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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | unitnz 33201 | 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 33202* | 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 33203* | 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 | elrgspnlem1 33204* | Lemma for elrgspn 33208. (Contributed by Thierry Arnoux, 5-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ 𝑁 = (RingSpan‘𝑅) & ⊢ 𝐹 = {𝑓 ∈ (ℤ ↑m Word 𝐴) ∣ 𝑓 finSupp 0} & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ 𝑆 = ran (𝑔 ∈ 𝐹 ↦ (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤))))) ⇒ ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝑅)) | ||
| Theorem | elrgspnlem2 33205* | Lemma for elrgspn 33208. (Contributed by Thierry Arnoux, 5-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ 𝑁 = (RingSpan‘𝑅) & ⊢ 𝐹 = {𝑓 ∈ (ℤ ↑m Word 𝐴) ∣ 𝑓 finSupp 0} & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ 𝑆 = ran (𝑔 ∈ 𝐹 ↦ (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤))))) ⇒ ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) | ||
| Theorem | elrgspnlem3 33206* | Lemma for elrgspn 33208. (Contributed by Thierry Arnoux, 5-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ 𝑁 = (RingSpan‘𝑅) & ⊢ 𝐹 = {𝑓 ∈ (ℤ ↑m Word 𝐴) ∣ 𝑓 finSupp 0} & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ 𝑆 = ran (𝑔 ∈ 𝐹 ↦ (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤))))) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝑆) | ||
| Theorem | elrgspnlem4 33207* | Lemma for elrgspn 33208. (Contributed by Thierry Arnoux, 5-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ 𝑁 = (RingSpan‘𝑅) & ⊢ 𝐹 = {𝑓 ∈ (ℤ ↑m Word 𝐴) ∣ 𝑓 finSupp 0} & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ 𝑆 = ran (𝑔 ∈ 𝐹 ↦ (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤))))) ⇒ ⊢ (𝜑 → (𝑁‘𝐴) = 𝑆) | ||
| Theorem | elrgspn 33208* | Membership in the subring generated by the subset 𝐴. An element 𝑋 lies in that subring if and only if 𝑋 is a linear combination with integer coefficients of products of elements of 𝐴. (Contributed by Thierry Arnoux, 5-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ 𝑁 = (RingSpan‘𝑅) & ⊢ 𝐹 = {𝑓 ∈ (ℤ ↑m Word 𝐴) ∣ 𝑓 finSupp 0} & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝑁‘𝐴) ↔ ∃𝑔 ∈ 𝐹 𝑋 = (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔‘𝑤) · (𝑀 Σg 𝑤)))))) | ||
| Theorem | elrgspnsubrunlem1 33209* | Lemma for elrgspnsubrun 33211, first direction. (Contributed by Thierry Arnoux, 13-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑁 = (RingSpan‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐸 ∈ (SubRing‘𝑅)) & ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝑅)) & ⊢ (𝜑 → 𝑃:𝐹⟶𝐸) & ⊢ (𝜑 → 𝑃 finSupp 0 ) & ⊢ (𝜑 → 𝑋 = (𝑅 Σg (𝑒 ∈ 𝐹 ↦ ((𝑃‘𝑒) · 𝑒)))) & ⊢ 𝑇 = ran (𝑓 ∈ (𝑃 supp 0 ) ↦ ⟨“(𝑃‘𝑓)𝑓”⟩) ⇒ ⊢ (𝜑 → 𝑋 ∈ (𝑁‘(𝐸 ∪ 𝐹))) | ||
| Theorem | elrgspnsubrunlem2 33210* | Lemma for elrgspnsubrun 33211, second direction. (Contributed by Thierry Arnoux, 13-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑁 = (RingSpan‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐸 ∈ (SubRing‘𝑅)) & ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝑅)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐺:Word (𝐸 ∪ 𝐹)⟶ℤ) & ⊢ (𝜑 → 𝐺 finSupp 0) & ⊢ (𝜑 → 𝑋 = (𝑅 Σg (𝑤 ∈ Word (𝐸 ∪ 𝐹) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤))))) ⇒ ⊢ (𝜑 → ∃𝑝 ∈ (𝐸 ↑m 𝐹)(𝑝 finSupp 0 ∧ 𝑋 = (𝑅 Σg (𝑓 ∈ 𝐹 ↦ ((𝑝‘𝑓) · 𝑓))))) | ||
| Theorem | elrgspnsubrun 33211* | Membership in the ring span of the union of two subrings of a commutative ring. (Contributed by Thierry Arnoux, 13-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑁 = (RingSpan‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐸 ∈ (SubRing‘𝑅)) & ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝑅)) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐸 ∪ 𝐹)) ↔ ∃𝑝 ∈ (𝐸 ↑m 𝐹)(𝑝 finSupp 0 ∧ 𝑋 = (𝑅 Σg (𝑓 ∈ 𝐹 ↦ ((𝑝‘𝑓) · 𝑓)))))) | ||
| Theorem | irrednzr 33212 | 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 33213 | 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 33214 | 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 33215 | Syntax for ring localization equivalence class operation. |
| class ~RL | ||
| Syntax | crloc 33216 | Syntax for ring localization operation. |
| class RLocal | ||
| Definition | df-erl 33217* | 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 33218* | 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 33219 | Ring localization is a proper operator, so it can be used with ovprc1 7385. (Contributed by Thierry Arnoux, 10-May-2025.) |
| ⊢ Rel dom RLocal | ||
| Theorem | erlval 33220* | 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 33221* | 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 33222 | 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 33223 | 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 33224* | 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 33225 | 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 33226 | 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 33227 | 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 33228* | 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 33229 | 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 33230 | 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 33231 | 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 33232 | 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 33233 | 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 33234 | 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 33235* | 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 33236 | 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 33237 | 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 | domnpropd 33238* | If two structures have the same components (properties), one is a domain iff the other one is. (Contributed by Thierry Arnoux, 13-Oct-2025.) |
| ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (𝐾 ∈ Domn ↔ 𝐿 ∈ Domn)) | ||
| Theorem | idompropd 33239* | If two structures have the same components (properties), one is a integral domain iff the other one is. See also domnpropd 33238. (Contributed by Thierry Arnoux, 13-Oct-2025.) |
| ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (𝐾 ∈ IDomn ↔ 𝐿 ∈ IDomn)) | ||
| Theorem | idomrcan 33240 | 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 33241 | Obsolete version of domnlcan 20634 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 33242 | Obsolete version of domnlcanb 20633 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 33243 | Obsolete version of idomrcan 33240 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 33244 | 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 33245 | 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 33246 | 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 33247 | 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 33248 | 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 33249 | Declare the syntax for the Euclidean function index extractor. |
| class EuclF | ||
| Definition | df-euf 33250 | Define the Euclidean function. (Contributed by Thierry Arnoux, 22-Mar-2025.) Use its index-independent form eufid 33252 instead. (New usage is discouraged.) |
| ⊢ EuclF = Slot ;21 | ||
| Theorem | eufndx 33251 | Index value of the Euclidean function slot. Use ndxarg 17104. (Contributed by Thierry Arnoux, 22-Mar-2025.) (New usage is discouraged.) |
| ⊢ (EuclF‘ndx) = ;21 | ||
| Theorem | eufid 33252 | Utility theorem: index-independent form of df-euf 33250. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ EuclF = Slot (EuclF‘ndx) | ||
| Syntax | cedom 33253 | Declare the syntax for the Euclidean Domain. |
| class EDomn | ||
| Definition | df-edom 33254* | 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 33255 | 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 33256* | 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 33257 | 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 | qfld 33258 | The field of rational numbers is a field. (Contributed by Thierry Arnoux, 19-Oct-2025.) |
| ⊢ 𝑄 = (ℂfld ↾s ℚ) ⇒ ⊢ 𝑄 ∈ Field | ||
| Theorem | subsdrg 33259 | A subring of a sub-division-ring is a sub-division-ring. See also subsubrg 20511. (Contributed by Thierry Arnoux, 26-Oct-2025.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ (𝜑 → 𝐴 ∈ (SubDRing‘𝑅)) ⇒ ⊢ (𝜑 → (𝐵 ∈ (SubDRing‘𝑆) ↔ (𝐵 ∈ (SubDRing‘𝑅) ∧ 𝐵 ⊆ 𝐴))) | ||
| Theorem | sdrgdvcl 33260 | 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 33261 | 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 33262 | 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 33263 | Syntax for the field of fractions of a given integral domain. |
| class Frac | ||
| Definition | df-frac 33264 | 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 33265 | Value of the field of fractions. (Contributed by Thierry Arnoux, 5-May-2025.) |
| ⊢ ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅)) | ||
| Theorem | fracbas 33266 | 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 33267 | 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 33268* | 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 33269 | 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 33270* | 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 33271 | Syntax for a function generating sub-fields. |
| class fldGen | ||
| Definition | df-fldgen 33272* | 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 33275). If the base structure is a field, this is a subfield (see fldgenfld 33281 and fldsdrgfld 20711). 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 33273* | 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 33274 | The field generated by a set of elements contains those elements. See lspssid 20916. (Contributed by Thierry Arnoux, 15-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝑆 ⊆ (𝐹 fldGen 𝑆)) | ||
| Theorem | fldgensdrg 33275 | 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 33276 | 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 33277 | Generated subfields preserve subset ordering. ( see lspss 20915 and spanss 31323) (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 33278 | 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 33279 | 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 33280 | 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 33281 | 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 33282 | 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 33283 | 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 | ||
| Theorem | rhmdvd 33284 | A ring homomorphism preserves ratios. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
| ⊢ 𝑈 = (Unit‘𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ / = (/r‘𝑆) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → ((𝐹‘𝐴) / (𝐹‘𝐵)) = ((𝐹‘(𝐴 · 𝐶)) / (𝐹‘(𝐵 · 𝐶)))) | ||
| Theorem | kerunit 33285 | If a unit element lies in the kernel of a ring homomorphism, then 0 = 1, i.e. the target ring is the zero ring. (Contributed by Thierry Arnoux, 24-Oct-2017.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑆) & ⊢ 1 = (1r‘𝑆) ⇒ ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑈 ∩ (◡𝐹 “ { 0 })) ≠ ∅) → 1 = 0 ) | ||
| Syntax | cresv 33286 | Extend class notation with the scalar restriction operation. |
| class ↾v | ||
| Definition | df-resv 33287* | Define an operator to restrict the scalar field component of an extended structure. (Contributed by Thierry Arnoux, 5-Sep-2018.) |
| ⊢ ↾v = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩))) | ||
| Theorem | reldmresv 33288 | The scalar restriction is a proper operator, so it can be used with ovprc1 7385. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ Rel dom ↾v | ||
| Theorem | resvval 33289 | Value of structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ 𝑅 = (𝑊 ↾v 𝐴) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩))) | ||
| Theorem | resvid2 33290 | General behavior of trivial restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ 𝑅 = (𝑊 ↾v 𝐴) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = 𝑊) | ||
| Theorem | resvval2 33291 | Value of nontrivial structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ 𝑅 = (𝑊 ↾v 𝐴) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)) | ||
| Theorem | resvsca 33292 | Base set of a structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ 𝑅 = (𝑊 ↾v 𝐴) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐹 ↾s 𝐴) = (Scalar‘𝑅)) | ||
| Theorem | resvlem 33293 | Other elements of a scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.) |
| ⊢ 𝑅 = (𝑊 ↾v 𝐴) & ⊢ 𝐶 = (𝐸‘𝑊) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅)) | ||
| Theorem | resvbas 33294 | Base is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.) |
| ⊢ 𝐻 = (𝐺 ↾v 𝐴) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐵 = (Base‘𝐻)) | ||
| Theorem | resvplusg 33295 | +g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.) |
| ⊢ 𝐻 = (𝐺 ↾v 𝐴) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → + = (+g‘𝐻)) | ||
| Theorem | resvvsca 33296 | ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Proof shortened by AV, 31-Oct-2024.) |
| ⊢ 𝐻 = (𝐺 ↾v 𝐴) & ⊢ · = ( ·𝑠 ‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → · = ( ·𝑠 ‘𝐻)) | ||
| Theorem | resvmulr 33297 | .r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.) |
| ⊢ 𝐻 = (𝐺 ↾v 𝐴) & ⊢ · = (.r‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → · = (.r‘𝐻)) | ||
| Theorem | resv0g 33298 | 0g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ 𝐻 = (𝐺 ↾v 𝐴) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → 0 = (0g‘𝐻)) | ||
| Theorem | resv1r 33299 | 1r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ 𝐻 = (𝐺 ↾v 𝐴) & ⊢ 1 = (1r‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → 1 = (1r‘𝐻)) | ||
| Theorem | resvcmn 33300 | Scalar restriction preserves commutative monoids. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ 𝐻 = (𝐺 ↾v 𝐴) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐺 ∈ CMnd ↔ 𝐻 ∈ CMnd)) | ||
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