HomeTheorem List (p. 333 of 497)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | erlcl1 33201 | 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 33202 | 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 33203* | 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 33204 | 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 33205 | 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 33206 | 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 33207* | 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 33208 | 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 33209 | 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 33210 | 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 33211 | 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 33212 | 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 33213 | 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 33214* | 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 33215 | 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 33216 | 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 33217* | 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 33218* | If two structures have the same components (properties), one is a integral domain iff the other one is. See also domnpropd 33217. (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 33219 | 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 33220 | Obsolete version of domnlcan 20679 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 33221 | Obsolete version of domnlcanb 20678 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 33222 | Obsolete version of idomrcan 33219 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 33223 | 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 33224 | 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 33225 | 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 33226 | 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 33227 | 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 33228 | Declare the syntax for the Euclidean function index extractor. |
| class EuclF | ||
| Definition | df-euf 33229 | Define the Euclidean function. (Contributed by Thierry Arnoux, 22-Mar-2025.) Use its index-independent form eufid 33231 instead. (New usage is discouraged.) |
| ⊢ EuclF = Slot ;21 | ||
| Theorem | eufndx 33230 | Index value of the Euclidean function slot. Use ndxarg 17213. (Contributed by Thierry Arnoux, 22-Mar-2025.) (New usage is discouraged.) |
| ⊢ (EuclF‘ndx) = ;21 | ||
| Theorem | eufid 33231 | Utility theorem: index-independent form of df-euf 33229. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ EuclF = Slot (EuclF‘ndx) | ||
| Syntax | cedom 33232 | Declare the syntax for the Euclidean Domain. |
| class EDomn | ||
| Definition | df-edom 33233* | 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 33234 | 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 33235* | 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 33236 | 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 33237 | The field of rational numbers is a field. (Contributed by Thierry Arnoux, 19-Oct-2025.) |
| ⊢ 𝑄 = (ℂfld ↾s ℚ) ⇒ ⊢ 𝑄 ∈ Field | ||
| Theorem | subsdrg 33238 | A subring of a sub-division-ring is a sub-division-ring. See also subsubrg 20556. (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 33239 | 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 33240 | 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 33241 | 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 33242 | Syntax for the field of fractions of a given integral domain. |
| class Frac | ||
| Definition | df-frac 33243 | 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 33244 | Value of the field of fractions. (Contributed by Thierry Arnoux, 5-May-2025.) |
| ⊢ ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅)) | ||
| Theorem | fracbas 33245 | 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 33246 | 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 33247* | 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 33248 | 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 33249* | 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 33250 | Syntax for a function generating sub-fields. |
| class fldGen | ||
| Definition | df-fldgen 33251* | 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 33254). If the base structure is a field, this is a subfield (see fldgenfld 33260 and fldsdrgfld 20756). 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 33252* | 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 33253 | The field generated by a set of elements contains those elements. See lspssid 20940. (Contributed by Thierry Arnoux, 15-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝑆 ⊆ (𝐹 fldGen 𝑆)) | ||
| Theorem | fldgensdrg 33254 | 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 33255 | 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 33256 | Generated subfields preserve subset ordering. ( see lspss 20939 and spanss 31275) (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 33257 | 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 33258 | 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 33259 | 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 33260 | 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 33261 | 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 33262 | 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 33263 | Extend class notation with the class of all ordered rings. |
| class oRing | ||
| Syntax | cofld 33264 | Extend class notation with the class of all ordered fields. |
| class oField | ||
| Definition | df-orng 33265* | 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 33266 | 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 33267* | 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 33268 | An ordered ring is a ring. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
| ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring) | ||
| Theorem | orngogrp 33269 | An ordered ring is an ordered group. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
| ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp) | ||
| Theorem | isofld 33270 | 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 33271 | 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 33272 | 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 33273 | 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)) | ||
| Theorem | orngrmulle 33274 | 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 → (𝑋 · 𝑍) ≤ (𝑌 · 𝑍)) | ||
| Theorem | ornglmullt 33275 | In an ordered ring, multiplication with a positive does not change strict comparison. (Contributed by Thierry Arnoux, 9-Apr-2018.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ oRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ < = (lt‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 < 𝑌) & ⊢ (𝜑 → 0 < 𝑍) ⇒ ⊢ (𝜑 → (𝑍 · 𝑋) < (𝑍 · 𝑌)) | ||
| Theorem | orngrmullt 33276 | In an ordered ring, multiplication with a positive does not change strict comparison. (Contributed by Thierry Arnoux, 9-Apr-2018.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ oRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ < = (lt‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 < 𝑌) & ⊢ (𝜑 → 0 < 𝑍) ⇒ ⊢ (𝜑 → (𝑋 · 𝑍) < (𝑌 · 𝑍)) | ||
| Theorem | orngmullt 33277 | In an ordered ring, the strict ordering is compatible with the ring multiplication operation. (Contributed by Thierry Arnoux, 9-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ < = (lt‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ oRing) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 0 < 𝑋) & ⊢ (𝜑 → 0 < 𝑌) ⇒ ⊢ (𝜑 → 0 < (𝑋 · 𝑌)) | ||
| Theorem | ofldfld 33278 | An ordered field is a field. (Contributed by Thierry Arnoux, 20-Jan-2018.) |
| ⊢ (𝐹 ∈ oField → 𝐹 ∈ Field) | ||
| Theorem | ofldtos 33279 | An ordered field is a totally ordered set. (Contributed by Thierry Arnoux, 20-Jan-2018.) |
| ⊢ (𝐹 ∈ oField → 𝐹 ∈ Toset) | ||
| Theorem | orng0le1 33280 | In an ordered ring, the ring unity is positive. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
| ⊢ 0 = (0g‘𝐹) & ⊢ 1 = (1r‘𝐹) & ⊢ ≤ = (le‘𝐹) ⇒ ⊢ (𝐹 ∈ oRing → 0 ≤ 1 ) | ||
| Theorem | ofldlt1 33281 | In an ordered field, the ring unity is strictly positive. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
| ⊢ 0 = (0g‘𝐹) & ⊢ 1 = (1r‘𝐹) & ⊢ < = (lt‘𝐹) ⇒ ⊢ (𝐹 ∈ oField → 0 < 1 ) | ||
| Theorem | ofldchr 33282 | The characteristic of an ordered field is zero. (Contributed by Thierry Arnoux, 21-Jan-2018.) (Proof shortened by AV, 6-Oct-2020.) |
| ⊢ (𝐹 ∈ oField → (chr‘𝐹) = 0) | ||
| Theorem | suborng 33283 | Every subring of an ordered ring is also an ordered ring. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
| ⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) ∈ oRing) | ||
| Theorem | subofld 33284 | Every subfield of an ordered field is also an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
| ⊢ ((𝐹 ∈ oField ∧ (𝐹 ↾s 𝐴) ∈ Field) → (𝐹 ↾s 𝐴) ∈ oField) | ||
| Theorem | isarchiofld 33285* | Axiom of Archimedes : a characterization of the Archimedean property for ordered fields. (Contributed by Thierry Arnoux, 9-Apr-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐻 = (ℤRHom‘𝑊) & ⊢ < = (lt‘𝑊) ⇒ ⊢ (𝑊 ∈ oField → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∃𝑛 ∈ ℕ 𝑥 < (𝐻‘𝑛))) | ||
| Theorem | rhmdvd 33286 | 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 33287 | 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 33288 | Extend class notation with the scalar restriction operation. |
| class ↾v | ||
| Definition | df-resv 33289* | 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 33290 | The scalar restriction is a proper operator, so it can be used with ovprc1 7442. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ Rel dom ↾v | ||
| Theorem | resvval 33291 | 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 33292 | 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 33293 | 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 33294 | 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 33295 | 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 33296 | 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 33297 | +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 33298 | ·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 33299 | .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 33300 | 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)) | ||
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