HomeTheorem List (p. 333 of 489)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | slmd0vlid 33201 | Left identity law for the zero vector. (hvaddlid 31055 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → ( 0 + 𝑋) = 𝑋) | ||
| Theorem | slmd0vrid 33202 | Right identity law for the zero vector. (ax-hvaddid 31036 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (𝑋 + 0 ) = 𝑋) | ||
| Theorem | slmd0vs 33203 | Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 31042 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑂 = (0g‘𝐹) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) = 0 ) | ||
| Theorem | slmdvs0 33204 | Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 31056 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · 0 ) = 0 ) | ||
| Theorem | gsumvsca1 33205* | Scalar product of a finite group sum for a left module over a semiring. (Contributed by Thierry Arnoux, 16-Mar-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐺 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ (𝜑 → 𝐾 ⊆ (Base‘𝐺)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝑊 ∈ SLMod) & ⊢ (𝜑 → 𝑃 ∈ 𝐾) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑄 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄)))) | ||
| Theorem | gsumvsca2 33206* | Scalar product of a finite group sum for a left module over a semiring. (Contributed by Thierry Arnoux, 16-Mar-2018.) (Proof shortened by AV, 12-Dec-2019.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐺 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ (𝜑 → 𝐾 ⊆ (Base‘𝐺)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝑊 ∈ SLMod) & ⊢ (𝜑 → 𝑄 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃 ∈ 𝐾) ⇒ ⊢ (𝜑 → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑃)) · 𝑄)) | ||
| Theorem | prmsimpcyc 33207 | A group of prime order is cyclic if and only if it is simple. This is the first family of finite simple groups. (Contributed by Thierry Arnoux, 21-Sep-2023.) |
| ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ ((♯‘𝐵) ∈ ℙ → (𝐺 ∈ SimpGrp ↔ 𝐺 ∈ CycGrp)) | ||
| Theorem | cringmul32d 33208 | Commutative/associative law that swaps the last two factors in a triple product in a commutative ring. See also mul32 11456. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 · 𝑌) · 𝑍) = ((𝑋 · 𝑍) · 𝑌)) | ||
| Theorem | ringdid 33209 | Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍))) | ||
| Theorem | ringdird 33210 | Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))) | ||
| Theorem | ringdi22 33211 | Expand the product of two sums in a ring. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝑇 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇)))) | ||
| Theorem | urpropd 33212* | Sufficient condition for ring unities to be equal. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑇 ∈ 𝑊) & ⊢ (𝜑 → 𝐵 = (Base‘𝑇)) & ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝑆)𝑦) = (𝑥(.r‘𝑇)𝑦)) ⇒ ⊢ (𝜑 → (1r‘𝑆) = (1r‘𝑇)) | ||
| Theorem | subrgmcld 33213 | A subring is closed under multiplication. (Contributed by Thierry Arnoux, 6-Jul-2025.) |
| ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝐴 ∈ (SubRing‘𝑅)) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐴) | ||
| Theorem | ress1r 33214 | 1r is unaffected by restriction. This is a bit more generic than subrg1 20610. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 1 = (1r‘𝑆)) | ||
| Theorem | ringinvval 33215* | The ring inverse expressed in terms of multiplication. (Contributed by Thierry Arnoux, 23-Oct-2017.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∗ = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑁 = (invr‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) = (℩𝑦 ∈ 𝑈 (𝑦 ∗ 𝑋) = 1 )) | ||
| Theorem | dvrcan5 33216 | Cancellation law for common factor in ratio. (divcan5 11996 analog.) (Contributed by Thierry Arnoux, 26-Oct-2016.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈 ∧ 𝑍 ∈ 𝑈)) → ((𝑋 · 𝑍) / (𝑌 · 𝑍)) = (𝑋 / 𝑌)) | ||
| Theorem | subrgchr 33217 | If 𝐴 is a subring of 𝑅, then they have the same characteristic. (Contributed by Thierry Arnoux, 24-Feb-2018.) |
| ⊢ (𝐴 ∈ (SubRing‘𝑅) → (chr‘(𝑅 ↾s 𝐴)) = (chr‘𝑅)) | ||
| Theorem | rmfsupp2 33218* | A mapping of a multiplication of a constant with a function into a ring is finitely supported if the function is finitely supported. (Contributed by Thierry Arnoux, 3-Jun-2023.) |
| ⊢ 𝑅 = (Base‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ Ring) & ⊢ (𝜑 → 𝑉 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝐶 ∈ 𝑅) & ⊢ (𝜑 → 𝐴:𝑉⟶𝑅) & ⊢ (𝜑 → 𝐴 finSupp (0g‘𝑀)) ⇒ ⊢ (𝜑 → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) finSupp (0g‘𝑀)) | ||
| Theorem | unitnz 33219 | In a nonzero ring, a unit cannot be zero. (Contributed by Thierry Arnoux, 25-Apr-2025.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝑋 ≠ 0 ) | ||
| Theorem | isunit2 33220* | Alternate definition of being a unit. (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ (∃𝑢 ∈ 𝐵 (𝑋 · 𝑢) = 1 ∧ ∃𝑣 ∈ 𝐵 (𝑣 · 𝑋) = 1 ))) | ||
| Theorem | isunit3 33221* | Alternate definition of being a unit. (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ ∃𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 1 ∧ (𝑦 · 𝑋) = 1 ))) | ||
| Theorem | irrednzr 33222 | A ring with an irreducible element cannot be the zero ring. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝐼 = (Irred‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → 𝑅 ∈ NzRing) | ||
| Theorem | 0ringsubrg 33223 | A subring of a zero ring is a zero ring. (Contributed by Thierry Arnoux, 5-Feb-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → (♯‘𝐵) = 1) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) ⇒ ⊢ (𝜑 → (♯‘𝑆) = 1) | ||
| Theorem | 0ringcring 33224 | The zero ring is commutative. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → (♯‘𝐵) = 1) ⇒ ⊢ (𝜑 → 𝑅 ∈ CRing) | ||
| Syntax | cerl 33225 | Syntax for ring localization equivalence class operation. |
| class ~RL | ||
| Syntax | crloc 33226 | Syntax for ring localization operation. |
| class RLocal | ||
| Definition | df-erl 33227* | Define the operation giving the equivalence relation used in the localization of a ring 𝑟 by a set 𝑠. Two pairs 𝑎 = ⟨𝑥, 𝑦⟩ and 𝑏 = ⟨𝑧, 𝑤⟩ are equivalent if there exists 𝑡 ∈ 𝑠 such that 𝑡 · (𝑥 · 𝑤 − 𝑧 · 𝑦) = 0. This corresponds to the usual comparison of fractions 𝑥 / 𝑦 and 𝑧 / 𝑤. (Contributed by Thierry Arnoux, 28-Apr-2025.) |
| ⊢ ~RL = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋(.r‘𝑟) / 𝑥⦌⦋((Base‘𝑟) × 𝑠) / 𝑤⦌{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑤 ∧ 𝑏 ∈ 𝑤) ∧ ∃𝑡 ∈ 𝑠 (𝑡𝑥(((1st ‘𝑎)𝑥(2nd ‘𝑏))(-g‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎)))) = (0g‘𝑟))}) | ||
| Definition | df-rloc 33228* | Define the operation giving the localization of a ring 𝑟 by a given set 𝑠. The localized ring 𝑟 RLocal 𝑠 is the set of equivalence classes of pairs of elements in 𝑟 over the relation 𝑟 ~RL 𝑠 with addition and multiplication defined naturally. (Contributed by Thierry Arnoux, 27-Apr-2025.) |
| ⊢ RLocal = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋(.r‘𝑟) / 𝑥⦌⦋((Base‘𝑟) × 𝑠) / 𝑤⦌((({⟨(Base‘ndx), 𝑤⟩, ⟨(+g‘ndx), (𝑎 ∈ 𝑤, 𝑏 ∈ 𝑤 ↦ ⟨(((1st ‘𝑎)𝑥(2nd ‘𝑏))(+g‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎))), ((2nd ‘𝑎)𝑥(2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ 𝑤, 𝑏 ∈ 𝑤 ↦ ⟨((1st ‘𝑎)𝑥(1st ‘𝑏)), ((2nd ‘𝑎)𝑥(2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑟)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑟)), 𝑎 ∈ 𝑤 ↦ ⟨(𝑘( ·𝑠 ‘𝑟)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑟) ×t ((TopSet‘𝑟) ↾t 𝑠))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑤 ∧ 𝑏 ∈ 𝑤) ∧ ((1st ‘𝑎)𝑥(2nd ‘𝑏))(le‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑤, 𝑏 ∈ 𝑤 ↦ (((1st ‘𝑎)𝑥(2nd ‘𝑏))(dist‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎))))⟩}) /s (𝑟 ~RL 𝑠))) | ||
| Theorem | reldmrloc 33229 | Ring localization is a proper operator, so it can be used with ovprc1 7487. (Contributed by Thierry Arnoux, 10-May-2025.) |
| ⊢ Rel dom RLocal | ||
| Theorem | erlval 33230* | Value of the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ − = (-g‘𝑅) & ⊢ 𝑊 = (𝐵 × 𝑆) & ⊢ ∼ = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )} & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑅 ~RL 𝑆) = ∼ ) | ||
| Theorem | rlocval 33231* | Expand the value of the ring localization operation. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ − = (-g‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ ≤ = (le‘𝑅) & ⊢ 𝐹 = (Scalar‘𝑅) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝐶 = ( ·𝑠 ‘𝑅) & ⊢ 𝑊 = (𝐵 × 𝑆) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ 𝐽 = (TopSet‘𝑅) & ⊢ 𝐷 = (dist‘𝑅) & ⊢ ⊕ = (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩) & ⊢ ⊗ = (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩) & ⊢ × = (𝑘 ∈ 𝐾, 𝑎 ∈ 𝑊 ↦ ⟨(𝑘𝐶(1st ‘𝑎)), (2nd ‘𝑎)⟩) & ⊢ ≲ = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1st ‘𝑎) · (2nd ‘𝑏)) ≤ ((1st ‘𝑏) · (2nd ‘𝑎)))} & ⊢ 𝐸 = (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1st ‘𝑎) · (2nd ‘𝑏))𝐷((1st ‘𝑏) · (2nd ‘𝑎)))) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑅 RLocal 𝑆) = ((({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), ⊗ ⟩} ∪ {⟨(Scalar‘ndx), 𝐹⟩, ⟨( ·𝑠 ‘ndx), × ⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), (𝐽 ×t (𝐽 ↾t 𝑆))⟩, ⟨(le‘ndx), ≲ ⟩, ⟨(dist‘ndx), 𝐸⟩}) /s ∼ )) | ||
| Theorem | erlcl1 33232 | Closure for the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝑈 ∼ 𝑉) ⇒ ⊢ (𝜑 → 𝑈 ∈ (𝐵 × 𝑆)) | ||
| Theorem | erlcl2 33233 | Closure for the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝑈 ∼ 𝑉) ⇒ ⊢ (𝜑 → 𝑉 ∈ (𝐵 × 𝑆)) | ||
| Theorem | erldi 33234* | Main property of the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ − = (-g‘𝑅) & ⊢ (𝜑 → 𝑈 ∼ 𝑉) ⇒ ⊢ (𝜑 → ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑈) · (2nd ‘𝑉)) − ((1st ‘𝑉) · (2nd ‘𝑈)))) = 0 ) | ||
| Theorem | erlbrd 33235 | Deduce the ring localization equivalence relation. If for some 𝑇 ∈ 𝑆 we have 𝑇 · (𝐸 · 𝐻 − 𝐹 · 𝐺) = 0, then pairs ⟨𝐸, 𝐺⟩ and ⟨𝐹, 𝐻⟩ are equivalent under the localization relation. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ − = (-g‘𝑅) & ⊢ (𝜑 → 𝑈 = ⟨𝐸, 𝐺⟩) & ⊢ (𝜑 → 𝑉 = ⟨𝐹, 𝐻⟩) & ⊢ (𝜑 → 𝐸 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝐻 ∈ 𝑆) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → (𝑇 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 ) ⇒ ⊢ (𝜑 → 𝑈 ∼ 𝑉) | ||
| Theorem | erlbr2d 33236 | Deduce the ring localization equivalence relation. Pairs ⟨𝐸, 𝐺⟩ and ⟨𝑇 · 𝐸, 𝑇 · 𝐺⟩ for 𝑇 ∈ 𝑆 are equivalent under the localization relation. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑈 = ⟨𝐸, 𝐺⟩) & ⊢ (𝜑 → 𝑉 = ⟨𝐹, 𝐻⟩) & ⊢ (𝜑 → 𝐸 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝐻 ∈ 𝑆) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝐹 = (𝑇 · 𝐸)) & ⊢ (𝜑 → 𝐻 = (𝑇 · 𝐺)) ⇒ ⊢ (𝜑 → 𝑈 ∼ 𝑉) | ||
| Theorem | erler 33237 | The relation used to build the ring localization is an equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ − = (-g‘𝑅) & ⊢ 𝑊 = (𝐵 × 𝑆) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) ⇒ ⊢ (𝜑 → ∼ Er 𝑊) | ||
| Theorem | elrlocbasi 33238* | Membership in the basis of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ (𝜑 → 𝑋 ∈ ((𝐵 × 𝑆) / ∼ )) ⇒ ⊢ (𝜑 → ∃𝑎 ∈ 𝐵 ∃𝑏 ∈ 𝑆 𝑋 = [⟨𝑎, 𝑏⟩] ∼ ) | ||
| Theorem | rlocbas 33239 | The base set of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ − = (-g‘𝑅) & ⊢ 𝑊 = (𝐵 × 𝑆) & ⊢ 𝐿 = (𝑅 RLocal 𝑆) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐿)) | ||
| Theorem | rlocaddval 33240 | Value of the addition in the ring localization, given two representatives. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ 𝐿 = (𝑅 RLocal 𝑆) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) & ⊢ (𝜑 → 𝐸 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝐻 ∈ 𝑆) & ⊢ ⊕ = (+g‘𝐿) ⇒ ⊢ (𝜑 → ([⟨𝐸, 𝐺⟩] ∼ ⊕ [⟨𝐹, 𝐻⟩] ∼ ) = [⟨((𝐸 · 𝐻) + (𝐹 · 𝐺)), (𝐺 · 𝐻)⟩] ∼ ) | ||
| Theorem | rlocmulval 33241 | Value of the addition in the ring localization, given two representatives. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ 𝐿 = (𝑅 RLocal 𝑆) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) & ⊢ (𝜑 → 𝐸 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝐻 ∈ 𝑆) & ⊢ ⊗ = (.r‘𝐿) ⇒ ⊢ (𝜑 → ([⟨𝐸, 𝐺⟩] ∼ ⊗ [⟨𝐹, 𝐻⟩] ∼ ) = [⟨(𝐸 · 𝐹), (𝐺 · 𝐻)⟩] ∼ ) | ||
| Theorem | rloccring 33242 | The ring localization 𝐿 of a commutative ring 𝑅 by a multiplicatively closed set 𝑆 is itself a commutative ring. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ 𝐿 = (𝑅 RLocal 𝑆) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) ⇒ ⊢ (𝜑 → 𝐿 ∈ CRing) | ||
| Theorem | rloc0g 33243 | The zero of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐿 = (𝑅 RLocal 𝑆) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) & ⊢ 𝑂 = [⟨ 0 , 1 ⟩] ∼ ⇒ ⊢ (𝜑 → 𝑂 = (0g‘𝐿)) | ||
| Theorem | rloc1r 33244 | The multiplicative identity of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐿 = (𝑅 RLocal 𝑆) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) & ⊢ 𝐼 = [⟨ 1 , 1 ⟩] ∼ ⇒ ⊢ (𝜑 → 𝐼 = (1r‘𝐿)) | ||
| Theorem | rlocf1 33245* | The embedding 𝐹 of a ring 𝑅 into its localization 𝐿. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐿 = (𝑅 RLocal 𝑆) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [⟨𝑥, 1 ⟩] ∼ ) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) & ⊢ (𝜑 → 𝑆 ⊆ (RLReg‘𝑅)) ⇒ ⊢ (𝜑 → (𝐹:𝐵–1-1→((𝐵 × 𝑆) / ∼ ) ∧ 𝐹 ∈ (𝑅 RingHom 𝐿))) | ||
| Theorem | domnmuln0rd 33246 | In a domain, factors of a nonzero product are nonzero. (Contributed by Thierry Arnoux, 8-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Domn) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (𝑋 · 𝑌) ≠ 0 ) ⇒ ⊢ (𝜑 → (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) | ||
| Theorem | domnprodn0 33247 | In a domain, a finite product of nonzero terms is nonzero. (Contributed by Thierry Arnoux, 6-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Domn) & ⊢ (𝜑 → 𝐹 ∈ Word (𝐵 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝑀 Σg 𝐹) ≠ 0 ) | ||
| Theorem | idomrcan 33248 | Right-cancellation law for integral domains. (Contributed by Thierry Arnoux, 22-Mar-2025.) (Proof shortened by SN, 21-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ (𝐵 ∖ { 0 })) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → (𝑋 · 𝑍) = (𝑌 · 𝑍)) ⇒ ⊢ (𝜑 → 𝑋 = 𝑌) | ||
| Theorem | domnlcanOLD 33249 | Obsolete version of domnlcan 20743 as of 21-Jun-2025. (Contributed by Thierry Arnoux, 22-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝑅 ∈ Domn) & ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑋 · 𝑍)) ⇒ ⊢ (𝜑 → 𝑌 = 𝑍) | ||
| Theorem | domnlcanbOLD 33250 | Obsolete version of domnlcanb 20742 as of 21-Jun-2025. (Contributed by Thierry Arnoux, 8-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝑅 ∈ Domn) ⇒ ⊢ (𝜑 → ((𝑋 · 𝑌) = (𝑋 · 𝑍) ↔ 𝑌 = 𝑍)) | ||
| Theorem | idomrcanOLD 33251 | Obsolete version of idomrcan 33248 as of 21-Jun-2025. (Contributed by Thierry Arnoux, 22-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → (𝑌 · 𝑋) = (𝑍 · 𝑋)) ⇒ ⊢ (𝜑 → 𝑌 = 𝑍) | ||
| Theorem | 1rrg 33252 | The multiplicative identity is a left-regular element. (Contributed by Thierry Arnoux, 6-May-2025.) |
| ⊢ 1 = (1r‘𝑅) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → 1 ∈ 𝐸) | ||
| Theorem | rrgsubm 33253 | The left regular elements of a ring form a submonoid of the multiplicative group. (Contributed by Thierry Arnoux, 10-May-2025.) |
| ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → 𝐸 ∈ (SubMnd‘𝑀)) | ||
| Theorem | subrdom 33254 | A subring of a domain is a domain. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ (𝜑 → 𝑅 ∈ Domn) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) ⇒ ⊢ (𝜑 → (𝑅 ↾s 𝑆) ∈ Domn) | ||
| Theorem | subridom 33255 | A subring of an integral domain is an integral domain. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) ⇒ ⊢ (𝜑 → (𝑅 ↾s 𝑆) ∈ IDomn) | ||
| Theorem | subrfld 33256 | A subring of a field is an integral domain. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ (𝜑 → 𝑅 ∈ Field) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) ⇒ ⊢ (𝜑 → (𝑅 ↾s 𝑆) ∈ IDomn) | ||
| Syntax | ceuf 33257 | Declare the syntax for the Euclidean function index extractor. |
| class EuclF | ||
| Definition | df-euf 33258 | Define the Euclidean function. (Contributed by Thierry Arnoux, 22-Mar-2025.) Use its index-independent form eufid 33260 instead. (New usage is discouraged.) |
| ⊢ EuclF = Slot ;21 | ||
| Theorem | eufndx 33259 | Index value of the Euclidean function slot. Use ndxarg 17243. (Contributed by Thierry Arnoux, 22-Mar-2025.) (New usage is discouraged.) |
| ⊢ (EuclF‘ndx) = ;21 | ||
| Theorem | eufid 33260 | Utility theorem: index-independent form of df-euf 33258. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ EuclF = Slot (EuclF‘ndx) | ||
| Syntax | cedom 33261 | Declare the syntax for the Euclidean Domain. |
| class EDomn | ||
| Definition | df-edom 33262* | Define Euclidean Domains. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ EDomn = {𝑑 ∈ IDomn ∣ [(EuclF‘𝑑) / 𝑒][(Base‘𝑑) / 𝑣](Fun 𝑒 ∧ (𝑒 “ (𝑣 ∖ {(0g‘𝑑)})) ⊆ (0[,)+∞) ∧ ∀𝑎 ∈ 𝑣 ∀𝑏 ∈ (𝑣 ∖ {(0g‘𝑑)})∃𝑞 ∈ 𝑣 ∃𝑟 ∈ 𝑣 (𝑎 = ((𝑏(.r‘𝑑)𝑞)(+g‘𝑑)𝑟) ∧ (𝑟 = (0g‘𝑑) ∨ (𝑒‘𝑟) < (𝑒‘𝑏))))} | ||
| Theorem | ringinveu 33263 | If a ring unit element 𝑋 admits both a left inverse 𝑌 and a right inverse 𝑍, they are equal. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → (𝑌 · 𝑋) = 1 ) & ⊢ (𝜑 → (𝑋 · 𝑍) = 1 ) ⇒ ⊢ (𝜑 → 𝑍 = 𝑌) | ||
| Theorem | isdrng4 33264* | A division ring is a ring in which 1 ≠ 0 and every nonzero element has a left and right inverse. (Contributed by Thierry Arnoux, 2-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (𝑅 ∈ DivRing ↔ ( 1 ≠ 0 ∧ ∀𝑥 ∈ (𝐵 ∖ { 0 })∃𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 1 ∧ (𝑦 · 𝑥) = 1 )))) | ||
| Theorem | rndrhmcl 33265 | The image of a division ring by a ring homomorphism is a division ring. (Contributed by Thierry Arnoux, 25-Feb-2025.) |
| ⊢ 𝑅 = (𝑁 ↾s ran 𝐹) & ⊢ 0 = (0g‘𝑁) & ⊢ (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁)) & ⊢ (𝜑 → ran 𝐹 ≠ { 0 }) & ⊢ (𝜑 → 𝑀 ∈ DivRing) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing) | ||
| Theorem | sdrgdvcl 33266 | A sub-division-ring is closed under the ring division operation. (Contributed by Thierry Arnoux, 15-Jan-2025.) |
| ⊢ / = (/r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝐴 ∈ (SubDRing‘𝑅)) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ≠ 0 ) ⇒ ⊢ (𝜑 → (𝑋 / 𝑌) ∈ 𝐴) | ||
| Theorem | sdrginvcl 33267 | A sub-division-ring is closed under the ring inverse operation. (Contributed by Thierry Arnoux, 15-Jan-2025.) |
| ⊢ 𝐼 = (invr‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝐴 ∈ (SubDRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ∈ 𝐴) | ||
| Theorem | primefldchr 33268 | The characteristic of a prime field is the same as the characteristic of the main field. (Contributed by Thierry Arnoux, 21-Aug-2023.) |
| ⊢ 𝑃 = (𝑅 ↾s ∩ (SubDRing‘𝑅)) ⇒ ⊢ (𝑅 ∈ DivRing → (chr‘𝑃) = (chr‘𝑅)) | ||
| Syntax | cfrac 33269 | Syntax for the field of fractions of a given integral domain. |
| class Frac | ||
| Definition | df-frac 33270 | Define the field of fractions of a given integral domain. (Contributed by Thierry Arnoux, 26-Apr-2025.) |
| ⊢ Frac = (𝑟 ∈ V ↦ (𝑟 RLocal (RLReg‘𝑟))) | ||
| Theorem | fracval 33271 | Value of the field of fractions. (Contributed by Thierry Arnoux, 5-May-2025.) |
| ⊢ ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅)) | ||
| Theorem | fracbas 33272 | The base of the field of fractions. (Contributed by Thierry Arnoux, 10-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝐹 = ( Frac ‘𝑅) & ⊢ ∼ = (𝑅 ~RL 𝐸) ⇒ ⊢ ((𝐵 × 𝐸) / ∼ ) = (Base‘𝐹) | ||
| Theorem | fracerl 33273 | Rewrite the ring localization equivalence relation in the case of a field of fractions. (Contributed by Thierry Arnoux, 5-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ ∼ = (𝑅 ~RL (RLReg‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐸 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (RLReg‘𝑅)) & ⊢ (𝜑 → 𝐻 ∈ (RLReg‘𝑅)) ⇒ ⊢ (𝜑 → (⟨𝐸, 𝐹⟩ ∼ ⟨𝐺, 𝐻⟩ ↔ (𝐸 · 𝐻) = (𝐺 · 𝐹))) | ||
| Theorem | fracf1 33274* | The embedding of a commutative ring 𝑅 into its field of fractions. (Contributed by Thierry Arnoux, 10-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ ∼ = (𝑅 ~RL 𝐸) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [⟨𝑥, 1 ⟩] ∼ ) ⇒ ⊢ (𝜑 → (𝐹:𝐵–1-1→((𝐵 × 𝐸) / ∼ ) ∧ 𝐹 ∈ (𝑅 RingHom ( Frac ‘𝑅)))) | ||
| Theorem | fracfld 33275 | The field of fractions of an integral domain is a field. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ (𝜑 → 𝑅 ∈ IDomn) ⇒ ⊢ (𝜑 → ( Frac ‘𝑅) ∈ Field) | ||
| Theorem | idomsubr 33276* | Every integral domain is isomorphic with a subring of some field. (Proposed by Gerard Lang, 10-May-2025.) (Contributed by Thierry Arnoux, 10-May-2025.) |
| ⊢ (𝜑 → 𝑅 ∈ IDomn) ⇒ ⊢ (𝜑 → ∃𝑓 ∈ Field ∃𝑠 ∈ (SubRing‘𝑓)𝑅 ≃𝑟 (𝑓 ↾s 𝑠)) | ||
| Syntax | cfldgen 33277 | Syntax for a function generating sub-fields. |
| class fldGen | ||
| Definition | df-fldgen 33278* | Define a function generating the smallest sub-division-ring of a given ring containing a given set. If the base structure is a division ring, then this is also a division ring (see fldgensdrg 33281). If the base structure is a field, this is a subfield (see fldgenfld 33287 and fldsdrgfld 20821). In general this will be used in the context of fields, hence the name fldGen. (Contributed by Saveliy Skresanov and Thierry Arnoux, 9-Jan-2025.) |
| ⊢ fldGen = (𝑓 ∈ V, 𝑠 ∈ V ↦ ∩ {𝑎 ∈ (SubDRing‘𝑓) ∣ 𝑠 ⊆ 𝑎}) | ||
| Theorem | fldgenval 33279* | Value of the field generating function: (𝐹 fldGen 𝑆) is the smallest sub-division-ring of 𝐹 containing 𝑆. (Contributed by Thierry Arnoux, 11-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝑆) = ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) | ||
| Theorem | fldgenssid 33280 | The field generated by a set of elements contains those elements. See lspssid 21006. (Contributed by Thierry Arnoux, 15-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝑆 ⊆ (𝐹 fldGen 𝑆)) | ||
| Theorem | fldgensdrg 33281 | A generated subfield is a sub-division-ring. (Contributed by Thierry Arnoux, 11-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝑆) ∈ (SubDRing‘𝐹)) | ||
| Theorem | fldgenssv 33282 | A generated subfield is a subset of the field's base. (Contributed by Thierry Arnoux, 25-Feb-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝑆) ⊆ 𝐵) | ||
| Theorem | fldgenss 33283 | Generated subfields preserve subset ordering. ( see lspss 21005 and spanss 31380) (Contributed by Thierry Arnoux, 15-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝑇 ⊆ 𝑆) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝑇) ⊆ (𝐹 fldGen 𝑆)) | ||
| Theorem | fldgenidfld 33284 | The subfield generated by a subfield is the subfield itself. (Contributed by Thierry Arnoux, 15-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ∈ (SubDRing‘𝐹)) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝑆) = 𝑆) | ||
| Theorem | fldgenssp 33285 | The field generated by a set of elements in a division ring is contained in any sub-division-ring which contains those elements. (Contributed by Thierry Arnoux, 25-Feb-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ∈ (SubDRing‘𝐹)) & ⊢ (𝜑 → 𝑇 ⊆ 𝑆) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝑇) ⊆ 𝑆) | ||
| Theorem | fldgenid 33286 | The subfield of a field 𝐹 generated by the whole base set of 𝐹 is 𝐹 itself. (Contributed by Thierry Arnoux, 11-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝐵) = 𝐵) | ||
| Theorem | fldgenfld 33287 | A generated subfield is a field. (Contributed by Thierry Arnoux, 11-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ Field) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ↾s (𝐹 fldGen 𝑆)) ∈ Field) | ||
| Theorem | primefldgen1 33288 | The prime field of a division ring is the subfield generated by the multiplicative identity element. In general, we should write "prime division ring", but since most later usages are in the case where the ambient ring is commutative, we keep the term "prime field". (Contributed by Thierry Arnoux, 11-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) ⇒ ⊢ (𝜑 → ∩ (SubDRing‘𝑅) = (𝑅 fldGen { 1 })) | ||
| Theorem | 1fldgenq 33289 | The field of rational numbers ℚ is generated by 1 in ℂfld, that is, ℚ is the prime field of ℂfld. (Contributed by Thierry Arnoux, 15-Jan-2025.) |
| ⊢ (ℂfld fldGen {1}) = ℚ | ||
| Syntax | corng 33290 | Extend class notation with the class of all ordered rings. |
| class oRing | ||
| Syntax | cofld 33291 | Extend class notation with the class of all ordered fields. |
| class oField | ||
| Definition | df-orng 33292* | Define class of all ordered rings. An ordered ring is a ring with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
| ⊢ oRing = {𝑟 ∈ (Ring ∩ oGrp) ∣ [(Base‘𝑟) / 𝑣][(0g‘𝑟) / 𝑧][(.r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))} | ||
| Definition | df-ofld 33293 | Define class of all ordered fields. An ordered field is a field with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 18-Jan-2018.) |
| ⊢ oField = (Field ∩ oRing) | ||
| Theorem | isorng 33294* | An ordered ring is a ring with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 18-Jan-2018.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ ≤ = (le‘𝑅) ⇒ ⊢ (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)))) | ||
| Theorem | orngring 33295 | An ordered ring is a ring. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
| ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring) | ||
| Theorem | orngogrp 33296 | An ordered ring is an ordered group. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
| ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp) | ||
| Theorem | isofld 33297 | An ordered field is a field with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
| ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing)) | ||
| Theorem | orngmul 33298 | In an ordered ring, the ordering is compatible with the ring multiplication operation. (Contributed by Thierry Arnoux, 20-Jan-2018.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ≤ = (le‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → 0 ≤ (𝑋 · 𝑌)) | ||
| Theorem | orngsqr 33299 | In an ordered ring, all squares are positive. (Contributed by Thierry Arnoux, 20-Jan-2018.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ≤ = (le‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ oRing ∧ 𝑋 ∈ 𝐵) → 0 ≤ (𝑋 · 𝑋)) | ||
| Theorem | ornglmulle 33300 | In an ordered ring, multiplication with a positive does not change comparison. (Contributed by Thierry Arnoux, 10-Apr-2018.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ oRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ ≤ = (le‘𝑅) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → 0 ≤ 𝑍) ⇒ ⊢ (𝜑 → (𝑍 · 𝑋) ≤ (𝑍 · 𝑌)) | ||
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