (0g‘𝑊) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ⊆ 𝐵) → 0 ∈ (𝑁‘𝑆)) | ||
| Theorem | 0nellinds 33331 | The group identity cannot be an element of an independent set. (Contributed by Thierry Arnoux, 8-May-2023.) |
| ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊)) → ¬ 0 ∈ 𝐹) | ||
| Theorem | rspsnid 33332 | A principal ideal contains the element that generates it. (Contributed by Thierry Arnoux, 15-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ (𝐾‘{𝐺})) | ||
| Theorem | elrsp 33333* | Write the elements of a ring span as finite linear combinations. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
| ⊢ 𝑁 = (RSpan‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝑁‘𝐼) ↔ ∃𝑎 ∈ (𝐵 ↑m 𝐼)(𝑎 finSupp 0 ∧ 𝑋 = (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑎‘𝑖) · 𝑖)))))) | ||
| Theorem | ellpi 33334 | Elementhood in a left principal ideal in terms of the "divides" relation. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑌 ∈ (𝐾‘{𝑋}) ↔ 𝑋 ∥ 𝑌)) | ||
| Theorem | lpirlidllpi 33335* | In a principal ideal ring, ideals are principal. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐼 = (LIdeal‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ LPIR) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐽 = (𝐾‘{𝑥})) | ||
| Theorem | rspidlid 33336 | The ideal span of an ideal is the ideal itself. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
| ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ 𝑈 = (LIdeal‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝐾‘𝐼) = 𝐼) | ||
| Theorem | pidlnz 33337 | A principal ideal generated by a nonzero element is not the zero ideal. (Contributed by Thierry Arnoux, 11-Apr-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐾‘{𝑋}) ≠ { 0 }) | ||
| Theorem | lbslsp 33338* | Any element of a left module 𝑀 can be expressed as a linear combination of the elements of a basis 𝑉 of 𝑀. (Contributed by Thierry Arnoux, 3-Aug-2023.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 0 = (0g‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ LMod) & ⊢ (𝜑 → 𝑉 ∈ (LBasis‘𝑀)) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ ∃𝑎 ∈ (𝐾 ↑m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣)))))) | ||
| Theorem | lindssn 33339 | Any singleton of a nonzero element is an independent set. (Contributed by Thierry Arnoux, 5-Aug-2023.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → {𝑋} ∈ (LIndS‘𝑊)) | ||
| Theorem | lindflbs 33340 | Conditions for an independent family to be a basis. (Contributed by Thierry Arnoux, 21-Jul-2023.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑂 = (0g‘𝑊) & ⊢ 0 = (0g‘𝑆) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑆 ∈ NzRing) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐼–1-1→𝐵) ⇒ ⊢ (𝜑 → (ran 𝐹 ∈ (LBasis‘𝑊) ↔ (𝐹 LIndF 𝑊 ∧ (𝑁‘ran 𝐹) = 𝐵))) | ||
| Theorem | islbs5 33341* | An equivalent formulation of the basis predicate in a vector space, using a function 𝐹 for generating the base. (Contributed by Thierry Arnoux, 20-Feb-2025.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑂 = (0g‘𝑊) & ⊢ 0 = (0g‘𝑆) & ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑆 ∈ NzRing) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐼–1-1→𝐵) ⇒ ⊢ (𝜑 → (ran 𝐹 ∈ (LBasis‘𝑊) ↔ (∀𝑎 ∈ (𝐾 ↑m 𝐼)((𝑎 finSupp 0 ∧ (𝑊 Σg (𝑎 ∘f · 𝐹)) = 𝑂) → 𝑎 = (𝐼 × { 0 })) ∧ (𝑁‘ran 𝐹) = 𝐵))) | ||
| Theorem | linds2eq 33342 | Deduce equality of elements in an independent set. (Contributed by Thierry Arnoux, 18-Jul-2023.) |
| ⊢ 𝐹 = (Base‘(Scalar‘𝑊)) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝐵 ∈ (LIndS‘𝑊)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐾 ∈ 𝐹) & ⊢ (𝜑 → 𝐿 ∈ 𝐹) & ⊢ (𝜑 → 𝐾 ≠ 0 ) & ⊢ (𝜑 → (𝐾 · 𝑋) = (𝐿 · 𝑌)) ⇒ ⊢ (𝜑 → (𝑋 = 𝑌 ∧ 𝐾 = 𝐿)) | ||
| Theorem | lindfpropd 33343* | Property deduction for linearly independent families. (Contributed by Thierry Arnoux, 16-Jul-2023.) |
| ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) & ⊢ (𝜑 → (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐿))) & ⊢ (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿))) & ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ (Base‘𝐾)) & ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) & ⊢ (𝜑 → 𝐾 ∈ 𝑉) & ⊢ (𝜑 → 𝐿 ∈ 𝑊) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑋 LIndF 𝐾 ↔ 𝑋 LIndF 𝐿)) | ||
| Theorem | lindspropd 33344* | Property deduction for linearly independent sets. (Contributed by Thierry Arnoux, 16-Jul-2023.) |
| ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) & ⊢ (𝜑 → (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐿))) & ⊢ (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿))) & ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ (Base‘𝐾)) & ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) & ⊢ (𝜑 → 𝐾 ∈ 𝑉) & ⊢ (𝜑 → 𝐿 ∈ 𝑊) ⇒ ⊢ (𝜑 → (LIndS‘𝐾) = (LIndS‘𝐿)) | ||
| Theorem | dvdsruassoi 33345 | If two elements 𝑋 and 𝑌 of a ring 𝑅 are unit multiples, then they are associates, i.e. each divides the other. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑉 ∈ 𝑈) & ⊢ (𝜑 → (𝑉 · 𝑋) = 𝑌) ⇒ ⊢ (𝜑 → (𝑋 ∥ 𝑌 ∧ 𝑌 ∥ 𝑋)) | ||
| Theorem | dvdsruasso 33346* | Two elements 𝑋 and 𝑌 of a ring 𝑅 are associates, i.e. each divides the other, iff they are unit multiples of each other. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ IDomn) ⇒ ⊢ (𝜑 → ((𝑋 ∥ 𝑌 ∧ 𝑌 ∥ 𝑋) ↔ ∃𝑢 ∈ 𝑈 (𝑢 · 𝑋) = 𝑌)) | ||
| Theorem | dvdsruasso2 33347* | A reformulation of dvdsruasso 33346. (Proposed by Gerard Lang, 28-May-2025.) (Contributed by Thiery Arnoux, 29-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝜑 → ((𝑋 ∥ 𝑌 ∧ 𝑌 ∥ 𝑋) ↔ ∃𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑈 ((𝑢 · 𝑋) = 𝑌 ∧ (𝑣 · 𝑌) = 𝑋 ∧ (𝑢 · 𝑣) = 1 ))) | ||
| Theorem | dvdsrspss 33348 | In a ring, an element 𝑋 divides 𝑌 iff the ideal generated by 𝑌 is a subset of the ideal generated by 𝑋 (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (𝑋 ∥ 𝑌 ↔ (𝐾‘{𝑌}) ⊆ (𝐾‘{𝑋}))) | ||
| Theorem | rspsnasso 33349 | Two elements 𝑋 and 𝑌 of a ring 𝑅 are associates, i.e. each divides the other, iff the ideals they generate are equal. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → ((𝑋 ∥ 𝑌 ∧ 𝑌 ∥ 𝑋) ↔ (𝐾‘{𝑌}) = (𝐾‘{𝑋}))) | ||
| Theorem | unitprodclb 33350 | A finite product is a unit iff all factors are units. (Contributed by Thierry Arnoux, 27-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐹 ∈ Word 𝐵) ⇒ ⊢ (𝜑 → ((𝑀 Σg 𝐹) ∈ 𝑈 ↔ ran 𝐹 ⊆ 𝑈)) | ||
The sumset (also called the Minkowski sum) of two subsets 𝐴 and 𝐵, is defined to be the set of all sums of an element from 𝐴 with an element from 𝐵. The sumset operation can be used for both group (additive) operations and ring (multiplicative) operations. | ||
| Theorem | elgrplsmsn 33351* | Membership in a sumset with a singleton for a group operation. (Contributed by Thierry Arnoux, 21-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 𝑍 = (𝑥 + 𝑋))) | ||
| Theorem | lsmsnorb 33352* | The sumset of a group with a single element is the element's orbit by the group action. See gaorb 19288. (Contributed by Thierry Arnoux, 21-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)} & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ⊕ {𝑋}) = [𝑋] ∼ ) | ||
| Theorem | lsmsnorb2 33353* | The sumset of a single element with a group is the element's orbit by the group action. See gaorb 19288. (Contributed by Thierry Arnoux, 24-Jul-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑥 + 𝑔) = 𝑦)} & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ({𝑋} ⊕ 𝐴) = [𝑋] ∼ ) | ||
| Theorem | elringlsm 33354* | Membership in a product of two subsets of a ring. (Contributed by Thierry Arnoux, 20-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ × = (LSSum‘𝐺) & ⊢ (𝜑 → 𝐸 ⊆ 𝐵) & ⊢ (𝜑 → 𝐹 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑍 ∈ (𝐸 × 𝐹) ↔ ∃𝑥 ∈ 𝐸 ∃𝑦 ∈ 𝐹 𝑍 = (𝑥 · 𝑦))) | ||
| Theorem | elringlsmd 33355 | Membership in a product of two subsets of a ring, one direction. (Contributed by Thierry Arnoux, 13-Apr-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ × = (LSSum‘𝐺) & ⊢ (𝜑 → 𝐸 ⊆ 𝐵) & ⊢ (𝜑 → 𝐹 ⊆ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐸) & ⊢ (𝜑 → 𝑌 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝑋 · 𝑌) ∈ (𝐸 × 𝐹)) | ||
| Theorem | ringlsmss 33356 | Closure of the product of two subsets of a ring. (Contributed by Thierry Arnoux, 20-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ × = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐸 ⊆ 𝐵) & ⊢ (𝜑 → 𝐹 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐸 × 𝐹) ⊆ 𝐵) | ||
| Theorem | ringlsmss1 33357 | The product of an ideal 𝐼 of a commutative ring 𝑅 with some set E is a subset of the ideal. (Contributed by Thierry Arnoux, 8-Jun-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ × = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐸 ⊆ 𝐵) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) ⇒ ⊢ (𝜑 → (𝐼 × 𝐸) ⊆ 𝐼) | ||
| Theorem | ringlsmss2 33358 | The product with an ideal of a ring is a subset of that ideal. (Contributed by Thierry Arnoux, 2-Jun-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ × = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐸 ⊆ 𝐵) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) ⇒ ⊢ (𝜑 → (𝐸 × 𝐼) ⊆ 𝐼) | ||
| Theorem | lsmsnpridl 33359 | The product of the ring with a single element is equal to the principal ideal generated by that element. (Contributed by Thierry Arnoux, 21-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ × = (LSSum‘𝐺) & ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐵 × {𝑋}) = (𝐾‘{𝑋})) | ||
| Theorem | lsmsnidl 33360 | The product of the ring with a single element is a principal ideal. (Contributed by Thierry Arnoux, 21-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ × = (LSSum‘𝐺) & ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐵 × {𝑋}) ∈ (LPIdeal‘𝑅)) | ||
| Theorem | lsmidllsp 33361 | The sum of two ideals is the ideal generated by their union. (Contributed by Thierry Arnoux, 21-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⊕ = (LSSum‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) & ⊢ (𝜑 → 𝐽 ∈ (LIdeal‘𝑅)) ⇒ ⊢ (𝜑 → (𝐼 ⊕ 𝐽) = (𝐾‘(𝐼 ∪ 𝐽))) | ||
| Theorem | lsmidl 33362 | The sum of two ideals is an ideal. (Contributed by Thierry Arnoux, 21-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⊕ = (LSSum‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) & ⊢ (𝜑 → 𝐽 ∈ (LIdeal‘𝑅)) ⇒ ⊢ (𝜑 → (𝐼 ⊕ 𝐽) ∈ (LIdeal‘𝑅)) | ||
| Theorem | lsmssass 33363 | Group sum is associative, subset version (see lsmass 19648). (Contributed by Thierry Arnoux, 1-Jun-2024.) |
| ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝑅 ⊆ 𝐵) & ⊢ (𝜑 → 𝑇 ⊆ 𝐵) & ⊢ (𝜑 → 𝑈 ⊆ 𝐵) ⇒ ⊢ (𝜑 → ((𝑅 ⊕ 𝑇) ⊕ 𝑈) = (𝑅 ⊕ (𝑇 ⊕ 𝑈))) | ||
| Theorem | grplsm0l 33364 | Sumset with the identity singleton is the original set. (Contributed by Thierry Arnoux, 27-Jul-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ({ 0 } ⊕ 𝐴) = 𝐴) | ||
| Theorem | grplsmid 33365 | The direct sum of an element 𝑋 of a subgroup 𝐴 is the subgroup itself. (Contributed by Thierry Arnoux, 27-Jul-2024.) |
| ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝐴) → ({𝑋} ⊕ 𝐴) = 𝐴) | ||
| Theorem | quslsm 33366 | Express the image by the quotient map in terms of direct sum. (Contributed by Thierry Arnoux, 27-Jul-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → [𝑋](𝐺 ~QG 𝑆) = ({𝑋} ⊕ 𝑆)) | ||
| Theorem | qusbas2 33367* | Alternate definition of the group quotient set, as the set of all cosets of the form ({𝑥} ⊕ 𝑁). (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ (SubGrp‘𝐺)) ⇒ ⊢ (𝜑 → (𝐵 / (𝐺 ~QG 𝑁)) = ran (𝑥 ∈ 𝐵 ↦ ({𝑥} ⊕ 𝑁))) | ||
| Theorem | qus0g 33368 | The identity element of a quotient group. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
| ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)) ⇒ ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → (0g‘𝑄) = 𝑁) | ||
| Theorem | qusima 33369* | The image of a subgroup by the natural map from elements to their cosets. (Contributed by Thierry Arnoux, 27-Jul-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁)) & ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺)) & ⊢ (𝜑 → 𝐻 ∈ 𝑆) & ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝐺)) ⇒ ⊢ (𝜑 → (𝐸‘𝐻) = (𝐹 “ 𝐻)) | ||
| Theorem | qusrn 33370* | The natural map from elements to their cosets is surjective. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑈 = (𝐵 / (𝐺 ~QG 𝑁)) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁)) & ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺)) ⇒ ⊢ (𝜑 → ran 𝐹 = 𝑈) | ||
| Theorem | nsgqus0 33371 | A normal subgroup 𝑁 is a member of all subgroups 𝐹 of the quotient group by 𝑁. In fact, it is the identity element of the quotient group. (Contributed by Thierry Arnoux, 27-Jul-2024.) |
| ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)) ⇒ ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝐹 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ 𝐹) | ||
| Theorem | nsgmgclem 33372* | Lemma for nsgmgc 33373. (Contributed by Thierry Arnoux, 27-Jul-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺)) & ⊢ (𝜑 → 𝐹 ∈ (SubGrp‘𝑄)) ⇒ ⊢ (𝜑 → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝐹} ∈ (SubGrp‘𝐺)) | ||
| Theorem | nsgmgc 33373* | There is a monotone Galois connection between the lattice of subgroups of a group 𝐺 containing a normal subgroup 𝑁 and the lattice of subgroups of the quotient group 𝐺 / 𝑁. This is sometimes called the lattice theorem. (Contributed by Thierry Arnoux, 27-Jul-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ} & ⊢ 𝑇 = (SubGrp‘𝑄) & ⊢ 𝐽 = (𝑉MGalConn𝑊) & ⊢ 𝑉 = (toInc‘𝑆) & ⊢ 𝑊 = (toInc‘𝑇) & ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) & ⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) & ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺)) ⇒ ⊢ (𝜑 → 𝐸𝐽𝐹) | ||
| Theorem | nsgqusf1olem1 33374* | Lemma for nsgqusf1o 33377. (Contributed by Thierry Arnoux, 4-Aug-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ} & ⊢ 𝑇 = (SubGrp‘𝑄) & ⊢ ≤ = (le‘(toInc‘𝑆)) & ⊢ ≲ = (le‘(toInc‘𝑇)) & ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) & ⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) & ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺)) ⇒ ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇) | ||
| Theorem | nsgqusf1olem2 33375* | Lemma for nsgqusf1o 33377. (Contributed by Thierry Arnoux, 4-Aug-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ} & ⊢ 𝑇 = (SubGrp‘𝑄) & ⊢ ≤ = (le‘(toInc‘𝑆)) & ⊢ ≲ = (le‘(toInc‘𝑇)) & ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) & ⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) & ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺)) ⇒ ⊢ (𝜑 → ran 𝐸 = 𝑇) | ||
| Theorem | nsgqusf1olem3 33376* | Lemma for nsgqusf1o 33377. (Contributed by Thierry Arnoux, 4-Aug-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ} & ⊢ 𝑇 = (SubGrp‘𝑄) & ⊢ ≤ = (le‘(toInc‘𝑆)) & ⊢ ≲ = (le‘(toInc‘𝑇)) & ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) & ⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) & ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺)) ⇒ ⊢ (𝜑 → ran 𝐹 = 𝑆) | ||
| Theorem | nsgqusf1o 33377* | The canonical projection homomorphism 𝐸 defines a bijective correspondence between the set 𝑆 of subgroups of 𝐺 containing a normal subgroup 𝑁 and the subgroups of the quotient group 𝐺 / 𝑁. This theorem is sometimes called the correspondence theorem, or the fourth isomorphism theorem. (Contributed by Thierry Arnoux, 4-Aug-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ} & ⊢ 𝑇 = (SubGrp‘𝑄) & ⊢ ≤ = (le‘(toInc‘𝑆)) & ⊢ ≲ = (le‘(toInc‘𝑇)) & ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) & ⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) & ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺)) ⇒ ⊢ (𝜑 → 𝐸:𝑆–1-1-onto→𝑇) | ||
| Theorem | lmhmqusker 33378* | A surjective module homomorphism 𝐹 from 𝐺 to 𝐻 induces an isomorphism 𝐽 from 𝑄 to 𝐻, where 𝑄 is the factor group of 𝐺 by 𝐹's kernel 𝐾. (Contributed by Thierry Arnoux, 25-Feb-2025.) |
| ⊢ 0 = (0g‘𝐻) & ⊢ (𝜑 → 𝐹 ∈ (𝐺 LMHom 𝐻)) & ⊢ 𝐾 = (◡𝐹 “ { 0 }) & ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)) & ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻)) & ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)) ⇒ ⊢ (𝜑 → 𝐽 ∈ (𝑄 LMIso 𝐻)) | ||
| Theorem | lmicqusker 33379 | The image 𝐻 of a module homomorphism 𝐹 is isomorphic with the quotient module 𝑄 over 𝐹's kernel 𝐾. This is part of what is sometimes called the first isomorphism theorem for modules. (Contributed by Thierry Arnoux, 10-Mar-2025.) |
| ⊢ 0 = (0g‘𝐻) & ⊢ (𝜑 → 𝐹 ∈ (𝐺 LMHom 𝐻)) & ⊢ 𝐾 = (◡𝐹 “ { 0 }) & ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)) & ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻)) ⇒ ⊢ (𝜑 → 𝑄 ≃𝑚 𝐻) | ||
| Theorem | lidlmcld 33380 | An ideal is closed under left-multiplication by elements of the full ring. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 𝑈 = (LIdeal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑈) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐼) | ||
| Theorem | intlidl 33381 | The intersection of a nonempty collection of ideals is an ideal. (Contributed by Thierry Arnoux, 8-Jun-2024.) |
| ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → ∩ 𝐶 ∈ (LIdeal‘𝑅)) | ||
| Theorem | 0ringidl 33382 | The zero ideal is the only ideal of the trivial ring. (Contributed by Thierry Arnoux, 1-Jul-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → (LIdeal‘𝑅) = {{ 0 }}) | ||
| Theorem | pidlnzb 33383 | A principal ideal is nonzero iff it is generated by a nonzero elements (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 ≠ 0 ↔ (𝐾‘{𝑋}) ≠ { 0 })) | ||
| Theorem | lidlunitel 33384 | If an ideal 𝐼 contains a unit 𝐽, then it is the whole ring. (Contributed by Thierry Arnoux, 19-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ (𝜑 → 𝐽 ∈ 𝑈) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) ⇒ ⊢ (𝜑 → 𝐼 = 𝐵) | ||
| Theorem | unitpidl1 33385 | The ideal 𝐼 generated by an element 𝑋 of an integral domain 𝑅 is the unit ideal 𝐵 iff 𝑋 is a ring unit. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ 𝐼 = (𝐾‘{𝑋}) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑅 ∈ IDomn) ⇒ ⊢ (𝜑 → (𝐼 = 𝐵 ↔ 𝑋 ∈ 𝑈)) | ||
| Theorem | rhmquskerlem 33386* | The mapping 𝐽 induced by a ring homomorphism 𝐹 from the quotient group 𝑄 over 𝐹's kernel 𝐾 is a ring homomorphism. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ 0 = (0g‘𝐻) & ⊢ (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻)) & ⊢ 𝐾 = (◡𝐹 “ { 0 }) & ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)) & ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)) & ⊢ (𝜑 → 𝐺 ∈ CRing) ⇒ ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐻)) | ||
| Theorem | rhmqusker 33387* | A surjective ring homomorphism 𝐹 from 𝐺 to 𝐻 induces an isomorphism 𝐽 from 𝑄 to 𝐻, where 𝑄 is the factor group of 𝐺 by 𝐹's kernel 𝐾. (Contributed by Thierry Arnoux, 25-Feb-2025.) |
| ⊢ 0 = (0g‘𝐻) & ⊢ (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻)) & ⊢ 𝐾 = (◡𝐹 “ { 0 }) & ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)) & ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻)) & ⊢ (𝜑 → 𝐺 ∈ CRing) & ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)) ⇒ ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingIso 𝐻)) | ||
| Theorem | ricqusker 33388 | The image 𝐻 of a ring homomorphism 𝐹 is isomorphic with the quotient ring 𝑄 over 𝐹's kernel 𝐾. This a part of what is sometimes called the first isomorphism theorem for rings. (Contributed by Thierry Arnoux, 10-Mar-2025.) |
| ⊢ 0 = (0g‘𝐻) & ⊢ (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻)) & ⊢ 𝐾 = (◡𝐹 “ { 0 }) & ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)) & ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻)) & ⊢ (𝜑 → 𝐺 ∈ CRing) ⇒ ⊢ (𝜑 → 𝑄 ≃𝑟 𝐻) | ||
| Theorem | elrspunidl 33389* | Elementhood in the span of a union of ideals. (Contributed by Thierry Arnoux, 30-Jun-2024.) |
| ⊢ 𝑁 = (RSpan‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑆 ⊆ (LIdeal‘𝑅)) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝑁‘∪ 𝑆) ↔ ∃𝑎 ∈ (𝐵 ↑m 𝑆)(𝑎 finSupp 0 ∧ 𝑋 = (𝑅 Σg 𝑎) ∧ ∀𝑘 ∈ 𝑆 (𝑎‘𝑘) ∈ 𝑘))) | ||
| Theorem | elrspunsn 33390* | Membership to the span of an ideal 𝑅 and a single element 𝑋. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝑁 = (RSpan‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ + = (+g‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝐼)) ⇒ ⊢ (𝜑 → (𝐴 ∈ (𝑁‘(𝐼 ∪ {𝑋})) ↔ ∃𝑟 ∈ 𝐵 ∃𝑖 ∈ 𝐼 𝐴 = ((𝑟 · 𝑋) + 𝑖))) | ||
| Theorem | lidlincl 33391 | Ideals are closed under intersection. (Contributed by Thierry Arnoux, 5-Jul-2024.) |
| ⊢ 𝑈 = (LIdeal‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐽 ∈ 𝑈) → (𝐼 ∩ 𝐽) ∈ 𝑈) | ||
| Theorem | idlinsubrg 33392 | The intersection between an ideal and a subring is an ideal of the subring. (Contributed by Thierry Arnoux, 6-Jul-2024.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 𝑈 = (LIdeal‘𝑅) & ⊢ 𝑉 = (LIdeal‘𝑆) ⇒ ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (𝐼 ∩ 𝐴) ∈ 𝑉) | ||
| Theorem | rhmimaidl 33393 | The image of an ideal 𝐼 by a surjective ring homomorphism 𝐹 is an ideal. (Contributed by Thierry Arnoux, 6-Jul-2024.) |
| ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑇 = (LIdeal‘𝑅) & ⊢ 𝑈 = (LIdeal‘𝑆) ⇒ ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵 ∧ 𝐼 ∈ 𝑇) → (𝐹 “ 𝐼) ∈ 𝑈) | ||
| Theorem | drngidl 33394 | A nonzero ring is a division ring if and only if its only left ideals are the zero ideal and the unit ideal. (Proposed by Gerard Lang, 13-Mar-2025.) (Contributed by Thierry Arnoux, 13-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑈 = (LIdeal‘𝑅) ⇒ ⊢ (𝑅 ∈ NzRing → (𝑅 ∈ DivRing ↔ 𝑈 = {{ 0 }, 𝐵})) | ||
| Theorem | drngidlhash 33395 | A ring is a division ring if and only if it admits exactly two ideals. (Proposed by Gerard Lang, 13-Mar-2025.) (Contributed by Thierry Arnoux, 13-Mar-2025.) |
| ⊢ 𝑈 = (LIdeal‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝑅 ∈ DivRing ↔ (♯‘𝑈) = 2)) | ||
| Syntax | cprmidl 33396 | Extend class notation with the class of prime ideals. |
| class PrmIdeal | ||
| Definition | df-prmidl 33397* | Define the class of prime ideals of a ring 𝑅. A proper ideal 𝐼 of 𝑅 is prime if whenever 𝐴𝐵 ⊆ 𝐼 for ideals 𝐴 and 𝐵, either 𝐴 ⊆ 𝐼 or 𝐵 ⊆ 𝐼. The more familiar definition using elements rather than ideals is equivalent provided 𝑅 is commutative; see prmidl2 33402 and isprmidlc 33408. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 14-Jan-2024.) |
| ⊢ PrmIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑎 ∈ (LIdeal‘𝑟)∀𝑏 ∈ (LIdeal‘𝑟)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑟)𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))}) | ||
| Theorem | prmidlval 33398* | The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))}) | ||
| Theorem | isprmidl 33399* | The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃))))) | ||
| Theorem | prmidlnr 33400 | A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃 ≠ 𝐵) | ||
| < Previous Next > |
| Copyright terms: Public domain | < Previous Next > |