HomeTheorem List (p. 339 of 509)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | imaslmod 33801* | 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 33802* | 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 33803* | 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 33804* | 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 33805* | 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 33806 | 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 33807* | 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 33808 | 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 | znfermltl 33809 | 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 33810* | 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 33811* | Variation on ellspd 22021. (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 33812 | 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 33813 | 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 | elrsp 33814* | 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 33815 | 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 33816* | 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 33817 | 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 | lbslsp 33818* | 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 33819 | 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 33820 | 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 33821* | 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 33822 | 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 33823* | 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 33824* | 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 33825 | 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 33826* | 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 33827* | A reformulation of dvdsruasso 33826. (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 33828 | 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 33829 | 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 33830 | 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 33831* | 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 33832* | The sumset of a group with a single element is the element's orbit by the group action. See gaorb 19440. (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 33833* | The sumset of a single element with a group is the element's orbit by the group action. See gaorb 19440. (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 | ringlsmss 33834 | 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 33835 | 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 33836 | 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 33837 | 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 33838 | 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 | lsmssass 33839 | Group sum is associative, subset version (see lsmass 19802). (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 33840 | 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 33841 | 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 33842 | 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 33843* | 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 33844 | 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 33845* | 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 33846* | 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 33847 | 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 33848* | Lemma for nsgmgc 33849. (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 33849* | 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 33850* | Lemma for nsgqusf1o 33853. (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 33851* | Lemma for nsgqusf1o 33853. (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 33852* | Lemma for nsgqusf1o 33853. (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 33853* | 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 33854* | 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 33855 | 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 | intlidl 33856 | The intersection of a nonempty collection of ideals is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 8-Jun-2024.) |
| ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → ∩ 𝐶 ∈ (LIdeal‘𝑅)) | ||
| Theorem | inlidl 33857 | The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by AV, 28-Jun-2026.) |
| ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅)) → (𝐼 ∩ 𝐽) ∈ (LIdeal‘𝑅)) | ||
| Theorem | pidlnzb 33858 | 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 33859 | 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 33860 | The ideal 𝐼 generated by an element 𝑋 of a commutative ring 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 ∈ CRing) ⇒ ⊢ (𝜑 → (𝐼 = 𝐵 ↔ 𝑋 ∈ 𝑈)) | ||
| Theorem | rhmquskerlem 33861* | 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 33862* | 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 33863 | 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 33864* | 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 33865* | 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 33866 | 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 33867 | 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 33868 | 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 | drngidlhash 33869 | 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 | cmxidl 33870 | Extend class notation with the class of maximal ideals. |
| class MaxIdeal | ||
| Definition | df-mxidl 33871* | Define the class of maximal ideals of a ring 𝑅. A proper ideal is called maximal if it is maximal with respect to inclusion among proper ideals. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
| ⊢ MaxIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = (Base‘𝑟))))}) | ||
| Theorem | mxidlval 33872* | The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (MaxIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝐵)))}) | ||
| Theorem | ismxidl 33873* | The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵))))) | ||
| Theorem | mxidlidl 33874 | A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅)) | ||
| Theorem | mxidlnr 33875 | A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ 𝐵) | ||
| Theorem | mxidlmax 33876 | A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝐼)) → (𝐼 = 𝑀 ∨ 𝐼 = 𝐵)) | ||
| Theorem | mxidln1 33877 | One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ¬ 1 ∈ 𝑀) | ||
| Theorem | mxidlnzr 33878 | A ring with a maximal ideal is a nonzero ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ NzRing) | ||
| Theorem | mxidlmaxv 33879 | An ideal 𝐼 strictly containing a maximal ideal 𝑀 is the whole ring 𝐵. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) & ⊢ (𝜑 → 𝑀 ⊆ 𝐼) & ⊢ (𝜑 → 𝑋 ∈ (𝐼 ∖ 𝑀)) ⇒ ⊢ (𝜑 → 𝐼 = 𝐵) | ||
| Theorem | crngmxidl 33880 | In a commutative ring, maximal ideals of the opposite ring coincide with maximal ideals. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
| ⊢ 𝑀 = (MaxIdeal‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → 𝑀 = (MaxIdeal‘𝑂)) | ||
| Theorem | mxidlprm 33881 | Every maximal ideal is prime. Statement in [Lang] p. 92. (Contributed by Thierry Arnoux, 21-Jan-2024.) |
| ⊢ × = (LSSum‘(mulGrp‘𝑅)) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (PrmIdeal‘𝑅)) | ||
| Theorem | mxidlirredi 33882 | In an integral domain, the generator of a maximal ideal is irreducible. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑀 = (𝐾‘{𝑋}) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ≠ 0 ) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) ⇒ ⊢ (𝜑 → 𝑋 ∈ (Irred‘𝑅)) | ||
| Theorem | mxidlirred 33883 | In a principal ideal domain, maximal ideals are exactly the ideals generated by irreducible elements. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑀 = (𝐾‘{𝑋}) & ⊢ (𝜑 → 𝑅 ∈ PID) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ≠ 0 ) & ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅)) ⇒ ⊢ (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ 𝑋 ∈ (Irred‘𝑅))) | ||
| Theorem | ssmxidllem 33884* | The set 𝑃 used in the proof of ssmxidl 33885 satisfies the condition of Zorn's Lemma. (Contributed by Thierry Arnoux, 10-Apr-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝)} & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) & ⊢ (𝜑 → 𝐼 ≠ 𝐵) & ⊢ (𝜑 → 𝑍 ⊆ 𝑃) & ⊢ (𝜑 → 𝑍 ≠ ∅) & ⊢ (𝜑 → [⊊] Or 𝑍) ⇒ ⊢ (𝜑 → ∪ 𝑍 ∈ 𝑃) | ||
| Theorem | ssmxidl 33885* | Let 𝑅 be a ring, and let 𝐼 be a proper ideal of 𝑅. Then there is a maximal ideal of 𝑅 containing 𝐼. (Contributed by Thierry Arnoux, 10-Apr-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼 ≠ 𝐵) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝐼 ⊆ 𝑚) | ||
| Theorem | drng0mxidl 33886 | In a division ring, the zero ideal is a maximal ideal. (Contributed by Thierry Arnoux, 16-Mar-2025.) |
| ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → { 0 } ∈ (MaxIdeal‘𝑅)) | ||
| Theorem | drngmxidl 33887 | The zero ideal is the only ideal of a division ring. (Contributed by Thierry Arnoux, 16-Mar-2025.) |
| ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → (MaxIdeal‘𝑅) = {{ 0 }}) | ||
| Theorem | drngmxidlr 33888 | If a ring's only maximal ideal is the zero ideal, it is a division ring. See also drngmxidl 33887. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑀 = (MaxIdeal‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝑀 = {{ 0 }}) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing) | ||
| Theorem | krull 33889* | Krull's theorem: Any nonzero ring has at least one maximal ideal. (Contributed by Thierry Arnoux, 10-Apr-2024.) |
| ⊢ (𝑅 ∈ NzRing → ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅)) | ||
| Theorem | mxidlnzrb 33890* | A ring is nonzero if and only if it has maximal ideals. (Contributed by Thierry Arnoux, 10-Apr-2024.) |
| ⊢ (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅))) | ||
| Theorem | krullndrng 33891* | Krull's theorem for non-division-rings: Existence of a nonzero maximal ideal. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → ¬ 𝑅 ∈ DivRing) ⇒ ⊢ (𝜑 → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝑚 ≠ { 0 }) | ||
| Theorem | opprabs 33892 | The opposite ring of the opposite ring is the original ring. Note the conditions on this theorem, which makes it unpractical in case we only have e.g. 𝑅 ∈ Ring as a premise. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → Fun 𝑅) & ⊢ (𝜑 → (.r‘ndx) ∈ dom 𝑅) & ⊢ (𝜑 → · Fn (𝐵 × 𝐵)) ⇒ ⊢ (𝜑 → 𝑅 = (oppr‘𝑂)) | ||
| Theorem | oppreqg 33893 | Group coset equivalence relation for the opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ⊆ 𝐵) → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼)) | ||
| Theorem | opprnsg 33894 | Normal subgroups of the opposite ring are the same as the original normal subgroups. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (NrmSGrp‘𝑅) = (NrmSGrp‘𝑂) | ||
| Theorem | opprlidlabs 33895 | The ideals of the opposite ring's opposite ring are the ideals of the original ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (LIdeal‘𝑅) = (LIdeal‘(oppr‘𝑂))) | ||
| Theorem | oppr2idl 33896 | Two sided ideal of the opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂)) | ||
| Theorem | opprmxidlabs 33897 | The maximal ideal of the opposite ring's opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) ⇒ ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘(oppr‘𝑂))) | ||
| Theorem | opprqusbas 33898 | The base of the quotient of the opposite ring is the same as the base of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (Base‘(oppr‘𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))) | ||
| Theorem | opprqusplusg 33899 | The group operation of the quotient of the opposite ring is the same as the group operation of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) & ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅)) & ⊢ 𝐸 = (Base‘𝑄) & ⊢ (𝜑 → 𝑋 ∈ 𝐸) & ⊢ (𝜑 → 𝑌 ∈ 𝐸) ⇒ ⊢ (𝜑 → (𝑋(+g‘(oppr‘𝑄))𝑌) = (𝑋(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌)) | ||
| Theorem | opprqus0g 33900 | The group identity element of the quotient of the opposite ring is the same as the group identity element of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) & ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅)) ⇒ ⊢ (𝜑 → (0g‘(oppr‘𝑄)) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))) | ||
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