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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | rrgsupp 20701 | Left multiplication by a left regular element does not change the support set of a vector. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Revised by AV, 20-Jul-2019.) |
| ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐸) & ⊢ (𝜑 → 𝑌:𝐼⟶𝐵) ⇒ ⊢ (𝜑 → (((𝐼 × {𝑋}) ∘f · 𝑌) supp 0 ) = (𝑌 supp 0 )) | ||
| Theorem | rrgss 20702 | Left-regular elements are a subset of the base set. (Contributed by Stefan O'Rear, 22-Mar-2015.) |
| ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ 𝐸 ⊆ 𝐵 | ||
| Theorem | unitrrg 20703 | Units are regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.) |
| ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝑈 ⊆ 𝐸) | ||
| Theorem | rrgnz 20704 | In a nonzero ring, the zero is a left zero divisor (that is, not a left-regular element). (Contributed by Thierry Arnoux, 6-May-2025.) |
| ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ NzRing → ¬ 0 ∈ 𝐸) | ||
| Theorem | isdomn 20705* | Expand definition of a domain. (Contributed by Mario Carneiro, 28-Mar-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )))) | ||
| Theorem | domnnzr 20706 | A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.) |
| ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) | ||
| Theorem | domnring 20707 | A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.) |
| ⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring) | ||
| Theorem | domneq0 20708 | In a domain, a product is zero iff it has a zero factor. (Contributed by Mario Carneiro, 28-Mar-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0 ∨ 𝑌 = 0 ))) | ||
| Theorem | domnmuln0 20709 | In a domain, a product of nonzero elements is nonzero. (Contributed by Mario Carneiro, 6-May-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Domn ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≠ 0 )) → (𝑋 · 𝑌) ≠ 0 ) | ||
| Theorem | isdomn5 20710* | The equivalence between the right conjuncts in the right hand sides of isdomn 20705 and isdomn2 20711, in predicate calculus form. (Contributed by SN, 16-Sep-2024.) |
| ⊢ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 )) ↔ ∀𝑎 ∈ (𝐵 ∖ { 0 })∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )) | ||
| Theorem | isdomn2 20711 | A ring is a domain iff all nonzero elements are regular elements. (Contributed by Mario Carneiro, 28-Mar-2015.) (Proof shortened by SN, 21-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸)) | ||
| Theorem | isdomn2OLD 20712 | Obsolete version of isdomn2 20711 as of 21-Jun-2025. (Contributed by Mario Carneiro, 28-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸)) | ||
| Theorem | domnrrg 20713 | In a domain, a nonzero element is a regular element. (Contributed by Mario Carneiro, 28-Mar-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝐸) | ||
| Theorem | isdomn6 20714 | A ring is a domain iff the regular elements are the nonzero elements. Compare isdomn2 20711, domnrrg 20713. (Contributed by Thierry Arnoux, 6-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) = 𝐸)) | ||
| Theorem | isdomn3 20715 | Nonzero elements form a multiplicative submonoid of any domain. (Contributed by Stefan O'Rear, 11-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑈 = (mulGrp‘𝑅) ⇒ ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ Ring ∧ (𝐵 ∖ { 0 }) ∈ (SubMnd‘𝑈))) | ||
| Theorem | isdomn4 20716* | A ring is a domain iff it is nonzero and the left cancellation law for multiplication holds. (Contributed by SN, 15-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑎 ∈ (𝐵 ∖ { 0 })∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ((𝑎 · 𝑏) = (𝑎 · 𝑐) → 𝑏 = 𝑐))) | ||
| Theorem | opprdomnb 20717 | A class is a domain if and only if its opposite is a domain, biconditional form of opprdomn 20718. (Contributed by SN, 15-Jun-2015.) |
| ⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ Domn ↔ 𝑂 ∈ Domn) | ||
| Theorem | opprdomn 20718 | The opposite of a domain is also a domain. (Contributed by Mario Carneiro, 15-Jun-2015.) |
| ⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ Domn → 𝑂 ∈ Domn) | ||
| Theorem | isdomn4r 20719* | A ring is a domain iff it is nonzero and the right cancellation law for multiplication holds. (Contributed by SN, 20-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ (𝐵 ∖ { 0 })((𝑎 · 𝑐) = (𝑏 · 𝑐) → 𝑎 = 𝑏))) | ||
| Theorem | domnlcanb 20720 | Left-cancellation law for domains, biconditional version of domnlcan 20721. (Contributed by Thierry Arnoux, 8-Jun-2025.) Shorten this theorem and domnlcan 20721 overall. (Revised by SN, 21-Jun-2025.) |
| ⊢ 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 | domnlcan 20721 | Left-cancellation law for 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 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝑅 ∈ Domn) & ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑋 · 𝑍)) ⇒ ⊢ (𝜑 → 𝑌 = 𝑍) | ||
| Theorem | domnrcanb 20722 | Right-cancellation law for domains, biconditional version of domnrcan 20723. (Contributed 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 ∈ Domn) ⇒ ⊢ (𝜑 → ((𝑋 · 𝑍) = (𝑌 · 𝑍) ↔ 𝑋 = 𝑌)) | ||
| Theorem | domnrcan 20723 | Right-cancellation law for domains. (Contributed 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 ∈ Domn) & ⊢ (𝜑 → (𝑋 · 𝑍) = (𝑌 · 𝑍)) ⇒ ⊢ (𝜑 → 𝑋 = 𝑌) | ||
| Theorem | domneq0r 20724 | Right multiplication by a nonzero element does not change zeroness in a domain. Compare rrgeq0 20700. (Contributed by SN, 21-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 })) & ⊢ (𝜑 → 𝑅 ∈ Domn) ⇒ ⊢ (𝜑 → ((𝑋 · 𝑌) = 0 ↔ 𝑋 = 0 )) | ||
| Theorem | isidom 20725 | An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.) |
| ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn)) | ||
| Theorem | idomdomd 20726 | An integral domain is a domain. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ (𝜑 → 𝑅 ∈ IDomn) ⇒ ⊢ (𝜑 → 𝑅 ∈ Domn) | ||
| Theorem | idomcringd 20727 | An integral domain is a commutative ring with unity. (Contributed by Thierry Arnoux, 4-May-2025.) Formerly subproof of idomringd 20728. (Proof shortened by SN, 14-May-2025.) |
| ⊢ (𝜑 → 𝑅 ∈ IDomn) ⇒ ⊢ (𝜑 → 𝑅 ∈ CRing) | ||
| Theorem | idomringd 20728 | An integral domain is a ring. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ (𝜑 → 𝑅 ∈ IDomn) ⇒ ⊢ (𝜑 → 𝑅 ∈ Ring) | ||
| Syntax | cdr 20729 | Extend class notation with class of all division rings. |
| class DivRing | ||
| Syntax | cfield 20730 | Class of fields. |
| class Field | ||
| Definition | df-drng 20731 | Define class of all division rings. A division ring is a ring in which the set of units is exactly the nonzero elements of the ring. (Contributed by NM, 18-Oct-2012.) |
| ⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})} | ||
| Definition | df-field 20732 | A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.) |
| ⊢ Field = (DivRing ∩ CRing) | ||
| Theorem | isdrng 20733 | The predicate "is a division ring". (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 2-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 }))) | ||
| Theorem | drngunit 20734 | Elementhood in the set of units when 𝑅 is a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ))) | ||
| Theorem | drngui 20735 | The set of units of a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑅 ∈ DivRing ⇒ ⊢ (𝐵 ∖ { 0 }) = (Unit‘𝑅) | ||
| Theorem | drngring 20736 | A division ring is a ring. (Contributed by NM, 8-Sep-2011.) |
| ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | ||
| Theorem | drngringd 20737 | A division ring is a ring. (Contributed by SN, 16-May-2024.) |
| ⊢ (𝜑 → 𝑅 ∈ DivRing) ⇒ ⊢ (𝜑 → 𝑅 ∈ Ring) | ||
| Theorem | drnggrpd 20738 | A division ring is a group (deduction form). (Contributed by SN, 16-May-2024.) |
| ⊢ (𝜑 → 𝑅 ∈ DivRing) ⇒ ⊢ (𝜑 → 𝑅 ∈ Grp) | ||
| Theorem | drnggrp 20739 | A division ring is a group (closed form). (Contributed by NM, 8-Sep-2011.) |
| ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp) | ||
| Theorem | isfld 20740 | A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.) |
| ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing)) | ||
| Theorem | flddrngd 20741 | A field is a division ring. (Contributed by SN, 17-Jan-2025.) |
| ⊢ (𝜑 → 𝑅 ∈ Field) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing) | ||
| Theorem | fldcrngd 20742 | A field is a commutative ring. (Contributed by SN, 23-Nov-2024.) |
| ⊢ (𝜑 → 𝑅 ∈ Field) ⇒ ⊢ (𝜑 → 𝑅 ∈ CRing) | ||
| Theorem | isdrng2 20743 | A division ring can equivalently be defined as a ring such that the nonzero elements form a group under multiplication (from which it follows that this is the same group as the group of units). (Contributed by Mario Carneiro, 2-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) ⇒ ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ Grp)) | ||
| Theorem | drngprop 20744 | If two structures have the same ring components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 28-Dec-2014.) |
| ⊢ (Base‘𝐾) = (Base‘𝐿) & ⊢ (+g‘𝐾) = (+g‘𝐿) & ⊢ (.r‘𝐾) = (.r‘𝐿) ⇒ ⊢ (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing) | ||
| Theorem | drngmgp 20745 | A division ring contains a multiplicative group. (Contributed by NM, 8-Sep-2011.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) ⇒ ⊢ (𝑅 ∈ DivRing → 𝐺 ∈ Grp) | ||
| Theorem | drngid 20746 | A division ring's unity is the identity element of its multiplicative group. (Contributed by NM, 7-Sep-2011.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) ⇒ ⊢ (𝑅 ∈ DivRing → 1 = (0g‘𝐺)) | ||
| Theorem | drngunz 20747 | A division ring's unity is different from its zero. (Contributed by NM, 8-Sep-2011.) |
| ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → 1 ≠ 0 ) | ||
| Theorem | drngnzr 20748 | A division ring is a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
| ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing) | ||
| Theorem | drngdomn 20749 | A division ring is a domain. (Contributed by Mario Carneiro, 29-Mar-2015.) |
| ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Domn) | ||
| Theorem | drngmcl 20750 | The product of two nonzero elements of a division ring is nonzero. (Contributed by NM, 7-Sep-2011.) (Proof shortened by SN, 25-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ (𝐵 ∖ { 0 }) ∧ 𝑌 ∈ (𝐵 ∖ { 0 })) → (𝑋 · 𝑌) ∈ (𝐵 ∖ { 0 })) | ||
| Theorem | drngmclOLD 20751 | Obsolete version of drngmcl 20750 as of 25-Jun-2025. The product of two nonzero elements of a division ring is nonzero. (Contributed by NM, 7-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ (𝐵 ∖ { 0 }) ∧ 𝑌 ∈ (𝐵 ∖ { 0 })) → (𝑋 · 𝑌) ∈ (𝐵 ∖ { 0 })) | ||
| Theorem | drngid2 20752 | Properties showing that an element 𝐼 is the identity element of a division ring. (Contributed by Mario Carneiro, 11-Oct-2013.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → ((𝐼 ∈ 𝐵 ∧ 𝐼 ≠ 0 ∧ (𝐼 · 𝐼) = 𝐼) ↔ 1 = 𝐼)) | ||
| Theorem | drnginvrcl 20753 | Closure of the multiplicative inverse in a division ring. (reccl 11929 analog). (Contributed by NM, 19-Apr-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ∈ 𝐵) | ||
| Theorem | drnginvrn0 20754 | The multiplicative inverse in a division ring is nonzero. (recne0 11935 analog). (Contributed by NM, 19-Apr-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ≠ 0 ) | ||
| Theorem | drnginvrcld 20755 | Closure of the multiplicative inverse in a division ring. (reccld 12036 analog). (Contributed by SN, 14-Aug-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ≠ 0 ) ⇒ ⊢ (𝜑 → (𝐼‘𝑋) ∈ 𝐵) | ||
| Theorem | drnginvrl 20756 | Property of the multiplicative inverse in a division ring. (recid2 11937 analog). (Contributed by NM, 19-Apr-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ((𝐼‘𝑋) · 𝑋) = 1 ) | ||
| Theorem | drnginvrr 20757 | Property of the multiplicative inverse in a division ring. (recid 11936 analog). (Contributed by NM, 19-Apr-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝑋 · (𝐼‘𝑋)) = 1 ) | ||
| Theorem | drnginvrld 20758 | Property of the multiplicative inverse in a division ring. (recid2d 12039 analog). (Contributed by SN, 14-Aug-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ≠ 0 ) ⇒ ⊢ (𝜑 → ((𝐼‘𝑋) · 𝑋) = 1 ) | ||
| Theorem | drnginvrrd 20759 | Property of the multiplicative inverse in a division ring. (recidd 12038 analog). (Contributed by SN, 14-Aug-2024.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ≠ 0 ) ⇒ ⊢ (𝜑 → (𝑋 · (𝐼‘𝑋)) = 1 ) | ||
| Theorem | drngmul0or 20760 | A product is zero iff one of its factors is zero. (Contributed by NM, 8-Oct-2014.) (Proof shortened by SN, 25-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0 ∨ 𝑌 = 0 ))) | ||
| Theorem | drngmul0orOLD 20761 | Obsolete version of drngmul0or 20760 as of 25-Jun-2025. (Contributed by NM, 8-Oct-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0 ∨ 𝑌 = 0 ))) | ||
| Theorem | drngmulne0 20762 | A product is nonzero iff both its factors are nonzero. (Contributed by NM, 18-Oct-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 · 𝑌) ≠ 0 ↔ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ))) | ||
| Theorem | drngmuleq0 20763 | An element is zero iff its product with a nonzero element is zero. (Contributed by NM, 8-Oct-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ≠ 0 ) ⇒ ⊢ (𝜑 → ((𝑋 · 𝑌) = 0 ↔ 𝑋 = 0 )) | ||
| Theorem | opprdrng 20764 | The opposite of a division ring is also a division ring. (Contributed by NM, 18-Oct-2014.) |
| ⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing ↔ 𝑂 ∈ DivRing) | ||
| Theorem | isdrngd 20765* | Properties that characterize a division ring among rings: it should be nonzero, have no nonzero zero-divisors, and every nonzero element 𝑥 should have a left-inverse 𝐼(𝑥). See isdrngrd 20766 for the characterization using right-inverses. (Contributed by NM, 2-Aug-2013.) Remove hypothesis. (Revised by SN, 19-Feb-2025.) |
| ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → · = (.r‘𝑅)) & ⊢ (𝜑 → 0 = (0g‘𝑅)) & ⊢ (𝜑 → 1 = (1r‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝑥 · 𝑦) ≠ 0 ) & ⊢ (𝜑 → 1 ≠ 0 ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ∈ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → (𝐼 · 𝑥) = 1 ) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing) | ||
| Theorem | isdrngrd 20766* | Properties that characterize a division ring among rings: it should be nonzero, have no nonzero zero-divisors, and every nonzero element 𝑥 should have a right-inverse 𝐼(𝑥). See isdrngd 20765 for the characterization using left-inverses. (Contributed by NM, 10-Aug-2013.) Remove hypothesis. (Revised by SN, 19-Feb-2025.) |
| ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → · = (.r‘𝑅)) & ⊢ (𝜑 → 0 = (0g‘𝑅)) & ⊢ (𝜑 → 1 = (1r‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝑥 · 𝑦) ≠ 0 ) & ⊢ (𝜑 → 1 ≠ 0 ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ∈ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → (𝑥 · 𝐼) = 1 ) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing) | ||
| Theorem | isdrngdOLD 20767* | Obsolete version of isdrngd 20765 as of 19-Feb-2025. (Contributed by NM, 2-Aug-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → · = (.r‘𝑅)) & ⊢ (𝜑 → 0 = (0g‘𝑅)) & ⊢ (𝜑 → 1 = (1r‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝑥 · 𝑦) ≠ 0 ) & ⊢ (𝜑 → 1 ≠ 0 ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ∈ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ≠ 0 ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → (𝐼 · 𝑥) = 1 ) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing) | ||
| Theorem | isdrngrdOLD 20768* | Obsolete version of isdrngrd 20766 as of 19-Feb-2025. (Contributed by NM, 10-Aug-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → · = (.r‘𝑅)) & ⊢ (𝜑 → 0 = (0g‘𝑅)) & ⊢ (𝜑 → 1 = (1r‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝑥 · 𝑦) ≠ 0 ) & ⊢ (𝜑 → 1 ≠ 0 ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ∈ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ≠ 0 ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → (𝑥 · 𝐼) = 1 ) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing) | ||
| Theorem | drngpropd 20769* | If two structures have the same group components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 27-Jun-2015.) |
| ⊢ (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 ∈ DivRing ↔ 𝐿 ∈ DivRing)) | ||
| Theorem | fldpropd 20770* | If two structures have the same group components (properties), one is a field iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.) |
| ⊢ (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 ∈ Field ↔ 𝐿 ∈ Field)) | ||
| Theorem | fldidom 20771 | A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015.) (Proof shortened by SN, 11-Nov-2024.) |
| ⊢ (𝑅 ∈ Field → 𝑅 ∈ IDomn) | ||
| Theorem | fldidomOLD 20772 | Obsolete version of fldidom 20771 as of 11-Nov-2024. (Contributed by Mario Carneiro, 29-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝑅 ∈ Field → 𝑅 ∈ IDomn) | ||
| Theorem | fidomndrnglem 20773* | Lemma for fidomndrng 20774. (Contributed by Mario Carneiro, 15-Jun-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Domn) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ { 0 })) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 · 𝐴)) ⇒ ⊢ (𝜑 → 𝐴 ∥ 1 ) | ||
| Theorem | fidomndrng 20774 | A finite domain is a division ring. Note that Wedderburn's little theorem (not proved) states that finite division rings are fields. (Contributed by Mario Carneiro, 15-Jun-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝐵 ∈ Fin → (𝑅 ∈ Domn ↔ 𝑅 ∈ DivRing)) | ||
| Theorem | fiidomfld 20775 | A finite integral domain is a field. (Contributed by Mario Carneiro, 15-Jun-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝐵 ∈ Fin → (𝑅 ∈ IDomn ↔ 𝑅 ∈ Field)) | ||
| Theorem | rng1nnzr 20776 | The (smallest) structure representing a zero ring is not a nonzero ring. (Contributed by AV, 29-Apr-2019.) |
| ⊢ 𝑀 = {〈(Base‘ndx), {𝑍}〉, 〈(+g‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉, 〈(.r‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉} ⇒ ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∉ NzRing) | ||
| Theorem | ring1zr 20777 | The only (unital) ring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) (Proof shortened by AV, 7-Feb-2020.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ ∗ = (.r‘𝑅) ⇒ ⊢ (((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {〈〈𝑍, 𝑍〉, 𝑍〉} ∧ ∗ = {〈〈𝑍, 𝑍〉, 𝑍〉}))) | ||
| Theorem | rngen1zr 20778 | The only (unital) ring with one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 14-Feb-2010.) (Revised by AV, 25-Jan-2020.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ ∗ = (.r‘𝑅) ⇒ ⊢ (((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 ≈ 1o ↔ ( + = {〈〈𝑍, 𝑍〉, 𝑍〉} ∧ ∗ = {〈〈𝑍, 𝑍〉, 𝑍〉}))) | ||
| Theorem | ringen1zr 20779 | The only unital ring with one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 15-Feb-2010.) (Revised by AV, 25-Jan-2020.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ ∗ = (.r‘𝑅) & ⊢ 𝑍 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1o ↔ ( + = {〈〈𝑍, 𝑍〉, 𝑍〉} ∧ ∗ = {〈〈𝑍, 𝑍〉, 𝑍〉}))) | ||
| Theorem | rng1nfld 20780 | The zero ring is not a field. (Contributed by AV, 29-Apr-2019.) |
| ⊢ 𝑀 = {〈(Base‘ndx), {𝑍}〉, 〈(+g‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉, 〈(.r‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉} ⇒ ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∉ Field) | ||
| Theorem | issubdrg 20781* | Characterize the subfields of a division ring. (Contributed by Mario Carneiro, 3-Dec-2014.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (𝑆 ∈ DivRing ↔ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴)) | ||
| Theorem | drhmsubc 20782* | According to df-subc 17856, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17885 and subcss2 17888). Therefore, the set of division ring homomorphisms is a "subcategory" of the category of (unital) rings. (Contributed by AV, 20-Feb-2020.) |
| ⊢ 𝐶 = (𝑈 ∩ DivRing) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCat‘𝑈))) | ||
| Theorem | drngcat 20783* | The restriction of the category of (unital) rings to the set of division ring homomorphisms is a category, the "category of division rings". (Contributed by AV, 20-Feb-2020.) |
| ⊢ 𝐶 = (𝑈 ∩ DivRing) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → ((RingCat‘𝑈) ↾cat 𝐽) ∈ Cat) | ||
| Theorem | fldcat 20784* | The restriction of the category of (unital) rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.) |
| ⊢ 𝐶 = (𝑈 ∩ DivRing) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) & ⊢ 𝐷 = (𝑈 ∩ Field) & ⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → ((RingCat‘𝑈) ↾cat 𝐹) ∈ Cat) | ||
| Theorem | fldc 20785* | The restriction of the category of division rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.) |
| ⊢ 𝐶 = (𝑈 ∩ DivRing) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) & ⊢ 𝐷 = (𝑈 ∩ Field) & ⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → (((RingCat‘𝑈) ↾cat 𝐽) ↾cat 𝐹) ∈ Cat) | ||
| Theorem | fldhmsubc 20786* | According to df-subc 17856, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17885 and subcss2 17888). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.) |
| ⊢ 𝐶 = (𝑈 ∩ DivRing) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) & ⊢ 𝐷 = (𝑈 ∩ Field) & ⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘((RingCat‘𝑈) ↾cat 𝐽))) | ||
| Syntax | csdrg 20787 | Syntax for subfields (sub-division-rings). |
| class SubDRing | ||
| Definition | df-sdrg 20788* | Define the function associating with a ring the set of its sub-division-rings. A sub-division-ring of a ring is a subset of its base set which is a division ring when equipped with the induced structure (sum, multiplication, zero, and unity). If a ring is commutative (resp., a field), then its sub-division-rings are commutative (resp., are fields) (fldsdrgfld 20799), so we do not make a specific definition for subfields. (Contributed by Stefan O'Rear, 3-Oct-2015.) TODO: extend this definition to a function with domain V or at least Ring and not only DivRing. |
| ⊢ SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing}) | ||
| Theorem | issdrg 20789 | Property of a division subring. (Contributed by Stefan O'Rear, 3-Oct-2015.) |
| ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing)) | ||
| Theorem | sdrgrcl 20790 | Reverse closure for a sub-division-ring predicate. (Contributed by SN, 19-Feb-2025.) |
| ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝑅 ∈ DivRing) | ||
| Theorem | sdrgdrng 20791 | A sub-division-ring is a division ring. (Contributed by SN, 19-Feb-2025.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝑆 ∈ DivRing) | ||
| Theorem | sdrgsubrg 20792 | A sub-division-ring is a subring. (Contributed by SN, 19-Feb-2025.) |
| ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝐴 ∈ (SubRing‘𝑅)) | ||
| Theorem | sdrgid 20793 | Every division ring is a division subring of itself. (Contributed by Thierry Arnoux, 21-Aug-2023.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → 𝐵 ∈ (SubDRing‘𝑅)) | ||
| Theorem | sdrgss 20794 | A division subring is a subset of the base set. (Contributed by Thierry Arnoux, 21-Aug-2023.) |
| ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑆 ∈ (SubDRing‘𝑅) → 𝑆 ⊆ 𝐵) | ||
| Theorem | sdrgbas 20795 | Base set of a sub-division-ring structure. (Contributed by SN, 19-Feb-2025.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝐴 = (Base‘𝑆)) | ||
| Theorem | issdrg2 20796* | Property of a division subring (closure version). (Contributed by Mario Carneiro, 3-Oct-2015.) |
| ⊢ 𝐼 = (invr‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ ∀𝑥 ∈ (𝑆 ∖ { 0 })(𝐼‘𝑥) ∈ 𝑆)) | ||
| Theorem | sdrgunit 20797 | A unit of a sub-division-ring is a nonzero element of the subring. (Contributed by SN, 19-Feb-2025.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑈 = (Unit‘𝑆) ⇒ ⊢ (𝐴 ∈ (SubDRing‘𝑅) → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ≠ 0 ))) | ||
| Theorem | imadrhmcl 20798 | The image of a (nontrivial) division ring homomorphism is a division ring. (Contributed by SN, 17-Feb-2025.) |
| ⊢ 𝑅 = (𝑁 ↾s (𝐹 “ 𝑆)) & ⊢ 0 = (0g‘𝑁) & ⊢ (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁)) & ⊢ (𝜑 → 𝑆 ∈ (SubDRing‘𝑀)) & ⊢ (𝜑 → ran 𝐹 ≠ { 0 }) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing) | ||
| Theorem | fldsdrgfld 20799 | A sub-division-ring of a field is itself a field, so it is a subfield. We can therefore use SubDRing to express subfields. (Contributed by Thierry Arnoux, 11-Jan-2025.) |
| ⊢ ((𝐹 ∈ Field ∧ 𝐴 ∈ (SubDRing‘𝐹)) → (𝐹 ↾s 𝐴) ∈ Field) | ||
| Theorem | acsfn1p 20800* | Construction of a closure rule from a one-parameter partial operation. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
| ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑎 ∩ 𝑌)𝐸 ∈ 𝑎} ∈ (ACS‘𝑋)) | ||
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