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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | imasmhm 33301* | 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 33302* | 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 33303* | 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 33304* | 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 33305 | 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 33306* | 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 33307 | 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 33308 | 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 33309 | The quotient set of a cartesian product is trivial. (Contributed by Thierry Arnoux, 16-Jan-2024.) |
| ⊢ (𝐴 ≠ ∅ → (𝐴 / (𝐴 × 𝐴)) = {𝐴}) | ||
| Theorem | qusxpid 33310 | 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 33311 | 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 33312 | Converse of qustriv 33311. (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 33313 | 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 33314* | 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 33315* | Variation on ellspd 21727. (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 33316 | 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 33317 | 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 33318 | 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 33319* | 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 33320 | 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 33321* | 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 33322 | 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 33323 | 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 33324* | 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 33325 | 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 33326 | 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 33327* | 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 33328 | 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 33329* | 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 33330* | 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 33331 | 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 33332* | 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 33333* | A reformulation of dvdsruasso 33332. (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 33334 | 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 33335 | 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 33336 | 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 33337* | 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 33338* | The sumset of a group with a single element is the element's orbit by the group action. See gaorb 19204. (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 33339* | The sumset of a single element with a group is the element's orbit by the group action. See gaorb 19204. (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 33340* | 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 33341 | 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 33342 | 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 33343 | 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 33344 | 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 33345 | 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 33346 | 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 33347 | 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 33348 | 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 33349 | Group sum is associative, subset version (see lsmass 19566). (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 33350 | 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 33351 | 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 33352 | 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 33353* | 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 33354 | 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 33355* | 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 33356* | 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 33357 | 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 33358* | Lemma for nsgmgc 33359. (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 33359* | 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 33360* | Lemma for nsgqusf1o 33363. (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 33361* | Lemma for nsgqusf1o 33363. (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 33362* | Lemma for nsgqusf1o 33363. (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 33363* | 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 33364* | 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 33365 | 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 33366 | 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 33367 | 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 33368 | 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 33369 | 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 33370 | 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 33371 | 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 33372* | 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 33373* | 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 33374 | 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 33375* | 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 33376* | 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 33377 | 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 33378 | 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 33379 | 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 33380 | 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 33381 | 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 33382 | Extend class notation with the class of prime ideals. |
| class PrmIdeal | ||
| Definition | df-prmidl 33383* | 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 33388 and isprmidlc 33394. (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 33384* | 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 33385* | 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 33386 | 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) | ||
| Theorem | prmidl 33387* | The main property of a prime 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)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) ∧ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃) → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃)) | ||
| Theorem | prmidl2 33388* | A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 38049 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (((𝑅 ∈ Ring ∧ 𝑃 ∈ (LIdeal‘𝑅)) ∧ (𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))) → 𝑃 ∈ (PrmIdeal‘𝑅)) | ||
| Theorem | idlmulssprm 33389 | Let 𝑃 be a prime ideal containing the product (𝐼 × 𝐽) of two ideals 𝐼 and 𝐽. Then 𝐼 ⊆ 𝑃 or 𝐽 ⊆ 𝑃. (Contributed by Thierry Arnoux, 13-Apr-2024.) |
| ⊢ × = (LSSum‘(mulGrp‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑃 ∈ (PrmIdeal‘𝑅)) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) & ⊢ (𝜑 → 𝐽 ∈ (LIdeal‘𝑅)) & ⊢ (𝜑 → (𝐼 × 𝐽) ⊆ 𝑃) ⇒ ⊢ (𝜑 → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃)) | ||
| Theorem | pridln1 33390 | A proper ideal cannot contain the ring unity. (Contributed by Thierry Arnoux, 9-Apr-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼 ≠ 𝐵) → ¬ 1 ∈ 𝐼) | ||
| Theorem | prmidlidl 33391 | A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.) |
| ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃 ∈ (LIdeal‘𝑅)) | ||
| Theorem | prmidlssidl 33392 | Prime ideals as a subset of ideals. (Contributed by Thierry Arnoux, 2-Jun-2024.) |
| ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅)) | ||
| Theorem | cringm4 33393 | Commutative/associative law for commutative ring. (Contributed by Thierry Arnoux, 14-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 · 𝑌) · (𝑍 · 𝑊)) = ((𝑋 · 𝑍) · (𝑌 · 𝑊))) | ||
| Theorem | isprmidlc 33394* | The predicate "is prime ideal" for commutative rings. Alternate definition for commutative rings. See definition in [Lang] p. 92. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃))))) | ||
| Theorem | prmidlc 33395 | Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 12-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ 𝐵 ∧ 𝐽 ∈ 𝐵 ∧ (𝐼 · 𝐽) ∈ 𝑃)) → (𝐼 ∈ 𝑃 ∨ 𝐽 ∈ 𝑃)) | ||
| Theorem | 0ringprmidl 33396 | The trivial ring does not have any prime ideal. (Contributed by Thierry Arnoux, 30-Jun-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → (PrmIdeal‘𝑅) = ∅) | ||
| Theorem | prmidl0 33397 | The zero ideal of a commutative ring 𝑅 is a prime ideal if and only if 𝑅 is an integral domain. (Contributed by Thierry Arnoux, 30-Jun-2024.) |
| ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ { 0 } ∈ (PrmIdeal‘𝑅)) ↔ 𝑅 ∈ IDomn) | ||
| Theorem | rhmpreimaprmidl 33398 | The preimage of a prime ideal by a ring homomorphism is a prime ideal. (Contributed by Thierry Arnoux, 29-Jun-2024.) |
| ⊢ 𝑃 = (PrmIdeal‘𝑅) ⇒ ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ 𝑃) | ||
| Theorem | qsidomlem1 33399 | If the quotient ring of a commutative ring relative to an ideal is an integral domain, that ideal must be prime. (Contributed by Thierry Arnoux, 16-Jan-2024.) |
| ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) ⇒ ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ∈ (PrmIdeal‘𝑅)) | ||
| Theorem | qsidomlem2 33400 | A quotient by a prime ideal is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.) |
| ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ IDomn) | ||
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